UNIVERSITY OF LJUBLJANA FACULTY OF MATHEMATICS AND PHYSICS DEPARTMENT OF PHYSICS Marko Uplaznik Transport properties of electrons in Mo6SxI9-x nanowire integrated chips Doctoral thesis ADVISER: prof. dr. Dragan Dragoljub Mihailovi´c Ljubljana, 2009 UNIVERZA V LJUBLJANI FAKULTETA ZA MATEMATIKO IN FIZIKO ODDELEK ZA FIZIKO Marko Uplaznik Transportne lastnosti elektronov v integriranih čipili z nanožicami M06 S xl9_x Doktorska disertacija MENTOR: prof. dr. Dragan Dragoljub Mihailovi´c Ljubljana, 2009 Abstract Basic property determination of novel nanomaterials is one of the main fields in nano-technology. The research includes along with the goal experiment also the material preparation, design and manufacture of measurement circuits, the integration of nanoparticles in the chips and the analysis of results. In this doctoral dissertation we report on the electron transport properties of Mo6SxI9-x nanowires. The synthesized material was dispersed in acetone using ultrasound and was later replaced with isopropanol due to di-electrophoretical integration of single bundles in measurement chips, that were annealed before the measurement at 700?C in vacuum, improving electrical connection between the bundle and the contacts. The circuit that was produced with electron beam lithography included a several 100 nm narrow gap, that was bridged by bundles integrating them into the circuit. The measured variables were the current versus voltage characteristics at different temperatures from room temperature till 18 K. The result analysis of four thin bundles was based on three main theoretical transport predictions: the Luttinger liquid, environmental Coulomb blockade and variable range hopping. Two bundles showed Lut-tinger liquid and variable range hopping, whereas the other two combined the hopping mechanism with the effects of environmental Coulomb blockade. We confirmed that the bundles are composed of single nanowires strands but included also high disorder and even insulated islands that act as quantum dots. For thicker bundles we observed the effect of cycling two times, where the conductivity changed for each temperature scan. We suggest an explanation of the phenomenon through Fermi glass theory by assuming the transformation of localized states to non-localized ones in such a way that the mobility edge passes the Fermi energy and thus fundamentally alters the transport mechanisms in the system. PACS: 73.63.Nm, 73.40.Cg Keywords: inorganic nanowires, electron transport properties, disordered nanowires, temperature annealing, variable range hopping, environmental Coulomb blockade, Luttinger liquid, nanowire dispersion Povzetek Določevanje osnovnih lastnosti nanomaterialov je eno temeljnih področij nanotehnologije. Raziskave poleg ciljnih eksperimentov vključujejo pripravo razpršin materiala, oblikovanje in izdelavo merlinih vezij, integracijo nanodelcev v čipe in analizo rezultatov. V tej doktorski disertaciji poročamo o meritvah elektronskih lastnosti nanosvežnjev Mo6SxIg-x-Sintetiziran material smo z ultrazvokom razpršili v acetonu in ga kasneje nadomestili z izopropanolom zaradi dielektroforetične integracije posameznih svežnjev v merilne čipe, ki smo jih pred meritvijo popuščali v vakuumu na temperaturi 700°C, s čimer smo izboljšali električno povezavo med kontakti in svežnjem. Vezje, ki smo ga izdelali s pomočjo elektronske nanolitografije, je v osnovi vsebovalo ozko režo širine nekaj 100 nm, ki so jo svežnji premostili ter se tako integrirali v vezje. Merilne opazljivke so bile karakteristike toka skozi merilni čip v odvisnosti od napetosti, merjene pri različnih temperaturah od sobne do 18 K. Analiza rezultatov meritev štirih tankih svežnjev je temeljila na teoretičnih napovedih glavnih transportnih mehanizmov: Luttingerjeve tekočine, impedančne Coulombove blokade in preskakovanja spremenljivega dosega. Pri dveh svežnjih smo pokazili na soobstoj Luttingerjeve tekočine in preskakovanja, pri preostalih dveh pa smo poleg preskakovanja sklepali še na impedančno Coulombovo blokado. Tako smo potrdili, da so svežnji sestavljeni iz posameznih nanožic, vendar pa so prepredeni z nehomogenostmi in celo električno izoliranimi strukturnimi otočki, ki delujejo kot kvantne pike. Pri debelejših svežnjih smo v dveh primerih opazili še efekt cikliranja, kjer se prevodnost spreminja ob vsakem temperaturnem ciklu. Pojav razložimo s teorijo Fermijevega stekla, kjer predvidevamo prehajanje lokaliziranih v nelokaliziranana stanja ob spreminjanju strukture med meritvijo tako, da rob mobilnosti prečka Fermijevo energijo, s čimer se fundamen-talno spremenijo transportni mehanizmi v sistemu. Stvarni vrstilec (PACS): 73.63.Nm, 73.40.Cg Ključne besede: anorganske nanožice, elektronski transport, nehomogene nanožice, temperaturno popuščanje, preskakovanje spremenljivega dosega, impedančna Coulombova blokada, Luttingerjeva tekočina, razpršina nanožic Acknowledgements - Zahvale This dissertation couldn’t be made without the devoted team of my colleagues and fellow researches. Non of the experiments would be possible without the material that was synthesized and processed by Damjan Vengust and Aleˇs Mrzel. The basis for the measurements was developed by Roman Yusupov, supported by priceless experience of Primoˇz Kuˇsar and Tomaˇz Mertelj. Many fundamental steps of nanowire handling and integration were established and developed by Boˇstjan Berˇciˇc. His work on thicker bundles and nanobundle networks was considered exemplary and enabled the difficult transition towards thinner nanobundles. The team was supervised by my mentor, professor Mi-hailovi´c, that guided the research through fruitful debates and with his persistence and patience facilitated reaching our goals. A thanks goes also to all other members of the department for complex matter, that followed me on my research path from the beginning till the end. Special thanks goes to Jure Strle that often sacrificed his time for theoretical debates, calculations and remarks on my work. Posebna hvala gre mojim kolegom, ki so me podpirali in vzpodbujali se posebej ob teˇzjih trenutkih. Hvala Andreju Vreˇckotu, Matjaˇzu Zemljiˇcu, Lukatu Ravniku, Mirkotu Kokoletu, Ireni Dolenc, Blaˇzu Zupanˇciˇcu in starejˇsim kolegom Danijelu Vrbaniˇcu ter Poloni Umek, ki sta z razmevanjem delila svoje bogate izkuˇsnje. K uspeˇsnem zakljuˇcku dela sta prispevala tudi moja stara prijatelja Miha Nemevˇsek in Andrej Pukˇsiˇc, ki sta vedno verjela vame. Zahvala pa gre tudi moji druˇzini in stricu Stankotu, ki me je usmerjal na moji akademski poti. Mostly I would like to thank Mihaela Ploscaru, that morally supported and helped me in the research. Multumesc pentru tot, Burtici¸ta mea! Contents 1 Introduction 13 2 Theoretical overview of one-dimensional electron transport 15 2.1 Basic consequences of strong 1D confinement ................. 15 2.1.1 Quantum wells - quantum dots .................... 15 2.1.2 Long range (dis)order in 1D structures ................ 18 2.2 Ballistic transport in 1D - Landauer formula ................. 20 2.3 Fabry-Perot segmentation ........................... 27 2.3.1 One and two scattering places ..................... 28 2.3.2 Three and n scattering places ..................... 28 2.4 Electron transport in quantum dots ...................... 34 2.4.1 Coulomb blockade oscillations ..................... 34 2.4.2 Environmental Coulomb blockade theory ............... 43 2.5 Variable range hopping ............................. 49 2.6 Luttinger liquid ................................. 56 3 Sample preparation and measurements 60 3.1 E-beam lithography ............................... 61 3.1.1 Substrate and e-resist spinning .................... 64 3.1.2 E-beam writing and developing .................... 65 3.1.3 Sputtering and lift-off ......................... 68 3.2 MoSIx Nanowires ................................ 70 3.2.1 The synthesis and the chemical structure of MoSIx nanowires ............................ 71 3.3 Dielectrophoretical attachment of single bundles over a narrow gap . . . . 77 3.3.1 Theoretical considerations ....................... 78 3.3.2 The attachment procedure ....................... 80 3.4 Measurements .................................. 88 3.4.1 Measurement setup ........................... 88 3.4.2 The process of temperature annealing ................. 89 3.4.3 Measured samples ............................ 98 12 CONTENTS 4 Results and discussion 100 4.1 The thin bundles................................100 4.1.1 The “S” curves .............................103 4.1.2 The “J” curves..............................107 4.2 The thick bundles................................114 4.2.1 The cycling effect............................118 5 Conclusion 125 Razširjeni povzetek v slovenščini 127 6 Uvod 128 7 Teoretične napovedi 130 7.1 Kvantne pike in (impedančna) Coulombova blokada.............130 7.2 Preskakovanje spremenljivega dosega.....................136 7.3 Teorija Luttingerjeve tekočine.........................138 8 Nanožice MoSIx in izvedba eksperimenta 140 8.2 Priprava razpršin in izdelava merilnih čipov .................142 9 Rezultati 147 9.1.1 Krivulje tipa “S” ............................149 9.1.2 Krivulje tipa “J” ............................150 9.1.3 Debelejši svežnji in efekt cikliranja ..................154 10 Zaključek 158 Chapter 1 Introduction Nanotechnology appears to be a novel promising route in many fields of scientific research, in medicine, electronics, physics, biochemistry,. . . and is expected to find new applications in the areas of information industry, diagnostics, even cancer treatment. What makes this new approach exciting and demanding are the scales and dimensions of objects involved in order to explore new possibilities and phenomena that arise when the macroworld collides with the micro- even nanoworld. These systems are often called mesoscopic and are considered to be one of the unexplored areas of science that utilizes highly developed technology and precise instruments, since the minute objects, that behave in practically the same way as molecules, need to be handled with the same precision as macro objects. This is the core of the nanotechnology: the structures are precisely designed on the atomic level and are intended to be controlled perfectly in terms of position and function. Such high expectations make nanotechnology a multidisciplinary science that includes particularly (bio)chemistry, physics, electronics that have to be combined in order to reach the set goals. The field already has a huge success in designing and controllably manufacturing different compounds and novel materials such as nanoparticles, nanowires, nanorods, nanohorns, ..., that are characterized under modern microscopy (transmission and scanning electron microscopy (TEM, SEM), atomic force microscopy (AFM), scanning tunneling microscopy (STM) and spectroscopic techniques (X-ray diffraction (XRD), X-ray absorption fine structure (XSAFS), Raman spectroscopy, UV-VIS spec-troscopy,. . . ). The integration of the compounds into the circuits was also demonstrated and many basic properties were explored, but the transition into high scale production hasn’t yet been established. Mostly the materials are used as additives and coatings, but functional mechanical or electronic devices are still in the phase of research. Promising materials for such devices are the nanowires and nanotubes that exhibit novel physical properties on the basis of quantum mechanics and can be at the same time handled with sufficient accuracy to integrate them into measurement chips that are produced using the e-beam lithography [1] with sufficiently high resolution in the nanometer range. Exemplary work on carbon nanotubes [2, 3, 4] confirms the onedimensional behavior of narrow wires through the presence of Luttinger liquid [5, 6]. Other low dimensional phenomena such as environmental Coulomb blockade have also been observed, confirming theoretical predictions for such systems [7, 8, 9]. In inorganic nanowires (niobium and molybdenum 14 1 Introduction selenide) similar Luttinger liquid behavior was measured along with charge density wave formation and variable range hopping transport [10, 11]. Such behavior is predicted in the theory of Fermi glasses developed by Mott and Anderson [12] and is closely related to the disorder in the systems. Research involving the inorganic compound based on molybdenum, sulfur and iodine with a general formula Mo6SxI9-x under the signature MoSIx [13, 14] was mainly concentrated on bulk samples in form of pressed pellets and included optical properties [15, 16, 17, 18], sound propagation studies [19] and electrical transport properties of nanowire sheets [20] or networks [21, 22], where also variable range hopping has been observed. Other studies involved sample preparation techniques and solubility properties in various solvents [23, 24] proving an intrinsic compatibility with a variety of chemicals, including water. The transport of individual, thick bundles (above 50 nm in diameter) has also been explored and again variable range hopping was observed [1, 25] whereas thinner bundle were studied on their self-assembling properties with different (bio)materials, setting the route to biochemistry [26]. In our work we attempted to measure basic electronic properties of single and thin MoSIx bundles, which proved to be quite a challenge. Our research included, apart form the material synthesis, all technological steps that lead to the final experiment. We were confronted with the majority of problems that also other researchers face: sample preparation, circuit production, single bundle integration and finally the transport measurement. In this thesis we begin with physical phenomena in one dimension and introduce the theories from the experimental point of view. After the basic description of our nanowires and dispersion preparation we turn our attention to the circuit production, discussing the process and challenges of the e-beam lithography, followed by dielectrophoretical bundle integration into a circuit, forming the measurement chip. Before describing the measurements, we report on the thermal annealing procedure and in the end we conclude with the analysis of results and general discussion. Chapter 2 Theoretical overview of one-dimensional electron transport One-dimensional systems are especially interesting for theoretical considerations since strong particle confinement lies in the area of the quantum mechanics thus having new physical properties which are often quite unexpected compared to the well explored bulk material properties. Our field of interest is electron current transport through such systems from theoretical and from experimental point of view by the exploration of single nanobundle measurement chips. In this chapter we first discuss the general properties of strong electron confinement in quantum mechanical terms, followed by some main transport theories that could be applied to a real measured system. We consider some 1D systems in terms of impurity and non-homogeneity content that have profound impact on electron travel along the system, especially if the confinement isn’t as strong and allows transport along other dimensions. We tried to present the issues so that the underlying physics comes upfront, sometimes even with basic mathematical treatments that lead to commonly familiar result. 2.1 Basic consequences of strong 1D confinement Before we begin with the overview of some theoretical models let us depict the fundamental consequences in 1D structures that arise solely due to the strong spatial confinement of fermions. They are driven by quantum mechanical effects that become significant at low temperatures and in confined dimensions. 2.1.1 Quantum wells - quantum dots Let us consider an independent and isolated system, where the electrons are trapped. We describe it as a quantum well with harmonic, step-like or other potential barrier shape. In every case, the calculated energy states become less dense due to the shrinkage of space. Let us take a look into basic properties on the example of 1D quantum harmonic oscillator [27] [28](Fig. 2.1), that describes a deep quantum well with a harmonic (quadratic) 16 2.1 Basic consequences of strong 1D confinement potential: 2 2 V = -mu x , 2 (2.1) where u denotes an equivalent to classical angular frequency in terms of (K/m)l/2 with K being the elastic constant of classical analogy with a mass on a spring, m mass and x the position of the particle inside the well. Figure 2.1: The energy levels of a 1D harmonic oscillator (left) and eigenstate probabilities of the first four energy levels (right); the colors denote the same n in both diagrams. To determine the wave functions and hence the probability for a particle to be found at the specific point in the oscillator, we need to solve the Schr¨odinger equation that includes also the energy levels for each wave function: H d l\) 1 2 2/ / -------— + - mu x ip = E ipi 2m dx2 2 (2.2) where rip denotes wave function and E its energy level. The solution of the problem is a family of eigenfunctions with proper boundary conditions (i/j(x —»• — oo) = 0 and ijj(x —> oc) = 0), each with its eigenenergy. Here we only write the result: tyn = —Z~ ) e 2h (2 Ti!) 2 Hn — X Tih ^J (2.3) where n denotes the index of each wave function and Hn the Hermite polynomials of order n. The eigenenergies are equally spaced and follow the expression: 1 En = hu(n +). 2 (2.4) 2.1 Basic consequences of strong 1D confinement 17 If we are interested in three dimensional confinement, as is often the case in nanotech-nology, the described 1D solutions can be easily expanded to include new dimensions, if the quantum states in each dimension remain uncoupled and the potential is isotropic, by adding new quantum numbers for each new dimension. For the harmonic oscillator in 3D we have three quantum numbers: nx, ny and nz. The solutions in this case have the same basic form as for the 1D: 3 Enx n nz = hu)(nx + Uy + Uz +----). 2 (2.5) Even though the spacing between the levels remains constant, the number of states with same energy grows quadratically with the energy, since more degenerate eigenstates are possible. The eigenstates are a product of 1D states for each dimension: Tnx,nv,nz *_ c ( ' tjx ! ' 'jy !''jz !) 7rn nx 7- ^ nv ^ y ^z 7- z h j h h (2.6) For the infinitely deep 1D square well [27] between 0 and xo we similarly get a family of eigenfunctions with eigenenergies but with the difference that now the energy levels are not equally spaced any more, but grow quadratically with the index of the wave function Fig. 2.2 [27] [28]: ipn En 2 2 nirx — sin(------), Xo Xo n27v2h2 2mxn (2.7) As with the harmonic oscillator we generalize by introducing a quantum number for each new (additional) dimension. The total energy is a sum over all dimensions and the corresponding wave functions become the product of the wave functions for each dimension: Ynx ,ny ,nz J-Jrix ,ny ,nz 2 2 2 2 2 xo Vo zo 7V2h2 2m ( n2 n2 n2 0 VI z o nx7TX nv7vy nz7vz sin(--------) sin(--------) sin(--------), Xq yo Zq (2.8) As we will see later, this quantization and energy steps between states above typical thermal energy kT lead to interesting transport phenomena in a very unexpected way. _ _ _ _ 18 2.1 Basic consequences of strong 1D confinement Figure 2.2: The energy levels in infinitely deep 1D square well (left) and eigenstate probabilities for the first four energy levels (right); the colors match in both diagrams. 2.1.2 Long range (dis)order in 1D structures The mechanisms of electron scattering in an arbitrary electron guide are diverse. Along with the common scattering on the lattice disorder, stoichiometric discrepancies and on impurities, more exotic phonon and magnon interactions with the electrons are present. For low temperatures the phonon scattering is negligible, however, as we briefly show in this subsection, the magnon population remains present even at absolute zero temperature. The last claim follows from the more fundamental Mermin-Wagner theorem 1. If we cite the abstract of their publication [29]: “It is rigorously proven that at any nonzero temperature, a one- or twodimen-sional isotropic spin-S Heisenberg model with finite range exchange interaction can be neither ferromagnetic nor antiferromagnetic. The method of proof is capable of excluding a variety of types of ordering in one and two dimensions.” That report deals rigorously with the theorem, while here we just illustrate the physical base for this phenomenon [30]. We will show that at a given non-zero temperature the magnon excitations destroy the magnetic order in one (or two) dimension. We start with writing the magnetization at some temperature as the difference between the magnetization at absolute zero (M(T = 0)) and the thermally excited magnon magnetization ?M(T): M(T)=M(T=0)-?M(T). (2.9) The reduction in magnetization is proportional to the number of excited states, obtained by the integration of the product between the density of states g(E) and the 1Also known as Mermin-Wagner-Hohenberg theorem or Coleman theorem. 2.1 Basic consequences of strong 1D confinement 19 probability for the state occupation according to the Bose-Einstein statistics: ?M (T) ~ g (E) f°° \ 1 1 9(E)ë-------- o ekBT — 1 dE. (2.10) The density of states is obtained by calculating the volume element for each state in k space. Naturally the result depends strongly on the dimensionality of the system. Let us consider the general case for a d dimensional system and the energy dispersion E oc |k|ra. For the limiting case of infinite system size the number of states on an interval (k, k + dk) or (E, E + dE) is quotient of the volume of the shell in d-dimensional k space and the volume for one single state : N = oo) limits the integral till zero, leaving us to explore the behavior at _ 20 2.2 Ballistic transport in 1D - Landauer formula E = 0. Now we expand the Bose-Einstein function around zero2 and rewrite the integral to: „() lrT1 f°° („ r 1 1 . ?o z00 1 ?MT ~ fcT o-E-)E------- = (fcT ) q(E) — . (2.15) o ekT - 1 o ^ Due to the power law connection between the density of states g(E) and the energy E the value of the last integral at E = 0 depends strongly on dimensionality of the system. Placing the expressions from the table 2.1 into (2.15) we find divergent behavior for d = 1 and d = 2 at vanishing E. Let us write the whole magnetization once more: r° 1 M(T) = M(T = 0) - const T g(E) —. (2.16) Jo E The consequence of the divergence is the destruction of the magnetization order in the system M(T = 0) for even the slightest non-zero temperatures thus prohibiting long range order in one- and twodimensional spin arrays. Practically that would mean that an electron traveling in strongly confined 1D system would encounter (at least) many magnons on its way resulting in scattering and thus preventing undisturbed transport. This result already gives the taste of unexpected phenomena in a system, especially a realistic one since even in perfect structures and strong confinement the traveling electron encounters scattering. 