Scientific paper
Atomic Scale Models for RBa2Cu3O6 5 and R1XPrXBa2Cu3O6.5 Compounds (R=Y and Lanthanides)
Alexander Chroneos,ab Ioannis L. Goulatis,c Ruslan V. Vovkc
aInstitute of Microelectronics, NCSR Demokritos, Aghia Paraskevi 15310, Greece. Tel.: +30 210 6503113, Fax: +30 210 6511723, E-mail: chroneos@imel.demokritos.gr.
b Department of Materials, Imperial College, London SW7 2BP, United Kingdom
c Kharkov National University, 4 Svoboda Square, 61077 Kharkov, Ukraine
Received: 01-12-2005
Abstract
Atomic scale simulation techniques based on energy minimization have been employed to study the structural parameters of a range of orthorhombic RBa^u3O65 and R1XPrXBa2Cu3O65 compounds. The new interatomic potential parameters have been derived by simultaneously fitting to the known structural parameters of a range of oxides, such as Cu-O, R2O3, RBa2Cu3O6 5 and R1XPrXBa2Cu3O6 5, a total of 62 compounds. The technological significance of the rare-earth cuprate superconductors has been briefly reviewed, whereas the predictions have been compared with previous experimental and theoretical studies. For all compounds the derived data yields excellent agreement compared to the experimental results. The aim is to generate transferable potentials that can be applied as the basis for future theoretical studies of the defect chemistry of this important set of compounds.
Keywords: rare-earth oxides, lanthanides, cuprate superconductor, atomistic simulation
1. Introduction
RBa2Cu3O7 (R= Y and lanthanides) compounds are of technological importance as high-temperature superconductors. The critical temperature (Tc) of about 90 K of these compounds only weakly depends on the nature of R; however, it should be noted that Ce and Tb do not form the orthorhombic structure,1 Pm is radioactive, and Pr-Ba2Cu3O7 is non-metallic and non-superconducting ("praseodymium anomaly"), even though it exhibits the orthorhombic unit cell.2 The investigation of the presence or absence of superconductive properties in compounds of identical crystal structure and the understanding of the conditions under which the phenomenon is not present can be important. This is highlighted by the amount of experimental work aiming at explaining the "praseodymium anomaly".3-6 It is evident that the structure-property relationships are technologically significant because of the application of high-temperature cuprate superconductors.3 For example, oxygen vacancies are modulating the hole doping of the Cu-O planes that in turn are critically important for the superconductivity of cuprates. There have previously been a number of atomistic simulation studies of cuprate superconductors including the structural and
defect properties of R2MCu2O6 (R = La, Nd, Y; M = Ca, Sr, Ba)7, the ionic and electronic defects in YBa2Cu3O7 (YBCO)8 and the defect chemistry in HgBa2Ca2Cu3O8+5.9
The purpose of this paper is to introduce a new transferable two-body potential model that can describe the structure of a range of oxides including CuO, R2O3, RBa2Cu3O6 5 and R1XPrXBa2Cu3O6 5. To illustrate the applicability of the atomic scale techniques, the dependence of the lattice parameters and interatomic distances of a range of R1XPrXBa2Cu3O6 5 (X = 0.25, 0.5 and 0.75) on the ionic radius of R3+ and the Pr content has been predicted. Even though pair potentials have limits, their great advantage is the ability to model large numbers of atoms. Consequently, the potential models developed in this study can be applied to simulate the defect chemistry of these materials. This is important, as point defects can influence the superconducting properties of cuprate superconductors. In this study calculations were performed using the GULP10 code.
1.1. Crystallography
RBa2Cu3O7 exhibits the orthorhombic structure (space group Pmmm, No. 47)11 with a range of lattice pa-
rameters depending on the rare earth ion. In this orthor-hombic structure there exist two independent Cu sites consisting of square- planar CuO3 chains and square-pyramidal CuO2 planes in the a-b plane. The R3+ and Ba2+ ions provide an effective framework that bounds the copper oxide. The orthorhombic unit cell of YBa2Cu3O7 is presented in Figure 1. The RBa2Cu3O6 5 considered has a closely related crystal structure and only differs by the partial occupancy of the O(1) oxygen ions. In R1-XPrXBa2Cu3O6 5 there is partial occupancy of the R3+ sites by the praseodymium ions.
