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Ab initio study of incommensurately modulated crystals
Computational Materials Science 22 (2001) 112±117
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Ab initio study of incommensurately modulated crystals Razvan Caracas *, Xavier Gonze Laboratoire de Physico-Chimie et de Physique des Mat eriaux, Universit e Catholique de Louvain, Bat. Boltzmann, 1, Place Croix du Sud, B-1348 Louvain-la-Neuve, Belgium Accepted 2 April 2001
Abstract We present an ab initio study of the average structure of some incommensurately (IC) modulated materials: K2 SeO4 , Mo2 S3 , AuTe2 (calaverite) and Pb2 MgTeO6 (elpasolite structure). All the calculations are done using the local density approximation (LDA) and/or the general gradient approximation (GGA) of the density functional theory (DFT). The electronic band structures, the corresponding density of states and the valence electron density distributions are determined for all the materials. Three-dimensional representations of the valence electron density distributions help to identify the important structural groups present in the structures and the chemical bonds between the atoms. Partial valence electron density maps are generated in order to link the electronic bands to dierent atomic orbitals or to linear combinations of atomic orbitals. K2 SeO4 and Pb2 MgTeO6 are insulators. For both materials the electron band structure is made of weakly dispersive electronic bands. Mo2 S3 and AuTe2 present a metallic character. Consequently for these two materials the Fermi surface has been also analyzed in order to ®nd the instabilities and possible nesting of the Fermi surface. Results of the ab initio determination of the average structure are also reported. The accuracy of these results, compared to the experimental data, is very material-dependent, the ®nal results varying within 1% and 10% of the experimental data. For AuTe2 the LDA gave 6±8% dierences, while the GGA improved considerably the quality of the results, reducing the dierences of up to 3±4%. For Mo2 S3 , LDA is within 2% of the experiment, while for the other materials the agreement lies between the one for AuTe2 and the one for Mo2 S3 . Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Incommensurately (IC) modulated crystals are aperiodic ordered crystals, presenting (at least) one modulation superposed onto the basic three-dimensional periodicity of the lattice with which it is incommensurate. Several types of modulations have been observed, like displacive, compositional, magnetic spin, etc. We deal only with materials exhibiting displacive modulations, for which we *
Corresponding author.
started an ab initio study based on the density functional theory (DFT). The choice of the analyzed systems is based on several considerations: the systems should be as representative as possible for the ensemble of the IC phases; should not, ideally, contain too many atoms (less than 30 is reasonable for the needed CPU time and memory used); should have a maximum of symmetry; and, very important, should have an IC phase at low-T. Our choice up to now is restrained to: K2 SeO4 , Mo2 S3 , AuTe2 (calaverite structure) and Pb2 MgTeO6 (elpasolite structure).
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 1 7 6 - 8
R. Caracas, X. Gonze / Computational Materials Science 22 (2001) 112±117
K2 SeO4 is a prototype of a whole family of A2 BX4 dielectrics with a large number of members [1,2]. It presents a rather complicated phase transition diagram passing through several IC phases. All the polymorphs are derived from the high-T hexagonal phase, P63 =mmc; Z 2 molecules in the unit cell. We will analyze two phases: the P63 =mmc hexagonal phase and the Pbnm orthorhombic phase. Mo2 S3 is a monoclinic phase [3], with metallic character of the average structure. A possible charge density wave mechanism for the IC transition is to be checked. AuTe2 (calaverite) is one of the best known IC phases. It set a lot of problems to the crystallographers who tried to describe its morphology [4,5]. A possible CDW mechanism [6] is also to be checked with respect to a Au valence ¯uctuation proposed earlier in the literature [7].