2.2 Ballistic transport in 1D - Landauer formula One of the basic models for transport through a 1D system is a direct, unscattered travel of an electron from one side of the system to the other. This transport is called ballistic since the electron passes the whole length of a 1D structure without being obstructed by any kind of mechanism, hence like a bullet. This model applies for the most ideal system of perfectly uniform and impurity free nanowires. Let us describe the system as a 1D wire of length L between two bulk electrodes (Fig. 2.3). We can also say that we have confinement in two dimensions, y an z whereas x remains unconstricted. As presented in the previous section the states in the system get separated in both confined dimensions y and z according to the given potential and the geometry of the wire, whereas in the x direction the electron states suffer no restrictions, thus being described as plane waves with the continuum of energy levels. If we deal with the problem similarly as presented for the case of the harmonic oscillator or the potential well, we solve the Schr¨odinger equation for the y and z direction and separately for the x direction. Since we have in mind only basic consequences for the transport we treat the confinement potential as a general function V(y, z). Writing the equation: h2 fd2tpn(y,z) d2tpn(y, z)\ ------------------------------------------------- + V(yi Z) fn(Vi z)(n) = cn fn(Vi z) , (2.17) 2m dy2 dz2 ex - 1 « x + ... + a(x ) 2.2 Ballistic transport in 1D - Landauer formula 21 / r" / \ ^L Channel T\E] Mr \ / / a) Figure 2.3: The transport channel with the transmission r[E] between two reservoirs with chemical potentials /il and /ir a) and an artistic rendering of a nanowire/nanotube suspended between two major gold reservoirs. The inside of the tube shows schematically the positional probability of an electron for the case of cylindric potential well; the solutions for this quantum system are the sombrero-shaped Bessel functions - here we show the first three states (I, II and III on the image). where index n denotes some excitation state, (pn(y, z) the adequate eigen function for the confinement in y and z and en the energy of the state. Combining this result with the plane waves in x direction we can write the whole wave function ipkx,n as: 4kxx ^rkx,n (fin(y, z). (2.18) The total energy En^x is the sum of the energies of the plane wave and the solution for the confined dimensions y and z: En,kx = tn + "7---- 2m (2.19) As we learned from dealing with harmonic potential and the 1D potential well the energy levels in transversal, confined direction become separated. Together with the continuum of plane wave energies the system consists of subbands of continuous states that begin with the discrete values of en and overlap in steps as the energy rises; naturally also the subbands carry the adequate index n. A scheme of subband structure is depicted in Fig. 2.4a). Having introduced the electronic levels into our system we deal now also with the electric current. As schematically depicted on figure 2.3a we consider two contact reservoirs with chemical potentials /il and /ir connected to our finite 1D system of length L. 22 2.2 Ballistic transport in 1D - Landauer formula Starting with the definition of the current density j inside a subband n for an interval dk around a specific k and having in mind also discrete energy distribution due to the finite length L we can write for zero temperature and the transmission set to 1: 2e jn k = e P v (k) = v (k), (2.20) L where the linear density of states p includes also the two possibilities for the spin of the electrons. For finite number of electrons and a finite temperature we need to add the probability factor for the electron to occupy the level with the energy E(k) in the form of the Fermi function f(E — fi). We get: 2e jnk = T vn\k)j\E — fiL). (2-21) L To get full current for one subband we make a sum over k and since we are interested in the right direction of the current we take only k > 0. Since for ID systems the current density j and the current / are interchangeable we write: 2e ^ n In = r Vn{k)f(E — fit), (2.22) fc>0 where the arrow over In points in the direction of current from left to right. In the continuous limit (the length if the channel is large compared to its width) we can write an integral instead of the summation: ^ 2e f°° , M~ In = r vn{k)f(E — fi^dn, (2.23) L 0 and by including the general expression for the k: 27T „ L k = —n =^ an = —dk (2.24) we can rewrite the (2.23) into: r^ e f°° W/r, ln = — vn(k)j{E — ßL)dk, (2.25) W0 which is then rewritten again via: h2k2 hk dE E =------, vn(k) = — =^- dk = —— (2.26) 2m m nvn(k) in the integral over the energy for the entire subband, starting at the beginning of the subband era: M 2e f°° F, , 1 =— j{E — fiL^dE. (2.27) h L 2.2 Ballistic transport in 1D - Landauer formula 23 Figure 2.4: a) A scheme of subbands for a system with equally spaced subbands (hco) and b) a diagram of x(E) function for a system with equally spaced energy levels - red curve (e.g. harmonic oscillator Eq. (2.4)) and for nonconstant spacing - blue curve (e.g. potential well Eq. (2.7)). Before we add all the contributions from each subband let us define a function x(E) that denotes the sum of Heaviside step functions H(E — en): x(E) = y H{E — en). (2.28) The step-like nature of x(E) is depicted on 2.4b). Now we summarize the (2.27) over all subbands and get the overall current from left to right electrode: T—> = y i = y — jiE — ßiAdE, (2.29) which is simplified by summarizing over the subbands inside of the integral and stretching the integral from —oc to oc by limiting the subbands with the Heaviside function H(E — en). The result now reads as: T—> = r / tl{E — en)j[E — ßhjdE = — j[E — ß^jXyEjdE. (2.30) '^ _ '^ Naturally we write the same expression for the current from right to left but this time with chemical potential /xr: 7~<— = — j[E — ß^i)x[E)dE} '^ (2.31) which gives the total current through the constriction as the difference between the left and right current: 24 2.2 Ballistic transport in 1D - Landauer formula T 7- T- 2e r\f(TP \ f(TP l (mM (0*0\ 1=1 — 1 = — [J (-h ~ ßh) — J (ti — /jR)\x(ti)arj. [Z.61) ,lj —oo To get the basic idea, let us, along with the transmission set to unity, pin also the temperature to 0. In this case the Fermi function is a step function and the function x(ti) is an integer x on tne interval |jWr,/il]- Now the Eq. (2.32) simplifies to: 2e 2e2 fi\, — ßR 2e2 1 = ^x(ßh — ßn) = TX-----------=iXV) (2.33) h hen from where we read the conductivity G as: 2e2 G = —Xi (2.34) h since V denotes the voltage ML~MR. This is the fundamental result of this system. We learned that in the best case scenario, when the electron on its path doesn’t encounter any kind of scattering, the conductance is limited by the quantum of conductance: Go =— = 7.75 • 10~ S or Ro = 12.897 kQ (2.35) h and takes the values of the multiples of Go- If the transmission isn’t perfect, set to unity, then we introduce a multiplying parameter r G (0,1) that lowers the current in Eq. (2.32), transforming the conductance in Eq. (2.33) to 2e2 G =—XT- (2.36) h This result carries the name the Landauer formula3. Adding the transmission t(E), that can be in general dependent on the energy of electrons, to the total current expression (2.32) we get the most general description of the electron transport through the system: 2e . . I = — [f(E — Hi) — j(E — ßR)\x(E)T(E)dE. (2.37) h - This expression can be simplified if we are interested in zero voltage limit conductance so that the difference between the chemical potentials of both electrodes remains small and we can write Taylor expansion of both Fermi functions. Let us set the values to: ßh = ß + S fi and /ir, = fi. (2.38) The expansion follows: df(E,fi) j(E — fii) — f(E — /Ir) = j(E — (fi + ofi)) — j(E — fi) « —ofi^, (2.39) oE 3Rolf Landauer; (1927 - 1999), an IBM physicist of German origin. 2.2 Ballistic transport in 1D - Landauer formula 25 which transforms the Eq. (2.37) to: 2e r f df(E,ß) 1 öß 1 =—---------------x(rj)T(hj)drj —, (2.40) h_oo dE e directly giving us the final result for conductance: 2g2 oo df(E,ß) G(ß) =—----- x(E)r(E)dE. (2.41) h _QO dE If we describe in words: the conductance for low voltage 5ß/e is the contribution of the subbands that overlap with the derivative of the Fermi function —gE . As the energy increases, the derivative travels towards higher subbands following the x(E) dependance as depicted on Fig. 2.4b. The derivative of the Fermi function is known also as the broadening function. For a concrete example let us take the harmonic oscillator in 2D with equally spaced energy levels (Eq. (2.5)) and with the x(E) function. For ballistic transport the transmission is set to unity and the oscillator parameter fko is set to 0.1 eV. To clarify the numeric calculation we consider the definition of x(E) from the Eq. (2.28). By putting it into the Eq. 2.41 and placing the sum over all subbands before the integral we lift the lower integration limit to era: 2e2 ^—\ f°° df(E,ß) 2e2 ^—\ f°° df(E,ß) G(ß) = — y---------------H (E — €n)dE = — y---------------dE. (2.42) h „ -oo aE h ^ hn aE Now we can easily integrate and by evaluating the values at the limits we get: 2e2 2e ^ G(ß) = — —f (E — ß) h /(era — ß)- (2.43) We can say that the Fermi function travels from one band to the next one with the contribution to the sum of all passed subbands and those that cross the function. If we draw the conductance we get the very well known staircase (Fig. 2.6a)) with the step size %-. For higher temperatures the edges are smeared out but the distances between the levels remain the same as for low temperatures. Considering the expression Eq. (2.43), this result is the direct consequence of the subband energy level structure and the step shape of the Fermi function. With the increasing voltage (ß) the step crosses more and more subbands thus enabling them to contribute to the conductance (Fig. 2.5a). Due to the spacing between the subbands in comparison to the the width of Fermi function edge the conductance for the energies inside of the subband remains constant and only changes on the transition between the subbands. As mentioned before, the expression (2.42) gives the same staircase only that in this case the broadening function — XF picks only narrow portions of the x(E) function to contribute significantly to the integral (Fig. 2.5b). In fact only those steps are integrated, around which the broadening function is centered thus making the steps smooth if the broadening function stands close to the edges, so that both neighboring steps get included _ 26 2.2 Ballistic transport in 1D - Landauer formula Figure 2.5: A graphical depiction of the numerical summarization/integration of the expressions (2.41) - b) and (2.43) - a) (in both cases transmission r was set to unity). It can be seen, that the broadening function (centered at /j, = 0.32 eV) for higher temperatures (300 K) on the diagram b) overlaps significantly with the neighboring subbands whereas for the lower ones (100 K) the whole hump remains confined inside of one subband, except at the edges. The alternative summarization of Fermi function a) shows the intersection of the step with different subbands; the colored vertical lines represent the contribution to the overall sum (violet for 300 K and green for 100 K). Here the contribution to the overall sum consists of the subbands deep inside of the Fermi function, thus before the step, and of minor cross sections of the function’s hump. Again for high temperatures the frontal subbands participation rises whereas for the lower ones the whole step drops practically inside of one subband. Figure 2.6: The temperature diagrams of conductance staircase a) for equidistant electron levels (fajj = 0.1eV) calculated according to Eq. (2.43) and of the broadening function b) with /j, = 0, thus centered at zero. For low temperatures the function gets close to Dirac delta function. The legend in the upper right corner holds for both graphs. 2.3 Fabry-Perot segmentation 27 in the integral. The broadening functions for some temperatures are presented in Fig. 2.6b); note that for low temperatures the function gets close to the Dirac delta function. Our brief demonstration of the 1D conductance gave an unfamiliar result. The conductance is limited! In bulk material theory and practical applications not only allow huge conductances like in superconductive materials. That is truly a remarkable twist in the story of transport measurements for small, confined but otherwise perfect current guides. 2.3 Fabry-Perot segmentation Perhaps that first phenomenon that comes in mind when thinking about non-uniform systems is segmented structure with barriers separating portions consecutively. This situation is equivalent to light passing through a series of semi-mirrors with finite reflectivity and transmitivity also known as the Fabry-Perot transmitter. Using the same terminology we can adopt the idea and predict a series of tunneling barriers with tunneling probability4 T standing for the light transmitivity. The situation is schematically shown in Fig. 2.7 where the incident electron stream encounters a sequence of barriers before reaching the other end. This model completely ignores quantum mechanical effects (except for allowing the electrons to tunnel through a barrier) in term of spin, wavefunctions, Pauli principle, interference, . . . and focuses on the total transmitivity Tall through such a system. Even though we are fully aware of the model’s inadequacy we demonstrate that also relatively modest inhomogeneity can cause even in this model the transmitivity to drop fast at a small number of barriers despite considerable barrier transmitivity. Figure 2.7: The particle stream from left encounters a sequence of n barriers with trans-mitivity T and reflectivity R. The fraction of the incident beam that passes through is denoted as Tall and depends on the number of barriers and the transmitivity of a single barrier. We discuss the situation with an incident current of particles from left -›I encounters n barriers with transmitivity T and reflectivity R = 1 - T. The goal is to obtain the 4The notations for T and R in this section shouldn’t be mistaken for temperature and resistance. 28 2.3 Fabry-Perot segmentation overall fraction Tall of the incident beam that passes through, formally written as: -›IR =Tall-›IL. (2.44) For introduction we solve the the problem for one and two barriers followed by a general solution for n number of barriers. 2.3.1 One and two scattering places In the case of one scattering place the result is trivial since the electrons don’t return back. The transmitivity T is simply its nominal value. For two barriers we need to take into account also the possibility of scattering many times before finally penetrating the second barrier. The contribution to the overall trans-mitivity can be divided into orders of scattering where we count the number of backscat-tering before leaving the system. First scattering order would then be just passing through without backscattering; TT = T2. In the second order an electron scatters back at the second barrier and in order to reach again the end of the system it must scatter backwards again at the first barrier. We write this as TRRT = T2R2. If we itemize first four scattering orders: 1. T2 2. T2R2 3. T2R4 4. T2R6 We recognize the pattern, enabling us to summarize over all orders: Tall = T2 +T2R2 +T2R4 +T2R6 +... = T2(1+R2 +R4 +R6 +...) = t y R% = i=0 T 2 T T' 1 — H2 (2.45) (2.46) (2.47) (2.48) 2.3.2 Three and n scattering places The counting of different scattering orders gets more complicated for three barrier system. In order to determine the contribution of each order we use the notation which will help us to count properly all possibilities. We follow the electron as it passes the barriers and write the product for each event. Of course now the electron can travel also backwards from the right segment to the left one which enriches the possibilities considerably. Not _ 2.3 Fabry-Perot segmentation 29 to loose the clarity we count the passing of the middle barrier that results as hopping between the segments and thus introduce the hopping order. For each scattering order we get similarly then for two barriers a table of possibilities. The hopping of order 1 gives the direct traveling from one segment to the other and prohibits the hopping between them but allows scattering inside of each segment: We can write the sum for this hopping 1. order - T3 2. order - T3R2 3. order - T3R4 4. order - T3R6 TTT TRRTT TTRRRRT TTRRRRRRT TTRRT TRRTRRT TRRTRRRRT TRRRRTT TRRRRTRRT TRRRRRRTT Table 2.2: The first four scattering orders for hopping order 1 in two segment (3 barrier) system. order: Thopi = T (1 + 2R + 3R + 4R + ...) oo = T3R-2J2*R2i (2.49) (2.50) i=\ We summarize using a standard trick of summation and rename R2 = x: oo s = y ix% i=\ Sx +x+x+x+x+... = S Sx + 1 1 — X — 1 = S S = x (1 — x)2 (2.51) (2.52) (2.53) (2.54) The summation then reads: -^ hopl Tc (1 — R2)2 (2.55) Similarly we write the possibilities for the second hopping order where we allow one transition between the segments; strictly speaking there are additional two passes of middle barrier of the electron since it has to return in order to leave the system in the correct direction. _ 30 2.3 Fabry-Perot segmentation Let us write the contributions for first several orders: 1. order - T5R2 2. order - T5R4 3. order - T5R6 4. order - T5R8 TTRTRTT TTRTRTRRT TTRTRTRRRRT TTRTRTRRRRRRT TTRTRRRTT TTRTRRRTRRT TTRTRRRTRRRRT TTRRRTRTT TTRRRTRTRRT TTRRRTRTRRRRT TRRTRTRTT TRRTRTRTRRT TRRTRTRTRRRRT TTRTRRRRRTT TTRTRRRRRTRRT TTRRRTRRRTT TTRRRTRRRTRRT TRRTRTRRRTT TRRTRTRRRTRRT TTRRRRRTRTT TTRTRRRRRRRTT TRRTRRRTRTT TTRRRTRRRRRTT TRRRRTRTRTT TRRTRTRRRRRTT TTRTRRRRRRRTT TTRRRTRRRRRTT TRRTRTRRRRRTT TTRRRRRTRRRTT TRRTRRRTRRRTT TRRRRTRTRRRTT TTRRRRRRRTRTT TRRTRRRRRTRTT TRRRRTRRRTRTT TRRRRRRTRTRTT Table 2.3: The first four scattering orders for hopping order 2 in two segment (3 barrier) system. Before we can write the sum over all scattering orders: 2~hop2 = T R (1 + 4R + 10R + 20R ...) (2.56) we need to determine the coefficients for each scattering order. We notice that they follow a general rule: /3\ /4\ /5\ /6\ 1,4,10,20... = , , , .... (2.57) 3 3 3 3 We can even test this rule. If we examine the sequences of Ts and Rs we observe that each possibility starts and ends with a T (not surprisingly since the electron enters and leaves the system with tunneling through the barrier) and that they can be divided into strings of two characters that follow combinations and have some restrictions. What shuffles in each sequence is: n =(scattering order - 1) number of RR, one T and two sets 2.3 Fabry-Perot segmentation 31 of RT. There is one constriction: T and both RTs can only be in sequence5 TRTRT with possibility of the RRs in between. To get the number of sequences we take non repetitional permutations of the number of all shuffling strings (number of RRs plus one T plus two RTs —> n— 1 + 1 + 2 = n + 2), divide it with the factors for non repetitional permutation for those strings that are repeating and finally divide also by three due to the sequence condition for T and both RTs. The general coefficient reads as assumed in (2.57): (n + 2)! fn + 2\ (2.58) 32!(n+2-3)! 3 The sum over all orders can be now written in a compact form: 2hop2 = T y R n . (2.59) ^-^ 3 n=l 3 We deal with the summation similarly as before (R2 = x) (2.51): s—\ fi + 2\ ^-^ 3 S = y x (2.60) 3 S = x + 4x + 10e + 20x + 35x + ... (2.61) S'a; + x + 3x + 6x + 10e + ... = S. (2.62) We get the new series which we summarize separately: M = x + 3x + 6x + 10x + ... (2.63) Mx + x + 2x + 3x + 4x + ... = M (2.64) x Mx +------- = M (2.65) (1 — x)2 x M =--------. (2.66) (1 — x) The sum S (2.60) is then: x S =-------, (2.67) (1 — x)4 and thus the Thop2 from (2.59) and (2.67): 5 R2 2hop2 = T------- . (2.68) (1 — ii2)4 We can estimate also other hopping order contributions and write the full sum over all hopping orders: Tau = T------- + T------- + T------- + .... (2.69) (1 — it2)2 (1 — it2)4 (1 — R2)b T 3 ^—\ / R \ Tau =------ / ------ • (2.70) (1 — ii2)2 ^—' 1 + -R 5Any other possibility allows the electron to leave the system prematurely. 32 2.3 Fabry-Perot segmentation The expression is geometrical series and can be summarized to: T TAll = . (2.71) 3 - 2T For the general case of n scattering barriers we have to look one last time at the series of transmissions for calculated systems: T T1 =T= (2.72) 1 - 0T T T2 = (2.73) 2 - 1T T T3 = (2.74) 3 - 2T . (2.75) T Tn = . (2.76) n - (n - 1)T This derivation used a brute force approach giving us the possibility to perform the summation over possible scattering and even hopping order. Another much simpler approach uses the derivation with the help of transformational matrices that describe the transition of the current over barriers. Similarly than before we start with one barrier with the transmittance T and reflectivity R = 1 - T, but this time we take the general case of currents incoming and reflecting on both sides of the barrier: Ir = Tli, + RIn (2.77) Il = RIl+TIr,. (2.78) To get the transformational matrix we write the current on each side as vectors with components denoting the left and right direction: ^r 1 T T — R R~\ \ li, \ j____ = ------- j____ Mr T -R 1 /L 1 T-R R L ‹-. (2.79) T -R 1 Since in our case the electrons come only from left li, we set the current from right Ir to zero. In order to get the total transmittance we write the transformation (2.79) inversely. The determinant of the transformational matrix is 1 and the inverse is written simply by switching the elements on the diagonal and changing the sign of the off diagonal elements: /L 1 r 1 —R i r /R 1 /L T R T — -R 0 11- IR . (2.80) T RT-R 0 Now we write the reverse transformation of -›IR to -›IL and extract the incoming current: 2.3 Fabry-Perot segmentation 33 h TL (2.81) which is of course obvious solution. For two barriers we simply multiply their transformational matrices. After simplifying using the connection between R and T we get new transformational matrix: T-R R 1 -R 1 T and writing it in the inverse form as in (2.80): 1 T T-R R R1 i 1 r --- t T -2R 2R 2R 1 + R (2.82) /L 1 r y_____ = --------- 1 + R 2R T 2R -2R HŽr 1 0 (2.83) Now the overall transmittance gets the form: / T R h T d T (2.84) 1+R L 2-T which we already know from previous derivation (2.48). We can already see the multiplying pattern in the new transformational matrix. We notice that simply R is subtracted or added to each matrix element. For the case of n barriers the matrix takes form: -^R 1 [ j____ = ------ -^R T T -nR -nR nR 1 + (n - 1)R ]\t] and from its inverse we finally get the familiar total transmitivity (2.76): I T R n - (n - 1)T Il =^ Ta\\ = T n - (n - 1)T (2.85) (2.86) To obtain the quantitative nature of such system we tested the total transmitivity on the single barrier transmitivity at some fixed n (for long wires the n can easily be 10) revealing rather strong decrease (for T = 0.91 Tall is 0.5 from Fig. 2.8a) suggesting that in such sequence of segments the electrons will need to struggle to pass, even if the tunneling through individual barriers isn’t strongly suppressed. Moreover also the number of segments reduces the total transmitivity for some fixed T. It turns out that for T = 0.95 the total fraction of electrons that travel through falls to ? 