Figure 1. The orthorhombic unit cell and atom labelling of YBa2Cu3O7.8
2. Computational Method
The atomic scale simulation technique is based on the classical Born model12 description of an ionic crystal lattice. Interatomic potential functions are defined to simulate the long-range attractive and short-range repulsive forces between the ions in the unit cell of the solid. Coulomb's law describes the long-range forces, whereas the short-range energy terms are approximated by using parameterized pair potentials of the Buckingham form. The lattice energy, EL, is given by
j>i i	---+AS exp 4xsq Fg	w I PIJ J	i] 'J
is the interionic separation between specific to ions i and j
of charge qi and qj respectively. In this relation the Cou'	J
lombic forces are summed via Ewald's method, whereas Buckingham pair potentials are summed directly up to a cut-off value of 20 A.
The perfect lattice is generated by assigning ions to a unit cell, which is repeated throughout space by the application of periodic boundary conditions as described by the crystallographic lattice vectors. Thermodynamically it is a constant pressure calculation, as allowing the ions in the unit cell and the lattice vectors to relax to zero strain, minimises the total energy of the system (Newton-Raph-son energy minimisation procedure).
The parameters used in this study were derived by simultaneously fitting to the experimental atomic positions of 62 oxides. This methodology enhances the trans-ferability of the potential models and its efficacy has been demonstrated in previous studies.13-17
The Dick and Overhauser shell model was used to describe the polarisability of oxygen ions.18 More specifically, in the Dick and Overhauser shell model electronic polarisation is described by the displacement of a mass-less charged shell connected to a massive charged core by an isotropic harmonic spring of force constant k [eV A 2]. The O2- ions have a shell charge of -2.04 e, a core charge of 0.04 e and a force constant of 6.3 eV A-2.
Table 1. Buckingham interatomic potential parameters.
Interaction	A (eV)	p (A-1)	C (eV A-6)
O2-- O2-	9547.96	0.21916	32.0
Cu2+- O2-	3859.2	0.245	15.5
Ba2+- O2-	905.7	0.3976	0.0
Y3+- o2-	1766.4	0.3385	0.0
La3+- O2-	2078.5	0.3467	15.55
Pr3+- O2-	2004.6	0.3415	14.2
Nd3+- O2-	1975.2	0.3404	13.8
Sm3+- O2-	1941.9	0.34	12.55
Eu3+- O2-	1888.6	0.34	12.2
Gd3+- O2-	1855.9	0.339	11.9
Tb3+- O2-	1838.2	0.3385	14.5
Dy3+- O2-	1787.4	0.338	10.94
Ho3+- O2-	1738.7	0.338	11.1
Er3+- O2-	1694.5	0.338	11.3
Yb3+- O2-	1624.2	0.338	13.5
Lu3+- O2-	1533.6	0.339	10.3
where Aj, pij and Cij are adjustable short-range potential parameters (Table 1), e0 is the permitivity of free space, rn
The experimental structural data for every R1-XPrXBa2Cu3O6 5 compound has been fitted separately by applying the 'relax' fitting technique and free energy minimization of GULP10. The internal variables have been optimised with respect to the internal energy and the strain variables with respect to the free energy (zero static internal stress approximation).
3. Results and Discussion
To extend the transferability, the potential model derived for the RBa2Cu3O6 5 and R1-XPrXBa2Cu3O6 5 has also been fitted to a number of rare-earth oxides (R2O3). In previous studies the potential model has reproduced the lattice parameters of Sc2O3, Y2O3 and La2O3 to within 0.1% of the experimental values.1517 Shannon19 has tabulated the ionic radii of the chemical elements for a range of possible coordination environments and oxidation states. In all the figures 2 and 3 Shannon's19 ionic radii (r) for oxidation number +3 and 8-fold coordination have been adopted.
3.1. YBa2Cu3O65, PrBa2Cu3O65 and ErBa2Cu3065	'
To demonstrate the efficacy of the computational methodology applied to study orthorhombic cuprates, the structural parameters of YBa2Cu3O6 5, PrBa2Cu3O6 5 and ErBa2Cu3O6 5 have been compared to previous studies. YBa2Cu3O6.5 has been selected, as YBCO is one of the most studied superconductors. The predictions for lattice parameters; Y-O and Cu-O distances for YBa2Cu3O65 have been compared with experimental and computational investigations20 (Table 2). It is evident that the derived parameters are in excellent agreement with the experimental values. Furthermore, the volume per unit cell of YBa2Cu3O6 5 is predicted to 0.05%. The equivalent structural parameters of ErBa2Cu3O6.5 have been compared to experimental results21 and the Er-O and Cu-O distances have been predicted to within 0.3%, whereas the volume per unit cell to 0.7% (Table 3). The structure-property relations of PrBa2Cu3O7 are important, because of the "praseodymium anomaly", and have been subject of a number of crystallographic studies.5 The experimentally determined22 structure of PrBa2Cu3O6 54 has been compared to the predicted values (Table 4). The derived volume per unit cell is underestimated by 1.13%, a fact that can be partially attributed to the difference in the oxygen concentration between the experimental and calculated unit cell. The Pr-O and Cu-O distances have been predicted to within 0.7%
cuprate block. As a consequence, the increase in the rare-earth radius results in an increase of the a (and b) unit cell parameters and a reduction of the height (lattice parameter c) of the unit cell (Figure 2). This is reinforced by the experimental results20-22 for YBa2Cu3O6.5, PrBa2Cu3O6.54 and ErBa2Cu3O6 5 (Tables 2-4).	.	.
