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Theoretical approach to initial growth kinetics of GaN on GaN(0 0 1)
ARTICLE IN PRESS
Journal of Crystal Growth 300 (2007) 62–65 www.elsevier.com/locate/jcrysgro
Theoretical approach to initial growth kinetics of GaN on GaN(0 0 1) Y. Kangawaa,, Y. Matsuoa, T. Akiyamab, T. Itob, K. Shiraishic, K. Kakimotoa a
Research Institute for Applied Mechanics, Kyushu University, 6-1 Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan b Department of Physics Engineering, Mie University, Mie 514-8507, Japan c Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan Available online 6 December 2006
Abstract We carried out theoretical analyses based on ab initio calculations incorporates in which free energy of the vapor phase is incorporated in order to determine the initial growth kinetics of c-GaN on GaN(0 0 1)-(4 1). The feasibility of the theoretical approach had been confirmed by calculations of Ga adsorption–desorption transition temperature and transition beam equivalent pressures on the GaAs(0 0 1)-(4 2)b2 surface in our previous work [Y. Kangawa, T. Ito, A. Taguchi, K. Shiraishi, T. Ohachi, Surf. Sci. 493 (2001) 178]. The results of calculations suggest that no Ga adsorption occurs on the initial surface under typical growth conditions but that a Ga adsorption site appears after N adsorption on GaN(0 0 1)-(4 1). That is, in the initial growth stage of c-GaN on GaN(0 0 1)-(4 1), a Nadsorbed structure is formed and then Ga adsorbs on the N adatom. r 2006 Elsevier B.V. All rights reserved. PACS: 31.15.Ar; 68.35.Bs; 68.47.Fg Keywords: A1. Adsorption; A3. Molecular beam epitaxy; B1. c-GaN
1. Introduction It is expected that cubic GaN (c-GaN) would have many advantages in physical properties such as phonon propagation since the crystals have higher crystallographic symmetry than that of hexagonal ones. However, there have been few studies on c-GaN compared with the number of studies on hexagonal GaN (h-GaN). This is because of its metastable nature and because of the lack of an appropriate substrate with stability at high temperatures. Recently, some researchers have succeeded in growing cGaN by controlling the reconstructed surface [1] and {1 1 1} facet formation which causes the h-GaN mixing [2]. Some physical properties have been characterized, and detailed reconstructed structures on GaN(0 0 1) are clarified [3,4]. In the literature [3,4], a GaN(0 0 1)-(4 1) reconstructed structure based on a linear Ga-tetramer model has been suggested, and typical MBE growth of cGaN has been performed near a 4 1 stable condition [1]. This is different from the case of GaAs growth. In the case Corresponding author.
E-mail address: [email protected] (Y. Kangawa). 0022-0248/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2006.10.203
of GaAs, the most typical reconstructed structure under usual growth conditions is GaAs(0 0 1)-(2 4) with Asdimer rows. That is, c-GaN was grown under a Ga-rich (group-III source rich) condition, while GaAs was grown under an As-rich (group-V source rich) condition. This implies that growth kinetics of c-GaN and that of GaAs are different. However, we cannot compare the growth kinetics of c-GaN with that of GaAs because the detailed growth kinetics of c-GaN has not been elucidated, whereas that of GaAs has been reported [5,6]. It is important to know the detailed growth kinetics of c-GaN from a scientific viewpoint as well as from the viewpoint of suppressing h-GaN mixing. In the present work, we carried out theoretical analyses based on ab initio calculations in which free energy of the vapor phase is incorporated [7,8] in order to determine the initial growth kinetics of c-GaN on GaN(0 0 1)-(4 1).
