University of Ljubljana Faculty of Mathematics and Physics Department of Physics Rok Žit ko Many-particle effects in resonant tunneling of electrons through nanostructures Doctoral thesis ADVISER: Prof. Dr. Janez Bonča COADVISOR: Prof. Dr. Igor Muševič Ljubljana, 2007 Univerza v Ljubljani Fakulteta za matematiko in fiziko Oddelek za fiziko Rok Žitko VEČDELČNI POJAVI PRI RESONANČNEM TUNELIRANJU ELEKTRONOV SKOZI NANOSTRUKTURE Doktorska disertacija MENTOR: prof. dr. Janez Bonča SOMENTOR: prof. dr. Igor Muševič Ljubljana, 2007 Abstract Effects of electron-electron interactions on the transport properties of nanostructures are explored, focusing on the conductance through systems of coupled quantum dots and the tunneling spectroscopy of magnetic adsorbates on surfaces; both systems can be modeled using quantum impurity models. The properties of impurity models are described in considerable detail and a new implementation of Wilson's numerical renormalization group is introduced. Double quantum dot systems of various coupling topologies are studied. In parallel double quantum dot, local moments order ferromagnetically and S =1 Kondo effect occurs. In side-coupled double quantum dot, Kondo screening proceeds in two stages. Triple quantum dot is studied using several numerical methods and a low-temperature phase diagram is proposed. A wide regime of non-Fermi liquid behavior is found at finite temperatures in the cross-over region between antiferromagnetic ordering regime and the two-stage Kondo regime. In the second part, construction of a low-temperature scanning tunneling microscope is described. The Kondo effect induced by magnetic adsorbates on surfaces is reviewed and a two-level model is proposed to describe the tunneling spectroscopy experiments performed on single magnetic adatoms. PACS: 73.63.Kv 72.15.Qm 73.23.Hk 71.10.Hf 68.37.Ef Keywords: quantum impurity models, Kondo effect, numerical renormalization group, coupled quantum dots, tunneling spectroscopy. Povzetek Interakcije med elektroni pomembno vplivajo na transportne lastnosti nanostruktur. Posebno zanimiva sta prevodnost sistemov sklopljenih kvantnih pik in tunelska spektroskopija magnetih nečistoč, adsorbiranih na površinah kovin. Oba sistema lahko opišemo z modeli kvantnih nečistoč, katerih lastnosti so podrobno opisane. Predstavljena je nova izvedba metode numerična renormalizacijska grupa. Proučeni so sistemi dvojne kvantne pike z različnimi topologijami sklopitve. V vzporedni dvojni kvantni piki se lokalna momenta uredita feromagnetno in pride do Kondovega pojava s spinom 1. V stransko sklopljeni dvojni kvantni piki Kondovo senčenje poteka v dveh korakih. Sistem trojne kvantne pike je obravnavan s komplementarnimi numeričnimi metodami in določen je fazni diagram sistema pri nizkih temperaturah. V prehodnem območju med režimoma antiferomagne-tnega urejanja in dvostopenjskega senčenja obstaja široko območje, kjer se sistem v nekem temperaturnem intervalu obnaša kot ne-Fermijeva tekočina. V drugem delu doktorskega dela je opisano sestavljanje nizko-temperaturnega vrstičnega tunelskega mikroskopa. Opisan je Kondov pojav na površinah, ki ga povzročijo magnetne nečistoče, in vpeljan nov dvonivojski model, ki opisuje poskuse s tunelsko spektroskopijo na posameznih magnetnih atomih. PACS: 73.63.Kv 72.15.Qm 73.23.Hk 71.10.Hf 68.37.Ef Ključne besede: modeli kvantnih nečistoč, Kondov pojav, numerična renormalizacijska grupa, sklopljene kvantne pike, tunelska spektroskopija. I would like to sincerely thank my advisors, prof. dr. Janez Bonča and prof. dr. Igor Muševič, for guiding, valuable advice, discussions and support, and especially for granting me a considerable extent of autonomy in my work. I also thank prof. dr. Albert Prodan and Erik Zupanič for sharing the joys and sorrows of building an STM. I acknowledge the very fruitful discussions on various topics related to Kondo physics and quantum transport that I have had with Jernej Mravlje, prof. dr. Anton Ramsak and dr. Tomaž Rejec. I also thank Ivan Kvasič for building many electronic gadgets and for troubleshooting, Stefan Fölsch at Paul Drude Institut for design tips and valuable advice on STM hardware, Sven Zöpfel of Createc for assembly instructions, and Peter Panjan for discussions on friction and for hard-coating some of the STM parts. Finally, I thank my parents for support and Tamara for patiently enduring my devotion to science. Contents 1 Introduction 11 1 Theory of tunneling through nanostructures 14 2 Theory of quantum impurity models 15 2.1 The concept of a quantum impurity model.................. 15 2.2 Kondo effect................................... 17 2.3 Symmetries in quantum impurity models................... 19 2.4 Fermi liquid and non-Fermi liquid systems.................. 25 2.5 Effective field theories of the Kondo problem................. 27 2.6 Spin-charge separation............................. 28 2.7 Conformal field theory............................. 29 3 Renormalization group 32 3.1 Renormalizability, universality and scaling theories ............. 33 3.2 Numerical renormalization group ....................... 35 3.3 Implementation overview............................ 36 3.4 Logarithmic discretization........................... 38 3.5 Symmetries and basis construction ...................... 43 3.6 RG transformation and iterative diagonalisation............... 48 3.7 Computable quantities............................. 53 3.8 Recursion relations for operators........................ 61 3.9 Density-matrix NRG.............................. 62 4 Other methods for impurity models 66 4.1 Green's function method for noninteracting problems............ 66 4.2 Gunnarsson-Schönhammer variational method................ 67 4.3 Quantum Monte Carlo method ........................ 70 5 Quantum transport theory 74 5.1 Conductance from phase shifts......................... 75 5.2 Sine formula................................... 79 5.3 Meir-Wingreen formula............................. 80 4 CONTENTS 5 II Systems of coupled quantum dots 85 6 Properties of single impurity models 86 6.1 Single-channel Kondo model.......................... 86 6.2 Two-channel Kondo model........................... 89 6.3 Anderson model................................. 92 7 Properties of two-impurity models 105 7.1 General properties of double quantum dot systems.............. 106 7.2 Double quantum dot: parallel configuration ................. 108 7.3 Double quantum dot: side-coupled configuration............... 132 7.4 Double quantum dots in magnetic field.................... 144 8 Properties of three-impurity models 147 8.1 Triple quantum dot: linear configuration................... 148 III Scanning tunneling microscopy and adsorbates 168 9 Scanning tunneling microscopy 169 9.1 Applications of the scanning tunneling microscopy.............. 169 9.2 Construction of the low-temperature STM.................. 171 9.3 Ultra-high vacuum system........................... 187 9.4 Preparation of clean surfaces and evaporation of materials......... 190 9.5 Sample STM images.............................. 193 10 Clusters of magnetic adatoms and surface Kondo effect 200 10.1 Review of experimental results on surface Kondo effect........... 201 10.2 Theory of the surface Kondo effect ...................... 206 10.3 NRG calculations................................ 212 IV Conclusion 220 11 Conclusions 221 12 Povzetek disertacije v slovenskem jeziku 231 V Appendices 241 A Tensor operators and Wigner-Eckart theorem 242 B Green's functions 243 6 CONTENTS C Generalized Schrieffer-Wolff transformation 246 D Scaling equations to second order in J 249 E Transformations of band and coupling Hamiltonians 250 F Majorana fermions 253 List of abbreviations and acronyms Abbreviation Full form ID one-dimensional 2CK two-channel Kondo 2D two-dimensional 2DEG two-dimensional electron gas 2IK two-impurity Kondo AES Auger electron spectroscopy AFM antiferromagnetic BA Betlie Ansatz CFT conformal field theory CPMC constrained-path Monte Carlo DFT density functional theory DMFT dynamic mean field theory DMNRG density-matrix NRG DOS density of states DQD double quantum dot d.o.f. degree of freedom e-e electron-electron ESCA electron spectroscopy for chemical analysis FF ferromagnetically frozen FL Fermi liquid FI frozen-impurity FM ferromagnetic FO free-orbital FP fixed point HF Hartree-Fock GS Gunnarson-Schönhammer (variational method) LDOS local density of states LEED low-energy electron diffraction LSDA local spin density approximation LHe liquid helium LM local-moment LN2 liquid nitrogen LT low-temperature MO molecular orbital NFL non-Fermi liquid NRG numerical renormalisation group NS Neveu-Schwarz p-h particle-hole (symmetry) 7 Abbreviation Full form QFT quantum field theory QIM quantum impurity model QMC quantum Monte Carlo QMS quadrupole mass spectrometer QPT quantum phase transition QD quantum dot R Ramon RG renormalization group RKKY Ruderman-Kittel-Kasuya-Yosida RT room temperature SC strong-coupling SEM scanning electron microscope SET single electron transistor SIAM single-impurity Anderson model STM scanning tunneling microscope SWT Schrieffer-Wolff transformation TEM transmission electron microscope TQD triple quantum dot TSK two-stage Kondo UHV ultra-high vacuum VF valence fluctuation 8 List of symbols Symbol Description Conventional unit Energy of conduction band electron eV Creation operator for band electron Crystal momentum, wavenumber A Magnetic quantum number, also a, sz Pauli matrices Number of conduction band states Hopping Hamiltonian creation operator Half bandwidth Spin-spin exchange coupling Spin operator Density of states Kondo exchange interaction Boltzmann's constant Inverse temperature, 1/ikßT) Kondo temperature Fermi wavenumber Fermi velocity Velocity of charge/spin modes Wilson's ratio Lattice site indexes Tunnel hopping, overlap integral eV Charge quantum number, <& = rii — 1 Isospin operator Nambu spinor Parity Linear size of the system m Energy of impurity orbital eV Deviation from p-h symmetric point, 5 = e^ + U/2 eV Creation operator for impurity electron Charge-charge coupling, Coulomb correlation energy eV Conductance quantum, G0 =2e2/h A/V= S Dimensionless conductance, g = G/Go Impurity-conduction-band hybridization eV Tunnel rate, resonance half-width eV Retarded Green's function Spectral function 1/eV k n a Nc f] j in D J S p(u) J, Jk kB ß TK kp vF V c, V s Rw i,J Q,Q I & p L td 5 S uiß u Go 9 Vk A,r Aij(uj) eV eV 1/eV eV = 86 ßeV/K 1/eV K Ä"1 m/s m/s 9 Symbol Description Conventional unit X charge Impurity charge susceptibility Xspin j Ximp Impurity magnetic susceptibility üimp Impurity entropy Z Partition function 7 Linear coefficient in heat capacity, C = 7T 9 Gyromagnetic ratio ßB Bohr magneton = eh/2me A NRG discretization parameter N NRG iteration number E Energy ev t Dimensionless energy, e = E/D b Spectrum broadening parameter LM Self-energy eV S kjU.kV Scattering matrix z, Z Quasiparticle weight n Electron density _3 m K Potential scattering amplitude eV Oqp Scattering phase-shift rad r Scattering time s T* Local-moment formation temperature K ^RKKY Effective spin exchange coupling eV Ferromagnetic ordering temperature K T (l) T(2) ^ ) 1K Upper, lower Kondo temperature K A Asymmetry parameter TA NFL-FL cross-over temperature K I (Tunneling) current A V (Bias) voltage V Q Fano form factor B Magnetic field T Liberal use of h = l and kß =1 system of units is made, especially in longer derivations. 10 Chapter 1 Introduction Advances in nanoscience and nanotechnology empower us with new tools for probing systems of increasingly small sizes. Nowadays one can, for example, measure transport properties of semiconductor quantum dots, single molecules, and even individual atoms adsorbed on a surface. The interest in such systems is twofold. On one hand, transport through nanostructures is of fundamental interest since a number of very characteristic quantum effects can be studied. On the other hand, nanostructures represent the ultimate degree of miniaturization of electronic devices and they are likely candidates for the building blocks of the circuitry of tomorrow. A notable phenomenon that is commonly at play on the nanoscopic scale is quantum tunneling - transmission of electrons through classically-forbidden energy barriers. The use of the electron tunneling as a spectroscopic technique eventually led to the invention of the scanning tunneling microscope (STM) in 1982. Since then, STM has become one of the most versatile tools in nanoscience. In addition to its most common use as a topographic microscope with atomic resolution, STM can be applied to perform lateral manipulations of adsorbed species and to transfer adsorbates from sample to tip and vice-versa. It can also induce dissociation, desorption and conformation changes in molecules and it is even possible to perform chemical synthesis. Finally, STM can be used to perform tunneling spectroscopy of the smallest magnetic objects - single magnetic atoms, clusters and molecules adsorbed on surfaces. Somewhat larger magnetic nanostructures can be built using quantum dots. Quantum dots are microscopic puddles of electrons which can be considered as artificial atoms, since the confined electrons are quantized and form orbitals much like the electrons in orbit around an atomic nucleus. Particularly interesting are semiconductor quantum dots patterned in high-quality heterostructures grown by molecular beam epitaxy. Lateral quantum dots, for example, are defined in AlGaAs/GaAs heterostructures with a subsurface layer of the high-mobility two-dimensional electron gas by patterning metallic gates on the sample surface.1 By adjusting the voltages on these gates, it is possible to control the number of the electrons trapped in the quantum dot and change the coupling of the dot with the 11 12 CHAPTER 1. INTRODUCTION electron gas. Quantum dots thus serve as tunable realizations of quantum impurity models - models of point-like impurities with internal degrees of freedom - in which on-site energy and hybridization strength can be easily swept in-situ. Quantum impurity models attract the interest of the solid state physics community both due to their unexpectedly complex behavior and intrinsic beauty, as well as due to their ubiquitous applicability to a vast array of physical systems such as bulk Kondo systems, heavy-fermion compounds and other strongly correlated systems,2 dissipative two-level systems,3 single magnetic impurities and quantum dots.4-6 In very small electronic devices the electron-electron interactions are strong and they induce interesting many-particle effects. The most notable is perhaps the Kondo effect which appears to be a relatively generic feature of nanodevices.7-10 The Kondo effect is a many-particle phenomenon arising from the interaction between a localized spin and free electrons that leads to increased spin-flip scattering rate of the electrons at low temperatures. This gives rise to various anomalies in the thermodynamic and transport properties, in particular to increased conductance through nanostructures. The conductance through a quantum dot in the Kondo regime as a function of temperature, gate and bias voltages is in agreement with theoretical predictions that such dots behave rather universally as single magnetic impurities11 and can be modelled using single impurity Anderson and Kondo models.11'12 Systems of multiple coupled impurities are realizations of generalized Kondo models where more exotic Kondo states may occur. The research in this field has recently intensified due to a multitude of new experimental results; the multi-impurity magnetic nanostructures under study are predominantly of two kinds: clusters of magnetic adsorbates on surfaces of noble metals (such as Ni dimers,13 Ce trimers,14 molecular complexes15) and systems of multiple quantum dots.1'16-19 Double quantum dot and dimers of magnetic atoms, for example, are realizations of the two-impurity Kondo model, which exhibits quantum criti-cality. This model has been applied to study the competition between magnetic ordering and Kondo screening.20'21 Fermi and non-Fermi liquid behaviors, ferromagnetic and anti-ferromagnetic correlations, and diverse behavior of heavy fermion systems22 are all believed to result from the competition between the Kondo effect and magnetic exchange interaction.16 Recently, several experimental realizations of the two-channel Kondo model using quantum dots have been proposed.23~27 The two-channel Kondo model exhibits non-Fermi liquid behavior and it has been used in the past to explain the unusual logarithmic temperature dependence of the magnetic susceptibility and linear vanishing of the quasiparticle decay rate in some Ce and U compounds at low temperatures.28 Research on systems of quantum dots thus also sheds light on the behavior of extended bulk correlated materials. NRG The method of choice to study the low-temperature properties of quantum impurity models is Wilson's numerical renormalization group (NRG).29-31 The NRG technique consists of logarithmic discretization of the conduction band, mapping onto a one-dimensional chain with exponentially decreasing hopping constants, and iterative diagonalization of the resulting Hamiltonian. NRG is one of the very few methods which give comprehensive information on the behavior of impurities both at zero and at finite temperatures. 13 The dissertation is structured as follows. Introductory Part I is largely devoted to the theory of tunneling through nanostructures and related topics. The concepts of quantum impurity models, Kondo effect, effective field theories, spin-charge separation, Fermi liquid and non-Fermi liquid systems are introduced in Chapter 2. Chapter 3 presents the renor-malization group approach to the quantum impurity models with a particular emphasis on Wilson's numerical renormalization group, while other numerical methods are briefly presented in Chapter 4. Finally, the transport formalism for calculating the conductance through a nanostructure is given in Chapter 5. Part II presents the main body of the results on the properties of the nanostructures described as single impurity models (Chapter 6), two-impurity models (Chapter 7) or three-impurity models (Chapter 8). Both thermodynamic and transport properties are described; often the knowledge of thermodynamic behavior is essential for proper understanding of the transport properties of a system. Part III is devoted to scanning tunneling microscopy. Chapter 9 describes the construction of a new low-temperature STM and gives some background information on technical aspects of STM operation. The magnetic properties of clusters of magnetic adatoms, with an emphasis on the surface Kondo effect, are described in Chapter 10. Some technical matters are relegated to Appendices. Part I Theory of tunneling through nanostructures 14 Chapter 2 Theory of quantum impurity models This chapter introduces the concept of quantum impurity models (Section 2.1) and the associated Kondo physics (Sec. 2.2). Section 2.3 is devoted to symmetries which play a central role in these models. The notions of Fermi liquids and non-Fermi liquids are presented in Sec. 2.4. It is shown that quantum impurity problems are one-dimensional field theories (Sec. 2.5) and that spin and charge degrees of freedom separate (Sec. 2.6). I conclude with a short introduction to the boundary conformal field theory which provides a simple account of the essence of the Kondo effect (Sec. 2.7). 2.1 The concept of a quantum impurity model Quantum impurity models (QIMs) describe interaction between a point-like impurity with internal degrees of freedom and a continuum of states. For example, the impurity may be a substitutional defect such as a magnetic impurity atom carrying local magnetic moment (the quantum degree of freedom is spin), the continuum may consist of itinerant conduction band electrons of the non-magnetic host material, and the interaction may be antiferromagnetic exchange interaction between the local moment and the conduction band electrons:2'32 a model with such characteristics belongs to the important class of the s-d exchange quantum impurity models. More generally, the impurity degree of freedom may also be orbital moment,33 orbital pseudo-spin,34~36 helicity,37'38 impurity charge (isospin)39-43 or a discrete coordinate (positional pseudo-spin).3 The continuum is typically fermionic (conduction band electrons), but may also be bosonic (collective electron excitations in the host,44 phonon modes45), and the interaction can be hybridization or some generalized exchange interaction. The renewed interest in QIMs is largely due to the fact that many nanostructures belong to the class of quantum impurity systems and that they may be easily characterized by transport measurements.46'47 The paradigmatic case of a QIM is the Kondo model of a spin-1/2 magnetic impurity interacting with a fermionic band with constant density of states p via antiferromagnetic 15 16 CHAPTER 2. THEORY OF QUANTUM IMPURITY MODELS exchange interaction with a coupling constant J that does not depend on energy nor on the direction in the momentum space.48 The Kondo Hamiltonian is H = J2^clßck, + Js-S. (2.1) k/i The first term describes the conduction band of non-interacting electrons: Ckß is the annihilation operator for conduction band electron with momentum k and spin \i. Operator s is the electron spin-density at the position of the impurity: s = E/oUl^')V. (2.2) Here JOß = nrr /_^ ckß (2.3) ViVc k is the combination of the conduction band operators that create an electron at the position of the impurity, i.e. the Wannier orbital centered at the impurity (Nc is the number of states in the band). S is the impurity spin operator; its components obey the SU(2) anti-commutation relation [Si,Sj] = itijkSk- In the conventional Kondo problem S = 1/2. but S > 1/2 generalizations are also studied. The coupling constant J is assumed to be "small". Considerable attention was devoted to the Kondo problem due to unexpectedly complex behavior of this seemingly simple problem. Even though J is small, the problem cannot be treated by the perturbation theory;49 it turns out, instead, that the system is strongly renormalized at temperatures below some characteristic Kondo temperature T k ~ exp(—1/pJ), where p is the density of states of the conduction band at the Fermi level. This unusual behavior is called the Kondo effect. The adjective quantum in quantum impurity model emphasizes that the scattering potential seen by the conduction-band electrons in QIMs is "non-commutative".28 This means that the scattering T-matrix contains terms where the divergent logarithmic terms are multiplied by a commutator of the interaction matrix elements. This commutator can generally be simplified to a commutator of the operators corresponding to the impurity degrees of freedom: for a quantum impurity this commutator is non-zero, while for an impurity with no internal degrees of freedom or for a classical impurity the commutator is zero. In the case of the Kondo problem, the relevant commutator is the spin SU(2) commutator [Si,Sj] = itijkSk- Since the bracket is non-zero, the model is indeed non-commutative and logarithmic terms appear. These terms indicate the break-down of the perturbation theory and the non-trivial behavior of the system. It should be noticed that in the 6*^00 limit, a spin behaves like a classical angular momentum; in the Kondo model with large impurity spin S, the effective interaction is indeed weakened, the Kondo crossover becomes broader and less pronounced.50'51 Other problems that belong to the QIM family are two-level systems like metallic glasses where an atom may tunnel between two positions possessing levels close in energy52 and the 2.2. KONDO EFFECT 17 spin-boson model where a two-level system interacts with harmonic oscillators that mimic the environment responsible for decoherence and dissipation.45. Related problems, though not real QIMs, are also the resonant level scattering problem and the X-ray absorption edge singularity problem.53'54 2.2 Kondo effect a) ooooo b) ooooo OO0OO ooooo ooooo S=l/2 S=0 Free local Kondo screened local magnetic moment magnetic moment Figure 2.1: Quantum impurity with magnetic moment and the Kondo effect, a) Prototype quantum impurity system: an impurity atom carrying net local magnetic moment embedded in a non-magnetic host material, b) At high temperature the impurity moment is nearly free and the system has magnetic response, c) At low temperature the conduction band electrons screen the local magnetic moment in the Kondo effect. The system is then non-magnetic. The Kondo effect in magnetic quantum impurity systems (Fig. 2.1) is a subtle many-electron effect in which conduction-band electrons in the vicinity of the impurity screen the local moment to form a collective entangled non-magnetic ground state at low temperatures. Alternatively, in the language of the boundary conformal field theory55, the Kondo effect consists of the disappearance of the impurity spin degree of freedom from the problem as it is swallowed by the conduction electron spin density (see Sec. 2.7. The Kondo problem was the first known asymptotically free theory:2'56 the local moment is essentially free at high energies (i.e. for high momentum exchange that probes short-distance behavior, Fig. 2.1b), but the system becomes strongly interacting at low energies (longdistance behavior) and the local moment loses its individuality, Fig. 2.1c . In this sense, Kondo physics is akin to color confinement in quantum chromodynamics: particles with color charge (such as quarks) cannot be isolated since the force between a pair of particles increases with the separation. In the context of magnetic impurities this translates into the absence of magnetic response at low temperatures. A characteristic feature of the Kondo effect is the emergence of the Abrikosov-Suhl (or Kondo) resonance, a narrow scattering resonance near the Fermi level, at temperatures less than the Kondo temperature. The Abrikosov-Suhl resonance is of many-particle origin and c) y,"—jp" ^ Kondo S 1 * \ screening / ö n Q ^ cloud / * Jl\ ft !•**¦'! 18 CHAPTER 2. THEORY OF QUANTUM IMPURITY MODELS appears due to correlated behavior of electrons. Since the thermodynamic and transport properties of systems at low temperatures are predominantly determined by electrons with energies close to the Fermi level, this resonance gives significant contributions to the specific heat, magnetic susceptibility and scattering-rate at low temperatures;57 the Abrikosov-Suhl resonance is thus the origin of the anomalies observed in experiments. This resonance is also directly involved in the enhanced conductance through quantum dots and molecules in the Kondo regime7'9. It also appears in the dl/dV measurements in scanning tunneling spectroscopy, albeit in more convolved form of a Fano resonance due to the interference between different tunneling channels4 (see Sec. 10.2 and 10.3). In the scaling picture (described more thoroughly in the next chapter, Sec. 3.1), the effective model of the Kondo model at lower temperatures again takes the form of the Kondo Hamiltonian, but with a temperature dependent exchange constant J{T). The effect of the high-energy electrons is thus to renormalize the exchange interaction, whose strength grows as the temperature is decreased: J(T) = l-pJ]n(kBT/D)' (2'4) where J is the bare exchange constant which appears in the original Hamiltonian. At the Kondo temperature kßTx = Dexp(—l/pJ), J(T) diverges: this indicates that on this temperature scale the electron scattering becomes strong. The scaling approach thus provides a useful qualitative definition of TK. For temperatures below T k the system is said to be in the strong-coupling regime [as indicated by the divergent J(T)] where half a unit of the impurity spin is screened by the electrons.58 As the temperature is increased, the system crosses over to the asymptotically-free local moment regime where the free spin behavior is logarithmically approached.59 Another interpretation of the Kondo cross-over is possible , according to which the Kondo effect is a way for the system to reduce its energy by "tunneling" through different degenerate impurity states. A "Kondo singlet" bound state is generated in this manner. Its bounding energy is on the scale of the Kondo temperature, which can be demonstrated using a simple variational wave-function.2'60'61 The "tunneling" point of view makes it clear that the degeneracy of the impurity ground states plays a very important role: the variational calculation shows that degeneracy enters as a factor in the exponential function of the expression for Tk- Increased degeneracy can thus strongly enhance the Kondo temperature. This implies that if the symmetry of the problem is extended by suitably tuning the model parameters (for example using gate voltages of a quantum dot system), the Kondo temperature can increase; in turn, this may lead to enhanced conductance at finite temperatures.62 It should be emphasized, however, that this may occur only if the symmetry of the entire system is increased; both the impurity and the host system must have matching symmetry2 In bulk impurity systems, for example, the orbital degeneracy may increase the Kondo temperature only if there is orbital exchange, i.e. when the effective model is of the Coqblin-Schrieffer kind with the degeneracy factor iV = 2(2/ + 1). If 2.3. SYMMETRIES IN QUANTUM IMPURITY MODELS 19 an impurity with orbital degree of freedom couples to a single orbital channel, there is no such enhancement.2 In principle, Kondo effect can occur whenever the impurity ground state is degenerate. In the familiar case of the S = 1/2 Kondo model, the two relevant states form a \S = |,Sz = ±|) magnetic doublet. In quantum dot systems, the degeneracy between different spin multiplets can be intentionally induced by tuning the magnetic field since the orbital energy strongly depends on the field, whereas the Zeeman energy is small due to the very small gyromagnetic ratio in GaAs. At the degeneracy point, large zero-bias resonances with increased Kondo temperature are observed.63'64 this is the singlet-triplet Kondo effect that has been a subject of intense studies in recent years.63~75 We may also obtain more exotic doublet-doublet and doublet-quadruplet Kondo effects.62 2.3 Symmetries in quantum impurity models This work mainly concerns quantum impurity models where one or several impurity sites are coupled to at most two single-mode conduction channels, i.e. to two non-interacting continua of states. Such models are appropriate for most systems where the interacting region is embedded between two conduction leads, as is the case in most experimental situations. We assume that each continuum state carries a spin index ß =T> j and that in the relevant low-energy range the dispersion may be linearized. For convenience we set the chemical potential in the middle of the band, which we also choose as our energy zero. The conduction band Hamiltonian is thus "band = 2_^ tk°kßaCkßa, (2-5J k,ß,a where the rescaled wave-number k ranges from —1 to 1, e^ = Dk where D is the half-bandwidth, and a is the channel index, a = 1,2. The density of states is constant, p = 1/(2D). Since tk = e_fc, the conduction band is particle-hole symmetric. In fact, the Hamiltonian (2.5) has very high degree of symmetry. By decomposing the complex fermionic operators c k into real and complex parts (Majorana fermions,) it can be shown that the symmetry group is SO(8).21'76 This symmetry group has same very peculiar properties (such as the triality symmetry between its representations) which play a role in the non-Fermi liquid behavior that this class of problems may exhibit.21'76 The total SO(8) symmetry of the conduction bands is reduced by the impurity-lead coupling to some product of subgroups of SO(8). We thus need to study the possible remaining symmetry of the total model Hamiltonian; we have to find the maximal set of mutually commuting operators that also commute with the Hamiltonian. In addition to the internal SO(8) symmetry of the particles, the field theoretic version of (2.5) exhibits conformal symmetry:21'55>77~81 the theory is rotation, translation and scale invariant in the two-dimensional (2D) space-time. In fact, this holds at low temperatures 20 CHAPTER 2. THEORY OF QUANTUM IMPURITY MODELS for all one-dimensional systems of free fermions.82 This behavior is a direct consequence of the fact that excitations of a Fermi liquid behave in the first approximation as massless particles. The conformal symmetry in 2D is very special because there is an infinite number of generators of the group of local conformal transformations: such high symmetry is extremely constraining and there can in general be only a very small number of free parameters in such conformal field theories (CFTs). In fact, many such theories may be solved exactly83 See also Sec. 2.7. We remark in passing that the symmetry of the Hamiltonian changes under the renormal-ization flow. In particular, symmetries may be restored at low energies and fixed point Hamiltonians generally have higher symmetry than the underlying microscopic model.84 For example, while -ffband is conformally symmetric, the conformal invariance is lost after the impurity is introduced into the system. One of the important insights of the boundary CFT approach is that as the system approaches a fixed point, the conformal symmetry is restored: Lie group symmetry is enhanced to Kac-Moody symmetry. 2.3.1 Spin symmetry SU(2)spin The local spin operator for a single orbital is defined as where a = {ax,ay,az} are the Pauli matrices. In component form, we find Si -= - ( ßj|ßj| — ailail ia iV. 6- — audit-, where we have defined Sf = Sx + iSy and S~ = Sx — iSy. In the absence of magnetic field, the system is isotropic in the spin space and it has SU(2)spin symmetry. The generators are the components of S = VS, (2.8) where index i ranges over all sites of the system (both impurity sites d\ and conduction band levels ck). Both S2 and Sz commute with H, therefore the invariant subspaces can be classified according to quantum numbers S and Sz. Furthermore, when the Hamiltonian is rotationally invariant in the spin space, Sz does not play any role in the diagonalization and it can be taken into account using the Wigner-Eckart theorem (see Appendix A). In the case of an XXZ anisotropy in the spin space (for example due to an anisotropic exchange interaction between the impurities and the conduction band), the SU(2) symmetry 2.3. SYMMETRIES IN QUANTUM IMPURITY MODELS 21 is reduced to 0(2) ~ U(l)spin x Z2 symmetry. U(l)spin corresponds to rotations around the z-axis: Sz remains a good quantum number but the Wigner-Eckart theorem no longer applies. Z2 corresponds to spin inversion, which is still a symmetry operation in this case. Spin inversion can be defined as the mapping ai>ß -> (2ß)ai-ß. (2.9) Here (2//) = ±1 for ß =|, {= ±1/2 is a conventional phase factor. Spin inversion operator U = exp(iirSy) is an element of the SU(2)spin. In the presence of magnetic field applied along the z-axis, the Z2 symmetry is lost and the remaining spin symmetry is just U(l)spin. It should be remarked that for a single site 3 3 Si = 4^T + nü - 2^,pair) = -^(Zrii - Hi), (2.10) where niß = a\ßaißl rti = n^ + n^ and ni)Pair = rt^rtii = rtiirti — l)/2. It thus suffices to calculate (ni) and (n2) to determine the local spin. In the case of two and more sites, we need to know the occupancies, squares of occupancies and scalar products of spin. For example, for two sites we have S2 = (Si + S2)2 = ^ (2m + 2n2 -n\- nj) + 2Si • S2. (2.11) 2.3.2 Charge conservation U(l)charge We define the charge operator Q as Q = J](^-1), (2.12) i where i again runs over all the sites of the problem. We subtract one so that the charge is defined as the excess charge with respect to a half-filled system. In all physically sensible problems, the total charge Q is a conserved quantity. This conservation is associated with a U(l)charge symmetry due to global phase (gauge) invariance. Note that the three operators Q, S2 and Sz are mutually commuting. 2.3.3 Particle-hole symmetry Z2 The particle-hole (p-h) transformation maps particle-like excitations above the Fermi energy into hole-like excitations below the Fermi energy. A system is said to be p-h symmetric when it is invariant with respect to the p-h transformation. This is only possible if all spectral functions of the system are symmetric with respect to the Fermi level, A(u) = A(—u). The p-h transformation is given by the mapping 4,-u - (-lY(2ß)ah-,. (2.13) 22 CHAPTER 2. THEORY OF QUANTUM IMPURITY MODELS The reason for the reversed spin on the right hand side of this expression is that an addition of a spin-up particle and removal of a spin-down particle both increase total spin by 1/2; in other words, the spin of a hole excitation in the ß = —1/2 Fermi sea is 1/2. Index i in Eq. (2.13) has a very special meaning. It must be defined so that its parity alternates between sites connected by electron hopping terms. This requirement is equivalent to demanding that the lattice be bipartite in the sense that each site is connected by hopping only to the sites of the opposite sublattice; in other words, the lattice must be a two-colorable graph, Fig. 2.2. On a simply connected lattice (an open chain of any length or a closed ring with even number of sites) i may simply be the site number. A slightly more complicated case is that of parallel quantum dots that are not inter-coupled by hopping: all parallel dots must have the same parity of index i. To demonstrate the necessity of the alternating sign in the definition, consider how a typical hopping term transforms: t [a\ai+x + a\+la^j ->¦ t ({-l)1 ai{-l)i+la\+l + (—l)i+1Oi+i(—l^aj^ = t (a\ai+x + a\+la^j . (2.14) Without sign alternation, the hopping term would flip sign under p-h transformation and the p-h symmetry of the Hamiltonian would be broken. Furthermore, it should be noted that while {rii — l)2 terms are p-h invariant, {rii — 1) maps to — {rii — 1). Figure 2.2: Lattice representation of the connectivity between the orbitals. a) Examples of bipartite lattices corresponding to Hamiltonians that may exhibit p-h symmetry, b) If the lattice is not a two-colorable graph, the system cannot be p-h symmetric. 2.3.4 Isospin symmetry SU(2)iso Under the p-h transformation, the charge Q —> —Q. In fact, the relation between the p-h transformation and the U(l)charge symmetry is analogous to the relation between the spin inversion (which maps Sz —> — Sz) and the U(l)spin symmetry. This suggest that in the presence of the p-h symmetry, the full symmetry in the charge space might be larger than U(l)charge if certain conditions are fulfilled. This is in fact the case;20 such extended SU(2)iso symmetry is called the isospin symmetry (sometimes also axial charge symmetry).20>85~88 2.3. SYMMETRIES IN QUANTUM IMPURITY MODELS 23 We first define the Nambu spinor42'89'90 by *- = ((-i)4ü ' (2'15) where i has the same special meaning as in the previous section on the p-h symmetry. The isospin up component (a = \ =|) is a particle creation operator, while the isospin down component (a = - \ =\) is a particle annihilation operator. Often only the spin up Nambu spinor L = rj^ is needed: ä=(, i'L ) ¦ (2-16) v(-i)Xi/ The Nambu spinors define local isospin operators20'91 I = ti(i N/2 there will be residual spin (under-screening), while for S < N/2 the electrons tend to overscreen the impurity. In fact, both for S > N/2 and for S < N/2 the residual spin is \S — N/2\. The important difference lays in that the residual exchange interaction is ferromagnetic and it scales to zero for the case of underscreening, while the residual interaction is antiferromagnetic in the case of overscreening. Mehta et al. have shown that while for S > N/2 the system is a FL, the concept of a FL needs to be refined.98'99 A S = N/2 system with complete compensation is a regular FL , a S > N/2 system with residual spin is a so-called singular FL. In a regular FL, the inelastic scattering vanishes quadratically and on the lowest energy scale <7totai(w) = ^elastic (w); where a is the scattering cross-section. In singular FL, however, the inelastic scattering falls off much more slowly. NFL systems, on the other hand, have characteristically different properties.28 For S< N/2 the residual interaction is relevant and the FL fixed point is unstable; this leads to non-trivial physics.99'100 The thermodynamic quantities of NFL systems diverge at low temperatures, the resistivity decreases as T1/2 and there is a finite residual entropy which is not a logarithm of an integer number . In this work we will meet two such models: the over-screened two-channel Kondo (2CK) model (with S = 1/2 and N = 2, see also Section 6.2) , and the two-impurity Kondo model (Sec. 7.4) . Some further examples of NFL systems are the two-channel spin-flavor Kondo model,92'101 the single-channel Kondo problem with J =3/2 conduction band,102 the compactified (a — r) Kondo model103-107 and the three-impurity Kondo problem.37'38 NFL behavior usually appears for special values of model parameters. It often occurs that two different FL phases are separated by a continuous quantum phase transition (QPT) and that precisely at the transition point the system has a NFL low-temperature fixed point. Near the phase transition, the system may exhibit NFL characteristics at finite temperatures, but as the temperature is decreased it eventually evolves into a FL ground state. The levels in the finite-size spectra of NFL systems are not equidistant, but can often be expressed as fractions. In the 2CK model, the lowest levels are, for example, 0, 1/8, 1/2, 5/8, 1+1/8, ...79'108 (see Sec. 8.1). In conformal field theory, excitation energies are determined by the scaling dimensions of operators characterizing the fixed point: one 2.5. EFFECTIVE FIELD THEORIES OF THE KONDO PROBLEM 27 operator for each eigenvalue. The fact that eigenvalues are not merely integers or half-integers is a direct proof that the system is a NFL, since its excitations cannot be expressed in terms of fermionic operators.108 Emery and Kivelson explained the NFL behavior in the 2CK model by the observation that only one of the impurity's Majorana degrees of freedom couples to the conduction electrons109. Since one Majorana fermion corresponds, roughly speaking, to one half of a physical Dirac fermion, the deviation from usual FL behavior is not surprising. The idea of obtaining NFL fixed point by "twisting" an odd number of Majorana fermions was further elaborated by J. Maldacena and A. W. W. Ludwig76 and J. Ye.92'110-112 Calculation of thermodynamic properties alone may not suffice to ascertain if the ground state is FL or NFL. Instead, correlation functions or excitation spectra must be determined. Using the NRG, both are easily accessible. 2.5 Effective field theories of the Kondo problem While models of physical systems that arise in the condensed matter theory naturally take the form of lattice models, effective models may be defined on the continuum. The field theoretic version of the Kondo Hamiltonian (2.1) is a one-dimensional (ID) relativistic quantum field theory (QFT). The theory is one-dimensional since the magnetic impurity is assumed to couple only to a one-dimensional continuum of conduction electron states with s symmetry about the impurity site:55 this holds more generally and, in fact, all QIMs are essentially one-dimensional. The theory is said to be relativistic since the electron and hole excitations in the vicinity of the Fermi level are dispersionless and therefore behave as ultrarelativistic massless particles. Due to the presence of an impurity interacting with the ID continuum, QIMs form a class of quantum models with properties in between 2D models, where critical behavior is possible, and true ID models where criticality does not occur.55 In fact, the Kondo effect was described as an "almost broken symmetry",113 a low-dimensional critical phenomenon involving long-time fluctuations at the magnetic site, but no critical fluctuations in space. In QIMs, low dimensionality of fluctuations prevents the development of a true broken symmetry; as a consequence, the transition to a Kondo correlated state is not a phase transition, but rather a cross-over. The validity of the effective QFT of the Kondo problem is limited to energies low enough that the linear approximation to the dispersion relation is possible. In real space, the Hamiltonian is expressed in terms of a field if)ß(x) defined on a continuous line parameterized by some fictitious position x: H = ivFJ2 I" dx^i(x)^^ + vF\Js(0) ¦ S. (2.28) J— OO ^^ /J, 28 CHAPTER 2. THEORY OF QUANTUM IMPURITY MODELS Here Vp is the Fermi velocity, A = p J is the dimensionless Kondo antiferromagnetic coupling constant and Js{x) is spin density Js(x) = J>i(x)±<7^VyM, (2-29) QFTs need to be regularized to remove the divergencies that appear due to the infinite number of quantum modes;82 in effective QFTs of lattice models a natural choice of the high-energy cut-off is provided by the lattice spacing. 2.6 Spin-charge separation Low-dimensional field theories have unique properties due to topological restrictions in reduced dimensionality; for example, fermions constrained to live on a ID line can scatter only forwards and backwards. A notable effect in one-dimensional systems is the separation of electron spin and charge which had been intensively studied in Luttinger liquids:114 fundamental low-energy excitations are not charged spin-1/2 Fermi-liquid quasiparticles, but rather spin-1/2 neutral particles (spinons) and charged spinless particles (holons). Such behavior has been found, for example, in one-dimensional solids such as SrCu02115 and ballistic wires in GaAs/AlGaAs heterostructures.116 Spin-charge separation also occurs in the Kondo problem. Using bosonization techniques,106'108'109'117'118 conduction band fermion fields can be described in terms of spin-up and spin-down boson fields. These bosons correspond to the particle-hole excitations in the conduction band and they can be recombined to form separate spin and charge fields which are essentially independent, but subject to a gluing condition 108>118 which is the only remnant of the charge-1, spin-1/2 nature of physical fermion particles. In the single-impurity spin Kondo problem, the impurity spin couples only to the spin field, while the charge field is decoupled.55'103'119 During the Kondo cross-over, the spectrum of charge excitations remains unchanged, only the spin sector is affected.55 In this sense, the spin and charge degrees of freedom are separated. In the absence of interactions, the Fermi velocity Vf is the only characteristic velocity scale of the problem. When interactions in a one-dimensional system lead to the spin-charge separation, two different velocities vs and vc appear; these are the velocity of spin and charge excitations.82 Specific heat coefficient depends both on spin and charge modes and is given by120 7/>=ift + f), (-0) while spin susceptibility only depends on the spin mode and is given by ^ = ^, (2.31) Xo vs 2.7. CONFORMAL FIELD THEORY 29 where 70 and Xo are specific heat coefficient and spin susceptibility of non-interacting electron gas. Wilson ratio is the ratio of spin susceptibility and the specific heat coefficient: it thus measures the fraction of the specific heat coming from the spin degrees of freedom.55'56 It is given by RW = *!p = ^- (2.32) lilo vc + vs If there is no spin-charge separation, this ratio is equal to one. Deviations form unity are a sign of spin-charge separation;120 in the single-channel Kondo problem, for example, we find Rw =2. 