113
Finally Pb2 MgTeO6 (elpasolite) [8,9] is a perovskite with a cubic high-T phase (Fm3m, Z 4 molecular units in the conventional unit cell) and a low-T IC phase (average structure: R3, Z 1) which has a very large range of Table 1 Comparison of the SeO4 tetrahedral geometry between the ab initio determination of the hexagonal K2 SeO4 and the experimental orthorhombic K2 SeO4 a Hexagonal structure (theoretical geometry) 1 bond Se±O of 1.6028 A 3 bonds Se±O of 1.6325 A 3 angles O±Se±O of 107.70° 3 angles O±Se±O of 111.18°
a
Orthorhombic structure (experimental geometry) 1 bond Se±O of 1.6351 A 2 bonds Se±O of 1.6427 A 1 bond Se±O of 1.6474 A
1 angle O±Se±O of 108.50° 1 angle O±Se±O of 108.65° 1 angle O±Se±O of 110.12° 1 angle O±Se±O of 110.88°
LDA gives good agreement with the experimental structure.
Fig. 1. Fermi surface of Mo2 S3 . The analysis of this rather complex Fermi structure is under way: we are looking for Fermi surface nesting.
114
R. Caracas, X. Gonze / Computational Materials Science 22 (2001) 112±117
stability. There is a series of other Sc- and Wbased perovskites [10], which follow the same phase transition sequence. A series of other IC compounds have been tested, but generally because they demand very long CPU computing times or very large memory capabilities they have been abandoned. Up to now the complete ab initio characterization of the average structures has been done for most of these compounds. The next step will be to pass on to the determination of the dynamical properties in order to understand what are the mechanisms of the IC transition. 2. Technical details All the calculations were based on the local density approximation (LDA) and /or the generalized gradient approximation (GGA) of the DFT [11,12]. We used the ABINITv2 (1999±2000) code [13]. The ABINIT software is based on pseudopotentials and planewaves. It relies on the adaptation to a ®xed potential of the band-by-band conjugate-gradient method [14] and on a potentialbased conjugate-gradient algorithm for the determination of the self-consistent potential [15]. As usual with planewave basis sets, the numerical accuracy of the calculation can be systematically improved by increasing the cut-o kinetic energy of the planewaves. These wave functions describe only the valence and the conduction electrons, while the core electrons are taken into account using pseudopotentials. We used Troullier±Martins pseudopotentials [16]. For the characterization of the electronic properties a set of convergence tests has been done in order to choose correctly the grid of special k points [17] and the planewave kinetic energy cuto. During these tests the k points grid density and the cut-o energy value have been consecutively and independently increased. The variation of the total energy has been monitored. Dierences of the order of 10 3 ha (1 hartree 27.211 eV) between two successive grids/cut-o energies has been considered a good indication of the convergence. For Mo2 S3 and AuTe2 regular 4 4 4 grids of
special k points, with 16 points in the irreducible part of the Brillouin zone and for K2 SeO4 and Pb2 MgTeO6 regular 2 2 2 grids of special k points have been ®nally adopted for the calculation of the electronic properties. The planewave kinetic energy cut-os of 25, 20, 40 and 35 hartree for Mo2 S3 , AuTe2 , K2 SeO4 and K2 SeO4 and Pb2 MgTeO6 , respectively, have been used in the calculations. The structural relaxation was conducted using the Broyden±Fletcher±Goldfarb±Shanno minimization (BFGS), modi®ed to take into account the total energy as well as the gradients (as in usual BFGS). 3. Results and discussion Valence electron densities, electronic band spectra and electronic density-of-states are calculated for the average structures of the chosen compounds. Then the average structures are Table 2 LDA ab initio structural determination of the average structure of Mo2 S3 a Experimental
LDA ab initio
Deviation
11.502 6.056 16.296 102.40 1108.838
11.625 6.070 16.362 102.16 1128.751
1.07% 0.24% 0.41% )0.23% 1.80%
Mol x Mol z Mo2 x Mo2 z
0.310 0.009 0.109 0.631
0.311 0.010 0.110 0.630
0.001 0.000 0.001 )0.001
S1 S1 S2 S2 S3 S3
x z x z x z
0.510 0.804 0.972 0.157 0.728 0.514
0.511 0.794 0.977 0.156 0.719 0.518
0.001 )0.009 0.005 0.000 )0.010 0.004
Mo1±S1 Mo1±S2 Mo1±S20 Mo2±S1 Mo2±S2 Mo2±S3
2.369 2.549 2.638 2.573 2.563 2.372
2.424 2.588 2.633 2.578 2.591 2.398
2.31% 1.54% )0.20% 0.20% 1.08% 1.08%
a b c b V
a
LDA calculations give excellent results,with very small deviations from the experimental values.
