- Email: [email protected]
Lattice dynamics of boron nitride
Materials Science and Engineering B59 (1999) 264 – 267
Lattice dynamics of boron nitride H.W. Leite Alves a,*, J.L.A. Alves a, J.L.P. Castineira b, J.R. Leite c b
a Departamento de Cieˆncias Naturais-FUNREI, Prac¸a D. Hel6e´cio, 74 C.E.P. 36.300 -000 Sa˜o Joa˜o del Rei MG, Brazil Departamento de Cieˆncias Fı´sicas-Uni6ersidade Federal de Uberlandia, C.P. 593 C.E.P., 38.400 -902 Uberlandia MG, Brazil c LNMS-Depto. de Fı´sica dos Materiais e Mecaˆnica, Instituto de Fı´sica, Uni6ersidade de Sa˜o Paulo, C.P. 66.318 C.E.P., 05.389 -970 Sa˜o Paulo SP, Brazil
Abstract Using the density-functional theory within the full potential linear augmented plane-wave (FP-LAPW) method, we have calculated ab initio the equation of state and the principal phonon modes in cubic boron nitride (c-BN), including their pressure dependence and the amplitude of the eigendisplacements. A good agreement with the experiments is obtained, whenever a comparison is possible: in fact, most of the results are predictions. A ten-parameter valence overlap shell model (VOSM) was constructed and we obtained the phonon dispersion curves, elastic constants and effective charges. Our results were compared with calculated theoretical data for c-BN and for other III – V materials and we found that the lattice dynamics properties for cubic boron nitride is very close to those of diamond. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Boron nitride; LAPW method; Lattice dynamics; Frozen phonons; Shell model
1. Introduction There has been considerable interest in the development of optoelectronic and microelectronic devices which work under hard conditions such as high temperatures or aggressive environment. One of the promising materials for this purpose is cubic Boron Nitride(c-BN), which satisfies these conditions due to its physical properties such as hardness, high melting point and resistance to corrosion [1,2]. Unfortunately, the amount of theoretical knowledge on this substance is limited to a handful of works [3 – 5] and, particularly, the data on phonon spectra and vibrational properties are rather scarce [6–8]. In the present work we perform total energy calculations in order to supply the missing information on the vibrational properties of c-BN. The phonon frequencies and the respective eigenvectors of the vibrations at the high symmetry points in the Brillouin-zone are calculated within the frozen phonon model [12]: the frequency of the phonon is obtained from the difference
* Corresponding author. Tel.: +55-32-3792489; fax: + 55-323792483. E-mail address: [email protected] (H.W. Leite Alves)
between the total energies of the displaced and the undisplaced patterns according the particular phonon mode. The procedures used are described elsewhere [11,12,15]. For the total energy calculations we used the ab initio full-potential linear augmented-plane-wave method(FP-LAPW) [9] together with the Ceperley– Alder data for the electron-gas exchange-correlation term [10]. The muffin-tin radii used were 1.37 a.u. for both boron and nitrogen atoms and the self-consistent total energy was converged to 10 − 4 eV. The cut-off angular momentum was l= 10 for the wave function inside the spheres and l=4 for the computational of non-muffin-tin elements.
2. Equilibrium properties and frozen phonons We have evaluated the total energy for various values of the lattice constant, at different plane-wave energy cut-offs and number of k-points sampled in the irreducible Brillouin Zone. Our results are summarised in Table 1. The results obtained for the lattice constant(a0= ˚ ), Bulk modulus (B0 = 3.71 Mbar) and its pres3.625 A sure derivative (B %0 = 3.6 Mbar) are in good agreement with the available experimental results [1,6], as well as
0921-5107/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 9 8 ) 0 0 3 2 7 - 4
H.W. Leite Al6es et al. / Materials Science and Engineering B59 (1999) 264–267
265
Fig. 1. Valence charge density for c-BN in static equilibrium calculated in the (110) plane. The unit of length is the lattice constant a0 and the contour interval is 0.05 electrons per cell. The symbols(X, c) indicates the boron and nitrogen atoms, respectively.
with previous calculations [3,8].On the other hand, our calculated full valence bandwidth of 20.8 eV, direct gap of 8.78 eV and indirect band gap of 4.5 eV at X compare well the corresponding values of 20.3, 8.6 and 4.2 eV obtained in previous pseudopotential calculations [3]. The real-space distribution of the eletronic charge density n(r) shown in Fig. 1 agrees with the previous calculations [3,8] as well: the valence charge distribution in BN behaves like a typical III – V semiconductor [14], viz. by showing a significant bond charge around the anion, i.e. a strong ionic bonding between B and N atoms. The frozen phonon results are summarized in Table 2, where we used as atomic displacements u =0.01a0 3, 0.01a0 and 0.015a0 2 for TO(G), longitudinal modes and transversal modes at X, respectively. The only data which can be compared to our calculations are the phonon frequency n and the Gru¨neisen parameter g for the TO mode at G obtained by Raman measurements[7].