2.7 Conformal field theory The boundary conformal field theory (CFT) for impurity models is a generalization of Noziere's local Fermi liquid approach. It is applicable to both Fermi liquid and non-Fermi liquid Kondo systems and provides a description of the ground state and the leading corrections that determine finite temperature behavior. The technique was developed in a series of papers by I. Affleck and A. W. W. Ludwig.21'55'56'77~81'121'122 It can be applied to multi-channel and higher-spin Kondo effect,28'80'81'121 two-impurity Kondo effects ,21,79 impurity assisted tunneling, and impurities in one-dimensional conductors or ID antifer-romagnets.56 An important feature of the approach is that the symmetry of the problem and the separation of the spin and charge sectors is directly exhibited. The essence of the CFT approach to impurity problems is the reformulation of the theory in terms of separate degrees of freedom using non-Abelian bosonization.56 In the case of a single conduction band, we introduce spin and charge (isospin) "currents" (densities) as (2.33) aß where Q(x) = is the real-space Nambu spinor. Normal ordering (double dots) has been introduced to remove divergences due to filled electron levels below the Fermi level. The conduction band Hamiltonian can then be rewritten in the Sugawara form TTVf 3^",n"-"' '3 #1 band I 1 ^ • JC ¦ JC ¦ +- V • Js ¦ Js n (2.34) Here Jn and Jn are the Fourier modes of the real-space currents J (x) and J (x) in a finite system of length 21. They each satisfy SU(2)i Kac-Moody commutation relations [J n ' ,J m ' \ — ^ abc'Jn+m ' "a,&"ra+m,077^'> {Z.ODj 30 CHAPTER 2. THEORY OF QUANTUM IMPURITY MODELS where tabc ig the antisymmetric tensor.21 The modes from different sectors commute: [JŽ*,J%b]=0. (2-36) This commutation relation embodies the (trivial) spin-charge separation of the free electrons. The full Kondo Hamiltonian is H = Hband + ^ (A J2 Jn • S j • (2.37) I draw attention to the fact that the impurity spin couples only to the spin degrees of freedom of conduction band electrons, while charge degrees of freedom are unaffected. At a special value A =1/3, we can introduce a new current Js = Js + S which satisfies the same Kac-Moody commutation relations as the old currents. The spin part of the Hamiltonian then becomes (up to a constant term) H(s) = ^J2-3 :&-JnS-, (2-38) n from which the spin impurity S has disappeared (it was "absorbed" by the conduction band). The charge sector remains unaffected by this change. The special value A = 1/3 is identified with the strong coupling fixed point of the problem.55 Even though the spin Kondo effect occurs in the spin sector, without involving the charge sector, the spin and charge degrees of freedom are not entirely decoupled; they are constrained by the gluing condition. The gluing condition declares which combinations of quantum numbers are allowed taking into account the charge-1 spin-1/2 nature of physical particles - electrons. In the present context the gluing condition depends on the boundary conditions (b. c.) imposed on the field if)(x). We first consider the case of anti-periodic b. c, rtp{l) = —rtp(—l). We can obtain half-integer spin only with an odd number of electrons (i.e. for half-integer isospin).55 Therefore 2IZ and 2SZ must have the same parity; this is the gluing condition. There are thus two Kac-Moody conformal towers with highest-weight states having (I,S) = (1/2,1/2) and (I,S) = (0,0), respectively. For periodic b.c. rtp{l) = if)(—l), and keeping in mind that the z-component of the isospin is defined with respect to half filling, we obtain half-integer spin for integer isospin and integer spin for half-integer isospin; 2IZ and 2SZ must then have different parity. There are two conformal towers, (I,S) = (1/2,0) and (I,S) = (0,1/2). Note that changing the b. c. from periodic to anti-periodic (or vice versa) amounts to imposing a phase shift of 7r/2 on the wave function.77 In the single impurity Kondo model, the finite-size spectrum of the strong coupling fixed point is obtained by a fusion in the spin sector.55'77 This means that the isospin sector remains intact, while the spin quantum number changes as S —> 1/2 — S. As a consequence 2.7. CONFORMAL FIELD THEORY 31 (1/2,1/2) —> (1/2,0) and (0,0) —> (0,1/2), i.e. the gluing conditions change from those for the anti-periodic b. c. to those for periodic b. c, and vice versa, which is equivalent to the characteristic ?/2 phase shift for quasiparticle scattering in the Kondo regime. Chapter 3 Renormalization group The renormalization group (RG) technique, in particular as applied in the field of condensed matter physics and many-body theory, is an important tool to study the effective behavior of systems at low energies and long wavelengths, i.e. their macroscopic response functions.29-123-129 RG is more than a computational tool that provides us with effective Hamiltonians; the concepts that emerged from RG (renormalization flow, scaling, running coupling constants, fixed points, criticality, critical exponents, operator content, relevance, irrelevance, marginality, universality, etc.) are the central notions of modern physics. K. G. Wilson's contribution in this domain was acknowledged by the Nobel prize awarded to him in 1982 "for his theory for critical phenomena in connection with phase transitions". Renormalization is a way of understanding the relation between the different ways a physical system behaves at different energy scales. To study a system at low energies, the irrelevant high-energy short-wavelength degrees of freedom are integrated out to obtain an effective description in terms of modified, "renormalized" coupling constants. Let us consider an example which - while not computationally practicable - illustrates well the perspective from which a system is considered from the renormalization group point of view. We focus on electrons in a piece of metal, neglecting the motion of nuclei (phonons), Fig. 3.1a. As a first step of the renormalization, we can imagine "tracing out" the core electron levels to obtain an effective description of the valence levels. This effective description could, for instance, take the form of a tight-binding model for orbitals near the Fermi level with electron-electron interactions, i.e. a Hubbard-like model, Fig. 3.1b. The next step might consist of "tracing out" the charge-fluctuation degrees of freedom (on the scale of U) to obtain the effective dynamics of spin degrees of freedom (on the scale of t2/U), i.e. a Heisenberg-like model, Fig. 3.1c. A final RG step might then involve "tracing out" the local moments to obtain a long-wavelength description in terms of collective modes such as the spin density waves on the basis of a Ginzburg-Landau functional approach, Fig 3. Id. We see that a simple Hamiltonian arises from more complicated ones; this is the origin of the universality. Renormalization is thus an essential ingredient of model building in many-body theory.123 32 3.1. RENORMALIZABILITY, UNIVERSALITY AND SCALING THEORIES 33 a) Microscopic Hamiltonian Figure 3.1: Renormalization as a way to proceed from complex microscopic models to simple universal effective models. In this chapter I present several aspects of the renormalization group theory. Section 3.1 is devoted to the concepts of renormalizability and universality; it also describes the RG approach to the Kondo model in the form of a simple scaling theory. Section 3.2 introduces Wilson's numerical renormalization group (NRG) and contains a brief review of its many applications, while Section 3.3 is an overview of my implementation of this technique for studying general multi-impurity multi-channel quantum impurity models. The following sections are devoted to the main elements of NRG: logarithmic discretization, hopping Hamiltonian (Sec. 3.4), symmetries and basis construction (Sec. 3.5), RG transformations, iterative diagonalization and truncation (Sec. 3.6), finite-size spectra, fixed points, expansions around fixed points, thermodynamic quantities, correlation functions and dynamic properties (Sec. 3.7), and recursion relations (Sec. 3.8). Finally, Section 3.9 introduces the density-matrix NRG technique and its implementation in the basis with well defined charge and total spin quantum numbers. 3.1 Renormalizability, universality and scaling theories The Kondo Hamiltonian, Eqs. (2.1) and (2.28), defines a renormalizable quantum field the-ory_ 130-132 This [s equivalent to saying that the field theory is fully determined by a limited number of running coupling (renormalization) constants. The notion of renormalizability is therefore related to the concept of universality: the existence of only a limited number of renormalization constants implies that apparently different Hamiltonians can be mapped into each other at low energy scales and will thus exhibit universal properties. Writing the dimensionless coupling constant in the Kondo model as A = pJ, we find that the Kondo model in the wide-band (D —> oo) limit is a scale-invariant theory, i.e. there appears to be no characteristic length scale. Nevertheless, a length scale given by L = vf/'(&b2V) is dynamically established due to many-particle corrections. The phenomenon where a dimensionless coupling constant (A) becomes dimensionful (TK) is known in particle physics as dimensional transmutation and it is closely related to the emergence of the scaling laws.133-136 In the context of the Kondo problem, the dimensional transmutation means that instead of a range of theories, parameterized by a dimensionless coupling A = pJ, we have a range of theories differing only in the value of a dimensional parameter, the Kondo temperature TK.133 This implies that the behavior of all problems in the universality class of the Kondo problem can be described by universal functions; for example, the magnetic ¦ooo ¦#¦ b) Hubbard model (tight-binding lattice) c) Heisenberg model (spin lattice) d) Effective model describing spin density waves 34 CHAPTER 3. RENORMALIZATION GROUP particle excitations hole excitations Figure 3.2: Cutoff renormalization: the particle and hole excitations from the hatched regions at the top and bottom of the conduction band are integrated out to obtain an effective Hamiltonian at lower energy scale. susceptibility \ is a universal function oiT/TK.29 The Kondo problem therefore has scaling property. Some of the first applications of the RG ideas to the Kondo problem are due to P. W. Anderson. 137~140 Among other results, he derived the scaling laws of the Kondo model in the perturbative regime using a simple ("poor man's") cutoff renormalization technique.140 Cutoff renormalization consists of tracing out the degrees of freedom in the two infinitesimal energy intervals near the top (electrons) and bottom (holes) of the conduction band and determining the effect these states have on the coupling constants in the Kondo Hamiltonian, see Fig. 3.2 and Appendix D. It is found that the effective model of the Kondo model is (at least approximately) the same Kondo model with modified coupling constants, i.e. in this so-called scaling regime the Kondo model is self-similar under the renormalization flow. In the first-order perturbative renormalization, the following scaling equation is obtained after some simplifications:2 dj d\nV = -pJ2 (3.1) with J{V = D) = J, where D and J are bare bandwidth and bare coupling, J is the running coupling constant, V is the running parameter (energy scale), and p is the spectral density (density of states) of the conduction band. As the energy scale is reduced, the coupling constant increases. Assuming constant p, a solution may be obtained in closed form: The physics on the energy (or, equivalently, temperature) scale V depends on the renor-malized parameter J7(T>) rather than on the bare parameter J: the renormalized parameter takes into account the effects of the high-energy intermediate states. This approach works as long as p J < 1, so that the use of the perturbation theory makes sense. The scale of the Kondo temperature is determined by the value of the running parameter V where the renormalized coupling constant becomes large, for example p J ~ 1. We obtain T-, K Dexp (------ pj (3-3) 3.2. NUMERICAL RENORMALIZATION GROUP 35 On this energy scale the perturbative renormalization fails. Higher-order scaling equation can be calculated:2 J^ = -PJ2 + P2J3 + 0(J4). (3.4) It gives a better estimate of the Kondo temperature, however pushing scaling calculations to higher orders does not reveal the behavior of the problem as the temperature tends to zero. The reason is simple: the behavior of the Kondo system is, in fact, qualitatively different for T ^$> Tk and for T Tk, the perturbative scaling approach gives adequate results, while for T 129 Being non-perturbative, it does not suffer from logarithmic singularities, as scaling approaches do. NRG builds upon the RG approach to the Kondo problem of Anderson, Yuval and Hamann,123,137-139 however in NRG the RG transformations are performed numerically. The essential advantage of this approach is that the calculation need not be guided by "physical intuition" and is therefore unbiased: however, by the same token there is no straight-forward description in terms of running coupling constants to provide a simple physical picture. Schematically, NRG consists of logarithmic discretization of the conduction band(s) and of iterative diagonalisation of a series of Hamiltonians. The method was expounded in K. G. Wilson's seminal paper "The renormalization group: Critical phenomena and the Kondo problem" (Rev. Mod. Phys., 1975)29 where it was applied to numerically solve the Kondo problem. This work represents a turning point in the field of impurity problems since an essentially exact solution for the temperature dependence of the thermodynamic quantities in the cross-over region between the high-temperature local-moment regime and the low-temperature strong-coupling regime was obtained for the first time. NRG has since then become the principal tool in the field of the quantum impurity physics. The approach was used to study the potential scattering in the Kondo problem 5132>141 the s-d problem with spin 1 ?98>142>143 the two-channel Kondo problem ;28>8o, 91,144,145 particle-hole symmetric30'146 and asymmetric31 Anderson model, orbitally degenerate Anderson model147-150 and models where different magnetic configurations are mixed.151 More complex multi-impurity problems are also tractable: significant effort was devoted to the two-impurity Kondo problem ?20>152-154 two-impurity Anderson model155'156 and, more recently, to clusters of three and more Anderson impurities.157'158 In addition, local phonon modes 36 CHAPTER 3. RENORMALIZATION GROUP can also be taken into account as in the Anderson-Holstein model.39'40' 159~161 New direction are applications of NRG to quantum impurity problems with bosonic continuum bath,162 non-trivial density of states (pseudo-gap) of the conduction band,163-168 non-Fermi liquid fixed points 5105>169-171 magnetic impurities in superconductors172'173 and quantum phase transitions .174>175 Recently, NRG has become widely applied to study conductance through single, 176~179 double,93' 180~188 triple157'158'189 and multiple quantum dots,51'190 including quantum dots attached to ferromagnetic leads,191'192 and to study singlet-triplet transitions.64'73'75'193 A number of exotic Kondo states were found, among them the SU(4) Kondo effect.194 NRG is increasingly often used as the impurity solver in the dynamical mean-field theory (DMFT) approach to lattice problems.129'195'196 DMFT builds on the observation that in the limit of infinite connectivity, lattice models can be mapped to effective impurity models subject to a self-consistency condition that relates the impurity Green's function to the hybridization function. Examples of using NRG as the solver are applications to the Hubbard model,197 the periodic Anderson model,198 the Hubbard-Holstein model199'200 and the two-band Hubbard model.201 3.3 Implementation overview For the purposes of this dissertation a new NRG code (named "NRG Ljubljana"202) was designed and implemented from scratch. The main design goals were flexibility in setting up new problems, ease of taking into account various symmetries, and speed. The code was implemented in a layered architecture, see Fig. 3.3. The cornerstone is a Mathematica package sneg for performing calculations with second quantization operators. This package is used, on one hand, in deriving the recursion relations for the NRG iteration and, on the other hand, for exactly diagonalizing the initial Hamiltonian and transforming the matrices of all operators of interest in the basis of eigenstates of the Hamiltonian in each invariant subspace. The NRG iteration routines are implemented in C++ for speed. Diagonalisations are performed using dsyev and dsyevr routines from the LAPACK library,203 while all other matrix and vector operations use the ublas library from the project boost. Standard Template Library containers are heavily used, which makes the code easy to read (and maintain) and helps avoid memory leaks. Additional information can be found on the "NRG Ljubljana" home page http://nrgljubljana.ijs.si/. The package was released freely for general use under the GNU Public license.204 3.3.1 Package sneg Package sneg is a collection of transformation rules for Mathematica, which simplifies calculations using the anti-commuting fermionic second quantization operators. The foundation is a definition of non-commutative multiplication with automatic reordering of opera- 3.3. IMPLEMENTATION OVERVIEW 37 Model Parameters Observables Input to NRG: eigenstates, irreducible matrix elements Output: eigenstates, expectation values spectra Results Problem setup Mathematica NRG iteration C++ Postprocessing Perl, Mathematica Figure 3.3: The three-step procedure from the problem definition to the results in "NRG Ljubljana" code. tors in a standard form (normal ordering with creation operators preceding the annihilation operators), which takes into account selected (anti-)commutation rules. Standard form reordering allows simplification of expressions and the choice of normal ordering permits efficient evaluation of matrix elements in a given basis. Some of the additional capabilities of the package that are relevant to the NRG code are: • Generation of basis states with well-defined number Q and spin S (or other quantum numbers). • Generation of matrix representations of operators (in particular of the Hamiltonian) in selected basis. • Collection of functions that generate various operator expressions, such as electron number, electron spin and isospin, one-electron and two-electron hopping, exchange interaction, etc. • Occupation-number representation of states and evaluation of operator-vector expressions. Among miscellaneous features of the package are manipulation routines for operator expressions (canonic conjugation, spin inversion), calculation of vacuum expectation values of operator expressions, transformations from product-of-operators to occupation-number representations of states and vice-versa, Dirac's bra-ket notation, simplifications using Wick's theorem, support for sums over dummy indexes (momentum, spin) and simplifications of such expressions, etc. Package sneg is useful beyond NRG calculations. It has been applied to perform exact diagonalizations on Hubbard clusters, perturbation theory to higher orders51 and calculation of commutators of complex operator expressions. It should also be 38 CHAPTER 3. RENORMALIZATION GROUP suitable for educational purposes, since it makes otherwise tedious calculations a routine operation: the nicest feature is perhaps that the use of the package prevents inauspicious sign errors when commuting fermionic operators. Package sneg was also released freely for general use under the GNU Public license (http://nrgljubljana.ijs.si/sneg). 3.4 Logarithmic discretization The essential element of the NRG approach is the logarithmic discretization of the conduction band whereby the infinite number of the continuum degrees of freedom is reduced to a finite number; this renders the numerical computation tractable. If we attempted to discretize the band linearly, we would obtain a single interval centered around k =0 that would contain an infinite number of different energy scales: this is undesirable, since it is known that in the Kondo problem excitations at each energy scale contribute equally. It is thus preferable to perform a discretization which divides the band into a set of different energy scales; in this manner the energy-scale separation - a known property of QIMs - is achieved explicitly. Viewed from another perspective, the logarithmic mesh gives a good sampling of the states near the Fermi energy which play an essential role in the Kondo problem. Wilson's logarithmic discretization consists of the following steps:29'30 1. The conduction band is divided into slices of exponentially decreasing width, for example into intervals /" =[-A~m, -K~{m+l)}D for holes and /+ = [A"(m+1), A~m]D for electrons with m > 0, see Fig. 3.4. A > 1 is called the logarithmic discretization parameter (parameter Lambda in "NRG Ljubljana"). An upper bound of A = 3 has been established for reliable computation of thermodynamic properties in this discretization scheme.29'205 2. Each interval is Fourier-transformed, i.e. we construct a complete set of wave func- tions rtpml inside each interval: 2^ exp(iumle), for eL/„ (3.5) 1 1 ~7Š~ 1 1 Figure 3.4: Original Wilson's division of the conduction band into bins of geometrically decreasing width. Each thick colored line corresponds to a state which represents the entire interval of conduction band states delimited by a pair of dashed lines. 3.4. LOGARITHMIC DISCRETIZATION 39 c) | Impurity |---(0)-(1)-(2)-(3)--- Figure 3.5: Various representations of the logarithmic discretization in quantum impurity problems, a) Discretized problem and coupling connectivity of wave-functions ip^. b) Onion-shell representation of Wannier orbitals around the impurity, c) Hopping or Wilson chain Hamiltonian. where um is the fundamental Fourier frequency in the mth interval A-1), and / > 0. Functions if)- are defined similarly for e G I'r wm = 27rA"7(l-" The first wave function (/ =0) in each interval is a constant. Only such "average states" t/j^0 couple to the impurity, while other Fourier components are localized away from it, Fig. 3.5a. We therefore retain only rip^0 and drop the remaining states from consideration. This is clearly an approximation, since the states ip^ couple to ipm0- It was shown that this coupling goes to zero as A —> 1 (i.e. in the continuum limit) and that even for moderately large A = 2 up to A =3 the approximation is good.29 Physically, we are neglecting those conduction band states that are localized far away from the impurity in the real space and, at the same time, far away from the Fermi surface in the reciprocal space.29'206 There is no a priori justification for this approximation; in the words of Wilson: "The only true justification for using the logarithmic division is that a successful calculation results." Unitary transformation to a tridiagonal basis is performed using the Lanczos algorithm. The initial state is the Wannier orbital about the impurity site; this is the orbital to which the impurity is coupled in the standard Kondo problem. Lanczos states correspond to creation operators /J, f{ and have a radial extent of A1/2//^, A3/2/fcp, ... about the impurity:29'91 they form "onion shells" around the impurity, Fig. 3.5b. The conduction band Hamiltonian rewritten in this basis takes the form of a one-dimensional tight-binding model with interacting impurity attached to its end, Fig. 3.5c. This tight-binding Hamiltonian is named the hopping Hamiltonian or the Wilson chain. The problem is thereby reduced to an effective one dimensional problem. In the A —> I limit, a continuum model is recovered. It may be noted that the low-energy levels for small A are approximately equidistant (as in the field theory defined on a finite-size system), while for moderate A the energies are spaced exponentially starting with the third level, Fig. 3.6. CHAPTER 3. RENORMALIZATION GROUP W4 Figure 3.6: Positive one-particle eigenenergies of the Wilson chain Hamiltonian with even number of sites as a function of the discretization parameter A. 4. The total Hamiltonian is defined on an infinitely long chain with exponentially decreasing coupling constants: H — Himp + Hc H chain #chain - /. /.6^ ^ (ll,ßafn+l,ßa + H.c) , (3.6) n=0 ßa where Himp is the impurity Hamiltonian, HG is the coupling Hamiltonian, and -ffcnain is the Wilson chain Hamiltonian. In the original Wilson's scheme Lra are correction factors 1 - A"(ra+1) [(l-A-(2n+l))(l-A-(2n+3))]V2 which rapidly tend to 1. The coupling Hamiltonian Hq must be rewritten in terms of the Wilson chain operators. In the simplest case of the single-impurity one-channel Kondo problem, it is equal to Hr JS- ß,ß j Jo,ß I 2 aw' J /. 0,n'; (3. where S is the impurity spin operator. It was found that to connect the numerical results at finite A to the A —> 1 limit, it is necessary to correct the coupling constant T (Anderson-model-like coupling) or J (Kondo-model-like coupling) by multiplying it by a correction factor30'207 AA = -r- 11+ A- 21-A- lnA. (3.9) This correction can be enabled in "NRG Ljubljana" by setting the option Alambda to true. While AA is typically small (A2 ~ 1.04, A3 f« 1.1, AA f« 1.16 ), it must be recalled that T or J enter the exponential function in the expression for the Kondo temperature, therefore AA has a significant effect. 3.4. LOGARITHMIC DISCRETIZATION 41 ill! iiiliii 1 I -1 - 1 Ä" 1 1 1 1 W A1" 1 Ä 1 z =1/4 z =1/2 z = 3/4 Figure 3.7: Yoshida's discretization scheme and the interleaved method. One should keep in mind that for A^ 1, the Hamiltonian obtained from the discretization is only an approximation to the original impurity model and that, strictly speaking, NRG is not an exact method. Nevertheless, by comparing results with known analytical solution (Bethe Ansatz), a remarkable agreement is found. The principal advantage of NRG is its applicability to more complex problems where analytical approaches fail. Improved discretization schemes are the interleaved method (also known as the "z-trick") 154,205,208 an(j an approacn based on an over-complete basis of states;209'210 the latter was found to give excellent results and was used in most of the calculations presented in this work. All three approaches are implemented in "NRG Ljubljana": the corresponding configuration options are disc=wilson, disc=yoshida and disc=campo, i.e. they are named after the first authors of the publication where they were introduced. In the interleaved method (disc=yoshida), the first positive-frequency interval is 1 > e > A~z, the others are A1-Z~m > f > _/\-*_m (m=1, 2, • • •), see Fig. 3.4; for z =1 this reduces to the original discretization. We then average over the sliding (twist) parameter z (z in "NRG Ljubljana") in the interval 0 1 limit in the case of both improved methods. In the case of Yoshida discretization, the correction factor A^ was used, while no such correction is necessary in the case of Campo discretization. Varying the sliding parameter z can also be used to assess numerical accuracy of the results by comparing eigenvalue spectra computed for different values of z. This is particularly important if the finite-size spectrum itself is the result of interest: the z-trick namely cannot be used to average the spectra in a meaningful manner. For large A, the spectra for 42 CHAPTER 3. RENORMALIZATION GROUP IIA 1/8 H m In 2 "~'—i—'—r U/D=0.5 r/u=o.o8 5=0 S i f-^' I i I i I i — A=2 ¦ - A=3 - A=at A=6 ¦ ¦ A=8 -5 -4 -3 -2 log10(T/D) -4 -3 -2 log10(T/D) (a) Campo's discretization, efFect of increasing A (b) Yoshida's discretization, effect of increasing A Figure 3.8: Comparison of magnetic susceptibility and entropy of the single-impurity Anderson model calculated using two different discretization types for different values of the discretization parameter A. The coarse results were post-processed by averaging over z (the z-trick) and even-odd effects were removed by averaging over two consecutive NRG iteration steps. different z can differ substantially, even though the z-averaged quantities (such as spectral functions) are an excellent approximation to the A —> 1 results. There is no good a priori recipe for choosing the value of A, the number of values of z, and the number of states retained in the NRG iteration; this depends on the number of impurities (i.e. the degeneracy), the values of model parameters and the quantities computed. For each new class of problems, a convergence study should be performed. The majority of computations in this work were performed with A = 4 and for 4 or 8 values of z. The density of states (DOS) in the conduction band is usually taken to be independent of energy, i.e. p = const, which is also known as the flat-band approximation. This choice is particularly convenient as it leads to analytic simplicity and some calculations can be performed in closed form.211 In addition, RG treatment of the problem has shown that in the case where all the energy scales of the problem are much smaller than the bandwidth, the form of the DOS at large energies is irrelevant in the RG sense.29 Nevertheless, NRG calculations can be setup for an arbitrary DOS of the conduction electrons.163'165'208'210'212 "NRG Ljubljana" has built-in support for flat bands (band=f lat) and for tight-binding bands with cosine dispersion (band=cosine) where the hynridisation function is T(e) oc \Jl — (e/D)2. There is also a stub for arbitrary DOS to be defined by the user (band=dmf t), which is required if NRG is used as the impurity solver in DMFT. 3.5. SYMMETRIES AND BASIS CONSTRUCTION 43 3.5 Symmetries and basis construction The efficiency of NRG calculations can be significantly improved if the symmetries of the problem are taken into consideration. In addition to the performance concerns, the implementation of symmetries is important on a more fundamental ground: if the conservation laws are not built-in, numerical round-off errors tend to induce accidental symmetry breaking which, if relevant, can lead to erroneous results. Continuous symmetries (such as SU(2)spin, SU(2)isospin, U(l)charge) can be taken into account by constructing the basis states using the Lie group representation theory (i.e. Clebsch-Gordan coefficients and the Wigner-Eckart theorem, App. A).30 Discrete symmetries (such as parity or particle-hole Z2 symmetries) can be taken into account by projecting basis states to invariant subspaces with well defined Z2 quantum number using suitable projection operators. By taking explicitly into account the full symmetry of the problem, a formerly intractable problem falls within the reach of modern computers. For example, while not so long ago it was deemed difficult to obtain anything but the NRG eigenvalue flows for the two-channel problems,28 it is now possible to perform calculations of thermodynamic and even dynamic properties of multi-impurity two-channel problems.158 In the current implementation of "NRG Ljubljana", the following symmetry types are supported: • U(I)charge x U(I)spin, i.e. good quantum numbers are charge Q and spin projection Sz (symmetry type QSZ in "NRG Ljubljana") - suitable for general quantum impurity models in the presence of the magnetic field: • U(I)charge x U(I)spin x Z2, i.e. good quantum numbers are charge Q, spin projection Sz and parity P (symmetry type QSZLR in "NRG Ljubljana") - suitable for models with reflection symmetry: • U(I)charge x SU(2)spin, i.e. good quantum numbers are charge Q and total spin S (symmetry type NRG) - suitable for general QIMs in the absence of the magnetic field: • U(I)charge x SU(2)spin x Z2, i.e. good quantum numbers are charge Q, total spin S and parity P (symmetry type QSLR): • SU(2)iso x SU(2)spin, i.e. good quantum numbers are total isospin I and total spin S (symmetry type ISO) - suitable for QIMs at the particle-hole symmetric point: • SU(2)iso x SU(2)spin x Z2, i.e. good quantum numbers are total isospin I, total spin S and parity P (symmetry type ISOLR). For each symmetry type, the basis and coefficients for various NRG transformations are derived symbolically using a Mathematica program that uses the sneg library. In the following, we will describe the (Q, S) and (I, S) basis; other symmetry types are conceptually similar. 44 CHAPTER 3. RENORMALIZATION GROUP (q, s) States (k) (q's) Stat6S(k) [-1, 1 ) b\,a\ [- , 1 ) 0) It (0,0) b\b\,2(a\b\-a\b\),a\a] (l|o) a{a| (0'1) b\a\, (a) One channel (2,0) aiaj&i&j (b) Two channels Table 3.1: Basis states for additional sites for (Q,S) basis represented by the corresponding electron creation operators that need to be applied on the empty vacuum state. Bold small-case q and s are the quantum numbers of charge and spin on the added site(s), while k indexes different states with the same (q,s). 3.5.1 Construction of (Q,S) basis - symmetry type NRG At the very least, all physically relevant models are charge-conserving. In the absence of the magnetic field, the problems are also rotationally invariant in the spin space; the total spin S and the component Sz are then also conserved. In addition, the component Sz can be eliminated from the problem by the use of the Wigner-Eckart theorem (Appendix A). It follows that we can classify states in subspaces according to quantum numbers Q and S. We first consider the case of a single conduction channel. For brevity, we denote by a)ß the creation operator for an electron of spin ß on the site of the Wilson chain that is added at the (N + l)th NRG iteration, i.e. a^ = f^+1^- The Fock space for the new site is composed of four states. Due to rotational invariance, states form spin-multiplets. A single state from each such multiplet needs to be retained, as all other members of the multiplet can be taken into account using the Wigner-Eckart theorem; by convention, in each multiplet with spin S we retain the state with the highest projection Sz = S. The four basis states for the additional site are thus represented by the three states given in Table 3.1a. In two-channel channel problems, two sites are added to the Wilson chain in each iteration, one from each conduction band. The creation operators for the second band are denoted by 6^ = f N+1ß2- The 16 states that form the Fock space of the two newly added sites are represented by the 10 states given in Table 3.1b. We also need a prescription for generating a basis with well defined Q and S for (iV+ l)-site Wilson chain given the eigenstates of the A-site Wilson chain from the previous iteration. This is easily accomplished using the angular momentum algebra (Clebsch-Gordan coefficients).29'30 Let F i (QS) denote the subspace QS at stage A used to construct states 3.5. SYMMETRIES AND BASIS CONSTRUCTION 45 \QSSzH)n+i with well defined Q, S, Sz at stage N + 1; index i numbers the possible ways of adding the angular momenta together (i = 1,..., 4 for one-channel case, i = 1,..., 16 for two-channel cases), while r numbers the consecutive eigenstates in the subspace QS at step N. For convenience, we also define f?(Sz) = Sz = Sz — ß, the spin projection Sz for /U-term in the expression, and g^(SSz) = SSZ. With this short-hand notation established, we are able to write the prescription as s(i) \QSSzri)N+1= Y, (g?(SSz);S(i),ß\SSz) \Ft(QS)f?(Sz)r)N ®\i,ß), (3.10) ß=-s(i) where (SiTOi;S2TO2|Sto) denotes the Clebsch-Gordan coefficient for joining spins Si and S'2 into spin S, \)n+i and \)n denote states for (N + l)-site and iV-site Wilson chain, \i,ß) are the states on the added site(s) tabulated in Table 3.1 and S(i) = s, the total spin quantum number of the \i,fj) state. The rules for forming the new subspaces are given in Table 3.2. In "NRG Ljubljana", these tables can be found in files coef/lch-In.cpp and coef/2ch-In.cpp or, generally, in coef/*-In.cpp. As an example, i = 2 and i =3 in the one-channel case correspond to two different ways of obtaining total spin S by adding a spin-\ object, either from S = S — \ or from S = S + \. It should be noted in passing that a singlet and a triplet never couple into a singlet state (that would be a violation of the triangle inequality). This must be taken into account when constructing state % = 10 in the two-channel case; S =0 is then forbidden. 3.5.2 Construction of (I,S) basis - symmetry type ISO When the U(l)charge conservation of charge symmetry can be extended to the full SU(2)iso isospin symmetry, an additional complication arises due to the phase factor in the definition of the isospin down component of the tensor operator (which corresponds to the annihilation operator). The brackets of creation and annihilation operators must be expressed using the irreducible matrix elements (I'I'zS'S'zr'\fl\IIzSSzr) = (lIz;^\I'I'z)(SSz;^\S'S'z)(I'S'r'\\MlSr) (3.11) and (I%S'%rf\fiJIIzSSzr) = (-l)\-2p)(lIz&-^ (3.12) Here i is the site index which takes even or odd values on the underlying bipartite lattice (see Sec. 2.3 on the isospin symmetry). As the problem is assumed to be spin and isospin isotropic, neither Sz nor Iz play any role in the diagonalization of Hn+i. The basis states for the added site of the Wilson chain are given in Table 3.3. The invariant subspaces are constructed in analogy to the case of (Q, S) basis in the previous subsection. Again Fi(IS) denotes the subspace IS at stage iV used to construct 46 CHAPTER 3. RENORMALIZATION GROUP „ „ i Q S (q,s,k) 1 Q +2 s (-2,0,1) 2 Q + l S — 2 (-l,il) 3 Q + l s + \ (-1,1,1) 4 Q + l Q + l Q 2 s + \ s (-1,1,2) (-1,1,2) (0,0,1) i Q Š (q,s,k) 5 6 1 Q + l S (-1,0,1) 7 Q s (0,0,2) 2 Q S-\ (o,|,i) 8 Q s (0,0,3) 3 Q S + ± (o,|,i) 9 Q S-l (0,1,1) 4 Q-l S (1,0,1) 10 Q s (0,1,1) 11 Q S+l (0,1,1) (a) One channel 12 Q-i S ~ 2 (1,5,1) 13 Q-l s + \ (l,il) 14 Q-l S ~ 2 (1,5,2) 15 Q-l s + { (l ± 21 l1, 2' L> 16 Q-2 s (2,0,1) (b)r rwo channels Table 3.2: Subspaces Fi(QS) = (Q,Š) and basis states for the additional site used to construct zth combination of basis states for the new iteration for (Q,S) basis (i,s) States (k) (i, s) States (k) (5,O) aU\ (0,|) a] (a) One channel (1,0) (- -) \2i 2> (0,1) (0,0) aja}6{6} CZ-t-1/1 t/-t-« CI \ Cl^U^ -52(a\a\ + b\b\),j-(a\b\- - <46l) (b) Two channels Table 3.3: Basis states for additional site in the isospin-spin (/, S) basis. Bold small-case i and s are the quantum numbers of charge and spin on the added site(s), while k indexes different states with the same (i,s). 3.5. SYMMETRIES AND BASIS CONSTRUCTION 47 Z I 5 (i, s, k) 1 I- 1 s (1,0,1) 2 / s (1,0,1) 3 4 5 6 7 8 9 10 11 12 /+1 I+\ I+\ I+\ I s S — 2 2 5 + ± s+\ S ~ 2 2 5 + 1 5 + ± 5-1 (1,0,1) '1 1 1) ^2> 2> ^ ^2> 2> ^ ^2> 2' > '± i 21 ^2> 2>^ '± i 21 ^2> 2>^ '± i 21 ^2> 2>^ '± i 21 ^2> 2>^ (0,1,1) i 1 2 3 4 I I + \ I I (a) 5 (i,s,k) S (1,0,1) 5 (|,o,i) 5-i (o,|,i) s + i (o,|,i) One channel 13 I 5 (0,1,1) 14 I 5+1 (0,1,1) 15 I 5 (0,0,1) 16 I 5 (0,0,2) (b) Two channels Table 3.4: Subspaces Fi(IS) = (I, S) and basis states for the additional site used to construct ith combination of basis states for the new iteration in the (/, 5) basis states \HzSSzri)N+i at stage N + 1. We also define f"(Iz) = Iz = Iz — a, f^(Sz) = Sz = Sz — /j, g"(IIz) = IIZ and g^(SSz) = SSZ. The new basis is then formed using a double application of the angular momentum addition rules: I(i) S(i) \IIzSSzi)N+l-- Y, E (g?(IIz);I(i),a\IIz)(g?(SS,);S(i),iJi\SS,) a=—I(i) ß=—S(i) x\Ft(IS)fT(Iz)ft(Sz)r)N®\i,ß,a). (3.13) I(i) and S(i) correspond to i and s quantum numbers of states \i,/j,,a). The rules for forming the states are summarized in Table 3.4. 48 CHAPTER 3. RENORMALIZATION GROUP 3.6 RG transformation and iterative diagonalisation In this section we describe how the hopping Hamiltonian is actually solved. We define a series of finite-size Hamiltonians of the form HN = AN/2 N Himp + HC + Y,Y, A'n/2tn ULafn+l,ß,a + H.c) n=0 ß,a (3.14) so that the full Hamiltonian is given by the limit H= lim (A-N/2Hn). (3.15) The factor AnI2 in the definition of Hn rescales the energy scale so that the smallest dimen-sionless excitation energy of Hn becomes of order 0(1). In some sense this is reminiscent of the rescaling of the fields in the momentum-space renormalization or rescaling of the free energy per site in the block-spin renormalization. The NRG iteration is then defined by the recursion relation Hn+1 = R{Hn} = VÄHn + L> (fn,,,Jn+i,,,a + H.c) . (3.16) The energies are rescaled by vA and one new site (one-channel problems) or two new sites (two-channel problems) from the Wilson chain are attached to the system, see Fig. 3.9. This recursion relation is the fundamental aspect of the NRG.29'30 Due to even-odd effects. the RG transformation is actually defined by two consecutive NRG iterations: Hn+2 = R2{Hn}, (3.17) so that the renormalization flow in the NRG is represented by the sequence of Hamiltonians ... —> Hn_2 —> Hn —> Hn+2 —> ... (3.18) A simple way of seeing that R by itself cannot be an RG transformation is that during an iteration step the energy levels are rescaled by VA, therefore R cannot have a fixed point. since two successive energies in the discretized conduction-band Hamiltonian are separated by a factor of A.213 R2, however, does have fixed points.29 It is also clear that the fixed point for even and odd N are generally different.28'29 The number of NRG steps performed is set by parameter Mmax. It should be noted in passing that the word "group" in renormalization group is actually inappropriate; in fact, it is a semi-group. There is namely no inverse transformation. This is related to the fact that information is "lost" (integrated out), either by doing coarse graining (real-space RG) or by truncation (NRG). An inverse NRG iteration is therefore impossible.126 3.6. RG TRANSFORMATION AND ITERATIVE DlAGONALISATION 49 a) One-channel case Impurity region J,r (HiR^- «-I Iteration 0 Iteration 1 Iteration 2 _ Iteration 3 b) Two-channel case Iteration 0 Iteration 1 Iteration 2 _ Iteration 3 a=l a=2 Figure 3.9: Hopping Hamiltonians and the successive iterations of the NRG procedure: one site from each channel is added during each RG step. As far as the NRG iteration is considered, the impurity region is a black box: all that is required are the eigenstates of the sum of the impurity and the coupling Hamiltonians, Himp + HC, and the irreducible matrix elements of the creation operators for an electron on the first (indexed as 0) site of the hopping Hamiltonian computed in the eigenbasis. 50 CHAPTER 3. RENORMALIZATION GROUP The Hamiltonian Hn describes the physics at the energy (temperature) scale T N oc ^-A~N/2/ß (3.19) Kb or, equivalently, at the length scale Ln oc ßAN/2/kF. (3.20) Here ß is a parameter of order 0(1); the corresponding setting in "NRG Ljubljana" is betabar. In my calculations I typically used ß = 0.46 or ß = 0.75. The exact definition of Tn depends on the discretization scheme: T N = ~(1 + A-^A"^-1)/2/^, for disc=wilson; Kb 2 T N = -^-1 (1 + A~l)A-{N-z)/2/ß} for disc=yoshida, (3.21) Kb 2 Tn = ^T^ A~{N~z)/2/ß, for disc=campo. kB m A From Eqs. (3.19) and (3.20) it follows that NRG iteration corresponds to L —> oo and T —> 0 at the same time, but in a way that the size of the system is finite at all times. From this it follows that NRG gives finite-size spectra. It should be kept in mind that the ground state degeneracy in a finite-size spectrum is obtained by taking the limit T —> 0 first, then L —> oo. By taking first L —> oo and then T —> 0, a different ground state degeneracy can be obtained. The two limits do not commute92'118! One should be aware of this when comparing with results obtained by means of quantum-field-theoretical methods in the L —> oo limit. One might expect that due to the exponential decrease of hopping parameters it might be possible to treat the successive sites in the Wilson chain by perturbation theory. This is not the case:29 when adding the (A + l)st site(s) to a chain of A sites, the coupling of order A~n^2 is a strong perturbation for the lowest eigenstates of the A-site chain which are also on the energy scale of A~n/2. We thus add a new site by performing an exact diagonalisation of a matrix Hn+\. One NRG iteration (3.16) consists of using the states from previous step to construct the Hamiltonian H^+i (nrg_makematrix routine), then diagonalizingit numerically (diagonalize routine). The full information about the system at step A is contained in the eigenstates of Hn and in the irreducible matrix elements (||/W||) (class Iterlnfo); this is clearly a much more detailed description compared to a small set of coupling parameters used in the conventional RG (scaling) approach. The Hamiltonian is written in the direct product basis \QSH)n+i ~ \QSr)(E)i [here we consider the case of (Q,S) basis, see Sec. 3.5 for other cases], therefore the Hamiltonian matrix takes the form of a block-matrix: diagonal blocks are diagonal matrices, the diagonal elements being the rescaled eigenvalues of the states \QSr) 3.6. RG TRANSFORMATION AND ITERATIVE DlAGONALISATION 51 channel coefficient ii channel coefficient 1 2 b 1 3 b 1 4 a 1 5 a 2 6 b 2 7 a 2 9 a ii' coefficient 2 3 10 6 a 1 2 1 b 1 3 1 3 7 a O A V2S q A - 2S'+2 3 3 4 10 11 7 a a b (a) One channel 4 8 a 4 9 b 4 10 b 5 7 b 5 8 a 5 10 b 5 11 b 2S v/1+25 v1+25 1 ^/1+2Š_ 1+2Š Vš 1+2Š 1 V2 +2 1+2š ¦1 v1+25 ¦/1+2Š V2(1+g) -1 6 6 7 7 7 7 9 9 12 a 13 a 12 b 13 b 14 a 15 a 14 b 15 b 12 b 14 a b b 10 12 10 13 10 14 a 10 15 a 11 13 b 11 15 a 12 16 a 13 16 a 14 16 b 15 16 b 1 1 1 1/2 V73?+ a/S a/S 1+Š Vl+S V1+s 1+s •S1+2Š - 2(1+g) ^1+2Š^ v2^ V1+2Š i- 2(1+g) (d) Two channels Table 3.5: Coefficients for off-diagonal blocks in the Hamiltonian matrix in the (Q, S) basis from the previous iteration; the out-of-diagonal blocks are constructed from the irreducible matrix elements {QSr\\f\\Q'S'r') weighted by coefficients that can be derived from the corresponding Clebsch-Gordan coefficients (see routine nrg_makematrix). These coefficients are given in Table 3.5 for the case of (Q,S) basis (see also Ref. 30). In "NRG Ljubljana". these coefficients can be found in coef /lch-offdiag. cpp and coef /2ch-offdiag. cpp or. generally, in coef /*-offdiag. cpp. We then diagonalize the Hamiltonian in each invariant sector separately to obtain the 52 CHAPTER 3. RENORMALIZATION GROUP coefficient 2 1 4 3 V2S'+1 ^/ŠF7+1 i i' coefficient 3 1 2 1 4 V2S"+1 (a) Q' = Q - 1, (b) Q' = g - 1, S" = 5 - i S" = S* + ^ Table 3.6: Irreducible matrix elements (QS'z||or."*"|\Q'S'i') for creation operator on the additional site in one-channel problems eigenstates \QSuj) = J2 Uqs(uj, ri)\QSri), (3.22) ri where Uqs is the unitary matrix which brings each Hamiltonian matrix in its diagonal form. Before proceeding to the next NRG iteration, the irreducible matrix elements of the newly added site(s), {QSi\\fN+1 \\Q'S'i'}, need to be recomputed from the irreducible matrix elements {QSi\\flf \\Q' S'i') from the previous iteration. The coefficients are given in tables 3.6 and 3.7 and can be found in files coef /lch-spinupa. cpp, coef /lch-spindowna.cpp, coef /2ch-spinupa. cpp, coef /2ch-spindowna. cpp coef /2ch-spinupb. cpp, coef /2ch-spindownb. cpp or, generally, in coef /Ich-* .cpp; the corresponding routines in "NRG Ljubljana" are recalc_f and recalc_irreduc. Since the total number of states is an exponential function of the iteration number N (ex 4N in the one-channel case and oc 16w in the two-channel case), Wilson proposed to simply truncate the number of states to some manageable size of the order of 1000 after each NRG iteration. The idea is that, since the coupling between consecutive sites of the chain decreases exponentially for increasing chain length, only the lowest-lying eigenstates are renormalized and the separation of scales is thus maintained iteration by iteration.29'214 This works because the matrix elements of fN are largest for similar eigenstates of Hn. while the matrix elements of fN between the low-lying eigenstates of Hn and the highly excited states that are truncated are small.29 In "NRG Ljubljana", truncation is controlled by parameters keep, keepmin and keepenergy. Parameter keep represents the maximum number of eigenstates that may be kept at each iteration; it should be increased as much as possible within the limits set by available computational resources. If parameter keepenergy is set to a positive value, the energy cut-off truncation scheme is used: only the eigenstates with the (rescaled) energy below the value of the parameter will be retained. The use of the energy cut-off truncation is recommended since a high number of states is kept when the degeneracy is high, and a low number when the degeneracy is low; in this fashion, the computational time is divided optimally between the iterations. Finally, keepmin sets the minimum number of states to be kept. It should also be remarked that eigenstates in NRG tend to be clustered. If 3.7. COMP UTABLE Q UANTITIES 53 i i' coefficient 5 1 1 7 0 V2S'+1 2^ 8 4 10 2 V2S'-1 2^ 11 3 -1 13 6 1 14 9 1^ 15 7 1 15 10 VS'+i WS7 16 12 VS'+^ i' coefficient ii' coefficient ii' coefficient V&Ti i i' coefficient 3 1 1 6 2 7 4 V2S'+1 2VŠ7 10 4 V2S'-1 2^ 11 5 1 12 9 13 7 i 13 10 VS'+i 15 8 1 16 14 4 11 3 11 2 11 2v|±T 6 2 ^ 6 3 -^-i 0 ° v^+1 ' 4 2VŠ7 ' ° 2^^+T 9 2-1 io 4 ^EI 9 4 1 10 3 -2^=3= ii c i2 10 5 ^^S 2^/8^+1 11 O 1 _____ ^ " 2^/ŠT+I 12 6 1 129 Vp 12 7 -^ 14 7 "73 n 7 -4! 12 10 -^ \/2VŠTT 14 10 A/2A/š5+I 13 10 ^^1 13 11 -^ A/fe ^2^ ^7^ 15 11 ^=# 15 8 1 14 ° 1 -^S 16 14 ^53 16 15 -^ ^/Š7+T VŠ7 \/S7+T (a)Q' = Q-l,S" = (b)Q' = Q-l,S" = (c) Q'= Q-l, S" = (d) Q'= Q-l, S" = S*+2 ^ ~ 2 ^+2 & ~ 2 Table 3.7: Irreducible matrix elements (QSiHatHQ'Si) for creation operator aJ (subtables a and b) and for creation operator bf (subtables c and d) on the additional site in two-channel problems the states are truncated in the middle of such a cluster, systematic errors and symmetry breaking may be induced. Parameter safeguard enforces retention of additional states, so that the "gap" between the highest retained and the lowest discarded state is at least safeguard. 3.7 Computable quantities While K. G. Wilson originally applied NRG to obtain the spectrum of excitations and the impurity contribution to the magnetic susceptibility,29 methods to calculate other quantities were soon introduced: one can determine specific heat,215'216 charge susceptibility,217 entropy,209 spin relaxation rates,218 and various zero-frequency response functions and equal-time correlation functions.219 NRG is likely the most versatile tool in the field of quantum impurity physics. 