R. Caracas, X. Gonze / Computational Materials Science 22 (2001) 112±117
determined ab initio and compared with the experimental data. Our main question is whether the LDA is suciently accurate or whether it is necessary to switch to GGA. K2 SeO4 . Two polymorphs have been analyzed, both generating average structures for the IC phases: the high-T P63 =mmc phase and the roomT Pbnm orthorhombic phase. The two structures are very similar from an electronic point of view. They are both insulators with a 3 eV gap. The electronic band structure is made of weakly dispersive bands. The valence electron density analysis reveals the ionic character of K2 SeO4 . Covalent Se±O bonds exist within the SeO4 tetrahedra. The K ions lie in the intertetrahedral void spaces ionically bonded to the tetrahedra. In the absence of experimental data, the ab initio determination of the structure of the hexagonal phase is started with the structural parameters of
115
the closely related compound K2 SO4 . The resulting geometry has been compared to the orthorhombic one. The agreement in bond lengths, bond angles and lattice parameters is fairly good, as can be seen from Table 1. Thus we expect that LDA is good enough to conduct dynamical calculations. Mo2 S3 . It has a monoclinic P2/m structure with a metallic character. The Mo and S atoms build some complex columns of MoS6 octahedra which host large rectangular void spaces. The electronic band structure is made of weakly dispersive bands grouped in several large regions. There are several electronic bands intersecting the Fermi level and thus several pieces of the Fermi surface. It presents a complex pattern, where a possible nesting vector may characterise parallel surfaces (Fig. 1). A detailed analysis of all the independent Fermi surfaces is necessary in order to ®nd the possible nesting vector responsible for the IC phase transition, and compare it
Fig. 2. Electronic band spectrum (a) and the density of states (b) of the average structure of calaverite. The CYCZC path is along the edges of the Brillouin zone. The structure possess a metallic character. We can further separate three relatively broad peaks in the DOS. The lowest one in energy corresponds mainly to the Te 6s orbitals, the next corresponds mainly to the Au 5d orbitals while the highest one in energy corresponds mainly to a hybridization of Au 6s, Au 5d and Te 6p orbitals. The Fermi energy is set to 0 eV.
116
R. Caracas, X. Gonze / Computational Materials Science 22 (2001) 112±117
with the experimentally determined modulation vector, q h0:5 0:469 0i. The agreement between the theoretical LDA values and the experimental data is excellent, with a maximum deviation of 2% (Table 2). AuTe2 . Calaverite presents a metallic structure. The electronic band structure may be split into three main regions, which correspond to the main peaks in the DOS. The peak at )10 down to )13 eV stems from the Te 6s orbitals, the peak at )4 down to )7 eV stems from the Au 5d orbitals and the peak intersecting the Fermi level stems mainly from Au 6s, 5d and Te 5p orbitals (Fig. 2). The ®rst ab initio structural determinations have been done using the C2/m, C-centered conventional lattice. The agreement between the experimental data and the theoretical results is rather wrong for the LDA, with deviations up to 9% in unit cell parameters and up to 8% in bond lengths.