60% (Fig. 2.8b). _ _ _ _ 34 2.4 Electron transport in quantum dots i 0.9 0.8 0.7 0.6 0.5 n-{n-l) T 0.92 0.94 0.96 0.! a) b) Figure 2.8: a) The transmitivity Tall decreases rapidly with decreasing T (n = 10). b) The transmitivity Tall decreases also for increasing n; the T was set to 0.95. Since this model ignores the basic quantum mechanical nature of electrons it becomes potentially useful if the measured systems are big enough to treat electrons as free projectiles without any interaction between each other. This demonstration shows that even though the ballistic transport governs the segments the barriers could decrease the conductance of a systems profoundly. Thus at this point we can already expect that the transport depends strongly on the system’s structure or actually on the inhomogeneities throughout the system even at low non-uniformities. In the following subsections we discuss several possibilities of non-uniform structures with the emphasis on the transport properties through it. 2.4 Electron transport in quantum dots Another interesting possibility for transport through confined structures is the case of a small island (quantum dot) weakly bound to the surrounding reservoirs. We introduced the basic description of such a dot in terms of quantum eigenstates in one of the previous sections 2.1.1 and here we deal with the transport through such a system coupled to external source, drain and the gate electrode. In general the island can be viewed locally or globally when connected to an external power source. In the first case the system is treated without the interaction to the rest of the world whereas in the latter one the charging energy rebalance is provided by external power supply widening the transport rules. In the first part of this section we discuss the basic Coulomb blockade phenomenon, followed by the environmental influence on the system’s transport behavior. 2.4.1 Coulomb blockade oscillations Now we discuss basic transport properties through a quantum dot, separated by a thin layer of insulator from the bulk reservoirs, weakly coupled via tunneling and unspoiled in 2.4 Electron transport in quantum dots 35 terms of the state separation on the dot itself. This model can be applied in the case of thin nanowires, weakly connected to the guides (Fig. 2.9a) or in the case of special wire segmentation (2.9b), where an insulated region is formed during the synthesis growth, due to random constrictions and structural inhomogeneities in the middle of the compound. a) b) Figure 2.9: a) A nanowire is weakly coupled to the reservoirs so that the electrons can hop on and off the wire-island only via tunneling. b) An insulated island can be formed in the middle of the wire due to physical constrictions (the bottom scheme) or due to structural inhomogeneities (the top scheme) creating a quantum dot. Theoretically we describe the system as a series of source reservoir, an island with discrete ladder states (quantum dot with the typical energy gaps ?E between the states) and drain reservoir ([8] chapter 5). To investigate this assembly a gate electrode is introduced in order to influence the states on the dot (Fig. 2.11). For basic exploration we confine ourselves to the low temperature regime, where the thermic energy doesn’t excite the electrons on the quantum dots to higher levels, thus AE ^ kßT, where AE denotes level spacing of the dot state ladder. Our interest lies in the transport properties, more specifically in the different conducting and non-conducting regimes. A simplified view, dot drain ^gate^ Figure 2.10: A quantum dot is connected to source and drain reservoirs. A gate electrode is placed in the vicinity to influence the position of the states in the dot. that ignores the charging effects6, suggests that the transport through the system exhibits resonant behavior - the transport is possible only when a level in the dot’s ladder aligns with the Fermi level of the source reservoir, controlled by source voltage. This simplified view depicted in Fig. 2.11 introduces the basic idea since real experiments become hard to illustrate as we show later. Continuously lifting the source voltage (gate voltage re- 6Also spin and electron interactions are disregarded. 36 2.4 Electron transport in quantum dots eV i E E FD source drain —•— eV a) i Es ~^» /~~ * E FD \ source drain -----•----- b) I[a.u.] 4 3 - 1 234 VS [?/e] c) Figure 2.11: The source voltage is altered with constant gate voltage. a) The EFS of the source lies between quantum dot states with the level spacing ? - the transport is prohibited. b) The EFS is aligned to an empty state in the dot allowing an electron to tunnel on the dot and finally of it to the drain reservoir - transport is enabled. c) The current increases in steps as the voltage increases, enabling higher levels to conduct. E FS EFD source drain —•— A ¦ E3 /- i* z' ^ E FD source drain t eV g a) b) G[G0 ] 1 1 234 Vg [?/e] c) Figure 2.12: Zero bias conductance is monitored as gate voltage is altered. a) The EFS of the source lies between quantum dot states - the transport is prohibited. b) The EFS is aligned to an empty state in the dot, allowing an electron to tunnel on the dot and of it to the drain reservoir - transport is enabled. c) The current vs. voltage conductance peaks. Transport is allowed when empty dot states align with the Fermi energy. mains constant), while disregarding the charging, the current would follow a step-like curve (known also as the Coulomb staircase) increasing according to the single level conductance with Landauer quantum (see section 2.2) disregarding the spin G0 = eh2 = 3.875 · 10-5 S, when higher unoccupied levels start to conduct (Fig. 2.11c). Another way to observe such a system is to monitor zero bias conductance G as the gate voltage changes (Fig. 2.12). 2 1 2.4 Electron transport in quantum dots 37 Here the states in the ladder pass the Fermi level of the reservoir only conducting at discrete points when they perfectly align. Current vs. gate voltage plot shows conductance peaks when a dot level aligns to Fermi energy (Fig. 2.12c). This simplified model should be understood as the underlying idea for the transport conditions under different investigation approaches. In this spirit we first introduce charging energy, which is actually the driving force of quantum dot conductivity phenomena, and in the end we conclude with the treatment of realistic measurement conditions that reveal the more complex nature of quantum dot transport. One way to determine the positions of the peaks, when charging energy is included, is through the equilibrium properties of electrons on the quantum dot in respect to the reservoirs. The grand canonical distribution gives us the probability to find N electrons on the quantum dot in equilibrium with the reservoirs: P(N) oc exp(—— [F(N) — NEf]), (2.87) kßT where F(N) denotes free energy, T the temperature and N the number of electrons on the dot. The transport will be governed by the N that maximizes the probability. In fact, since our interest lies in the close-to-zero temperature regime, only one N gives non-zero P(N), namely the one that minimizes the thermodynamic potential ?(N) = F(N) — NEp. When speaking about the transport it can be shown [7] that G —> 0 when T —> 0. Moreover the transport is possible only when P(N) and P(N + 1) are non-zero for the same N. With other words, the dot is found in the thermodynamic equilibrium with respect to the reservoirs for N and N + 1 at the same time, thus allowing an electron to tunnel on (N + 1) and again off the dot (N) creating current through the system via dot occupancy N^N+1^N^N+1^N^---. Formally speaking we have coexistence of two global minima7 in the thermodynamic potential ?(iV) for N and N + 1 from where we get the condition relation for both free energies at N and N + 1: ?(N + 1) = ?(N) —> F(N + 1) — F(N) = Ep. (2.88) This convenient relation for conductance peaks demands the determination of the free energy for the system. We can write it as a sum of charging energy U(N) and single electron levels Ep: N F(N) = U(N) + y Ep, (2.89) where the charge imbalance between the reservoirs and the dot is taken into account macroscopically through the potential difference between the dot and the reservoir including the contribution of nearby charges and particularly of the gate electrode. The dot is capacitively coupled to the reservoirs and the gate electrodes with the macroscopic capacitance8 C. The potential difference for charge Q reads classically as a sum of charge 7In general there could be also more minima, but we confine ourselves to the most probable case of two minima. We assume that the capacitance remains constant in respect to N. 38 2.4 Electron transport in quantum dots difference potential and the external field (gate electrode): Q 4>(Q) = 7^ + 0ext- (2.90) 6 U(N) is now the integral of the potential (2.90) over the charge9: rNe l„. w (A^e)2 A7 , U(N) = ç>(6j )a6j =-----iVeç>ext- (2.91) o 26 Inserting (2.89) in (2.88) and relabeling N by N — 1 we get: -E-W + U(N) — U(N — 1) = LV, (2.92) that gives with (2.91) the final condition for conductance peaks: / 1\ e2 En + iV----- — = LV + eçext- (2.93) 26 One way of analyzing this condition is to take the bare ladder with N — 1 electrons and count the charging energy of one electron tunneling onto the dot with the lowest free state at En. We write this as: e2 En +pz = Ep + e(f)ext(N — 1), (2.94) 26 again with N referring to the lowest unoccupied level. In other words: the lowest unoccupied state has to be positioned one half of the charging energy below the Ep level (Fig. 2.13a). As the electron jumps onto the dot, the equality now includes the newly occupied state at En. The number of electron is now of course A^ — 1 —> A^ yielding: e2 En-----7^ = Ep + e(pext(N), (2.95) 26 now with A^ referring to the highest occupied state. In words: the highest occupied level has to be placed one half of the charging energy above the Ep level (Fig. 2.13b). Afterwards the electron tunnels from the dot and the situation resets to initial configuration (Fig. 2.13c). Let us summarize the tunneling of an electron through the dot in a compact form: an electron tunnels onto the unoccupied state of the dot that is positioned e2/26* below the Ep, adds the charging energy (e2/6*) to the dot and thus lifts the newly occupied level to e2/26* above the Ep before finally tunneling to the other side resetting the potentials to initial situation. In literature a continuous “externally induced charge” Qext = C'/'ext is often defined next to quantized -Qe 2C Q as a purely theoretical description. Now the potential reads: U(N) = (Ne-2QCext)2 + constant. y 2.4 Electron transport in quantum dots 39 1 S 1 ------*- e or* --------*- e -------r 2C -eÖL^ B m 9 B K i M edtt Î a) b) e) Figure 2.13: a) In order for an electron to tunnel onto the dot the lowest unoccupied state has to be e2/2C below the EF surface. b) Electron adds the charging energy e2/C and the newly occupied level is now e2/2C above the EF surface. c) The final electron tunneling into the drain resets the system into initial situation. The left side of (2.93) can be understood also as renormalized energy levels EN? with lifted spin degeneracy by the charging energy e2/C that is added to level spacing ?? = ?E + e2/C; the redefined levels are depicted on Fig. 2.14. In this sense we can also implement the charging into the simplified picture from Fig. 2.11 or Fig. 2.12, setting the period ?/e to e/C. a) b) E. e; Figure 2.14: a) Bare quantum dot ladder is b) renormalized when charging energy is included. The scheme depicts the case where e2/C ~ 2(AE), where () denotes the average. In experiments the dot is capacitively coupled in form of junctions to source, drain and gate electrode and requires additional treatment [9]. Fig. 2.15 depicts such a system with V1 being the voltage across the source junction (C1), V2 across the drain junction and Vg (C2) the voltage on the gate, measured from drain electrode potential. For the moment we observe the case without the gate electrode and treat the system classically, comparing the Helmholtz free energy (the difference between the total energy and the work done by power sources) for electron tunneling through the junctions with rate n1 and n2 trough the first and second junction respectively. Basic electronics gives us the 40 2.4 Electron transport in quantum dots v v -?, Q 1 +?, V gate f2 -1, drain Figure 2.15: Double junction circuit with gate. The electrons are allowed to tunnel through capacitors from source and towards drain, whereas the gate voltage induces continuous additional charge on the island. charge on the junctions10: Qi = C\Vi, Ç2 = C2V2, q = Q2 — Qi + Ço = —^e + Ço> (2.96) where the n = n1-n2 denotes the net number of electrons and q0 the background charge11 on the island. Using the voltage drop equality: together with (2.96) we get: Vi = C2VS + ne — qo V2 = Vs = V\ + V2, C1VS - ne + q0 , C (2.97) where C? = C1 +C2. (2.98) If we now consider n1 electrons to tunnel through the first junction on the island we observe according to (2.98) the voltage V1 increase by n1e/C? consequently resulting as a voltage drop on V2 for -n1e/C? due to (2.97) that has to be provided from the external power. Similar result is obtained for n2 electron tunneling through the second junction on the island and the external work W1,2 for both cases reads as: W\ = n\eVsC2 and W2 = n2tVsC\ C?2 C? Total energy is of course the sum of the capacitor energies in both junctions: EC = 1C1V12 + 1C2V22 = C1C2VS2 + (ne - q0)2 . 22 2C? (2.99) (2.100) 10The electron charge becomes e › -e0 and e > 0. 11The background charge is induced by stray capacitances and always present impurities on the island. 2.4 Electron transport in quantum dots 41 The free energy is a difference between (2.100) and (2.99): j?() jp 1 (1 (nnr-2 2\ ,r(n A ^ ^i, ^2 = -c/c — W = — - \L1L2Vs + ( ne ~~ Co) ) — eKs^i^2 + C2rrii) • (2.101) 61 2 To get the conditions for an electron to tunnel onto the island and off it, we write the difference in free energies: AF = F(n\ ± 1, n2) — F(rii, /12) = 77 ( — =F (C^^s + ne — qo) ) , (2.102) 61 v 2 / AF = F (ni, ii2 ± 1) — F (ni, TI2) = 77 ( — =F (CiVs — ne + go) ) • (2.103) 61 2 The transport ni + 1 and ri2 — 1 from source to drain is possible if the difference in free energy falls below zero. For the case of n = 0 and qo = 0 we get: e e Vs1 > 77 from (2.102) and Vs <-----from (2.103). (2.104) 262 2Ci For symmetric junctions Ci = C2 the condition reads in the compact form \Vs\ > e/Cz. If we want to include the gate, the (background) charge on the island has to be modified since the gate electrode additionally polarizes the dot: q = —ne + go + Cg(Vg — V2) or q0 —> q0 + Cg(Vg — V2). (2.105) This addition transforms the voltages over the junctions from (2.98) into: (C2 + C„)Vs — C„V„ + ne — qo CiVs + C„V„ — ne + qo v 1 =-------------------7;--------------------, y2 =--------------7-;---------------, (2.106) where Ci_ = Ci + C2 + Cfl. The energy differences from (2.102) and (2.103) become also gate voltage dependant and take the form: AF = 77 ( - =f ((C2 + Cg)Vs — CgVg + ne — go) ) , (2.107) 61 V2 AF = — ( - =f (CiVs + CoT/g — ne + go) ) • (2.108) 61 2 The conductance conditions are the same as before and include lowering of the free energy and thus negative values of (2.107) and (2.108) to enable transport. We get a functional dependance for Vs and Vg in form of boundary lines with different slopes for tunneling through the first junction onto and through the second junction off the island. It reads: Cg e ne Vs > pz-----^0 +77;----- ~~ 7^-----) (2.109) ^2 + Cg 2(Ü2 + Cg) 62 + Cg Cg e ne Vs < ~~ 77 ^g----- + 77. (2.110) Gl 2(61) Gl This family of curves is usually depicted on Vs — Vg plot also known as the Coulomb diamonds. A scheme is shown on Fig. 2.16 for the case where Ci = 30% Cz. The white 42 2.4 Electron transport in quantum dots conducting regime non-conducting regime ici n _____^. Points: A—^ T^2^(q2q) Figure 2.16: A V^ vs. Ug plot for the family of the conduction conditions from (2.109) and (2.110). The white parallelograms denote the non-conducting regimes with corresponding number of electrons on the dot n, whereas their shaded complement stands for the conducting conditions. Two slopes k\ and Jc2 describe the linear boundaries of each n for tunneling onto the island through the first and off it through the second junction respectively, with coincidental zeros for same family with given n positioned at —e/2Cg+ne/Cg. parallelograms are the regions where the transport gets suppressed, each off them carrying n electrons onto the dot ascending in integers from left to right. The shaded areas stand for the conductance regime according to (2.109) and (2.110). Also the zero-bias oscillations are present with the period e/Cg along with the other characteristic points (intersections A, B, T) and both slopes k\ and Jc2 from (2.109) and (2.110) respectively. This rich result gives first clues that the transport in minute and disordered systems may yield a rich palette of possibilities, resulting as voltage regions of suppressed conduction. An isolated island can thus produce regions in parametric space that prohibits electrons to pass, which has to be taken into account when experiments for familiar systems are attempted. Unfortunately the story doesn’t end here when real measurements are performed since one has to take into account also the realistic measurement equipment that places the measured system into an environment that affects the transport as well. In the following subsection we discuss this problem in detail. 2.4 Electron transport in quantum dots 43 2.4.2 Environmental Coulomb blockade theory In the previous subsection we discussed the transport properties from the transport blockade point of view. Even though obtaining the rich Coulomb diamond behavior demanded taking into account the environmental influence more closely, the dynamics of charge equilibrium reestablishment through the capacitive junctions still remained hidden. The treatment of the ultrasmall junction dynamics coupled to an external power circuit with finite impedance is know under the name Environmental Coulomb blockade theory. The theory’s formalism is rather tedious and in most cases the final results aren’t analytically solvable, often leaving the field for further theoretical exploration. For this reason only the basic steps are introduced, omitting the detailed derivations and giving only the core results. An ultrasmall junction under investigation is composed of metal-insulator-metal series with finite tunneling rates V and V across it as depicted on Fig. 2.17a ([8] chapter 2). Such an element is embedded into an electric circuit with the impedance Z^, and a power source V (Fig. 2.17b). r,r Z((o) metal metal C, RT V the symbol a) b) Figure 2.17: a) Schematic drawing of a metal tunnel junction. The arrows indicate forward and backward tunneling through the barrier. b) An ultrasmall tunnel junction with capacitance C and tunneling resistance RT coupled to a voltage source V via the external impedance Z?. The current vs. voltage characteristic is obtained as a difference between the charge tunnel rates in the opposite directions: i(v) = e( r (v) — r (v)). (2.111) The Hamiltonian for the whole system includes the quasi particles in electrodes, the environmental part and the tunneling Hamiltonian: H = Hqp + Henv + HT. (2.112) 44 2.4 Electron transport in quantum dots The Hqp and the Ht include more convenient variables for phase difference p and charge Q in the treatment: p(t) = p(t) - %Vt (2.113) n and We can write them in form: Q = Q-CV. (2.114) Hw = J2(ek + eV)clacka + J2tqclacqa (2.115) ko qa HT = ^T^c^e-^ + tf.c., (2.116) kqa where k and q denote the wave vectors with spin a on the left and right electrode. In the (2.115) the sums correspond to the electrons over the energies tk and tq for the left and right one, respectively. In the (2.116) the term describes the annihilation of the electron with the wave vector k and spin a in left and the creation of the electron with wave vector q and same spin on the right one. The T\.qa denotes the matrix element for such left-to-right tunneling event. If the tunneling resistance Rt is large compared to the natural resistance scale in terms of resistance quanta Rq = h/e2 then the states in the electrodes mix weakly and the term Ht can be taken as a perturbation. Moreover if we assume that the charge equilibrium is established before tunneling (the time between two tunneling processes in larger than the charge relaxation) the tunneling rate from (2.111) can be obtained through the Fermi golden rule: 27t ~ 2 rw = T (f\HT\t) b(Et-Ef) (2.117) for tunneling from the initial state \i) to final state |/). To obtain the matrix element they are written as a product of quasiparticle state and a charge state that becomes coupled to the environment (they are called the reservoir states): \i) = \E)\R) and |/) = \E')\R'). Now the factors in tunneling Hamiltonian act separately on quasiparticle space and on defined reservoir states and the matrix element reads: {f\HT\i) = {E'\Hf\E)(^\e-^\R) + {E'\Hf\E){R'\e^\R}, (2.118) with: Hf = YJTkqc)qacka. (2.119) kqa The total tunneling rate is a sum over all initial states weighted with the probability to find this states and over all final states. The expression to be evaluated now reads: r(V) = 2L dEdE'Y,\{E'\HTP\E)\2\{R'\e-^\R)\2 " -^ R,R> x Pß(E)Pß(R)b(E + eV + ER-E>- E'R). (2.120) 2.4 Electron transport in quantum dots 45 Further derivation analyzes this expression in detail, finally presenting it in the form: T^t^ ! f°° # n I (v/) =----- dE---------ST-T(eV — E), (2.121) e2RT-co l-e~ßE where Rt gathers all the constants and denotes the tunneling resistance. The P(E) may be interpreted as the probability to exchange the energy between the tunneling electron and the environmental modes that can become excited in the resonances. Basically the tunneling electron excites environmental modes, that form according to the impedance and this is the core mechanism of environmental influence on the transport. Formally the P(E) is the Fourier transform of the phase-phase correlation function J it): 1 f°° P(E) = — / dte *- '+li , (2.122) 27tft y.oo that is formally defined as: J if) = {[ 1 eV T (V) =------------s, (2.130) e2RT 1 - e~ßeV giving the current vs. voltage characteristic through (2.127) in Ohmic law form of voltage-biased tunnel junction: V 1(V) = —. (2.131) Rt In consequence the Coulomb gap can not be reached even at highest voltages. High impedance environment In contrast to low impedance, high impedance environment allows the electron to easily excite the modes. It turns out that the spectral density is peaked at ui = 0, making Ohmic damping, i.e. Z(uS) = R, most suitable to consider this limit. The total impedance is given by R/(1 + (ojRC)2), but takes the form (n/C)b(u)) for the limit ui —> 0. The correlation function J(t) gets the form: 7t / 1 2\ J(t) = — it + —t , (2.132) CRq nfj that gives P(E) from (2.122): (E-EC) 2 P(E) = e 4EckT } (2.133) \4rtEckT where Ec denotes the transfer energy to the environment. In the low temperature limit kT <^i Ec the (2.133) simplifies to: P(E) = b(E — Ec) (2.134) and the current vs. voltage characteristic can be obtained: eV — Ec I ( V ) =------—-----\(eV — En), (2.135) where x(E) is the step unit function. Since according to (2.134) a tunneling electron always transfers the energy Ec to the environment, tunneling becomes possible only if the energy eV at disposal exceeds Ec. Here we observe the Coulomb gap that corresponds to the charging energy similarly as in the isolated island case only that here the charging energy of the junction coupled to the environment causes the suppression of the transport. 