Table 1 Convergence of the calculated static properties of zinc-blende BN with energy cut-offs of the plane-wave expansion and with the number of k-points in the Brillouin zone sampling Epw (Ry)
k-points
˚) a0 (A
B0 (Mbar)
B %0
49.7 54.2 60.1 49.7 49.7 Experiment [1,6]
3 3 3 10 22
3.695 3.696 3.696 3.627 3.625 3.616
3.43 3.42 3.42 3.71 3.71 3.82
3.5 3.5 3.5 3.6 3.6 –
For the longitudinal phonons, our results confirm that the displacement patterns consist in oscillations of either the B or N sublattices alone [15]. However, the difference between the masses of B and N is very small. This means that the difference between the frequencies of LO(X) and LA(X) modes are also small, as shown in Table 2. Comparing our calculated g values for c-BN with those of BP [11], the former have smaller values, which is consistent with the higher value obtained for the Bulk modulus of c-BN compared with BP. For the transverse phonons at X we know that we can treat these modes as coupled oscillations of displacement patterns S1 and S2 which correspond to the respective phonon modes in the diamond structure. We proceed, then, as the previous works [12,15], and evaluate the right linear combination of these displacement patterns by diagonalizing the 2× 2 matrices of the coupling coefficients. As observed also for BP [11], our results showed that the TA(X) is a soft mode and its Table 2 Summary of the most important phonon frequencies with their respective Gru¨neisen parameter g calculated ab initioa Phonon mode
n (THz)
g
n 0 k-points
TO(G) LO(X) LA(X) TO(X) TA(X) Exp.: TO(G) [7]
31.53 33.09 29.74 26.69 20.73 31.62
1.40 0.11 0.27 1.53 −0.10 1.50
20 18 18 24 24 –
a The results given here were obtained with the cutoff energy of 60.1 Ry in the plane-wave expansions.
H.W. Leite Al6es et al. / Materials Science and Engineering B59 (1999) 264–267
266
Table 3 Parameters of the VOSM for c-BN: (a) the initial guess obtained by mass approximation on the parameters of GaP [13]; (b) the final set of parameters a
(a) (b) a
l
ku
k %u
kru
k %ru
Z1
Y1
Y2
K1
K2
47.5 46.9
−1.15 0.89
−0.31 −1.71
4.61 4.84
−7.06 −3.28
2.00 2.43
6.03 5.31
−1.87 −2.93
371.2 270.7
82.00 82.33
All parameters are in e 2/Vcell, with Vcell = a 3/4 and a0 =3.627, except the charges Z1, Y1 and Y2 which are in atomic units.
frequency decreases with increasing pressure: the Gru¨neisen parameter g has a negative value. However, both S1 and S2 modes have a very small coupling coefficient: the TO(X) and TA(X) phonons of c-BN have similar displacement pattern as TO(X) and TA(X) modes in the diamond structure.
3. Shell model The particular version of the shell model we used is known as valence overlap shell model (VOSM) [13]. In the present work, the 10-parameters model was fitted (by a non-linear least-squares method with constrained parameters and weighting) to all available information about phonon frequencies, elastic constants and eigenvectores at X. In order words, to the set of phonon frequencies evaluated by the frozen phonon method in the previous section, we added the experimental values of the LO(G) frequency and of the elastic constants c11, c12 and c44. We also included in the minimized sum of squares the data on the eigendisplacements for the transverse modes at X. The fitting process follows closely the procedure developed in [11] in which the same ideas were applied to BP. In Table 3 we summarize the model parameters obtained by the fitting procedure: the initial set of parameters (row a) and the final set (row b). The calculated phonon dispersion (Fig. 2) shows the same general features as other III – V semiconductor compounds [16] and, also shows the same features as observed by Karch et al. [8]. Finally, our evaluated elastic constants c11, c12, c44 were 81.74, 19.76 and 49.86×1011 dyn cm − 2 agrees well with both experimental and previous theoretical results described by Kim et al. [17].
