Stability of silicon carbide structures: from clusters to solid surfaces Rafael Gutierrez," Thomas Frauenheim," Thomas Kohler" and Gothard Seifertb "Theoretische Physik III, Institut fur Physik, Technische Universitat, D-09107 Chemnitz, Germany bTechnische Universitat, Institut fur Theoretische Physik, Mommenstrasse 13, D-01062 Dresden, Germany We present a density-functional based non-orthogonal tight-binding (DF-TB) Hamiltonian in application to silicon carbide. The Kohn-Sham orbitals of the system are represented by a linear combination of atomic orbital (LCAO) equation with respect to a minimal basis of the localized valence electron orbitals of all atoms. Within a two-centre approach all Hamiltonian and overlap matrix elements are derived in a parameter-free way via the construction of pseudo-atomic orbitals and potentials by self- consistent single-atom calculations using the local-density approximation (LDA).This is in favour of a tabulation of the corresponding Slater-Koster integrals us. distance. Making use of a non-self-consistent solution of the Kohn-Sham equations for the many-atom structure and an adjustment of the universal short-range repulsive two-particle potentials with respect to self- consistent field (SCF)-LDA results in the method becoming sufficiently accurate to obtain the total energy of all-scale silicon carbide structures and this is transferable and efficient for predictive molecular-dynamics simulations. We present results for the energetic stability and properties of various microclusters and molecules including interactions with hydrogen.We give proof of the stability of the solid-state modifications and calculate the vibrational density of states for the most stable zinc blende structure. In addressing further applications to surface properties, we discuss the (1 x 1) reconstruction of the (1 10)Sic surface. For a theoretical description of the ground-state properties of clusters, molecules and solid-state modifications, different approaches have been developed. Self-consistent field (SCF) Hartree-Fock (HF)lq2 and density-functional theory (DFT) calculation^^.^ including correlation effects at different levels of approximation [many-body perturbation theory, local-density approximation (LDA)] yield very accurate results. However, owing to the increasing computing time necessary, they become unpracticable at larger particle numbers.In particular, for implementation into molecular-dynamics (MD) simulations investigating structure formation at surfaces and interfaces, more efficient approaches are highly desirable. In contrast to the aforementioned ab initio methods the empirical potential approaches are, unfortunately, in many cases not transferable to a wider class of systems than they have been derived for. They become inappropriate for most structural simulations of predicting character. The present method belongs to a third class of structural simulation methods, which has received increasing attention in recent years: a hybrid between an ab initio method and a parametrized tight-binding (TB) It differs from the empirical TB approaches, where the Hamiltonian and overlap matrix elements are derived from fitting to an equilibrium structure data base, in that they are calculated straightfor- wardly, using a basis of atomic-like orbitals and potentials derived from DFT-LDA calculations on contrasted pseudo- atoms.Similar to the parametrized TB schemes, only two- centre integrals are retained and the total energy is composed of a 'band-structure energy' (sum of occupied Kohn-Sham orbitals) and a universal short-range repulsive two-particle potential, which additionally has to be adjusted to the differ- ence of the band-structure energy and SCF-LDA cohesive energies of proper reference systems.In two recent publication^^*^ this method has been proven to be highly transferable for complex C(H) and Si(H) struc- tures. In an extension of this work, the method will be firstly applied to various scale silicon carbide structures (Sic), ranging from small clusters to crystalline solids and solid surfaces. We will benchmark our method by comparison with recent studies on small Si,C, clusters and molecules within HF theory including correlation effects through Moller-Plesset (MP) perturbation theory and configuration interaction (CI) Furthermore, the properties of solid-state modifi- cations are compared with ab initio pseudo-potentials21 and SCF-TB calculations.22 The