2. Computational methods It is known that a GaN(0 0 1)-(4 1) reconstructed structure based on a linear Ga-tetramer is stable at typical
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growth conditions for c-GaN during MBE [1,3,4]. Therefore, we assume the -(4 1) reconstructed surface as an initial substrate and investigated the relationship between adsorption–desorption behavior of Ga and N on the surface and growth conditions such as temperature and V/ III ratio by an ab initio-based approach incorporating free energy of the vapor phase [7,8]. Here, adsorption–desorption behavior can be described by comparing the free energy of ideal gas per one particle, i.e., chemical potential (m), with the adsorption energy (Ead). That is, adsorption of atoms occurs when Ead is smaller than m, whereas desorption occurs when m is smaller than Ead. The chemical potential and adsorption energy can be calculated by the following methods. The chemical potential m for the ideal gas is given by m ¼ kB T lnðgkB Tztrans =pÞ,
(1)
ztrans ¼ ð2pmkB T=h2 Þ3=2 ,
(2)
where ztrans is the partition function for the translational motion. Here, kB is Boltzmann’s constant, T the temperature, g the degree of degeneracy of the electron energy level, p the BEP (beam equivalent pressure) of the particle, m the mass of one particle, and h is Planck’s constant. In the calculation, we consider atomic Ga and N as group-III and -V sources, respectively. In the case of atomic Ga and N, the degrees of degeneracy of electron energy level g are 2 and 4, respectively. The adsorption energies of Ga and N on a GaN(0 0 1)-(4 1) surface are obtained by ab initio calculations. In the present work, the calculations were performed by using the generalized gradient approximation (GGA) in density functional theory [9,10]. For the exchange-correlation energy, we used a functional form proposed by Perdew et al. [11]. Norm-conserving pseudopotentials [12] were used for Ga and fictitious H atoms, and an ultrasoft pseudopotential [13] was used for N atoms. The conjugate-gradient minimization technique is applied both for ionic and electronic degrees of freedom [14,15]. The valence wave functions are expanded by the planewave basis set with a cutoff energy of 64 Ry. Conventional repeated slab models consisting of four atomic layers of GaN, an atomic layer of fictitious H atoms, adatoms, and 22 A˚ of vacuum region in thickness were used for the calculations. 3. Results and discussion First, we studied the adsorption–desorption behavior of atomic Ga and N on the GaN(0 0 1)-(4 1) surface. Fig. 1(a) and (b) show a schematic drawing of the GaN(0 0 1)(4 1) surface and the adsorption energy on the sites denoted by the capital letters (A–E), respectively. Here, we give the energy of the model where the Ga or N for adsorption is still located in the vacuum region, i.e., 5 A˚ from the top surface, as zero. In Fig. 1(b), we can see that the most favorable sites for adsorption are ‘‘E’’ site for Ga and ‘‘A’’ site for N. In the case of Ga adsorption, the
Fig. 1. (a) Schematic drawing of GaN(0 0 1)-(4 1) surface. (b) Adsorption energies of Ga and N on sites A–E shown in (a).
adsorption energy on the surface is 2.1 eV. This value is larger than the formation energy of a Ga droplet (2.8 eV [7]). Therefore, the formation of a Ga droplet is easier than the formation of a Ga-adsorbed structure in this case. In the case of N adsorption, the adsorption energy is 6.4 eV. Using these values of adsorption energy, we can obtain phase diagrams related to the surface structures as shown in Fig. 2. In Fig. 2, it is found that no Ga-adsorbed structure would be formed, though Ga droplets seem to be formed on the surface at high Ga BEP and low temperature region. Here, the boundary line satisfies the conditions of mGa ¼ E Gadroplet where EGa-droplet is formation energy of a Ga droplet (2.8 eV). On the other hand, the N-adsorbed structure is stable at typical growth conditions as shown in the phase diagram. These results suggest that an N-adsorbed structure appears instead of a Ga-adsorbed structure in the case of adsorption of the first atom. Next, we investigated the adsorption–desorption behavior of the second atom on the N-adsorbed GaN(0 0 1)(4 1) surface. Fig. 3(a) and (b) show a schematic drawing of the N-adsorbed GaN(0 0 1)-(4 1) surface and the adsorption energy on each site, respectively. In Fig. 3(a), it can be seen that the most favorable sites are ‘‘D’’ for Ga adsorption and ‘‘B’’ for N adsorption. However, if a N atom is adsorbed near the N atoms on the surface, i.e., the case of adsorption on ‘‘A’’ and ‘‘D’’ sites, N atoms would desorb as an N2 molecule instead of N-dimer formation
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Fig. 2. p–T dependence of adsorption–desorption behavior of (a) Ga and (b) N on the initial surface.