BN, we have calculated its phonon frequencies and eigenvectors at high-symmetry points at the Brillouin zone (BZ). At the G point of the BZ, our results agree well with those obtained by Raman Scattering [7] and with previous results [8]. At X point of the BZ, our calculated data are predictions and are meant to supply the missing information and show similarities with the lattice dynamics properties of Diamond. Finally, we fitted all the available information by a 10-parameter VOSM and obtained the complete phonon dispersion curves in terms of this interpolation scheme. The calculated elastic constants agree with the experimental ones within : 5%, and the overall shape of our obtained phonon dispersion is in good agreement with previous LDA results [8].
4. Conclusions We have studied the lattice dynamics of c-BN using the density functional theory, within the framework of FP-LAPW. This procedure describes correctly the undistorted structure of c-BN in static equilibrium, and our results for lattice constant and Bulk modulus are in good agreement with the available experimental results. Having a good description of the static equilibrium of
Fig. 2. Phonon dispersion curves for c-BN calculated by VOSM. The symbols indicate the experimental data from Sanjurjo et al. [7].
H.W. Leite Al6es et al. / Materials Science and Engineering B59 (1999) 264–267
Acknowledgements Work supported by the Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico(CNPq), Brazil, and by the Fundac¸a˜o de Amparo a` Pesquisa do Estado de Minas Gerais(FAPEMIG). The computer resources were provided by the Centro Nacional de Processamento de Alto Desempenho em Minas Gerais & Regia˜o Centro-Oeste(CENAPAD-MG/CO). Grateful to M.A. Roma˜o and L.C. de Carvalho for help with the manuscript.
References
[7] [8]
[9] [10] [11] [12]
[13] [14]
[1] J.H. Edgar, Properties of group III Nitrides, INSPEC, London, 1994. [2] R.T. Paine, C.K. Narula, Chem. Rev. 90 (1990) 73. [3] R.M. Wentzcovitch, K.J. Chang, M.L. Cohen, Phys. Rev. B 34 (1986) 1071. [4] R.M. Wentzcovitch, M.L. Cohen, P.K. Lam, Phys. Rev. B 36 (1987) 6058. [5] J.L.P. Castineira, J.R. Leite, L.M.R. Scolfaro, R. Enderlein, J.L. A. Alves, H.W. Leite Alves, Mater. Sci. Eng. B 51 (1998) 53. [6] Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, New series, Vol. 17a, O.
.
[15]
[16]
[17]
267
Madelung (Ed.), Springer-Verlag, Berlin, 1982; O. Madelung (Ed.), Vol. 22a, Springer-Verlag, Berlin, 1989. J.A. Sanjurjo, E. Lo´pez-Cruz, P. Vogl, M. Cardona, Phys. Rev. B 28 (1983) 4579. K. Karch, F. Bechstedt, P. Pavone, D. Strauch, Physica B 219 – 220(1996) 445; K. Karch, F. Bechstedt, Phys. Rev. B 56 (1997) 7404. P. Blaha, K. Schwarz, P. Dufek, R. Augustyn, Computer Code Wien95, Technical University of Vienna, Vienna, 1995. D.M. Ceperley, B.I. Alder, Phys. Rev. Lett. 45 (1980) 566; J. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. H.W. Leite Alves, K. Kunc, J. Phys. Condens. Matter 4 (1992) 6603. K. Kunc, in: J.T. Devreese, P.E. Van Camp (Eds.), Eletronic Structure, Dynamics and Quantum Structural Properties of Condensed Matter, Plenum Press, New York, 1985, p. 227. K. Kunc, H. Bilz, Solid State Commun. 19 (1976) 1027. M.L. Cohen, J.R. Chelikowsky, in: M. Cardona, P. Fulde, K. Von Klitizing, H.-J. Queisser (Eds.), Electronic Structure and Optical Properties of Semiconductors, Springer-Verlag, Berlin, 1998. K. Kunc, R.M. Martin, in: J.T. Devreese, V.E. Van Doren, P.E. Van Camp (Eds.), Ab-initio Calculation of Phonon Spectra, Plenum Press, New York, 1983, p. 65. H. Bilz, W. Kress, in: M. Cardona, P. Fulde, H.-J. Queisser (Eds.), Phonon Dispersion Relations in Insulators, Springer-Verlag, Berlin, 1979, p. 101. K. Kim, W.R.L. Lambert, B. Segall, Phys. Rev. B 53 (1996) 16310.