considered surface reconstruction on Sic( 110) is related to recent results of Bechstedt et Sabisch et and Mehandru and Anderson.25 The paper is organized as follows.First, we give a short outline of the theoretical method. Next, we present the results on small clusters, molecules and solids, emphasising the dis- cussion of low-energy configurations and relative stabilities. As first application to the study of the surface properties, we determine the stable surface reconstruction of Sic (110) and finally give a short summary. Density-functional tight-binding method Since the method will be addressed to calculations of total energy and interatomic forces in MD structure simulations, it will be based on first principle concepts. Instead of solving the Kohn-Sham equations self-consistently, we construct a non- orthogonal TB Hamiltonian within the LCAO-LDA frame-work of DFT.We will give only a brief outline of the main ideas and refer the reader for more detail to refs. 7, 26 and 27. Contracted atomic-like valence orbitals #p (v-Rk) and potentials are generated by an SCF solution of the Kohn-Sham equations within LDA for a single pseudo-atom. The pseudo- atoms are constructed by incorporating an additional con-finement potential (~/r~)~in the effective single-particle poten- tials. The confinement radius ro is related to the covalent radius of the considered atom type and does not involve further parametrization. The minimal valence basis thus generated is used in an LCAO equation for representing the wavefunctions of the extended system: $i(v)=C c;#p(v-Rk) (1) pc.k Within a two-centre approximation all Hamiltonian and overlap matrix elements HpV,S,, are then calculated straight- forwardly within this basis.Since the three-centre and crystal- field integrals are neglected, all matrix elements depend only on the interatomic separation. They are calculated once for each atom type combination in favour of the tabulation of J. Muter. Chem., 1996, 6( lo), 1657-1663 1657 Slater-Koster integrals,? which may be used for many-atom calculations of a large variety of different-scale structures. Within our non-self-consistent treatment for the many-atomic structure the Kohn-Sham equations can be trans-formed into an algebraic system leading to a general eigenvalue problem: Using simple matrix diagonalization techniques we determine the single particle energies E~ and the eigenstate expansion coefficients.As is commonly accepted,28 the total energy can be written as the sum of a 'band-structure energy' Ebs and a repulsive energy Erep: occ Etot=EbsfErep=xEZ+ 1U(IRZ-RkI) 1 l<k The latter term as a universal short-range repulsive two- particle interaction is determined as the difference of the band- structure energy and the SCF cohesive energies of the diatomic molecule and the crystalline zinc blende Sic modification. In Fig. 1 the dependency of the matrix elements and the repulsive potential for Sic on the interatomic separation is shown. Corresponding results for carbon and silicon have been pub- lished re~ently.~?' Results Molecules As a first test of our method we have investigated several carbon-substituted silane molecules.In general, the structural parameters and symmetries of the investigated systems are in good agreement with ab initio calculations. The Si-Co bond length in all cases 1,s slightly overestimated (0.01-0.03 A) Fnd ranges from 1.663 A in monosubstituted silenes to 1.946 A in cyclopolysilane. The bond lengths and angles of the considered molecules are summarized in Table 1; the molecules are shown in Plate 1. HSiCH, H,CSiH,. There is no experimental evidence for the existence of the silene molecules HSiCH. CI calculation^'^ give a bent structure as the most stable form, followed by a linear chain about 9 kcal mo1-I higher in energy (1 cal= 4.184 J).The DF-TB method yields the same energetic order but a slightly reduced energy djfference of AE ~7 kc$ mol-I. The Si-C bond length (1.663 A) is enlarged by 0.03 A relative to the ab initioovalue. Compared with H2CSiH2 it is shortened by about 0.08 A, probably owing to the formation of a partial double bond in the latter compound. The bond angles in bent HSiCH, however, show larger deviations from the ab initio values, AZZ 15". In Table 2 the normal mode vibrational frequencies for H2CSiH2 are shown and compared with CI(d,+p) re~u1ts.l~In the last column we give the relative error taking as reference the experimental frequencies. The largest deviations are found for the high-frequency stretching modes, while lower lying modes are better reproduced.H,SiCH,. The structural parameters for this molecule agree quite well with SCF/3-21G16 calculations. The bond angles at the silicon atom are closer to the typical value of 109" for sp3 hybridization than found by ub initio calculations and the Si-C bond is lengthened slightly. Si3C3H6. This molecule forms a ring structure of D3,, symmetry, with alternating Si and C atoms, see Plate 1. The calculated bond lengths and angles agree very well with the t The Slater-Koster tables for Si-C, C-C, C-H, Si-H, H-H and Si-Si are available upon request from the authors. 