3.7.1 Finite-size spectra and fixed points The most easily obtained result in NRG is the spectrum of excitations above the ground state as a function of the temperature. An important amount of information may be 54 CHAPTER 3. RENORMALIZATION GROUP Running coupling constants —-----1—^i oo, where L is the system size).108 Some important quantities, such as the ground state entropy, depend on whether the system size is finite or not when the T —> 0 limit is taken92 (see also Sec. 3.6). If the lowest lying eigenstates for successive (N —> N +2) NRG transformations remain (nearly) unchanged, we say that a fixed point has been reached. More accurately, fixed-point Hamiltonian H* is defined as H* = R2{H*}. (3.23) With NRG, one can study the various fixed points of a given QIM, deviations from the universal spectra (determined by the operator content of the fixed point108), and the crossovers between the different fixed points, Fig. 3.10. If the excitation spectrum of a fixed point is in a one-to-one relation to the excitations of free electron gas, such fixed point is called Fermi liquid fixed point (see Section 2.4 on Fermi liquid and non-Fermi liquid systems). The spectra of Fermi liquid fixed points are composed of excitations that change particle number (particle and hole excitations) and 3.7. COMP UTABLE Q UANTITIES 55 "oo b)oc c)oo d)oo oo • o oo • o OO oo o# o# mm • • • • • • Q=0, S=0 Q=l,S=l/2 Q=-l, S=l/2 Q=0, S=0,1 Ground state Particle excitation Hole excitation Particle-hole excitation Figure 3.11: Pictorial representation of the ground state and excitations of a Fermi liquid. excitations where a particle is promoted to a higher level (particle-hole excitations), and combinations thereof, Fig. 3.11. The excitations at non-Fermi-liquid fixed points do not always have comparably simple interpretation. For Hamiltonian H^ near a fixed point H*, the NRG recursion relation can be expanded in powers of the deviation from the fixed point and linearized.29'30 Defining 5Hn = Hn — H*. we write 5HN+2 = R2{H* + 5HN} - H* « C*6HN, (3.24) where C* is a linear transformation. Like any linear operator, C* can be diagonalized C*Oi = XiOi, (3.25) and we expand 8Hn as 5HN = J2 Q\f/2Ol (3.26) i For A* > 1, the contribution of corresponding eigenoperator O* will grow with N: we say that such operators are relevant. For A* < 1, the contribution will vanish and we name such operators irrelevant. Finally, if A* =1, the operator O* is called marginal and its effect must be studied more carefully by considering non-linear corrections to the RG transformation.220 The operator content of a fixed point determines its stability with respect to perturbations:30 a fixed point with relevant operators is called unstable, while a fixed point with only irrelevant operators is stable. Fixed point with marginal operators can be either stable or unstable, or may lead to the emergence of lines of fixed point and to the breakdown of the universality.220 The knowledge of the leading eigenoperators (i.e. those with the highest eigenvalues A*) is instrumental in establishing effective Hamiltonians given by the fixed point Hamiltonian H* plus correction terms:30 Hff = H*+ u^-^HH, + u2A^-^26H2 + ... (3.27) where u\ and U2 are some coefficients which can be determined by analyzing the NRG spectrum. For example, finite-temperature corrections to the Fermi-liquid T =0 behavior of the Kondo model are determined by the leading irrelevant operators.29 Saving of eigenvalue results in "NRG Ljubljana" is controlled by parameters trace, dumpenergies and dumpannotated. 56 CHAPTER 3. RENORMALIZATION GROUP 3.7.2 Static thermodynamic quantities: susceptibilities, entropy, specific heat Static thermodynamic quantities, such as magnetic susceptibility, heat capacity and entropy, are determined primarily by the energy level splittings of order kßT. Energies much higher than kßT above the ground state are exponentially suppressed, while excitations with much lower energy can be considered thermally washed out; this turns out to be a good approximation. Thermodynamic quantities at temperature Tn can thus be calculated from the energy spectrum at the iVth stage of the NRG iteration.29'30'129 In the following, brakets denote grand-canonical averaging (O) = Tr [O exp(-ßH)] /Z, (3.28) where Z is the partition function Z =Tr(e_/3H) and ß = 1/kßT. In practice, the traces are computed in the truncated basis of NRG eigenstates at a given iteration step N, i.e. from a finite-size spectrum. In "NRG Ljubljana", the temperature dependence of static thermodynamic quantities is output to file chi which must be postprocessed to obtain presentable results. Magnetic and charge susceptibility The temperature-dependent impurity contribution to the magnetic susceptibility XimP(T) is defined as Ximp(T)= ^g? «S 2> - (S%) (3.29) where Sz is the total spin and the subscript 0 refers to the situation when no impurities are present (i.e. H is simply the band Hamiltonian ii/band), 9 is the electronic gyromagnetic factor, ßß the Bohr magneton and kß the Boltzmann's constant. It should be noted that the combination kßTximp/igßB)2 can be considered as an effective moment of the impurities, /ieff- In the presence of the magnetic field applied only to the impurity site, (S*2) needs to be replaced by (S*2) — (Sz)2 in accordance to the fluctuation-dissipation theorem. It may also be remarked that while K. G. Wilson originally proposed to calculate (Sz)o analytically, I find that it is more practical to actually perform a NRG calculation of Sz for a problem without impurities. This has an added benefit in that similar artefacts appear in (S*2) and (Sf)o and they cancel when subtraction is performed. By analogy, charge susceptibility is defined as Xcharge(T) = -^ {{ID - (J2)o) , (3.30) where Iz is the total isospin (recall that Iz = Q/2). In the absence of the particle-hole symmetry, (J2) needs to be replaced by (J2) — (Iz)2- 3.7. COMP UTABLE Q UANTITIES 57 Specific heat and entropy Defining energy as E = (H) = Tr (i7e_/3H), the heat capacity can be calculated from energy fluctuations as BE C(T) = ^ = kBß2 [(H") - (H)"} , (3.31) and we may define the impurity contribution to the heat capacity (impurity specific heat) as Cimp(T) = C iT) — Co (T), where C0 is the heat capacity of the conduction band without impurities. Furthermore, we have ßF = — In Z and E = F + TS, therefore T- = 4-F- = ßE-ßF = ßE +ln Z, (3.32) kB kBT and we may define the temperature-dependent impurity contribution to the entropy as Simp CO = S (T) — So(T). From the quantity S[mp/kB we can deduce the effective degrees-of-freedom v of the impurity as Simp/fcß ~ lnz/. In the following the suffix "imp" in Ximp; Simp, etc. will often be dropped if no confusion can arise, but one should keep in mind that impurity contribution to the quantity is always implied. We also set kß =1. The convergence with the number of states retained in the NRG iteration depends on the quantity being computed. For example, the energy accuracy required for a specific-heat calculation is considerably higher than that for the susceptibility.215 3.7.3 Correlation functions To characterize the state of a quantum impurity system it is often useful to calculate various correlations functions, i.e. thermodynamic expectation values of operators such as the on-site occupancy (ni), local charge-fluctuations ((5n)2) = ((^— (^i))2) = (n2) — {rii}2. local-spin (S2) and spin-spin correlations (Sj • Sj). In turn, these can be used to compute more complex quantities such as the concurrence which measures the entanglement between two qubits.221 In "NRG Ljubljana", the operators of interest are specified by writing the corresponding expression in terms of the second quantization operators. A number of auxiliary routines are available to simplify this process and the most commonly occurring operators are already built in the program (configuration setting ops). During the problem setup step, the operators are transformed in their matrix forms and rotated into the eigenbasis of the initial Hamiltonian by performing suitable unitary transformations: all these steps are performed automatically "behind the scenes" by the Mathematica part of the NRG package. This approach turned out to be extremely flexible, since the user can focus on physics rather than hand-code low-level routines and to worry about implementational details. To be able to make full use of the symmetries of the problem, the operators need to be expressed in the form that makes them singlets with respect to the symmetry group. For 58 CHAPTER 3. RENORMALIZATION GROUP example, nf is a spin-singlet, and can be directly computed in the (Q,S) basis. It is not, however, an iso-spin singlet, but (rii — l)2 = g2 is. For a computation in the (/, S) basis, one therefore performs a calculation for q\ and adds 2{rii) — 1 = 1 to the results (recall that (rii) = 1 due to the p-h symmetry). In the presence of mirror Z2 symmetry, it must be taken into account that operators may be even or odd with respect to the reflection. In the case of two impurities embedded in a series between two conduction leads, for example, ne = ri\ + U2 is even, while n0 = ri\ — ri2 is odd. In a calculation where reflection symmetry is explicitly taken into account, the expectation values (n\) and (n^) can be obtained by calculating suitable combinations of (ne) and (n0) after the NRG run. More generally, "NRG Ljubljana" supports operators that are singlet, doublet or triplet with respect to spin and singlet or doublet with respect to isospin. This is sufficient for all calculations of interest, but support for more general symmetries may in principle be easily added. 3.7.4 Dynamic quantities A major extension of the NRG was a method to calculate dynamic properties such as the spectral functions. 147'208'222~225 While T =0, uj =0 conductance of Fermi-liquid systems may be obtained from finite-size spectra alone, finite-temperature and finite-frequency conductance, as well as the conductance of non-Fermi-liquid systems can only be computed if spectral functions are known .213>226-230 Using NRG, one can determine local single-particle (spectral function, ((dß; djj})^), magnetic (dynamic spin susceptibility, ((Sz; Sz))u) and charge excitations (dynamic charge susceptibility, ((n;n))w). It is also possible to distinguish between elastic and inelastic contributions to the scattering cross-section.231 The conventional approach to the NRG spectral function calculations is based on the observation of Sakai et al.147 that as we proceed from one iteration to the next, the lowest few eigenstates split due to the interaction with the added shell states, while the intermediate lower levels do not show any essential change. The intermediate states thus form a good approximation of the eigenstates of the Hamiltonian in the N —> 00 limit and are thus used to compute the excitation energies and the transition matrices. The spectral function matrix for multi-impurity problems is defined as (see Appendix B) A, = -l/(27r)Im(GL + GrJt), (3.33) where G^-(cj) = ((diß; dj^}}^ is the (out-of-diagonal for i =L j) retarded Green's function of the impurity. It can be computed using standard NRG techniques from matrix elements of the creation operators using the following spectral decompositions: 1 ___ r- / \ ^ Aitj(u > 0) = - J^ Re ^(m\dl\n0)J (m|4l^o)J x 5(u - Em), T" r/ a* 1 (3-34) Aitj(uj < 0) = - J2 Re [{{mo\dl\n)J {mo\d]\ri}\ x 6(u + En). mo,n 3.7. COMP UTABLE Q UANTITIES 59 Indices m0,n0 with subscript 0 run over (eventually degenerate) ground states and indices to, n without a subscript over all states. Calculations can be improved by directly calculating the one-particle self-energy X(cj);214 this approach leads to more accurate results and it is especially suitable for applications of NRG as an impurity solver in the DMFT197 . Further improvements include a better approach to merge partial spectral information from consecutive NRG iterations232 (in :NRG Ljubljana" the result of the conventional spectrum calculation is output to files spec_*_pts_* .dat, while the result obtained with the N/N+2 trick is output to files spec_*_dens_* .dat). For problems where the high-energy spectral features depend on the low-energy behavior of the system, the spectral function has to be computed taking into account the reduced density matrix obtained from the density matrix of the low-temperature fixed-point: this is the density-matrix NRG (DMNRG) developed by W. Hofstetter233 (see Sec. 3.9). This approach is needed, for example, in the case of the Anderson impurity in magnetic field,233 or for the side-coupled double quantum dot near the points of ground state level crossing (see Sec. 7.3). Recently, a time-dependent NRG was introduced234 by generalizing the idea behind the DMNRG: time-dependent NRG makes possible to study the effects of sudden changes of the parameters and the ensuing relaxation to the steady state solution. In this approach, a density matrix in full Fock space is introduced by judiciously using the information from the discarded part of the NRG eigenstates. This idea has led to new approach for calculation of equilibrium spectral functions: the "full density matrix" NRG235 or "complete Fock space" NRG.236 This method does not suffer from over-counting of excitations, it fulfills sum rules and correctly reproduces spectral features at energies below the temperature. An important observation for practical calculations is that as the number of states retained is increased, the calculated spectra do not suddenly change; they rather gradually improve and converge toward the true spectrum.147 This implies that even rough spectra are qualitatively correct. 3.7.5 Spectrum broadening and smoothing Since QIMs are represented in NRG by hopping Hamiltonians of a finite size, the computed spectral functions are represented as a sum of delta peaks. To obtain a meaningful continuous function, these peaks need to be broadened. The original approach to obtaining a smooth curve was by Gaussian broadening, followed by separate spline interpolation of results in odd and even steps, and by the averaging of the two curves.147 A better approach is the logarithmic-exponential broadening:232 each data point (delta function peak at u0) is smoothed into e"b 2 /4 / (lnw-lnwo)2' Fb(u,uJo) = —^exp (----------------------J, (3.35) i.e. a Gaussian function on a logarithmic scale, where b is a broadening parameter, typically b = 0.3. One should keep in mind an important feature regarding the broadening procedure. 60 CHAPTER 3. RENORMALIZATION GROUP Namely, due to broadening the spectral resolution at energy u is always limited to f(u)= d(Jof(u)o)Fb((j,(j0). (3.37) Let us consider its effect on a narrow Lorentzian of width A centered at u =0: A2 O^ + A2" f{uj)=-- I duo 2 | A9Fb(u),u)0) *+°°, A2 e-fc2/4 / {y_yoy dy0-----—— A9 ,- exp DC exp(2y0)+ A2 b^ l \ b2 (3.38) where we performed substitution lncj0 = Do and introduced y = lna;. The limit u -^ 0 corresponds to y -^ — oo. The integrand is a Gaussian-like function centered at y with a j/o dependent weight oc l/(exp(y2) + A2). For small enough uj (to be concrete, u <€. A), this weight becomes a constant and the integral can be evaluated exactly. We find limw^o/(^) = exp(—62/4). For b = 0.3, this gives 0.98. In other words, even in the absence of any other approximations, the logarithmic broadening at b = 0.3 introduces an error of few percent in the Kondo peak weight. In addition, the Lorentzian is narrowed, see Fig. 3.12. These facts must be taken into account when quantitative details in the results are important. In that case b should be reduced as much as possible. Typically, the value of b is chosen to be 0.3 or less (parameter loggauss_b). 3.8. RECURSION RELATIONS FOR OPERATORS 61 0.8-0.6-0.4-0.2- -::^ 1 ' 1 — Lorentzian, width A=0.1 — Lorentzian broadened with b=0.3 - i,i, - 0.1 0.2 ffl Figure 3.12: Effect of logarithmic broadening on a Lorentzian curve. 3.8 Recursion relations for operators After each iteration, expectation values of operators of interest are computed and the irreducible matrix elements of these operators recomputed in the new eigenbasis for the next iteration. It is important to note that it is only possible to consider operators that transform as tensor operators with respect to the symmetry group that is taken into account in the NRG implementation. As an illustration, we consider the case of the (Q,S) symmetry and a tensor operator operator O of rang M with respect to the spin SU(2) group. The information about the operator O at iteration N is entirely contained in a matrix On of irreducible matrix elements {QSr\\0\\Q'S'r'). The non-zero subspaces for singlet operators have Q = Q' and S = S', for doublet (creation) operators we must have Q = Q' +1 and S = S' ± 1/2 and for triplet operators (such as spin) we must have Q = Q' and S = S' or S = S' ± 1. In the basis of eigenvalues \QSu) of the (N + l)st iteration, we write (QSu\\Ö\\Q'S'u/)N+1 {QSSzu\ÖAQ'S'S'z(u/,r'i'){QSSri\ÖM\Q'S',S - M,r'i')N+1 (S',S-M;MM\SS) We then take into account the definitions of \QSSzri) states, (3.10), and write Y^ v-^ c(QS,Q'S',ii',aß) (3.40) „^V<^-M;MM|^) (3.41) x UQS{u)ri)UQ,s,{u')r'i'){Fl{QS)f:{S)r\ÖM\Fl,{Q'S')^{S - M)r')N+l) where c(QS,Q'S',ii',aß) is a scalar product between (q,s,k) states on the added sites of the Wilson chain and a,ß are the corresponding sz components of these states. We rewrite 62 CHAPTER 3. RENORMALIZATION GROUP this as y> c{QS, Q'S', ii', aß)(gßt,(S',S - M);MM\g?{SS)) '"~hß (S>,S-M;MM\SS) xY,Uqs(oj,ri)Uq,s>(oj\r'i')(Ft(QS)r\\6\\Ft,(Q'Sy)N rr' After taking into account which subspaces Fi(QS) and Fi>(Q'S') are connected by operator O and performing the sums over a and ß, it turns out that for given (QS) and (Q'S') subspaces, only a small number of (ii1) combinations contribute. We finally write (Q^||6||Q^V>W+1 = J]CXQS,^ ii' rr' (3.43) The coefficients C qs,q>s>,h> are computed using a computer algebra system and they are tabulated in the manual (Ref. 237). The corresponding routines in "NRG Ljubljana" are recalc_singlet, recalc_doublet and recalc_triplet; they all call a low-level routine recalc_general which performs the actual computation. The coefficient tables can be found in files *-singlet.cpp (for singlet operators), *-doublet* .cpp (for doublet operators) and *-triplet* • cpp (for triplet operators). 3.9 Density-matrix NRG At iteration N, the information about the behavior of the system at temperatures much lower than Tn is yet unknown; in particular, the true ground state of the system is not yet determined. Since the zero-temperature spectral function is defined by the matrix elements between the ground state and the excited states, there is no guarantee that the high-frequency part of the spectral function calculated by the conventional NRG procedure will be correct. Discrepancies appear, for example, in the presence of the magnetic field: for H ^$> Tk, the impurity spin is polarized and the true spectral functions are asymmetric. The asymmetry of functions calculated by the conventional NRG approach is strongly underestimated, since at the temperature (energy) scale above H, the magnetic field does not yet significantly affect the finite-size spectrum.233 Density-matrix NRG technique188'206'233 remedies the shortcomings of the conventional approach. It was originally implemented by W. Hofstetter for NRG calculations in (Q, Sz) basis for studying the effect of the magnetic field on spin-projected spectral functions. In the absence of a magnetic field, it is advantageous to use the (Q,S) basis. The improvement in numerical efficiency is sufficient to enable consideration of more complex systems, such as the double quantum dot. Density-matrix NRG technique consists in running the NRG calculations in two runs. The first run is a usual NRG iteration, with the sole exception that the Uqs matrices are stored for later use. After the last iteration, the zero-temperature density matrix is estimated (3.42) 3.9. DENSITY-MATRIX NRG 63 using the truncated basis as P=\Y, eM-PEQs.)\QSSzoj){QSSzoj\, (3-44) QSSzco where uj enumerates different states in each (Q,S) subspace and the grand-canonical statistical sum is Z = Tr [eM-ßH)] = J] exp(-/?LQSw). (3.45) QSSzw The reduced density-matrices are then computed for higher temperatures. During the second NRG run, the traces are computed with respect to these reduced density-matrices. rather than using the grand-canonical density-matrix p = exp(—ßH)/Z. We now illustrate how the recursion rules for computing the reduced density matrix are derived in the (Q,S) basis for the single-channel case. Unitary transformation of states from the Nth to (N + l)th stage is (see Eq. (3.22)) \QSSzuj)n+1 = ^2 UQS(uj\ri)\QSSzri)N+1, (3.46) ri where the \QSSzH)n+1 states are defined by Eq. (3.10). They may be expanded as:30 \QSSzrl)N+1 = \Q + 1, SSzr)N, \QSSzr2)N+1 = uflN+1)1\Q,S - -,SZ- 12,r)N + vf{N+1)i\Q,S - 12,Sz + 12,r)N) 1 (3.47) \QSSzr3)N+1 = wf(N+1^\Q,S + 12,SZ — 2-,t)n + yf(N+1)i\Q>S + 2'^ + 2>r)w; \QSSzrA)N+1 = flN+1)if(N+1)i\Q - l,SSzr)N., where fJN+1)ß is the creation operator for electrons on the (N + l)th site of the hopping Hamiltonian and u,v,w, and y are the Clebsch-Gordan coefficients s + sz\1/2 fs-sz\1'2 fS-Sz + l\1'2 [S + Sz + l 1/2 -2S~) 'V=\-2S-) >W= -{ 2S +2 ) >v=^lsX+V) ¦ (3.48) Density matrix in the basis of \QSSzri)j^+1 states is P= Z~2, exP(-/3LW0 y^ UQs{u\ri)UQs{u\r'i')\QSSzri)N+1{QSSzr'i'\N+1. (3.49) QSSzu! ri,r'i' We now perform a partial trace over the states on the additional (N + l)th site to obtain projector operators defined on the chain of length N. Diagonal projectors (those with 64 CHAPTER 3. RENORMALIZATION GROUP i = i) are: Trw+1 (\QSSzrl)(QSSzr'l\) = \Q + 1, SSzr)N(Q + 1, SS.r'|w Trw+1 (\QSSzrA)(QSSzr'A\) = \Q - l,SSzr)N(Q - 1, SS,r|w TtN+1(|QS'^r2)(QS'(S'y2|)=U2|Q,(S'-i,^-i,r>N(Q,(S'-i,^-i,r,U + w2|Q, 5 - |, S^ + |, r)N(Q, S -\,Sz + \, r'\N TrN+l (\QSSzr3){QSSzr3\) =w2\Q,S+ \,SZ - \,r)N{Q,S + \,SZ - \,r'\N + y2\Q,S+\,Sz + \,r)N{Q,S + \,Sz + \,r'\N The i = i' =2 terms can be simplified after summing over Sz: s Y, TrN+1(\QSSzr2)N+1(QSSzr'2\N+1) sz=-s s = Y fez + vs,sz-i) \Q,S- \,SZ- ±,r)N{Q,S- \,Sz-\,r'\N Sz = -S+1 +(.s-h 2S+1 (3.50) (3.51) 7 J \Q, S — g, Sz)n{Q, S — g, Sz,r \n 2S SZ = -(S-^) The spin multiplicity of (QS)n+i space is 2S +1, while the spin multiplicity of (Q, S — ^)n is 2S. The factor (2S + 1)/(2S) is therefore merely a normalization factor. In the last line we emphasize that in the iV-site space the Sz runs over all permissible values for spin S — \. An analogous simplification can be performed for % = %' = 3. Out-of-diagonal terms which correspond to different charge on the additional site are clearly zero, while other out-of-diagonal terms such as i =2, i' =3 give zero when summed over. Non-zero partial traces of projector operators are therefore: Y Tr N+l\QS Szri) N+i{QS Szr'i'\N+i = ö^c^S) J^ \QiSiSzir) N{QiSiSzir'\N (3.52) with ci = c4 = 1, c2 = 2f±i, c3 = ||±|, Qi = Q +1, Q2 = Qa = Q, Qi = Q - 1, Si = 5*4 = S, 5*2 = S — |, S3 = 5+ \ and corresponding S.^ ranges over all possible values for a given Si. The reduced density matrix is diagonal in its (QS) subspace index. In a sense, it has similar symmetry properties as the singlet tensor operators. In general we therefore have Preduced = Y / y Cu,J \QSSzLü)N+i {QSSzLü'\N+1 QSSZ ujuj' -- Y Y,C^;N+1 Y UQs{u}\ri)UQS{u}'\r'i')\QSSzri)N{QSSzr'i'\N QSSZ ujuj' ri,r'i> = Y Y CS5'N+l Y UQsMri)UQS(cj'\r'i)ct(S) J] \QlSlSzlr)N(QlSlSzlr'\N. QS ujuj' ri,r' Szi (3.53) 3.9. DENSITY-MATRIX NRG 65 This is to be compared with P^duced =EE C^N\QSSzr)N(QSSzr'\N. (3.54) qssZ rr> We finally obtain the recursion relation for calculation of coefficients C^r,' in the reduced density matrix: C?rf>N = ^t-^N+1Uq.lA^\rl)Uq.lA^r'l) 'jJUl' + L C^s'N+1Uq+l,s^\r^Uq+l,s(uyi) 'jJUl' L25+2 ^ o,s+\,n+i (3-55) ujuj' 2S ^ q,S-\,N+\ , , ujuj' This is the main result of the derivation. Using known Uqs matrices, recursion is applied after the first NRG run to calculate density matrices for all chain lengths. In the DM-NRG scheme the spectral function is evaluated in the second NRG run using the reduced density matrix preduced as A"M = E (0'M»014N)P-Uced + (j\dl\m){H\m)pfuced) 5(u-(E3-Em)). (3.56) ijm Both terms contribute at positive and negative frequencies. Chapter 4 Other methods for impurity models In this chapter, I briefly describe methods that are used in this work - in addition to NRG - to solve impurity models. Section 4.1 describes the Green's function method for non-interacting models, while sections 4.2 and 4.3 are devoted to the variational methods and Quantum Monte Carlo methods, respectively. I remark at this point that the results of these methods (when used in their respective domains of validity) agree very well with results obtained using NRG; for a direct comparison see, for example, page 154. I will also make use of the Bethe Ansatz (BA) method, which provides exact solution of the Kondo model.50'52'59'238'239 Using BA, the thermodynamics of the many-body problem is reduced to a system of infinitely many non-linear integral equations239'240 which are easily solved numerically. While BA does not clarify the physics behind the Kondo effect, it is extremely useful since it provides exact results that can be compared with NRG calculations to ascertain their reliability. 4.1 Green's function method for noninteracting problems While the topic of this work are effects of interactions, it is sometimes sufficient to describe physical systems using effective non-interacting models. Since the electron spin then plays no role, it may be dropped from consideration and the model usually takes the form of a tight-binding Hamiltonian for spinless particles. In transport problems, the scatterer is described using #imp = X^y|«)0'l, (41) ij 66 4.2. GUNNARSSON-SCHONHAMMER VARIATIONAL METHOD 67 where i,j range over the lattice sites of the scattering region. The scatterer couples to the conduction leads via hopping terms of the form H' = - Y, ^ (N) (Li+H-c-) - Y, **.< (i*) (R\ + H-c-) • (4-2) i i For simplicity, let us assume that only single site (i =1) is coupled to both the left and right lead (this will, in fact, be the case when this method is applied to study the Fano effect in side-coupled double quantum dot in Sec. 7.3). The retarded self-energy matrix due to the coupling to the leads is Tf = El + ER, (4.3) Pl]h = t2gTLL-, other components zero, (4.4) Er]ii = t29RR> other components zero, (4.5) where grLL and grRR are retarded Green's function of uncoupled left and right semi-infinite lead, respectively. The effective Hamiltonian i7eff = -ffimp + Sr is then used to obtain the retarded Green's function of the impurity region: Gr =(el - Fes)"1. (4.6) Spectral function is defined as A=-(Gr -Ga)= --lmGr, (4.7) 7T 7T while the spectral function is simply p = TrA Finally, we define the transmission function: tLR = Tr(rLGrrRGa) (4.8) with r^ = z(EL — LL) and Tr = i(ErR — ER), from which follows the conductance: G = G0tLR{t = 0). (4.9) 4.2 Gunnarsson-Schönhammer variational method Variational method is an approximate technique to calculate ground state energy of a quantum system. It can be shown that as the variational Ansatz is improved, the variational energy will tend toward the true ground state energy. In general, however, the variational (or "trial") wavefunction will not necessarily tend toward the true ground state wavefunc-tion; this implies that there is no guarantee that the calculated correlations approach the actual values (but often they do). Nevertheless, variational methods are valuable in that they allow very complex problems to be solved in a simple analytical manner, often with results in closed form. The earliest application to the Kondo problem is Yosida's wave 68 CHAPTER 4. OTHER METHODS FOR IMPURITY MODELS function for the s-d model,60 which was later generalized for the Anderson model.61 This variational approach correctly reproduces the Kondo energy oc exp( —1/pJ). The variational method used in this work (the "Gunnarsson-Schönhammer variational method") was originally developed for the Anderson model by K. Schönhammer to study hydrogen atom chemisorption on metal surfaces.241'242 Solution of this problem requires proper description of electron correlation effects because charge fluctuations on the hydrogen atom are strongly suppressed due to electron-electron repulsion. The method was later extended to N-fold degenerate Anderson model by Gunnarsson and Schönhammer242-244 to study photoemission and absorption spectra of rare earth and actinide compounds. Variational calculation consists of finding a trial wavefunction with free parameters, determining the ground state energy as a function of these parameters and performing a minimization. The wavefunction with the lowest energy is then the best approximation to the true wavefunction from the subspace of the trial wavefunctions. A good trial wave-function must therefore have the characteristics that we expect the true ground state to have. In particular, the exact ground state of the Anderson model is known to be a singlet, so a good starting point for a variational calculation is the restricted Hartree-Fock approximation with identical orbitals for both spins. In the Hartree-Fock solution |^hf) the charge fluctuations cannot be properly suppressed (since electrons always come in pairs), therefore K. Schönhammer proposed the following Ansatz:241 |$o) = (AoPo + A1TP1T + XnPn + A2P2) |#hf>, (4.10) with variational parameters A» and projection operators Pi which project onto the subspaces with i electrons on the impurity atom. In the case of a single orbital we can write the projection operators as: Po = (l-nT)(l-ni) (4.11) Pi)T = nT(l-7H) (4.12) Pu = ni(l-n}) (4.13) P2 = n^ni- (4-14) The motivation for Ansatz (4.10) is obvious: if X\ß are large compared to A0 and A2, the charge fluctuations in the Kondo regime are suppressed. Improved results can be obtained if, instead of I^hf), one uses a generalized Hartree-Fock state I^hf), constructed from non-interacting eigenstates (pk of an effective single-particle Hamiltonian i7eff:242 + ^5>lA + h.c). (4.15) If n0 = riHF and V = V, then |^hf) g°es °ver to the restricted Hartree-Fock ground state |^hf)- in general, n0 and V are additional free variational parameters. 4.2. GUNNARSSON-SCHONHAMMER VARIATIONAL METHOD 69 The variation of the A» leads to a 4 x 4 eigenvalue problem of the form det(H — eS) = 0. Here S is the 4x4 over-lap matrix, since the projected states \^a) = P^I^hf) do not form an orthonormal basis: Sßa = (^ß\^a). The lowest eigenvalue gives the ground-state energy as a function of n0 and V. A minimization with respect to n0 and V is then performed in order to obtain the variational ground-state energy. Schönhammer's variational Ansatz for the single-impurity Anderson model becomes exact in three limiting cases: (a) noninteracting case, U =0, (b) uncoupled case, T = 0, and (c) zero band-width case, D =0. The Ansatz is also known to give a very good interpolation for all values of parameters.241 In fact, the test results for a single Anderson site match very well the exact Bethe Ansatz results .245>246 It can be shown that this variational method always gives ground-state energy which is lower or equal to the corresponding Hartree-Fock ground-state energy.245 Like all variational methods, this technique is, however, limited to zero temperature calculations. The procedure can be generalized to interacting systems with M sit,es.157'221'245'247'248 In the auxiliary noninteracting Hamiltonian, the arbitrary parameters are matrix elements describing the hopping between the leads and the central region and all the matrix elements of the central region C. The variational Hilbert space is spanned by a set of a = 1,..., 4M basis functions |*«) = P«|*HF> = nPil*HF>, (4-16) iec where P^ are projection operators on unoccupied, singly occupied and doubly occupied site i = 1,...,M. Variation of corresponding parameters \%n again leads to a diagonalisation of the original Hamiltonian in the reduced basis: Hßada = ESßada, (4-17) with Hßa = (i^\H\^a) and Sßa = {ipß\ipa), where \if)a) = P^hf)- To obtain the variational ground state we minimize the lowest eigenstate of this eigenvalue problem with respect to the variable parameters of the auxiliary Hamiltonian. Finally, the variational technique can also be extended to cases where phonon degrees of freedom are present in the interacting region.41'249 In this case, |^hf) is the ground state of the effective electron-phonon system which is solved numerically exactly, while Kondo interaction are treated variationally, as previously described. To calculate expectation value of a local operator (for example the total occupancy iV = Y^ Tii), we write the variational wavefunction as * = J]6W«, (4.18) a where da are the variational coefficients of the ground state. We know that, by construction, r(pa is associated with some definite occupancy Na. We thus have N\iftß) = Nß[tpß). We 70 CHAPTER 4. OTHER METHODS FOR IMPURITY MODELS write N = Y,d*a(iPa\N\ijß)dß (4.19) a, ß = YJdldß^Mß)Nß (4.20) a,ß T. a,ß d*JßSaßNß. (4.21) In this manner we can obtain the expectation values of all operators that can be expressed in terms of occupancies n»; such operators are, for example, charge fluctuation nf and local spin Sf = 3/4(2rij — nf). Expectation values of more general operators, such as (ipa^o^ipß), need to be evaluated using Wick's theorem. 4.3 Quantum Monte Carlo method Quantum Monte Carlo (QMC) methods are in many cases the only reliable numerical tool for exploring the properties of quantum systems with very strong correlations between the particles. Lattice fermion problems can be tackled with the auxiliary field (determinantal) QMC method where the fermion degrees of freedom are integrated out of the problem at the expense of introducing classical scalar auxiliary fields. 4.3.1 Projection to the ground state The ground state of a quantum system can be projected out of an (almost) arbitrary trial wavefunction ^T by applying an exponential operator e~@H and taking the 6 —> oc limit.250 For simplicity we disregard the normalization and we write |*o> = lim e-@H\^T), (4.22) where ^o is the ground state. A necessary condition for the method to work is that the trial wavefunction has some overlap with the ground state, (^oI^t) Y" 0- In practice this requirement is easy to satisfy. All measurable physical quantities can be obtained from the partition function Z = Tr [e-@H] = J](V>|e-0H|V>>. (4.23) At zero temperature, 6 goes to infinity and the ground state is projected out of any state rtp. The expression above then reduces to Z= {^T\e-@H\^T}, (4.24) where ^t is again an arbitrary trial wavefunction. The expectation value Z of the evolution operator e~@H is therefore the quantity of central interest. 4.3. QUANTUM MONTE CARLO METHOD 71 4.3.2 Trotter decomposition We consider the evolution operator for a Hamiltonian which can be decomposed into hopping and interaction part, H = Ht + Hi: e-@H = e-e(Ht+H1)_ (425) An exponential of a sum of commuting operators x can be exactly decomposed in the form of a product of exponentials using the Glauber formula251 eA+B = eAeBe-[A'B]/2, if [A, B]= const. (4.26) Kinetic and electron-electron repulsion terms do not commute, so this simple expression does not apply to interacting cases. Nevertheless, the two terms can still be approximately separated. We divide the imaginary time 6 into small time steps (time slices): . m e"0" = (e -*TH . (4.27) Here m is the number of the time slices and Ar is the length of a time step, so that 0 = mAr. If Ar is small, we can perform the approximate Trotter decomposition:252 The dots denote a remainder of the order of Ar. Putting (4.27) and (4.28) together, we obtain the final expression , m By decreasing the step size Ar, the systematic error can be made arbitrarily small, but then the number of time steps needs to be increased. In reality, a compromise is made between the number of steps and the numerical stability of the method. 4.3.3 Hubbard-Stratonovich transformation Electron-electron interaction terms are quartic in operators: Hi = U (nT - J) (ni -\) = U (d\d} - J) [d\d{ - J) . (4.30) It is possible to transform quartic form x2 = yA into quadratic ones x = y2 using Gaussian identity in the reverse: e^2 = -)= fd(j) e-^2+4>x (4.31) V 2"7T J 1 Commuting in the broader sense that the commutator is a number and not an operator. 72 CHAPTER 4. OTHER METHODS FOR IMPURITY MODELS The exponential of a quartic form x2 is then found to be an integral over an auxiliary variable (p of the exponential of a quadratic form x = y2, weighted by the Gaussian factor e~2^ . If x is an operator quantity, this is called a Hubbard-Stratonovich transformation,253 and (p is called the auxiliary field conjugated to the operator x. In the case of the Anderson Hamiltonian one can form a square of quadratic forms by writing / 1 \ / 1 \ 1 „1 (4.32) n,j n\ l)=-\(n,-ni) 2 + \; and we obtain -Art/( nT-I)( ni-i) e-ArU/4 d4>e -^2 e\/A-uM^-ni) . (4.33) There is also a discrete version of the Hubbard-Stratonovich transformation due to Hirsch254 where instead of the continuous integral one has a sum over two terms: -Ar (7/4 -Ar{7(nT-i)(n;-i) = L_____ V^ as^-r^) 2 ^ s=±l (4.34) Here a is a numerical constant defined by cosh(cü)= eArU^2. The Hubbard-Stratonovich transformation can be considered as a replacement of the two-particle part Hj of the Hamiltonian by an effective single-particle Hamiltonian Hj(s), which depends on the stochastic spin variable s. We rewrite the projector e-0 as -OH I [e -ArH'e -ATHt] = ] n=l n=l E e-ArHj(sn)e-ArHt E n e -ArHj(s„)e-Ari?^ (4.35) We introduce the evolution operator USiß) for the effective single-particle Hamiltonian H' = Ht + Hi(S) in a given configuration S (S denotes the set of field variables sn for all time slices): ^S(e) = ] J (e -ArH(s„) e -ArHtJ _ (4.36) n=l Operator US(@) makes the independent particles evolve for an imaginary time 6 in a varying magnetic field S. Using the evolution operators, the projector can be rewritten as -en L>(e). (4.37) We have split the interacting problem into a large number of non-interacting ones (one for each realization of S) and we will have to sum over all possible configurations of the auxiliary fields to recover the physics of e-e interactions. 4.3. QUANTUM MONTE CARLO METHOD 73 4.3.4 Integration over fermion degrees of freedom The exponentials of single-particle operators that appear in the expression (4.36) can be integrated exactly (recall that these are basically Gaussian "integrals"). Trial wavefunctions are usually Slater determinants represented by matrices of coefficients P. It can be shown that an exponential of a single-particle operator simply transforms a Slater determinant into another Slater determinant, therefore (*T|C/s(e)|*T> = det[PtßsP], (4.38) where L>s is the matrix representation of the evolution operator Us- The fermion degrees of freedom then no longer appear in the problem, but now we have to sum over all possible configuration of the auxiliary field: (*T|e"0H|*T) = J]det[ptßsP]- (4-39) s This problem can, in principle, be simulated with classical Metropolis algorithm.255 Unfortunately, simulations of fermion systems are unavoidably256 plagued by the so-called minus sign problem257-259 caused by the anti-commuting nature of the fermion fields. Here the minus sign problem shows up in Eq. (4.39), since the determinant is not necessarily positive definite. 4.3.5 Constrained Path Quantum Monte Carlo In the Constrained Path Quantum Monte Carlo method,157' 260~264 the ground state wave function |^0) is projected from a known trial function \^t) using a branching random walk that generates an over-complete space of Slater determinants \(p): |*o> = 5>^>> (44°) where c^ > 0. To completely specify l^o), only determinants satisfying (^t\4>) > 0 are needed, because |^o) resides in either of two degenerate halves of the Slater determinant space separated by a nodal plane. In this manner, the minus-sign problem is alleviated. Extensive testing has demonstrated a significant insensitivity of the results to reasonable choices of |\[/T}.260>261>265 In the context of quantum dots, this method has been applied to the single-impurity Anderson model246 and to the chain of three Anderson impurities157 (Section 8.1). Chapter 5 Quantum transport theory Mesoscopic effects occur when the coherence length of electrons exceeds the size of the device through which the electric current flows; electrons then travel coherently through the system and behave in a wave-like manner so that quantum mechanical interference effects can occur. As the electrons scatter only off the boundaries (walls) of the device, rather than on the defects or phonons, we say that the transport is ballistic. The conductance through nanoscopic constrictions is often found to be quantized in units of the conductance quantum G0 = ^ = ^T^[12.9kQ}-1 . (5.1) a Tin This is the conductance of a single-mode conduction channel taking into account both spin orientations.266'267 In lateral quantum dots, for example, the tunnel barriers from the two-dimensional electron gas to the quantum dot are obtained by successively pinching off the propagating channels using the gate electrodes. When the last channel is nearly pinched off, the Coulomb blockade regime develops. In this regime, only one channel in each lead is coupled to the dot.69'70 Similar situation naturally occurs in the scanning tunneling microscopy when the tip is controllably brought into atomic contact with the surface or an adsorbate;268'269 the number of channels is then related to the chemical valence of the contact atoms. It is interesting to note in passing that thermal conductance also becomes quantized when the thermal wavelength Ath = ZitKcKkBT) exceeds the size of the device; in ballistic one-dimensional channel the thermal conductance approaches iT2k2BT/3h per phonon mode.270-271 We focus on systems where a very small (essentially zero dimensional) scatterer is embedded between two metallic contacts, i.e. on systems that can be described by quantum impurity models. The two types of systems mentioned above are particularly relevant: quantum dots and adsorbed atoms (or molecules) on surfaces of metals, Fig. 5.1, al and bl. Both systems can be modeled using discrete lattice models as an impurity in contact with two semi-infinite tight-binding chains. The main difference is that typically a quantum dot is 74 5.1. COND UCTANCE FROM PHASE SHIFTS 75 al) bi) a2) <5><2XT>-0- a=L a=R STM tip Adsorbed atom Substrate b2) a=tip a=substrate Figure 5.1: al) Typical conductance measurement setup for probing transport properties of quantum dots, bl) STM measurement of the tunneling current through an adsorbed impurity atom. a2, b2) Tight-binding model representations of both systems. coupled to the conduction leads with comparable strengths, while adatom electron levels are strongly hybridized with the substrate conduction bands and only weakly to the STM tip unless the STM tip is brought from the tunneling regime to the atomic contact regime with individual adatoms.269 It was observed that the temperature, source-drain bias, and magnetic field dependences of the conductance through a magnetic impurity are qualitatively (but not necessarily quantitatively) similar to each other.75 Finite-temperature and finite-bias conductance can therefore be inferred from the magnetic field dependence of the T =0 zero-bias (i.e. linear) conductances which is easily calculated using several different approaches. In this chapter I describe calculation of conductance by extracting quasiparticle scattering phase shifts from Fermi liquid fixed point NRG eigenvalue spectra (Sec. 5.1), by subtracting total energies for periodic and anti-periodic boundary conditions of an auxiliary ring into which the interacting region is embedded (Sec. 5.2) or using impurity spectral functions in conjunction with Meir-Wingreen formula (Sec. 5.3). All three methods are used later on in this dissertation. 5.1 Conductance from phase shifts The low-temperature fixed point of the majority of quantum impurity models that are relevant to describe transport through nanostructures is Fermi liquid; in fact, observing non-Fermi liquid behavior is exceedingly difficult and is a subject of significant experimental efforts.272 Fermi liquid systems are particularly simple, since they are essentially non-interacting free fermion systems with twisted boundary conditions which determine scattering phase shifts. At zero temperature, the transport properties of a scatterer can be related to the phase shifts of electrons at the Fermi surface; the conductance is then computed using the Landauer-Büttiker formula. The advantage of this approach is that the quasiparticle scattering phase shifts can be extracted directly from the Fermi-liquid fixed 76 CHAPTER 5. QUANTUM TRANSPORT THEORY point NRG eigenvalue spectra .54>69'75>79'98>189>190>194>273>274 For a single-channel problem. the conductance is fully determined by two phase shifts, öj^ and 8^p, which are equal in the absence of the magnetic field. Two-channel problems can be characterized by four phase shifts, #qp en'^, cqp611"'', #qpd'^ and #qpld'-''. These phase shifts encode all information about the physics at zero temperature.94 This approach is also useful at finite temperatures as long as the system is near the Fermi liquid fixed point, i.e. for T 275-277 The scattering of electrons near the Fermi level is described by the scattering matrix S. Its elements are the amplitudes of scattering S^a, of electrons with spin ß from lead a E {L, R} to lead a': cm— ( ^ll bRL\ _ (r L t R\ , . j — I qn qn I — I ,ii a I , yo.t) \ÖLR ÖRR/ \lL 'R/ where we have introduced alternative notation in terms of the reflection and transmission matrices r and t. The scattering matrix can be diagonalized by rotation in the R — L space to the new basis of channels a and 669 US»W=[ o _mJk). (5.3) The unitary transformation is U = exp(i9ry) exp(«0r2:), where t% are the Pauli matrices in the R — L space. Angle 9 mixes left and right conduction channel: while (f> changes the phase (it is zero for problems where wavefunction can be made real). Note also that the phase shifts are defined only modulo it, so that 5qp and #qp + 7r are equivalent. According to Landauer formula , the zero-temperature conductance is determined by the transmission probability tR = SRL:266,276 G(T = 0) = G0^2\Srl\2- (5-5) ß In terms of the scattering phase shifts this equals69 G = Go sin2(2Ö)| J2 sin2(C - O- (5-6) ß The expression does not depend on (f>, while 9 sets the maximal conductance: the unitary limit can be achieved only when 9 = ±7r/2, which corresponds to a symmetric (or antisymmetric) problem with respect to reflection symmetry. 5.1. COND UCTANCE FROM PHASE SHIFTS 77 In the case of particle-hole symmetry, we have SßS~ß =1 for the scattering matrix, so that 5^ + 5^ =0, where 7 G {a, b}.11 Even in the presence of the magnetic field we may then write, for example, G = Go sin2(2#) sin2(^J - #L), (5.7) or an equivalent expression in terms of 8 1 p(e) dt{e) de (5.13) where p(e) is the density of states in the leads and t(e) is the energy dependent transmission function. The method becomes exact in the thermodynamic N —> oo limit, while in finite rings the errors scales approximately as iV-1.245 The sine formula can be proved rigorously for a non-interacting system by relating the conductance to persistent currents in the auxiliary ring.245 The reason this approach also works for interacting systems with Fermi liquid zero-temperature fixed point is, basically, that the fixed-point effective Hamiltonian is non-interacting, therefore the same proof goes through unchanged for the quasiparticle Hamiltonian. The method does not seem to be suitable for singular Fermi liquid systems , such as the problem of two parallel quantum dots (Section 7.2), when the ground state energies are calculated using the variational method (Section 4.2).279 Since at T =0 this problem is equivalent to a Fermi liquid system and an entirely decoupled spin-1/2 degree of freedom, in principle the presence of the residual spin should not affect the energy difference in any way and the conductance formula should apply. It seems likely that the problem is either the use of the variational method for a problem with degenerate ground state or the very slow (logarithmic) approach to the T =0 fixed point;98 in either case, the problem seems to be computational in nature and not a weakness of the formalism per se. Finally, it should be remarked that 80 CHAPTER 5. QUANTUM TRANSPORT THEORY charge transfer between the conduction bands destabilizes known non-Fermi liquid fixed points, 93>158 therefore all systems with finite conductance from left to right conduction lead are necessarily Fermi liquids. This implies that the sine formula holds for any two-lead system in which conductance calculations makes any sense at all, i.e. it is fairly general. Unfortunately, in a system with very small energy scales or very slow approach to the FL fixed point, calculations of energy are impracticable since an exceedingly large auxiliary ring system is required. The calculation of the quasiparticle phase shifts using NRG is the method of choice in such cases. 