Lattice dynamics of boron nitride H.W. Leite Alves a,*, J.L.A. Alves a, J.L.P. Castineira b, J.R. Leite c b
a Departamento de Cieˆncias Naturais-FUNREI, Prac¸a D. Hel6e´cio, 74 C.E.P. 36.300 -000 Sa˜o Joa˜o del Rei MG, Brazil Departamento de Cieˆncias Fı´sicas-Uni6ersidade Federal de Uberlandia, C.P. 593 C.E.P., 38.400 -902 Uberlandia MG, Brazil c LNMS-Depto. de Fı´sica dos Materiais e Mecaˆnica, Instituto de Fı´sica, Uni6ersidade de Sa˜o Paulo, C.P. 66.318 C.E.P., 05.389 -970 Sa˜o Paulo SP, Brazil
Abstract Using the density-functional theory within the full potential linear augmented plane-wave (FP-LAPW) method, we have calculated ab initio the equation of state and the principal phonon modes in cubic boron nitride (c-BN), including their pressure dependence and the amplitude of the eigendisplacements. A good agreement with the experiments is obtained, whenever a comparison is possible: in fact, most of the results are predictions. A ten-parameter valence overlap shell model (VOSM) was constructed and we obtained the phonon dispersion curves, elastic constants and effective charges. Our results were compared with calculated theoretical data for c-BN and for other III – V materials and we found that the lattice dynamics properties for cubic boron nitride is very close to those of diamond. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Boron nitride; LAPW method; Lattice dynamics; Frozen phonons; Shell model
1. Introduction There has been considerable interest in the development of optoelectronic and microelectronic devices which work under hard conditions such as high temperatures or aggressive environment. One of the promising materials for this purpose is cubic Boron Nitride(c-BN), which satisfies these conditions due to its physical properties such as hardness, high melting point and resistance to corrosion [1,2]. Unfortunately, the amount of theoretical knowledge on this substance is limited to a handful of works [3 – 5] and, particularly, the data on phonon spectra and vibrational properties are rather scarce [6–8]. In the present work we perform total energy calculations in order to supply the missing information on the vibrational properties of c-BN. The phonon frequencies and the respective eigenvectors of the vibrations at the high symmetry points in the Brillouin-zone are calculated within the frozen phonon model [12]: the frequency of the phonon is obtained from the difference
* Corresponding author. Tel.: +55-32-3792489; fax: + 55-323792483. E-mail address: [email protected] (H.W. Leite Alves)
between the total energies of the displaced and the undisplaced patterns according the particular phonon mode. The procedures used are described elsewhere [11,12,15]. For the total energy calculations we used the ab initio full-potential linear augmented-plane-wave method(FP-LAPW) [9] together with the Ceperley– Alder data for the electron-gas exchange-correlation term [10]. The muffin-tin radii used were 1.37 a.u. for both boron and nitrogen atoms and the self-consistent total energy was converged to 10 − 4 eV. The cut-off angular momentum was l= 10 for the wave function inside the spheres and l=4 for the computational of non-muffin-tin elements.
2. Equilibrium properties and frozen phonons We have evaluated the total energy for various values of the lattice constant, at different plane-wave energy cut-offs and number of k-points sampled in the irreducible Brillouin Zone. Our results are summarised in Table 1. The results obtained for the lattice constant(a0= ˚ ), Bulk modulus (B0 = 3.71 Mbar) and its pres3.625 A sure derivative (B %0 = 3.6 Mbar) are in good agreement with the available experimental results [1,6], as well as
0921-5107/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 9 8 ) 0 0 3 2 7 - 4
H.W. Leite Al6es et al. / Materials Science and Engineering B59 (1999) 264–267
265
Fig. 1. Valence charge density for c-BN in static equilibrium calculated in the (110) plane. The unit of length is the lattice constant a0 and the contour interval is 0.05 electrons per cell. The symbols(X, c) indicates the boron and nitrogen atoms, respectively.
with previous calculations [3,8].On the other hand, our calculated full valence bandwidth of 20.8 eV, direct gap of 8.78 eV and indirect band gap of 4.5 eV at X compare well the corresponding values of 20.3, 8.6 and 4.2 eV obtained in previous pseudopotential calculations [3]. The real-space distribution of the eletronic charge density n(r) shown in Fig. 1 agrees with the previous calculations [3,8] as well: the valence charge distribution in BN behaves like a typical III – V semiconductor [14], viz. by showing a significant bond charge around the anion, i.e. a strong ionic bonding between B and N atoms. The frozen phonon results are summarized in Table 2, where we used as atomic displacements u =0.01a0 3, 0.01a0 and 0.015a0 2 for TO(G), longitudinal modes and transversal modes at X, respectively. The only data which can be compared to our calculations are the phonon frequency n and the Gru¨neisen parameter g for the TO mode at G obtained by Raman measurements[7].