1658 J. Muter. Chem., 1996, 6(10), 1657-1663 -0.4 h ---' -0.6 t 0.8 -(b) ___. s, s,,\ \ s, \ \ s,\ -\0.3 \ __ s, \ \ l .Y' 1 -0.7 2 4 6 8 0 bond length/a, Fig. 1 (a)Hamiltonian (HMv)and (b)overlap S,,, matrix elements, and (c) repulsive potential, Erep,us. interatomic separation ab initio re~u1ts.I~ Only the C-H bond is found to be slightly longer in the present approach. Silabenzene, SiC,H,. Our results are compared with SCF/3- 21G*18 and ST02G19 calculations, see Table 1. The bond lengths and angles are described reasonably well. The C-Si-C bond angle is underestimated by about 6%. Cyclopolysilane,(SiH,),CH, and 1,3-substituted cyclotetrasil- ane, (SiH2)* (CH,),. These compounds form isoscale and square configurations, respectively, with hydrogen pairs bonded out-of-plane to each atom.The Si-Si bond length in cyclopolysilan~ (see Table 1) is shorter compared with that of disilane (2.34 A). The structural parameters of these molecules agree well with the ab initio result^.^^^^^ The 1,3-substituted tetrasilane molecule is stretched slightly along the Si-Si direction according to our method, yieldigg an overestimation of the Si-C bond length by about 0.02 A. Table 1 Geometric properties of different C-substituted silane molecules (see Plate 1) compared with ab initlo calculations bond length/A, bond angleldegrees molecule symmetry HSiCH (silene) cm S1-Hs1-c C-H bent-H Sic H c, s1-c S1-H C-H H-Si-C H-C-H H,CSiH, C21 si-c (silaethylene) Si-H C-H H-Si-H H-C-H H3SlCW3 si-c S1-H C-H H-C-H H-SI-H S13C3Hh s1-c Si-H C-Hc-s1-c s1-c-Sl SiC5H6 s1-c (silabenzene) Si-H C-H c-c c-c-c c-s1-c S1-C-H si-c-c (SlH,),CH, s1-c (cyclopolysilane) Si-H C-H s1-Sl H-Si-H s1-c-Sl ( S1Hz)2(CHd2 s1-c ( 1,3-~ubstltuted) S1-H cyclotetrasilane) C-Hsi-c-s1c-s1-c H-C-H H-Si-H Small silicon carbide clusters In this section we present results for small silicon carbide clusters Si,C, We will discuss the minimal energy configur- ations and the energetic differences to different low-energy metastable configurations For comparison we choose as refer- ence recent high-level ab znitzo Hartree-Fock calculations including correlation effects on the MP2 level and by CI of refs 9-14 Conjugate gradient relaxation techniques have been per- formed to obtain the minimal energy clusters In Table 3, characteristic bond lengths and angles of the ground-state geometries found with the DF-TB method are given, the Sic clusters are shown in Plate 2 For comparison the ab znztzo data of refs 9-14 have been included In general, the symmetry of the investigated clusters and their geometric parameters are in good agreement with the ab znztio calculations The energetic differences of various metastable structures yield the same trends as obtained on the ab znztzo level (see Fig 2) DF-TB ab initto ref 1459 1451 1614 1587 1108 1075 1663 1635 1471 1467 1127 1082 144 9 128 8 135 17 150 1 1 74 1703 1475 1459 1127 1083 118 1 114 5 109 115 1946 1917 16 1484 1 49 112 1082 104 6 108 3 109 56 107 8 1777 1766 17 1482 1483 1140 1094 119 8 118 7 120 1 121 3 1775 176 19 178 1472 1465 1477 111 1073 1092 1394 1391 1 400 124 9 101 3 125 2 107 4 116 6 107 5 126 6 125 4 124 1 118 2 125 9 118 2 1946 1901 15 1482 1464 1127 1084 2 266 2 251 114 1 112 6 71 2 72 6 1946 1923 15 1484 113 91 5 85 9 88 53 94 0 103 7 111 1 Si,C.Ab initio calculations9 using double-zeta plus double polarisation basis set show that the minimum on the potential- energy surface corresponds to a bent structure wit! Czv symmetry and geometrical parameters r(Si-C)= 1 670 A and Si--S-Si=133" With our method we obtain r(Si-C)= 171 A and Si-C-Si=136", in good agreement with the results of the more sophisticated calculations However, the consideration of electron correlation within CI gives a smaller Si-C-Si bond angleJ120") and a slightly increased Si-C bond