5.3 Meir-Wingreen formula Meir and Wingreen considered the problem of transport through an interacting region embedded between two conduction leads in full generality.280 The system is described by the generic Hamiltonian: H= Y, cLcka + Hint({dl},{dn})+ Y, (vka>ncladn + h.c.y (5.14) k,adL,R n,k,adL,R Operator cka creates an electron with momentum k in channel a in either the left (L) or the right lead (R), while operators Sn form a complete orthonormal set of creation operators in the interacting region. Here a and n are multi-indexes which include all quantum numbers that are necessary to uniquely specify the state (in addition to momentum k for lead electrons). In other words, a may include spin and orbital quantum numbers, as well as a lateral mode quantum number indicating the quantization level of the state in the lateral confinement potential of the lead or of the contact area. Index n typically includes site index and spin. Using Keldysh formalism,281 the expression for current is derived to be280 I=^Jde{Tr{ [fL(e)TL - fR(e)TR] (G r - G a )} +Tr { (TL - TR) G < }) , (5.15) with ^n,m = 2vr J>«(e)Va'n(e) [Va'm(e)Y . (5.16) G", Ga and G< are retarded, advanced and lesser Green's function matrices, respectively (see Ref. 281 and Appendix B), pa(e) is the density of states in channel a and Va>n{t) equals Vka,n f°r e = efc«- Fermi-Dirac distribution functions are given by /l(c) = [1 + exp((e — ß^/kßT)]-1 and an analogous expression for fR; the source-drain bias voltage V is defined as ßL — ßR = eV. In equilibrium /l(c) = /ß(e) = /eq(e) and, furthermore, G< = — /eq(Gr — G"), therefore the current vanishes. Usually the tunnel coupling is spin-conserving and the hopping elements Va'm are diagonal in spin indexes. Then n and m indexes of T matrices must be of the same spin and the 5.3. MEIR-WINGREEN FORMULA 81 coupling matrices T have block-diagonal structure with respect to spin: o o (5.17) with Fn = Fn = 0. In Meir-Wingreen formulas the T matrix always left-multiplies a Green function matrix and a trace is performed. We find Tr o o GIT GU = Tr MT^TT MT-U = Tr (rnGTT)+Tr (ruGu) . (5.18) The important conclusion one can draw from this calculation is that in the case of spin-conserving tunneling, we only need consider the Green's functions that are diagonal in their spin index. If the problem is, in addition, spin-isotropic, it suffices to consider a single spin projection and then multiply the final result by 2. For the purposes of this work, I consider the specialization to single-mode channels with spin degree of freedom (no orbital degeneracy) and spin-conserving tunnel coupling described by the general Hamiltonian H= Y, %4«^ + Hint({<,},KM})+ Y, (VkZcl^ + h.c.). (5.19) k,a={L,R},/i n,k,a={L,R},/i Now the channel index a takes only two values, L and R, and n indexes sites in the interacting region; spin ß has been factored out and it now appears explicitly in the expressions. The Meir-Wingreen formula for current is rewritten as 1 = ü IdtY. (TriLML)rLM - Me)rß1 (GrM - G"M)} + Tr{(rLM - vRß)G<ß}) ¦ (5.20) VLn% = 27rpL(e)VL^n(e) [VL^m{e)] *. (5.21) with It should be appreciated that Meir-Wingreen approach applies under very general conditions (for example, the system need not be in a Fermi-liquid ground state). 5.3.1 Proportionate coupling We now consider the case of proportionate coupling where the couplings to the leads differ only by a constant factor, so that rL(e) = Arß(e) for all energies. The lesser Green's function G< then cancels from a symmetrized expression for current which takes a simpler form: /= ~ /de[/L(e)-/ß(e)]VlmTr[^G^], (5.22) h ß 82 CHAPTER 5. QUANTUM TRANSPORT THEORY where Y^ = YL^YRß/(rLß + TRß) = FLß (5 23) 1+ A Note that in the symmetric case TL = Tß, i.e. A =1, we have T = TL/2. The zero-bias differential conductance G{T) = (dI/dV)v=o is given by G(T) = "G° /de Tn \ ^ImTr[r^G^(e)]. (5.24) J 2T(1 + cosh(e/fcBT)) *-^ Taking the zero-temperature limit and noting that lim—----------——------ = 5(e), (5.25) r^o 2T(1 + cosh(e/kBT)) w' v ' we obtain (for real T) G(T = 0) = -Go J2 Tr [r"Im G^(e = °)] • (5-26) Application: single-impurity Anderson model If a single dot is coupled to the leads, the matrix TM reduces to a scalar quantity Tß and the current is determined by the spectral function Aß = — ^ImC7* of the dot at e = 0: G = Gott [rTAT(e = 0) + T^e = 0)]. (5.27) For spin-isotropic problems (i.e. in the absence of the magnetic field), Tf = Tj_ = T and A| = Aj_. Defining the spectral function as A = A^ + Aj_, we obtain G = G07TTA(e = 0). (5.28) For a Fermi-liquid system A(e = 0) = sin2 5qp/(7rT), and we recover the expected result G = G0 sin2 5qp. In Fig. 5.4 I compare conductance calculated from the spectral function to that obtained using the phase shift method from Section 5.1. While the results agree qualitatively, there are nevertheless noticeable quantitative discrepancies due to systematic errors in NRG calculations of spectral functions; in particular, the conductance does not reach unitary limit as it should. Phase shift method is therefore more accurate if T =0 results are sought. On the other hand, spectral functions provide additional information about the behavior of the system at finite frequencies from which one can also infer behavior at finite temperatures. 5.3. MEIR-WINGREEN FORMULA 83 Figure 5.4: Conductance through a quantum dot described by the single impurity Anderson model as a function of the gate voltage 8. NRG discretization parameter is A =4 and averaging over 8 values of the sliding parameter z was used for the spectral function method. The extracted phase shifts overlap for different values of z. Figure 5.5: Three parallel quantum dots. Application: parallel quantum dots If several quantum dots (say N =3) are coupled in parallel between two conduction leads. Fig. 5.5, and there is no magnetic field, the components of the hybridization matrix are r, np(e)Vn(e)V^(e); (5.29) where Vn is the hopping amplitude from the impurity orbital n = 1,...,N to the conduction band. In a simplified model we assume a constant density of states p0 and an energy-independent hybridization strength T = 7rp0\V\2 which is the same for both orbitals. All components of the hybridization matrix are then the same: r„)TO = T. Equation (5.26) simplifies to G = -G0nT 2_j Im (-----G r nm I . nrn >¦ ' (5.30) The quantity in the parenthesis is related to the spectral matrix for all orbitals, Anm = - l/(27r)Im(G^TO + G^ra). We are particularly interested in the symmetrized and normalized spectral density function g(e) defined by 9 (e)=7rry^ AraTO(e) (5.31) 84 CHAPTER 5. QUANTUM TRANSPORT THEORY where n,m = 1,...,N. This quantity appears in the final expression for the T = 0 conductance: G = Gog(e = 0). (5.32) In a simple approximation, the temperature dependence of the conductance through the quantum dots can be deduced from the energy dependence of the function g(e). 5.3.2 Strongly asymmetric coupling In the case where an atom is chemisorbed on a surface and it is being probed by the STM tip in the tunneling regime, the hybridization of the atom is much stronger to the substrate than to the tip. Let index a = L stand for the substrate, and a = R for the tip. Assuming that the voltage drop occurs between the adatom and the STM tip (i.e. in the tunneling junction), and that the current is not too high, the atom must be in thermodynamic equilibrium with the substrate and we may assume G< = —/^(G" — G"). The expression for current then simplifies to 1 = 1 E / de (^e) - ^)) Tr[r^(G^ - G"*)], ß (5.33) = I E j de (^ - ^)) Trdini Gn Like in the case of proportionate coupling, only the spectral function is needed. It can be computed for an effective equilibrium problem in which the STM tip is absent. The current may then be computed even at finite bias voltage, since we assumed that the tip does not probe the atom intrusively. Of course, the physical content of Eq. (5.33) is fully equivalent to the Tersoff-Hamman approach which relates the tunneling current to the local density of states.282 This is the approach I will use in later chapters to study the scanning tunneling spectroscopy of magnetic impurities on surfaces of noble metals. Part II Systems of coupled quantum dots 85 Chapter 6 Properties of single impurity models Familiarity with the properties of simple single-impurity models is a necessary background for understanding more complex models discussed in the following chapters. I thus present the results of NRG calculations for the Kondo model (Sec. 6.1), the two-channel Kondo model (Sec. 6.2), and the Anderson model (Sec. 6.3). In fact, these results alone demonstrate most of the unusual behavior of QIMs; multi-impurity models differ mostly in that Kondo screening may compete in a non-trivial way with other effects, in particular with magnetic ordering. 6.1 Single-channel Kondo model The single-channel S = 1/2 Kondo model was already introduced in Sec. 2.2 on the Kondo effect. Its Hamiltonian is H = J2^clßck, + Js-S. (6.1) k/i Henceforth we assume a constant density of states p = 1/(2D) in the conduction band (unless explicitly stated otherwise), where D is the half-bandwidth. The thermodynamic properties for a range of values of the Kondo exchange constant J are shown in Fig. 6.1. In Fig. 6.1a we observe the Kondo screening of the impurity spin degree of freedom from the local moment value of 1/4 to the strong coupling value of 0. At the same time, the dissolution of the impurity degree of freedom into the conduction band can be observed as the decrease of entropy from In 2 to 0, Fig. 6.1b. Note also that the decrease of the entropy occurs in a much narrower temperature range than the reduction of the susceptibility. Finally, the Kondo effect is associated with a peak in the impurity contribution to the heat capacity, Fig. 6.1c; this peak is yet another manifestation of the disappearance (freezing-out) of a degree of freedom in the system. (Since the specific heat is simply the temperature derivative of the entropy, it contains the same physical information.) The specific heat 86 6.1. SINGLE-CHANNEL KONDO MODEL 87 -12 -10 -6 -4 -2 log10(T/D) Figure 6.1: Impurity contribution to a) magnetic susceptibility, b) entropy and c) heat capacity Parameters used are A = 4, ß = 0.46, energy cutoff 15un, 8 values of slide parameter z. Symbols on the curve for pj = 0.1 correspond to the exact Bethe Ansatz results for S = 1/2 Kondo model magnetic susceptibility. peak occurs somewhat below Tk215 The Kondo temperature itself can be accurately extracted by fitting the NRG results for the magnetic susceptibility with the universal Kondo magnetic susceptibility curves obtained using the exact Bethe Ansatz approach; for pj = 0.1 we thus determine TK/D = 1.07 x 10~5, which agrees well with the estimate TK = JDeffVp7exp(-l/pJ) « 1.14 x 10"5, where I took D& = 0.85D215 These plots reproduce known results for the strong-coupling regime at T eff is an effective bandwidth.29 The first terms in the expansion of $ are Hv) = --l-\n\y\ + 0{V). (6.7) y ^ The first two terms are usually retained, which gives kBTK = DeiiVfple-7J. (6.8) The effective bandwidth is D ~ 0.85L».215 The Kondo resonance can be observed in the spectral function of the first site of the Wilson chain, i.e. of the Wannier orbital to which the impurity orbital couples via the exchange interaction, Fig. 6.2. It should be noticed that the presence of the impurity affects the conduction band spectral density even at energies above the scale of the Kondo temperature. 6.2. TWO-CHANNEL KONDO MODEL OS \ **=- S. V 0.4 0.3 0.2 0.1 - - pj=0 - pj=0.1 - pJ=0.2 - * i .5 -0.25 0 co/D 0.25 0 Figure 6.2: Spectral function of the first site of the Wilson chain for the Kondo model. The increase of the spectral function at high frequencies is due to the band-edge effects. 6.2 Two-channel Kondo model The two-channel Kondo (2CK) model was introduced by Nozieres and Blandin in their paper on the Kondo effect in real metals (Ref. 100) where they considered various extensions of the original S = 1/2 single-channel Kondo model. The 2CK model consists of a magnetic impurity with spin S = 1/2 coupled to two equivalent channels. It was the first QIM with a non-trivial NFL fixed point and it was studied by a variety of methods: NRG,79'91'144 bosonization and refermionization,108'109 Bethe-Ansatz52'239'283,284 and CFT.55'77'285 The 2CK model describes the quadrupolar Kondo effect in some cerium and uranium heavy-fermion materials.286 Since a single QD embedded between two non-interacting leads couples only to the symmetric combination of the electrons from both leads,46 such a system is governed by the single-channel Kondo model. To observe 2CK behavior, more elaborate setups with several QDs need to be used.23'24'26'158 A related two-level system model was proposed to explain zero-bias anomalies seen in tunnel junctions and point contacts where two-level tunneling systems couple to the conduction electrons.287~291 An experimental realization of the 2CK model has been proposed in the form of a modified single electron transistor (SET) with a large side-coupled quantum dot playing the role of an additional "lead".23 By finely tuning the gate voltages, an equal coupling of the leads to the S = 1/2 local moment on the SET can be achieved.25'26 The two-channel Kondo effect in this system has been recently observed.272 The 2CK Hamiltonian is 100 kaß a (6.9) where a = 1,2 is the channel index and sa is the electron spin-density at the impurity site of channel a: s = 2^ footß (2VJ foav (6.10) fW 2CK system is said to be channel symmetric if Ji = J2 = J. For channel asymmetric models 90 CHAPTER 6. PROPERTIES OF SINGLE IMPURITY MODELS we define the average exchange constant Javg = |(J1+J2) and parameter A = p(Ji — J2); the asymmetry may be quantified by the asymmetry parameter A = A/(pJavg)2 (see below). The two channels are strongly coupled through the impurity spin operator; the spins of conduction electrons at the impurity site tend to be glued together to form a total spin S = l.100 For infinite coupling J, the impurity spin would therefore be over-screened. What happens instead is that the model scales into a strong-coupling fixed point with a finite value of the coupling constant, where the screening is perfect.91'100'108 A fixed point with finite coupling leads to critical behavior and to power-law dependences in thermodynamic properties as H and T tend to zero (the critical point is H = T =0). The ground state is not a singlet and the entropy at zero temperature is finite S(T = 0) = In 2/2,77'283'284'292 see Fig. 6.3. 1/4 m •59 P H 1/8 S 0 ln2 ln2/2 - --¦ 0 — 0.0001 — 0.0002 — 0.0003 — 0.0005 — 0.001 - 0.005 - 0.01 -10 -8 -6 -4 login(T/D) Figure 6.3: Impurity contribution to magnetic susceptibility and entropy in the two-channel Kondo model for different values of the parameter A = p(Ji — J2). NRG parameters are A = 4, ß = 0.75, cutoff 8uN or at most 2000 states. SU(2)spin x SU(2)iso NRG code was used. Perturbative scaling estimate of the Kondo temperature is 1 28 TK ~ DeffpJexp (------ (6.II; Using NRG, it can be verified that this expression gives a correct description for small pj, Fig. 6.4. For large pJ, the Kondo temperatures decreases exponentially with I/pJ: 6.2. TWO-CHANNEL KONDO MODEL 91 moreover, small J and large J regimes are related by a duality transformation up to higher-order logarithmic corrections.293 The self-duality point occurs for J*/D f« 0.7, where the Kondo temperature is of the order of the bandwidth, TL ~ 0.5L>.293 5 0.05 0.1 0.15 0.2 pJ Figure 6.4: Dependence of Tk on the Kondo exchange constant J. I find _Deff ~ 3-D. Resistance (scattering rate) due to a 2CK impurity goes as R(T)/R(0) f« 1 — AT1/2,28 which is to be compared with the familiar T2 law for Fermi liquid systems. Such temperature dependence has been observed in point contact experiments.28 The NFL strong-coupling fixed point has 0(3) x 0(5) symmetry and its finite-size spectrum is 0,1/8,1/2, 5/8,1,1 + 1/8.79'111 Such non-integer succession of energies can be observed in the NRG eigenvalue spectra, Fig. 6.5. two-channel Kondo model N (odd) Figure 6.5: NRG eigenvalue flow of the 2CK model (A = 4,z = 1/6). The black strips correspond to the 2CK NFL fixed point spectrum 1/8,1/2, 5/8,1 as predicted by the con-formal field theory (after rescaling by 1.71781). The model needs not be particle-hole symmetric to have a NFL fixed point. The potential scattering only affects the charge sector which remains noninteracting, while the Kondo physics occurs in the spin sector without affecting the charge sector.81 The potential scattering is an exactly marginal operator, just like in the single-channel Kondo model; we thus obtain a line of stable NFL fixed points.81 Interestingly, the zero-temperature scattering matrix is universal even in the presence of p-h symmetry breaking;81 this implies that the zero-temperature resistivity is independent of the phase shift if the channel symmetry is maintained. 92 CHAPTER 6. PROPERTIES OF SINGLE IMPURITY MODELS The NFL fixed point is unstable with respect to channel symmetry breaking. If A = p(Jo — J\) is not strictly zero, another cross-over occurs at some low temperature Ta26 T a oc T k x A2 (pj^y (6.12) and the system ends up in the stable Fermi-liquid fixed point of the conventional single-channel Kondo model. The criterion for observability of the NFL regime is clearly Ta <^C TK, or A = A/(pJavg)2 < 1. 0.006 ,0.004 0.002 Figure 6.6: Dependence of T/^/Tk on the asymmetry parameter A (left plot) and the average exchange constant p,Javg (right plot). The expected dependence is T/^/Tk = xA2 with A = A/(pJavg)2. I find consistently x f« 0.14. When magnetic field is applied, the system crosses over to a FL ground state. The NFL fixed point is, however, stable with respect to the exchange anisotropy.79'91'108 The broken spin SU(2) symmetry of the original model is restored in the vicinity of the NFL fixed point.108 6.3 Anderson model The single-impurity Anderson model (SIAM) was introduced by P. W. Anderson as a model of the formation of local moments in solids.294 It consists of a single impurity orbital with electron-electrion (e-e) repulsion U, hybridized with a band of conduction electrons. The Hamiltonian is H = H] band H, imp Hc, with #i band X/fc4 kßCkß; kß Himp = edn + Un1nl, ^c = J]n(4^ + H.c.); (6.13) kß 6.3. ANDERSON MODEL 93 are the band Hamiltonian, the impurity Hamiltonian and the coupling Hamiltonian, respectively. The number operator nß is defined as nß = d)ßdß and n = n^ + nj_, e Ck-^j dj^ —> —d-ß. Parameter 8 thus represents the measure for the departure from the particle-hole symmetric point. Equivalently, gate voltage 8 is the isospin-space analog of the magnetic field in spin space. To cast the model into a form that is more convenient for a NRG study, we make two approximations. We assume a constant density of states, p = 1/(2D). Second, we approximate the dot-band coupling with a constant hybridization strength, T = 7rp\VkF\2. Neither of these approximations affects the results in a significant way. The strong interaction regime in SIAM is defined by U/(Tn) ^$> l.295 At this value of the U/T ratio the Hartree-Fock theory fails and gives an unphysical magnetic ground state with broken symmetry, (ridtß) ^ (n^-^).2'296 Sometimes it is said that SIAM becomes non-perturbative in U for U > Tit. In fact, the perturbation theory is valid for all values of U as evidenced by the Bethe Ansatz solution:2 the impurity susceptibility has the form of a power series in argument U/T which is absolutely convergent for all U.297 This is, however, of limited use, since an infinite number of diagrams would need to be summed. It should be noted, however, that the Kondo problem is clearly not perturbative in T. For every U and T, SIAM will end up in the strong-coupling (SC) fixed-point (FP) below some temperature. If U ^$> T, the cross-over temperature is the Kondo temperature Tk(T,U); for r> [/, the threshold temperature is of the order T, assuming T < D.30 If we are at T =0, we can't tell from the fixed point properties alone whether we are in the Kondo regime or not. At T =0, there is no difference between the Kondo screened system and the strongly hybridized (effectively non-interacting) impurity system as far as the adiabatic measurements are concerned (magnetic susceptibility, specific heat, linear conductance), as long as U ^ 0. Only high-energy probes (photoemission) can detect the difference. To talk about Kondo effect, there really must be a local moment regime at intermediate temperatures. The fixed points are identified by considering special values (namely 0 and oo) for the parameters 8, U and T and comparing the resulting Hamiltonian H^ in the limit iV —> oc with the free-electron Hamiltonian.31 Depending on the values of parameters 6, U, T and T, the system can be in a number of different regimes which are associated with different 94 CHAPTER 6. PROPERTIES OF SINGLE IMPURITY MODELS fixed points of the model:31 1. FO: free-orbital regime: the high-temperature regime where spin and charge fluctuations occur and states |0), | |), | j),|2) are equally probable. The entropy is clearly S'imp = hi 4, while the effective moment /ieff = kßTx/(gßB)2 is 1/8. 2. VF: valence-fluctuation regime, characteristic of the asymmetric Anderson model: the regime where fluctuations between two values of charge occur. The low-energy impurity states are, |0), | |) and | j) for 8 > 0, or | f), I I) an(i |2) for 8 < 0. The entropy is S'imp = hi3, the effective moment is /ieff = 1/6- 3. LM: local-moment regime, where the system behaves like a Kondo model, eventually with potential scattering if 8 ^ 0. The impurity states are | f) and | J,); the magnetic susceptibility is Curie-Weiss like. The entropy is S'imp = hi2, and /ieff = 1/4- 4. SC: strong-coupling regime, this is the regime below the Kondo temperature where the impurity magnetic moment is screened. The impurity spin and the conduction band electrons form a singlet state; there are only residual interactions between renormalized conduction-electron degrees of freedom. The magnetic susceptibility is constant and the specific heat is linear: the system is a Fermi liquid. 5. FI: frozen-impurity regime, this is essentially the same regime as SC The two are related by potential scattering, i.e. there is a line of fixed points joining FI and SC fixed points. The various regimes are schematically outlined in the "phase diagram", Fig. 6.7, for constant U and T such that U/Ttt ^$> 1. We consider only 8 > 0, since due to the symmetry of the Anderson model with respect to simultaneous particle-hole and 8 —> — 8 transformation, the phase diagram is symmetric about the 8 = 0 axis. For 8/U < 1, cross-over from FO to VF occurs at Tj* = U/a, where a denotes a numeric value of order 1 (typically a ~ 5).31 On this scale the excitations to the doubly occupied n =2 states freeze out. In the VF regime, the hybridization-induced virtual transitions from the n = 1 to the n =0 subspace are still important; they renormalize the impurity on-site energy td which becomes temperature dependent:31'211 Ed(T)^ed + -ln^. (6.15) 7T 1 If Ed(T) < 0 and \Ed(T)\ >T asT decreases, the system is unstable with respect to LM regime.31 The cross-over occurs at T2* given by T* ~ -Ed(T*) = -E*d. (6.16) or Ej ~ td + - In (—U/Ed). The condition for the existence of the LM regime is 6.3. ANDERSON MODEL 95 where 7 is some constant. In the LM regime, the impurity behaves like a Kondo impurity and undergoes Kondo screening at the Kondo temperature Tk that will be defined below in the section on the Schrieffer-Wolff transformation. For increasing 6/U, the effective exchange coupling pj becomes of order unity and Tk rises to become of order T2*. If Ed(T) goes positive, the system crosses over from VF to FI regime at a temperature31 TZ~Ed{T3)* = E*d*. (6.18) When \Ed(T)\ < T, there is a regime with non-universal properties where LM, VF and FI regions of the phase diagram meet. Figure 6.7: "Phase diagram" for the single-impurity Anderson model. All transitions are smooth cross-overs, hence the boundaries and cross-over temperatures are "fuzzy". Only ö > 0 half-plane is shown due to symmetry. The behavior of thermodynamic quantities in the symmetric case (5 =0) is shown in Fig. 6.8. For T (6-26) vr \-ed U + edJ~ vr \%-5 § + 8) "ttUI-A(8/U) and K is the potential scattering amplitude PK = TT" (— - 77^— 27T \-ed U + td Here Y = irpV2. These expressions hold for 8 ^C U and for a fiat density of states p = 1/(2D). In the VF regime (8 —> U/2), the impurity-orbital energy is renormalized to E* « ed + (r/vr)ln(-[//L*).31'207'211 Then we have T 2r / i i pK 7T V-.K ^ + ^d/ ; d d' (6.28 r / i i 27TV-S* [/ + L* Whereas in the Kondo model only spin fluctuations are possible, in SIAM with U <^ D and \8\ ^C D real charge fluctuations occur on the impurity at high temperatures. They freeze out on the temperature scale U; at lower energy scales only virtual charge fluctuations are possible and the Anderson impurity behaves as a spin (Kondo) impurity. If SIAM is to be mapped to the Kondo model, the effective bandwidth in the Kondo problem must therefore be associated with the energy scale [J.2-207 In general, the effective particle and 98 CHAPTER 6. PROPERTIES OF SINGLE IMPURITY MODELS hole effective bandwidths Dx and D2 may be different. Haldane argued that the effective bandwidth L>eff is given by the geometric mean, since half the processes generating In T behavior are particle-like and half are hole-like:207 ßeff = VDiD2 (6.29) with In A = ln(Wi/4) + 1/4 and Wx = >y\U/2 + 8\, W2 = >y\U/2 - 8\, where 7 = (2/71") expC ~ 1.13387 and C is Euler's constant. We obtain 7el/4 öeff = ±—Uy/l-4{8/U)*, (6.30) o which gives in the particle-hole symmetric point (8 =0) Deff = 0.182f/. (6.31) This compares well with the NRG calculations for U/D CO.l. For U ^$> D, De^ is given by the scale of the bandwidth D (recall that the effective bandwidth for the S = 1/2 Kondo impurity is L>eff ~ 0.85D, Ref. 215). In the intermediate U ~ D case, D^/D is some smooth function / of the argument L = U/D with asymptotic behavior /(L) -^ 0.182^ for ^ —> 0 and /(^) -^ const for ^ -^- 00. In this work, I will often approximate the Kondo temperature using the following expression rK = 0.182[/v/^ JKexp (-----l—), (6.32) with pJx = ST/ttU. This expression is valid for U <^ D and 8 = 0, i.e. for the symmetric model in the wide-band limit. 6.3.2 Correlation functions Additional insight in the behavior of SIAM can be obtained by computing correlation functions, for example as a function of hybridization T for fixed U, Fig. 6.9. The first panel presents charge fluctuations (q2) = (n2) — (n)2 on the impurity site. Charge fluctuations decrease linearly in T for T ^ U, exponentially on the scale T ~ U, while the low-temperature asymptotic behavior is linear in T/U and is given by a universal function (q2) = (4/7r)(r/[/).302 The second panel presents charge fluctuations (q2) on the first site of the Wilson chain. As this site is part of the non-interacting conduction band, (q2) ~ 1/2 for all T/U. The deviation from 1/2 is proportional to U/T and drops to zero exponentially on a scale set by T. The increase of the deviation with the interaction strength U indicates that e-e repulsion on the impurity site induces correlated electron behavior also in the conduction band, as expected. The third panel present the charge-charge correlation function (qqf) = 6.3. ANDERSON MODEL 99 ((n — l)(uf — I)). A positive value of this correlation function indicates that occupancy tends to be the same on both sites, i.e. both impurities tend to be either both empty or both occupied; conversely, a negative value indicates that the impurity is doubly occupied when the first site on the chain is empty and vice versa. For large value of T, (qqf) is strongly negative. This is not unexpected: for large T, only short-distance behavior is important. Taking into consideration only the impurity site and the first site on the Wilson chain, and considering both sites half-filled on the average, the only charge fluctuation mechanism is the transfer of one electron from one site to the other (if this is allowed by the Pauli principle), resulting in one of the sites being empty and the other doubly occupied: such fluctuations lead to negative charge-charge correlations, as explained above. On the scale of T ~ U, the charge-charge correlations become positive since for small T large-distance behavior becomes important - there is a small energy gain if the impurity and the first site on the chain are both empty or both occupied at the same time. Finally, for T <^ U the charge-charge correlations drop to zero. The fourth panel presents spin-spin correlation function between the impurity site and the first site of the Wilson chain, (S • S/). It is negative for all T/U, as expected from the sign of the effective Kondo exchange interaction as obtained using the Schrieffer-Wolff transformation. It should be noted that for small Y/U ratio, the spin-spin correlation function tends to zero in spite of the fact that the Kondo effect must occur at some low temperature Tk- This seems to be in contradiction with the common description of the strong coupling fixed point as consisting of the first site of the Wilson chain strongly bound into a spin-singlet state with the impurity site, while the rest of the chain is decoupled. In reality, the Kondo screening cloud is an extended object of size oc 1/Tk and for that reason the short-range spin-spin correlation (S • S/) may be small. There is, however, a sum-rule /*oo o J (S-s(r))dr=--, (6.33) where (r) is the spin-density at point r.303 6.3.3 Effect of magnetic field The effect of the magnetic field applied to the quantum dot depends on the direction of the field. If the field is applied in the plane of the 2DEG, there will be Zeeman splitting of the spin states in the dot, while the orbital energies will be only slightly affected.69 Conversely, for a perpendicular field the orbital energies will be strongly shifted and Zeeman splitting can be in the first approximation neglected. Therefore in-plane field can be modeled as a "Zeeman field", H' = gßBBSz (6.34) while perpendicular field can be absorbed in the definition of the on-site energies. In my calculations, the gyromagnetic ratio g and the Bohr magneton ßB will be absorbed in the field B, so that the perturbation term is H' = BSZ and the field is measured in units of energy. 100 CHAPTER 6. PROPERTIES OF SINGLE IMPURITY MODELS -4 -3 -2 -1 10 10 10 10 r/D Figure 6.9: Charge fluctuations, charge-charge and spin-spin correlation functions in SIAM. Quantities without a subscript refer to the impurity, while quantities with a subscript / refers to the zero-th site of the Wilson chain, i.e. the localized conduction-band Wannier orbital that the impurity hybridizes with. The effect of the Zeeman field on impurity spin and charge fluctuations is demonstrated in Fig. 6.10. For B ^C Tk, the magnetic field has no effect, while for B ^$> Tk the impurity spin is completely polarized. The transition occurs on the scale B ~ Tk, but the approach to asymptotic values is slow (logarithmic). There is also some effect on the charge fluctuations (An2) for moderate B, but it only becomes appreciable when B becomes comparable to U (such fields are mostly of academic interest; the highest laboratory continuous magnetic fields are of the order of 45T (National High Magnetic Field Laboratory, Tallahassee), which corresponds to ~ 5meV for g = 2). 6.3.4 Spectral functions and conductivity Systems of coupled quantum dots and magnetic impurities on surfaces are mainly characterized by measuring their transport properties. Conductance can be determined by calculating the spectral functions or the quasiparticle scattering phase shifts (see Chapter 5). Since in quantum dots the impurity level 8 (or ed) can be conveniently controlled using gate voltages, we study the conductance as a function of 8. If the coupling to the left and right electrode of a single impurity is symmetric, it can be shown that the dot couples 6.3. ANDERSON MODEL 101 ¦yT-1/4 Figure 6.10: (Color online) Operator expectation values for an Anderson impurity in a magnetic field. U/D = 1, T/U = 0.05, TK = 2.14 x 10-5D. only to the symmetric combination of conduction electron wave-functions from left and right lead, while the antisymmetric combinations of wave-functions are totally decoupled and are irrelevant for our purpose.46 We use Meir-Wingreen's formula for conductance in the case of proportionate coupling; at zero temperature, the conductance is G = GonrA(0); (6.35) where Go = 2e2/h, A(u) is the spectral function (local density of states) of electrons on the impurity site and T = irpV2. In Fig. 6.11 I plot the spectral function A(u) and the corresponding conductance of the Anderson model for a range of values of the "gate voltage" 8. The conductance is high for a range of 8 where the system is in the Kondo regime. This wide conductance peak is sometimes called the Kondo plateau; it is delimited approximately by \8/U\ < 0.5. The uj =0 spectral function is related to the quasiparticle phase-shift through A(c = 0) = ^E. LIT (6.36) According to Friedel sum rule, we have 8qp = |(n) + a correction term which vanishes in the infinite bandwidth limit due to Anderson's compensation theorem.2'75'304 In SIAM, conductance can thus also be deduced from the impurity occupancy. Unfortunately, this approach cannot always be generalized to multi-impurity models. The Kondo (or Abrikosov-Suhl) resonance for 8 Lorentzian peak: A(u)= ^Re 0 can be approximately described iT K 'jj iT K (6.37) where the Kondo resonance width YK is proportional to the Kondo temperature TK, YK = fcßTV/O^Oövr.213'215 It might be noticed that since 1/0.2067T ~ 1.5, extracting Kondo 102 CHAPTER 6. PROPERTIES OF SINGLE IMPURITY MODELS -0.5 -0.25 0 0.25 0.5 Figure 6.11: Spectral function A(u) and conductance through a quantum dot described by SIAM for a range of parameters 8 with U/D = 0.5, T/U = 0.08. Color of each spectral function corresponds to the value of the conductance. temperature from the peak width by equating YK = kßTK leads to a significant error: nevertheless, this procedure is helpful when one is interested only in the scale of Tk- It should also be observed that the weight of the quasiparticle peak in the spectral function is low (of the order Tk/T). Finally, I remark that the Lorentzian form is found only asymptotically for uj ^C TK; for uj > TK the tails are long and very slowly (logarithmically) decaying.305 For 8 7^ 0, the peak is displaced from the Fermi level by approximately 2TKcoiaxi8qp.2 For 8 —> ±C//2, the Kondo peak merges with Hubbard satellites at uj = e 3Tk- The effect of the Zeeman field on the conductance is demonstrated in Fig. 6.13: as the field B increases past the Kondo temperature Tk, the Kondo resonance splits in two so that the value of the spectral function at uj =0 decreases and drives the conductance to small values. It can be observed that the effect of the magnetic field to the zero-temperature conductance is similar to the effect of non-zero temperature to the conductance in the absence of the magnetic field. 6.3. ANDERSON MODEL 103 -0.04 -0.02 0 m/D 0.02 0.04 Figure 6.12: Effect of the magnetic field on the impurity spectral function for the single-impurity Anderson model. In the subfigures, the field B stands for gßB, while the temperature T k stands for ksTx- i 0.8 0.6 0.4 0.2 0 WD=0.5 /C~ ' ~~"\ r/u=o.o8 / \ / \ — B=0 "B/D-10"4 "B/D=10"3 ~B/D=10"2 — B/D-IO"1 /a \ / \\ -0.5 0 S/U 0.5 Figure 6.13: Conductance through a QD described by the SIAM in magnetic field of increasing strength. Conductance is computed from extracted spin-dependent quasiparticle scattering phase shifts 8qp(T. 104 CHAPTER 6. PROPERTIES OF SINGLE IMPURITY MODELS Comparison of different conduction bands In Fig. 6.14 we plot the conductance as a function of the gate voltage for three different hybridization strengths T, and for a) linear dispersion tk = Dk (r(e) = Y) and b) (co)sine dispersion tu = D sink (T(e) = T\/l — e2). Differences are clearly minimal. The results agree well in the Kondo plateau and for large 8, while the differences are largest in the VF regime (the flanks of the Kondo plateau) where the relevant energy scale is high. In the Kondo regime, the Kondo temperature is set by the hybridization strength on the scale of the Kondo temperature itself, i.e. essentially by the hybridization strength at the Fermi level T(e = 0). For the same T(e = 0), Tk will be roughly the same, since only the effective bandwidth L>eff will be affected by the shape of the conduction band dispersion. i 0.8 0.6 o o ü 0.4 0.2 "0 0.1 0.2 0.3 0.4 8/D Figure 6.14: Comparison of the conductance sweeps obtained for two different types of conduction band: linear dispersion ("flat-band") and cosine dispersion (tight-binding chain). Chapter 7 Properties of two-impurity models Magnetic and transport properties of systems of several magnetic impurities, such as multiple quantum dots or clusters of magnetic adatoms, depend on several elements. At low temperatures, these systems tend to lower their energy and entropy by at least two different mechanisms: by magnetic ordering and by Kondo screening. If both processes occur on the same temperature scale, the resulting competition may lead to non-trivial physics. In the two-impurity Kondo (2IK) model, for example, a quantum critical phase transition at J ~ Tk separates two different regimes: strongly bound local magnetic singlet for J 3> Tk and two separate Kondo singlets for J Y and U ^$> t, the occupation of each dot is n ~ 1 and low-energy excitations are spin fluctuations.326 The effective Hamiltonian takes the form of a 108 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS two-impurity Kondo model: H = JSi • S2 + se • (JelSi + Je2S2) + s0 • ( Jo1Si + Jo2S2). (7.4) The Sj are spin operators associated with the dots and se/° = Z^ /o,e/o,At(2CrW,)/o,e/0,At' (7-5) are the spin operators of the Wannier orbitals in even and odd channels to which the impurities couple. To leading order, we have J = 4t2/U and p,Ja,i = 8ra>i/irU. In general, the Hamiltonian also features a channel-changing scattering terms which involve the operator se-o = 2_^ /o.e^l^wO/lW- (7-6) These terms are especially important in the case of the double quantum dots coupled in series between two conduction leads. Such systems, however, are not discussed at length in this dissertation. We may also rewrite the Hamiltonian in terms of the eigenstates of -f/imp:188'327 Himp = Y,E(a)\a){a\ (7.7) a He = 22 t eMß^a \a)(ß\ fo,e/o/ß +tl/o,ß\ß^afo,e/o,ß \ß)(a\- (7-8) e/o,ß,a,ß Multi-indexes a and ß stand for quantum numbers (Q, S, Sz, r) and the effective hopping coefficients te/o,ß\ß^a = ^te/osialdljß) (7.9) i correspond to electrons hopping from the conduction band to the dots. 7.2 Double quantum dot: parallel configuration In this section we study systems of parallel QDs coupled to the same single-mode conduction channel. Since a unified treatment is possible for any number of dots, we will discuss the general iV-impurity case whenever possible and specialize to N =2 where needed. The motivation for such models comes primarily from experiments performed on systems of several QDs connected in parallel between source and drain electron reservoirs; these systems can be modelled in the first approximation as several Anderson impurities embedded between two tight-binding lattices as shown schematically in Fig. 7.1. Assuming that the coupling to the left and right electrode is symmetric, only symmetric combinations of conduction electrons play a role (see Sec. 7.1) and the use of a single channel model is justified. 7.2. DOUBLE QUANTUM DOT: PARALLEL CONFIGURATION 109 In fact, our model is relevant more generally for any system where RKKY interaction is ferromagnetic, for example to clusters of magnetic adatoms on metallic surfaces.14'328'329 We model the parallel QDs using the following Hamiltonian, which we name the "N-impurity Anderson model": H = -ffband + fldots + Hc. (7-10) Here Hhand = ^kß tkc\ßckß is the conduction band Hamiltonian. Hdots = Y,?=1 Hdotti with U 2 Hdat,i = 5(ni-l) + — (rii - 1) 1 = edrii + Un^n^ is the iV-impurity quantum dot Hamiltonian. Finally, H< = -jw L (^i^V + H.c.) (7.12) c kßi is the coupling Hamiltonian, where Nc is the normalization constant (number of states in the conduction band). We assume a constant density of states in the band, p = 1/(2D), and a constant hybridization strength I\ Note that for N =1 this model coincides with the SIAM discussed in Sec. 6.3. Figure 7.1: Systems of parallel QDs. The tight-binding hopping parameter t determines the half-width of the conduction band, D =2t, while parameter t' is related to the hybridization r by Y/D = [t'/tf. The discussion of this model is structured as follows: 1) using the generalized Schrieffer-Wolff transformation we show that the effective Hamiltonian in the local moment regime is the AMmpurity S = 1/2 Kondo model; 2) we show that the ferromagnetic RKKY interaction locks the spins into a state of maximal total spin, S = N/2] 3) we study the single-channel S = N/2 Kondo screening of this collective spin in which half a unit of spin is screened; 4) we demonstrate the robustness of this behavior by studying the stability of the system with respect to various perturbations and we explore the quantum phase transitions driven by these perturbations. 110 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS 7.2.1 Low-temperature effective models We will demonstrate that the low-temperature effective model for the multiple impurity system is the S = N/2 Kondo model: H = Hband + JKs ¦ S, (7.13) where s is the local-spin density in the Wannier orbital in the conduction band that couples to all N impurities, S is the collective impurity S = N/2 spin operator and Jk is the anti-ferromagnetic spin-exchange interaction that can be derived using the Schrieffer-Wolff transformation. The value of Jk will be shown to be independent of N. Let us consider the different time scales of the iV-impurity Anderson problem, focusing on the (nearly) symmetric case 5 <^. U well within the Kondo regime, U/(Ttt) ^$> 1. The shortest time scale, tu ~ h/U, represents charge excitations. The longest time scale is associated with the Kondo effect (magnetic excitations) and it is given by tk ~ h/Tx where TK is the Kondo temperature of single-impurity Anderson model, rK = 0.182[/v/^7exp(-l/pJ), (7.14) with pj = 8T/ttU. As we will show later, there is an additional time scale tj ~ /i/Jrkky; originating from the ferromagnetic RKKY dot-dot interactions: 64 r2 ^RKKY ~ U{pJK) 2 = — —. (7.15) Tlz U From the condition for a well developed Kondo effect, U/(Tir) ^$> 1, we obtain Jrkky ^ U. We thus establish a hierarchy of time scales tjj <*C t j <^tk- Based on the three different time-scales, we predict the existence of three distinct regimes close to the particle-hole symmetric point. The local moment regime is established at T ~ Tj*, where Tj* = U/a and a is a constant of the order one.30 In this regime the system behaves as N independent spin S = 1/2 impurities. At T ~ TJ., where TJ. = Jrkky//? and ß is again a constant of the order one, spins bind into a high-spin S = N/2 state. With further lowering of the temperature, at T ~ TK the S = N/2 object experiences the Kondo effect which screens half a unit of spin (since there is a single conduction channel) to give a ground-state spin of S — 1/2 = (N — l)/2. 7.2.2 Schrieffer-Wolff transformation for multiple impurities For T < Tj*, SIAM can be mapped using the Schrieffer-Wolff transformation (SWT) to the Kondo model, see Sec. 6.3.1. In this subsection we show that for multiple impurities a generalized SWT can be performed and that below Tj*, the AMmpurity Anderson model maps to the iV-impurity S = 1/2 Kondo model. Furthermore, the exchange constant is shown to be the same as in the single impurity case, i.e. it does not depend on N. 7.2. DOUBLE QUANTUM DOT: PARALLEL CONFIGURATION 111 In the first approximation, each impurity can be considered independent due to the separation of time scales. Therefore, we choose the generator S in the SWT to be the sum S = J2i^i °f generators Si, where Si for each impurity has the same form as in the single-impurity case: Si = E 7-TT<-/x4A - B.c. (7.16) kßa with e± = 8± U/2 and n~l_ = rii-ß, n~_ß =1 — ni-ß. The resulting effective Hamiltonian given by HeS = H0 + ^[S,Hc] (7.17) features 0(V^) effective interactions with the leading terms that can be cast in the form of the Kondo antiferromagnetic exchange interaction Hex = J2Jks-S1} (7.18) i where Si is the S = 1/2 spin operator on impurity i defined by Si = J2aa, dia(l/2cr)dia> and the exchange constant is JK = 2\VkF\2[ ,r TTlnl + ,r , TTlnl ). (7.19) 1 1 \5 -Uj2\ + ]5 + U/2\ This result is identical to Jk obtained for a single impurity, see Sec. 6.3.1. In addition to the expected impurity-band interaction terms, SWT produces impurity-impurity interaction terms. In the p-h symmetric 5 =0 case, these additional terms can be written as W I2 / N \ Ai/eff =2^- r^m-NJ hhop, (7.20) where ft-hop = 2_^ [difj.djfj, + djßdißj . (7-21) Since the on-site charge repulsion favors states with single occupancy of each impurity, the term in the parenthesis in Eq. (7.20) is on the average equal to zero. Furthermore, if on each site charge fluctuations are small, (n2) — (ni)2 ~ 0, hopping between the sites is suppressed and the term hhop represents another small factor. The Hamiltonian AH is thus not relevant: impurities are indeed independent. On departure from the p-h symmetric point (5^0), Ai7efr generalizes to AHeff =2^t f(J2 n% - N j - 2N^\ hhop. (7.22) 112 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS For moderately large 8/U this Hamiltonian term still represents only a small correction to Eq. (7.18). However, for strong departure from the p-h symmetric point, close to the VF regime (i.e. 8 —> U/2), Ai7eff becomes comparable in magnitude to Hex and generates hopping of electrons between the impurities. The above discussion leads us to the conclusion that just below Tj* the effective Hamiltonian close to the p-h symmetric point is Heff = -ffband + 2_^ ^kS ' ^i- (7.23) i If the dots are described by unequal Hamiltonians H dot,i or have unequal hybridizations VL, then the mapping of the multi-impurity Anderson model to a multi-impurity Kondo model still holds, however with different effective exchange constants Jlk. 7.2.3 RKKY interaction and ferromagnetic spin ordering The RKKY interaction is expected to be ferromagnetic as shown by the following qualitative argument. We factor out the spin operators in the effective Hamiltonian Eq. (7.23): Hefi = -ffband + JrS ¦ ^, Sj. (7.24) i Spins Si are aligned in the ground state since such orientation minimizes the energy of the system. This follows from considering a spin chain with N sites in a "static magnetic field" Jks. The assumption of a static magnetic field is valid due to the separation of relevant time scales, tk ^> tj. Impurity states with S < N/2 are clearly excited states with one or several "misaligned" spins. The inter-dot spin-spin coupling is a special case of the RKKY interaction, therefore a characteristic functional dependence given by T TTl T V 64r2 16V& ,70v Jrkky oc U(pJK) = — — = -jTF , 7.25 is expected. The factor U in front of (pJx)2 plays the role of a high-energy cut-off, much like the 0.182Č7 effective-bandwidth factor in the expression for Tk, Eq. (7.14); this is due to the fact that Anderson impurities behave as local moments only for temperatures below T* oc U. Using the Rayleigh-Schrödinger perturbation theory we calculated the singlet and triplet ground state energies Es and Et to the fourth order in Vk for the two-impurity case (see the Appendix in Ref. 51). The RKKY exchange parameter is defined as Jrkky = Es —Et; positive value of Jrkky corresponds to ferromagnetic RKKY interaction. For U/D < 0.1, the prefactor of (pJx)2 m the expression (7.25) is indeed found to be linear in U. Together with the prefactor the perturbation theory leads to51 Jrkky = 0.62U(pJK)2 for [//L> 2 the exchange interaction between each pair of impurities has the same strength as in the two impurity case. Therefore, for temperatures just below Tj, the effective Hamiltonian for the iV-impurity Anderson model becomes Htfl = -ffband + Jks ' 2_^ S* ~~ ^RKKY 2_^ Si ' Sj- (7.27) i i 1. The relevant energy scales are then well separated (Tk <^Tf <^i TJ) which enables clear identification of various regimes and facilitates analytical predictions (see also Section 7.2.1). In Fig. 7.2 we show temperature dependence of magnetic susceptibility and entropy for N = 1,2,3 and 4 systems. As the temperature is reduced, the system goes through the following regimes: 114 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS 1. At high temperatures, T > Tj*, the impurities are independent and each is in the free orbital regime (FÖ) (states |0), | f), | I) an(i |2) on each impurity are equiprobable). Each dot then contributes 1/8 to /xeff = Tx/(gßß)2 for a total of /xeff = A/8. The 30 entropy approaches Simp = Aln4 since all possible states are equally probable. 2. For Tp < T < Tj* each dot is in the local-moment regime (LM) (states | f) and | J,) are equiprobable, while the states |0) and |2) are suppressed). Each dot then contributes 1/4 to ßefi for a total of A/4. The entropy decreases to Simp = A In 2. 3. For TK < T < TF and A > 1 the dots lock into a high spin state S = N/2 due to ferromagnetic RKKY coupling between local moments formed on the impurities. This is the ferromagnetically frozen regime (FF)152 with /xeff = S(S + l)/3 = A/2(A/2 + l)/3. The entropy decreases further to S[mp = hi(2S + 1) = ln(A + 1). 