2.4 Electron transport in quantum dots 47 Ohmic impedance The case of finite and frequency-independent impedance Z^ = R is often understood under the term of environmental Coulomb blockade theory, since their results, even though in general analytically non-solvable, become in limiting terms most applicable when compared to measured data. We restrict ourselves to the case of zero temperature and consider the limits of low and high energies in P(E). For that we utilize the relation from (2.129) that is valid for zero temperature. To evaluate the integral the ratio between the real part of total impedance and the resistance quantum Rq has to be obtained: HeZt 1 T 1 1 1 1 = —Re------- =---------, (2.136) Rq Rq iujC + l/R g 1 + (uj/ujr) where the dimensionless parameter g and the frequency uir stand for: Rq g =— (2.137) R 1 g Ec ujr = -----=-------. (2.138) RC 7t h The P(E) is obtained using the differential equation that arises after the derivation of (2.129): dP(E) 2 P(E) — 1, (2.139) 2 g dE g E with the solution: P(E) OC ti 9 (2.14U) for small positive energies. For negative energies P(E) vanishes since we consider the case of zero temperature. With a more detailed analysis of J(t) and P(E) one may determine also the normalization constant, completing the (2.140): g-2y/fl -y [jih]~\g P(E) =— — ^ , (2.141) l(2/g) E g Ec where y denotes Euler constant12. To finally calculate the I(V) dépendance the expression (2.126) gets simplified for zero temperature and assuming V > 0 into: I(V) = dE(eV — E)P(E), (2.142) ôRt o eRx before inserting the (2.141) that yields: g-2y/fl y [ttelV^l 1 B I(V) =-------- —F— at T = 0 K and V —> 0. (2.143) r(2 + 2/g) Rrp g Ec Now also the experimentally obtainable zero-bias anomaly of the conductance at low temperatures can be expressed: /2 \ e-2y/g 1 rTt e|V| 1 ^ (7(low T, low V) = (----hi] ^---------------------— • (2.144) g r(2 + 2/g) Rx g Ec Y = 0.577... 48 2.4 Electron transport in quantum dots This is one of the main results of environmental Coulomb blockade theory and importantly remains valid in terms of power law behavior (the prefactor depends on the high-frequency behavior of the impedance) even for general environments as long as the zero-frequency impedance Z(0) stays finite (e.g. Z = y/L/C). The power exponent changes accordingly into: 2 2Z(0) g Rq For high energies the P(E) in general behaves according to: 2ReZt(E/h) P(E) =--------------, E Rq that gives with (2.136): 2g EC2 P(E) = ?2 E3 for E › ?. Inserting it into (2.142) gives the current vs. voltage: i(y) = RT v — g e2 1 2C 7t2 4C2 V + for V ›?. (2.145) (2.146) (2.147) (2.148) As expected for higher voltages the behavior approaches Ohm’s law, with the shift in /(V)[ 2G./iT dI(V) a) V[e/2C] b) 2 V[e/2C] Figure 2.18: Zero-temperature a) current-voltage characteristics for the Ohmic model and b) the derivative dI/dV that approaches the constant value 1/RT for high voltages at g = ?,20,2,0.2 and 0 from top to bottom. charging energy. The numerical investigation from Fig. 2.18 depicts the situations for different values of parameter g (or resistance R). The curves show the transition from low- to high impedance environment demonstrating that quantum fluctuations destroy the Coulomb blockade. As a criterion for the occurrence of the Coulomb blockade one may require that for vanishing voltages the curvature for the current-voltage characteristic goes to zero or current derivative as a function of voltage that starts at some initial value _ 1 e 2.5 Variable range hopping 49 (close to zero for low g values) and approaches a constant value for high voltages (Fig. 2.18b). One can see the parameter g as a function of temperature since the environmental resistance could be connected to the coupling between the system and the current leads. In this case the Fig. 2.18 could indicate behavior for different temperatures, but with addition, that the high voltage slopes (and the asymptotic values for the derivative) could also change with temperature since the Rt gets affected as well. 2.5 Variable range hopping Until now systems with rather specific scattering sites have been introduced that resulted in specific transport characteristics based on the nature of electron interaction with the scattering entity (barriers from 2.3, quantum dot from 2.4). A more general approach [12] on the other hand deals with a non-crystalline system as a whole, treating imperfections as possible electron traps acting as localization places overlooking its local nature (whether its an impurity, local defect, and alien island, a lattice deformation). If we take a look at the possibilities regarding scattering we get three scenarios observing the uncertainty Ak compared to k: 1. Ak/k < 1- scattering is weak and surfaces of constant energy are spherical (e.g. liquid metals). 2. Ak/k ~ 1 - scattering is strong and k is not a good quantum number for describing eigenstates and the concept of Fermi surface (for metals) is no longer valid. 3. Yet stronger interaction yields a localized wavefunction ipE at some given energy E. We are particularly interested in the last case where strong scattering suppresses the electron transport. Here the wavefunction ipE with quantized energy is confined to a small region of space, falling off exponentially with distance13 as exp(—aR). What is surprising is that even though one can have a finite and continuous density of states N(E), all states are localized, although there can be strong overlap between the wavefunctions of neighboring states. Moreover if the states are filled up to a limiting Fermi energy in the range where states are localized, the conductivity (g(E)) vanishes as the temperature tends to zero. This is very different from the crystals where the insulating behavior occurs when the Fermi energy lies in the region with vanishing N(E). In other words the non-crystalline materials can become insulators even with the finite value of N(Ep). The materials that exhibit such a property are called Fermi glasses. The statement of complete suppression conductivity at zero temperature can be manifested in vanishing of the total wavefunction contribution to conductivity at given E obtained as configurational average: {(Te)- (2.149) In fact this criterium is used as satisfactory condition of localization for wavefunctions of energy E. To get ge, commonly an electromagnetic wave F cos ut is used to get ce(cj) and 13This localization is known as the Anderson localization. For consistency with the literature we use the symbol R for the distance. 50 2.5 Variable range hopping then by limiting the ui towards zero the ce(0) can be deduced. The approach is convenient because we can grab the problem with Fermi golden rule and write for some tbE(x,y, z) in a volume CI the probability per unit of time that an electron makes a transition for a state with energy E to any of the states with energy E + fko: 1 9 <,27T. 9 ^„T/-^ *. -er —Ixe+he lgiiiV(-fc + fko). (2.150) An ë The matrix elements are averaged over all states with the energy near E' = E + fko and are obtained by: XE+hw,E = / rtpE,x(rtpE)d3x. (2.151) By redefining the matrix element from (2.151) to h f d o h XE+hw,E =----- rtpE+Tiw^rWE)d x =-----DE+hE (2.152) mu ox mu and by defining the conductivity (JE(u) so that <7e(cj)öF2 is the mean rate loss of energy per unit volume we integrate the conductivity over all energies multiplied by the number of occupied states per unit volume in the energy range dE and taking into account the probability of (un)occupied states for up- and downward jumps along with the spin factor 2, we get the expression: 2ne2h2Q. fri, (JE(uj) = ------7:----- [j(E){l — f(E-\- nu)) — m2uj — f(E + Hcj){l — f(E)}]\D\N(E)N(E + Hcj)dE. (2.153) After limiting the expression at zero temperature for ui —> 0 the final result known also as Kubo-Greenwood formula arises: 2ne2h3Q. with ce(0) =--------\DE\av„{N(E)} , (2.154) m2 d o . . De = WE'^~vpE)d x [E = E ). (2.155) ox Again the avg represents an average over all states E and all states E' such that E = E', so that at T = 0 the conductivity 2r(p2- 2.5 Variable range hopping 51 b) N(E) Figure 2.19: a) Potential wells for a crystalline lattice with corresponding energy band of width B. b) Crystalline lattice is randomly altered within Vq to create a non-crystalline lattice - Anderson lattice. The localized states are usually described in the Anderson model where crystalline potential wells separated by a with band-width B (Fig. 2.19a) are randomly modified within the spread of energies Vq to create a non-uniform potential (Fig. 2.19b). Using the tight-binding approximation, many studies have been made to observe the transition between crystalline and non-crystalline structure in terms of functions becoming localized. The Anderson criterion uses the ratio Vq/B as the limit that makes the states localized together with the coordination number z that is connected to14 B. We can say that this parameter describes the non-crystallinity of the system transforming a crystal into a random lattice. First calculations proposed the value 5.5 for the coordination 6 but further calculations tend to lower this value. What is important from our point of view is the result for one-dimensional systems that predicts all states to be localized. The more general case deals with situations in which the states are non-localized in one range of energies and localized in another. We also believe that this is most likely the case for a realistic system such as our own, quasi-onedimensional objects. Also the theoreticians explored the phenomenon first proving that the localized and non-localized states (when the Anderson criterion is not satisfied) cannot coexist for a given configuration, in fact they proved the existence of the critical energy EC that separates non-localized and localized states (Fig. 2.20a). They discovered that localized states are gathered near the extremities of the energy band without any discontinuity in N(E) nor in any of its derivations. EC is also known as the mobility edge. The position of this edge regarding the Anderson ratio was of great interest since the band structure and the basic properties depend on it. A schematic depiction in Fig. 2.20b shows that EC lies (measured from the middle of the band) on the limit of the band for Vq = 0 - perfect crystalline structure, then the band broadens and the edge moves outwards, reaches its peak and finally falls in the middle of the band after overpassing the Anderson criterion (in this case Vq/B = 2) entering fully In tight binding approximation B = 2zl where / denotes the transfer integral. 52 2.5 Variable range hopping localized case of non-crystalline lattice. a) b) Figure 2.20: a) Density of states in the Anderson model where the non-localized states are in the center of the band, separated by Ec and E'c from the localized ones near the band extremities. b) The plot of Ec against Vo/B measured from the middle of band. If the Fermi energy Ep lies within localized states the system resembles a semiconductor or better, a doped semiconductor where a gap is formed between the filled states (valence band for pure semiconductor or states of impurities) and the conductive continuum band. The conductivity can now be written in the same form as for the semiconductor by replacing the Eg with the Ec — Ep: e c -e f c(0) = <7mine kT , (2.157) with (Tmin denoting the the conductivity of continuum band. This form of conduction is predominant at higher temperatures or when E p lies close to Ec. Moreover if in the system of Fermi glass type the Fermi energy at zero temperature can move from below to above Ec (e.g. the change in composition/disorder of the system) there should be a sharp change in the DC conductivity from zero to a finite value as schematically depicted on Fig. 2.21a. Such a change is called an Anderson transition. Another theoretical investigation involves the temperature dependance of resistivity as a function of Anderson ratio Vo/B. Not surprisingly when the E p upon the Vq/B change passes the Ec into the localized states the gap opens and conductivity follows the law from (2.157)15 as schematically depicted on Fig. 2.21b. The second conductivity mechanism is called thermally activated hopping conduction and includes electrons close to the Ep. As illustrated on Fig. 2.22 the conduction rate is determined by the hopping of an electron from the state A below the E p to one above B. The probability p per unit of time that this occurs is determined by three factors: a) the Boltzman factor e~w'kT, where W denotes the difference between the two states, Here the pre-factor is taken to be constant towards temperature. 2.5 Variable range hopping 53 — 4 — 3 — 2 — 1 4 X/B a) b) Figure 2.21: a)The D.C. conductivity up as a function of E. b) The plot of resistivity p aginst T for values of Vo/B increasing from curves 1 to 4, evoking the conduction of the Fermi glass by energy excitations above Eq. Curve 2 shows the value of p for E p at Ec so that 1/p = (Jm-m. b) a factor z/ph depending on the phonon spectrum, c) a factor e~2aR containing the overlap of wavefunctions16, giving the expression: P = ^phC a W kT (2.158) In the external field F and at finite temperature T the current j can be obtained by multiplying the hopping probability with the number of electrons at Fermi energy 2N(Ep)kT followed by charge e, hopping distance R and finally taking into account the possibility of hopping in two directions regarding the field: j = 2eRkTN(Ep)i>vhe e kT = 2eRkTN(Ep)i'pile kT sinh — eRF kT (2.159) This is the most general result that is valid also for higher fields and temperatures. In weaker fields, eRF <^i kT the expression can be approximated and the conductivity gets the form: a = j F = 2e R N(Ep)^^ — 2aR— -j-F (2.160) For low temperatures the hopping distance R increases and the conductivity expression reforms. We need to calculate the maximum hopping probability knowing the activation energy W for the states in the range R (for 3D): W = 3 4nR3N(Ep) (2.161) 16This is called “nearest neighbor” or “Miller - Abrahams” hopping. 54 2.5 Variable range hopping EL B C Figure 2.22: The mechanism of hopping conduction. Two hops are shown from an occupied state A to B and from B to C. and taking into account the average hopping distance R: r ar OR ------------- = ---- j r2dr 4 R = —5-------= —. (2.162) The hopping probability (2.158) using R instead of R has maximum when17: 3 9 —a =----------------, (2.163) 2 47ti?4 N(EfjkT giving the optimum value for R: o 1/4 R=------- . (2.164) {2na N (Ep ) k T }i/4 Now the probability (2.158) reads as: __ B p = Up^e t1/4 } (2.165) where B equals: q 1/4 ol/4 a a B = Bo and Bo = 2 — . (2.166) kN(Ep) 27t Q 1/4 o a 6 and Bo = 2 — kN(Ep) 27t The conductivity (2.160) is therefore: _ B C* o o .B a = 2e R N^Ep)^^ t1/4 = —e R N^Ep)^^ t1/4 . (2.167) 8 Other treatments give similar results with the difference of numerical pre-factor Bo that can vary upon the method used. Assuming that z/ph varies little with R and T. EF 2.5 Variable range hopping 55 The same derivation can be performed also for two and one dimension, revealing the same behavior as (2.167) thus following general law, in theory often written in form: __ B a = Ae t^ (2.168) with adequate constants A, B and exponent A for each dimension. We gather the results in table 2.4. Perhaps a more convenient form of (2.168) from an experimental point of dimension A [e2R2 N(Ep ) i/ph] B S0 T0 X 1 2 3 i 2 8 3 9 8 Bo \ ,„?„ ,} B0 L "I 1 1 klNyhiF) I Bo i L ml A w klNyhiF) v2 / 3 "\ ^/^ \7t 2M\l/4 2a kN(Ep) Sa TtkN(Ep) 24a TCkN(Ep) 1 2 1 3 1 4 Table 2.4: The constants of (2.168) and (2.169) for 1D, 2D and 3D. view would be: (7 = ( o ) (2.169) with To also in the table 2.4. More recent work observing closely [31] the impurity distances and the effects of strongly anisotropic screening of the Coulomb potential yields yet richer possibilities for conductivity. The same general conductivity temperature dependance has been proposed but with additional possibilities taking into account also different cases of the density of states that arise due to 3D Coulomb interactions. The main result has the same form as (2.169): (TVRH)A ß + 1 (T = eoe t and A = (2.170) ß + d+ 1 where ß denotes the power-law exponent in the density of states N(E) oc Eß and d the hopping dimension. The values are gathered in table 2.5. dimension 1 2 3 N(E)=const ant N(E) oc \E\ N(E) oc E2 1 2 3 2 3 4 1 1 3 3 2 5 1 2 1 4 5 2 Table 2.5: The exponents A from (2.170). Variable range hopping appears to be applicable for systems with randomly distributed scattering places, covering all dimensions making the theory useful even in the nanoworld where the control over structural homogeneity is very limited. If this issue doesn’t play A 56 2.6 Luttinger liquid a major role for more regular structures (such as carbon nanotubes) the wide palette of familiar structures may cause the production of very disordered compounds that are more related to non-crystalline materials than to crystalline structures in terms of long range symmetry. 2.6 Luttinger liquid Strongly one-dimensional systems often exhibit different physics compared to three-dimensional, bulk material. We pointed out in the subsection 2.1.2 the spontaneous vanishing of long range order in 1D only due to space dimensionality. Similar peculiarities are observed when electronic states and excitations are explored implying very different behavior compared to the well known Fermi liquid that governs the macro world [6, 5]. Here the combination of the Pauli principle with low excitation energy (e.g. kT 0 regime, where the range of allowed excitations shrinks to a one-parameter spectrum uinu ~ vv\q\ (Fig. 2.23). In addition low-energy particle-hole pairs with momenta between 0 and 2kp are not allowed. The route leads -2kF 2k a) b) Figure 2.23: a) Particle-hole excitations in 1D. b) The spectrum of allowed states has no low-energy states with 0 < \q\ < 2kp. through the Hilbert space of states and includes utilization of bilinear forms in bosons for free fermions, describing the excitations. We just write the general form of the Hamilto-nian but we don’t follow the derivations any further: V~^ V~^ \ \ ( 1 + 1 H = y y vv\q\ bvqbl + ^—' ^—' ' 'y 2 (2.171) v=p,a q k 58 2.6 Luttinger liquid where vv = Vf. Theoreticians were dealing also with the calculations of basic charge transport properties and via the mentioned bosonization they obtained the result [3]18: / = I0T +a sinh e^ v 2kT r a ¦ eV I 1 +------- %---- 2 2"7TkT (2.172) where a depended on the Luttinger parameter g = vp/vp. Since experiments inevitably included macroscopic leads, the results include also the charge transfer from Fermi liquid to Luttinger liquid (FL-LL) and also realistic cases of impurities inside of wires with adequate Luttinger liquid - Luttinger liquid connections (LL-LL). Both cases give different a(g) dependence: «LL-LL = (g 1 - 1)/4, «fl-ll = (g + g~l - 2)/8. (2.173) In realistic cases g proves to be small and the exponents appear to be connected via «LL-LL = 2cüpL-LL. From the experimental point of view a closer inspection of the curve (2.172) is needed to successfully extract the parameters. The basic characteristics emerge: 1. all IV curves for different temperatures collapse to a single curve with typical knee when I/Ta+l is plotted against eV/kT (Fig. 2.24), 1000 100 10 1 0.1 0.01 10 100 eV/kT [a.u.] 1000 Figure 2.24: The collapsed curve with typical knee emerges when I/Ta+l is plotted against eV/kT. 2. at low temperatures the LL-LL junctions become most resistive governing the conduction power-law in the low-voltage regime as G oc TaLL~LL, 3. at high voltages the FL-LL junctions dominate and / oc \/apL_PL+1. 