Table 1 Convergence of the calculated static properties of zinc-blende BN with energy cut-offs of the plane-wave expansion and with the number of k-points in the Brillouin zone sampling Epw (Ry)
k-points
˚) a0 (A
B0 (Mbar)
B %0
49.7 54.2 60.1 49.7 49.7 Experiment [1,6]
3 3 3 10 22
3.695 3.696 3.696 3.627 3.625 3.616
3.43 3.42 3.42 3.71 3.71 3.82
3.5 3.5 3.5 3.6 3.6 –
For the longitudinal phonons, our results confirm that the displacement patterns consist in oscillations of either the B or N sublattices alone [15]. However, the difference between the masses of B and N is very small. This means that the difference between the frequencies of LO(X) and LA(X) modes are also small, as shown in Table 2. Comparing our calculated g values for c-BN with those of BP [11], the former have smaller values, which is consistent with the higher value obtained for the Bulk modulus of c-BN compared with BP. For the transverse phonons at X we know that we can treat these modes as coupled oscillations of displacement patterns S1 and S2 which correspond to the respective phonon modes in the diamond structure. We proceed, then, as the previous works [12,15], and evaluate the right linear combination of these displacement patterns by diagonalizing the 2× 2 matrices of the coupling coefficients. As observed also for BP [11], our results showed that the TA(X) is a soft mode and its Table 2 Summary of the most important phonon frequencies with their respective Gru¨neisen parameter g calculated ab initioa Phonon mode
n (THz)
g
n 0 k-points
TO(G) LO(X) LA(X) TO(X) TA(X) Exp.: TO(G) [7]
31.53 33.09 29.74 26.69 20.73 31.62
1.40 0.11 0.27 1.53 −0.10 1.50
20 18 18 24 24 –
a The results given here were obtained with the cutoff energy of 60.1 Ry in the plane-wave expansions.
H.W. Leite Al6es et al. / Materials Science and Engineering B59 (1999) 264–267
266
Table 3 Parameters of the VOSM for c-BN: (a) the initial guess obtained by mass approximation on the parameters of GaP [13]; (b) the final set of parameters a
(a) (b) a
l
ku
k %u
kru
k %ru
Z1
Y1
Y2
K1
K2
47.5 46.9
−1.15 0.89
−0.31 −1.71
4.61 4.84
−7.06 −3.28
2.00 2.43
6.03 5.31
−1.87 −2.93
371.2 270.7
82.00 82.33
All parameters are in e 2/Vcell, with Vcell = a 3/4 and a0 =3.627, except the charges Z1, Y1 and Y2 which are in atomic units.
frequency decreases with increasing pressure: the Gru¨neisen parameter g has a negative value. However, both S1 and S2 modes have a very small coupling coefficient: the TO(X) and TA(X) phonons of c-BN have similar displacement pattern as TO(X) and TA(X) modes in the diamond structure.
3. Shell model The particular version of the shell model we used is known as valence overlap shell model (VOSM) [13]. In the present work, the 10-parameters model was fitted (by a non-linear least-squares method with constrained parameters and weighting) to all available information about phonon frequencies, elastic constants and eigenvectores at X. In order words, to the set of phonon frequencies evaluated by the frozen phonon method in the previous section, we added the experimental values of the LO(G) frequency and of the elastic constants c11, c12 and c44. We also included in the minimized sum of squares the data on the eigendisplacements for the transverse modes at X. The fitting process follows closely the procedure developed in [11] in which the same ideas were applied to BP. In Table 3 we summarize the model parameters obtained by the fitting procedure: the initial set of parameters (row a) and the final set (row b). The calculated phonon dispersion (Fig. 2) shows the same general features as other III – V semiconductor compounds [16] and, also shows the same features as observed by Karch et al. [8]. Finally, our evaluated elastic constants c11, c12, c44 were 81.74, 19.76 and 49.86×1011 dyn cm − 2 agrees well with both experimental and previous theoretical results described by Kim et al. [17].