The use of GGA the results reduced the deviation from the experiment down to 4% for the unit cell parameters and down to 6% for the bond lengths, but which remains relatively large. In both LDA and GGA there is a tendency to reduce the difference in the two Au±Te bonds, which correspond to the equatorial (multiplicity 4) and apical (multiplicity 2) dimensions of the AuTe6 octahedron (Table 3). Pb2 MgTeO6 . Both high-T Fm3m and low-T R3 average structure have been analyzed. Elpasolite is an insulator with a 1.5 eV gap. The electronic band structure is made of weakly dispersive bands. The corresponding peaks of the DOS correspond to the dierent atomic/molecular orbitals. The theoretical structure agrees relatively well with the experimental data. The rhombohedral structure exhibits a stronger distortion than the
Table 3 Ab initio determination of the average structure of calaveritea Experimenal
Theory LDA
a b c b V Te x Te z 4 bonds Au±Te 2 bonds Au±Te a
7.182 4.402 5.056 89.0 159.846 0.6890 0.2888 2.9699 2.6680
GGA
6 6 6 k points grid
Deviation
4 4 4 k points grid
Deviation
6.928 4.018 4.722 90.11 131.445
)3.54% )8.72% )6.61% 1.11 )17.77%
7.4371 4.2248 5.1377 89.82 161.431
3.55 % )4.03% 1.62% 0.82 0.99%
)0.0329 0.0679 )7.9% 2.5%
0.6513 0.2946 2.9986 2.8336
)0.0547 0.0201 0.97% 6.21%
0.6663 0.3084 2.7351 2.7346
LDA largely underestimates the experimental values, while the GGA improves our results.
Table 4 LDA ab initio determination of the structure of the R3 and Fm3m polymorphs of Pb2 MgTeO6 (elpasolite)a R3 Experimental [1]
Deviation
This work
Experimental [2]
10.4521 60.27
10.6675 59.9
)2.01% 0.61%
14.8021
14.8035
Ox Pbx
0.2669 0.2507
0.2408 0.25
0.0261 0.0007
0.2645
0.26
Mg±O bond (bohr) Te±O bond (bohr)
3.9390 3.4935
3.8342 3.7473
2.73% )6.77%
3.9154 3.4856
3.9325 3.6112
a0 (bohr) a
a
Fm3m
This work
The theoretical cubic structure is better than the rhombohedral one with respect to the experiment.
Deviation 0.00 0.0045 )0.43% )3.47%
R. Caracas, X. Gonze / Computational Materials Science 22 (2001) 112±117
cubic one, expressed as a contraction and an oblation. The Te±O bonds shortens in both structures (Table 4).
4. Conclusions The results of the ab initio calculations of the average structures of some IC materials are presented. The electronic properties are investigated and the assignment of the dierent peaks in the DOS to the dierent atomic and/or molecular orbitals is done. The results of the structural determination of the average structure are discussed with respect to the experimental data. The discussion is centered onto the use of LDA or GGA for the pursuit of the calculation. LDA is good in the case of K2 SeO4 , Mo2 S3 and Pb2 MgTeO6 , while for AuTe2 , the use of GGA is needed. The next step of the project is the determination of the dynamical properties of these materials, in order to ®nd the mechanisms responsible for the IC transitions.