18In literature one may find different manifestations of the expression, sometimes even mistyped. The correct formulation includes in the imaginary part of the Gamma function ? also in the fraction, otherwise the curve exhibits very different behavior. i 2.6 Luttinger liquid 59 Often [10, 11] the exponents are renamed, cüll-ll to simply a and cüfl-ll to ß. Since the dominant junction prevails, the voltage V in the expression (2.172) denotes the actual voltage drop over such a junction and not the high bias pressed over a system. For this reason their ratio in form of parameter 7 is introduced to the characteristic so that the final formula reads: 2 (2.174) This is also the core result for Luttinger liquid applications in terms of charge transport along 1D confined systems. With this rather exotic theory we conclude our journey through the theories that apply to confined and nonuniform systems. In the following chapters we focus on the experimental part i.e. the manufacture of the measurement chips, sample preparation, post-production treatment and the measurement itself. Finally the results are discussed in the spirit of the theories presented here, thus deducing the nature of the electron transport along our nanowires. / = IqT sinh jeV 2kT ß ¦ leV r 1 + + i— 2 2tikT Chapter 3 Sample preparation and measurements Before we describe the experimental work in detail let us first present the aim of our research in terms of practical realization. As mentioned in the introduction of this report our goal was to determine/measure the electrical conductivity properties of thin MoSIx nanowire bundle as schematically depicted on (Fig. 3.1a). This basically means that we have to plug ends of the device under test to an electrometer and measure the current vs. voltage (IV) characteristics of the system. In the macro word this presents a rather trivial task since the standard measurement equipment can be used along with the macroscopic connections to the measured sample in form of different pins, crocodile clamps or we can even solder the electrodes to the sample. I V T sample a) Figure 3.1: a) A simplified sketch of IV measurement as a function of temperature for some arbitrary sample and b) the artistic expression of electrodes introduced over a thin, single nanowire bundle and c) a bundle (dielectrophoretically) attached over a narrow prefabricated gap between two electrodes. The situation changes severely as the sample size decreases to nanoscale especially when the objects can be observed only with modern microscopy techniques such as elec- 3.1 E-beam lithography 61 tron microscopy or atomic force microscopy. In the nanoworld the question of contact introduction to the measured structure becomes a technological issue accompanied with a myriad of peripheral factors that affect the whole process. We have two possibilities of wiring minute structures: the selected object is located under a microscope, followed by a targeted placement of metal contacts over it (Fig. 3.1b) or reversely, the object is guided and pinned over prefabricated electrodes with a narrow gap between them (Fig. 3.1c). Also the handling of the sample demands an epopee of efforts since such minute objects need special attention even to be observed, seen and defined as chemical compounds not to mention the manipulation and targeted pinning of structures needed in order to connect them to the macroelectrodes. In this chapter we focus on the issues of minute electrode/device manufacture, processing of the sample and the procedures for bundle attachment to the circuit electrodes that lead to a successful measurement. Let us first summarize all the steps needed to reach the goal of measuring current - voltage characteristic of a thin, single nanowire bundle: • nanolithographic manufacture of microchip devices, • sample preparation, • attachment of the bundle to the electrodes, • temperature annealing, • measurement. 3.1 E-beam lithography The manufacture of simple microchip devices that served as the contact electrodes of nanowire bundles is step one of our experimental procedure since without the skill to produce electrodes the measurements aren’t possible. We addressed this problem with standard e-lithographic procedure widely used across the world in other research facilities. Depending on the budget the whole technique, along with the accompanying equipment and the know-how, can be commercially acquired. This high performance instruments are specially developed for the task of minute structure production with high accuracy, stability and quality. In many cases such systems are modified for the purpose of similar electronic measurements and even nanoparticle manipulation [32, 33]. Moreover such tasks demand special research laboratories and facilities in order to reach the highest nanoscale performance along with the quality and reproducibility. Custom built vibra-tionally isolated rooms and clean rooms with controlled air flow that require strict working policies for the researches are required for such quality. Due to limited funds we started with a modest 40 000 EUR second hand electronic microscope (Fig. 3.2) that operated fully analogically without the built-in lithography electronics. Moreover the whole lithographic procedure needed to be introduced and developed from scratch: the resists and accompanying chemicals, e-beam writing, chemical treatment, metal introduction and the finalization of the process. Also the laboratories weren’t optimal since we had only a very 62 3.1 E-beam lithography general chemical laboratory as the center for lithographic techniques. All these factors predetermined our best resolution but for structures of several hundreds of nanometers in size (length) it was sufficient and we carried on with the development. Even though our pre-graduate work [1] made first steps and roughly introduced the technique, the goal of measuring one single and thin bundle by far exceeded the performance and skills of the acquired procedures. We spent most of the research time tinkering, modifying and refining the whole lithographic procedure in order to reach the desired performance level. Let us stress that according to before-mentioned lithographic and system limitations we didn’t pursue developing the approach of contacts over a bundle but we concentrated solely on the attachment of a bundle over a gap between the electrodes. We mentioned both cases since the first one is dominantly used. Figure 3.2: The electron microscope JEOL JXA-840a modified to perform e-beam writing. We used software that guides the electron beam through an interface whereas other settings are adjusted manually over the control panel. What lithography actually is? The word comes from German “die Lithographie” that is coined from Greek “lithos” stone + “graphein” write and can be thus translated as “to write or draw on stone”. The technique was used in 19th century for fast printing and the reproduction of images and is practiced even today by many artists [34] to imprint the features into a stone using hydrophobic ink followed by a special chemical treatment. When the ink is applied afterwards with a roller it attaches only to the pre-drawn areas 3.1 E-beam lithography 63 whereas unspoiled areas remain pure1. If a paper is pressed against the surface the ink leaves a trace of the image on the paper (the mirror image, of course). You can think of it as a very sophisticated and heavy stamp. This differentiation between written and unwritten areas is the basic concept also in electron- or photolithography. Figure 3.3: a) A layer of an e-beam resist is spun on a substrate and a predesigned pattern is written with an e-beam over it. b) In the process of development the electron treated areas are dissolved leaving trenches of exposed substrate c) so that the sputtered metal attaches directly to it in the desired and pre-written shape d) that remains after the lift-off. The process namely starts with the writing of desired circuit patterns over an electron or photon sensitive resist layer on top of a nonconductive substrate (Fig. 3.3a). Afterwards the chemically changed written areas are dissolved in appropriate solvents producing trenches in the shape of the desired circuit in the resist (Fig. 3.3b). In those trenches the substrate is totally exposed whereas elsewhere it remains covered with the resist. The following process of metal deposition covers the entire substrate with a thin layer of desired metal thus depositing metal inside the trenches as well as on top of the resist (Fig. 3.3c). During the lift-off the entire remaining resist is dissolved along with the metal deposited on top and washed away, leaving only the metal in the trenches which remained unaffected by the lift-off process. The end result are the metal electrodes shaped in the form of the pre-written pattern (Fig. 3.3d). The smallest feature limit using such 1Children utilize similar but reverse technique for fast painting of background with water based colors on pre-painted wax crayon picture. 64 3.1 E-beam lithography a procedure is determined by the resolution of the trench production, which is directly related to the method used. In the case of photolithography the lower limits are at several hundreds of nanometers (related of course to the wavelength of the light), whereas in the electron beam writer the smallest feature can reach even several decades of nanometers with the best equipment. Let us itemize the e-lithographic steps we needed to acquire: 1. e-resist handling and spinning, 2. e-beam writing, 3. developing, 4. metal sputtering, 5. the lift-off. Since we used this wide spread and standard procedure as a tool to serve the prime goal of single bundle measurement we discuss only the relevant points that affected the shape and quality of the circuits and consequently the measurements. 3.1.1 Substrate and e-resist spinning Successful circuits must be fabricated on a nonconductive substrate in order to prevent the obvious short circuit between the features via the substrate. On the other hand e-beam writing is most successful on better conducting substrates so we have to make a compromise. A good balance can be achieved by using a silicon wafer with a thicker oxide layer to make the surface non-conductive, but with the sufficiently conductive core to prevent high charging that disables accurate e-beam writing. The plates can be bought with arbitrarily thick oxide layer2. We selected 600 nm of oxide on top of 300 µm thick wafer, extra polished to reduce the surface roughness to minimum so that small bundles would be clearly spotted on the surface. For controllable and even reproducible production of circuits a uniform layer of well defined thickness must be deposited on the substrate. One of the best ways to achieve this is spin casting. A droplet of resist dissolved in an appropriate solvent is put on a substrate and spun for several minutes. The surplus liquid is then (centrifugally) ejected from the surface leaving a trail of thin layer over the whole substrate area. The thickness can be controlled and is proportional the spinning frequency and inversely proportional to the concentration of the e-resist. When we mastered the basics of the lithography we preliminarily modified a general centrifuge into a spinner, which worked satisfactorily for early tests and also for bigger circuits/bundles later on. As we approached the scales relevant for our nanoscale objects it became clear that a professional spin coater was required3 (Fig. 3.4). We acquired an adequate unit that not only simplified the resist deposition but also enabled full control and perfect reproducibility. 2The web page of the distributor: www.universitywafer.com. 3The unit was bought from Laurell Technologies corporation: www.laurell.com. 3.1 E-beam lithography 65 Figure 3.4: WS-400B-6 NPP/LITE spin coater was used in order to reproducibly deposit a thin and uniform layer of e-resist over the silicon wafer. The e-beam resists were picked according to the suggestions of our e-beam lithography chemicals provider4. Commonly researchers use two different layers of e-resist on top of each other, the upper one slightly less sensitive then the lower one. For the bottom layer we used a 2% solution of poly(methyl methacrylate) (PMMA) MAA/33% in ethyl acetate spun at 4000 rpm for two minutes (the resulting thickness was ?190 nm) followed by 10 minutes of baking on a hot plate at 200?C and 1% solution of PMMA 950k in chlorobenzene spun at 6000 rpm for two minutes (the resulting thickness was ?90 nm) followed by 10 minutes of baking on a hot plate at 160?C. As we discuss in the section 3.1.3 this choice was necessary and even inevitable since the deposition of metal via sputtering isn’t as successful as one might hope or expect. 3.1.2 E-beam writing and developing In the e-beam illumination step the stage is set for the production of the final circuit. As described above the electrons bombard the substrate/e-resist in the predesigned pattern in the form of the desired circuit (Fig. 3.3a). We used a scanning electron microscope modified with the help of the company LPKF5, and succeeded to directly implement 4The web page of the e-resist and the accompanying chemicals provider: www.allresist.de. 5LPKF Laser & Elektronika d.o.o., Polica 33, SI-4202, Naklo, Slovenia 66 3.1 E-beam lithography their laser guidance software (Fig. 3.5) used in laser cutters through the development of an adequate computer-microscope control panel interface that converts signals from the software output into analog signals that directly control the position of the e-beam in the microscope. We also needed to learn how to use our system and to determine basic operational parameters along with the basic procedures. Even though we spent a considerable amount of time acquiring the new technology we will not discuss the details further since they are rather technical and specific. Figure 3.5: A CAD program (SCAPS) enables to draw arbitrary patterns and to transfer them directly to the e-beam guidance of the electron microscope. Additional features such as beam speed and hatching greatly simplified the design and the writing steps, thus lowering the production time of circuits. In the process of writing the backscattered electrons play a decisive role in trench production since the resist gets illuminated also from beneath (Fig. 3.6a). This effect actually dominates the final width of the channel making it roughly proportional to the e-beam current; the phenomenon is known as the proximity effect[35, 36] and must be carefully taken into account when we deal with the chemical development step. The chemically altered illuminated areas are more soluble in developer (in our case a mixture of methyl isobutyl ketone and isopropanol in ratio 1 : 3) and dissolve faster leaving behind the voids that form the trenches. Since also the unspoiled areas dissolve slowly a treatment in stopper (in our case pure isopropanol) is needed. This fact along with the poorly defined limits between the illuminated and unspoiled areas due to the proximity effect makes the time of development crucial for the successful production of channels; in our case 50 ± 2 s. Let us stress at this point that the step exhibits extraordinary sensibility, narrowing the parameter margins of previous processes (spinning, writing, developing, stopping). 3.1 E-beam lithography 67 Since our machine along with the rest of the procedures shows some intrinsic instability accompanied with a residual uncertainty, the overall yield of successful trench production gets reduced. If everything goes well the resulting channel has the edges of the form of a roof edge that sticks outwards from a supporting wall Fig. 3.6b. top resist bottom resist SiO2 silicon wafer a) b) Figure 3.6: a) The electrons scatter backwards illuminating the e-resist also from beneath. b) A successful development gives a channel with roof-like edges. The proximity effect doesn’t have an important role in the production of big structures, whereas for the smaller ones the relative discrepancy between designed and actual structure grows with the size reduction of the features. The effect especially starts to become a problem when we would like to create two trenches close together. In this case the proximity effect stretches also to the nearby feature illuminating the wall between them two times as much. This often results as the overexposure of the separation e-resist wall in many cases also as its collapse sometimes even fusing together both channels. This issue affects our circuit production directly since the electrode design includes a narrow channel between the metal contacts and consequently writing of two wide trenches close together. The production yield is reduced even more in this case since the optimal time of development for normal edges differs from the one for nearby trenches. Practically this means that the separation walls collapse before the other edges get adequately developed. On Fig. 3.7 the AFM amplitude images of successful trench production along with typical failed attempts are presented. In most cases the top e-resist layer of the wall structure gets dissolved partially or completely (Fig. 3.7b and 3.7c), or the sharp edges get rounded raising the probability of electrodes getting fused in the process of metal deposition (Fig. 3.7d). This combined with poor stability of the electron microscope writer makes the production of circuits extremely demanding and pain staking already at this point stretching our research time more then expected. 68 3.1 E-beam lithography Figure 3.7: The AFM images of typical trenches carved in the e-resists: a) successful, b) collapsed top e-resist layer of the wall in the middle, c) e-resist collapse throughout the entire length leaving only a narrow stripe of bottom layer and d) rounded edges of the top layer that can result as contact fusion after metal deposition. 3.1.3 Sputtering and lift-off In previous steps we were able to optimize the procedures in order to manufacture adequate trenches in the layers of e-resist. Since we didn’t possess our own metal deposition unit we were forced to seek collaborations in the group for thin metal coatings at our institute6. The group has an accurate, reliable and diverse metal sputtering system that enabled us to perform uniform and reproducible metal deposition over our carved e-resist layers. Unfortunately their procedure is optimized so that the metal gets deposited at every possible angle so that even holes, indentations or complex shaped objects get uniform coating without any shades or missed spots. This however was a serious obstacle for our 6Department for thin films and surfaces. Their webpage: http://www.ijs.si/ctp/ijs-dept-f3A.html-l2. 3.1 E-beam lithography 69 deposition since we were hoping for a perpendicular introduction of the metal in order to create confined depositions in the middle of the trenches with metal attached to the edges. As depicted in Fig. 3.8a the metal nicely forms a narrow and well defined stripe if the metal falls perpendicular to the surface (as in the case of metal evaporation), whereas in the case of omnidirectional sputtering the metal attaches even to the edges, widening the features to larger size defined by the proximity effect (Fig. 3.8b). metal top resist bottom resist Si02 silicon wafer nominal width c) ¦ metal M top resist bottom resist SiO2 silicon wafer actual width d) Figure 3.8: a) The perpendicularly deposited metal falls in the middle of the trenches without touching the edges whereas b) for omnidirectional sputtering the metal is thrown also in the cavities beneath the wall’s roof-like top reducing the gap between two electrodes from the c) nominal width of the upper e-resist wall to d) the actual width determined by the base size of the e-resist wall. Alternatively one could propose to reduce the cavities created by this effect but this would backfire immediately since the metal deposition would create steep, almost vertical edges at the contact ends endangering the successful circuit production or in the case of narrow channel production it would result as fused contacts. In truth not even the last process of lift-off, when the rest of the e-resist is dissolved in remover (in our case N-Methyl-2-pyrrolidone for 12 hours) and disposed together with the overlying metal, 70 3.2 MoSIx Nanowires influences much the end result so we were forced to use this inadequate but uniform and accurate sputtering technique. Fortunately we had more freedom in the choice of the sputtering metal. Due to limited lithography performance we were forced to sputter only thin, up to 50 nm thick layers of preferably softer metal to ensure successful lift-off. The metal adhesion to the surface also needed to be strong enough to withstand all the processes during the later bundle integration procedures and measurements. Titanium has good adhesion and can be sputtered in a thin layer but it is rather hard thus obstructing the lift-off. Moreover it shows high chemical reactivity especially with oxygen resulting in non-conductive oxides lowering the conductivity of contacts in the circuit itself. One alternative is gold that appears soft and inert, but unfortunately it sticks extremely poorly to the silicon oxide wafer making it necessary to use a sticky layer of nickel, titanium or chrome to act as a form of glue for better adhesion of gold. In this way the overall thickness increases and the contacts show only limited durability since the gold layer wears down when mechanically stressed. Best results were achieved by using nickel or palladium. Both exhibit high chemical inertia, high adhesion to the substrate surface and also show great resilience towards mechanical stress. The difference between them is that the palladium contacts appear to have a higher low-limit of contact thickness than nickel. We observed that palladium shows a strong tendency to wear off when sputtered in layers thinner than 50 nm whereas nickel preserves its integrity to the limits of sputtering procedures; the thinnest layers were 15-20 nm thick. For this reason we chose nickel to produce measurement circuits7. Despite the described drawbacks we continued with the production of the circuits with relatively low yield of 10 - 20% and the production time of roughly one week. The minority of the produced contacts had a well defined gap between both electrodes and were considered successful (Fig. 3.9a) whereas the majority carried the legacy of poorly created trenches that resulted as extremely narrow gaps with possibility of connection between the electrodes (Fig. 3.9b) or even more unexpected features when metal was pushed beneath the upper layer creating a soft transition from one electrode to another with some well defined areas in between (Fig. 3.9c). Often the electrodes were surrounded by a metallic crown (Fig. 3.9d) that could be washed away with a cotton stick, making the circuit workable but only if the contacts weren’t fused at any point. 3.2 MoSIx Nanowires Armed with adequate circuits we focused our attention to the sample preparation. We were hoping to get a monodispersed solution of the thinnest nanowire bundles or perhaps even only single nanowires. In this section we introduce the chemical structure of our MoSIx nanowires along with a brief introduction to their synthesis and a recipe for the successful preparation of the nanowire dispersion in acetone and isopropanol. 