4. Finally, for T < Tk, the total spin is screened from S = N/2 to S = S — 1/2 = (N — l)/2 as we enter the strong-coupling regime (SC) with /xeff = S(S + l)/3 = (N — 1)/2[(A — l)/2 + l]/3. The remaining S — 1/2 spin is a complicated object: a S = N/2 multiplet combination of the impurity spins antiferromagnetically coupled with a spin-1/2 cloud of the lead.152 In this regime, the entropy reaches its minimum value of Sjmp = In(2,5 + 1) = In N. N Kondo temperature Tk/D LM-FO temperature TF/D 1 1.20 xlO-12 2 1.23 xlO-12 1.87 xl(-5 3 1.29 xlO-12 2.11 x 10-5 4 1.32 xlO-12 2.32 x 10-5 Table 7.2: Kondo temperatures for different numbers of quantum dots iV corresponding to plots in Fig. 7.2. In Fig. 7.2, atop the NRG results we additionally plot the results for the magnetic susceptibility of the S = N/2 Kondo model obtained using the Bethe-Ansatz (BA) method. For T < TJ. nearly perfect agreement between the A-impurity Anderson model and the corresponding S = N/2 SU(2) Kondo model are found over many orders of magnitude. This agreement is used to extract TK of the multiple-impurity Anderson model. The fitting is performed numerically by the method of least-squares; in this manner very high accuracy of the extracted Tk can be achieved. The results in Table 7.2 point out the important result of this work that the Tk is nearly independent of N, as predicted in Section 7.2.3. In this sense, the locking of spins into a high-spin state does not, by itself, weaken the Kondo effect;18'324 however, it does modify the temperature-dependence of the thermodynamic and transport properties.98'99 It is instructive to follow transitions from FO to L M and FF regimes through a plot combining the temperature dependence of the magnetic susceptibility and of other thermodynamic 7.2. DOUBLE QUANTUM DOT: PARALLEL CONFIGURATION 115 a) 2 1.5 P5 H "S H 0.5 T" ------ 1 dot ------ 2 dots ------ 3 dots ------ 4 dots _________.—-** ?2V FF T=TT, •"" «*-"" T=TV U/D=0.01 r/U=0.02 LlvT FO T=T, -15 -10 log10(T/D) -5 b) 0.5, - 3/«4--------- - 2 In 4 T=T T=T * K * *F ___ (lm / - 3 In 2 /' / / 1 1 - In 4 .. .. // 1 1 1 _/ • J * T=Tj ln 3 ____-¦ ¦-.---'(JF) -y-, In 2 —_--'" '"' U/D=0.01 r/u=o.o2 ........ /:..... 20 -15 -10 log10(T/D) -5 Figure 7.2: a) Temperature-dependent susceptibility and b) entropy of the iV-dot systems calculated using the NRG. The symbols in the susceptibility plots were calculated using the thermodynamic Bethe Ansatz approach for the corresponding S = N/2 SU(2) Kondo models (• S = 1/2, ¦ S = 1, ¦ S = 3/2, ^ S = 2). 116 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS quantities, as presented in Fig. 7.3. Charge fluctuations ((5n)2) show a sudden drop at T ~ TJ representing the FO-LM transition. In contrast, the magnitude of the total spin S increases in steps: S = 1/2, (\/7 — l)/2 and 1; these values are characteristic for doubly occupied DQD in the FO, LM and FF regime, respectively. U/D=0.01 r/U=0.02 5=0 Figure 7.3: Temperature-dependence of susceptibility, charge fluctuations ((5n)2), total spin S and the spin-spin correlations (Si • S2) of the 2-dot system. The LM-FF transition temperature TF can be deduced from the temperature dependence of the spin-spin correlation function. In the FF regime the spins tend to align, which leads to (Si • S2) —>~ 1/4 as T —> 0, see Fig. 7.3. The transition from 0 to 1/4 occurs at T ~ TF. We can extract TF using the (somewhat arbitrary) condition (Si • S2)(TJ) = l/2(Si • S2)(T ^ 0). (7.29) In section 7.2.7 we show that this condition is in very good agreement with TF = Jrkky/' ß obtained by determining the explicit inter-impurity AFM coupling constant Ji2, defined by the relation Jrkky + J12 = 0 that destabilizes the high-spin S = N/2 state. The extracted T p transition temperatures that correspond to plots in Fig. 7.2 are given in Table 7.2. We find that they weakly depend on the number of impurities, more so than the Kondo temperature. The increase of TF with N can be partially explained by calculating TF for a spin Hamiltonian H = — Jrkky Si<-/ S • Sj for N spins decoupled from leads. Using Eq. (7.29) we obtain TF « 1.18 JRKKy for N = 2, TF « 1.36 JRKKy for iV = 3 and TJ « 1.55 Jrkky for N = 4. By performing NRG calculations of TF for other parameters U and V and comparing them to the prediction of the perturbation theory, we found that the simple formula (7.26) for Jrkky agrees very well with numerical results. The effect on thermodynamic properties of varying U while keeping T/U (i.e. pJk) fixed is illustrated in Fig. 7.4 for 2- and 3-dot systems. Parameters T and U enter expressions for TF = Jrkky Iß and TK only through the ratio T/U, apart from the change of the effective 7.2. DOUBLE QUANTUM DOT: PARALLEL CONFIGURATION 117 log10(T/D) U/D Figure 7.4: a) Temperature-dependent susceptibility of the 2 and 3-dot systems with the same T/U ratio. Open (filled) symbols are Bethe-Ansatz results for the S = 1 (S = 3/2) Kondo model, b) Comparison of LM-FF transition temperature TF with predictions of the perturbation theory, Eq. (7.26). c) Comparison of calculated Tk with the Haldane's formula, Eq. (7.14). bandwidth proportional to U, see Eq. (7.14) and (7.26). This explains the horizontal shift towards higher temperatures of susceptibility curves with increasing U, as seen in Fig. 7.4a. NRG results and BA results for the Kondo models with S = 1 and S = 3/2 show excellent agreement for T < TF. In Figs. 7.4b and 7.4c we demonstrate the nearly linear [/-dependence of Tp and Tk, respectively. In Fig. 7.5 we show the effect of varying T/U while keeping U fixed. In this case, Tj* stays the same, Tp is shifted quadratically and Tk exponentially with increasing T/U. Fig. 7.5b shows the agreement of Tp with expression (7.26), while Fig. 7.5c shows the agreement of the extracted values of Tk with formula (7.14). We note that for N > 2, eventual coupling to an additional conduction channel would lead to screening by additional half a unit of spin152'154 and the residual ground state spin would be S — 1 = N/2 — 1. This may occur, for example, due to a small asymmetry in the coupling to the source and drain (left and right) electrodes; the two screening channels are formed by the even and odd linear combinations of the conduction electrons from both electrodes. For N > 3 and three channels (due to weak coupling to some third electrode), three half-units of spin would be screened, and so forth. These additional stages of Kondo screening would, however, occur at much lower temperatures; all our findings still apply at temperatures above subsequent Kondo cross-overs. In systems of multiple QDs, an additional screening mechanism is possible when after the first Kondo cross-over, the residual interaction between the remaining spin and the Fermi liquid quasi-particles is antiferromagnetic.174 This leads to an additional Kondo crossover at temperatures that are exponentially smaller than the first Kondo temperature, as 118 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS a) 0.6 ^0.5 oo P * 0.4 H 0.3 0.2, ------1--------•— I7U=0.01 r/u=o.oi5 r/u=o.o2 r/u=o r/u=o 25 -20 -15 loa U/D=0.01 -10 -5 „(T/D) lOo.Ol 0.02 0.03 0.04 r/u Figure 7.5: a) Temperature-dependent susceptibility of the 2-dot system for equal e-e repulsion U/D = 0.01 and for different hybridization strengths I\ Symbols represent the Bethe-Ansatz susceptibility for the S =1 Kondo model with corresponding TK. b) Comparison of calculated and predicted TF, Eq. (7.26). c) Fit of Tk to the Haldane's formula, Eq. (7.14). discussed in Sec. 7.3.3. In parallelly coupled systems, the residual interaction between the remaining spin and the Fermi liquid quasi-particles is, however, ferromagnetic as can be deduced from the splitting of the NRG energy levels in the SC fixed point:99 the SC fixed point is stable. 7.2.5 Stability of N =2 systems with respect to various perturbations The following subsections are devoted to the effect of various physically relevant perturbations for the N =2 system. We generalize the Hamiltonians to Himp = Y, H**,i + Un{m - l)(ra2 - 1) + ii2 Y, (dipd*P + H-c-) + Ji2Si • S2 Hdot,i = Si(rii -l) + —(rii- l) 2 #c = ]>> (4Am + H-c.) . (7.30) i,n We will study different perturbation terms separately. It will be shown that FF regime and the ensuing S =1 Kondo effect are fairly robust against perturbation of increasing strength; very strong perturbations, however, lead to quantum phase transitions (QPTs) from the S = N/2 state. QPTs are triggered by the competition between various effects 7.2. DOUBLE QUANTUM DOT: PARALLEL CONFIGURATION 119 (such as magnetic ordering and Kondo screening) when the two relevant energy scales become comparable. Some of the results will be easier to understand if H[mp and Hc are transformed to symmetric and antisymmetric (gerade/ungerade) basis states [for J\2 =0]: ß — ^2 I M ~^~ 2ß) ' Uß ~ y/2 HC 2,1* J • (' -31y 9, We define the asymmetry in coupling of dots to the band by At by t\ = t+A and t2 = t—At and the difference in on-site energies A8 by 8\>2 = 8 ± A8. The transformed Hamiltonian reads: #imp + Hc = (8 - t12)(ng -I)+ (5 + t12)(nu - 1) -----4-----J K - 1) + I -----4-----J (^ - 1) U + 3C/i2 (nfl - l)(nu - 1) + (-[/ + f/12) Sfl • SM / (7.ozJ ^+fA2\ /tt , tt -2t(/et)/i^ + h.c.) - 2At (fl^Ufj, + h.c.) + Ac (flr^iv + h.c.) + const. The hopping t\2 leads to hybridization between the atomic levels and results in the formation of "molecular orbitals" with energies tu>g = 8 ± ti2. We observe that for At = 0, only symmetric (gerade) level is coupled to the leads. In addition, for A8 = 0, the two levels are "decoupled", i.e. there is no one-electron hopping from g to u. 7.2.6 Variation of the on-site energy levels Deviation from the particle-hole symmetric point A small departure from the p-h symmetric point does not destabilize the S =1 Kondo behavior: the magnetic susceptibility curves still follow the BA results even for 8/U as large as 0.4, see Fig. 7.6a. The triplet state is destabilized for some critical value 8C, beyond which the S =1 Kondo effect does not occur. In the asymmetric SIAM, the VF regime is characterized by /ieff = 1/6, see Sec. 6.3. For two uncorrelated dots in the VF regime, we expect /ieff ~ 1/3. In Fig. 7.6a we plotted a number of susceptibility curves for parameters 8 in the proximity of the singlet-triplet transition. While there is no clearly-observable VF plateau, the value of /xeff is indeed near 1/3 between T* and T*(8C). In Fig. 7.6b we compare calculated TK with analytical predictions based on the results for the single impurity model.31 For 8 ^ 0, we use the SWT results for exchange Jk and 120 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS a) —i-----------'-----------1-----------'-----------1-----------'-----------r U/D=0.01,r/U=0.02 T,(SJ! 0O1 6.2 6.3 6.4 0.5 5/U c) 2x10"" e .5 * B.1X10 H 1 data -fit -10 -8 -6 log10(T/D) 0.1 0.2 0.3 0.4 0.5 8/U Figure 7.6: a) Temperature-dependent susceptibility of the 2-dot systems on departure from the p-h symmetric point. Symbols are fits to the universal susceptibility obtained using the Bethe-Ansatz method for the S =1 Kondo model, b) Calculated and predicted Kondo temperature (fit 1), Eq. (7.34). For comparison we also plot TK given by Eq. (7.14), which shows expected discrepancy for large 8/U (fit 2). exponential function. Calculated TF and the fit to an potential scattering K, see Sec. (6.3.1). The effective Jk that enters the expression for the Kondo temperature is31 Jk = Jk [I + (vrpX)2] , (7.33) and the effective bandwidth 0.182Č7 is replaced by 0.182|L'^|. The Kondo temperature is now given by (7.34) TK = 0.182\E*d\VpJKexp (-l/(pJK)J . This analytical estimate agrees very well with the NRG results: for moderate 8/U, the results obtained for asymmetric single impurity model also apply to the multi-impurity Anderson model. In Fig. 7.6c we show the ^-dependence of the LM-FF transition temperature TF. Its value remains nearly independent of 8 in the interval 8 < 0AU and then it suddenly drops. More quantitatively, the dependence on 8 can be adequately described using an exponential function .'S-Ör 1 — exp T*p(8) = T*p(0) X (7.35) where TF(0)/D = 1.8 x 10-5 is the transition temperature in the symmetric case, 8C/D = 0.45 is the critical 8 and X/D = 2.1 x 10-2 is the width of the transition region. Exchange interaction Jrkky does not depend on 8 for U/D = 0.01 ^C 1, which explains constant value of Tp(8) for 8 < 0AU. At a critical value 8C, TF goes to zero and for still higher 8 the spin-spin correlation becomes antiferromagnetic. Since the ground-state spins are different, the triplet and singlet regime are separated by a QPT at 8 = 8C. This transition is induced 7.2. DOUBLE QUANTUM DOT: PARALLEL CONFIGURATION 121 by charge fluctuations which destroy the ferromagnetic order of spins as the system enters the VF regime. The exponential dependence arises from the grand-canonical statistical weight factor exp[8(n — 2)/(kßT)], where n is the number of the electrons confined on the dots. The transition is of the first order, since for equal coupling of both impurities to the band there is no mixing between the n =2 triplet states and the n =0 singlet state.174 For 8 slightly lower than the critical 8C, the effective moment /xeff = Tx(T)/(gßB)2 shows a rather unusual temperature dependence. It first starts decreasing due to charge fluctuations, however with further lowering of the temperature the moment ordering wins over, ßef{ increases and at low-temperatures approaches the value characteristic for the partially screened S = 1 moment, i.e. /xeff = 1/4. :l____________i_____________I_____________i_____________LI________i_____________I_____________i_____________I_____________i_____________I_____________i_____________I 0 0.2 0.4 0.6 0.8 1 1.2 Figure 7.7: "Phase diagram" for the two-impurity Anderson model. Compare with the diagram for the single-impurity model, Fig. 6.7. The "phase diagram" for the two-impurity model that schematically represents the flow between the fixed points is shown in Fig. 7.7. In comparison with the analogous diagram for the single-impurity model (Fig. 6.7), there are two notable differences: an additional ferromagnetically-frozen fixed point region FF appears, and the strong coupling SC and frozen impurity FI regimes no longer correspond to the same line of fixed points, but are, instead, separated by the quantum phase transition that was described above. Note that in general, there may exist several qualitatively different intermediate-temperature high-spin and low-temperature strong-coupling fixed points. For N > 3, there might in principle exist (for 8 ^ 0) additional ferromagnetically-frozen regimes with S = n/2, where n < N is the occupancy of the impurities near such fixed-points. The system would then make a transition from such FF(n) regime to a SC(n) regime with residual spin S = (n — l)/2. Since the SC(n) fixed points have different spin, the system will go through cross-overs or true quantum phase transitions as 8 is increased toward U/2 at zero-temperature. There can be at most N — 1 such transitions, since there are N SC fixed points with residual spin ranging from 0 to (N — l)/2 in steps of 1/2, where SC(N — 1) should be identified with the FI fixed point. It turns out (results not shown), that for N =3 there are indeed two 122 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS Figure 7.8: a) Temperature-dependent susceptibility of the 2-dot system with unequal (detuned) on-site energies, 5\ = A, #2 = —A. Full symbols present Bethe-Ansatz results of the equivalent S = 1 Kondo model, while empty symbols are BA results of a S = 1/2 Kondo model, b) Comparison of calculated and predicted Kondo temperature, see Eqs. (7.14) and (7.36). c) The Kondo temperature of the S=l/2 Kondo screening on the singlet side of the transition and a fit to Eq. (7.37). quantum phase transitions and, surprisingly, the same holds for N =4 and N =5 as well. The conductance through N parallel quantum dots is shown in Subsection 7.2.10. Splitting of the on-site energy levels We next consider the 2-dot Hamiltonian with unequal on-site energies 8i and focus on the case 5\y2 = ±A. This model is p-h symmetric under a generalized p-h transformation ck —> Ck-ß-, d\ß —> d,2,-ß, d2ß —> di-fj,; the total occupancy of both dots is thus exactly 2 for any A. We can therefore study the effect of the on-site energy splitting while maintaining the p-h symmetry. This perturbation induces hybridization between gerade and ungerade levels, see Eq. (7.32). Susceptibility curves are shown in Fig. 7.8a for a range of values of A. For A up to some critical value Ac f« 0.47 the 2-dot Anderson model remains equivalent to the S =1 Kondo model for T 0. As seen from Fig. 7.9, for Ji2 above a critical value Jc, the RKKY interaction is compensated, local moments on the dots form a singlet state rather than the triplet which in turn prevents formation of the S =1 Kondo effect. The phase transition is of the first order.174 Using Eq. (7.29), we obtain T*F/D » 1.87 x 1C-5 for the non-perturbed problem with the same U and T, while Jc/D f« 1.68 x 10-5. Taking into account the definition Tp = Jrkky/ß, where ß ~ 1, we conclude that Jrkky agrees well with the critical value of Jc, i.e. Jc = Jrkky- The perturbation theory prediction of Jrkky/-E> = 1.6 x 10-5 also agrees favorably with numerical results. As long as Ji2 TK, the S =1 Kondo effect survives and, moreover, TK remains unchanged, determined only by the value of pJk as in the Ji2 = 0 case. The 124 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS 0.6 0.5 0.2 0.1 -20 -15 -10 -5 0 log10(T/D) Figure 7.9: Temperature-dependent susceptibility of the 2-dot systems for different anti-ferromagnetic inter-impurity couplings Ju- Circles are Bethe-Ansatz results for the susceptibility of the S =1 Kondo model with the Kondo temperature which is equal for all cases where Ji2 , Jrkky/D « 0.62 x (64/vr2) {V2/U), therefore the critical tu,c is given by tu,c ~ T and it does not depend on U. 7.2.8 Inter-impurity electron repulsion We study the effect of the inter-impurity electron repulsion (induced by capacitive coupling between the two parallel QDs) by turning on the Uu term: 2 ffdots = Yl H^ + ^ (ni - l)(n2 - 1). (7.39) i=\ This perturbation breaks SU(2)iso symmetry to U(l)charge, see Sec. 2.3.4. Results in Fig. 7.11 show that the inter-impurity repulsion is not an important perturbation as long as Uu U the electrons can lower their energy by forming on-site singlets and the system enters the charge-ordering regime.184 The system behaves in a peculiar way at the transition point Uu = U where Uu and U terms can be combined using isospin operators as U/2 (4(/02 + 4(/2)2) + C/1247f/| = 2U(IZ)2. (7.40) Note also that exchange and two-electron hopping terms drop of the Hamiltonian in the gerade/ungerade basis, see Eq. (7.32). We now have an intermediate temperature fixed U/D=0.01,17U=0.02 126 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS 0.6 1 ' 1 ' - ¦ — U12/U=0.9 /,',¦' \ - 0.5 - ._._ U,„/U=0.99 /// V ¦ —V1^1 /// >< \ \\ ja0.4 - .... U,,/U=1.01 // / / \ \ \ \ i / ' /' f P 0.3 ^s*'*'' p "~rKA _ H 0.2 .¦ 0.1 - U/D=0.01, r/U=0.02 •' 1,1,1 " -20 -15 -10 log10(T/D) 3 2.5 2 i 1.5 1 0.5 0. 2/«4 --------U12/U=0.99 / _ - — u12/u=i.o / _._. u12/u=i.oi ln 6~ _ ln 4- fl± ln 3 ' — i In 2 . - U/D=0.01, I7U= =0.02 i - , I i / i ¦ 15 -10 log,„(T/D) Figure 7.11: a) Temperature-dependent susceptibility of the 2-dot systems for different inter-impurity electron-electron repulsion parameters U\2- Circles are the Bethe-Ansatz results for the S = 1/2 Kondo model which fit the NRG results in the special case U\2 = U. b) Temperature-dependent entropy of the 2-dot systems for different inter-impurity electron-electron repulsion Uu- point with a six-fold symmetry of states with Iz =0 as can be deduced from Eq. (7.40) and the entropy curve in Fig. 7.11b). While the SU(2)iso isospin symmetry is broken for any Uu ^ 0, a new orbital SU(2)orb pseudo-spin (approximate) symmetry appears at the special point U^ = U. For two impurities we can define an orbital pseudo-spin operator as O = E E 4(1/2^)^, (7.41) ß i,J=l,2 where cr is the vector of the Pauli matrices. Note that the orbital pseudo-spin and isospin operators do not all commute, therefore the full orbital pseudo-spin and isospin SU(2) symmetries are mutually exclusive. The quantum dots Hamiltonian Hdots commutes for Uu = U with all three components of the orbital pseudo-spin operator; the decoupled impurities thus have orbital SU(2)orb symmetry. Furthermore, pseudo-spin O and spin S operators commute and the symmetry is larger, SU(2)spin ® SU(2)orb. In fact, the set of three Sl, three Ol and nine operators StO:' are the generators of the SU(4) symmetry group of which SU(2)spin ® SU(2)orb is a subgroup. The six degenerate states are the spin triplet, orbital singlet and the spin singlet, orbital triplet149 which form a SU(4) sextet (the notation is \S,Sz,0, Oz)): |i,i,o,o> = | u>, |o,o,i,i) = |u,o>, |1,0,0,0> = 1^/2(1 u> + I U» , 10,0,1,0) = 1^/2(1 u> - I U» ; 11,-1,0,0} = ||,|}, |0,0,1,-1> = |0,U>. It should be noted that the orbital pseudo-spin eigenstates |0,0,1,±1) can be combined 7.2. DOUBLE QUANTUM DOT: PARALLEL CONFIGURATION 127 into an isospin triplet eigenstate \S = 0, / = 1,Iz = 0) = l/-\/2(| Tl>0) + |0,TI)) and an isospin singlet eigenstate \S = 0,1 = 0) = l/-\/2(| Tl>0) — |0,|J,)). This recombination is possible because Iz (f/(l)charge charge operator) commutes with both the Hamiltonian (see Eq. (7.40)) and the orbital pseudo-spin operators. The coupling of impurities to the leads, however, breaks the orbital symmetry. Unlike the model studied in Ref. 184, our total Hamiltonian H is not explicitly SU(4) symmetric, and unlike in the model studied in Ref. 194, in our system this symmetry is not dynamically (re)established on the scale of the Kondo temperature. No SU(4) Kondo effect is therefore expected. Instead, as the temperature decreases the degeneracy first drops from 6 to 4 and then from 4 to 2 in a S = 1/2 SU(2) Kondo effect (see the fit to the BA result in Fig. 7.11). There is a residual two-fold degeneracy in the ground state. Perturbation theory (Appendix in Ref. 51) shows that the sextuplet splits in the fourth order perturbation in the coupling to the band, 14. The spin-triplet states and the state \S = 0, / = 0) form the new four-fold degenerate low-energy subset of states, while the states \S = 0,1 = 1,IZ = 0) and \S = 0,0 = l,Oz = 0) have higher energy. The remaining four states can be expressed in terms of molecular-orbitals u and g: \l,l,0,0) = g\u\\0), \1, 0,0,0) = l/V2(u\g\ + g\uty\0), ,-1,0,0) = g\u\\0), (7.42) \S = 0,1 = 0) = l/v/2 (u\g\ - g\u\} |0>. The four remaining states are therefore a product of a spin-doublet in the gerade orbital and a spin-doublet in the ungerade orbital. Due to the symmetry of our problem, only the gerade orbital couples to the leads, while the ungerade orbital is entirely decoupled. The electron in the gerade orbital undergoes S = 1/2 Kondo screening, while the unscreened electron in the ungerade orbital is responsible for the residual two-fold degeneracy. 7.2.9 Unequal coupling to the continuum ri 12 We finally study the Hamiltonian that allows for unequal hybridizations Tj = np\V/.F\ . We set VL = oiVl, i.e. T2 = Cü2ri. The effective low-temperature Hamiltonian can be now written as 2 Heft = -ffband + s • ^^ Jk,Si ~ ^RKKYSl ' S2. (7.43) i=\ with Jk,2 = o? Jk,\ and with the effective RKKY exchange interaction given by a generalization of Eq. (7.25) ^rkky = 0.62Up2JKA JK)2 = a2JRKKY, (7.44) where Jrkky is the value of RKKY parameter at a =1. In our attempt to derive the effective Hamiltonian we assume that in the temperature regime T < Jrkky the two 128 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS moments couple into a triplet. Since the two Kondo exchange constants J^,i are now different, we rewrite i7eff in Eq. (7.43) in the following form ffeff = Hband + S • ^ + J*>* (Sl + S2)) + S • ^ ~ J*>* (Sl - S2)) - jfKKYSl ¦ S2. (7.45) Within the triplet subspace, Si + S2 is equal to the new composite spin 1, which we denote by S, Si — S2 is identically equal to zero, and Si • S2 is a constant —1/4. As a result, the effective Jk is simply the average of the two exchange constants: JK1 + JK)2 , ^K,eff =---------------• (7.46) Susceptibility curves for different a are shown in Fig. 7.12. Note that Tk determined using Eq. (7.14) combined with the naive argument given in Eq. (7.46) fails to describe the actual Kondo scale for a < 0.4 as seen from Fig. 7.13. This is due to admixture of the singlet state, which also renormalizes Jk, even though the singlet is separated by Jrkky ^ ^k from the triplet subspace. Note however, that J rkky *s wei^ described by the simple expression given in Eq. (7.44) as shown in Fig. 7.13. By performing a second-order RG calculation (see Appendix D), which takes the admixture of the singlet state into account, we obtain Tk as a function of a which agrees very well with the NRG results, see Fig. 7.13. The following effective low-temperature Kondo Hamiltonian is used: ffeff = tfband + Jis • Si + J2s • S2 - jfKKY (Si • S2 - 1/4) , (7.47) where I have added a non-significant additive constant Jrkky/4. Introducing spin-1 operator S defined by the following Hubbard operator expressions: Sz = X^ — X^, S+ = 1/2(Xfo + Xoi) and S~ =(S+Y, we obtain H = -ffband + Js • S + JrkKY^SS + A(sz(Xos + Xso)+ s +{Xis - Xs1) (7.48) + s - (Xsi-Xis)), where index S denotes the singlet state and we have Ji + J2 _ T /-, , 2> A = ^^ = Jo(l-«2 )/2. (7.49) Equations (D.4) reduce to two equations for J and A P A2 6J = po\6D\ D D 4- Tefl 5A = -2p0^AJ. (7.50) 7.2. DOUBLE QUANTUM DOT: PARALLEL CONFIGURATION 129 a) 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 P 0.25, 1 ' ' 1 ' ' ' ¦ — ? = 1.4 1 ' ' ' I ' ' ' I ' ' - ------? = 1.2 - - — ? = 1 A . — ? = 0.85 // — ? = 0.7 // — ? = 0.5 // / ? = 0.1 1 / - ?=0.1 / / / III ?=1-4 \ - - '".' ¦ ¦ i==T. . / / \ -^ U/D=0.01, ?/U=0.02 \ 1 , , , 1 , , , 1 , , s, 20 -16 -12 log „(T/D) b) 0.6 '0.5 0.4 0.3 0.2, _ ? = i i i i = 0.1 i ' , i — ? = = 0.05 — ? = = 0.01 — ? = = 0.005 — ? = = 0.001 — ? = = 0.0005 ;" ? = = 0.0001 i a ; - ?=0.1 i J ' ? =0.0001 \ - s= =1 Kondo */ // \ " y /s= =1/2 Kondo U/D=0.01, ?/u= i , , , i , =0.02 , I , , , I -25 -20 -15 -10 log10(T/D) Figure 7.12: Temperature-dependent susceptibility of the 2-dot system with unequal coupling to the leads, ?2 = a2?1. a) The range of a where TK is decreasing, b) The range of a where TK is increasing again. Circles (squares) are BA results for the S = 1 (S = 1/2) Kondo model. The arrows indicate the evolution of the susceptibility curves as the parameter a decreases. Figure 7.13: Comparison of calculated and predicted Kondo temperature Tk and effective exchange interaction J rkky- Average J k corresponds to Eq. (7.46), scaling result is obtained numerically as described in the text. Fit to J rkky corresponds to Eq. (7.44). 130 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS from which ensue the following scaling equations dJ ~2 A2 D -Poeff = 0.182Č7 for the Anderson model and we take J (I = logL>eff) = J and A(/ = logL>eff) = A with J and A taken from Eq. (7.49). We integrate the equations numerically until J starts to diverge. The corresponding cut-off D defines the Kondo temperature. For extremely small a, J rkky eventually becomes comparable to the Kondo temperature, Fig. 7.13. For that reason the ferromagnetic locking-in is destroyed and the system behaves as two S = 1/2 doublets, one of which is screened at T^ = Tk(Jk,i) as shown in Fig. 7.12b. 7.2.10 Conductance of parallel quantum dot systems Using the spectral-function and the quasiparticle-scattering phase-shift methods, I have computed the conductance through the system of N parallel quantum dots. The results for N = 1,2,3 were computed using both methods and they agree up to the expected errors associated with spectral function calculations in NRG. For N =4 and N =5, only the phase-shift method was used. The results are shown in Fig. 7.14. The results for N =2 reflect the quantum phase transition discussed in Subsection 7.2.6. The conductance at the transition point drops in a discontinuous manner for ~ 0.2G0- At this point, the occupancy changes only slightly, however the spin-spin correlations flip from positive to negative (not shown). In the 8 < 8C regime, the low-temperature fixed points lie on a line of S = 1 Kondo model strong-coupling fixed points with S = 1/2 residual spin, while for 8 > 8C, the low-temperature fixed points lie on a line of free-impurity fixed points. In the 8 > 8C regime, the conductance goes through a maximum of conductivity near 8 = U/2. When magnetic field is applied, it has a strong effect on the Kondo plateau for 8 < 8C, while it affects the narrow peak for 8 > 8C only weakly unless the field is extremely large. The results for N =3 reflect two different quantum phase transitions. The conductance exhibits a discontinuous jump by ~ 0.2G0 at 8 = 8C\, then the conductance decreases in the narrow interval 8C\ < 8 < 8c2 to 0. This is followed by a sudden jump from zero conductivity to unitary conductivity at #2c, then the conductance slowly decreases toward 0. The first transition point at 8C\ is similar to the transition in the N =2 case: occupancy is affected only slightly, while the spin-spin correlations change sign. The second transition point at 8C2 is quite different: the occupancy changes by exactly 1 electron, while the spin-spin correlations appear continuous. Since a change of electron occupancy by 1 corresponds, by the Friedel sum rule, to a tt/2 phase shift in electron scattering, this explains the change of conductance from 0 to G0. 7.2. DOUBLE QUANTUM DOT: PARALLEL CONFIGURATION 131 Even number of dots Odd number of dots B=10 D B=10 D ¦ B=10 D O O o 0 0.2 0.4 0.6 0 8/U 0 0.2 0.4 0.6 0.8 8/U Figure 7.14: Conductance through the system of N parallel quantum dots as a function of the gate voltage for a range of magnetic fields. Results shown are obtained using the phase-shift method. ?= 4; 200 NRG iterations were performed to reach the zero temperature limit. For the most demanding N = 5 calculation, up to 5000 states were retained in the NRG truncation. 132 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS For both N =2 and N =3, the phase transitions are replaced by smooth cross-overs when the impurity-band coupling constants are made unequal. The conductances of N =4 and N =5 are similar to N =2 and N =3, respectively. There are no additional discontinuities in the conductance as a function of the gate voltage. It turns out that for all N > 3 with symmetric coupling to the band, there are precisely two phase transitions of the same types that occur for N =3. The second transition corresponds to a change of total electron number by N — 2. Since for N = 4 the occupancy changes by two electrons, the phase shift is ir and the quantum phase transition is not mirrored by a discontinuity in the conductance, thus the conductance is similar to that of N = 2. For N = 5, the occupancy change by three electrons leads to a 37r/2 phase shifts and there is again a conductance jump from 0 to G0, like for N =3. These results can be generalized for arbitrary N ^ 1: for odd N, there are two quantum phase transitions and two conductance discontinuities, while for even N there are also two quantum phase transition, but a single conductance discontinuity. 7.3 Double quantum dot: side-coupled configuration We now study the DQD system in the side-coupled configuration. The first QD is embedded between source and drain electrodes while the second QD is coupled to the first through a tunneling junction; there is no direct coupling of the first QD to the conduction bands. The Hamiltonian is H = -ffband + -ffdots + Hc (7.52) where i^band = l>2ku ekCkßCkß and ii/dots and Hc are the impurity and coupling Hamiltonians 2 #dots = 2_^ Hdot>i — 12_^ {dlßd2ß + H.c.J , i=l i" (7.53) Hc = I/VNcYjVM^ + H.c.), where Hdat,i = S(rii — 1) + U/2(n — l)2. For simplicity, we choose the on-site energies and Coulomb interactions to be equal on both dots (the effect of the on-site energy splitting is studied in Ref. 188). Coupling between the dots is described by the inter-dot tunnel coupling t. By changing the gate voltage and the inter-dot tunneling rate, the system can be tuned to a non-conducting spin-singlet state, the usual Kondo regime with odd number of electrons occupying the dots, the two-stage Kondo regime with two electrons, or a valence-fluctuating state associated with a Fano resonance; these regimes are shown in the phase diagram of the system in Fig. 7.15. We will first study the case of large t with wide regimes where the conductance is enhanced due to the Kondo effect, separated by low-conductance regimes 7.3. DOUBLE QUANTUM DOT: SIDE-COUPLED CONFIGURATION 133 t/D 0 0.25 / 0.5 0.75 1 Two-stage 8/D Kondo Figure 7.15: The phase diagram of the side-coupled DQD. U/D = 1, T/D = 0.03. Grey area represents Kondo regime where S ~ 1/2, (n) ~ 1 and G/G0 ~ 1. In shaded area. called spin-singlet regime, where S ~ 0 and (n) ~ 2, there is strong antiferromagnetic spin-spin correlation and G/G0 ~ 0. The two-stage Kondo regime is discussed in Subsection 7.3.3. where localized spins on DQD are antiferromagnetically (AFM) coupled. In the case of small t we will study the two stage Kondo regime where the two local moments are screened at different Kondo temperatures.158'174'183'188 7.3.1 Strong inter-dot coupling Conductance and correlation functions The conductance through the DQD at different values of t is shown in Fig. 7.16 as a function of the gate voltage 8. The conductance is enhanced for a wide range of 8 due to the Kondo effect. To better understand multidot problems in the case of strong inter-dot coupling, it is helpful to rewrite the total Hamiltonian in terms of the eigenstates. The eigenvalue diagram in Fig. 7.17 represents the gate-voltage dependence of the multiplet energies E(Q,S,r) of the isolated DQD. From this diagram we can read off the ground state and the excited states for each parameter 8. Such eigenvalue diagrams are a very useful tool to study multi-impurity and multi-level systems. The Kondo effect tends to occur whenever the ground state of the system is degenerate and there are excited states with Q' = Q ± 1, S' = S ± 1/2. Ranges where these conditions are fulfilled appear in the form of triangular level crossings such as ABC in Fig. 7.17. Here, these intervals are given approximately by 81 < \8\ < 82, where 81 = t{2y/l + {U/U)2 - 1) and 82 = U/2 + t. These estimates are obtained from the lowest energies of states with zero, one and two electrons on the isolated DQD, see Table 7.1. The widths of conductance plateaus (measured at G/G0 = 1/2) are therefore approximately 134 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS Figure 7.16: Conductance and correlation functions of DQD. Besides different values of t, indicated in the figure, other parameters of the model are T/D = 0.03 and U/D = 1. Temperature is T/D = 10-9, which for all parameters used corresponds to a zero temperature limit. In particular, in the Kondo plateaux T <^i TK for all 8. -1 ^05 0 05 1 <5/D Figure 7.17: Eigenvalue diagram for isolated DQD system. The diagram is symmetric, since for 8 —> —8, E(Q,S,r) —> E(—Q,S,r). Points A and B correspond to valence-fluctuation regions when the charge on the dot changes, while point C corresponds to the center of the Kondo regime, when the Kondo temperature is the lowest. Parameters are U/D = 1 and t/D = 0.3. 7.3. DOUBLE QUANTUM DOT: SIDE-COUPLED CONFIGURATION 135 I U.J (a) ^- — 0.4 y _/ 0.3 ß / / / 0.2 - / 4 1 0.1 -1 1 n 1 ......... 0 0.2 0.4 0.6 0.8 1 t/D 10" 10"' & 10 10 (b) t/D=03 ?/D=0.7 r \ W 0.4 0.6 0.8 1 8/D 1.2 1.4 Figure 7.18: a) The width of conductance peak A vs. t as obtained from NRG calculations (full circles), compared with the analytical result as given in the text (dashed line), b) Kondo temperatures Tk vs. 8 as obtained from the widths of Kondo peaks (full circles). Analytical estimate, Eq. 7.55, is shown using dashed lines. The rest of parameters are identical to those in Fig. 7.16. given by A = U/2 + 2t(l - V1 + (U/4:t)2). (7.54) Note that in the limit of large t, A ~ U/2, and in the limit of large U, A ~ 2t. Comparison of conductance-peak widths with the analytical estimate A is shown in Fig. 7.18a. We now confirm the presence of the Kondo effect by considering various correlation functions. In Fig. 7.16b we show S, calculated from expectation value (S^ot) = S(S + 1), where Stot = Si + S2. S reaches value 1/2 in the regime where G/Go = 1 which indicates that high conductance is associated with the presence of local moment on the DQD. The average occupancy (n) in this regime approaches odd-integer values 1 and 3, Fig. 7.16c. Transitions between regimes of nearly integer occupancies are rather sharp; they are visible as regions of enhanced charge fluctuations measured by An2 = (n2) — (n)2, as shown in Fig. 7.16d. Finally, we show in Fig. 7.16e spin-spin correlation function (Si • S2). Its value is negative between two separated Kondo regimes where conductance approaches zero, i.e. for —81 < 8 < 81, otherwise it is zero. This regime further coincides with (n) ~ 2. Each dot thus becomes nearly singly occupied and spins on the two dots form a local singlet due to effective exchange coupling J f« 4t2/U. In Fig. 7.18b we present Kondo temperatures TK vs. 8 extracted from the widths of Kondo peaks in spectral functions. NRG results fit well the expression obtained using the generalized SWT (see Sec. 6.3.1 and Appendix C): TK = 0.182[/v/pJexp[-l/pJ] with p.] 2Y a ß TT \|L(-1±0)-L(-2,0,0)| |L(-U,0)-L(0,0, (7.55) (7.56) 136 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS DM-NRG A((0) 0.5 0 0.5 co/D Figure 7.19: Zero-temperature spectral function Ai(uj) sweeps for t/D = 0.3. a) Spectral function calculated using the conventional NRG approach, b) Spectral function calculated using the DM-NRG approach. Note that the vertical line, representing the Kondo peak. has been artificially broadened. Its true width is Tk- where a = K-l, |, l,0|d|| 2,0,0,0)|2 = l/2; /3=|(0,0,0,0|4|-l,y,0)|2 (it + u + ^Jm2 + u2) (7.57) 8 (I6t2 + U(U + Vl6t2 + [/2)) Spectral function Spectral function calculations using the conventional NRG approach fail for this model: the spectral functions manifest spurious discontinuities and the normalization sum rule is violated for some choices of model parameters.188 Correct results can be, however, obtained using the DM-NRG technique188,206 presented in Sec. 3.9. In Fig. 7.19 we present sweeps of Ai(lü) calculated using both approaches. In vast regions of the plot the results are in perfect agreement. The differences appear for those values of 6 where the ground state changes. Three characteristic spectral functions calculated using the DM-NRG are shown in Fig. 7.20. Features in the spectral function sweeps can be easily interpreted using eigenvalue diagram in Fig. 7.17. At low temperatures and for constant Ö, spectral function A{u) will be high whenever the energy difference AE = E\ — E0 between the ground state (0) and an excited state (1) is equal to +u; (particle excitations, Q\ = Qq + 1, S\ = So ± ^) or to —uj (hole excitations, Q± = Q0 — 1, Si = S0 ± |). At 6 =0 two broad peaks are seen located symmetrically at lü ~ ±<$i (see Fig. 7.19 and Fig. 7.20 at 6 = 0). At this point the model is p-h symmetric and therefore E(Q, S, r) = E(—Q, S, r) for all Q, S, r. Consequently, the spectrum is also symmetric, Ai{u) = Ai(-uj). With increasing Ö, the particle excitation energy E(l, \, 0) — L"(0, 0, 0) increases and the corresponding peak quickly washes out. The hole excitation energy E(—l, \, 0) — E(0, 0, 0) decreases and the peak gains weight. 7.3. DOUBLE QUANTUM DOT: SIDE-COUPLED CONFIGURATION 137 5 4 ^3 3 1 0 Figure 7.20: Three zero-temperature spectral functions A1(u) = Pd(cj) for t/D = 0.3: at the p-h symmetric point 5 = 0, in the Kondo regime 5/D = 0.62 and in the empty-orbital regime 5/D = 1. Inset: Scaling of spectral functions A1(u/TK) in the Kondo region 0.52 < 5 < 0.76. Parameters are as in Fig. 7.16. At 5 = 51, L(0, 0, 0) = E(-1, 12, 0) (point A in Fig. 7.17) and the system enters the Kondo regime: a sharp many-body resonance appears which remains pinned at the Fermi level throughout the Kondo region (see Fig. 7.19 and Fig. 7.20 at 5/D = 0.62). Kondo effect occurs since the ground state is a doublet, S = 1/2, and there are excited states with S' = 0,1, Q' = Q ± 1. The high-energy peaks at u = L(0,0,0) - L(-1,12,0) > 0 and 'jj = E(-1,12,0) - L"(-2,0,0) < 0 in the spectral function are also characteristic: they correspond to particle and hole excitations that are at the heart of the Kondo effect. In the case of the DQD we also see additional structure for 51 < 5 < 52: a broad peak at uj = L(0,1,0) - L(-1, 12,0) which corresponds to virtual triplet excitations from the ground state. These excitations could also be taken into account in the calculation of the effective exchange interaction, Eq. (7.56), however due to their high energy, they only lead to an exponentially small difference in the Kondo temperature, which may be neglected. In the inset of Fig. 7.20 we show scaling of Kondo peaks vs. uj/TK. In the case of perfect scaling, all curves should exactly overlap. However, Kondo temperatures of different peaks differ by almost four orders of magnitudes, as seen in Fig. 7.18b. Moreover, Kondo peaks become asymmetric near the edges of the Kondo region, i.e. for 5 > 51 and 5 < 52. Note also that for each point in Fig. 7.18b there is a respective spectral function presented in the inset of Fig. 7.20. -1-------1-------1-------1-------1-------1-------1-------1-------1- -1-------1-------1-------1-------1-------1-------1-------1-------r- "T 5/D=0 5/D=0.62 5/D=l 138 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS O Č5 1.0 0.5 0.0 0.8 ^ 0.5 0.2 0.0 A 2.0 v 1.0 0.0 0.3 ^0.1 0 A 0.6 oTo.4 % 0.2 1 0.0 ¦p (a) t/D=0.001 (b) -^ C. v (d) P\ (e) i V . i i 1.0 - 0 0.25 0.5 0.75 1 S/D 0.0 0.1 0.2 0.3 0.4 0.5 b/D Figure 7.21: Conductance and correlation functions at t/D = 0.001 (a,...,e), t/D = 0.01 (f,and g) and a blow-up of f) in the inset of g). Dashed lines in a) represent G/G0 and b) S of a single quantum dot with otherwise identical parameters. Dashed line in c) represents (n\) of DQD, and finally dashed line in the inset of g) represents the semi-analytical model described in the text. In h) a schematic plot of different temperatures and interactions is presented as explained in the text. NRG values of the gap in Ai(u) at u) =0 and T (7-58) where e = (E — E0)/(T/2) is a rescaled energy, E0 is the resonance position, T its width, while q is the Fano parameter that determines the form of the resonance. Physically, q is the ratio between resonant and direct scattering probability. In the limit q —> oo, the Fano resonance goes into the Breit-Wigner (Lorentzian) resonance, while in the q —> 0 limit it has the appearance of a symmetric dip. Fano resonances have been detected in QDs and SETs,334'335 in QDs embedded in Aharonov-Bohm interferometers336 and in quantum wires with side-coupled QD.337 In our case, the Fano resonance is a consequence of a sudden charging of the nearly decoupled dot 2, as its energy e crosses the Fermi level at tp =0. Meanwhile, the electron density on the dot 1 remains a smooth function of 8, as seen from (n\) in Fig 7.21c. This can be qualitatively understood from a simple noninteracting model F = ei|l)(l| + e2|2)(2|+t(|l)(2| + h.c). (7.59) Using the Green's function method (Sec. 4.1) we obtain the following expression for conductance G = G\22(e2 + ^tr)-2^ + t^ (7-60) 140 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS where t' is the coupling of impurity 1 to the leads in the tight-binding description. Conductance is clearly zero when the on-site energy e2 of the side-coupled dot crosses the Fermi level. If the levels are shifted using the gate-voltage (ei = Vg and e2 = Vg — A, where A is the detuning parameter), we find that G = 0 at Vg = A and G = Go at Vg = ±(A + VA2 +4t2). The width of the Fano resonance is proportional to the energy difference between these extrema: t2 w = |(VA2 +4t2 - A) ~ - + ö(t4). (7.61) In the interacting case, we may improve this calculation using a simple model. We will consider the total Hamiltonian H in the t =0 limit exactly and then couple the two subsystems using perturbation theory. For t =0, the exact Green's function of impurity 1 at uj =0 is ro ,/rr r ± a\ cos (f> sin