length of 1686 A The bent structure is energetically followed by a linear chain at a slightly higher energy (AEw 0 95 kcal mol-I), in reasonable agreement with the ab znrtzo results (AEwO6-2 1 kcal mol-l) Sicy The energies of the different Sic3 isomers obtained with our method are displayed in Fig 2(u) in comparison with ab inztio results lo We obtain a good agreement in the relative stabilities of the different configurations However, while the J Mater Chew, 1996, 6(lo), 1657-1663 1659 Table 3 Geometric properties of the SIC clusters (shown in Plate 2) in comparison with ab initio data" (only the structural parameters of the minimal energy configurations found with the present method are given) bond length/A, bond angleldegrees molecule symmetry DF-TB ab initio" CSi2 c2, Si-C 1.71 1.67 Si-C-Si 130 120 Sic, (IV) CZc C(2)-C(3) 1.3 10 1.298 C(3)--C(4) 1.275 1.307 Si( 1)-C(2) 1.751 1.722 Si,C, (11) C, Si(1)-C(2) 2.11 I .92 C(2)--C(4) 1.24 1.23 C(4)-Si(6) 1.86 1.88 C(2)-Si( l)-C(3) 101.4 101.8 C(4)-Si( 6)-C( 5) 81.99 82.2 Si( 1)-Si(2) 2.55 3.1 1 Si(2)-C( 5) 1.84 1.67 C(5)-C(6) 1.33 1.32 Si( 3)-C( 5) 1.807 1.83 C(5)-C(4) 1.30 1.28 Si( 3)- Si( 1) 2.56 2.47 C(S)-Si( 1) 2.32 2.19 -~Si(6)-Si(3)-Si( 1) 59.69 -Si(6)-Si(2)-Si( 1) 59.69 C( 1 )-C(2) 1.335 1.38 C(l)-Si(3) 2.16 1.92 C(1)-Si(5) 1.727 1.738 Si( 3)- Si(5) 2.56 2.563 "Refs 9-14.Plate 1 Structures of C-substituted silane molecules in Table 1. Si, yellow circles; C, grey circles; H, green circles. Table 2 Vibrational frequencies of silaethylene; the corresponding ab initio results [CI (d, +p)] and experimental values are given as reference molecule rep. DF-TB CI(d, +p)" exp." error (%)b ~ CH2SiH, a, 958 989 927 3 (7) 1191 1041 985 20 (6) 1526 1476 1350 13 (10) 2733 2388 2229 18 (4) 2994 3254 ----a2 640 750 --bl 495 438 72 1 801 741 -2 (8) Plate 2 Structures of the Sic clusters in Table 3. Si, yellow circles; C, b2 286 501 green circles.834 876 817 2 (7) 2757 2404 2239 23 (7) 3059 3352 corresponding to isomer IV (all roman numerals correspond a Ref. 15.b Relative errors with respect to the experimental results to the notations in ref. 10). This strong competition between (values in parentheses correspond to the ab inito calculations). the linear and ring structures has also been observed in the ab initio calculations." The ground state obtained at the ah initio ab initio calculations give a strong stabilization of a four- level," within the present calculation scheme, is determined to membered ring, having Czvsymmetry, with a transannular be 30 kcal mol-' less stable than the linear chain coming carbon-carbon bond (111), we find the linear structure with fourth in the energetic order [see Fig.2(a)]. Interestingly, the the Si atom at the end of the chain as the energetically lowest- authors report that a linear arrangement would be the ground lying isomer, 1. This is followed by a ring structure without state using only a double-zeta basis, while including polariz- C-C cross bonding at about 2 kcal mol-' higher energy, ation function the ring structure is favoured. This indicates 1660 J. Muter. Chem., 1996, 6(lo), 1657-1663 150 100 50 0 -100 1 t-200 8 Y 12 150 100 50 0 'd X xv -50 0 Fig. 2 Relative stabilities, AE/kcal mol-' (1 cal=4.184 J) of Sic clusters calculated within the DF-TB method (---); (-) correspond to the ab initio calculations. (a) Sic,; (b) Si,C,; (c) Si,C,; (d) Si,C,; (e) Si,C,.The energy differences of the Sic isomers refer to the ground states found within ab initio calculations. that the energetic order can be affected strongly and even reversed by changing the size of the basis set. Si2C4. The ab initio calculations'' including correlation effects at the MP2 level yield a linear chain as the most stable configuration followed by a chair-like ring structure with C, symmetry. Within our calculations, the latter is not a stable point on the potential-energy surface, but it relaxes into a planar ground-state configuration with C, symmetry (see Table 3). However, the bond lengths and angles of this structure correspond quite well with those of Miihlhauser et al." with the exception of the Si( 1)-C(3) bond length, see Table 3.Note that this structure is relatively 'soft' with respect to out- of-plane distortions. A planar structure with D,,symmetry (I) is determined with almost the same energy, AEz0.16 kcal mol-I whereas the linear chain [I11 in Fig. 2(b)], is about 25 kcal mol less stable than structure 11. Three-dimensional