BN, we have calculated its phonon frequencies and eigenvectors at high-symmetry points at the Brillouin zone (BZ). At the G point of the BZ, our results agree well with those obtained by Raman Scattering [7] and with previous results [8]. At X point of the BZ, our calculated data are predictions and are meant to supply the missing information and show similarities with the lattice dynamics properties of Diamond. Finally, we fitted all the available information by a 10-parameter VOSM and obtained the complete phonon dispersion curves in terms of this interpolation scheme. The calculated elastic constants agree with the experimental ones within : 5%, and the overall shape of our obtained phonon dispersion is in good agreement with previous LDA results [8].
4. Conclusions We have studied the lattice dynamics of c-BN using the density functional theory, within the framework of FP-LAPW. This procedure describes correctly the undistorted structure of c-BN in static equilibrium, and our results for lattice constant and Bulk modulus are in good agreement with the available experimental results. Having a good description of the static equilibrium of
Fig. 2. Phonon dispersion curves for c-BN calculated by VOSM. The symbols indicate the experimental data from Sanjurjo et al. [7].
H.W. Leite Al6es et al. / Materials Science and Engineering B59 (1999) 264–267
Acknowledgements Work supported by the Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico(CNPq), Brazil, and by the Fundac¸a˜o de Amparo a` Pesquisa do Estado de Minas Gerais(FAPEMIG). The computer resources were provided by the Centro Nacional de Processamento de Alto Desempenho em Minas Gerais & Regia˜o Centro-Oeste(CENAPAD-MG/CO). Grateful to M.A. Roma˜o and L.C. de Carvalho for help with the manuscript.
References
[7] [8]
[9] [10] [11] [12]
[13] [14]
[1] J.H. Edgar, Properties of group III Nitrides, INSPEC, London, 1994. [2] R.T. Paine, C.K. Narula, Chem. Rev. 90 (1990) 73. [3] R.M. Wentzcovitch, K.J. Chang, M.L. Cohen, Phys. Rev. B 34 (1986) 1071. [4] R.M. Wentzcovitch, M.L. Cohen, P.K. Lam, Phys. Rev. B 36 (1987) 6058. [5] J.L.P. Castineira, J.R. Leite, L.M.R. Scolfaro, R. Enderlein, J.L. A. Alves, H.W. Leite Alves, Mater. Sci. Eng. B 51 (1998) 53. [6] Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, New series, Vol. 17a, O.
.
[15]
[16]
[17]
267
Madelung (Ed.), Springer-Verlag, Berlin, 1982; O. Madelung (Ed.), Vol. 22a, Springer-Verlag, Berlin, 1989. J.A. Sanjurjo, E. Lo´pez-Cruz, P. Vogl, M. Cardona, Phys. Rev. B 28 (1983) 4579. K. Karch, F. Bechstedt, P. Pavone, D. Strauch, Physica B 219 – 220(1996) 445; K. Karch, F. Bechstedt, Phys. Rev. B 56 (1997) 7404. P. Blaha, K. Schwarz, P. Dufek, R. Augustyn, Computer Code Wien95, Technical University of Vienna, Vienna, 1995. D.M. Ceperley, B.I. Alder, Phys. Rev. Lett. 45 (1980) 566; J. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. H.W. Leite Alves, K. Kunc, J. Phys. Condens. Matter 4 (1992) 6603. K. Kunc, in: J.T. Devreese, P.E. Van Camp (Eds.), Eletronic Structure, Dynamics and Quantum Structural Properties of Condensed Matter, Plenum Press, New York, 1985, p. 227. K. Kunc, H. Bilz, Solid State Commun. 19 (1976) 1027. M.L. Cohen, J.R. Chelikowsky, in: M. Cardona, P. Fulde, K. Von Klitizing, H.-J. Queisser (Eds.), Electronic Structure and Optical Properties of Semiconductors, Springer-Verlag, Berlin, 1998. K. Kunc, R.M. Martin, in: J.T. Devreese, V.E. Van Doren, P.E. Van Camp (Eds.), Ab-initio Calculation of Phonon Spectra, Plenum Press, New York, 1983, p. 65. H. Bilz, W. Kress, in: M. Cardona, P. Fulde, H.-J. Queisser (Eds.), Phonon Dispersion Relations in Insulators, Springer-Verlag, Berlin, 1979, p. 101. K. Kim, W.R.L. Lambert, B. Segall, Phys. Rev. B 53 (1996) 16310.