0. The conductance G = G07rr(—l/7rlm(?i) is r = r _______________2r2e2sin20_______________ 0 z2t4 + 2r2e2 - zH" cos(20) - 2zt2tY sin(20)' l " ] Results of the NRG calculation are compared to the prediction from Eq. 7.67 in the inset of Fig. 7.21g. We see that general features are adequately described, but there are subtle 7.3. DOUBLE QUANTUM DOT: SIDE-COUPLED CONFIGURATION 141 differences due to non-perturbative electron correlation effects. Numerically calculated Fano resonance is wider than the semi-analytical prediction and G does not drop to zero. In particular, from Eq. 7.67 it follows that G = 0 at 8 = U/2 (or e = 0) and G = Go at 8 = U/2 + t2 tan (p/Y. These details are not corroborated by NRG results which show, for example, maximal conductance at e = 0. With increasing t, the width of the resonance increases, as shown for t/D = 0.01 in Fig. 7.21f. For t > 0.1, the resonance merges with the Kondo plateau and disappears (see Fig. 7.16a). 7.3.3 Weak inter-dot coupling: the two-stage Kondo effect The two-stage Kondo effect is a generic name for successive Kondo screening of the impurity local moments at different temperatures.69-71'75'152'174'183'188 This term has been used in two different (but closely related) contexts: 1) two step screening of S =1 spin in the two-channel case,152 2) two step screening of two local moments in the single-channel case.70'71'174 In the first case, the first-stage Kondo screening is an underscreened S =1 Kondo effect which reduces the spin to 1/2, while the second-stage Kondo screening is a perfect-screening S = 1/2 Kondo effect which leads to a spin singlet ground state.70'233 In the second case, at a higher Kondo temperature TK the Kondo effect occurs on the more strongly coupled impurity; the Fermi liquid quasiparticles associated with the Kondo effect on the first impurity participate in the Kondo screening of the second impurity on an exponentially reduced Kondo temperature scale T^- .174:,183,188 The first case is relevant when the ground state is a triplet, while the second case occurs when the ground state is a singlet, but there is a nearby excited triplet state.71 In fact, the two cases are connected through a quantum phase transition which occurs at the degeneracy point between the singlet and triplet state;71'174 this also demonstrates the connection between the two-stage Kondo effect and the "singlet-triplet" Kondo effect.70 In the case of the side-coupled DQD, the inter-impurity exchange interaction is antiferro-magnetic and it is given by the superexchange expression u\ At2 1 ~ r^j --------- t u J=7;\ \/ t +16"T «TT- (7-68) The two-stage Kondo effect of the second type occurs when J < TK, where TK = T^' is the Kondo temperature of the SIAM that describes impurity 1 (without impurity 2).183'188 The second Kondo crossover then occurs at 42) = ^ exp(-c141)/J). (7.69) Constants C\ and c2 are of the order of 1 and they are problem-dependent. The spectral function Ai(u>) of impurity 1 increases at u ~ TK\ but then drops at u ~ TK\ i.e. there 142 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS Tk, the impurity spins bind in a singlet before any Kondo effect can occur. There is again a gap of width J in the spectral function and conductance is zero for T < J. It should be noted that from the zero-temperature conductance plots alone we cannot determine if the system underwent two-stage Kondo screening or if a local singlet was formed; in fact, the T =0 fixed point is the same in both cases and, moreover, the two regimes are continuously connected.174 Equipped with this theory, we now return to the description of the results presented in Fig. 7.21a for 8 < U/2. Just below 8 < U/2, J< TK' and the system is in the two-stage Kondo regime. In the range of 8 where T falls in the interval given by T^2) < T < T™, the conductance is high. For still lower 5, T< TK' and the conductance decreases. Finally, for lowest 8, J> TK' and the system is in the local singlet regime where the conductance is zero. NRG results for the gap in the spectral function Ai(uS) calculated at T = 0 (open circles and squares) follow analytical results for TK (8) when J< TK and they approach J when J> TK\ as expected, see Fig. 7.21h. As shown in Fig. 7.21a for 0.3D < 8 < U/2, G/G0 calculated at T = 10-9D follows results obtained in the single quantum dot case and approaches value 1. The spin quantum number S in Fig. 7.21b reaches the value S ~ 0.8, consistent with the result obtained for a system 7.3. DOUBLE QUANTUM DOT: SIDE-COUPLED CONFIGURATION 143 1/2 ^ 3/8 Bc 1/4 'S. ^" Hm 1/8 mi 0 1.5 0.5 0 log10(T/D) Figure 7.23: Spin susceptibility and specific heat of the side-coupled double quantum dot for a range of values of the interdot hopping parameter t. We begin to enter the two-stage Kondo regime for t/D < 1.8 10-4. of two decoupled spin-1/2 particles, where (S2) = 3/2. This result is also in agreement with (n) ~ 2 and the small value of the spin-spin correlation function (Si • S2), presented in Fig. 7.21c and 7.21e respectively. With further decreasing of 8, G/G0 suddenly drops to zero at 8 < 0.3D. This sudden drop is approximately given by T ~ T^'(8), see Figs. 7.21a and h. At this point a gap opens in Ai(u) at uj = 0, which in turn leads to a drop in the conductivity. The position of this sudden drop in terms of 8 is rather insensitive to the chosen T, as apparent from Fig. 7.21h. Below 8 < 0.25D, which corresponds to the condition J ~ T^- (8), also presented in Fig. 7.21h, the system crosses over from the two-stage Kondo regime to a regime where spins on DQD form a singlet. In this case S decreases and (Si-S2) shows strong antiferromagnetic correlations, Figs. 7.21b and e. The lowest energy scale in the system is J, which is supported by the observation that the size of the gap in Ai(u) (open circles in Figs. 7.21h) is approximately given by J. The main difference between t/D = 0.001 and t/D = 0.01 comes from different values of J =4t2/U. Since in the latter case J is larger, the system enters the AFM singlet regime at much larger values of 8, as can be seen from comparison U/D=l,r/U=0.03, 8=0 -a) t/D , -s 10 ...... 10"4 ---------1.2 10"4 ---------1.4 10"4 --------- 1.6 10"4 ---------1.8 10"4 - 2 10" - 3 10" ,n-3 10 144 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS of Figs. 7.21g and f. Consequently, the regime of enhanced conductance shrinks. 7.4 Double quantum dots in magnetic field In Fig. 7.4 I plot the conductance through the double quantum dot systems of different coupling topologies as a function of the inter-dot exchange coupling J and of the applied magnetic field B. The parameters entering the corresponding quantum impurity models are chosen so that U and Y are the same in all cases. The effect of the magnetic field on the conductance through the double quantum dot depends strongly on the type of the Kondo effect which occurs in the system. In the side-coupled DQD, two-stage Kondo screening occurs when J < TK . In this regime. the conductance is unitary if TK <5 TK . This is clearly visible in Fig. 7.4a. For the smallest values of J shown. Tft is essentially zero and the conductance is high for B below TK ~ 10~4D. For J of the order of TK , non-monotonous field-dependence of the conductance can be observed with a high-conductance plateau in the range TK Cß< TK . This plateau evolves for J> TK into increasingly narrow peak centered near B ~ J. For B = J, the singlet and the lowest triplet state are degenerate and the system conducts due to the singlet-triplet Kondo effect. This plot therefore clearly establishes the relation between the two-stage Kondo effect and the singlet-triplet Kondo effect : they are the two extreme limits of the same type of behavior related to the near-degeneracy of singlet and triplet states. In the serially-coupled DQD, the conductance is high in the absence of the field only when J ~ TK . For smaller J, each impurity tends to form a separate Kondo correlated state with the neighboring conduction lead, and the conductance is low. For larger J, the impurities form a strongly-bound inter-impurity spin-singlet state, which again leads to low conductance. For J ~ T^', the system crosses over between these two regimes and the conductance exhibits a unitary peak. It is interesting to note that the related two-impurity Kondo system undergoes a true quantum phase transition with non-Fermi liquid properties at the critical point at J ~ TK , however the charge transfer between the two channels destabilizes this NFL fixed point93 in the case of the serially-coupled DQD where the exchange interaction is generated by the superexchange mechanism due to the electron hopping ; similar behavior is found in the triple quantum dot system presented in the following chapter. The lowest value of B shown in Fig. 7.4b is much lower than TK . therefore the behavior is similar to the one in the absence of the field, with the exception that the highest conductance does not quite reach the unitary limit. For stronger fields, we obtain the singlet-triplet Kondo effect as in the case of the side-coupled DQD. The parallel DQD undergoes S =1 Kondo screening for J < Jrkky- The expected very slow (logarithmic) approach to the unitary conductance limit is clearly visible in Fig. 7.4c. For small B, the conductance drops abruptly to zero when J is increased past Jrkky, 7.4. DOUBLE QUANTUM DOTS IN MAGNETIC FIELD 145 a) Side-coupled DQD G/Go °-5 o b) Serial DQD c) Parallel DQD g!0(J/U) o(J/U) G/Go g!0(J/U) Figure 7.24: Conductance of double quantum dot systems of different coupling topologies as a function of the inter-dot antiferromagnetic exchange interaction J and of the applied magnetic field B. The parameters are the same in all three cases: U/D = 1, ?/D = 0.05. ö =0. Conductance is calculated by extracting quasiparticle phase shifts. In the plots for side-coupled and serial DQD, data points with G/G0 = 1 have been added to the singlet-triplet Kondo peaks a posteriori for presentation purposes, therefore the widths of these narrow peaks are, in fact, largely exaggerated. 146 CHAPTER 7. PROPERTIES OF TWO-IMPURITY MODELS since the spins bind into a singlet and the Kondo effect no longer occurs. For large B. the decrease in conductance is less pronounced. It should be noticed that there is no singlet-triplet Kondo effect in this system. Chapter 8 Properties of three-impurity models While the two-impurity models were intensely studied in the past (and still are), the attention is recently shifting to more complex three-impurity models. Near the particle-hole symmetric point (or, equivalently, at half filling), systems with even or odd number of QDs have radically different behavior due to the distinct properties of integer and half-integer spin states. The half-integer spin states are always degenerate and QD systems with such ground states tend to exhibit some form of the Kondo effect for any coupling strength; the zero-temperature conductance of systems of an odd number of dots will be high. In systems with an even number of QDs, however, the range of half filling is generally associated with Mott-Hubbard insulating behavior.273 The prototype three-impurity model describes the triple quantum dot (TQD) system in the linear configuration, usually modelled as a three-site Hubbard chain embedded between two non-interacting electron reservoirs (conduction leads). TQD structures have been manufactured in recent years and the analysis of their stability diagrams demonstrates that a description with a Hubbard-like model is indeed a good approximation.339'340 Models of three Anderson-like impurities have already been studied by a variety of techniques in different temperature regimes.24'189'190'327'341-347 While many features were previously known, detailed understanding of the underlying microscopic mechanisms emerged only recently.157'158 In this chapter I study linear TQDs. The special feature of this system is the presence of two equivalent screening channels (as in the 2CK model) combined with two-stage Kondo screening and/or magnetic ordering. The main message conveyed by this chapter is that the three-dot structures are a promising system for both theoretical and experimental study of NFL physics.24'158 The relevance of these results is reinforced by the recent detection of the 2CK effect in a system of a QD coupled to leads and to a quantum box:272 exotic Kondo effects can indeed be detected in semiconductor nanoelectronic devices. 147 148 CHAPTER 8. PROPERTIES OF THREE-IMPURITY MODELS Figure 8.1: a) Model I: the triple quantum dot embedded between two leads, b) Model II: a related system with exchange interaction between the impurities. 8.1 Triple quantum dot: linear configuration In this section, I consider two different but closely related systems of three Anderson impurities coupled in series between two conduction channels, Fig. 8.1. They are described by the Hamiltonian H = i^band + -^imp + Hc, where #band = 22 ekcikßCvkß (8.1) v,k,ß describes the left and right conduction lead (u = L,R), and HC = J2 Mc[kßdlß + c^ds, + H.c.) (8.2) k/i describes the coupling of the bands to the left and right impurity (numbered 1 and 3, while 2 is the impurity in the middle, Fig. 8.1). In model I, Him> is the Hubbard Hamiltonian 3 2 ^Lp = 2^ Hdot,i + /_^/_^t \ditldi+i:ß + H.c.J , (8.3) i=\ i=l ß with ^dot,i = 5(ni-l) + ^(ni-l) 2 , (8.4) where, as before, 8 is the on-site energy (gate voltage), U is the on-site e-e repulsion, ni = S« d\ dip 1S the- electron number on site i and t is the inter-impurity hopping. Model II is the exchange-only variant of the model I, with 3 ^imp = 2_^ ^dot,i + ^Si • S2 + JS2 • S3, (8.5) i=\ where S» = ^2ßß/ diß{l/1(Tßßi)dißi is the spin operator on site z, and J is the exchange constant. We set J to the superexchange value of J =4t2/U to relate the two models for t <^ U. In two-channel QIMs, two different types of the particle-hole symmetry may be realized, see Sec. 2.3.5 and Refs. 21,28. For 8 =0, models I and II both have p-h symmetry of the first type. As I show below, this has important consequences on the zero-temperature conductance through the QD molecules. 8.1. TRIPLE QUANTUM DOT: LINEAR CONFIGURATION 149 The ground state of model I is Fermi liquid (FL) for any choice of parameters.158'189'190 If. however, the impurities are coupled only by exchange interaction, as in model II, the system has a non-Fermi liquid (NFL) ground state of the 2CK type with a residual In 2/2 zero-temperature entropy.28'80 Replacing spin exchange interaction with hopping (i.e., going from model II to model I) enables charge transfer between the left and right conduction channels, thereby inducing channel asymmetry24'93 which drives the system to a FL ground state.28 Nevertheless, model I exhibits NFL properties in an interval of finite temperatures where the system approaches the (unstable) 2CK fixed point. It will be shown in the following (Sec. 8.1.5) that there is in fact a range of hopping parameters t where this temperature interval is particularly wide. As this interval is entered from above, the conductance through the side dots increases to a half of the conductance quantum, while the conductance through the system remains small. At lower temperatures the conductance through the system increases to the unitary limit as the system crosses over to the FL ground state. While our main tool will again be NRG, in this section comparisons with other methods are also made. In particular, the zero-temperature conductance will be computed using the sine formula (Sec. 5.2) with energies of the auxiliary ring system obtained using the constrained-path quantum Monte Carlo (CPMC), Sec. 4.3, and with the Gunnarson-Schönhammer variational method (GS), Sec. 4.2. The CPMC calculations were performed on a ring of 100-180 sites. As the number of sites with interaction is small, CPMC produces ground state energies with excellent precision, typically of the order of AE/E = 10-6. On the other hand, the constrained number of sites in the ring limits the energy resolution of this approach. With GS, the size of the ring can be increased up to a few thousands sites. While this improves the energy resolution by an order of the magnitude, none of these two methods can compete with NRG which can in principle handle arbitrarily small energy scales. In this section, I discuss both dispersionless bands (NRG calculations) and cosine bands (NRG, CPMC, GS calculations). In the latter case, T =2t'2/t, where if is the coupling between sites 1 and 3 of the TQD and the first sites in left and right tight-binding chains. 8.1.1 Mapping to molecular-orbital levels For large t/U, the TQD system described by model I behaves as an artificial molecule composed of three atoms (QDs). In the non-interacting L7 —> 0 limit, the molecule has three distinct non-interacting levels: non-bonding (0), bonding (-) and anti-bonding (+) molecular-orbitals (Fig. 8.2) with energies E±f0 = ±VZt,0 (8.6) and effective hybridizations r±)0 = ^r,ir. (8.7) 150 CHAPTER 8. PROPERTIES OF THREE-IMPURITY MODELS When interactions are switched on, the non-bonding level remains at E0 =0, while the other two are symmetrically shifted to E± ~ ±(V2t+^U)+ 0(U2 /t) where the [/-term is a consequence of the inter-orbital repulsion. The effective intra-orbital e-e repulsion is given by U± ~ \U + 0(U2/t) and U0 ~ \U + 0[U3/t2}. These molecular orbitals are spatially extended throughout the TQD system even in the presence of interactions. It should be noted that bonding and anti-bonding wavefunctions are symmetric (even parity), while the non-bonding wavefunction is antisymmetric (odd parity), Fig. 8.2. Bonding (-) Non-bonding (0) Anti-bonding (+) even odd even Figure 8.2: Molecular-orbitals in a linear artificial molecule The energy-level structure of the system can be conveniently represented in the form of the eigenvalue diagram (see also Sec. 7.3.1). In Fig. 8.3 I show the case of both large t/U and small t/U. In the latter case, the description in terms of molecular-orbital levels breaks down. The features in these diagrams will be commented in more detail in the following. a) t/U=\ E Figure 8.3: The energy-level structure for the decoupled TQD system in the case of a) strong and b) weak inter-impurity tunnel coupling. 8.1.2 Phase diagram and physical regimes The zero-temperature phase diagram of the TQD system, Fig. 8.4, features several phases with enhanced conductance which will be discussed in this section. •/•\* Q, S 3,0 2,1/2 1,1 1,0 0,3/2 0,1/2 -1,1 -1,0 -21/2 -3,0 8.1. TRIPLE QUANTUM DOT: LINEAR CONFIGURATION 151 TSK .4 0.6 (ed+U/2)/U Figure 8.4: Ml, M3: MO Kondo regime with (n) ~ 1,3. M2: non-conductive even-occupancy state. L3: local Kondo regime with (n) ~ 3. TSK: two-stage Kondo regime. Due to the p-h symmetry, the diagram is mirror-symmetric with respect to the 8 =0 axis: for negative 8 < 0 we thus find M4 non-conductive regime and M5 MO Kondo regime. For large t/U, the molecular-orbital levels are filled consecutively by electrons as the gate voltage is swept; this is clearly visible in the diagram a of Fig. 8.3, where the total number of electrons in the ground state N = Q +3 increases in steps of 1 at well separated gate voltages 8. Each molecular orbital can accommodate two electrons, which bind into a spin singlet (S = 0 for N even). The ground state is thus a spin doublet (S = 1/2) only when the total occupancy N is odd, i.e., when there is a single electron in one of the molecular orbitals. Model I then maps to an effective SIAM and we say that the system is in the molecular-orbital (MO) Kondo regime (shaded regions labelled 'Ml' and 'M3' in Fig. 8.4). The unpaired electron develops local moment for T < U^, where [Jeff is U0 for Q = 0 (N = 3) or U± for Q = ±2 (N = 1,5). The local moment is then Kondo screened in the conventional single-channel Kondo effect by the electrons in the leads. Depending on the symmetry of the relevant orbital wave-function (even for bonding and anti-bonding levels and odd for non-bonding level), the Kondo effects occurs either in the even or odd conduction channel formed by the symmetric or antisymmetric combination of conduction band electrons. The Kondo temperatures T±0 for levels (±,0) can be determined from Eq. (6.32) with effective parameters Ue^ and ?eff, where ?eff is ?0 or ?±. Just like in the case of a single-impurity Anderson model, in MO Kondo regime the conductance approaches G = Go for low temperatures, T 0 limit by exact diagonalization and expansion to the lowest non-trivial order. The expressions are in excellent agreement with numerical results for conductance presented in the following section. It should be noted that all transitions are smooth cross-overs; there are no abrupt phase transitions. Phase 1 Phase 2 Condition empty Ml M2 M3 L3 Table 8.1: Definitions of boundaries and cross-over regions between various phases in Fig. 8.4. Here J = At2/U. Model II has AFM and TSK regimes separated by the crossover regime. There is clearly no MO regime; instead, the AFM regime extends to the region of high J, where the two models describe very different physical systems. Since the left and right conduction channels are not communicating (in the sense that there are no L « i? cotunneling processes), the channel symmetry is maintained and a stable 2CK NFL ground state is expected for all J. Ml 5 ~ U/2+ y/2t 5 ~ U/2+3t2/U, t < U 5 ~ U/8+ y/2t, t > U 5 ~ U/2 - y/2t +3t2/U, t < U 5~U{l/A + i-2(U/t)2), t>U L3 V2t ~ J TSK J ~ 2TlK M2 M3, L3 154 CHAPTER 8. PROPERTIES OF THREE-IMPURITY MODELS 8.1.3 Zero-temperature conductance In model I with t =0, left and right conduction channels become decoupled and the TQD is clearly not conducting. In contrast, for large t this system can be mapped to SIAM (see above) and is fully conducting at the p-h symmetric point.157'190'348 The p-h symmetry of the first type restrains the phase shifts for the even and odd channel, 8^n and 8°pd, to either 0 or 7r/2;21 the conductance, given by G = G0sin2(C-Cn)> (8-9) can thus be either G = Go or G =0. Consequently, there are only two possibilities: either the conductance is G = Go down to the t —> 0+ limit, or there is a quantum phase (2) transition to a zero conductance state at some small t. I will show that for T <^ TK the TQD is also fully conducting in the TSK regime for any i ^ 0 and that the transition from AFM to TSK regime is not a phase transition. Weak perturbations from the p-h symmetric point are marginal, therefore the TQD is expected to have an extended region of high T =0 conductance as a function of the gate voltage. In Fig. 8.6 I present the zero-bias conductance G along with the total occupancy (n) as a function of the gate voltage for a range of t and for fixed U/D = 0.5 and T/D = 0.09. The conductance is calculated with various methods, as explained in the introduction to this section. For t/U > 0.2, the system is in the MO regime. As the occupancy monotonically decreases from 6 (full TQD) to 0 (empty TQD), the conductance exhibits well resolved peaks when occupancy is odd and valleys when occupancy is even.7'349 In this regime, the conductance obtained with CPMC and GS methods shows good agreement, and the Hartree-Fock method gives reasonably good results. The local regime emerges for t/U < 0.2. The conductance is unitary near the p-h symmetric point, while in the charge transfer region, 8 ~ U/2, the conductance exhibits humps separated by dips. In this range the CPMC method is no more applicable since due to the computational restrictions on the system size, its energy resolution is insufficient to describe the small Kondo scale. It should be noted that the TQD is fully conducting at the p-h symmetric point for any t and as t/U is reduced the system goes continuously through three different Kondo regimes. 8.1.4 Correlation functions In Fig. 8.7 I show the ground state expectation values of charge fluctuations (8ni)2 = n? — {rii)2 and spin-spin correlations between neighboring Si-S2 and between side impurities Si • S3 at the p-h symmetric point as a function of t. These calculations were performed using NRG (A = 4, Campo-Oliveira discretization, truncation with energy cut-off at \2ujN or at most 2000 states kept). 8.1. TRIPLE QUANTUM DOT: LINEAR CONFIGURATION 155 Figure 8.6: Conductance G/G0 and occupancy (n) as a function of the gate voltage for various inter-dot hopping parameters t. Note that the energy scale in panels (d) and (e) is different. Error bars of CPMC are smaller than the size of circles. For comparison the Hartree-Fock (HF) results are also shown. For model I, the smooth cross-over from MO to AFM regime, predicted to occur on the scale of t\ ~ U/2\[2 (Table 8.1) is reflected in the decrease of charge fluctuations and the increase of spin-spin correlations. The cross-over from AFM to TSK regime occurs when J ~ Tj(' or t2 ~ \JTk'U/2: as t decreases past L2 the spin-spin correlations tend toward zero as the spins decouple. For model II, the results in the TSK regime match closely those of model I, while in the AFM regime near t ~ t\ the differences become notable. Large values of J 3> T suppress charge fluctuations on side-dots, (5n\)2 —> 0, while local moments on impurities tend to form a well developed AFM spin-chain (for comparison, in isolated three-site spin chain (Si • S2) = —1/2 and (Si • S3) = 1/4). If U/T is smaller, the local moments on sites 1 and 3 are less well developed since the charge fluctuations are larger, Fig. 8.7b. The characteristic values of the spin correlation functions in the AFM regime then cannot be attained. Consequently, the boundaries between various 156 CHAPTER 8. PROPERTIES OF THREE-IMPURITY MODELS t/D or (JU)1/2/2D t/D or (JU)1/2/2D Figure 8.7: Charge fluctuations and spin-spin correlations of model I (lines without symbols) and model II (lines with symbols) as a function of the inter-dot coupling t (for model I) or corresponding J = At2/U (for model II) at the p-h symmetric point. Left panel corresponds to U/D = 1, right panel corresponds to U/D = 0.5. The molecular-orbital (MO) regime is characterized by large on-site charge fluctuations, the antiferromagnetic spin-chain regime (AFM) by negative spin correlations of neighboring spins 1-2 and positive correlation of spins 1-3, and the two-stage Kondo regime (TSK) by vanishing spin correlations. regimes are more fuzzy. We now focus on the local regime with double occupancy, Fig. 8.8 (for reference, results at td = 0 are shown in panel b). For small t, I find {c\ßC3ß} ~ 0 (not shown) and Si • S3 ~ 0, which implies that sites 1 and 3 are uncorrelated: the second electron equally occupies symmetric and antisymmetric orbitals \/\[2{c\il ± c3/U). This implies that both left and right sites are independent and both form a valence fluctuating state with the neighboring lead. The electron on the central site then undergoes Kondo screening by coupling antiferromagnetically with the quasiparticles from each valence fluctuating state. The screening of spins thus occurs in two stages. The first stage corresponds to the behavior found in the strongly asymmetric Anderson model with 8 —> U/2, thus the screening has exponential temperature dependence and is not Kondo-like. The second stage, however, is Kondo-like screening: although 8 is near U/2 on the central site also, the effective hybridization is very small. 8.1.5 Thermodynamic properties In Fig. 8.9a we plot the impurity contribution to the magnetic susceptibility and entropy at the p-h symmetric point for U/D = 1 and a range of t calculated using NRG (A = 4, ß = 0.75, Campo-Oliveira discretization, typically the truncation energy cut-off is set at 12cjjv and up to 3500 states are kept). The ground state of model I is nondegenerate, 8.1. TRIPLE QUANTUM DOT: LINEAR CONFIGURATION 157 0 0.002 0.004 0.006 0.008 0.01 10-3 10-2 10-1 10 - t/U t/D (a) (b) Figure 8.8: a) Properties of the TQD at double occupancy, (ntot) = 2. b) Correlation functions as a function of the inter-dot coupling t/D for ta = 0. Simp =0, and the impurities are fully screened for all t. In MO regime the system undergoes single-channel Kondo screening with TK that increases with t and becomes constant for t ^$> U, see Fig. 8.10. In this regime the system is well described by effective SIAM with parameters Uq and T0. In AFM regime, the binding of spins is most clearly discernible in the curves calculated at t/D = 0.05 which show a kink in S[mp at 3In2 (local moment formation), followed by an exponential decrease to S[mp = In 2 at T ~ At2/U. The Kondo screening in AFM regime is of single-channel type for t/D > 0.02. Between t/D = 0.02 and t2 there is a cross-over regime with NFL-like properties. Here magnetic ordering competes with the single-channel Kondo screening of left and right impurity. The magnetic moment is rapidly quenched at T ~ Tscr ~ J, yet the entropy does not go to zero but exhibits a In 2/2 NFL plateau.. At still lower temperature Ta, NFL fixed point is destabilized by the channel asymmetry and the system crosses over to the FL ground state characteristic of the conventional Kondo model. Note that in this regime Tscr is high while Ta is low (Fig. 8.10), making this range suitable for experimental study of NFL physics. In TSK regime, the left and right impurity are screened by the single-channel Kondo effect at temperature T^' that is nearly the same for all t < t2 (Fig. 8.10). The susceptibility is reduced from ~ 3/4 to 1/4 and the entropy from 3 In 2 to In 2. The central impurity (2) is screened by the 2CK effect at T^ , below which the system is near the NFL fixed 158 CHAPTER 8. PROPERTIES OF THREE-IMPURITY MODELS ^ 1/2 OB H H 1/4 3 In 2 S In 2 In 2/2 0 —p-i----,----p-1----r t/D=0.0001 — t/D=0.0008 — t/D=0.001 — t/D=0.002 — t/D=0.005 — t/D=0.05 -- t/D=0.2 _ -12 -9 -6 log10(T/D) 12 -9 -6 log10(T/D) -3 0 Figure 8.9: Impurity magnetic susceptibility and entropy at 5 =0 for model I (full lines) and model II (dashed line) with J = 4t2/U. Left panel corresponds to U/D = 1, right panel to U/D = 0.5. Lozenges ¦ are a fit to Bethe-Ansatz results for one-channel Kondo model (multiplied by two and shifted by 1/4) and squares ¦ are a fit to NRG results for 2CK model. Inset: TK = aTK exp(—bTK /J) scaling of the second Kondo temperature of model I. a = 0.97, b = 4.4, T^]/D w 1.0 10-5. cross-over AFM t/Dor(JU) /2D Figure 8.10: Cross-over scales of models I and II as functions of the inter-dot coupling. The magnetic screening temperature Tscr is defined by Tscrx(Tscr)/\gßB)2 = 0.07; it is equal to the Kondo temperature when screening is due to the single-channel Kondo effect. T^ is here defined as s^iT^/kB = In2/4. 8.1. TRIPLE QUANTUM DOT: LINEAR CONFIGURATION 159 point with In 2/2 entropy. In the inset to Fig. 8.9 we show that T^ scales as T^ oc T$ exp(-cT^Vj), as expected for the TSK effect.174'183'188 Model II has a stable NFL ground state. For low J, it has a TSK regime where the the (2) Kondo temperature TK\ determined by J, is lower than that of the corresponding model I, set by max{ Ji,J2} = J\ > J (Fig. 8.10). In the crossover regime physical properties of model II for T > T& match closely those of model I. In AFM regime, the Kondo temperature is a non-monotonous function of J. The energy required to break the doublet spin-chain state increases with J and the effective Kondo exchange constant Jk oc T/J decreases. Tk therefore decreases exponentially with increasing J. In contrast, the Kondo temperature of model I in MO regime increases monotonically with t and becomes constant for very large t. The spin-1/2 degree of freedom responsible for NFL behavior is either the collective spin of the three impurities forming a magnetic chain (in AFM regime) or the spin of the central impurity (in TSK regime). In the cross-over regime, the distinction between the two cases is lost. For smaller e-e repulsion U, Fig. 8.9b, the cross-over regime is not well developed. Furthermore, the difference between models I and II is accentuated. As previously commented, the models differ in that in model I electrons can cotunnel between one lead to another. The smaller U is, the more likely such tunneling events are, and the larger the channel asymmetry is. Hence, the In 2/2 plateau in the entropy curve disappears as Ta becomes comparable to TK ¦ For constant T, U thus controls the degree of the channel symmetry breaking. 8.1.6 Fixed points NRG eigenvalue flows show that for any t ^ 0 (i.e. for MO, AFM and TSK regimes alike), the model I at 8 =0 ultimately flows to the same strong coupling FL fixed point. The spectrum is a combination of two FL spectra: one for odd-length and one for even-length free electron Wilson chain.29 By performing the calculation in a basis with well defined parity, it can be ascertained that odd channel gathers a tt/2 phase shift, while even channel has zero phase shift. The fixed point eigenvalues of a single chain for A =2 are q* = 0.6555129,1.976002, 3.999881, A4, A5,..., A odd; fjj = 1.296385, 2.825966, 5.656852, A7/2,A9/2,..., A even. The predicted fixed point spectrum fits very well the computed NRG energy levels. The finite-size spectrum of the unstable intermediate-temperature fixed-point of model I is in agreement with the boundary conformal field theory predictions for the 2CK model80 (Fig. 8.11). The same fixed point is obtained as the stable zero temperature fixed point for all J in model II. In Fig. 8.12 we plot the NRG eigenvalue flow of model II in the 160 CHAPTER 8. PROPERTIES OF THREE-IMPURITY MODELS 0 10 20 30 40 50 60 70 80 N (odd) or T~A-N/2 Figure 8.11: NRG eigenvalue flow for odd-length Wilson chains of model I (A =2, z =1) in the cross-over regime (below) and the corresponding impurity entropy (above). The states are classified according to the total isospin and total spin quantum numbers, (I,S). The low-energy fixed point is a combination of one odd-length and one even-length Wilson chain free-electron spectrum (black full stripes). The black full strips are at energies 0.655 = rjl, 1.296 = f)l, 1.311 = 2rjl, 1.952 = Ja, the second stage Kondo effect occurs in the even channel. As the first stage Kondo effect led to tt/2 phase shifts in both channels, the additional shift of tt/2 in the even channel gives total zero phase shift in this channel (recall that the phase shifts are defined modulo 7r), while odd channel quasiparticles still experience tt/2 phase shift. This is consistent with the NRG eigenvalue flows presented in the previous subsection. The asymmetry parameter A = A/J2 5.17) 8.1. TRIPLE QUANTUM DOT: LINEAR CONFIGURATION 163 with A = p(Js — Ja) = 2pJLR and J = p(Js + Ja)/2 = pJavg determines the cross-over scale11 TA/TK « A2. (8.18) Estimating Js and J4 for t/D = 0.005 using the Schrieffer-Wolff transformation we obtain A2 ~ 10~6, to be compared with T&/TSCT ~ 10~5 determined by the NRG calculation. The discrepancy appears due to competing magnetic ordering and Kondo screening (and emerging two-stage Kondo physics); simple scaling approach fails in this case. log10(T/D) log10(T/D) Figure 8.14: Effect of various perturbation terms on the impurity contribution to the entropy, (a) Particle-hole symmetry breaking 5 ^ 0. (b) Parity breaking H' = Vi-3(n\—n3) (note that the channel asymmetry is not affected by this form of parity breaking), (c) Unequal e-e repulsion U\ = U3 ^ f/2- (d) "Dangerous" perturbation that breaks channel symmetry H' = V\n\. It is instructive to study the effect of various additional perturbation terms on the stability of the In 2/2 NFL plateau in the temperature dependence of the impurity contribution to the entropy. We focused on the cross-over regime, which occurs for U/D = 1 and Y/U = 0.045 around t/D = 0.005. We find a high degree of robustness with respect to the p-h symmetry breaking up to 5/U ~ 0.2, Fig. 8.14a, left-right symmetry (parity) breaking up to V1-3/U ~ 0.2, Fig. 8.14b, and unequal e-e repulsion parameters, Fig. 8.14c and more detailed 8.15. In this last case we notice that when U\ = U3 is decreased, the system is pushed towards the AFM regime,158 since the fluctuations on sites 1 and 3 increase and TK decreases below J. If, however, U\ = U3 is increased, the system goes into the two-stage Kondo regime since TK is higher than J. In this latter case, there will still be a In 2/2 NFL plateau, however the relevant temperature interval is shifted to considerably lower temperatures and becomes narrower.158 The most dangerous perturbation is the channel- 164 CHAPTER 8. PROPERTIES OF THREE-IMPURITY MODELS m zl 00 H PQ Mi 3/4 1/2 1/4 0 31n2 PQ < 21n2 6 C/3 ln2 0 'uju,...... - U/D=l, r/U=0.045 t"/U=0.005 6 -14 -12 -10 -8 -6 -4 log10(T/D) Figure 8.15: Impurity magnetic susceptibility and entropy for the TQD for a range of U1 = U3, while I/2 is held constant. 8 =0 __L__L___L__L_J ___L__i___1___i_ middle side U/D=l t/D=0.005 I7U=0.045 -1 1 1 1 II II - , 1 , 1 — srs3 — srs2 - , 1 , - 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 8/U 8/U Figure 8.16: Site occupancy, charge fluctuations and spin correlations as a function of the on-site energy 8. symmetry breaking, Fig. 8.14d, which rapidly wipes out the NFL plateau. It should be noted, however, that those asymmetries of the device that break the left-right symmetry can be corrected using gate voltages in experimental realizations of the three-impurity model where the on-site energies and inter-dot tunneling parameters can be controlled independently. It is interesting to follow the behavior of correlation functions as the gate voltage 8 is swept, Fig. 8.16. We observe that the occupancy of side (left and right) dots decreases and the charge fluctuations increase as we move towards the valence fluctuation regime of the side 8.1. TRIPLE QUANTUM DOT: LINEAR CONFIGURATION 165 dots; the occupancy of the central dot is hardly affected due to its weak effective coupling. An important consequence is the reduction of the spin correlations: the rigid spin chain breaks near 8 = 0.3U and the system enters the two-stage Kondo regime, as is visible in Fig. 8.14a. 8.1.8 Temperature dependence of conductivity Measuring the differential conductance in a three-terminal configuration (see the insets in Fig. 8.17) provides an experimental probe into the NFL behavior. The qualitative temperature dependence of the zero-bias conductance through the system can be inferred in a very rough approximation from the frequency dependence of the spectral functions. The conductance through the system is given by Gseriai/Go ~ 4(7rrA13)2 (Ref. 350) and the conductance through a side dot in three-terminal configuration by Gside/Go ~ Trr^.351 The appropriately normalized spectral densities are shown in Fig. 8.17 for the cases of cross-over regime with a NFL region and AFM regime with no discernible NFL behavior. In the NFL region (t/D = 0.005 and Ta < T < Tscr), the conductance GSide ~ l/2Go-while Ggeriai ~ 0. The increase of the conductance through the system at T < Ta is concomitant with the cross-over from NFL to FL fixed point, since charge transfer (or, equivalently, channel asymmetry) destabilizes the NFL fixed point like in the two-impurity case.93 In the AFM regime with no NFL region, both conductances increase below the same temperature scale, i.e. at T < Tscr. Measuring GSide and Gseriai could therefore serve as an experimental probe for observation of NFL physics. In Ai(u), the Hubbard peak at U/2 corresponds to adding an electron to the site, while the "magnetic-excitation" peak at J appears when, after adding an electron, the electron with the opposite spin hops from the impurity into the band. This breaks the AFM spin chain, increasing the energy by J, see Fig. 8.18. The magnetic peak evolves into a "molecular-orbital" peak at the energy of the non-bonding orbital (for t in MO regime) or into the Kondo peak of the side dot [for t in TSK regime, see Fig. (8.19)]. It may also be observed that the approach to the uj =0 limit is different in FL and NFL cases. 8.1.9 Conclusion In a wide range of gate voltages around the p-h symmetric point, the TQD system has a FL ground state with high conductance at T =0. The different regimes exhibit different approaches to this fixed point. At finite T, the system has NFL properties which can be detected by measuring Gside and Gseriai in a three-terminal configuration. The most likely candidate for observing this 2CK behavior is the cross-over regime with competing magnetic ordering and Kondo screening, J ~ Tk- In this regime the NFL behavior occurs in a wide temperature range and it is fairly robust against various perturbation that do not additionally increase the channel asymmetry. 166 CHAPTER 8. PROPERTIES OF THREE-IMPURITY MODELS 10" 10 co/D Figure 8.17: Dynamic properties of model I in the AFM (dashed lines) and in the cross-over regime (full lines). Upper panel: on-site spectral function Ai(u) of the left dot. Lower panel: out-of-diagonal spectral function Ais(u) squared. Temperature TA is of order TA, T % is of order Tscr. a b) c m4<] +-+4^ (HWKf Figure 8.18: The schematic representation of the process leading to a peak at uj = J in the spectral function Ai(u). 8.1. TRIPLE QUANTUM DOT: LINEAR CONFIGURATION 167 jcrAjCro) 4[7trA13(co)]2 io"15 io"12 io"9 io"6 io"3 co/D co/D U/D=l, r/U=0.045, 6=0 Figure 8.19: The evolution of the spectral functions from the TSK regime through crossover to the AFM regime. Part III Scanning tunneling microscopy and adsorbates 168 Chapter 9 Scanning tunneling microscopy The scanning tunneling microscope (STM) was invented by G. Binnig and H. Rohrer in 1982.352 It brought unprecedented resolution in imaging of surfaces of conducting samples and quickly became one of the prominent tools in surface science. In this chapter I briefly review some of the applications of the scanning tunneling microscopy (Sec. 9.1), describe the construction of a low-temperature STM (Sec. 9.2) and give some background information on ultra-high vacuum systems (Sec. 9.3) and sample preparation (Sec. 9.4). Finally, in Section 9.5 1 show some examples of images obtained with the new instrument. 9.1 Applications of the scanning tunneling microscopy A number of good comprehensive review articles and books on scanning tunneling microscopy and related topics were published in the past.353-360 These cover general aspects of imaging of clean and adsorbate-covered surfaces, tunneling theory and interpretation of STM images, scanning tunneling spectroscopy and STM hardware. In addition, there is a number of reviews of more specialized subject matters such as imaging of metal surfaces,361 tunneling theory,362 interaction of surface states with nanostructures,363 high resolution image interpretation,364'365 chemical identification via inelastic tunneling spectroscopy,366'367 manipulation of atoms368 and molecules369 and single-molecule chemistry370 In this section I thus only review topics that I personally find particularly interesting. An important research field where STMs find use is surface chemistry and catalysis of molecules. This domain of research has many immediate applications; some notable examples are organic light emitting diodes, thin film transistors and molecular electronics (the use of single molecules as building blocks of electric circuits371'372). In Table 9.1 I list some combinations of organic molecules and surfaces where successful imaging was performed. Topographical STM images of molecules sometimes exhibit atomic scale features. A particularly good example are planar molecules where the molecular "height" is typically 2 A and the internal structure is visible with ~ 0.4 A corrugation.373 Such 169 170 CHAPTER 9. SCANNING TUNNELING MICROSCOPY planar molecules are highly interesting for molecular electronics; they typically have a central aromatic ir board and spacer-groups ("legs") that lift the molecular backbone from the surface.374'375 Generally, internal structure features agree well with simple Hückel molecular-orbital calculations:373'376 image contrast is similar to calculated valence charge density In addition, STM can be used to determine a number of important properties377 such as adsorbate orientation and binding site,378'379 diffusion rates,380~386 molecular conformations,375'376'387'388 conductance of molecules389-391 and growth and crystallization of molecules on surfaces.375'392 Molecule Surface Reference C2H2 Cu(211) 393 C2H2 Cu(100) 394,395 butene Pd(110) 396 CO Cu(211) 397 CO Cu(lll) 398,399 C6H6, CO Rh(lll) 400 benzene Ag(110) 401 benzene Pt(lll) 378 naphthalene Pt(lll) 379 azulene Pt(lll) 388 xylene Pd(lll) 402 anthracene Ag(110) 403 pentacene Cu(lll) 404 sexiphenyl Ag(lll) 405 Molecule Surface Reference pyridone Cu(110) 406 pyridine Cu(100) 394 pyrollidine Cu(100) 407 Cu-PC Cu(100) 373 Cu-PC Al2O3/NiAl(110) 408 /3-carotene Cu(lll) 409 decacyclene Cu(110) 375 HtBDC Cu(110) 410 coronene Ag(lll) 411 TB-PP Au(lll) 412 PTCDA Ag(HO) 413 Lander Cu(110) 375,414 C60 Au(lll) 415 C60 Au(110) 416 C60 Ag(100) 417,418 Table 9.1: Observation of molecules on different surfaces Using magnetic tips, one can study magnetic ordering on atomic scale,419'420 magnetic vortex cores,421 and magnetic hysteresis.422 An emerging field is combining STM with electron .423,424 in spin resonance spectroscopy to detect the presence of localized spins on surfaces this type of experiments, the STM is operated in a magnetic field and the high-frequency components of the tunneling current are monitored to detect spin precession. From measured Larmor frequency, the gyromagnetic ratio of the spin entity can be inferred.424 Especially important is the tunneling spectroscopy which provides information on local density of states (spectral functions)425 with high spatial and energy resolution. It was applied to systems which exhibit the Kondo effect,4'5'269'426~429 superconductors,430~435 and artificial atomic chains,436 to name just a few. It is even possible to study vibrational properties of single molecules395'401'407'437'438 and magnetic properties of single atoms,439 spin chains440'441 and magnetic islands.442 A new trend is to study molecules on a thin oxide layer grown on metal surfaces, for example AI2O3 on NiAl(llO) surface.408'443 Since 9.2. CONSTRUCTION OF THE LOW-TEMPERATURE STM 171 the oxide is a tunneling barrier, the molecule is coupled to the metal conduction band very weakly. This decreases the hybridization broadening of molecular-orbital electronic levels; in addition, vibrational states become better defined.443 Very accurate tunneling spectroscopy can then be performed.408 The tip of the STM can also be used to perform manipulations. It is possible to perform lateral and vertical manipulations of single atoms393'397'444~449 and molecules,369' 397>450~453 and to controllably induce reversible conformation changes of adsorbed molecules.387 Atoms can be extracted from the native substrate by controlled tip crash.454 Finally, using STM surface reactions can be studied at the single molecule level.370'394'396'429'438'455~457 I conclude this short review section by remarking that atomic-scale imaging is also possible using atomic force microscope (AFM)458~461 which also allows measuring of chemical-bonding forces between two atoms.462 9.2 Construction of the low-temperature STM We have undertaken the construction of an ultra-high-vacuum (UHV) liquid-helium-cooled low-temperature (LT) STM (Fig. 9.1) to complement an existing room-temperature (RT) instrument Omicron UHV STM-1. The design constraint for the new LT STM was to accept the Omicron sample plates in order to retain compatibility with the existing equipment, and that the LT STM be housed in the same vacuum system. The long-term goal is to achieve capabilities comparable to those of similar systems developed at the Freie Universität Berlin463 and commercialized by SPS-Createc GmBH, i.e. vertical and horizontal stability of ~ 1pm at 6K. Figure 9.1: Schematic drawing of the modified Besocke type STM scanner. Piezo tubes are shown in yellow, the ramp disk in green, the tip holder and the tip in red, the sample plate in dark blue and the sample clamping support in light blue. 172 CHAPTER 9. SCANNING TUNNELING MICROSCOPY 9.2.1 The cryostat In low-temperature STMs, either the sample or the entire STM head with the sample can be cooled down to low temperatures. The advantage of the latter approach is enhanced stability, reduced transfer of impurities from hot to cold surfaces (i.e. to the sample), and, most importantly, better energy resolution in tunneling spectroscopy. Several cryostat types are used. Liquid-helium (LHe) bath cryostats are commonly used,463 but it is also possible to cool down an STM using flow cryostats.464~467 We have used a bath LHe cryostat (Fig. 9.2a) based on the original design by G. Meyer.463 It consists of a 41 LHe reservoir suspended on a long thin-walled stainless-steel tube (neck) which reduces losses through thermal conduction. The STM head itself is suspended on three springs from the cold copper plate on the bottom of this reservoir. The head is surrounded by a nickel-plated copper radiation shield which is solidly bolted to the LHe reservoir. At the bottom of the radiation shield chamber, there is a gold-plated copper plate that faces oppositely mounted plate which holds magnets; this assembly enables the system of the tube, reservoir and radiation shield to play the role of a large magnetically damped pendulum which reduces the transfer of the environmental horizontal vibrations to the STM head. Liquid helium reservoir is enclosed in a larger (5.31) liquid nitrogen (LN2) shroud whose main purpose is to reduce losses via radiation transfer from the vacuum chamber walls at room temperature to the LHe part of the system. In addition, there is a thermal link (copper braid) from the LN2 reservoir to the top of the LHe neck to provide heat sinking to LN2 temperature; in this manner, the temperature gradient on the longer portion of LHe reservoir neck is significantly smaller (furthermore, thermal conductivity coefficients are lower at reduced temperatures). An additional nickel-plated copper radiation shield is bolted to the bottom of the LN2 reservoir which provides another layer of radiation shielding for the STM head. Care was taken to reduce any openings in the shields as much as possible: radiation losses radiation losses are one of the main limiting factors both for the ultimate working temperature of the STM head and for the cryogen (in particular LHe) consumption. To reduce radiation losses in the vertical direction (i.e. radiation from the RT top wall of the UHV chamber to the LHe reservoir), a number of radiation plates are soldered to the LHe neck. Finally, there is a thermal link (three 100//m high-purity Au wires) between the cold finger and the baseplate of the STM. This link proved to be crucial to reduce the ultimate temperature of the STM head and to provide thermal stability in time. It is very important how the thermal link wires are installed: at low temperatures, the thermal conductivity of very pure gold is high and an important source of thermal resistance is the contact resistance. Contact resistance depends on materials on both sides of the junction; it is lowest in the case of homogeneous junctions (the same material on both sides) when the thermal impedance is matched. Even for homogeneous junctions, the differences are large: the conductance of gold-gold contacts is 20 times larger than that of copper-copper contacts.468'469 It is interesting to note that the majority of heat is carried across the interface by thermal waves (phonons) rather than by electrons.468'470 Thermal conductance increases linearly with the applied pressure so 9.2. CONSTRUCTION OF THE LOW-TEMPERATURE STM 173 the contacts must be tight. GHe recovery line port Neck of LHe reservoir vacuum chamber wall radiation baffles LN2 reservoir 5.3l LHe reservoir 4l LHe cold plate LHe radiation shield LN2 radiation shield GHe recovery line port vacuum flange Neck of LHe reservoir radiation baffles vacuum chamber wall LN2 reservoir 20l LHe reservoir 6l LHe cold plate LHe radiation shield LN2 radiation shield (a) Old cryostat (b) New cryostat Figure 9.2: Schematic drawings of the currently used and newly constructed LHe bath cryostat. The holding times in the fully assembled system with this cryostat are 14 h for LN2 and 26h for LHe (provided that LN2 reservoir is never empty). The ultimate temperature of the STM head is 5.9 K as measured on the sample holder stage using a Si diode (LakeShore DT471). The cool-down time of the STM from RT to LN2 temperature is approximately one day, and it takes further 15 h to go down from LN2 to the lowest attainable temperature. In 2005, the construction of a new improved LHe bath cryostat was undertaken (Fig. 9.2b). It was designed to solve a number of problems that we experience with the existing cryostat. To assist design decisions, I have developed a simple Mathematica package Krio (http: //auger. i j s . si/nano/krio) for building thermal models of cryogenic systems. As input, it takes a list of parts of the system and thermal links (radiation, conduction) between them. The equations are then set up automatically and numerically solved. Temperature dependencies of thermal conductivity coefficients are taken into account. As output, we obtain ultimate temperatures of all parts and thermal currents between them. Using this package, one can easily identify critical elements which can then be optimized. One of the interesting results is that the surface emissivity of reservoirs and radiation shields is the most important element. Emissivity can be reduced by polishing and plating of large 174 CHAPTER 9. SCANNING TUNNELING MICROSCOPY surfaces, or by wrapping them in gold-plated mylar sheets (as long as this does not spoil the ultimate pressure in the vacuum system and reduce the pump-down time excessively). In the new cryostat, the cryogen reservoirs are significantly larger (61 for LHe and 201 for LN2), which should improve holding time. The volume inside the LHe cryoshield is enlarged to allow easier mounting of the STM head, better access for maintenance and neater routing of the instrumentation wiring. The cryostat is assembled from separate LHe insert and an external vacuum chamber with the LN2 reservoir; in this manner, LHe part of the cryostat can be pulled out for installation of wiring and repairs (should the need arise). There are additional vacuum flanges to improve flexibility of the system and the possibility of adjusting the tilt of the LHe neck has been added. The cryostat was manufactured by Vacutech, Ljubljana. At the time near the completion of this dissertation, the first tests of the cryostat are being planned. 