Si,C, configurations IV and V are found to be highly unstable and confirm the ab initio results. Si3C3' This cluster shows a very complex potential-energy surface with a plenty of stable isomers, see Fig. 2(c). Since many of these structures have almost the same energy, the energetic stability is expected to be very sensitive to the size of the basis set and approximation used.According to our results, the structure with C, symmetry, X, displayed in Table 3, is determined as the ground state, followed by an edge-capped trigonal bipyramid, XII, also having C, symmetry. The ground state derived from ab initio calculations," a tetrahedral Sic, system faced-capped by two Si atoms, I, comes third in our calculations, about 12 kcal mol-' higher in energy. Si4C,. As expected, by increasing the Si/C ratio in the clusters, three-dimensional structures become favourable over linear or planar arrangements. A non-planar structure with C,, symmetry [I1 in Table 3 and Fig. 2(d)],is clearly the most stable structure within our treatment as well as in the ab initio calculations." Also the energetics of the other isomers agree quite well with the ab initio results.Only the extreme high energy of structure IV could not be confirmed in our calcu- lations. The bond angles and bond lengths determined by the DF-TB calculations agree well with those of the ab initio calculations for nearly all the structures considered. In some cases the Si-Si and the Si-C bond lengths are overestimated by a few percent in the present scheme, see Table 3. Si,C2. For this cluster we again can confirm the ab initio prediction~.'~The ground state, displayed in Table 3, is a J. Muter. Chem., 1996,6(lo), 1657-1663 1661 planar structure with CZVsymmetry Isomer I1 is higher in energy by about 20 kcal mol-', and structure 111 is even less stable, in good agreement with the ub initio calculations, see Fig 2(e) Solid modifications and surface reconstruction To address applications of the present method to bulk systems we have performed total energy calculations as a function of the Si-C bond length for the zinc blende (zb) and rocksalt (rs) structures The calculations make use of a r-point Brillouin-zone sampling, which is a valid approximation for the considered large periodic supercells including 216 atoms The cohesive energy curver per atom $re shown in Fig 3(u) The equilibrium bond length o,f 1 887 A agrees very well with the experimental value of 1 89 A 29 The cohesive energy Ecoh= 6 93 eV atom -matches quite well the experimental (6 34 eV atom ') data and values obtained with an ah znztzo pseudo-potential (666eV atom 1)21 and SCF-TB (634eV atom-') calculations 22 In calculating the cohesive energies we have chosen the spin-polarised atoms as reference, ze the spin- polarisation energies of 0 64 eV for Si and 1 12 eV for C have been subtracted from the LDA values The rocksalt modifi- cation is obtained to be metastable and about 1 2 eV atom higher in energy than the zb structure This may be compared to the very high value of ca 4 eV atom reported by SCF-TB calculations 22 In addition, the phonon spectrum within the harmonic approximation of the zinc blende modification has been obtained by solving the eigenvalue problem HY=02MY, where H is the dynamical matrix, M denotes a matrix with -45 rocksan 0 ' .0.0 zinc blende vlcm-' Fig.3 (u) Cohesive energy per atom for the zinc blende and rocksalt silicon carbide structures us bond length obtained with respect to the spin-polarized Si and C atoms (b) Vibrational density of states for zinc blende (3C) silicon carbide The insert is the ab inztio density of states taken from ref 31 1662 J Muter Chem, 1996, 6(10), 1657-1663 the atomic masses on the main diagonal and CL) and Y are the eigenvalues and the corresponding eigenvectors, respectively The vibrational density of states for a 216-atom supercell of the zinc blende modification was then calculated The obtained spectrum of eigenvalues, which was broadened by a Lorentzian of constant width 30 cm , is plotted in Fig 3(h) The comparisons with recent ab ~izztiodata obtained by Bechtedt et a1 30 yields a good qualitative and partly quantitative agreement The frequency region around 750cm contains two dominant peaks which correspond to SirC stretching vibrations The peak positions and relative intensities are quite well described The gap in the spectrum between about 600 and 730 cm-' is slightly widened within the DF-TB calculations compared with the ah