117
References [1] H.Z. Cummins, Phys. Rep. 185 (1990) 211. [2] R. Caracas, Incommensurate crystal structures, http:// www.mapr.ucl.ac.be/crystal/, 2000. [3] W.J. Schutte, F. Disselborg, J.L. de Boer, Acta Cryst. B 49 (1993) 787. [4] B. Dam, A. Janner, J.D.H. Donnay, Phys. Rev. Lett. 55 (1985) 2301. [5] A. Janner, B. Dam, Acta Cryst. A 45 (1989) 115. [6] B.C.H. Krutzen, J.E. Ingles®eld, J. Phys.: Condens. Matter 2 (1990) 4829. [7] W.J. Schutte, J.L. de Boer, Acta Cryst. B 44 (1988) 486. [8] G. Baldinozzi, Ph. Sciau, A. Bulou, J. Phys.: Condens. Matter 9 (1997) 10531. [9] G. Baldinozzi, D. Grebille, Ph. Sciau, J.-M. Kiat, J. Moret, J.-F. Berar, J. Phys.: Condens. Matter 10 (1998) 6461. [10] G. Baldinozzi, G. Calvarin, Ph. Sciau, D. Grebille, E. Suard, Acta Cryst. B 55 (1999) 1. [11] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [12] W. Kohn, L.J. Sham, Phys. Rev. B 140 (1965) 1133. [13] ABINITv2.x, a common project of the Universite Catholique de Louvain, Corning Incorporated and other contributors, http://www.pcpm.ucl.ac.be/abinit (1999±2000). [14] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. Mod. Phys. 64 (1992) 1045. [15] X. Gonze, Phys. Rev. B 54 (1996) 4383. [16] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993.
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Ab initio study of incommensurately modulated crystals Razvan Caracas *, Xavier Gonze Laboratoire de Physico-Chimie et de Physique des Mat eriaux, Universit e Catholique de Louvain, Bat. Boltzmann, 1, Place Croix du Sud, B-1348 Louvain-la-Neuve, Belgium Accepted 2 April 2001
Abstract We present an ab initio study of the average structure of some incommensurately (IC) modulated materials: K2 SeO4 , Mo2 S3 , AuTe2 (calaverite) and Pb2 MgTeO6 (elpasolite structure). All the calculations are done using the local density approximation (LDA) and/or the general gradient approximation (GGA) of the density functional theory (DFT). The electronic band structures, the corresponding density of states and the valence electron density distributions are determined for all the materials. Three-dimensional representations of the valence electron density distributions help to identify the important structural groups present in the structures and the chemical bonds between the atoms. Partial valence electron density maps are generated in order to link the electronic bands to dierent atomic orbitals or to linear combinations of atomic orbitals. K2 SeO4 and Pb2 MgTeO6 are insulators. For both materials the electron band structure is made of weakly dispersive electronic bands. Mo2 S3 and AuTe2 present a metallic character. Consequently for these two materials the Fermi surface has been also analyzed in order to ®nd the instabilities and possible nesting of the Fermi surface. Results of the ab initio determination of the average structure are also reported. The accuracy of these results, compared to the experimental data, is very material-dependent, the ®nal results varying within 1% and 10% of the experimental data. For AuTe2 the LDA gave 6±8% dierences, while the GGA improved considerably the quality of the results, reducing the dierences of up to 3±4%. For Mo2 S3 , LDA is within 2% of the experiment, while for the other materials the agreement lies between the one for AuTe2 and the one for Mo2 S3 . Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Incommensurately (IC) modulated crystals are aperiodic ordered crystals, presenting (at least) one modulation superposed onto the basic three-dimensional periodicity of the lattice with which it is incommensurate. Several types of modulations have been observed, like displacive, compositional, magnetic spin, etc. We deal only with materials exhibiting displacive modulations, for which we *
Corresponding author.
started an ab initio study based on the density functional theory (DFT). The choice of the analyzed systems is based on several considerations: the systems should be as representative as possible for the ensemble of the IC phases; should not, ideally, contain too many atoms (less than 30 is reasonable for the needed CPU time and memory used); should have a maximum of symmetry; and, very important, should have an IC phase at low-T. Our choice up to now is restrained to: K2 SeO4 , Mo2 S3 , AuTe2 (calaverite structure) and Pb2 MgTeO6 (elpasolite structure).