9.2.2 The STM head The STM head we have built is based on modified Besocke-beetle type STM head:471'472 the modification consists in having a fixed sample, while the main piezo tube is mobile on a ramp disk that can be coarsely positioned both horizontally and vertically using three supporting piezoelectric tubes,463'473 while in the original design the scanner was fixed and the sample plate mobile.471 The main parts for the head were purchased from Createc GmBH, however the STM head was heavily adapted to accommodate samples mounted on Omicron-type sample holders (Fig. 9.1). In addition, we used a different design for magnetic eddy-current damping of vertical vibrations. The fully assembled STM head is shown in Fig. 9.3. Figure 9.3: Pictures of the assembled low-temperature STM head (left) and the ramp disk (right). Piezoelectric tubes with longitudinally quartered electrodes on the outside and a single continuous electrode on the inside wall are used.474'475 The coarse piezo tubes are Staveley Sensors EBL 7^2 (PZT-5A), L = 18.5mm long, D = 6.4mm wide, t = 0.5mm wall thickness; the central piezo is Staveley Sensors EBL #2 (PZT-5A), L = 10.3mm, D = 6.4mm, 9.2. CONSTRUCTION OF THE LOW-TEMPERATURE STM 175 t = 0.5mm.476 Some properties of this piezoceramic are given for reference purposes in Table 9.2. To drive the piezotubes laterally, bipolar symmetric voltages on opposing segments of the outer (X, Y) electrodes are used, while the tubes are extended by applying voltage to the inner electrode (Z). For this purpose we use a high-voltage amplifier manufactured by Createc that is based on high-slew rate (up to 1000 V///s) Apex PA85 MOSFET operational amplifiers. To move the main piezo scanner with respect to the sample surface and to perform the coarse approach of the tip to the surface, stick-slip technique is used to move the ramp disk.471'473'477 Saw-tooth like voltage profiles are typically used, but cycloidal477 and parabolic473 profiles are more effective. Both the amplitude and the frequency of the driving signal are important473 and they must be carefully determined for different temperatures at which the system is operated. We repoled the piezo tubes after observing that the ramp disk plate was running askew during rotation. This is achieved by applying high voltage (600 V, positive on outside electrodes, negative on inside electrodes) for a long time (several hours), while the piezos are moderately heated using an incandescent bulb held in close proximity. No appreciable deviations during rotation were observed after repoling. It should be noted that piezo tubes can depole with time due to aging and thermal cycling and even during normal use since scanning is equivalent to applying an ac field to the piezoelectric; taking the depoling field value from Table 9.2, we obtain a depoling voltage value of 350 V rms for our tubes. Property Value Units ^31 -1.73 Ä/V at 293 K d33 3.80 Ä/V at 293 K d3i -0.31 Ä/V at 4.2 K dss 0.69 Ä/V at 4.2 K Y, Young's modulus 6.3 1010N/m2 AC depoling field 7 kV/cm rms Dielectric constant K% 1725 Curie temperature 350 °C Thermal conductivity 1.5 W/mK Density 7.5 g/cm3 Table 9.2: EBL #2 material properties. Source: Staveley Sensors. During normal operation, the outer coarse piezos are used to scan over the surface area in X and Y directions, while the main central piezo is used for vertical displacements; in this mode of operation, the vertical and horizontal displacements are approximately decoupled. The vertical expansion of a piezotube is given by AZ d™ L, ¦VZ, (9.i; 176 CHAPTER 9. SCANNING TUNNELING MICROSCOPY where d3i is the piezoelectric coefficient, L tube length and t wall thickness. This gives 35 A/V piezo constant in Z direction at room temperature and 6.4 A/V at LHe temperature for the central piezo tube. Horizontal displacement is given by353'475 AX,AY=™gf-VX, (9-2) where D is the diameter of the tube. For the central piezo this gives 52 A/V piezo constant at RT and 9.2 A/V piezo constant at LHe temperature, while for the coarse piezos we obtain 170 Ä/V at RT and 30Ä/V at LHe. The piezoelectric behavior of the tubes, as well as the mechanical resonance frequencies of the tubes themselves and of the scanner as a whole, can be studied by measuring the double piezo response: one electrode (or a pair) is used to excite the piezo, another to measure the response.478~480 An example of such measurements is shown in Fig. 9.4. The lowest resonance frequency is found to be ~ 900 Hz. This lowest mechanical resonance frequency limits the highest scanning speed attainable.478 Capacitance of a piezo tube is _ 2K3 TwL ° InOD/lD (9-3) where ID = d and OD = d + t are inner and outer diameter, K3 T is the dielectric constant and eo is the vacuum permittivity. This gives C = 4.2 nF for the central piezo and C = 7.5 nF for coarse piezos. This is more than the capacitance of the wires, so measuring the capacitance is a reliable test to determine if the wires make contact to the electrodes when the system is already assembled. As a quick test, one can manually apply a high voltage ramp to a pair of electrodes (e.g., 200 V in one second) and observe the transient current which should rise to a few //A. For assembling parts that need to be glued together, we used special two-component epoxy glue which is sold as leak sealant by vacuum components companies ("epoxy patch" from Caburn). Once cured, this glue has very low degassing rate and is suitable for ultra-high vacuum applications. Excellent adhesion to sufficiently rough surfaces can be achieved and the joints are very strong, especially if the curing process is performed at elevated temperatures. Cured glue can withstand bake-out temperatures up to 125 °C; at higher temperatures it becomes softer. The STM tip holder is attached at the center of the central piezo tube using a magnet that is glued in a metal plated ceramic receptacle which, in turn, is glued to the end of the main piezo tube. The tips are made of etched tungsten wire (see below); the wire segment is inserted into a syringe tube and crimped. We first attempted to wire the head using 50 /iin thick polyimide-insulated (HML) high-purity copper wire (California Fine Wire Company) but this proved to be troublesome, especially in the initial stages of our work when the system had to be repeatedly disassembled for modifications and tuning. The wires tend to break near the soldering points where 9.2. CONSTRUCTION OF THE LOW-TEMPERATURE STM 177 I/V (A V Freq (kHz) Freq (kHz) (a) A2-C2 (b) Al-Cl Freq (kHz) (c) C1-A2 (d) Electrical wiring of a Besocke type STM Figure 9.4: Double piezoelectric response for various electrode combinations. The first electrode given in subfigure captions is being excited, the second is used to take measurements. 178 CHAPTER 9. SCANNING TUNNELING MICROSCOPY (a) Glued solder joints (b) PTFE assembly 1 (c) PTFE assembly 2 (d) Connector on the cold plate (e) Omicron sample plate holder (f) STM head (g) Heat in cryoshield sink Figure 9.5: Close-up pictures of some elements of the STM system and of wiring. 9.2. CONSTRUCTION OF THE LOW-TEMPERATURE STM 179 the insulation is stripped and the mechanical strength of the wire reduced; in addition. we observed that the wires tend to thin down during soldering, perhaps by the action of the soldering flux. Later we opted for 75 /iin and 100//m 0.45Ni-0.55Cu (constantan) wires which are much more robust. Furthermore, we developed the practice of putting a small drop of epoxy glue on the soldering points (Fig. 9.5a); the glue embeds the uninsulated part of the wire and plays the role of a stress relief. Due to small confines of our present LHe cryoshield, we had to be meticulous about the routing of the wires from the STM head to the connector at the cold plate to avoid tangling of the wires and to prevent that the wires touch the shield and bring in more environmental vibrations. We decided to use multiple stages: the signal wires first go from attachment points (piezos, Si diode, sample holder) to pins on a PTFE ring tightly attached to the guarding ring of the STM head (Fig. 9.5b), from there to pins on another PTFE ring fixed to the bottom of the cold plate (Fig. 9.5c), and from this ring to the connector (Fig. 9.5d). Using pins and sockets provides for easier maintenance. In addition, the wires are neatly coiled and, in a sense, become part of the spring suspension system (see below). From the connector at the cold plate, 150 /an polyimide-insulated stainless steel (SS316) wires take the signals to the RT connector at the top of the cryostat; stainless steel wires are chosen for their low heat conductivity and (relatively) low electrical resistance of ~ 150 Q. We test all cables for mutual shorts at > 400 V and for shorts to the ground at > 300 V. All signal wires must be properly thermally anchored. In the presently used cryostat, the wires are anchored on the cold plates of the LN2 and LHe reservoirs: they are glued in epoxy and embedded between two very thin mica plates over ~ 1cm length (Fig. 9.5g). These plates are then glued to the cold plates. There is no heat sink at the top of the cryostat as we do not have access to that area. In the new cryostat, the wires will be thermally anchored to LN2 temperature at the top of the LN2 reservoir using gold-plated copper bobbins (LakeShore Cryotronics, Inc.) and at the LHe cold plate at the bottom of the cryostat using beryllium oxide heat sink blocks (LakeShore), so that the temperature gradient from LN2 to LHe temperature will occur over a much longer length of wires (~ 1m). For soldering we use lead-free silver-tin alloy Castolin 157 in conjunction with Castolin 157N liquid flux. This combination is useful not only for soldering wires, but also for assembling metallic parts, in particular stainless steel parts where excellent bond strengths are achieved. It is very important to remove flux residue as it is etching. In addition, we observed that the flux reacts at high temperatures with the epoxy glue to produce insoluble conducting black goo which can lead to short circuits. 9.2.3 Bandwidths, resolution and vibrations For an estimated ~ 10 nF capacitance of the piezotube and the internal and external wires, and ~ 300 Q of resistance of a pair of wires, we estimate the RC time constant to be of the order of 3/xs. For a 512 x 512 STM image recorded in one minute, assuming equal forward 180 CHAPTER 9. SCANNING TUNNELING MICROSCOPY and backward scanning speed, the point-to-point mean time is ~ 100 ßs. The RC constant is thus not a limiting factor. Neither is the 60 kHz bandwidth of the high-voltage amplifier that drives the piezo tubes, nor the < 5 ßs conversion period of the D/A converters. We are therefore limited essentially by the bandwidth of the tunneling current preamplifier (for example, 1.2kHz at 109 V/A in the low-noise mode) and by the mechanical resonances of the scanner (which are also of the order of 1kHz).478 There is no intrinsic resolution limit in STM comparable to the diffraction limit of optical and electron microscopes. STM is essentially a counting apparatus: we measure the number of electrons that tunnel through the barrier in a unit of time. The wavelength of electrons concerned plays no role in this regard (beyond the fact that transmission probability is a function of electron energy/momentum). It is useful to remember that 1nA roughly corresponds to ~ 109 electrons per second. At ~ 1kHz bandwidth, the relevant time interval is 1ms, i.e. 106 electrons. Considering the tunneling as a Poissonian process, the shot noise is ~ 1/VlO6 = 10-3 of the signal amplitude, i.e. 1pA. Johnson-Nyquist noise at 1GQ impedance at 1kHz is lower by an order of the magnitude even at room temperature. Finally, the intrinsic noise of the current amplifier (Femto DLPCA-200) is declared to be 4.3fA/vHz, i.e. ~ 0.1pA at ~ 1kHz bandwidth. In usual circumstances, electronic noise is not a limiting factor since it is much smaller than the set-point tunneling current (provided, of course, that the system is properly shielded from external electromagnetic interference). Instead, lateral resolution is limited by the width of the tunneling current flux, i.e. by the atomistic details of the STM tip over which one has little control. Vertical resolution is chiefly limited by the tip-sample vibration amplitude, which also affects lateral resolution. For good results, stable tip and stable STM scanner are thus essential. In order to reduce the noise due to external electromagnetic interference in the tunneling signal, it is important to place the current amplifier as close to the tunneling junction as possible. A convenient choice is to connect it directly to the BNC feed-through at the top of the cryostat. The connection between the feed-throughs and the STM head is currently made using shielded twisted-pair cable; one wire is used for the tunneling current, the other for the bias voltage applied to the sample. A possible improvement would be to use two separate insulated coaxial cables for these two signals and to make a common ground at the input of the current amplifier. The two coaxial cables can then be twisted to eliminate the 50 Hz power line noise. It is recommended to wire the tunneling signal separately from the high-level signals used to drive the piezo-tubes to reduce noise coupling. External cables for high-level signals also need to be properly shielded to prevent that they pick up high-frequency electromagnetic interferences from the environment and radiate them in the STM head. Shielded paired data transmission cables (liycy) are a good choice. A pair of wires is used to carry a pair of signals that are connected to the opposite electrodes of a piezo tube. Another important aspect is correct grounding. Shields on cables should be grounded at one point only to prevent the noise currents from flowing.481 For ungrounded circuits (such as the STM junction), the best shield connection point is at the current amplifier 9.2. CONSTRUCTION OF THE LOW-TEMPERATURE STM 181 (amplifier common terminal).481 Shields around amplifiers should also be connected to amplifier common terminals.481 All grounds should be solidly connected (soldered, bolted) to avoid hard-to-diagnose problems later on. The vacuum chamber, pumps and other vacuum equipment is always grounded to a safety ground. It is appropriate to use separate signal grounds for all STM wiring and to connect the two grounds at a single point at the main power inlet box of the laboratory, thus avoiding possible ground loops and, more importantly, common ground impedance coupling problems.481 Unfortunately, this is not always practicable nor economical, since floating vacuum electrical feed-throughs are more expensive. By trial and error, we largely succeeded in eliminating the 50 Hz power line noise. We also find that it helps significantly to power the STM electronics through an insulation transformer; even better solution would perhaps be a dual conversion online uninterruptible power supply (online UPS) which is an excellent filter for removing line noise. To monitor noise levels, it is very useful to perform real-time spectral analysis of the tunneling current signal. Since the frequencies of interest are in the audible frequency range, the signal can be fed to an audio input of a personal computer via a high-impedance buffer (since audio inputs typically have relatively low input impedance). While this provides a convenient and inexpensive way to perform spectral analysis, one should nevertheless note that the frequency response in the consumer-grade audio cards is non-flat and that such inputs typically have a high-pass filter which prevents to observe noise below 10 Hz which also affects the operation of an STM in a significant manner. Typically, we observe that spectra contain some 50 Hz noise and higher harmonics, as well as sharp peaks attributed to mechanical resonances. Using a loudspeaker directed to the UHV chambers which is driven by a frequency generator, it is also possible to study the transfer of acoustic vibrations from the environment to the STM junction. Most importantly, the stability of the system can be quantified by imaging flat crystalline surfaces such as Cu(lll) (Fig. 9.6a) and calculating the standard deviation of the topographical z signal along a line profile in the fast scanning direction (Fig. 9.6b, first three rows) and along the slow scanning direction (Fig. 9.6a, last two rows). The heights can be calibrated accurately from the known step height, 210pm (Fig. 9.6b, first row). The standard deviation of the height on the lower part is er = 1.2 pm and a = 1.5pm. To calculate these values, the image was leveled using the three-point procedure482 and a polynomial background of second order was subtracted from each segment of the line profile (Fig. 9.6b, second and third row). The line profile along the slow scanning profile (Fig. 9.6b, fourth line) exhibits more irregularities. The standard deviation is higher, a = 4 pm. After performed additional image processing (step line correction482 and subtraction of the polynomial background of third order from the line profile, see fig. 9.6b, last line), the calculated standard deviation is a = 2 pm. 182 CHAPTER 9. SCANNING TUNNELING MICROSCOPY -10 -20 15 10 Lj 0 N -5 -10 -15 0 10 20 30 40 x or y [nm] 50 G=1.2 pm G=1.5 pm G=4 pm ) 10 20 30 40 50 6 _ 1 1 1 1 1 foil.. ,iil _ || 1 Wtfi ~ G=2 pm 60 Figure 9.6: Determination of the stability of the system from line profiles on flat surfaces. Cu(lll), U = 943mV, I = 0.38nA, image size (108nm)2, 328ms per line. 9.2. CONSTRUCTION OF THE LOW-TEMPERATURE STM 183 9.2.4 Vibration isolation To reduce transfer of environmental vibrations to the STM chamber, the entire vacuum system is suspended on pneumatic vibration isolators (Melles Griot self-leveling vibration isolators) which provide both vertical and horizontal isolation. Their resonance frequencies are around 1Hz and transmissibility is below 6% and 10% for horizontal and vertical vibrations, respectively, at 5Hz which is the typical frequency of vibrations in buildings. The STM head is suspended on stainless steel springs and its oscillations are damped by eddy current damping.353 This setup very effectively damps the high frequency (acoustic) range of environmental vibrations that are transferred through the springs. The relation between the axial load F, and the deflection A of a spring with coil diameter D, number of coils n, and wire diameter d is F = SA, where S is the stiffness or rate of the spring: S = -^- (9.4) 8nD3 v ' where G is the shear modulus of elasticity (modulus of rigidity). Elastic properties of some materials used in STM applications are listed in Table 9.3. For packed springs the length of the spring L = nd and S = Gd5/8LD3. Material Modulus of rigidity Young modulus Thermal expansion coefficient G [1010 N/m2] Y [1010 N/m2] a [10-6 K-1] Aluminum 2.6 Platinum 6.1 Gold 2.7 Silver 3 Copper 4.8 Iron 8.2 Music wire 8 Stainless steel 6.7 Tungsten 16.1 Table 9.3: Shear modulus (rigidity modulus), and Young modulus of elasticity for a selection of wire materials. The values are given for room temperature. Sources: CRC Handbook of chemistry and physics, and http://www.matweb.com. Let us consider the force exerted on the base plate of the STM if the attachment point of the spring oscillates as A = A0 sin(cjt). It is clear that a hard string will transfer vibrations more efficiently than a soft spring. The transfer of vibrations depends on the damping of the STM head using the eddy-current damper. We define the Q factor as Q - L• (9.5) 7 23 16.8 9 7.8 14 8.3 19.6 13 17 21.1 12 21 17.3 41.1 4.5 184 CHAPTER 9. SCANNING TUNNELING MICROSCOPY Here u)0 = \fSjm is the eigenfrequency of the vertical vibrations of the STM head of mass m, while 7 is the damping rate. The transfer function of this mechanical system is (Ref. 353; note that the power of cj0 in Eq. (10.17) in the book must be corrected to 4) n04 + 472n;2 (w0 - oj2)2 + 472cj2 im=j,r0.r„:" . (9.6) If there is no damping, the transfer function at high frequencies is K(u) ~ (u0/u)2. With damping, we have K(u0) ~ Q and K(u) ~ (l/Q)(u0/u) for cj > Qu0.353 A Q value of 3 — 10 is usually chosen as a compromise between the suppression of resonance and the suppression of high-frequency vibrations.353 Clearly, harder spring means higher Q if damping is left unchanged. Spring rate of typical coils used to suspend an STM head (d = 0.5mm, D = 4 mm, L = 10 cm, stainless steel) is S ~ 300 N/m. For three coils, we have a total stiffness of roughly 1000 N/m. It may be noted that a set of 25 copper wires of d = 100 //m diameter, each rolled into a spring of diameter D = 3 mm with n = 20 loops has a stiffness of S ~ 30 N/m. For a thicker wire of d = 200 /an, the stiffness is S ~ 450 N/m, which is comparable to the stiffness of the suspension springs, but probably still acceptable. The idea here is to use the wires as part of the suspension system. The wires must be firmly mechanically attached to the baseplate of the STM and only then routed to the final electrical connection point (piezo electrodes, leads of thermometric diode, etc.). It is not a good idea to connect the wires directly to their final connection point since in that case different forces are exerted to various parts of the instrument; if all wires are fixed on the baseplate, the differences are effectively compensated and the STM head tends to vibrate as a whole. The worst case scenario is if the wires are routed directly to the scanner, since environment vibrations then translate into sample-tip distance oscillations. 9.2.5 Tip preparation and characterization Regarding the tip, three factors are important for reliable STM operation: the tip must have high flexural resonant frequencies to allow high scanning rate, it must have a sharp apex (ideally a single site of closest atomic approach) and it must be clean so that a series resistance is not present.483 There is a large number of STM tip preparation recipes. Two types of tips are most commonly used: etched tungsten (W) tips and mechanically cut platinum/iridium (0.9Pt/0.1Ir) tips. W tips can be made very sharp, they are suitable for tough samples due to their hardness which prevents deformations and erosion during imaging, but their lifetime in air is short because of W oxidation.484'485 The resistance of a tungsten oxide layer can easily be much higher than the desired tunnel gap resistance; such a tip would crash before the set tunneling current can be obtained.483 In addition, tunneling spectroscopy of metal surfaces using oxidized tips shows a spectrum characteristic 9.2. CONSTRUCTION OF THE LOW-TEMPERATURE STM 185 of a semiconductor.486 Pt/Ir tips, on the other hand, are oxide free and can be used in air for very fiat surfaces, but they are not suitable for highly corrugated surfaces.484'487 These tips are prepared by cutting the wire with scissors. The reproducibility is low: the method "relies on operator's skill rather than on controllable parameters".487 Other types of tips are also sometimes used. Ni tips are oxide free,487 while Co tips form a homogeneous thin oxide layer which prevents further oxidation.488 Both can be reproducibly prepared by etching. Furthermore, they are ferromagnetic, so they can be used to perform spin-polarized tunneling microscopy. We mostly used W tips. We prepare them from polycrystalline tungsten wire of 99.95% purity with a diameter of 0.25 mm or 0.38 mm. We electrochemically etch485'489 the wire in a (typically) 2 N solution of KOH at a depth of a few millimeters beneath the surface. W wire is the anode and we use a ring of Au wire as the cathode. With an etching current of 10 mA, the process takes around 10 min. We use a home-made electronic device to accelerate the shutdown of the tunneling current when the wire starts to break apart since a short cutoff time is necessary to obtain a sharp tip.489'490 As mechanical cutting of W introduces defects, it is a good idea to remove the damaged material by etching away a few millimeters of the wires.491 W tips are robust with respect to dipping in liquids and even dropping if the very end of the tip is untouched.491 It was found, however, that ultrasonic cleaning485 or exposing the tip to a jet of water from a perpendicular direction bends the tip apex. It is important that etching and rinsing be performed using clean and dust-free liquids; we observed that tips are often covered with dust particles, often even near the tip apexes, Fig. 9.7. As etched, the tips are always chemically contaminated. Using ESCA, we have detected K containing crystallites, while other groups reported existence of thin carbonaceous phases.484'485'492 Above all, the tip is always coated with a layer of tungsten oxide. These contaminants can be removed by etching in concentrated HF.486 Since a new oxide layer would form on a time scale of several days483'484'492 (a contradicting report of Ekvall et al.491 maintains that no traces of oxides were formed for more than one month), it is best to perform the HF treatment just prior to introducing the tip in the UHV system. The tip is exposed for 10 s to concentrated HF, then immediately rinsed with deionized water and dried with pure isopropanol (IPA). All tungsten oxides are soluble in concentrated HF, while tungsten itself should be inert to attack by HF.486 Nevertheless, it is found that the tip sharpness is reduced after prolonged HF etching.485 On the other hand, short HF treatments do not remove the insulating layer entirely491 It is thus very important to perform further in-situ preparation of the tip using either ion bombardment483 or annealing;484'485'493 unfortunately such treatments are not currently possible in our system since we do not have the possibility to change tips in situ using a manipulator. Finally, and most importantly, the tips can be improved by intentionally crashing them in a soft metal surface (such as Cu or Ag) in a controlled manner and applying a large potential for a short time;486 the tip apex is then expected to be covered by sample atoms. We found that such crashing treatment was required for all our tips to achieve stable operation and 186 CHAPTER 9. SCANNING TUNNELING MICROSCOPY Figure 9.7: SEM images of two different electrochemically etched W tip with contamination: large-scale overview (left) and zoom to the apex region (right). atomic resolution. It is characteristic for bad tips to show an oscillating tunneling current when the tip is held stationary. Often the oscillations have the appearance of a series of current spikes between regions where the current is close to zero: a spike appears when a break is produced in an oxide layer and metallic region is exposed.486 The control of STM tip shape, bulk and surface composition is of extreme importance. To characterize a tip, an optical microscope can be used to eliminate tips that are bent on a large scale, while SEM and TEM can be used to obtain information on the apex region. With the resolution of the SEM, it is not really possible to distinguish a sharp tip from a blunt one.491 Only using TEM is it possible to study the very apex of the tip; one can even detect a possible insulating layer.491 A cheap alternative to TEM characterization is the use of the field-emission from the tip apex to determine the effective apex curvature.492 9.2.6 Sample transfer mechanism Due to budget constraints, instead of using standard but costly manipulators, a specialized sample transfer mechanism has been constructed (Fig. 9.8). Its purpose is to convey the 9.3. ULTRA-HIGH VACUUM SYSTEM 187 sample from the central chamber (where the sample storage carousel is located) to the STM head. It has three degrees of freedom: linear motion to insert the sample in the STM, vertical motion to let go the sample once it is properly fixed by the pair of clamps in the STM head, and rotation to suitably orient the sample. The mechanism is affixed in the narrow space of the flange neck in the UHV chamber where the STM is housed. The mechanism is operated using a wobble stick, as well as by linear and rotary feed-throughs that are located in other parts of the vacuum system and connected to the mechanism by thin stainless steel braided wires. Figure 9.8: Sample transfer mechanism A major difficulty in mechanical assemblies operated in UHV systems is friction which increases quickly as the oxide layers wear off, resulting in cold welds. Often even a single motion cycle can lead to a grinding halt. It is interesting to note that similar problems arise in the space technology. The solution consists in using light-load designs that rely on UHV-compatible ball bearings; direct rubbing of components must be avoided. If this is not possible, hard coating improves the situation considerably. We have experienced excellent results with TiN coating which eliminated cold welding; in addition, it has low coefficient of friction that does not increase with motion cycles in a perceptible manner. Vacuum-related mechanical problems can easily remain undetected during prototype construction at ambient conditions, therefore early testing in UHV is recommended. 9.3 Ultra-high vacuum system An important development in vacuum technology was the invention of copper gasket seals by Wheeler in 1962;494 this was the first reliable seal for achieving ultra high vacuum 188 CHAPTER 9. SCANNING TUNNELING MICROSCOPY (UHV), defined as pressure below 10~7Pa (or 10~9mbar). Pressures as low as 10~14 Pa have since been measured in the laboratory.495 It is interesting to note that this is still several orders of magnitude higher than the vacuum in the deep space.496 Use of UHV is important to study reactive surfaces with strong chemisorption of molecules: at the limit pressure of 10~9mbar it takes 103s to cover a surface with a monolayer of contaminants if the sticking coefficient cs is unity (oxygen sticking coefficient for poly-crystalline Cu at RT is quite low, ~ 10~3,497 it is even lower for smooth single crystal Cu surfaces;498 for more reactive metals and for semiconductors, it can be considerably higher). Obtaining UHV is rather demanding: the experimental set-up needs to be constructed from suitable materials, be extremely clean and have small surface area, and vacuum pumping must be performed in multiple stages (rotation or diaphragm fore pumps, turbomolec-ular pumps, and gettering ion and sublimation pumps when high vacuum is attained). Furthermore, the entire vacuum systems needs to be "baked" to remove adsorbed water, hydrocarbons, CO and C02.499 These species have a surface lifetime in the range of seconds to hours and therefore hinder rapid pump-down; in contrast, species with longer lifetime remain stuck to the surfaces, while species with shorter lifetime are quickly desorbed and pumped out of the system. Mean free path (for example of air molecules) at p = 10_9mbar is A ~ 50 km, therefore gas molecules collide with chamber walls more often than they do with each other. The corresponding Knudsen number Kn = X/d ^$> 1 (here d is the characteristic size of, e.g., a tube) and the flow characteristics are well within the free molecular flow regime.500 The pressure can vary from chamber to chamber and even from point to point inside a single chamber; it is important where the pressure reading is taken. It is worth noting that the number of gas molecules adsorbed on the walls of the chambers is higher than the number of flying molecules; at low pressure the surface area of the system is thus more important than the volume. Conservation of mass equation for gas in the chamber is494 V^ = L + QA- Sp, (9.7) where V and A are the system volume and surface area, L is the leak rate, Q is the out-gassing rate per unit area and S is the pumping speed. To obtain UHV, it is essential that L 1A0(E) 1 1 2vr -Re ,2i<5 le 2ikr 2i kr (9.12) where k is the energy dependent wave-number of the surface-state electrons, and A0(E) is the spectral function of the unperturbed 2DEG. This equation is valid in the far field of the scatterer. 9.5.4 Molecules on surfaces On many occasions, we have imaged Cu(lll) terraces covered with individual molecules, Fig. 9.16. Each molecule is imaged as a depression surrounded by a protruding ring. The molecules were not deposited intentionally; these are rather molecules which are difficult to eliminate in the sample cleaning process, most likely CO molecules which are a notorious 198 CHAPTER 9. SCANNING TUNNELING MICROSCOPY contaminant species on Cu(lll) surfaces. CO adsorbs on the top site with carbon atom down515 and it is imaged as a depression by clean metallic tips. The origin of the protruding ring are Friedel oscillations around the adsorbed molecule. (a) U = 211mV, I = InA, (b) U = 636mV, I = InA, T = 25K T = 25K Figure 9.16: Cu(lll) covered with a low concentration of CO molecules. We also observed in some cases that the molecules are displaced under the effect of the tip, Fig. 9.17. The molecules appear to be pushed away from the tip during successive line scans, which leads to appearance of worm-like linear structures; each such line corresponds to a single molecule being displaced. It should be noticed that not all molecules seem to be moved by the tip; some are immobile and imaged as depressions (black spots) without any artifacts. Later, when we attempted to perform controlled manipulations, it was confirmed that some molecules are indeed more easily displaced than others. It is likely that some molecules are trapped by subsurface impurities which lead to stronger binding, or that there are in fact two different molecular species on the surface. 9.5. SAMPLE STM IMAGES 199 Figure 9.17: Cu(lll) covered with a low concentration of CO molecules. U = 501mV. I = 0.96nA, T = 25K. Chapter 10 Clusters of magnetic adatoms and surface Kondo effect Magnetic nanostructures are attractive candidate systems for data recording applications (information storage in a magnetic medium) and for spintronics (information processing using spin degrees of freedom). Studies of magnetic properties of clusters of magnetic impurities elucidate how magnetism appears as the system size is increased.516'517 Properties of small clusters adsorbed on surfaces depend strongly on their size and geometry, as well as on their coupling to the substrate. Due to surface interactions the inter-atomic distances and cluster shapes may be very different from free clusters in vacuum;518'519 the adatoms in the smallest clusters (2-10 atoms) are often found to be planar and in registry with the surface atoms.517 Furthermore, in addition to bonding interaction (roughly equivalent to electron hopping in the context of quantum dots ) and direct exchange interaction, there will always be some substrate-mediated long-range indirect interaction between constituent atoms (indirect hopping, RKKY exchange interaction). Co, Fe, Mn and their dimers on NiAl(llO) were studied using STS by Lee et al.520 Single adatoms have spin-split single-electron resonances of sp character that can be easily observed in STS spectra. Spin-splitting originates from the exchange interaction with the spin-polarized, partially filled 3d states. When two such adatoms are brought together, the hybridization of these sp resonances depends of the sign of the magnetic interaction between the atoms. In the case of AFM coupling two spin-degenerate states are expected, while in the case of FM coupling four exchange-split resonances should appear. The spectral resolution in STS is sufficient to discern between the two possibilities: all three dimers were found to be ferromagnetic. This demonstrates that STM can be used to study internal magnetic coupling in metal clusters even without a spin-polarized tip. The Kondo effect that occurs in the systems of magnetic impurities on metal surfaces is of special importance; using a low-temperature STM it is possible for the first time to study a single magnetic adatom in interaction with a continuum of states in the substrate.4'5 In bulk Kondo systems we are in fact considering an ensemble average over a macroscopic 200 10.1. REVIEW OF EXPERIMENTAL RESULTS ON SURFACE KONDO EFFECT201 number of impurities with random inter-impurity interactions. The Kondo resonance has been observed, for example, using high-resolution photo-electron spectroscopy that averages over a typical surface area of 1mm2.521 An STM, however, has atomic resolution in space and < 1meV resolution in energy: with STM, magnetic impurities can be characterized both spectroscopically and in space. By measuring the difference of the dl/dV signal at the minimum and the maximum of the Fano resonance resulting from the Kondo effect, it is possible to locate spin centers in magnetic molecules with atomic precision.15 In Co(CO)4 complexes, for example, the Fano spectral signature is well visible when the tip is positioned above the center where the cobalt atom is located and it decays within a radius of 2.5 Ä.15 Another field of study focuses on the effects of magnetic impurities on correlated materials such as superconductors. It has been shown, for example, that a magnetic adatom (Fe) on a conventional BCS superconductor (Nb) breaks Cooper pairs, as expected.430 In this chapter I focus on the Kondo effect induced by magnetic adatoms and clusters. In Section 10.1 I review known experimental results, while in Section 10.2 I briefly present effective phenomenological and microscopic models in common use. Finally, in Section 10.3 I present results of my NRG calculations of the tunneling current affected by the presence of impurities; this section also provides the link to the first part of this dissertation. 10.1 Review of experimental results on surface Kondo effect Following the seminal work of Madhavan et al.4 and Li et al.,5 the surface Kondo effect has been detected in a large number of adsorbate/surface systems in which the magnetic moment is carried either by single atoms or by magnetic ions embedded in molecules; an overview of some measured Kondo temperatures in various systems is given in Table 10.1. It should be noted that experimentalists define the Kondo temperature as the half-width of the Fano resonance. The half-width is only proportional (but not equal) to the Kondo temperature. Since the coefficient of proportionality varies from case to case, published values only indicate the scale of the true Kondo temperature. Another complication is that extracting the half-width of the Fano resonance by fitting depends on the background-subtraction procedure. These facts should be taken into account when using published values of TK in theoretical work. Kondo temperatures are seen to cover a wide interval from much above the room temperature to temperatures below the experimentally achievable range; this merely reflects the exponential dependence of Tk on the microscopic parameters. 202 CHAPTER 10. CL USTERS OF MAGNETIC ADATOMS AND SURFACE KONDO EFFECT Surface Adatom/complex Kondo temperature Reference Au Co in bulk 300 - 700 K 4,522 Au(lll) Co 70 K 4,523 Au(lll) Co2 <6K 523 Ag(lll) Co 92 K 514,524 Cu Co in bulk ~500K 525 Cu(lll) Co 53±3K 426 Cu(lll) Co embedded 405 ± 35 K 526 Cu(lll) Co 54±2K 525 Cu(100) Co 88±4K 525 Cu(100) Co(CO)2 165 ±21 K 15 Cu(100) Co(CO)3 170 ±16 K 15 Cu(100) Co(CO)4 283 ±36 K 15 Cu(100) (Co(CO)2)2 138 ±21 K 15 Cu(100) (Co(CO)2)2 176 ±13 K 15 Au(lll) CoPC <5K 429 Au(lll) d-CoPC 208 K 429 Cu(100) Fe <60K 15 Cu(100) Fe(CO)4 140 ±23 K 15 Au(lll) Ti 70 K 527 Au(lll) Ni 100 ±20 K, mixed valent 13,527 Au(lll) Ni2 42±5K 13 Ag(100) Ti 40 K 528 Ag(lll) Ce 580 K 5 Au(lll) V, Cr, Mn, Fe <6K 527 Au(lll) Cr <7K 14 Au(lll) Cr3, equilateral 50 K 14 Table 10.1: Kondo temperatures for surface Kondo effect on metals. 10.1. REVIEW OF EXPERIMENTAL RESULTS ON SURFACE KONDO EFFECT203 10.1.1 Kondo effect in a single magnetic adatom A magnetic impurity (transition metal such as Co or lanthanide such as Ce) adsorbed on the surface of a normal metal (usually a noble metal such as Cu, Ag or Au) induces a characteristic narrow (~ 10 mV for Co/Cu) anti-resonance-like structure near the Fermi level in the electronic surface local density of states (LDOS). The characteristic asymmetric line shape resembles that of a Fano resonance;4 it should be contrasted with resonances of non-magnetic adatoms which are typically much broader (hundreds of mV). The resonance is spatially centered on the impurity atom and decays over a lateral distance of ~ 10 A. where the spectrum becomes identical to the one obtained on a clean surface.426 The resonance line shape tends to become more symmetric as the tip is moved radially outward: for Co/Au(lll) a symmetric dip is observed at a distance of 4 A.4 The observed features do not depend on the tip used.4'5 The simplest interpretation of these results is that the adatom magnetic moment is Kondo screened and that the observed asymmetric feature in the tunneling spectrum is related to the Kondo resonance. Since the d-orbitals of magnetic adsorbates are both spatially well localized within the adatom and far away in energy from the Fermi level, it is reasonable to assume that the majority of the tunneling current at small bias is carried by the sp-like states resulting from the hybridization of the substrate states with the adatom electronic levels (in particular adatom valence s levels). A good indication that this is true is the independence of the tunneling spectra on the tip-sample distances.4'426'427'525 What the STM probes is thus, in a very simple approximation, the LDOS of the conduction band electrons at the position of the impurity local moment (or, more accurately, at the position of the tip apex), and not the LDOS of the impurity d-level. The hybridization of an impurity site featuring a resonance in its spectral function with a continuum band induces a sharp anti-resonance in the LDOS of the continuum band at the position of the impurity: it is this anti-resonance that we observe in the experiments with magnetic adatoms. Using NRG (see Ref. 529 and Sec. 10.3), it can be shown that for asymmetric single-impurity Anderson model, the anti-resonance indeed has an asymmetric shape similar to that of the Fano model. This point of view is more than a restatement of the conventional description of the origin of the Fano resonance as an interference between two tunneling channels;4'530 I find it conceptually cleaner to describe the anti-resonance as a feature of the substrate LDOS with the effects of the impurity d-orbital taken into account, rather than as an interference phenomenon between the tunneling in unperturbed substrate band and the indirect tunneling through the perturbing impurity. The distinction between the two perspectives becomes unmistakable when the impurity d-level is modeled as a Kondo impurity. In this case, there is no hybridization to the d-level at all, only exchange interaction; nevertheless, a Kondo anti-resonance appears in the spectral function of the conduction band (see Fig. 6.2). Of course, if part of the tunneling current actually does flow directly through the impurity d level, the asymmetry is enhanced even further as described by the Fano model.529 Recently, the Kondo resonance has also been "directly" observed in the system of a Mn 204CHAPTER 10. CLUSTERS OF MAGNETIC ADATOMS AND SURFACE KONDO EFFECT adatom on an A1203 island formed on NiAl(llO) surface:439 a narrow conductance peak with Lorentzian shape was observed near zero bias. The magnetic ion is not directly interacting with the substrate electrons; instead, the interaction is mediated by an oxide film. More importantly, the tunneling mechanism is most likely different: in this case, the electrons tunnel from the STM tip to the Mn adatom, and then they tunnel anew through the oxide layer. This two-step tunneling process strongly depends on the impurity spectral function and is, in fact, very similar to the transport phenomena in quantum dots weakly coupled to two electron reservoirs that were the subject of the first Part of this dissertation. When a magnetic field is applied, the peak decreases in amplitude and then splits:439 this is a conclusive demonstration that the origin of the resonance is the Kondo effect. Kondo temperature increases with the number of atoms surrounding the impurity.525'526'531 For example, for Co/Cu(lll) the coordination number Z =3 and the Kondo temperature is Tk = 54 K, while for Co/Cu(100) Z =4 and Tk = 88 K.525 Furthermore, impurities in the bulk have significantly higher Tk than adatoms, while Tk = 405 K of an impurity embedded in the surface layer (Z =9) interpolates between TK for adsorbed Co atoms and the bulk Tk of Co/Cu [Z = 12).526 A higher number of neighbors implies stronger hybridization and thus higher Tk- Knorr et al.525 have proposed a phenomenological model J p ex Z which provides a reasonable fit. Sharp features in LDOS at one focus of an elliptical quantum corral can be coherently projected by the surface-state electrons to the other focus: this effect has been nicknamed "quantum mirage".426'532 The experiment is a clear demonstration of the wave nature of electrons. From a semi-classical point of view, the mirage effect occurs due to the property of an ellipse that the sum of the path lengths from the foci to an arbitrary point on the ellipse is constant; if a scatterer is placed at one focus, all scattered waves will interfere constructively at the other focus.426 The corral behaves as a resonant cavity and the sharp spectral feature projected is the Kondo/Fano resonance of the magnetic Co atom in the focus. As expected, the mirage vanishes when the magnetic adatom is displaced from the focus. A scattering theory that accounts quantitatively for these results was developed by Fiete et al.533'534 10.1.2 Dimers of Kondo impurities Dimers of magnetic adatoms are in many respects similar to the systems of double quantum dots that were the subject of the first part of this dissertation. The insight developed in studying quantum dots can also be applied to understand magnetic structures on surfaces. We saw the important role played by the various inter-impurity interactions. Dimers provide a convenient playground where the pair-wise interactions between Kondo impurities can be studied function of their separation.3 9 In quantum corrals formed by Co atoms, the Kondo effect signature is present around each adatom;426 at the mean inter-atom distance of 10 A the adatoms thus behave to a large extent as independent Kondo impurities. Other studies of Co/Au(lll) show that 10.1. REVIEW OF EXPERIMENTAL RESULTS ON SURFACE KONDO EFFECT205 the spectral line shape does not change in the presence of other Co atoms for inter-atom separation down to as low as 6 A,4 however the Kondo resonance abruptly disappears for Co-Co separations less than 6 A.523 In another type of experiment, interaction effects between Co impurities on Au(lll) remained small as Co coverage was increased to 1 ML,427 however the fraction of atoms that exhibited the Kondo effect was reduced. This experiment indicates that long-range RKKY interactions between Co atoms are weak. On the other hand, two focal magnetic adatoms in quantum mirage experiments interact quite strongly with one another (the Kondo resonances were perturbed)426 which hints that the surface-state electrons mediated RKKY interaction can be significantly amplified by constructing a suitable resonant cavity. Calculations of inter-impurity interaction between two magnetic atoms located at the foci of an elliptical quantum corral indicate that the quantum corral eigenmode mediated exchange interaction is ferromagnetic.535'536 Madhavan et al. have studied the evolution in electronic properties of Ni/Au(lll) as two Ni atoms are merged to form a dimer.13 In free Ni2 molecules, the inter-atomic exchange interaction is ferromagnetic.537 For inter-atom distance d > 12 A the adatoms do not interact. Hybridization effects become observable at d ~ 7 A. For d > 4 A the Kondo temperature is 100 ± 20 K, but it drops sharply at the closest separation of 3.4 A. Using LSDA calculations, they explained their findings by impurity energy level shifts; the system is moved from valence-fluctuation regime to a regime with stronger local moment and strongly reduced Tk- Recently, a study of Co dimers on Cu(100) was performed329 which sheds more light on this problem. It was shown that for d > 5.7 A, the resonance features recover their d —> oc form, while for d = 2.56 A (compact dimer) the tunneling spectrum is featureless; finally, for intermediate d= 5.12 A a strongly perturbed anti-resonance was found, which is broadened compared to the spectrum of the isolated Co atom. An antiferromagnetic coupling J ~ 16meV at d = 5.12 A was extracted, which should be compared with kßT^ = 7.6meV for Co/Cu(100). This system is thus near the critical point at J ~ 2kßT^. For the compact dimer, the RKKY interaction is ferromagnetic, binding the spins to S =1, and it appears that the (upper) Kondo temperature is pushed below the temperature range of the experiment. 10.1.3 Trimers of Kondo impurities Jamneala et al.14 have studied trimers of Cr atoms, a frustrated antiferromagnetic system. Cr has 3d5Asl configuration with large atomic magnetic moment and strong inter-atomic bonding.14 Cr3 clusters can be reversibly switched between two configuration with different electronic behavior. One has a resonance near ep, the other a featureless spectrum; in the original experiment, the atomic positions of atoms could not be resolved. It was proposed that the isosceles trimer shows the Kondo effect since it has magnetic moment, while a compact symmetric trimer does not since the spins sum up to zero.14 This description in 206 CHAPTER 10. CL USTERS OF MAGNETIC ADATOMS AND SURFACE KONDO EFFECT terms of classical spins is questionable: quantum mechanically, in a triangle we have an octuplet of states which split into a quartet and two degenerate doublets. The original interpretation has been superseded by different theories.37'38'538 It appears possible that in the symmetric trimer exotic Kondo effect with non-Fermi liquid behavior leads to an increased TK.37'38 10.1.4 Kondo effect in adsorbed molecules Kondo effect also occurs in adsorbed molecules containing a magnetic ion, for example in Co(CO)ra molecular complexes.15 The spectral features were related to the Kondo effect by observing that the resonance remained pinned at the Fermi level when the number of ligands was changed. Similar behavior was observed in binuclear cobalt carbonyls, which consist of two cobalt atoms and a number of carbonyl groups. The coupling between the spin of individual cobalt adatoms with their surroundings can be controlled by attachment of molecular ligands (here CO molecules). A reasonable fit was obtained using J p = J0 + cn, where n is the number of ligands. The exchange interaction J and the Kondo temperature increase due to two factors: increased hybridization between the orbital which carries spin and the conduction band, and delocalization of the d electrons which decreases the on-site Coulomb repulsion. This implies that by choosing appropriate ligands, the Kondo temperature can be controlled by modifying the chemical environment of the spin center.15 A similar experiment was performed on cobalt phthalocyanine (CoPC) molecules on Au(lIf ).429 In free CoPC molecule, the Co atom has unpaired d electron with magnetic moment. As 8 hydrogen atoms were removed from the PC backbone, the molecule became more strongly chemically bound to the surface and the Kondo temperature increased. 