GaN growth on the GaN(0 0 1)-(4 1) surface proceeds via the following process during the initial growth stage. First, a N-adsorbed structure is formed instead of a Ga-adsorbed structure. Subsequently, Ga adsorption on the N adatom occurs. Here, the N adatom is stabilized by being covered with a Ga adatom. This is different than the GaAs case, i.e., group-III atoms are incorporated in the crystal by being covered with group-V atoms in the case of GaAs. 4. Summary
Fig. 3. (a) Schematic drawing of N-adsorbed GaN(0 0 1)-(4 1) surface. (b) Adsorption energies of Ga and N on sites A–E shown in (a).
because a N-dimer is unstable on the surface. Phase diagrams were made using the calculated adsorption energies, i.e., 3.7 eV for Ga and 6.0 eV for N, as shown in Fig. 4. The results imply that the stable region of the Gaadsorbed structure appears in the phase diagram after the first N adsorption. That is, a Ga-adsorbing site appears after adsorption of the first N adatom. On the other hand, it is found that N adsorption might occur under the condition shown in the diagram; however, N adatoms seem to desorb as an N2 molecule if a N adatom migrates to the sites on an another N adatom. These results suggest that c-
We carried out theoretical analyses based on ab initio calculations in which free energy of the vapor phase is incorporated [7] in order to determine the initial growth kinetics of c-GaN on GaN(0 0 1)-(4 1). The results of calculation imply that a Ga adsorption site appears after N adsorption, though no Ga adsorption occurs on the initial surface under typical growth conditions. Furthermore, it was found that a N-dimer is unstable on the surface. This suggests that N adatoms might desorb as an N2 molecule if two N atoms meet together during the migration process. That is, the stabilizing process of N adatoms by covering with Ga adatoms is important to grow c-GaN during the initial growth stage. This process is different from the GaAs growth process [5,6]. That is, Ga atoms are incorporated in the crystal by being covered with Asdimers in the case of GaAs growth. Though more detailed examinations are required, these processes seem to correspond to the growth conditions, i.e., c-GaN is grown under a Ga-rich condition, while GaAs is grown under an As-rich condition. We are now performing calculations for the adsorption of charged N and for the growth of h-GaN on GaN(1 1 1) faceted plane to find an optimum growth condition of c-GaN. Acknowledgment Very helpful discussions with Dr. Akihito Taguchi are acknowledged.
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Fig. 4. p–T dependence of adsorption–desorption behavior of (a) Ga and (b) N on an N-adsorbed surface.
References [1] H. Okumura, K. Ohta, G. Feuillet, K. Balakrishnan, S. Chichibu, H. Hamaguchi, P. Hacke, S. Yoshida, J. Crystal Growth 178 (1997) 113. [2] R. Kimura, T. Suzuki, M. Ouchi, K. Ishida, K. Takahashi, J. Crystal Growth 278 (2005) 411. [3] J. Neugebauer, T. Zywietz, M. Scheffler, J.E. Northrup, C.G. Van de Walle, Phys. Rev. Lett. 80 (1998) 3097. [4] H.A.H. AL-Brithen, R. Yang, M.B. Haider, C. Constantin, E. Lu, A.R. Smith, N. Sandler, P. Ordejo´n, Phys. Rev. Lett. 95 (2005) 146102. [5] T. Ito, K. Shiraishi, Surf. Sci. 357–358 (1996) 486. [6] T. Ito, K. Shiraishi, Surf. Sci. 386 (1997) 241.
[7] Y. Kangawa, T. Ito, A. Taguchi, K. Shiraishi, T. Ohachi, Surf. Sci. 493 (2001) 178. [8] Y. Kangawa, T. Ito, Y.S. Hiraoka, A. Taguchi, K. Shiraishi, T. Ohachi, Surf. Sci. 507–510 (2002) 285. [9] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. [10] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [11] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3365. [12] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [13] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [14] J. Yamauchi, M. Tsukada, S. Watanabe, O. Sugino, Phys. Rev. B 54 (1996) 5586. [15] M. Tsukada, et al., Computer program package TAPP, University of Tokyo, Tokyo, Japan, 1983–2002.