7Some aspects of metal choices are presented also in the sections of dielectrophoresis and the annealing procedure. 3.2 MoSIx Nanowires 71 X 1» r .'; Figure 3.9: The AFM images of typical final circuits: a) successful, b) extremely narrow gaps with possible fused points, c) soft transitions of metal from one contact to another and d) typical metal crown surrounding the metal contacts. 3.2.1 The synthesis and the chemical structure of MoSIx nanowires MoSIx is actually a trademark name for this type of nanowires produced and distributed by the company Mo6 d.o.o.8. These wires form a structural family of inorganic nanowires composed of three different elements: molybdenum, sulfur and iodine under the formula: Mo6SxI9-x. The structure is closely related to Chevrel phases with the general formula MxMo6X8 where M stands for a metal and X for a chalcogen. In fact at the discovery there was a debate whether these wires weren’t just simply a new species in this rich family of structures. It turned out [14, 13] that indeed there was a close resemblance between the atomic arrangement in the nanowires and in most related crystal structures 8Mo6 d.o.o., Tehnoloˇski park Ljubljana, Teslova 30, SI-1000 Ljubljana, Email: info@mo6.com 72 3.2 MoSIx Nanowires such as Mo6S6I8, Mo6S2I8 and Mo6S8. The basic structural cells are closely related but with a major difference that in the case of MoSIx nanowires the lateral cross link bonds between the molybdenum octahedra via sulfur or iodine are missing (Fig. 3.11). Thus an interesting compound of many chemically separated wires is formed, held together only by weak van der Waals forces making them easily dispersable in various solvents. The individual wires on the other hand have a stable but soft structure since the molybdenum octahedra get longitudinally connected via sulfur planes surrounded by iodine ions. With other words: the MoSIx nanowires have the molybdenum backbone, that consists of molybdenum octahedra clusters connected via 3 sulfurs in the linkage plane altogether surrounded by iodine or sulfur ions (Fig. 3.10). Their close resemblance to Chevrel phases makes them come in greater packs of nanowires, in thicker bundles of various sizes. Figure 3.10: The MoSIx nanowire structure consists of molybdenum backbone formed by the octahedra clusters with 3 sulfurs in linkage plane, surrounded by iodine. a) The side view reveals the linkage planes and b) the cross section depicts the iodine arrangement around the molybdenum octahedra. Figure 3.11: a) The basic structural cell of separated MoSIx nanowires in a bundle and b), c), d) of most similar Chevrel phases with pointed cross links. Basic properties have also been studied using density function theory [37]. The calculations reveal finite density os states at EF for longitudinal direction (3.12a) thus predicting the material to be conductive. For the Drude peak damping of ? = 0.1eV the conductivity at room temperature is estimated to be around 5 × 1000 S/cm, which is two orders of magnitude lower than for carbon nanotubes. Soft bonds between the Molybdenum atoms 3.2 MoSIx Nanowires 73 and the linkage planes suggest possible distance modulation that can be denoted as the accordion effect. The calculations confirmed that assumption and determined two stable minima even in a flawless nanowire (3.12b). Figure 3.12: a) The density of states calculated using density function theory. b) The free energy calculation predicts two stable positions for the linkage sulfur towards the molybdenum octahedron. The described resemblance between various structures induced considerable uncertainty and sensitivity to the synthesis and dispersion preparation. Even though the synthesis follows a rather simple procedure, defining, controlling or even pinning the relevant parameters appear to be rather slippery tasks. Basically all the elementary ingredients in targeted stoichiometry mass ratio9 are put in a quartz ampoule, vacuumized (pressure 2 · 10-5mbar), sealed and put in an oven with homogeneous temperature10 of 720?C for 3 days. The resulting material has a puffy, wool-like appearance that contains residual iodine that evaporates spontaneously in 24 hours after the extraction from the ampoule. As pointed out the similarities of different stoichiometries and even to Chevrel phases make it rather hard to determine the proper structure of the resulting material. We can speculate that the thermodynamic growth conditions allow multiple structures (Mo6S6I8, Mo6S2I8 and Mo6S8) to be formed making the end material a multi-phase mixture. Unfortunately also the X-ray analysis cannot distinguish sharply between them so the material type is never completely certain. Also the growth mechanism and the environmental conditions aren’t well understood, thus reducing the reproducibility of synthesis. We were not directly part of material production research and had very little influence on the synthesis outcome so we focused intensely on sample dispersion in different solvents. Other groups dealt also with material post processing trying to determine the basic solubility properties of our material [24, 23]. Despite their elaborate work we couldn’t reproduce 9This sets the x in the Mo9-xSxI9-x formula. 10An oven without temperature gradients. 74 3.2 MoSIx Nanowires the dispersions according to their recipes at least not in the desired terms of small mean bundle diameter and of proper concentration. In addition to that the dispersions showed vivid agglomeration dynamics that expelled the wires from the solution to the bottom of the reservoir. After many failed attempts and the synthesis crisis we came up with a procedure that appears to have overcome all indicated problems. We have to keep in mind that our final goal of single bundle/nanowire integration demanded high quality dispersion, much higher than needed for average solutions suitable for population studies (UV-vis spectra, R¨ontgen analysis, sedimentation studies). Let us itemize the basic dispersion properties we needed to achieve in order to use it for single bundle integration: • a monodisperse solution, • low mean diameter of bundles, • few impurities, • high concentration, • low agglomeration and high stability of solution, • a dispersion in a solvent suitable for single bundle trapping over prefabricated cir-cuits11. The final recipe, that overcomes the itemized terms may sound rather simple, but it is based on many dead-end procedures. We discovered that we have to use the whole amount of material from the ampoule without any washing, dispersing, homogenizing or any kind of pre- preparation. Our initial attempts namely included all facts about the poor reproducibility of syntheses so we tried to use small amounts of given sample generation, learn the material handling for some specific synthesis and then to use it to prepare final solution for the experiment. As we discovered later using small amounts (< 5 mg) inhibited our sample preparation from the start since the population of targeted small bundles appears in the solution in rather small quantities. Using larger amounts (or as stated above, all of it) pays off since the results truly fulfill the demands by a large margin. The best solvent for initial dispersion of the bundles turned out to be acetone with a special emphasis on its purity. We believe that commonly used chemical reservoirs contain many impurities that influence the agglomeration and stability of the solution so we suggest to use fresh, perhaps even filtered or distilled chemicals if possible. For the dispersion we used an ultrasonic tip (small tip at a maximum allowed power for 10 minutes) to thoroughly disintegrate the initial material in a plastic reservoir with 50 ml of acetone to smaller meshes and single bundles (Fig. 3.13a). Afterwards the dispersion almost immediately separated into two phases: soluble and stable dispersion and the completely non soluble sediment that formed again a puffy phase on the bottom of the 11We discuss this later in the section of dielectrophoresis. 3.2 MoSIx Nanowires 75 reservoir (Fig. 3.13b). The upper dispersion had a yellow-orange appearance and was left for several days (7 days) to separate from the remaining non soluble phase. Afterwards the upper dispersion was carefully poured (10 ml) into a fresh glass reservoir and tightly sealed. We used glass beaker because we observe that the material tends not to stick heavily to the glass walls as in the case of the plastic (polypropylene) ones. Figure 3.13: a) The ultrasonic tip (Cole Parmer CP750 750W) was used to disperse as-synthesized material in acetone. b) The resulting solution clearly shows the soluble and non-soluble phases sharply separated. As we explain in 3.3 the proper solvent for the bundle incorporation into the circuit wasn’t acetone but isopropanol. To change the solvent we dried the nicely sedimented dispersion on a hot plate and immediately poured the isopropanol (10 ml) into the cooled reservoir. To redissolve the dried material stuck on the walls we used an ultrasonic bath at 100 % power for 2 minutes (Fig. 3.14a). The result was again a yellow-orange solution without sediments that appeared stable for months. We emphasize the color of the solution since we believe that it’s a strong indicator of dispersed bundles in the solution. The resulting yellowish tan (Fig. 3.14b) indicates thin and long bundles whereas more greyish, dark blue color (Fig. 3.14c) marks solutions with small amounts of single bundles, filled with crystalline objects that sediment extremely fast leaving the solution almost empty in terms of desired single bundles. Naturally we allow the possibility that the thinnest bundles exist even in this solutions but unfortunately they are useless for our circuit integration due to extremely small bundle concentrations. Along with the sample preparation we derived a more accurate test to determine the consistence of the dispersions. We simply measured the UV-vis spectrum of a solution and analyzed the peaks. We observed that at least one of the peaks appears to be connected to the small bundle population. As presented on the graph from (Fig. 3.15) the position 76 3.2 MoSIx Nanowires Figure 3.14: a) The ultrasonic bath (Transsonic digitalS ELMA T490HD, 2x40W, 40 kHz) was used to redisperse dried material in isopropanol. b) The successful final solution has yellow-orange color whereas c) dispersion with lower concentration of single thin bundles appears grey or dark blue. of the peak around 700 nm gives the information about the sample consistence. If the peak appears at lower wavelengths (between 690 nm and 700 nm) then thin bundles are present in strong concentration in the solution. The alternative cases of the peak appearing at higher wavelengths (greater then 700 nm) give smaller concentrations of the thinnest bundles. We have to admit that we constructed this test on the experience basis of also other researchers and we didn’t look more into it since we still had a long way to go till the successful measurement. a) 0.20 0.15 0.00 thin bundle dispersion (yellowish color) 454 nm / 2.73 eV thick bundle dispersion (gray color) 732 nm / 1.69 eV 1.0 483 nm / 2.56 eV 1.5 2.0 2.5 E [eV] 3.0 3.5 Figure 3.15: Successfully prepared samples (black curve) appear to have the most left peak of UV-vis spectrum pushed above 1.77 eV (700 nm) and don’t shift in time, whereas the failed ones (red curve) show the peak at 1.69 eV (732 nm) or lower. Also the upper peak follows same shift - 2.73 eV vs. 2.56 eV. 3.3 Dielectrophoretical attachment of single bundles over a narrow gap 77 Let us summarize the whole recipe once more: 1. the whole content of the synthesis ampoule is sonicated in 50 ml of acetone in an elongated plastic reservoir with a thin ultrasonic tip for 10 minutes (Fig. 3.13), 2. the dispersion is let to sediment for seven days, 3. the upper stable dispersion is carefully poured into a glass reservoir (10 ml), 4. to replace the acetone with target solvent the dispersion is dried on a hot plate (60?C) immediately followed by the introduction of the final solvent, 5. the material stuck to the glass walls of the reservoir is redissolved in an ultrasonic bath at 100% power for 2 minutes (Fig. 3.14). 3.3 Dielectrophoretical attachment of single bundles over a narrow gap The integration of a single, thin bundle into a circuit became the core of our research since the final measurements could have been performed by using standard measurement techniques. We were interested in electronic properties, particularly in the direct transport measurements that are perhaps the most demanding to perform since individual bundles have to be connected to the macro-electrodes. With this approach the preparation of the measurement chip is the hardest part since the technological skills to manufacture such circuit hybrids are pushed to a new level. The interpretation of the measurements on the other hand is quite straight forward and requires very little data post-processing in order to get the raw results for further analysis. In contrast to this individual approach there are population, statistical techniques that count on homogeneity of a sample since a collective response is measured. In the past we tried to determine the conductivity properties of a bulk pellet [22, 21], other researches tried to measure the transport through a thin foil of compressed sample [20]. Bulk sample approach is widely used also in other basic property measurements such as SQUID magnetic susceptibility scans of a sample capsule, laser absorption of a sample foil for the determination of electronic states [15, 16, 18, 19, 17]. The sample preparation for these experiments is rather simple but the measurement results demand careful analysis since the properties are tested on a group, burying the individual response. This problem widens even more if the sample contains a palette of different phases as in our case, since the overall response becomes a sum of all different structures making it impossible to distinguish directly between them. This was actually the main reason for our ambitious goal to manufacture a single, thin bundle connected to macro-electrodes so that the conductivity measurements bring us as close as possible to the true transport properties of our material. As discussed at the beginning of this chapter the two basic strategies involve the introduction of the contacts a over prelocated bundle (Fig. 3.1b) on a substrate or the attachment of a bundle from a dispersion over a prefabricated electrodes with appropriately 78 3.3 Dielectrophoretical attachment of single bundles over a narrow gap narrow gap (Fig. 3.1c). Our lithographic techniques together with poor repositioning capabilities enabled us to use the underlying-bundle technique only for the thickest bundles [1] whereas for the thinnest ones the most promising alternative was the overlying-bundle approach. In this section we discuss the technique of dielectrophoretical attachment of an individual bundle from a dispersion over two electrodes. 3.3.1 Theoretical considerations Perhaps more natural approach of particle attraction in a solution would be the elec-trophoresis where constant potential is pressed over the contacts gathering the charged particles form the solution (Fig. 3.17a)[38]. Our initial attempts showed that the resulting depositions resulted as huge amounts of non-bundle material that totally covered both electrodes without a trace of separated single bundles. This together with the underlying work of our colleges [26] pushed us to utilize the alternating potential or the dielectrophoresis (Fig. 3.17b) as the source of the attraction force. In contrast to the electrophoresis the force is governed by the electric field gradient rather than by the field itself and for that it naturally requires a non-uniform electric field, generated around the contacts as simulated on the Fig. 3.16. Figure 3.16: a) The 3D simulation of the non-homogenous electric field around two oppositely charged stripes b) the top view of and c) the cross section in the middle of the stripes, perpendicular to the gap. A dipolar moment p(?) is induced in neutral objects when inserted in an alternating electric field E (?) resulting as a dipolar force F (?) in the present field gradient (Fig. 3.17b). We can write the force as: F (?) = (p(?) ¦ x/)E(?). (3.1) The dipolar moment depends on the material polarizability per unit of volume ?(?): p(?) = V ?(?) ¦ E(?), (3.2) 3.3 Dielectrophoretical attachment of single bundles over a narrow gap 79 a) charged particle / ------------/- + + r _____Ql__ + + + + + + + + I& + + + + + + ---------\- + + ---------\ + + neutral paa rticle b) charged particle neutral particle net force Figure 3.17: a) In electrophoresis the charged particles are strongly pulled in constant electric field towards the opposite charged electrodes whereas neutral particles remain ignored. b) In an alternating, non-homogeneous field of dielectrophoresis the charged particles oscillate around a fixed point, hardly moving in the solution, allowing only neutral particles to feel the dielectrophoretical force. that further depends on the complex permittivities (e* = e + i—) of media e* on the particle e* and on the shape of the particle manifested in the factor f{e*mie*): a(uj) = emf(e^n,e*). (3-3) Now we modify the expression (3.1) to: F{uS) = V emRe{f(e*m,e*)(E(u)) ¦ V)L,(cj)}, (3-4) which can be rewritten to the final expression: F{u) = t;v Lm^{f(e*m,e*)VE(u)2}. 2 (3.5) Theoreticians calculated the factor f{s*mis*p) for some general shapes including rods and elongated, cylindrical objects that adequately describe our bundles with lengths /, radii r and the volume V = nr2l: fis S ) —— This gives the result for our bundles in form: ^3 LP F[uj) = —r I em Re 2 VE(u) i (3.6) (3.7) The described phenomenon conveniently annuls the dominant electric force by utilizing an alternating signal instead of a constant potential leaving the smaller charged particles to oscillate around a fixed position, hardly traveling in the solution (Fig. 3.17b). The bigger and elongated objects carrying small charge such as our nanowires are being pulled by the field gradient toward the electrodes eventually ending up bridging them. This scenario was the core of our dielectrophoretical attachment approach. 80 3.3 Dielectrophoretical attachment of single bundles over a narrow gap 3.3.2 The attachment procedure Gathering all the ingredients and setting the strategy we reached the point of actual bundle attachment to the circuit - bridging a pre-manufactured gap between two electrodes (Fig. 3.1c). Let us stress that dielectrophoretical attachment was optimistically attempted at every intermediate research results (different circuits, different syntheses and dispersion strategies) but with only limited success. We concluded that well defined and narrow gaps (200?300 nm in width) along with stable dispersion containing sufficient amount of thin single bundles accompanied by few impurities are the imperative before realistically expecting successful single bundle attachment. As introduced in the brief theoretical consideration all we had to do was to press an external alternating electric field to the gap electrodes, pour a droplet of solution for couple of seconds over it, blow dry the surplus solvent and check the circuit under the AFM (Fig. 3.18). Figure 3.18: An artistic rendering of a dispersion droplet over the two electrodes with signal pins pressed onto the circuit pads. Other researches have been studying this phenomenon more thoroughly from the theoretical point of view, revealing the complex nature of such attachment attempts [39]. Many effects are present that we will not discuss here since we used the technique as a tool to produce the circuit, not studying the process itself. The core dielectrophoresis is accompanied by the mechanisms such as gravity, electrothermal and light-electrothermal heating, buoyancy effects, Brownian displacement, electrolysis and AC-osmosis. Even though this rather simple technique hides a variety of processes we boldly proceeded with bundle attachments, having in mind only successful single bundle attachment. As depicted on Fig. 3.19 we constructed a simple mechanical press contacts that were controlled with micro-manipulator (Fig. 3.20b). The contact pins were made out of a 100 µm thin wire that can be found in some fine electronic cables. Thicker or more robust contact pins turned out to scratch the circuit, destroying it during the attachment process. The whole assembly from Fig. 3.20b) was placed under the magnifying glass mount (Fig. 3.20a) so that the circuit pads and the contact pins could be precisely aligned and the approaching monitored until the contact was established. Due to the statistical nature of the procedure we were expecting numerous attachment attempts before succeeding and 3.3 Dielectrophoretical attachment of single bundles over a narrow gap 81 Figure 3.19: The pins are pressed on the circuit pads manufactured on a silicon wafer. due to the shortage of circuits we were forced to repeatedly use the same ones. That is why we needed to design a fast, accurate and simple method to perform many depositions and to check them under the AFM. Such press contacts from Fig. 3.20b) proved to be the right way. The alternative of permanently attaching contacts onto the circuit pads wasn’t practical since the silver paste, that was used to stick the wire to the circuit, turned out to slowly dissolve in the nanowire solvent leaving a messy trail over the circuit making it useless. Moreover the minute objects that we were targeting could have been observed only under the AFM and a macro-circuit with attached macro-wires became very difficult to be mounted into the microscope. The first attempts quickly showed that even though all the necessary ingredients were present, along with a promising single bundle strategy, the integration was still far from reaching the goal and actually producing a working circuit. First we encountered the issue of selecting a suitable solvent for dielectrophoresis. Being totally dependent on of statistics and laws of probability for a successful utilization of dielectrophoresis the dispersions had to contain a sufficient amount of target material (the thinnest bundles), low amount of impurities that would be naturally deposited as well and low agglomeration tendencies that would result as big deposition clumps and would destabilize the solution over longer periods of time. Moreover it turned out that a quickly drying droplet disables any kind of deposition control since the material becomes denser as the liquid volume decreases with evaporation enabling only extremely dense depositions. From this point of view the acetone even though being most suitable solvent for sample dispersions as mentioned in section 3.2 isn’t the right way to go. Since we were familiar with the work of other researches [24, 23, 26] we tried to dissolve the material in water as an opposite to the acetone regarding droplet formation on the wafer and regarding the evaporation rate. Unfortunately the necessary high density, low impurity content and stability were not achieved, thus failing all dielectrophoretical attempts. The most suitable candidate turned out to be isopropanol that appeared to possess both qualities of acetone and water: 82 3.3 Dielectrophoretical attachment of single bundles over a narrow gap Figure 3.20: a) The dielectrophoretic setup consisted of a magnifying glass mount with an in-built illumination and b) the press-electrodes controlled by a vertical micromanipulator. the dispersion could be produced with same quality as for acetone but with slower evaporation rate. Thus with proper procedures the adequate dispersion could be prepared (see section 3.2) and the favorable physical properties enabled dielectrophoresis in a relatively controllable way. The next step was to determine the proper frequencies and amplitudes of the AC signal along with the appropriate timing. At the beginning we plugged in a sinus signal generator with the aim to control the depositions. Unfortunately it turned out that regardless of the signal settings the depositions appeared to be very dense. Remarkably the density didn’t decrease even when the generator was totally unplugged. After closer look we noticed that all our setup appeared to have been catching the background electro-magnetic (EM) signals from different electric instruments and even the electric wiring. By a simple process of elimination we succeeded to isolate the magnifying lamp illumination, or better, its power supply transformer as the major cause for the EM-signals. Amazingly even after total disconnection of all possible EM-sources the signal on the contact pins of the press contact still had enough power to attract single bundles form the dispersion at the most optimal rate. The final step was of course to ground the mount and the manipulator which killed the EM signals and no attachments were observed. In the process we tried also the chance deposition meaning that a droplet was poured over the circuit without the press contact in floating mode12 and no depositions were observed. Intrigued by the situation we tested the background EM-signals and determined the expected quasisinus signal of 50 Hz and the amplitude of 100 mV for the plugged and 10 mV for the unplugged illumination to be present; the signal vanished after the grounding. We also tried to simulate this favorable signal via signal generator with grounded mount but with no success. We concluded that the grounding of the whole mount influences the signal 12The mount was not grounded. 