10.1.5 Kondo effect in metal clusters on nanotubes Odom et al.539 have studied small cobalt clusters on metallic single-walled carbon nanotubes (CNTs) with STM. On small nanotube pieces, quantum box effects were detected. A peak near Ep was observed above the center of Co clusters which disappeared over a distance of 2nm from the cluster. Control experiments were performed with non-magnetic Ag atoms and on semi-conducting CNTs: no resonance was visible in either case, which is consistent with the hypothesis of the Kondo physics. 10.2 Theory of the surface Kondo effect STM probes Fermi-level LDOS of the sample at the position of the tip,425 i.e. the extended s'p wave functions rather than the well localized d or / orbitals,5'513'540 see Fig. 10.1. In a 10.2. THEORY OF THE SURFACE KONDO EFFECT 207 first approximation, we may thus assume that the STM current is due only to tunneling into the conduction band and we may neglect direct tunneling into the impurity d or / level.427 This holds approximately even when the tip is directly above the atom; in this case the adatom sp orbitals that protrude into the vacuum are strongly hybridized with the conduction band of the substrate.541 To understand the origin of the Fano resonances in the tunneling spectra measured over magnetic impurities, we must thus study the effect of the impurity on the continuum states. 0.01 0.1 1 10 r [bohr] Figure 10.1: Radial wavefunctions of free Co atom in [Ar]4s23d7 configuration computed using Hartree-Fock code.542 Note that the horizontal (radial) scale is logarithmic. The orbital that extends the furthest outward is 4s. 10.2.1 Effective quantum impurity models Magnetic impurities can be modeled as asymmetric Anderson model in the Kondo regime, but not far from the valence-fluctuation regime (see Sec. 6.3). This model can be mapped at low temperatures onto the Kondo model (Sec. 6.3.1 and Ref. 298). A simple approximation for the effective Kondo exchange interaction J is30 min(|ed|, |ed + C/|)" Here e \td\-, thus td sets the scale of J. The effective (half-)bandwidth must be appropriately defined. It must be emphasized that the effective bandwidth is not necessarily related to the width of either the bulk conduction 208 CHAPTER 10. CL USTERS OF MAGNETIC ADATOMS AND SURFACE KONDO EFFECT band or the surface-state band. In fact, we have De{{ ~ mm(Db,Ds, U, E*j), where Db is the half-bandwidth of the bulk band and Ds the half-bandwidth of the surface-state band (if both are at play), U is the Coulomb repulsion and Ed is the renormalized impurity energy which is usually the relevant energy scale in the valence-fluctuation regime (Sec. 6.3). E*d is the solution of transcendental equation31 r E*d = ed-------In 7T u (10.2) There are some indications that magnetic impurities on surfaces indeed are mixed valent (for example Ni/Au(lll)527). One must thus be very careful when extracting L>eff and J from experimentally determined Tk- Measured temperature dependence of the resonance-width agrees with theoretical predictions of broadening for Kondo impurities in the Fermi liquid (FL) regime.429'528 In the FL regime, one can make the following approximation, which holds in the particle-hole symmetric case: .94,228,528,543 A(uj,T) =----ImG>,T) = L7M 7T vr(cc;-(e(i + SßM))2 + S^M2; EÄ(w,T) SV,T) - + 7(w,T), (10.3) -y(u,T) = cT 'jj kBTK TV T where A(u, T) is the impurity spectral function at temperature T, L is the self-energy, and the coefficient c is 7r2/32 f« 0.3. A simple approximation for the resonance line-width is then528 FWHM = 2v/(7rkBT)2 + l/c(kBTK)2. (10.4) Fermi liquid theory is actually valid only for u,T Tk- A better test is thus the splitting of the Kondo resonance in strong magnetic field.439 The experimental spectra can be fitted with a Fano line 1+e2 ~i+ 1+ e2 UU-5J where e = (E — E0)/(T/2) is a rescaled energy, E0 is the resonance position, T its width, with parameter q that is typically between 0 and ~ l.541 The Fano line-shapes are plotted in Fig. 10.2 for a range of parameter q. Fits to Fano line shape reproduce the dip structure, the asymmetry and the shift in the minimum from EV.426 Small q <^. 1 corresponds to small coupling of the discrete state to the tunnel-current carrying continuum states, while large 5> 1 would imply strong tip-impurity coupling, either directly (tunneling of electrons into the localized d-orbital) or indirectly (tunneling of electrons to the d-orbital via the conduction band).427 The intermediate value q ~ 1 means that such coupling is small, yet significant. As we move away from the impurity, q decreases,4 and the line shape becomes more symmetric (for q =0, the line shape is a symmetric dip). Presently used models have difficulties in reproducing this feature.541'544'545 10.2.3 Microscopic theory Systems described by the quantum impurity models exhibit universality: their low temperature properties are described by a small number of parameters, often only two - the Kondo temperature and the quasiparticle scattering phase shift. In contrast, the line shapes found in the tunneling spectra measured over magnetic adatoms on surfaces are markedly dissimilar. It turns out that they depend on the details of the substrate-electronic-structure induced quantum interference phenomena. Phenomenological impurity models discussed in the previous section account well for the observed features. To gain more insight, theoretical predictions of the effective parameters starting from microscopic theories were performed.522'530'544~546 In particular, sophisticated 210CHAPTER 10. CLUSTERS OF MAGNETIC ADATOMS AND SURFACE KONDO EFFECT 0.4 0.3 f(q,e) 0.2 0.1 -3-2-10 1 2 3 e Figure 10.2: Fano resonance lines for a range of parameter q. I plot function f(q, e) = c ixS"' wnere c is some g-dependent normalization constant. Note that f(—q, —e) = /((?, e). therefore the Fano lines for negative q are mirror reflections of those for positive q. methods to calculate the hybridization matrix elements Vd,k were sought for;544'545 it was found early that these parameters are very sensitive to the details of the band structure,522 which unfortunately reduces the predictive power of simple theories. In addition, the d-level is shifted in energy by the surface potential; due to exponential dependence of Tk on td, a modest atomic level shift can lead to a big change in the Kondo temperature. To complicate matters further, in addition to the Kondo effect, the LDOS around the impurity is affected by the Friedel oscillations induced by the adatom.522 When a cobalt atom is adsorbed on a metal surface, its outmost s-wave electrons either gets transferred to the metal conduction band or to its own d orbital, therefore Co tends to form [Ar]3544 In addition, there are good reasons to believe that the relevant d-orbital is d 3z2_r2 and that the orbital motion is quenched by the broken symmetry at the surface (incidentally, this implies that the gyromagnetic ratio is g ~ 2). From this follows that the Co adatom can indeed be roughly modeled using the single impurity Anderson model. Ujsaghy et al. have studied the parameters that enter the effective model for cobalt on Au(lll).522 These can be used as simple modeling parameters and are given for reference purposes in Table 10.2. On-site Coulomb repulsion was taken to be proportional to the LSDA Stoner splitting. For the band cutoff (bulk and (111) surface bands of Au) an assumed value of D = 5.5eV can be used (the effective bandwidth is controlled by U and td anyhow). With these parameters, we get TK which is of the expected order of the magnitude. 10.2. THEORY OF THE SURFACE KONDO EFFECT 211 Parameter Value Broadening A = n\V\2p 0.2eV On-site Coulomb repulsion U 2.84 eV Orbital energy ta -0.84 eV Band cutoff D 5.5 eV Table 10.2: Effective parameters for the Anderson model of Co/Au(lll).522 10.2.4 Surface-state electrons or bulk conduction band electrons One of the main difficulties in microscopic modelling of the surface Kondo effect is determining the hybridization constants that enter the expression for the Kondo exchange interaction. Both bulk Bloch waves and surface-state electrons can play a role and, generally, both do. Most experiments are performed on the (111) facets of noble metals where bulk electrons coexist with Shockley surface-state electrons.547 In recent years, the question about which are predominant was widely debated. Clearly, the presence of the adatom induces mixing between the bulk and surface states525'545 and the d-level couples to the hybridized mixture of bulk and surface states; we may still be interested in the predominant character (bulk vs. surface) of these hybridized states. The available experimental results suggests a more important role of bulk states even on (111) surfaces.524'525'547'548 Recent studies based on DFT calculations also indicate much larger contribution from bulk states in Co/Cu(lll) system.545 The problem of a magnetic impurity on a surface is a two-band, but single channel problem, since only one effective channel couples to the impurity. In this respect it is similar to the quantum dot coupled to two distinct leads, but to a single channel of symmetric combination of wave-functions. If there are two bands coupled to a single S = 1/2 impurity orbital, a suitable effective model would be H = 2_^ ek c6,k/uCfc,k/u + 2_^ ekCs,k/uC*,k/u + "impurity k" k" _ (10-6) + E K {dW, + H.c.) + J2 Vk (dleko + H.c.) . kjt* kju with, for example, -ffjmpurity = E» ^4^ ~^~ ^n1ni- For definiteness we have assumed that the first band b corresponds to the substrate bulk band (3D), while the second band s corresponds to the surface-state band (2D). We assume that both bands are isotropic, the first in three-dimensions and the second in two-dimensions, i.e. ek depends only on k = |k|. It should be noted that k in 3D and k in 2D are not comparable. In particular, the Fermi moments are different, kbF ^ ksF. We can furthermore safely assume that Vk depends solely on the magnitude of k, but not on its direction, i.e. only the s-wave symmetric combination of surface-state electrons 212CHAPTER 10. CLUSTERS OF MAGNETIC ADATOMS AND SURFACE KONDO EFFECT couples to the dot. For VL such an approximation cannot be made since the hybridization clearly strongly depends on the direction of k. Nevertheless, we can compute the suitable angular averages of VL and label them by VL. The impurity only couples to the one-dimensional manifold of direction-averaged combinations of states. We then transform from momentum-space to energy-space by defining ae = {dtk/dk)~l/2ak when e = efc. We obtain (sums over repeated spin indexes are implied) r-Db r-Ds tl — / Ld{, e„^6,e/iCie + / tQ,s^€^j(Xs^ß(lt "T" ^impurity J-Dh J-Ds -Db rDs -Db fDb fDs + / [pb(e)]l/2Vb(e) (dlab>eß + H.c.) de + / [ps(e)]l/2Vs(e) (^aw + H.c.) de. J-Db J-Ds (10.7) Now comes the crucial step: at each energy e, the impurity couples only to the "symmetric" combination f _ Pb Vbab,eß + pJ Vsas,eß while it is decoupled from the "antisymmetric" combination: (10.8) 1/2T, 1/2T, VpWTp^v? { ) We choose a new cut-off D = m&x(Db,Ds), define pV2 = psV2 + pbVb2, and write the effective single-channel Anderson model: /D rD tflUde + Himpmity + / v7^ (dlU + H-c-) de- (10-10) D J-D Thus p\V\2 and consequently pJ are additive quantities. 10.3 NRG calculations Numerical renormalization group is a powerful technique for computing spectral functions (local densities of states) in problems where many-particle effects are important and need to be properly taken into account. It may be applied in the field of tunneling spectroscopy whenever the problem can be reduced to a quantum impurity problem with a small number of impurity orbitals and one or two conduction channels. NRG is particularly suited for studying the physics of Kondo impurities adsorbed on surfaces where the relevant magnetic orbital is typically a single d-level and there may be a small number of broad sp-derived levels near the Fermi level, while all other atomic levels away from the Fermi level 10.3. NRG CALCULATIONS 213 are irrelevant: such atoms can be accurately modeled by some multi-level Anderson-like impurity model. To my knowledge, the impurity sp orbitals are entirely neglected in all phenomenological as well as most microscopic theories that had been applied to the surface Kondo problem, even though these orbitals may modulate the density of states that the d-level couples to. Furthermore, these orbitals can carry a portion of the tunneling current when the STM tip is located directly above the impurity atoms. The closer the energy of these orbitals is to the Fermi level, the more important role they play. In this section I show by NRG calculations that a) the anti-resonance in the differential conductance is a direct consequence of the resonance on the d-level, b) the anti-resonance diminishes and splits in applied magnetic field, c) the form of the anti-resonance depends in an essential way on the proportions of the tunneling current that flows in the adsorbate s-level and in the substrate band, d) the s-level modulates the density of states to which the impurity couples, therefore the changes in the s-level energy can drive the Kondo temperature to extremely small values. The tunneling current is given by (see Sec. 5.3.2) /= ^y"de(/substrate(e) - /tip(e))Tr[rtip'"(e)ImGr"(e)]. (10.11) Here Ttip is the hybridization matrix between the STM tip states and the adsorbate levels, while Gr is the retarded Green's function matrix of the adsorbate. At zero temperature, the Fermi-Dirac distribution functions become /substrate (e) = 0{tp — t), /tiP(e) = 9(ep + eV — e), where tp is the Fermi level and V the voltage drop in the gap between the STM tip and the adsorbate; for V > 0 electrons flow from the tip to the sample. We then obtain d//dF=-Go/2j]Tr[rti^(e)ImG^(e)]e=eF+ey. (10.12) ß The imaginary part of the Green's function Gr is essentially orbitally-resolved local density of states of the impurity, while the hybridization matrix Ttip encodes both the density of states in the tip and the tunneling probability to various orbitals. The tunneling probability depends on the orbitals at play: the tunneling to the d-level is expected to be negligible or small, while the tunneling to the s-level is substantial. In addition, part of the tunneling current inevitably flows directly into the substrate states. This possibility is not taken into account in Meir-Wingreen formula and therefore neither in Eqs. (10.11) and (10.12). In Bardeen's theory, the tip-substrate current is proportional to a square of the surface integral over a separation surface between the tip and the sample353 m=~^L J (x*w" ^Vx*]'ds' (io-i3) where if) is the sample conduction band wave-function and \ the tip wave-function. The simplest approximation consists of assuming that the tip-substrate wave-function overlap 2UCHAPTER 10. CLUSTERS OF MAGNETIC ADATOMS AND SURFACE KONDO EFFECT is largest at the position of the impurity atom. From the perspective of NRG calculations, this approximation corresponds to assuming that the tip probes the Wannier orbital /o of the Wilson chain529 . The idea here is to consider /0 as part of the impurity system: Eq. (10.12) then still applies, but the trace over impurity orbitals is extended to include the /o orbital. The tunneling current then flows to some linear combination of orbitals, described by an operator such as h) oc tsd\ß + tf0fl , where ts and tf0 are the tunneling amplitudes for tunneling in either the s-level or in the substrate electron band.529'530 Equation (10.12) simplifies to dl/dV = Go/2 J]7r[rtip(e)AJe=LF+ey, (10.14) ß where Ttip(e) = 7rptiP(e)(^ + */0)- We take ts ano- tf0 to be energy-independent in the relevant energy range around the Fermi level, but this assumption may be relaxed. It should be emphasized that in this formalism the sum over different tunneling paths is performed coherently and that all quantum interference effects are taken into account. The quantity of central interest is thus the spectral function of the effective "orbital h", Before focusing on the two-level impurity model, we first study the conventional description of the surface Kondo effect using a single d-level Anderson model:30 U ~2 (2TD\l/2 (10.15) #imp = 5(nd -l) + —(nd- if By writing the coupling Hamiltonian Hc in this form, we assumed that the d-level hybridizes with the conduction band orbital centered at the impurity site, /0. Furthermore, we assume that when the STM tip is directly above the adsorbate the tunneling current flows only to /o, i.e. k^ = /o^. The impurity d-level spectral functions and the conduction band LDOS (or, equivalently, the spectral function of the /o orbital) are shown in Fig. 10.3. In the absence of the impurity, the conduction band LDOS is flat near the Fermi level, Af0(u) = 1/(2D). When the impurity is introduced, the impurity spectral function is mirrored in the conduction band LDOS. The following relation holds for a flat band at j° _ Q.548 Af0(uj) = A0(l-7rTAd(uj)). (10.16) The conduction band LDOS is thus exactly zero at the Fermi level in the case of symmetric Anderson model (5 =0). When the Anderson impurity is asymmetric (6 ^ 0), the Kondo resonance is also asymmetric and so is the anti-resonance, similar to what is found in the experimental tunneling spectra. It is known that applied magnetic field splits the Kondo resonance. This is expected to be reflected in a diminished Fano anti-resonance for small fields, and a double dip anti-resonance for strong fields, Fig. 10.4. To this date no experimental results of such splitting 10.3. NRG CALCULATIONS 215 0.5 3 ^0.5 U «f 0.5 (a) No impurity 1 1 (b) 8=0 (c) 8/U=0.2 U/D=0.5 r/U=0.04 0.5 0.4 0.3 0.2 0.1 0 0.5 0.4 0.3 0.2 0.1 0 0.5 0.4 0.3 0.2 0.1 -0.5 -0.25 0 ed,U. While it might appear that everything is already known about single-impurity models, I would like to point out that, for example, models where an impurity is coupled to bosonic bath (or even both fermionic and bosonic bath at the same time) only recently came under 226 CHAPTER 11. CONCLUSIONS scrutiny by bosonic generalization of NRG. Two-impurity models Systems of two impurities such as double quantum dots (DQD) are the simplest systems where the competition between magnetic ordering and Kondo screening can be studied. Parallel double quantum dots (and iV-dot generalizations) can be described by the multi-impurity Anderson model. At low temperature, this model maps to a multi-impurity Kondo model. It is found that the conduction-band-mediated RKKY exchange interaction is ferromagnetic, Jrkky ~ U(pJr)2 = (64/7r2)(r2/[7), therefore the impurity spins order and the system effectively behaves as a single-impurity S = N/2 Kondo model which undergoes S = N/2 Kondo effect. The Kondo temperature is the same irrespective of the number of the impurities N. The residual spin at zero-temperature is N/2 — 1/2 if there is no coupling to additional screening channels. The ferromagnetically ordered regime and the ensuing S = N/2 Kondo effect are fairly robust against various perturbations. Very strong perturbations lead, however, to quantum phase transitions (QPT) of different kinds, which have been meticulously studied for the N =2 case. If the parallel DQD system is moved from the particle-hole symmetric point by equally increasing both on-site energies, a QPT of the first kind between S = 1/2 and S = 0 ground states occurs. This transition is triggered by charge fluctuations which compete with RKKY ferromagnetic ordering; the ferromagnetic ordering temperature drops to zero as an exponential function of the on-site energies. If symmetric splitting between the two on-site energy levels is induced, a singlet-triplet QPT of the Kosterlitz-Thouless kind occurs. This is a transition between an inter-impurity triplet and a local spin-singlet on one of the impurities. The cross-over temperature is an exponential function T* oc exp[—Tk/Ju] where Ju is an exchange-interaction between two fictitious spins which itself is an exponential function of the energy splitting. The ferromagnetic ordering can be compensated by direct antiferromagnetic inter-impurity exchange interaction. The value of the RKKY interaction can be extracted in this manner and it is found to agree with the analytical estimates derived by fourth-order perturbation theory in hybridization parameters. Hopping between the impurities has similar effect via the superexchange mechanism. The critical hopping parameter t12)C is of the order of the hybridization T and does not depend on the value of U. It is interesting that the effect of the level splitting oc t is less pronounced than the effect of the induced exchange interaction oct2. If the inter-impurity electron repulsion Uu is increased to U, the decoupled impurities have SU(4) symmetry, since an additional SU(2)orb orbital pseudo-spin symmetry is established. The SU(4) symmetry is broken by the coupling to the conduction channel, therefore there is no SU(4) Kondo effect. Instead, the effective level degeneracy decreases from 6 (SU(4) 227 sextuplet) to 4 (product of two spin-doublets) on the symmetry-breaking scale, then from 4 to 2 in a S = 1/2 SU(2) Kondo effect; there is a double degeneracy of the ground state due to an effectively decoupled electron in the ungerade molecular orbital. In the case of unequal coupling of the two impurities to the conduction channel, the high energy singlet state admixes with the ground triplet states; second order scaling calculation gives a good estimate of the Kondo temperature in this case. For very strong asymmetry, the ferromagnetic locking-in is destroyed and the single-impurity S = 1/2 Kondo effect occurs with the more strongly coupled impurity while the other spin remains uncompensated. The behavior of the double quantum dot system in the side-coupled configuration strongly depends on the value of the inter-impurity hopping. For strong hopping, the system maps to an effective single-impurity Anderson model where the role of the impurity orbital is played by the bonding or anti-bonding molecular orbital. The use of eigenvalue diagrams for studying the low-temperature behavior was demonstrated: the Kondo temperatures can be accurately estimated from the energies of the lowest excited states of the double dot. Two wide Kondo plateaus are expected in the conductance as a function of the on-site energies in this regime. The spectral functions must be computed using the density-matrix NRG in this model; the conventional approach leads to spurious discontinuities and to the violation of the spectral sum rule. Density-matrix NRG in the basis with well defined charge Q and total spin S quantum numbers has been implemented for this purpose. For weak coupling, the two-stage Kondo effect occurs: the spin on the directly coupled impurity is screened at a higher Kondo temperature T^), while the spin on the side- (2) coupled impurity is screened at an exponentially reduced lower Kondo temperature TK'. The conductance can be high at finite temperatures even in the vicinity of the particle-hole symmetric point if TK e^,U pri ed — gßß\B\/2 in ed + U + gßß\B\/2. Čeprav bi nemara lahko pričakovali, da o modelih ene nečistoče vemo domala vse, to ne drži. Izpostavil bi rad, da se je, denimo, šele nedavno pričelo temeljito proučevanje modelov, pri katerih je nečistoča sklopljena z bozonsko kopeljo (ali celo sočasno s fermionsko in bozonsko kopeljo), z uporabo bozonske posplošitve NRG 236 POGLAVJE 12. POVZETEK DISERTACIJE V SLOVENSKEM JEZIKU Modeli dveh nečistoč Sistemi dveh nečistoč, kot je dvojna kvantna pika (DQD), so najpreprostejši sistemi, kjer lahko proučujemo tekmovanje med magnetnim urejanjem in Kondovim senčenjem. Vzporedno dvojno kvantno piko (in posplošitve na N vzporednih pik) lahko opišemo z Andersonovim modelom več nečistoč. Pri nizkih temperaturah se ta model preslika na Kondov model več nečistoč. Izkaže se, da je efektivna izmenjalna interakcija med nečistočami, ki se prenaša prek prevodniškega pasu (RKKY), feromagnetna, Jrkky ~ U(pJx)2 = (64/7T2)(r2/[7). Spini nečistoč se zato uredijo in sistem se efektivno obnaša kot Kondov model ene same nečistoče z velikim spinom, S = N/2, pri katerem pride do S = N/2 Kondovega pojava. Kondova temperatura je vselej enaka, ne glede na število nečistoč. Rezidualni spin je N/2 — 1/2, če le ni sklopitve na dodatne senčitvene kanale. Režim fe-romagnetne urejenosti in posledični S = N/2 Kondov pojav sta dokaj robustna napram različnim motnjam. Zelo močne motnje vseeno vodijo h kvantnim faznim prehodom različnih vrst, ki smo jih bolj natančno proučili v primeru N =2. Če v vzporedni dvojni kvantni piki odpravimo simetrijo med delci in vrzelmi tako, da obema nečistočama povišamo energijo v enaki meri, pride do kvantnega faznega prehoda prve vrste med osnovnima stanjema s spinom S = 1/2 oziroma S = 0. Ta prehod povzročijo fluktuacije naboja, ki tekmujejo s feromagnetnim urejanjem preko mehanizma RKKY: temperatura feromagnetne ureditve pada proti nič kot eksponentna funkcija energij nečistoč. Če vzpostavimo simetričen razcep energij na obeh nečistočah, pride do Kosterlitz-Thoulessovega kvantnega faznega prehoda med singletom in tripletom. To je prehod med kolektivnim tri-pletnim stanjem elektronov na obeh nečistočah in lokalnim singletnim stanjem na nečistoči z nižjo energijo. Temperatura prehoda je eksponentna funkcija T* oc exp[—Tk/Ju], kjer je Ji2 izmenjalna interakcija med namišljenima spinoma, ki je eksponentno odvisna od energijskega razcepa. Če odboj med elektroni na obeh nečistočah (kapacitivna sklopitev) povečamo do Uu = U. imata izolirani nečistoči simetrijo SU(4), saj se vzpostavi dodatna orbitalna psevdospinska simetrija SU(2)orb. Simetrijo SU(4) zlomi sklopitev nečistoč na prevodniški pas, zato ne pride do Kondovega pojava vrste SU(4). Namesto tega pade efektivna degeneracija s 6 (sekstuplet SU(4)) na 4 (produkt dveh spinskih dubletov) na skali zloma simetrije, nato pa s 4 na 2 ob Kondovem pojavu tipa SU(2) s spinom S = 1/2. Osnovno stanje ostane dvakrat degenerirano, saj v lihi molekularni orbitah ostane nezasenčen elektron, ki je povsem razklopljen od preostalega sistema. V primeru neenakega sklapljanja obeh nečistoč na prevodniški pas, se višje-energijsko sin-gletno stanje primeša k osnovnemu tripletnemu stanju. V tem primeru dobimo dober približek za Kondovo temperaturo z uporabo skaliranja drugega reda. Pri zelo močni asimetriji se feromagnetna ureditev poruši in pride do Kondovega pojava z S = 1/2 na nečistoči, ki je močneje sklopljena, medtem ko ostane drugi spin nezasenčen. 237 Lastnosti stransko sklopljene dvojne kvantne pike so močno odvisne od sklopitve med nečistočama, torej od vrednosti parametra skakanja. Pri močni sklopitvi se sistem preslika na efektiven Andersonov model za eno nečistočo, pri čemer igra vlogo orbitale vezavna ali anti-vezavna molekularna orbitala. Predstavljena je bila uporaba diagramov lastnih stanj za napovedovanje obnašanja pri nizkih temperaturah: Kondovo temperaturo lahko dokaj natančno določimo iz energij najnižjih vzbujenih stanj dvojne kvantne pike. V tem režimu pričakujemo dve široki Kondovi planoti v prevodnosti kot funkciji energije nečistoč. Spektralne funkcije moramo v tem modelu računati z metodo density-matrix (DM) NRG, saj običajni pristop vodi k nefizikalnim nezveznostim in h kršitvi vsotnega pravila. V ta namen sem izpeljal DM NRG v bazi dobro določenega naboja Q in celotnega spina S. V primeru šibke sklopitve med pikama pride do dvostopenjskega Kondovega pojava: spin na neposredno sklopljeni nečistoči je zasenčen pri višji Kondovi temperaturi T^), medtem ko je spin na stransko sklopljeni nečistoči zasenčen pri eksponentno nižji Kondovi temperaturi (2) TK ¦ Prevodnost je lahko pri končnih temperaturah visoka tudi v bližini točke, kjer obstaja simetrija med delci in vrzelmi, če je le izpolnjen pogoj TK U) v režim antiferomagnetne spinske verige (J ~ t), in nato v režim dvostopenjskega Kondovega pojava (J < TK ). V režimu antiferomagnetne spinske verige (AFM) se trije spini uredijo pri T ~ J v togo dubletno stanje, pri nižjih temperaturah pa je ta kolektivni spin senčen v običajnem S = 1/2 Kondovem pojavu. V režimu dvostopenjskega Kondovega pojava (TSK) se spina na prvem in tretjem mestu zasenčita pri višji Kondovi temperaturi T^', spin na sredinskem mestu pa pri eksponentno nižji drugi Kondovi temperaturi Tk oc Tk exp(—cTk /J), kjer je J = At2/U. Režima AFM in TSK sta ločena s prehodnim območjem, v katerem se sistem pri končnih temperaturah približa fiksni točki dvokanalnega Kondovega modela, ki ima lastnosti ne-Fermijeve tekočine (NFL). Režim NFL je robusten napram različnim motnjam, kot so, denimo, zlom simetrije med delci in vrzelmi, zlom parnosti in neenak odboj med elektroni na različnih mestih. Edine "nevarne" motnje so tiste, ki dodatno povečajo asimetrijo med kanaloma. V točki, kjer obstaja simetrija med delci in vrzelmi, prevodnost naraste na unitarno mejo pri T =0 za poljubno od nič različno vrednost parametra skakanja med pikami t. Poleg tega se izkaže, da je stabilna nizkotemperaturna fiksna točka enaka za vse t: v lihem kanalu je fazni premik ir/2, v sodem kanalu pa faznega premika ni. Fiziko ne-Fermijeve tekočine lahko zaznamo s primerjanjem prevodnosti skozi eno izmed stranskih pik in skozi celoten sistem. Prevodnost skozi stranski piki naraste na G0/2, medtem ko je prevodnost skozi sistem še vedno nizka, ko temperatura pade pod Kondovo temperaturo. Ko preidemo iz fiksne točko NFL v fiksno točko Fermijeve tekočine, naraste prevodnost skozi sistem na Go- To je v skladu z opažanjem, da je prenos naboja iz enega kanala v drugega natanko tista motnja, ki destabilizira fiksno točko ne-Fermijeve tekočine. Spreminjanje spektralnih funkcij v odvisnosti od parametra skakanja t prikazuje, kako se spektralni vrh molekularne orbitale razvije v vrh pri u = J (režim AFM), ta pa nato v prvi Kondov vrh (režim TSK). Režim dvostopenjskega Kondovega pojava se razteza od območja trojne zasedenosti do območja dvojne zasedenosti. V bližini dvojne zasedenosti je spin elektrona na sredinskem mestu senčen preko sklopitve na kvazidelce, ki izvirajo iz stanj valenčne fluktuacije na obeh stranskih mestih. Natančno proučevanje lastnosti bolj splošnih modelov treh nečistoč je šele v povojih in prostor modelskih parametrov, ki ga moramo raziskati, je ogromen. Zelo zanimiv sistem je gotovo trikotnik iz treh kvantnih pik, sklopljen na dva prevodniška kanala. V odvisnosti od parametrov se lahko ta sistem obnaša, denimo, kot dvokanalni Kondov model ene nečistoče, kot Kondov model dveh nečistoč, ali pa kot frustriran antiferomagnet. 239 Nizkotemperaturni vrstični tunelski mikroskop Vrstični tunelski mikroskop (STM) je vsestransko orodje na področju nanotehnologije. Lahko ga uporabimo za sestavljanje in karakterizacijo nanostruktur: tunelska spektroskopija nudi vpogled tako v elektronske kot v vibracijske lastnosti adsorbatov. Sestavili smo s tekočim helijem hlajeni nizkotemperaturni STM, ki deluje pri 5.9 K. Instrument doseže atomsko ločljivost na površinah kovin z zelo nizko korugacijo, kot je Cu(lll). Glavo mikroskopa smo priredili tako, da lahko vanjo vstavimo vzorce, pritrjene na Omi-cronove nosilce. Tako smo ohranili združljivost z že obstoječo opremo. Sestavljanje instrumenta sem opisal s številnimi podrobnostmi in podal sem dve metodi za preizkušanje in karakterizacijo vibracijskega in električnega obnašanja skenerja: meritev dvojnega piezo odziva in spektralno analizo tunelskega toka. Najnižjo mehansko resonanco najdemo pri 900Hz, kar je primerljivo z vrednostmi podobnih skenerjev, ki jih uporabljajo drugod. Konice pripravljamo z elektrokemijskim jedkanjem volframove žice v raztopini KOH s pomočjo doma narejenega elektronskega vezja, ki hitro prekine tok, ko je jedkanje končano. Konice nato očistimo v fluorovodikovi kislini, da odstranimo volframove okside, in jo dodatno izboljšamo in-situ z nadzorovanim zaletavanjem konice v mehko površino. Sestavili smo mehanizem za prenos vzorca iz komore za shranjevanje vzorcev v glavo mikroskopa. Da se prepreči hladno privarjenje, mehanizem uporablja vakuumsko združljive kroglične ležaje in trde prevleke na površinah, kjer prihaja do trenja. Da opazovana površina ostane čista, je mikroskop vgrajen v ultravisoko vakuumski sistem. Ultravisoki vakuum (UHV) dobimo z večstopenjskim črpanjem in s pregrevanjem sistema na 105 °C. Da bi bil končni pritisk čim nižji, je ključnega pomena kar se da zmanjšati degasiranje z uporabo primernih materialov in čimbolj omejiti notranjo površino sistema. Prostornina sistema vpliva predvsem na čas črpanja, da dosežemo UHV. Čisto in urejeno površino vzorca dobimo z izmeničnim ionskim sputtranjem in popuščanjem vzorca. Ta postopek smo računalniško avtomatizirali, stanje površine pa nadzorujemo z uklonom nizkoenergijskih elektronov in spektrostopijo Augerjevih elektronov. Pri zelo kvalitetnih vzorcih dobimo površino, ki je zadosti dobra za tunelsko mikroskopijo, v manj kot desetih ciklih čiščenja. Različne adsorbate nato naprašimo na površino z uporabo Knudsenove celice. Površine Cu(lll) in Cu(211) smo slikali z atomsko ločljivostjo. Opazovali smo stoječa valovanja površinskih elektronov (energijsko razločene Friedelove oscilacije) in difuzijo molekul CO na Cu(lll). Dinamične spremembe stanja konice so razvidne iz sprememb kvalitete slik med skeniranjem. Pred kratkim smo konstruirali nov kriostat na helijevo kopel, ki bo imel daljši čas pred potrebnim dotakanjem kriogenih tekočin. Poleg tega bo mikroskopska glava dosegla nižje temperature, saj bo sistem bolje izoliran, žice pa bodo bolje toplotno sidrane. Kristat bo imel boljše možnosti za nastavljanje in ga bo lažje vzdrževati. 240 POGLAVJE 12. POVZETEK DISERTACIJE V SLOVENSKEM JEZIKU Skupki magnetnih nečistoč in Kondov pojav na površinah Magnetne lastnosti skupkov adsorbatov lahko proučujemo s tunelskim mikroskopom tudi brez spinsko polarizirane konice. Možen pristop je, denimo, zaznavanje značilne posledice Kondovega pojava, ki se pojavi v obliki ozke antiresonance v bližini Fermijevega nivoja v diferencialnem tunelskem spektru dl/dV. Do Kondovega pojava na površinah lahko pride v posameznih magnetnih adatomih, di-merih in trimerih, v adsorbiranih molekulah, ki vključujejo magnetni ion, ter v skupkih kovinskih atomov na ogljikovih nanocevkah. Interakcije med nečistočami lahko proučujemo tako, da dva adatoma postopno približujemo, dokler se ne povežeta v skupek, ob tem pa spremljamo, kako se spreminja oblika antiresonance v spektrih. Pri primerjavah meritev s teoretičnimi izračuni je potrebno najti povezavo med širino antiresonance ter pravo Kondovo temperaturo sistema. Antiresonanco v tunelskem spektru najlaže razumemo, če upoštevamo, da so valovne funckije orbital d močno lokalizirane, zato večina tunelskega toka teče v orbitale sp ad-sorbata, ki so močno hibridizirane s pasom prevodniških elektronov v podlagi. Tunelski spekter je zato v prvem približku sorazmeren kar s spektralno funkcijo prevodniškega pasu na mestu nečistoče. Še boljši približek je, če nečistočo opišemo z dvonivojskim modelom, ker upoštevamo tako orbitalo d kot orbitalo s; orbitala s je močno hibridizirana s konti-nuom, medtem ko se orbitala d sklaplja z orbitalo s predvsem preko izmenjalne interakcije. Še vedno primanjkujejo eksperimentalni podatki pri spremenljivih temperaturah, predvsem v območju milikelvinov, ter v spremenljivih magnetnih poljih. Ti podatki bi lahko nudili odgovore na številna odprta vprašanja. Na področju teorije bi bilo vredno temeljito proučiti ustrezne večnivojske modele kvantnih nečistoč, predvsem njihove razširitve na dve nečistoči. To je težaven dvokanalni štirinivojski problem, ki pa je verjetno v dosegu zmogljivosti današnjih računalnikov in implementacij metode NRG. Part V Appendices 241 Appendix A Tensor operators and Wigner-Eckart theorem We consider a spherical tensor operator O of rank M with 2M +1 components indexed by ß = —M,..., -\-M. By definition, it satisfies the following relations [Jz,Off]=ßOff [j+,Off]=A(M,ß)Off+1 (A.l) [j.,OJf]=A(M,-fJL)OJf_1 where J are the generators of an SU(2) symmetry group and A(M, ß) = \J[M — ß)[M + ß + 1). The eigenstates of a system with SU(2) symmetry then satisfy the Wigner-Eckart theorem: M,3z\0"\otj,3z) = (ffz;Mß\jjz)(a,j\\0M\\a',f), (A.2) where j,f are spin quantum numbers, jz,j'z are projection (magnetic) quantum numbers and a, a' are additional non-angular quantum numbers. {j1jz1]J2Jz2\JJz) are SU(2) Clebsch-Gordan coefficients. The theorem is proved by inserting the defining commutators (A.l) between bras and kets and operating with the generators J to the left on bras and to the right on kets. The system of equations thus obtained is identical to the generating equations for the Clebsch-Gordan coefficients. 242 Appendix B Green's functions This appendix recapitulates essential formulas from many-particle physics and sets notation for the main text. An operator in the Heisenberg representation is defined by 0{t) = etHtO(0)e-tHt (B.l) or by the equation of motion il0(t) = [0(t),H]. (B.2) In general (and in particular for non-equilibrium problems) we need to define several Green's functions:280'550 G>(xi,x2) = —i(ip(xi)ip^ (x2)) G<(xi,x2) = i{4>Jf(x2)i>(xi)) G\xux2) = 6(ti -t2)G>(xi,x2) + 0(t2-ti)G<(xi,x2) Gt(xi,x2) = 0(t2-ti)G>(xi,x2) + 0(ti-t2)G<(xi,x2) - (B*3) = 9(h - t2) (G>(x1,x2) - G<(x1,x2)) = -lOitr - t2){tp(x1)^(x2) + ^2)^1)) Gr {Xl,X2) = Gr — Gr = Gr — Gr . G7> is greater Green's function (GF), G7< is lesser GF, G7* is time-ordered GF, G7* is anti-time-ordered GF, Gr is retarded GF and Ga is advanced GF. Here X\ stands for time t\. position ri and any other quantum numbers such as spin. Note that the Green's functions are defined in the Heisenberg representation. At nonzero temperatures, the retarded Green's function for systems in steady state can be expressed as: G\ß - t') = -i6(t - t')Tr (e-ftK-V[ci(t)c]tf) + ct(i')ci(i)]) (B.4) 243 244 APPENDIX B. GREEN'S FUNCTIONS Here and in the following i, j stand for any quantum numbers, in particular for the site number and the spin of electron. K = H — /iN, Ci(t) = elKtCi(0)e~lKt and Q is the grand-canonical partition function. We set t' =0, so that Gr depends only on t and we perform a Fourier transform to the energy (frequency) space: /DO dte^Glß). (B.5) ¦oo The spectral representation in terms of the eigenstates defined by K\m) = Em\m) can be derived as follows: GIß) = -iO{t)eßQ^{n\e-ßKCi{t)\m){m\c){ti)\n) + {n\e-ßKc){ti)\m){m\ci{t)\n) n,m = -i0(t)eßnJ2e~ßE" (^t(En~Em){n\ci\m){m\c]\v) + ea(Em-En){n\c]\m)(m\d\n) n,m = -ie(t)eßn^e(En-Em^n\ci\m){m\c]\n) (e~ßE" + e~ßEm) n,m (B.6) and after Fourier transforming we obtain ____ / j \ * j p—ßEn I p — ßEm G"» = «*> L (He!|n>) H^^ + tf. (B.7) n,m For i = j, we have f (m|q|n)) (m|ct|n) = |(m|q|n)|2. Plemelj formula —j^j = P^:—in5{x) can then be used to obtain the imaginary part: Imq» = -7reßn J] |a{uj). In the zero-temperature case we obtain: A(u > 0) = i Yl \{m\ct\no)\26 (u - Em) m,no A(uj < 0) = - Y, \{mo\c]M)\25 (to + En) (B.10) mo,n Z = Z> 245 where indexes TO0,n0 indicate summation over the eventually degenerate ground states, while indexes m and n correspond to sums over all states. Z is the partition function. For i ^ j, we consider the symmetrized function GL • + G^. Denoting the braket product f(m|ct|n)j {m\Cj\n} by a, we see that the replacement i <^> j yields a*. The symmetric sum of Green's functions then features a real factor a + a* = 2Rea and we can again use the Plemelj formula to obtain Im (GJ>)+ GJ») = e8n Y^ 2Re \((m\4\n})* Hcjirc)] (e~ßEn + e~ßEm) 5 (u + En - Em). (B.n; We define mixed spectral function MA") = "^Im (G r Gli) ¦ Specializing to T =0 we obtain: 1 Aij(uj > 0) = — J^ Re Um\cl\n0}) (m\c]\nQ) 5 (u - Em) m,no r- / \ if. Aitj(uj < 0) = — Y^ Re ({mo\cl\n)) {m0\c]\n) 5 (u + En) (B.12) (B.13) Note that there is a sum rule dujAijiuj) = 5. S3: (B.14) which is a consequence of the anticommutation relation for fermionic operators CiCj + CjCi = kj- Appendix C Generalized Schrieffer-Wolff transformation In this section I derive the Schrieffer-Wolff transformation to the Kondo model applicable to the systems of multiple impurities. The derivation is based on the approach of Ref. 551. We separate the Hamiltonian into a Hamiltonian of the isolated dots and conduction band H0 and a coupling Hamiltonian H1\ H0 = ^2^kcltack}a + ^2Ea\a){a\ (C.l) k,a a H1= Yl *H/^° \a)(ß\ ck,v + t*ko\ß^Jko \ß)(a\. (C.2) k,a,a,ß where multi-indexes a and ß stand for the quantum numbers, for example the set (Q, S, Sz,u). Coefficient t ka\ß^a correspond to electron excitations on the dots, while t*k(T^^a correspond to hole excitations on the dots. We perform a unitary transformation so that the transformed Hamiltonian will contain no operators linear in coupling coefficients t. H = epHe-p = H + [P, H]+ i [P, [P, H}} + • • • (C.3) H = H0 + H1 + [P, H0] + [P, H1} + 0(V3). (CA) We require [P, H0] = —H1, so that we will obtain H = H0 + [P, H1]. (C.5) Let I a) and |6) be two eigenstates of H0. We get (a\H0P - PH0\b) = {a\H1\b}, (C.6) 246 247 hence (a\P\b) = -^f, (C.7) where Ea and L& are eigenenergies. By premultiplying by \a) and postmultiplying by (b\ and summing, we obtain p = L#%i^i- (c-8) a,b Hamiltonian H\ connects states that differ by one electron in the dot region, AQ = ±1 and AS = ±|. For transitions where an electron is added to the dots, we write a = (a, 0) and b = (ß, ka), where multi-indexes ß = (Q,S,Sz,u) and a = (Q + 1, S + r, Sz + a, u/) correspond to eigenstates of isolated dots, r = ±1/2, and the energy difference is Ea — Eß — tk- For transitions where an electron is removed from the dots, we write a = (a, ka) and b = (ß, 0) with a = (Q — 1,S — r, Sz — a, u/) and the energy difference is Ea + tk — Eß. Taking into account the definition of H\, we then obtain \i t kalß^Ea -Eß- tk°^ + t k°^Ea -Eß + ekC P- 2_^ t ka\ß^a-^-------^T-----—c k,a + t*ka\a_ß—-------F_ , _ Ck,cf (C-9) a,ß,ko Note that some |cü)(/3| projectors contain all-in-all an odd number of fermionic operators. because the electron number changes by ±1. Then cka\a)(ß\ = — \a)(ß\cka. Now we need to compute the commutator [P,Hi\. Outside charge fluctuation regions, we may disregard two-electron hopping terms such as ------Ck'a'Cka, (CIO) L>a — Üß — Lfc because they only contribute as high-order corrections. The only relevant terms are those that correspond to electron/hole hopping on and off the dots which are initially in the ground state. These terms are v' ß Ivt h'rr> \ @ '- 1 '—' v n/ß — En — tk Ea — En — tu Ck'o'Cko -tk^afk,a,^a,\a)(a>\ ( Eß-Ea + tk + Eß-Ea + t'k ) '»"^ (en; The states \a) and \a!) correspond to degenerate ground states, which may in some cases correspond to different (Q,S,u) multiplets. These clearly cannot be mapped to a simple Kondo-like problem, so from here on we assume that there is a single ground multiplet. We're especially interested in terms with a = —a'. Then Sz(a') = Sz(a) + la and la')(a raises or lowers spin by 1. This corresponds to S+a~ + S~a+ term in the Kondo Hamiltonian. Due to spin isotropy, there must also exist corresponding Szaz terms for a = a' which lead to S • er expression. 248 APPENDIX C. GENERALIZED SCHRIEFFER-WOLFF TRANSFORMATION There are also terms with a = a', which correspond to mere energy renormalization of the ground and excited states, and, in the absence of the particle-hole symmetry, potential scattering terms, that we disregard. Finally, the effective Hamiltonian is Heft = Ho + 2_^ Jkk'S • (cfc'fi'2(T/''mllCfcf (C.12) kk' ßß' At low temperatures, only scattering near the Fermi energy is important, and we may use Jkk> ~ J = JkF,kF, where J= 8\tkF*\ a^ß\\ (c_13) Eß — Ea The final expression is then #eff = #o + JS ¦ s. (C.14) Appendix D Scaling equations to second order in J In this section Hubbard's X-operator notation is used. This notation is defined by Xp;q = \p)(q\, where \p) and \q) are states in a complete set of many particle states. The diagonal elements Xp;p are projection operators for the state \p). Products of operators can be contracted: X p;q X p/;q/ = 5p/yq X p;q/. (D.l) We consider an effective Hamiltonian of the form H = 2_^ CkCkaCka + 2_^ EmXmm + 2_^ Jmm'Xmm'CkaCk'a', (D-2) fccr m mm',kk',aa' where J™m> are generalized exchange constants. We write2 H11 + H12 (E - H22)-1 H21 + H10 (E - H00)-1 tf01]^ 1 == Ef1, (D.3) where subspaces 2 corresponds to states with one electron in the upper \5D\ edge of the conduction band, 0 corresponds to states with one hole in the lower \5D\ edge of the band. and 1 corresponds to states with no excitations in the edges that are being traced-over. Furthermore, Hij = PiHPj, where Pi are projectors to the corresponding subspaces i. After some tedious algebra, we find that to second order the coupling constant are changed by *—* h — u + tk — h,n — n0 *—* h — u — ty — h,n — n0 nr nr (D.4) 249 Appendix E Transformations of band and coupling Hamiltonians In this appendix I map the conduction-band and coupling Hamiltonians in discrete form (sum over wave-numbers k) to a continuous Hamiltonian in the energy space that is used in NRG. The derivation roughly follows that of Ref. 30. The main difference is that I specialize from the outset to a one-dimensional conduction channel such as a tight-binding chain. Furthermore, the wave-number k ranges from 0 to ir, whereas in bulk each component of the wave vector ranges from —n to n. The Hamiltonian of a one-dimensional single-mode conduction lead with a coupling term is H' = Hhand + HC = Y, ejfeC^Cfc/i + Y, Mclßdß + H.c). (E.l) k k Operators ck are normalized so that {ck,ck>} = §k,k'- We first go from discrete Ho a continuum k in the standard way. Let ak be the continuum operators normalized so that {ak, ay} = 8(k — k'). The replacement rule for sums over k is 5>-/V (E.2) k n J* where iVc is the number of sites in the chain. The operators ck transform as Cfc -»¦ J-j^ak- (E-3) This follows from the requirement that J2k k,{ck,ck} = Nc corresponds to J dkdk'{ak,ak} = 7T. We obtain ^2ekc{ck -> / dk ek a\ak; (E.4) k J Y.VkCk^(-f) dkV^k- (E.5) 250 251 Next we introduce energy representation by defining aE = (dE(k)/dk) ll2ak, where E(k) = tk, which makes {a)E) aE,} = (^jPj S(k - k')= S(E - E'). (E.6) This gives and dktka\ak = / dktk l j aEaE = dEEaEaE, (E.7) Nc ^ ^ TT dkVkak (E.8) Here D is the band edge (or half-bandwidth). As the density of states per spin is k dfc ^ ' (E.11) we finally obtain ,D Hc -+ / dE[p(E)]1/2V(E)aE. (E.12) J-D The energy-representation version of H' is therefore H'= f dEEaEßaEß+[ dE[p(E)]l/2V(E)(aEßdß + H.c). (E.13) J-D J-D In NRG calculations it is customary to measure all energies relative to the band edge D by working in terms of the dimensionless variable e = E/D and operators aL/U = \[~DaEil. We also define the hybridization function as T(e) = 7rp(e)V(e)2. It should be noted that all information about the energy dependence of the density of states and of the coupling V is contained in the function T alone.165 The final expression is thus / deeo^+Z de (^jjj (alßdß + H.c.) . (E.14) Let us consider the example of a one-dimensional tight-binding chain with hopping parameter t and dispersion Ek = — 2tcos(k). The bandwidth is D = It and the density of states is 252APPENDIX E. TRANSFORMATIONS OF BAND AND COUPLING HAMILTONIANS Furthermore, we take Hc = —t' ^ (d^Ci/i+H.c), i.e the coupling consists of simple hopping from the first site of the chain to the impurity site. The hybridization matrix elements are Vk = ~t'ywTisink' (E'16) The hybridization function is thus r(E)= vr (— , l \ (t'2—^— sin2 k) , (E.17) which for large Nc becomes r(e) = r0Vl - e2 (E.18) where T0 = It' j D = \{t! jt)2D. If a single quantum dot is coupled to two such tight-binding chains, T is twice as large, i.e. T0/D = (t'/t)2. Appendix F Maj orana fermions Let us consider a single fermion mode described by creation-annihilation operator pair c1,c with {c\c\ = 1 and {c,c} = 0 and {c^c*"} =0. These operators span a two-level Fock space so that c*"|0) = |1) and c|l) = |0). We decompose the annihilation operator c in real and imaginary parts by defining c = Rec + ümc = x + *L, (F-l) where \ and L are real, in the sense that y} = \ and Lt = L. From this it follows c) = x- it (F.2) The anti-commutation relations then lead to L2 = 1/4, \2 = 1/4 and {x,L} = 0. Two operators (c, c1) are needed to span a two-level Fock space in the case of complex fermions. Likewise, both \ and L need to be present to generate a two-level space; a single real Majorana fermion by itself can only be associated with a trivial single-level space. The degeneracy of a system with N Majorana modes is thus 2^N/2\ where [x] denotes the integer part of x. Spin operators for S = 1/2 can also be expressed in terms of two Majorana fermions a and b: ax = -4=,