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 1 7 6 - 8
R. Caracas, X. Gonze / Computational Materials Science 22 (2001) 112±117
K2 SeO4 is a prototype of a whole family of A2 BX4 dielectrics with a large number of members [1,2]. It presents a rather complicated phase transition diagram passing through several IC phases. All the polymorphs are derived from the high-T hexagonal phase, P63 =mmc; Z 2 molecules in the unit cell. We will analyze two phases: the P63 =mmc hexagonal phase and the Pbnm orthorhombic phase. Mo2 S3 is a monoclinic phase [3], with metallic character of the average structure. A possible charge density wave mechanism for the IC transition is to be checked. AuTe2 (calaverite) is one of the best known IC phases. It set a lot of problems to the crystallographers who tried to describe its morphology [4,5]. A possible CDW mechanism [6] is also to be checked with respect to a Au valence ¯uctuation proposed earlier in the literature [7].
113
Finally Pb2 MgTeO6 (elpasolite) [8,9] is a perovskite with a cubic high-T phase (Fm3m, Z 4 molecular units in the conventional unit cell) and a low-T IC phase (average structure: R3, Z 1) which has a very large range of Table 1 Comparison of the SeO4 tetrahedral geometry between the ab initio determination of the hexagonal K2 SeO4 and the experimental orthorhombic K2 SeO4 a Hexagonal structure (theoretical geometry) 1 bond Se±O of 1.6028 A 3 bonds Se±O of 1.6325 A 3 angles O±Se±O of 107.70° 3 angles O±Se±O of 111.18°
a
Orthorhombic structure (experimental geometry) 1 bond Se±O of 1.6351 A 2 bonds Se±O of 1.6427 A 1 bond Se±O of 1.6474 A
1 angle O±Se±O of 108.50° 1 angle O±Se±O of 108.65° 1 angle O±Se±O of 110.12° 1 angle O±Se±O of 110.88°
LDA gives good agreement with the experimental structure.
Fig. 1. Fermi surface of Mo2 S3 . The analysis of this rather complex Fermi structure is under way: we are looking for Fermi surface nesting.
114
R. Caracas, X. Gonze / Computational Materials Science 22 (2001) 112±117
stability. There is a series of other Sc- and Wbased perovskites [10], which follow the same phase transition sequence. A series of other IC compounds have been tested, but generally because they demand very long CPU computing times or very large memory capabilities they have been abandoned. Up to now the complete ab initio characterization of the average structures has been done for most of these compounds. The next step will be to pass on to the determination of the dynamical properties in order to understand what are the mechanisms of the IC transition. 2. Technical details All the calculations were based on the local density approximation (LDA) and /or the generalized gradient approximation (GGA) of the DFT [11,12]. We used the ABINITv2 (1999±2000) code [13]. The ABINIT software is based on pseudopotentials and planewaves. It relies on the adaptation to a ®xed potential of the band-by-band conjugate-gradient method [14] and on a potentialbased conjugate-gradient algorithm for the determination of the self-consistent potential [15]. As usual with planewave basis sets, the numerical accuracy of the calculation can be systematically improved by increasing the cut-o kinetic energy of the planewaves. These wave functions describe only the valence and the conduction electrons, while the core electrons are taken into account using pseudopotentials. We used Troullier±Martins pseudopotentials [16]. For the characterization of the electronic properties a set of convergence tests has been done in order to choose correctly the grid of special k points [17] and the planewave kinetic energy cuto. During these tests the k points grid density and the cut-o energy value have been consecutively and independently increased. The variation of the total energy has been monitored. Dierences of the order of 10 3 ha (1 hartree 27.211 eV) between two successive grids/cut-o energies has been considered a good indication of the convergence. For Mo2 S3 and AuTe2 regular 4 4 4 grids of