Journal of Crystal Growth 300 (2007) 62–65 www.elsevier.com/locate/jcrysgro
Theoretical approach to initial growth kinetics of GaN on GaN(0 0 1) Y. Kangawaa,, Y. Matsuoa, T. Akiyamab, T. Itob, K. Shiraishic, K. Kakimotoa a
Research Institute for Applied Mechanics, Kyushu University, 6-1 Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan b Department of Physics Engineering, Mie University, Mie 514-8507, Japan c Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan Available online 6 December 2006
Abstract We carried out theoretical analyses based on ab initio calculations incorporates in which free energy of the vapor phase is incorporated in order to determine the initial growth kinetics of c-GaN on GaN(0 0 1)-(4 1). The feasibility of the theoretical approach had been confirmed by calculations of Ga adsorption–desorption transition temperature and transition beam equivalent pressures on the GaAs(0 0 1)-(4 2)b2 surface in our previous work [Y. Kangawa, T. Ito, A. Taguchi, K. Shiraishi, T. Ohachi, Surf. Sci. 493 (2001) 178]. The results of calculations suggest that no Ga adsorption occurs on the initial surface under typical growth conditions but that a Ga adsorption site appears after N adsorption on GaN(0 0 1)-(4 1). That is, in the initial growth stage of c-GaN on GaN(0 0 1)-(4 1), a Nadsorbed structure is formed and then Ga adsorbs on the N adatom. r 2006 Elsevier B.V. All rights reserved. PACS: 31.15.Ar; 68.35.Bs; 68.47.Fg Keywords: A1. Adsorption; A3. Molecular beam epitaxy; B1. c-GaN
1. Introduction It is expected that cubic GaN (c-GaN) would have many advantages in physical properties such as phonon propagation since the crystals have higher crystallographic symmetry than that of hexagonal ones. However, there have been few studies on c-GaN compared with the number of studies on hexagonal GaN (h-GaN). This is because of its metastable nature and because of the lack of an appropriate substrate with stability at high temperatures. Recently, some researchers have succeeded in growing cGaN by controlling the reconstructed surface [1] and {1 1 1} facet formation which causes the h-GaN mixing [2]. Some physical properties have been characterized, and detailed reconstructed structures on GaN(0 0 1) are clarified [3,4]. In the literature [3,4], a GaN(0 0 1)-(4 1) reconstructed structure based on a linear Ga-tetramer model has been suggested, and typical MBE growth of cGaN has been performed near a 4 1 stable condition [1]. This is different from the case of GaAs growth. In the case Corresponding author.
E-mail address: [email protected] (Y. Kangawa). 0022-0248/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2006.10.203
of GaAs, the most typical reconstructed structure under usual growth conditions is GaAs(0 0 1)-(2 4) with Asdimer rows. That is, c-GaN was grown under a Ga-rich (group-III source rich) condition, while GaAs was grown under an As-rich (group-V source rich) condition. This implies that growth kinetics of c-GaN and that of GaAs are different. However, we cannot compare the growth kinetics of c-GaN with that of GaAs because the detailed growth kinetics of c-GaN has not been elucidated, whereas that of GaAs has been reported [5,6]. It is important to know the detailed growth kinetics of c-GaN from a scientific viewpoint as well as from the viewpoint of suppressing h-GaN mixing. In the present work, we carried out theoretical analyses based on ab initio calculations in which free energy of the vapor phase is incorporated [7,8] in order to determine the initial growth kinetics of c-GaN on GaN(0 0 1)-(4 1).