A previous experimental study5 investigating the relation between the unit cell parameters and the R3+radius of R0 5Pr0 5Ba2Cu3O7 compounds (R = Er, Ho, Y, Gd, Nd, Pr and La) has concluded that Pr induces distortion in the unit cell, which is evident in the shortening of the Pr-O bond length. In this study, the dependence of the R3+-O(2) bond length on the rare-earth radius for a range of R1-XPrXBa2Cu3O6 5 compositions has been determined and is in agreement with the experimental results.5 Both R3+-O(2) and Cu(1)-O(1) obey the lanthanide contraction rule and are strongly dependent on the average rare-earth radius of the unit cell. The predicted volumes per molecule for the RBa2Cu3O6.5 compounds, as a function of the atomic number, are consistent with the lanthanide contraction rule.23
In experimental structural studies of the RBa2Cu3O7 phases there is often disagreement regarding the relative position (z) of O(2) and O(3) in the orthorhombic unit cell. For this reason the relevant fractional coordinates of both these oxygen atoms have been tabulated for all the R1-XPrXBa2Cu3O6 5 considered in this study (Tables 5-6). These can be useful as a starting point for future structural determinations.
The average rare-earth radius influences the structure of the mixed rare-earth compounds. For example, Yb and Pr have an average Shannon19 8-fold ionic radius of 1.055A, which is close to the equivalent radius of Gd (1.053A). The lattice parameters of GdBa2Cu3O65 and Yb05Pr05Ba2Cu3O65 differ by only 0.1%. The dependence on the average rare-earth radius is also evident on the lattice energy (Figure 3). For the previous example, the lattice energies of GdBa2Cu3O6 5 and Yb0 5Pr0 5Ba2Cu3O6 5 differ by only 0.04%. This structural relation has been verified for different combinations of rare-earth ions and a range of Pr concentrations.
3.2. R1XPrXBa2Cu306 5 (X = 0.0, 0.25, 0.5 and 0.75)	.
In this study, the dependence of the lattice parameters, R3+-O(2) and Cu(1)-O(1) distances on the rare-earth radius and the Pr content has been investigated systematically for several concentrations of Pr in R1-XPrXBa2Cu3 O6.5 (Figure 2). The larger rare-earth atoms increase the a (and b) unit cell parameters (Figure 1) and effectively lead the Ba atoms to reposition within the cuprate block. This is enhanced, because of the partial occupancy of the O(1) atoms that in turn increases the available space within the
Table 2. The determined structure of YBa2Cu3O6 5 compared with previous experimental and calculated data.20
	Experiment	Previous Calc.	This Study
a(Â)	3.842	3.797	3.812
b(A)	3.878	3.872	3.907
c(Â)	11.747	11.710	11.747
Cu(1)-O(1)	1.939	1.936	1.942
Y-O(2)	2.406	2.387	2.404
V(A3)	175.02	172.16	174.95
Table 3. The experimentally determined and calculated structure
Table 4. The experimentally determined and calculated structure
a(A) b(Â) c(A) Cu(1)-O(1) Er-O(2) V(A3)
Experiment
3.835 3.879 11.751 1.918 2.378 174.81
This Study
3.785 3.877 11.831 1.924 2.372 173.61
a(A) b(A) c(A) Cu(1)-O(1) Pr-O(2) V(A3)
Experiment
3.914 3.919 11.727 1.960 2.437 179.88
This Study
3.850 3.948 11.701 1.967 2.456 177.85
oí ErBa2Cu3O65.
oí PrBa2Cu3O6 5.
Table 5. The relative coordinates z/c of O(2) in a range of R1XPrXBa2Cu3O65 compounds.