special k points, with 16 points in the irreducible part of the Brillouin zone and for K2 SeO4 and Pb2 MgTeO6 regular 2 2 2 grids of special k points have been ®nally adopted for the calculation of the electronic properties. The planewave kinetic energy cut-os of 25, 20, 40 and 35 hartree for Mo2 S3 , AuTe2 , K2 SeO4 and K2 SeO4 and Pb2 MgTeO6 , respectively, have been used in the calculations. The structural relaxation was conducted using the Broyden±Fletcher±Goldfarb±Shanno minimization (BFGS), modi®ed to take into account the total energy as well as the gradients (as in usual BFGS). 3. Results and discussion Valence electron densities, electronic band spectra and electronic density-of-states are calculated for the average structures of the chosen compounds. Then the average structures are Table 2 LDA ab initio structural determination of the average structure of Mo2 S3 a Experimental
LDA ab initio
Deviation
11.502 6.056 16.296 102.40 1108.838
11.625 6.070 16.362 102.16 1128.751
1.07% 0.24% 0.41% )0.23% 1.80%
Mol x Mol z Mo2 x Mo2 z
0.310 0.009 0.109 0.631
0.311 0.010 0.110 0.630
0.001 0.000 0.001 )0.001
S1 S1 S2 S2 S3 S3
x z x z x z
0.510 0.804 0.972 0.157 0.728 0.514
0.511 0.794 0.977 0.156 0.719 0.518
0.001 )0.009 0.005 0.000 )0.010 0.004
Mo1±S1 Mo1±S2 Mo1±S20 Mo2±S1 Mo2±S2 Mo2±S3
2.369 2.549 2.638 2.573 2.563 2.372
2.424 2.588 2.633 2.578 2.591 2.398
2.31% 1.54% )0.20% 0.20% 1.08% 1.08%
a b c b V
a
LDA calculations give excellent results,with very small deviations from the experimental values.
R. Caracas, X. Gonze / Computational Materials Science 22 (2001) 112±117
determined ab initio and compared with the experimental data. Our main question is whether the LDA is suciently accurate or whether it is necessary to switch to GGA. K2 SeO4 . Two polymorphs have been analyzed, both generating average structures for the IC phases: the high-T P63 =mmc phase and the roomT Pbnm orthorhombic phase. The two structures are very similar from an electronic point of view. They are both insulators with a 3 eV gap. The electronic band structure is made of weakly dispersive bands. The valence electron density analysis reveals the ionic character of K2 SeO4 . Covalent Se±O bonds exist within the SeO4 tetrahedra. The K ions lie in the intertetrahedral void spaces ionically bonded to the tetrahedra. In the absence of experimental data, the ab initio determination of the structure of the hexagonal phase is started with the structural parameters of
115
the closely related compound K2 SO4 . The resulting geometry has been compared to the orthorhombic one. The agreement in bond lengths, bond angles and lattice parameters is fairly good, as can be seen from Table 1. Thus we expect that LDA is good enough to conduct dynamical calculations. Mo2 S3 . It has a monoclinic P2/m structure with a metallic character. The Mo and S atoms build some complex columns of MoS6 octahedra which host large rectangular void spaces. The electronic band structure is made of weakly dispersive bands grouped in several large regions. There are several electronic bands intersecting the Fermi level and thus several pieces of the Fermi surface. It presents a complex pattern, where a possible nesting vector may characterise parallel surfaces (Fig. 1). A detailed analysis of all the independent Fermi surfaces is necessary in order to ®nd the possible nesting vector responsible for the IC phase transition, and compare it
Fig. 2. Electronic band spectrum (a) and the density of states (b) of the average structure of calaverite. The CYCZC path is along the edges of the Brillouin zone. The structure possess a metallic character. We can further separate three relatively broad peaks in the DOS. The lowest one in energy corresponds mainly to the Te 6s orbitals, the next corresponds mainly to the Au 5d orbitals while the highest one in energy corresponds mainly to a hybridization of Au 6s, Au 5d and Te 6p orbitals. The Fermi energy is set to 0 eV.