znztzo results,30 see also the insert in Fig 3(b) Finally, as a first application to surface structures, we have investigated the reconstruction of the clean (110) surface in the zinc blende modification of Sic using commonly accepted periodic surface-slab techniques 31 The model consists of 7 (110) layers including 9 C and 9 Si atoms each The bottom layer during the relaxation was held fixed in order to simulate the effect of an infinite crystal substrate and dangling bonds were removed by hydrogen saturation The results are com- pared with ah rnztto calculation^^^ 24 as well as with atomic superposition and electron delocalization (ASED) band tech- niques 25 The most relevant structural parameters are displayed in Table4 and the (110) surface layer geometry is shown in Plate 3(u) and (b) In accord with the predicitons of the aforementioned studies we find a buckling of the first surface layer The top layer Si-C bond is contracted in tomparison with the unrelaxed (I x 1) surface, 1 82 and 1 89 A, respectively The determined bond length is slightly larger than that predicted in refs 23 Table 4 Structural parameters of the reconstructed Sic( 110) surface (see Plate 3 for notation of atoms) bond lengt h/A, bond angleldegrees sic ( 110) DF TB u6 initio method r( 1)-(2) 1826 1767 DFT LDA" 41)-(4) 2 09 1761 186 189 ASEDb r(2)-(3) 1832 1823 1 84 (2)-(1)-(4) 9651 99 8 106 C-Si-c 1153 1202 122 (1)-(2) -(3) 119 1 1164 112 "Ref 24 bRef 25 Plate 3 1 x 1 ( 110) surface reconstruction of zinc blende silicon carbide (a)side view (b)top view and 24 [r(Si-C)= 1 76 A] The silicon and carbon atoms in the first surface layer relax parallel as well as perpendicular to the surface While both atom types are displaced outwards, the C atoms are more strongly pulled out of the surface, yielding the observed buckling Considering the substrate, we find a relatively strong relaxation of the subsurface layer, too Silicon atoms of the second layer relax slightly inwards and the carbon atoms move slightly outwards Aso a consequence, a quite long C,,, -Si,,b bond length of 2 07 A is established When compared with the ab znztzo results, this effect causes an enlarged twist-angle and an increased byckling height, +z, normal to the surface, AzDF TB =O 397 A (AZSCF =O 234 A), which is responsible for the shear distortion in the surface layer The Csub -Si,,, -C,,, and C,,, -Si,,, -C,,, bond angles of about 119 and 115", respectively, indicates sp2 hybridization of the Si atoms in the surface layer In general, the bond angles agree well (within 2-4%) with the ab znztzo results The energy gain of 0 72 eV dimer-' after the relaxation is only slightly larger than the values obtained by Bechstedt et al 23 (063 eV dimer '), Sabisch et (064 eV dimer-') and the ASED result of Mehandru and Anderson25 (064 eV dimer -') Summary We have presented a scheme for the determination of non- orthogonal tight-binding Hamiltonian and overlap matrix elements for SIC on the basis of density-functional theory Despite its simplicity the method has been proven to be transferable to all-scale silicon carbide modifications without addiitonal parametrization of the total energy By applying the method we obtain reliable results for the ground-state configurations of small clusters, molecules and bulk crystalline modifications The energetic differences between different meta- stable states of small clusters as well as their structural parameters and symmetries are described well In a first application to the study of Sic surface reconstruc- tions we confirm the buckled dimer 1 x 1 reconstruction on the Sic( 110) surface found in ab znztzo calculations Various other applications to the study of crystalline defect structures and the ( loo), ( 111) surface reconstructions of silicon carbide as well as amorphous a-SIC H modifications are now in progress and the results will be published elsewhere We gratefully acknowledge support from the Deutsche Forschungsgemeinschaft References 1 K Raghavachari, J Chem Phys ,1986,84,5672 2 K Raghavachari and C M Rohlfing, J Cheni Phys 1988, 89, 2219 3 M R Pederson and K A Jackson, Phys Rev B, 1990,41 74.53 4 K Laasonen and R M Nieminen, J Phys 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Phys Re0 B, 1994,50,13401 31 Th Frauenheim, U Stephan, P Blaudeck, D Porezag, H-G Busmann, W Zimmerling-Edling and S Lauer, Phys Rer B, 1993,48,18189 Paper 6/00593D, Recezued 25th January, 1996 J Mater Chem, 1996, 6(10), 1657-1663 1663