	Pr = 0	Pr = 0.25	Pr = 0.50	Pr = 0.75
Y	0.375	0.374	0.372	0.371
La	0.365	0.366	0.367	0.368
Pr	0.370	0.370	0.370	0.370
Nd	0.371	0.371	0.370	0.370
Sm	0.372	0.371	0.371	0.370
Eu	0.373	0.372	0.371	0.370
Gd	0.374	0.373	0.372	0.371
Dy	0.377	0.375	0.373	0.371
Ho	0.378	0.375	0.373	0.371
Er	0.379	0.376	0.374	0.372
Yb	0.382	0.378	0.375	0.372
Table 6. The relative coordinates z/c of O(3) in a range of R1XPrXBa2Cu3O65 compounds.
	Pr = 0	Pr = 0.25	Pr = 0.50	Pr = 0.75
Y	0.376	0.374	0.373	0.371
La	0.366	0.367	0.368	0.370
Pr	0.370	0.370	0.370	0.370
Nd	0.371	0.371	0.371	0.370
Sm	0.372	0.372	0.371	0.371
Eu	0.373	0.372	0.372	0.371
Gd	0.375	0.373	0.372	0.371
Dy	0.377	0.375	0.373	0.372
Ho	0.378	0.376	0.374	0.372
Er	0.380	0.377	0.374	0.372
Yb	0.382	0.378	0.375	0.372
Figure 2. Dependence of the (a) lattice parameters a and (b) lattice parameters c on the Shannon radii (8-fold coordination) and the Pr content for the R1XPrXBa2Cu3O65 compounds.
4. Conclusions
The results suggest that the average rare-earth ionic radius is important for the structure and energetics of RBa2 Cu3O6 5 and R^P^Ba^u^ 5 compounds. The differences in the lattice parameters between the R1-XPrXBa2Cu3 O6 5 compounds are reduced with the increase of the Pr content (Figure 2). The lanthanide contraction rule23 has been verified for all compounds considered.
Energy minimization techniques can provide structural data comparable to the experimental determinations (Tables 2-3), and can predict structural parameters (Tables 4-5) in excellent agreement with the experimental studies. Additionally, these methods allow the systematic analysis of the complex behaviour and defect chemistry of oxide materials at the atomic level. For example, in recent studies similar potential models had been used to predict the intrinsic defect chemistry and the extrinsic defect pro-
Figure 3. Dependence of the lattice energy per unit cell on the Shannon19 radii (8-fold coordination) and the Pr content of the R1-XPrXBa2Cu3O65 compounds.
cesses associated with the solution of divalent and tetrava-lent ions in a range of rare earth oxides.24'25
As the interatomic potential model introduced in this study has been fitted to both RBa2Cu3O7 superconductor compounds and rare earth oxides, it can be applied to model interface structures. This is technologically important as RBa2Cu3O7 thick films are grown on buffered nickel tapes. The buffer layer is used between the superconductor and the metal to limit the diffusion of metal ions and to retard the oxidation of nickel. It is also significant for the buffer layer to be lattice matched with the superconductor, as it provides the crystal template for the growth of the consecutive layers.26 The lattice mismatch between the buffer layer and the superconducting layer limits the current density of the superconducting film. Engineering the buffer layers with almost zero lattice mismatch on the high-temperature superconductor layer is feasible with the application of mixed rare earth oxides. For example, in a recent experimental study, Yb-Ba2Cu3O7-x was almost perfectly matched with an (Gd0 9Er01)2O3 buffer layer.27
There have been fewer atomic scale simulation studies of superconductor oxides, than other oxides. The existence of excellent previous modelling studies7-9 proves the ability of such techniques, to contribute to the understanding of these materials.
5. Acknowledgements
Prof. Julian Gale of Curtin University is gratefully acknowledged for providing the GULP code.
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Povzetek
Simulacije z minimizacijo energije na atomskem nivoju so bile uporabljene pri raziskavi strukturnih parametrov nekaterih ortorombskih spojin tipa RBa2Cu3O65 in R1-XPrXBa2Cu3O65. Parametri medatomskih potencialov so bili pridobljeni s proučevanjem znanih parametrov skupno 62 oksidov, kot so CuO, R2O3, RBa2Cu3O6 5 in R1-XPrXBa2Cu3O6 5. Zajeta je predstavitev tehnološke pomembnosti superprevodnih kupratov redkih zemelj, obenem so bili izračuni primerjani s prejšnjimi eksperimentalnimi in teoretičnimi izsledki. Pri vseh spojinah so izračunani podatki primerljivi z eksperimentalnimi opažanji. Namen dela je razviti uporabne potenciale, ki so osnova nadaljnjih teoretičnih raziskav na področju defektnih struktur lantanidnih kupratov.