116
R. Caracas, X. Gonze / Computational Materials Science 22 (2001) 112±117
with the experimentally determined modulation vector, q h0:5 0:469 0i. The agreement between the theoretical LDA values and the experimental data is excellent, with a maximum deviation of 2% (Table 2). AuTe2 . Calaverite presents a metallic structure. The electronic band structure may be split into three main regions, which correspond to the main peaks in the DOS. The peak at )10 down to )13 eV stems from the Te 6s orbitals, the peak at )4 down to )7 eV stems from the Au 5d orbitals and the peak intersecting the Fermi level stems mainly from Au 6s, 5d and Te 5p orbitals (Fig. 2). The ®rst ab initio structural determinations have been done using the C2/m, C-centered conventional lattice. The agreement between the experimental data and the theoretical results is rather wrong for the LDA, with deviations up to 9% in unit cell parameters and up to 8% in bond lengths.
The use of GGA the results reduced the deviation from the experiment down to 4% for the unit cell parameters and down to 6% for the bond lengths, but which remains relatively large. In both LDA and GGA there is a tendency to reduce the difference in the two Au±Te bonds, which correspond to the equatorial (multiplicity 4) and apical (multiplicity 2) dimensions of the AuTe6 octahedron (Table 3). Pb2 MgTeO6 . Both high-T Fm3m and low-T R3 average structure have been analyzed. Elpasolite is an insulator with a 1.5 eV gap. The electronic band structure is made of weakly dispersive bands. The corresponding peaks of the DOS correspond to the dierent atomic/molecular orbitals. The theoretical structure agrees relatively well with the experimental data. The rhombohedral structure exhibits a stronger distortion than the
Table 3 Ab initio determination of the average structure of calaveritea Experimenal
Theory LDA
a b c b V Te x Te z 4 bonds Au±Te 2 bonds Au±Te a
7.182 4.402 5.056 89.0 159.846 0.6890 0.2888 2.9699 2.6680
GGA
6 6 6 k points grid
Deviation
4 4 4 k points grid
Deviation
6.928 4.018 4.722 90.11 131.445
)3.54% )8.72% )6.61% 1.11 )17.77%
7.4371 4.2248 5.1377 89.82 161.431
3.55 % )4.03% 1.62% 0.82 0.99%
)0.0329 0.0679 )7.9% 2.5%
0.6513 0.2946 2.9986 2.8336
)0.0547 0.0201 0.97% 6.21%
0.6663 0.3084 2.7351 2.7346
LDA largely underestimates the experimental values, while the GGA improves our results.
Table 4 LDA ab initio determination of the structure of the R3 and Fm3m polymorphs of Pb2 MgTeO6 (elpasolite)a R3 Experimental [1]
Deviation
This work
Experimental [2]
10.4521 60.27
10.6675 59.9
)2.01% 0.61%
14.8021
14.8035
Ox Pbx
0.2669 0.2507
0.2408 0.25
0.0261 0.0007
0.2645
0.26
Mg±O bond (bohr) Te±O bond (bohr)
3.9390 3.4935
3.8342 3.7473
2.73% )6.77%
3.9154 3.4856
3.9325 3.6112
a0 (bohr) a
a
Fm3m
This work
The theoretical cubic structure is better than the rhombohedral one with respect to the experiment.
Deviation 0.00 0.0045 )0.43% )3.47%
R. Caracas, X. Gonze / Computational Materials Science 22 (2001) 112±117
cubic one, expressed as a contraction and an oblation. The Te±O bonds shortens in both structures (Table 4).
4. Conclusions The results of the ab initio calculations of the average structures of some IC materials are presented. The electronic properties are investigated and the assignment of the dierent peaks in the DOS to the dierent atomic and/or molecular orbitals is done. The results of the structural determination of the average structure are discussed with respect to the experimental data. The discussion is centered onto the use of LDA or GGA for the pursuit of the calculation. LDA is good in the case of K2 SeO4 , Mo2 S3 and Pb2 MgTeO6 , while for AuTe2 , the use of GGA is needed. The next step of the project is the determination of the dynamical properties of these materials, in order to ®nd the mechanisms responsible for the IC transitions.
117
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