3.3 Dielectrophoretical attachment of single bundles over a narrow gap 83 from the signal generator and no depositions are observed. Since we had been under huge time pressure with the core of our research still ahead of us and since the successful attachment conditions were achieved we consciously abandoned further modifications of the dielectrophoresis setup. Let us stress that the seemingly trivial grounding actually becomes a nightmare especially when the background influences the experiments in such a profound matter. We believe that special measures would be required in order to fully separate the background EM radiation and the external signals from the signal generator and since we found the proper parameters we devoted our research to the production of the circuits for the final measurement. As mentioned, many attempts were needed to obtain one successful single bundle attachment. In order to use the same circuit several times we cleaned it carefully after each attempt with cotton, dipped in isopropanol followed by flash drying with compressed nitrogen gas. Such a circuit was then placed under the magnifying glass, aligned with the contact pins followed by careful descent of the pins via a micro-manipulator until the contact was confirmed. As described, the illumination was turned off afterwards and a 5 µl droplet was poured with a micro-pipette extremely cautiously over the circuit avoiding the delicate wire pins. After 10 seconds the liquid was flash dried with compressed nitrogen gas and the circuit was ready for the AFM examination. To summarize once more the procedure in the compact form: 1. a circuit is gently cleaned with isopropanol dipped cotton and flash dried using compressed nitrogen gas; this step is repeated after each failed deposition, 2. the circuit is aligned under the magnifying glass, followed by monitored pins’ descent until the contact with the circuit is confirmed, 3. the illumination is disconnected from the power socket, 4. a 5 µl droplet of a dispersion is carefully poured over the circuit, 5. a 50 Hz signal of 10 mV amplitude is generated in the circuit, 6. after 10 seconds the solution is flash dried with compressed nitrogen gas, 7. after raising the contact pins the circuit is ready for AFM examination. Before we discuss the outcome of the procedure let us briefly comment on the metal used to manufacture the circuits. As described in the section 3.1.3 the choice of metal facilitated successful circuit production along with reliable measurements. It is clear that the mechanical stress resilience turns out to be more of a necessity than a virtue. Having in mind the several nanometer thick bundles also the upper thickness limit of the metal becomes an issue. Thinner layers are preferable on one hand due to the AFM imaging since the minute nanowire strands get lost quickly in the high and vivid topography of the circuit gorge and on the other hand due to the tendency to keep the bundle as straight as possible. Naturally, soft bundles tend to accurately follow the underlying surface topography thus not literally bridging the gap between the metals but bending 84 3.3 Dielectrophoretical attachment of single bundles over a narrow gap over both edges, touching the bottom of the gorge. These bends, also known as kinks, are believed to affect the electron transport through similar systems [40] so we wanted to keep these bends as flat as possible by keeping the metal thickness to a minimum. We came to the conclusion that that 20-25 nm thick nickel contacts would optimally satisfy all the mechanical stress resilience here and in the lithography section. Regarding the performance of our single bundle attachment procedure we realized that even under the best possible conditions the attachments still resulted in a wide palette of possible outcomes. Most often (?50 %) we observed multiple bundle depositions (Fig. 3.21a) that varied in density, mean diameter and the position of the bundles; sometimes not even a single one bridged the gap (Fig. 3.21b). Also agglomerates (Fig. 3.21c) and extremely thick bundles (Fig. 3.21d) found the way to our electrodes (?25 %) only proving that probability and statistics dominate the process. Luckily for us in quite some cases (?20 %) only several bundles bridged the gap (Fig. 3.21e) enabling tedious postprocessing using the AFM to cut specific wires leaving only one wire intact as described below. Our preferred outcome was present only in the minority of cases (up to 5 %) (Fig. 3.21f) and the rest were cases of impurities bridging the gaps (Fig. 3.21g and 3.21h or random junk attachment (Fig.3.21 i). The impurity bridges often lead to the destructions of the circuits since they couldn’t be cleaned and recycled again; this was also the end of many circuits. As pointed out before, images like Fig. 3.21e caused a great deal of 3.3 Dielectrophoretical attachment of single bundles over a narrow gap 85 Figure 3.21: The AFM images of typical depositions: a) most common dense deposition of many wires b) sometimes without bridging. c) Also frequently present deposition of agglomerates and d) extremely thick bundles, e) only few thin single bundles bridging the gap, f) the desired single bundle attachment, g) and h) the intriguing impurity bridges, i) the mixed deposition of all possible junk. frustration, since by finding a way to cut the surplus wires we would considerably widen the amount of favorable outcomes. The opportunity lies of course in the utilization of the AFM tip as the precise cutting tool. From the experiences with scanning the surfaces with the AFM tip we knew that our wires showed high adhesion to the surface and appeared to be rather soft, so we needed to scan the samples in tapping mode. In contact mode we were namely cutting or destroying the integrity of the bundles. Even the attempts of other colleagues to manipulate them on the surface ended the same way. So it was only natural to use the contact mode as the possible bundle cutting procedure or a route to destroy the particle bridges such as on the Fig. 3.21g, 3.21h and especially 3.21e where on the upper left corner a bridge distorts otherwise perfect deposition. More different outcomes were present here as well. 86 3.3 Dielectrophoretical attachment of single bundles over a narrow gap Figure 3.22: The AFM images of typical AFM tip cleaning: a1) › a2) a successful disconnection of three features, b1) › b2) a frequently observed gathered material after cleaning and c1) › c2) a undesired deformation of the selected bundle along with the slight material gathering. The most valuable and desired scenario contained a clean and precise cut of a bundle/feature in a specific location without interfering with other entities. A successful cut is presented on the Fig. 3.22a1 and 3.22a2, before and after the operation. The bundle on the right of the images was picked and the features marked with black arrows on Fig. 3.22a1 were recognized to have possible unwanted bridging between the contacts and were disconnected as presented on Fig. 3.22a2. In many cases the material after cleaning appeared to have gathered at the end of scanning lines making the situation even worse, especially when the features formed close to the picked bundle as on Fig. 3.22b1 and 3.22b2 or 3.22c1 and 3.22c2. Interestingly the amount of material appeared to have surpassed the material quantity in the features suggesting that some parts could have come from the AFM tip while roughly scratching the area. As also noticed in the macroscopi-cal cotton cleaning of the circuit some bridges exhibited extraordinary resilience towards mechanical scratching and AFM cleaning procedure only confirmed that, as depicted on the image sequence of Fig. 3.23 where the nasty bridge from Fig. 3.21e was attempted to be cleaned. We can conclude that even though the AFM cleaning process was shown to be a 3.3 Dielectrophoretical attachment of single bundles over a narrow gap 87 Figure 3.23: The AFM images sequence of unsuccessful bridge cleaning: a1) the initial situation with the target bridge on the left side of the gap, a2) the first attempt and a3) the widened cleaning area with bridge still intact. promising way to improve the single bundle attachment yield, the overall success turns out to be rather modest. Upon reflecting the outcomes we could establish a correlation between the AFM tips used but systematic study on this subject hasn’t been done. We noticed that the common silicon nitride (the product name OTESPA13) tips tend to get worn out easier producing additional deposits on the surface whereas platinum/iridium (the product name OSCM-PIT) covered tips not only didn’t leave any traces, the residual material of the cleaned features disappeared. We believe that an intrinsic potential is applied on the tip through the AFM apparatus attracting and cleaning particles from the surface. Such a case is depicted on Fig.3.21 a) where after cleaning the material simply vanishes. Perhaps the best demonstration of the procedure is presented on Fig. 3.24 where we succeeded to cut two out of three bundles. Figure 3.24: The sequence of AFM images of exemplary bundle cutting: a) the initial situation, b) the blowup from the previous image clearly showing the initial three bundles and c) the selected bundle on the right after cleaning is completely intact whereas the other two are disconnected. 13http://www.veeco.com 88 3.4 Measurements Dielectrophoresis turned out to be the best way to attach a single bundle over a gap thus integrating it into a measurement circuit. The technique combines a number of phenomena that can be tempered in the desired manner only if all important parts exhibit adequate properties. We have to emphasize especially the dispersion’s quality and the choice of the solvent. To fully understand and utilize the technique, also the pin design, signal introduction and especially grounding must be taken into account. We saw the dielectrophoretical attachment only as a route to achieve our goal of measuring electronic properties a single bundle and we were not going into a deeper investigation of the concept itself. 3.4 Measurements After laying a pathway for manufacturing single bundle circuits we were ready to perform conductivity measurements. As announced in the beginning of this chapter our interest lies in the current vs. voltage characteristics as a function of temperature. In this section we describe the measurement setup followed by the unforseen temperature pre-treatment of the circuits, which was not planned initially. 3.4.1 Measurement setup Current vs. voltage is perhaps the most basic measurement in the determination of electronic properties and gives a profound view into the mechanisms that govern the transport of electrons. As widely debated in the theoretical sections many effects might affect the electrons traveling along the wire and our goal was to experimentally measure our system and compare the results to the theoretical possibilities. All we needed in order to perform such a measurement was a voltage source and a current measurement unit. Since the temperature plays an important role in the electron dynamics a thermostat was included; a schematics of the setup is depicted on Fig. 3.25. We also needed to construct a custom measurement setup, but this time we had a very good standard measurement equipment at our hands. The core conductivity measurement was performed by Keithey 238 electrometer, a source-measure unit. For temperature setting and control we used a cryostat (Oxford Instruments) connected to a cryodrive (Edwards cryodrive 1.5) and driven by ITC 503 temperature controller. All components were controlled by the computer program that was specifically designed for the task of temperature controlled current vs. voltage characteristic measurements. The setup is presented on the photograph Fig. 3.26. The program had full control of the measurement scan and at the same time enabled custom data patterns designed for a specific system measurement (Fig. 3.27). Let us add a full description of the measurement procedure along with measurement conditions even though the whole technique hardly needs a deeper description. The silicon plate carrying the sample circuit was glued onto a copper cryostat holder to ensure high temperature conductance and proper control of the temperature in the sample. The circuit pads and the external socket pins were connected with a 25 µm thick gold wire that was 3.4 Measurements 89 Figure 3.25: A voltage is pressed over the sample circuit and the resulting current is measured. The sample is closed in a cryostat under high vacuum with controllable temperature. glued onto the pads using silver paste and carefully soldered to the connector plug (Fig. 3.28a). Before mounting it in the cryostat (Fig. 3.28b) the holder was left for at least 12 hours to dry otherwise the glue gets swollen resulting in an unreliable temperature control. After plugging the external wires to the holder plug the cryostat was covered with a radiation shield and finally sealed with the external cover. A two stage vacuum system composed of rotational and turbo-molecular pump was engaged to depressurize the cryostat chamber below 10-3 mbar. Finally the program had been set and the measurement could begin. The temperature range was swept step-by-step while stabilizing below the tolerance of 0.1 K at each temperature point before measuring the current vs. voltage curve. During the conductance scan the prescribed source voltage was pressed over the sample for 5 seconds before reading a sequence of the current values with 20 ms integration time and exporting the average value along with standard deviation error into a digital data file. The temperature scan interval stretched form the room temperature of 295 K till 18 K and backwards to get a comparison between cooling and heating data14. Surprisingly the setup was able to measure with the accuracy down to several pA, reliably measuring resistances up to 50 G? with the leakage resistance of about 100 G?. As presented in the results in the following subsections the stability and accuracy of the measurement also surpassed our expectations thus making the data gathered with the system credible and accurate. 3.4.2 The process of temperature annealing The story so far was presented as a sequence of research stages that were encountered before finally reaching the point of the measurement. That is entirely true for the desired thinnest samples but for the thicker ones several studies have been made revealing 14The duration of a cycle was roughly 14 hours. 90 3.4 Measurements Figure 3.26: The measurement setup, controlled by the custom made computer program, consisted of Keithley 238 measurement unit, Oxford Instruments cryostat, driven by the Edwards cryodrive 1.5 and two step vacuum system (rotational and turbo-molecular pump). the full perspective of the goal assignment. As mentioned earlier the measurements were performed at some intermediate level mainly with the purpose to test the manufacture techniques (circuits, contact metals) and to become acquainted with the measured material. Let us stress that we could abandon the idea of a thin bundle measurement and obtain some results using the thicker bundles of several 100 nm, but we continued in the direction of thinnest ones since we believe that the basic nanowire nature could be revealed much better with only a few nanowires in a bundle. Nevertheless the experiences gathered by dealing with bigger, even macroscopical bundles were priceless since many aspects of the final measurement procedure have been revealed, while unfortunately opening yet another research battlefield. Looking at similar systems like carbon nanotubes and molybdenum selenide nanowires [10, 11, 2, 4] we were expecting relatively low resistances in the range of resistance quantum for some arbitrary bundle, even more so since many nanowires in the bundle would conduct in parallel reducing the common conductance roughly by a factor of nanowire number in the bundle compared to a single nanowire conductance. The first results were 3.4 Measurements 91 Figure 3.27: The program sets the measuring sequence on the individual level for the voltage as well as for the temperature. Figure 3.28: a) The silicon plate carrying the circuit was glued onto the copper cryostat holder and connected to the in-built connector socket via gold wires. b) The holder was screwed onto the cryomount and plugged to the measurement wires. shocking since the values of resistance proved to be in the order of several G?, far from 4 92 3.4 Measurements the most optimistically expected fraction of 1.3 k? in ballistic transport through one nanowire. Repeated experiments only confirmed this behavior even more dramatically for the thinnest bundles. Comparing the measurement procedures we discovered that the post-productional treatment is needed in order to evoke current through the sample. The process is called annealing and is usually performed simply by baking the circuit in the oven. We wanted to go a step further since the process is not well understood and we can only speculate about what is happening. One can look at the phenomenon from two different points of view: the contact between the bundle and the metal from the circuit guides changes (improves) or the bundle itself undergoes a structural change. From the basic chemical properties we knew that the material appears stable in the atmospheric conditions up to 300?C (the material tends to get disintegrated due to oxygen) and in vacuum up to 900?C [13, 21]. Relying on that, we performed an annealing effect survey on the circuit resistance for different circuit production techniques, different contact metals and for different MoSIx stoichiometries[25]. The circuits were produced and measured accordingly in several different ways: a macro bundle suspended on a glass plate (1 µm diameter and 2 mm in length) directly connected with silver paste (Fig. 3.29a), four probe measurement for the contacts over a 220 nm thick, 2 µm long bundle and two probe measurement of dielectrophoretically attached bundles (DEP) of different stoichiometries. Desperately trying to reach thinnest bundles we even designed a measurement using special conductive atomic force microscopy (CAFM) techniques provided by our AFM microscope. As depicted on the artistic expression on Fig. 3.29b a deposition of bundles over a silicon oxide wafer was lithographically covered with a stripe contact hoping to partially imbed single bundles beneath. The metal contact acted as one measurement pole and the conductive (platinum/iriduim covered) AFM tip as the other. This experiment turned out to be quite demanding and in the end destructive. The most challenging was as expected the positioning of the tip over the bundle, obstructed by the intrinsic tip drift, and the contact formation. Initial mechanical and voltage treatment was necessary in order to evoke current through the bundle, often resulting in a total destruction of the bundle. This was the main reason why we couldn’t compare the effect of annealing on the same circuit and used two for each condition. Since the structural integrity needed to be preserved the annealing temperature was set to 700?C for an hour in vacuum. The circuits got annealed after the raw material was integrated into a circuit except for the macro-bundle where the raw material got thermally treated and afterwards connected directly with