2. Computational methods It is known that a GaN(0 0 1)-(4 1) reconstructed structure based on a linear Ga-tetramer is stable at typical
ARTICLE IN PRESS Y. Kangawa et al. / Journal of Crystal Growth 300 (2007) 62–65
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growth conditions for c-GaN during MBE [1,3,4]. Therefore, we assume the -(4 1) reconstructed surface as an initial substrate and investigated the relationship between adsorption–desorption behavior of Ga and N on the surface and growth conditions such as temperature and V/ III ratio by an ab initio-based approach incorporating free energy of the vapor phase [7,8]. Here, adsorption–desorption behavior can be described by comparing the free energy of ideal gas per one particle, i.e., chemical potential (m), with the adsorption energy (Ead). That is, adsorption of atoms occurs when Ead is smaller than m, whereas desorption occurs when m is smaller than Ead. The chemical potential and adsorption energy can be calculated by the following methods. The chemical potential m for the ideal gas is given by m ¼ kB T lnðgkB Tztrans =pÞ,
(1)
ztrans ¼ ð2pmkB T=h2 Þ3=2 ,
(2)
where ztrans is the partition function for the translational motion. Here, kB is Boltzmann’s constant, T the temperature, g the degree of degeneracy of the electron energy level, p the BEP (beam equivalent pressure) of the particle, m the mass of one particle, and h is Planck’s constant. In the calculation, we consider atomic Ga and N as group-III and -V sources, respectively. In the case of atomic Ga and N, the degrees of degeneracy of electron energy level g are 2 and 4, respectively. The adsorption energies of Ga and N on a GaN(0 0 1)-(4 1) surface are obtained by ab initio calculations. In the present work, the calculations were performed by using the generalized gradient approximation (GGA) in density functional theory [9,10]. For the exchange-correlation energy, we used a functional form proposed by Perdew et al. [11]. Norm-conserving pseudopotentials [12] were used for Ga and fictitious H atoms, and an ultrasoft pseudopotential [13] was used for N atoms. The conjugate-gradient minimization technique is applied both for ionic and electronic degrees of freedom [14,15]. The valence wave functions are expanded by the planewave basis set with a cutoff energy of 64 Ry. Conventional repeated slab models consisting of four atomic layers of GaN, an atomic layer of fictitious H atoms, adatoms, and 22 A˚ of vacuum region in thickness were used for the calculations. 3. Results and discussion First, we studied the adsorption–desorption behavior of atomic Ga and N on the GaN(0 0 1)-(4 1) surface. Fig. 1(a) and (b) show a schematic drawing of the GaN(0 0 1)(4 1) surface and the adsorption energy on the sites denoted by the capital letters (A–E), respectively. Here, we give the energy of the model where the Ga or N for adsorption is still located in the vacuum region, i.e., 5 A˚ from the top surface, as zero. In Fig. 1(b), we can see that the most favorable sites for adsorption are ‘‘E’’ site for Ga and ‘‘A’’ site for N. In the case of Ga adsorption, the
Fig. 1. (a) Schematic drawing of GaN(0 0 1)-(4 1) surface. (b) Adsorption energies of Ga and N on sites A–E shown in (a).
adsorption energy on the surface is 2.1 eV. This value is larger than the formation energy of a Ga droplet (2.8 eV [7]). Therefore, the formation of a Ga droplet is easier than the formation of a Ga-adsorbed structure in this case. In the case of N adsorption, the adsorption energy is 6.4 eV. Using these values of adsorption energy, we can obtain phase diagrams related to the surface structures as shown in Fig. 2. In Fig. 2, it is found that no Ga-adsorbed structure would be formed, though Ga droplets seem to be formed on the surface at high Ga BEP and low temperature region. Here, the boundary line satisfies the conditions of mGa ¼ E Gadroplet where EGa-droplet is formation energy of a Ga droplet (2.8 eV). On the other hand, the N-adsorbed structure is stable at typical growth conditions as shown in the phase diagram. These results suggest that an N-adsorbed structure appears instead of a Ga-adsorbed structure in the case of adsorption of the first atom. Next, we investigated the adsorption–desorption behavior of the second atom on the N-adsorbed GaN(0 0 1)(4 1) surface. Fig. 3(a) and (b) show a schematic drawing of the N-adsorbed GaN(0 0 1)-(4 1) surface and the adsorption energy on each site, respectively. In Fig. 3(a), it can be seen that the most favorable sites are ‘‘D’’ for Ga adsorption and ‘‘B’’ for N adsorption. However, if a N atom is adsorbed near the N atoms on the surface, i.e., the case of adsorption on ‘‘A’’ and ‘‘D’’ sites, N atoms would desorb as an N2 molecule instead of N-dimer formation
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Fig. 2. p–T dependence of adsorption–desorption behavior of (a) Ga and (b) N on the initial surface.