silver paste to the measurement contacts (Fig. 3.29b). The results gathered in the table 3.1 clearly show that for the dielectrophoretically deposited bundles the annealing improved the conductance by roughly three orders of magnitude for three different stoichiometries; the comparison of data before and after annealing is presented on Fig. 3.30b. This huge conductance improvement implies that indeed the connection between the bundle and the metal plays an important role in the electron transport. Also the four probe measurement supports this assumption since the measurement by design pushes out the contact resistances revealing the bare conductivity of the sample; the comparison of data before and after annealing is presented on Fig. 3.30a, where only modest modification of the conductance improvement was noticed. We can draw the same conclusion from the CAFM measurements since the conductivity for 3.4 Measurements 93 Figure 3.29: a) A suspended macro-bundle is glued directly to the measurement wires with silver paste. b) Partially covered thin bundle is tested for conductance at several points using a conductive AFM tip as one pole and the metal stripe contact as the other. both cases ranged in the same magnitude even though the conductivity on the titanium changed dramatically (point B on Fig. 3.31b and point 4 on Fig. 3.31c). Before annealing [Sm-1] After annealing [Sm-1] Material (measurement method) (contact metal) (contact metal) Mo6S3I6 (2-probe freestanding) Mo6S3I6 (4-probe lithography) Mo6S3I6 (CAFM) Mo6S3I6 (DEP) Mo6S4.5I4.5(DEP) Mo6S2I8 (DEP) / 0.07 (Ti) - 0.3 (Pd) 0.37 (Ti) - sample 1 1.3 x 10-4 (Ti) 3.7 x 10-5 (Ti) 2.3 x 10-5 (Ti) 9.5 (Ag) 2.5 (Pd) 0.52 (Ti) - sample 2 0.135 (Ti) 0.057 (Ti) 0.048 (Ti) Table 3.1: Single bundle conductance at 295 K. As briefly pointed out earlier this survey addressed also the issue of metal choice for the production of circuits. A thin metal layer can exhibit special and unforeseen properties when introduced to high temperatures. Under such conditions material migration can be expected especially for softer metals or metals with poor adhesion. Also chemical properties can play an important role since many substances otherwise inert at room temperatures become vividly reactive when heated. We totally abandoned the idea of using gold in combination with some adhesive layer (nickel, chromium, titanium) mainly due to poor lithographic properties (see section 3.1.3). The alternative was pure titanium or more inert palladium. We knew that titanium oxidizes fast at high temperatures but we were hoping that our vacuum (low amount of oxygen) was good enough to reduce this effect so that the measurements wouldn’t be affected. If titanium worked after annealing for the 94 3.4 Measurements Figure 3.30: a) A current vs. voltage characteristics for the four probe measurement of a bundle before and after annealing; the conductivity raises by roughly one order of magnitude b) whereas for the DEP circuits three orders of magnitude improvement was observed. The insets are the AFM images of the circuits with the scale bar of 2 µm length for the left and of 250 nm for the right image. DEP circuits it proved to be totally inadequate for the sensitive CAFM measurements. As depicted on the resistance vs. length (the distance between the contact point between the AFM tip and the bundle) diagram in Fig. 3.32b the linear curve includes contribution from the bundle and from the metal. The extreme value of R0 = 4.2T? gives the raw contribution of the titanium contact the alarming range of T?. Even direct measurement on the titanium (point 4 in Fig. 3.31c) confirmed that value. We believe, that the metal oxidized at high annealing temperatures covering otherwise good conducting electrode with a non-conductive oxide film. One might argue that to the total resistance also the narrow AFM tip contributes along with its touching point but as seen from the conductance curve of pure titanium before annealing (point B on Fig. 3.31b) the resistance is estimated at around 20M? thus negligible compared to overall resistance of the whole circuit (G? for the point A on Fig. 3.31b or T? for annealed sample from Fig. 3.31c). 3.4 Measurements 95 Figure 3.31: a) The conductance AFM cross section scheme of the experiment. b) The measurement of the non-annealed (70 nm diameter) and c) of the annealed circuit (6.5 nm diameter). Figure 3.32: a) Resistance as a function of distance form the titanium contact for the annealed circuit from Fig. 3.31b). b) Current vs. voltage characteristic of the titanium contact and the bundle for the non-annealed sample from Fig. 3.31c. We concluded that the titanium contacts oxidize severely transforming otherwise metallic and conductive material into poor conductor even into an insulator; the oxide can even be identified with the ball-shaped structures on the surface of the metal see on the AFM images from Fig. 3.31c. Let us add that for thin bundles the CAFM technique proved not to be an adequate method for conductance measurements partially due to hard positioning and mechanical destruction hazard but mainly due to extremely difficult current awakening in the bundle with or without annealing. In fact the activation stress turned out to be too violent and the bundles got often destroyed even before performing any measurements. These results convinced us to return to the path of dielectrophoretical attachment as the only strategy with good prospects for the measurement of the thin bundles. Once more we changed the metal for the production of the contacts from titanium to nickel since thin contacts required for thin bundles cannot be produced with favorable 96 3.4 Measurements palladium (see section 3.1.3). While performing first measurements on thin bundles we were confronted with similar problems as for the CAFM since the current through the circuits wasn’t detectable with our setup (upper resistance limit was several hundreds of G?) and the process of annealing became a necessity. As presented in Fig. 3.33 ideally after the treatment the bundle survives and the metal contact doesn’t show severe feature formation. Moreover we can even observe bundles submerging into the metal (Fig. 3.33c) establishing a better electric connection. This unique and exciting effect is present only on nanoscales since thin metal films possess new and unfamiliar properties compared to the bulk material. Most relevant in our case is of course the lowering of the melting point of thin films [41], that most probably in combination with surface tension and atmospheric conditions mediates the reformation of topographic structure of the layer. Figure 3.33: The sequence of AFM images of ideal annealing outcome: a) the initial situation, b) the blowup from the previous image and c) the circuit after temperature treatment shows slight buckling of the metal surface with the unspoiled bundle submerging under the metal. Unfortunately this favorable annealing property had also the downside since the perilous nature of high temperature treatment caused the destruction of many circuits and even bundles (Fig. 3.34a1 › 3.34a2). Mostly we observed vivid buckling of the metallic surface manifested extremely high topographic features sometimes causing even the contacts to fuse and create short connections (Fig. 3.34b1 › 3.34b2). Even strange crystalline formations (Fig. 3.34c1 › 3.34c2) were observed only proving yet again another poorly controllable and understood circuit preparation. This final bottleneck pushed the whole experiment to a new level. Many times we got excited over finally being able to produce a bundle integrated into a circuit but faced a cold shower such as in the case of bundle exemplary cutting, where perfect circuit had been produced but got completely destroyed; the sad ending of that circuit is depicted on Fig. 3.35. 3.4 Measurements 97 Figure 3.34: The AFM images of typical annealing downsides: a1) › a2) the bundle disintegrated , b1) › b2) contacts were fused together creating short connection and (c1) › (c2) crystalline formations on top of metal layer. Figure 3.35: The sequence of AFM images of a circuit destruction: a) the initial situation, b) the destruction of the bundle and c) a view of the whole buckled contacts with extreme topography. 98 3.4 Measurements 3.4.3 Measured samples The necessary annealing process comes in package with great circuit losses but we were still pushing towards a successful measurement knowing that we couldn’t expect many repetitions. Let us itemize the finale recipe of required steps to produce and post-treat the circuits in order to make successful measurements: 1. the whole amount of sample from the synthesis ampoule is sonicated for 5 minutes in acetone using an ultrasonic tip, left to sediment for seven days, dried and redispersed in isopropanol in glass reservoir using ultrasonic bath, 2. with e-beam lithography 25 nm thick nickel electrodes with 200 - 500 nm wide gaps are produced on a silicon wafer with 600 nm of silicon oxide on top, 3. a thin bundle is deposited from a droplet over the gap using 50 Hz signal with the amplitude 10 mV for 10 seconds before flash drying it with nitrogen gas, 4. the circuit is annealed in vacuum (in a sealed ampoule vacuumized till 2 · 10-5) mbar at 700?C for an hour, 5. finally the circuit is glued with heat conductive resin to a cryostat and the pads are connected using silver paste to 25 µm thick gold wires that are soldered to the electronic socket. Countless attempts finally paid off and we produced and successfully measured four samples. On the Fig. 3.36 are presented the AFM images of the samples with their diameter and length gathered in table 3.2. The nomenclature was left original and is composed of the prefix “na”that denotes the abbreviation for“nanos”( in English “deposition”), and the identification number. sample signature diameter[nm] length[nm] na12 5 530 na23 4.2 265 na27 4 200 na28 12.5 190 Table 3.2: The diameter and the length of the measured samples. 3.4 Measurements 99 Figure 3.36: The gallery of all successfully measured samples: a) na12, b) na23, c) na27 and d) na28. These four samples represent the core of our work and the results of their current-voltage measurement are presented and discussed in the next chapter. All other measurements on thicker bundle chips were manufactured and prepared by the procedures described in this chapter. Chapter 4 Results and discussion In the previous section we described in detail the experimental background that lead to the successful current vs. voltage measurements of four nanobundle integrated chips at various temperatures. Despite reaching our predetermined goal of testing the thinnest wires, the obtained results don’t promise reliable statistics. On the other hand in the case of homogeneous sample with stable and uniform structures of single nanowires one can expect reproducible results with slight discrepancies among them, making only few measurements sufficient to explore the properties of target structures. Unfortunately when dealing with real systems this is hardly the case. As already debated in the chemical structure section 3.2 our material shows multi-phase stabilities in terms of stoichiometries, allowing also a great number of structural defects in an individual wire. Each nanowire is thus a unique system with unique transport properties that can vary widely as debated in theoretical section 2. This awareness was our main motivation to continue with painstaking sample production in order to get as many repetitions as possible, since in the worse case scenario the measured chips would exhibit unrelated characteristics, which would be exciting from a theoretical point of view. As presented further in this section our palette of the thinnest four samples makes a statistical compromise by separating them roughly into two groups according to the basic shape of the IV curves at low voltages also exhibiting different underlying transport laws. Later on we add results of thick bundle measurements that again appear to form a separate group. We conclude the chapter with the discussion of the annealing effect and the overall comparison of measurements, justifying the term “integrated chip” over simply a “nanowire circuit”. 4.1 The thin bundles We consider nanowires thin if their diameter falls below 15 nm. Since these minute objects cannot be directly examined in terms of their composition and fine chemical structure, the data about their shape comes only from their AFM images. Our nanobundles were individually introduced in Fig. 3.36 of the previous section, with their sizes gathered in table 3.2. The raw IV characteristics for each sample are presented in Fig.4.1. The close-to-linear curves at higher temperatures tend to bend as the temperature drops. Plotting 4.1 The thin bundles 101 the normalized curves (Fig. 4.2), we observe two distinct shapes: 1. the “S” shape - the VI curves remind of the letter “S” - the samples na12 and na27, 2. the “J” shape - the high voltage regime appears linear with smooth step leading towards zero voltage slope - the samples na23 and na28. 300 200 100 0 -100 -200 -300 -1.0 a) -0.5 0.0 V [V] 0.5 1.0 500 na23 y^Sß 250 0 -250 ^aÉESEEEEm H3 jjp -500 w -1.0 b) -0.5 0.0 V [V] 0.5 1.0 200 100 0 -100 -200 0^^S= -1.0 c) -0.5 0.0 V [V] 0.5 1.0 400 200 0 -200 -400 -1.0 -0.5 d) 0.0 V [V] 0.5 1.0 Figure 4.1: The current vs. voltage characteristics from the room temperature till 18 K (from top till bottom): a) na12; the error bars depict high accuracy (0.5 % - 1 % ) of the measurements and are omitted in all other graphs, b) na23, c) na27 and d) na28. We introduced many transport mechanisms in the theoretical section but here we confine ourselves to the most likely ones also suggested and discussed in literature [10, 11, 4]: variable range hopping (section 2.5), environmental Coulomb blockade (sections 2.4 and 2.4.2) and the Luttinger liquid behavior (section 2.6). Mathematical expressions for each law can be tested on measurement data: the Luttinger liquid behavior law (2.174) allows all curves over all voltage ranges to collapse onto a single curve. Unfortunately the equivalently general result for the variable range hopping 2.159 has to be tested in low voltage limit that gives the known hopping behavior 2.170. The most tedious is the environmental Coulomb blockade theory that cannot be solved analytically for the general case, but demands temperature and voltage limitations in order to utilize quantitative analysis 2.144. Qualitatively the properties emerge only through numerical treatment, 102 4.1 The thin bundles 1.0 0.5 0.0 -0.5 -1.0 na12 jü /^éE^ J^ jT -1.0 a) 1.0 0.5 0.0 -0.5 -0.5 0.0 V [V] 0.5 -1.0 -1.0 -0.5 c) 0.0 V [V] 0.5 1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 -1.0 b) 1.0 0.5 0.0 -0.5 -1.0 -1.0 d) -0.5 0.0 V [V] 0.5 1.0 -0.5 0.0 V [V] 0.5 1.0 Figure 4.2: The normalized current vs. voltage characteristics from the room temperature till 18 K (from top till bottom). The curves’ development with dropping temperature are clearly seen. a) na12, b) na23, c) na27 and d) na28. revealing linear dependencies for high voltages and smooth transition of the slope from zero voltage to high voltage regime (numerical depiction on Fig. 2.18). Let us describe the numerical treatment we follow in the analysis: Luttinger liquid The collapsing diagram of underlying law 2.174 is obtained by plotting I/Ta+l against eV/kT (2.24). The a is the slope of zero voltage conductivity against temperature in the loglog plot. The ß is the exponent for the high voltage (eV ^ kT) limit since the general law exhibits power law behavior / oc V@+1. The 7 stands for the fitting parameter and adjusts the voltage drop over the circuit. Variable range hopping The plot ln(G) (for low voltage) against T~x yields curves that become linear for the correct hopping exponent (table 2.5). To extract the most adequate mechanism the fits are tested for Pearson’s correlation (values close to unity prove the best fits). Environmental Coulomb blockade Qualitatively the IV curves should exhibit linear dependence for high voltages with a smooth transition to zero voltage slope and the 4.1 The thin bundles 103 derivative di/dV approaches an asymptotic value. For low voltages and temperatures the dependance follows a power law behavior / oc V2'9 (2.144). 4.1.1 The “S” curves The curves remind strongly of the Luttinger liquid sinh dependency. The loglog plot G vs. T for the na12 reveals only remote resemblance to a straight line. We still fit a linear curve extracting the a for three instances: low temperatures, high temperatures and over all points (Fig. 4.3a). The curves for the overall slope with a = 2 collapse relatively well, especially since we plot throughout the entire temperature range (Fig. 4.3b). We tested the collapse resilience of the other values of parameter a for low and high temperatures (1.3 and 2.7), but the curves diverge severely for both cases (Fig. 4.3c and d). To obtain the Luttinger law fit the parameter ß was extracted from the curve 1000 100 10 14 10 a) 1E-11 1E-12 1 1E-13 1E-14 100 T [K] na12 C) 10 eV/kT 100 1E-12 1E-13 1E-14 1E-15 1E-16 b) 1E-12 1E-13 1E-14 1E-15 1E-16 1E-17 1E-18 d) -----data nal2, /3 = 1), since it should be the closest approximation to the correct value. It can also be seen that the zero voltage slopes remain constant and close to 1, thus in agreement with Ohm’s law. The plot Fig. 100 10 1 0.1 0.01 2.25n 2.00 1.75 1.50 1.25 1.00 0.75 na12 ». 0.01 a) 0.1 V [V] 50 100 150 200 250 300 b) T [K] Figure 4.4: a) The current vs. voltage characteristics are gathered on a LogLog plot, suggesting values for /3 + 1 in the form of slopes at high voltages. b) The slopes /3+1 are increasing with decreasing temperature as a result of departing from high voltage limit (eV ^ kT), where the power law / oc V@+1 holds. The arrow points the value of /3+1 = 2 that was selected for further analysis. 4.3b with relatively good fit suggests that Luttinger liquid behavior probably contributes to the electron transport. Remarkably even the a = 2/3 (Eq. (2.173)) prediction holds, reinforcing the suspicion that Luttinger liquid could be the most likely candidate for the electron transport through the system. Of the remaining possibilities only variable range hopping stands out, since the linear behavior for high voltages in the IV curves isn’t observed, dismissing the environmental Coulomb blockade as the governing transport phenomena. In Fig. 4.5a the conductance G is plotted against T~x for various exponents (from the table 2.5), together with the linear function fits in order to test the behavior. The values at low temperatures were not taken into account since noise-to-signal ratio increases for the lowest voltages and lowest temperatures yielding uncertain conductance values. Since different A lie close together, a correlation1 value between the fits and the data is compared in the histogram Fig.4.5b for each exponent, suggesting onedimensional hopping with the exponent 1/2, if the density of states is taken to be constant. Since this mechanism is also in a reliable agreement with the theoretical prediction, a definite answer cannot emerge. Following the described analysis also the na27 can be tested in a similar way. In this case the graph in Fig. 4.6a, yielding the Luttinger liquid parameter a = 2.3, follows a straight line much closer to that in the previous na12. Moreover, the IV curves overlap even closer (Fig. 4.6b) suggesting that the Luttinger liquid could be the dominant transport mechanism. More disperse plots for other testing values of a = 1.8 and a = 2.8 (Fig. 1High correlation yields the value closer to 1. 1 0 4.1 The thin bundles 105 100 io na^ 0.0 0.1 a) 0.2 0.3 0.4 T - A,=l/4 A,=l/3 ^=2/5 A,= l/2 ^=3/5 ^=2/3 ^=3/4 A,= l 0.5 1.000 0.998 0.996 0.994 0.992 0.990 0.988 0.986 b) nal2 1/4 1/3 2/5 1/2 3/5 2/3 3/4 Figure 4.5: a) The G against T~x plots in a logarithmic scales shows tests of is different A by comparing the linear fits for each curve. b) For quantitative comparison the Pearson’s correlation between the fits and the data is compared. The values closest to 1 reveal the closest fits, in this case 1/2, suggesting 1D transport. 4.6c and d respectively) similarly prove the sensibility of curve collapsing, supporting the theory since the fitted a = 2.3 gives the best results. The loglog plots of the IV curves also show the Ohmic behavior at zero voltage and the increase of slopes for high voltages. In this case the value ß + 1 is taken to be 2.6 and thus ß = 1.6. Here the equality a = 2ß isn’t followed any more. When comparing the data to the variable range hopping the characteristic plots in Fig. 4.8a exhibit different behavior, since in this case the closest fit exponent is 1/4 according to Fig. 4.8b, suggesting 3D hopping as in opposite to 1D hopping found in the case of the na12. Similar results were reported also by other researchers when measuring the transport properties of the molybdenum selenide nanowire bundles [10]. Their analysis of several samples comparable in size and structure2 to ours shows two types of transport when compared to Luttinger liquid: a ~ 2/3 and a ~ ß. The cases are related to the presence of the defects in the bundle; for a uniform structure the second equality holds, whereas breaking of the conducting channels may result closer to the first case. Our na12 appears to be closer to the disrupted channel structures, whereas the na27 contains fewer irregularities. Also variable range hopping could be a part of transport mechanism since the characteristic plots follow theory’s predictions. The dimensionality that is the main issue here appears to be different for both samples: na12 follows 1D and na27 3D hopping. This coexistence of both theories doesn’t appear to be intrinsic for other nanowire samples. A report [11] on transport measurements of NbSe3 nanowires dismisses the variable range hopping as a satisfactory possibility and focuses on the Luttinger liquid approach. Their samples on the other hand exhibit also the Peierls transitions with charge density wave properties, that yield falling conductance as a function of temperature; this was also the main reason for omitting the Wigner crystal theories in this report. We can speculate that in the mesoscopic systems the transport is governed by many effects resulting as a mixture of different mechanisms. Moreover the underlying phenomena exhibit qualitatively very 2Both have molybdenum octahedra forming the backbone. i i 106 4.1 The thin bundles 1000 100 10 0.1 10 a) lE-11 lE-12 1E-13 1E-14 1E-15 C) 100 T [K] na27 a=1.8 / / y y ^^Bn^dg ^^ _^5g ^ 10 100 eV/kT b) d) - data na27,