GaN growth on the GaN(0 0 1)-(4 1) surface proceeds via the following process during the initial growth stage. First, a N-adsorbed structure is formed instead of a Ga-adsorbed structure. Subsequently, Ga adsorption on the N adatom occurs. Here, the N adatom is stabilized by being covered with a Ga adatom. This is different than the GaAs case, i.e., group-III atoms are incorporated in the crystal by being covered with group-V atoms in the case of GaAs. 4. Summary
Fig. 3. (a) Schematic drawing of N-adsorbed GaN(0 0 1)-(4 1) surface. (b) Adsorption energies of Ga and N on sites A–E shown in (a).
because a N-dimer is unstable on the surface. Phase diagrams were made using the calculated adsorption energies, i.e., 3.7 eV for Ga and 6.0 eV for N, as shown in Fig. 4. The results imply that the stable region of the Gaadsorbed structure appears in the phase diagram after the first N adsorption. That is, a Ga-adsorbing site appears after adsorption of the first N adatom. On the other hand, it is found that N adsorption might occur under the condition shown in the diagram; however, N adatoms seem to desorb as an N2 molecule if a N adatom migrates to the sites on an another N adatom. These results suggest that c-
We carried out theoretical analyses based on ab initio calculations in which free energy of the vapor phase is incorporated [7] in order to determine the initial growth kinetics of c-GaN on GaN(0 0 1)-(4 1). The results of calculation imply that a Ga adsorption site appears after N adsorption, though no Ga adsorption occurs on the initial surface under typical growth conditions. Furthermore, it was found that a N-dimer is unstable on the surface. This suggests that N adatoms might desorb as an N2 molecule if two N atoms meet together during the migration process. That is, the stabilizing process of N adatoms by covering with Ga adatoms is important to grow c-GaN during the initial growth stage. This process is different from the GaAs growth process [5,6]. That is, Ga atoms are incorporated in the crystal by being covered with Asdimers in the case of GaAs growth. Though more detailed examinations are required, these processes seem to correspond to the growth conditions, i.e., c-GaN is grown under a Ga-rich condition, while GaAs is grown under an As-rich condition. We are now performing calculations for the adsorption of charged N and for the growth of h-GaN on GaN(1 1 1) faceted plane to find an optimum growth condition of c-GaN. Acknowledgment Very helpful discussions with Dr. Akihito Taguchi are acknowledged.
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Fig. 4. p–T dependence of adsorption–desorption behavior of (a) Ga and (b) N on an N-adsorbed surface.
References [1] H. Okumura, K. Ohta, G. Feuillet, K. Balakrishnan, S. Chichibu, H. Hamaguchi, P. Hacke, S. Yoshida, J. Crystal Growth 178 (1997) 113. [2] R. Kimura, T. Suzuki, M. Ouchi, K. Ishida, K. Takahashi, J. Crystal Growth 278 (2005) 411. [3] J. Neugebauer, T. Zywietz, M. Scheffler, J.E. Northrup, C.G. Van de Walle, Phys. Rev. Lett. 80 (1998) 3097. [4] H.A.H. AL-Brithen, R. Yang, M.B. Haider, C. Constantin, E. Lu, A.R. Smith, N. Sandler, P. Ordejo´n, Phys. Rev. Lett. 95 (2005) 146102. [5] T. Ito, K. Shiraishi, Surf. Sci. 357–358 (1996) 486. [6] T. Ito, K. Shiraishi, Surf. Sci. 386 (1997) 241.
[7] Y. Kangawa, T. Ito, A. Taguchi, K. Shiraishi, T. Ohachi, Surf. Sci. 493 (2001) 178. [8] Y. Kangawa, T. Ito, Y.S. Hiraoka, A. Taguchi, K. Shiraishi, T. Ohachi, Surf. Sci. 507–510 (2002) 285. [9] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. [10] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [11] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3365. [12] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [13] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [14] J. Yamauchi, M. Tsukada, S. Watanabe, O. Sugino, Phys. Rev. B 54 (1996) 5586. [15] M. Tsukada, et al., Computer program package TAPP, University of Tokyo, Tokyo, Japan, 1983–2002.