- Email: [email protected]
Phase stability of Mn-doped GaInAs alloys
Journal of Crystal Growth 318 (2011) 360–362
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Phase stability of Mn-doped GaInAs alloys Kohji Nakamura , Masahiro Miyake, Toru Akiyama, Tomonori Ito Department of Physics Engineering, Mie University, Tsu, Mie 514-8507, Japan
a r t i c l e i n f o
a b s t r a c t
Available online 18 November 2010
Phase stability of pseudo-ternary (Ga,In,Mn)As alloys with a zincblende structure is investigated by means of the full-potential linearized augmented plane-wave method. Calculated results predict that the ordered and disordered states are energetically less favorable with respect to the phase separation. When the formation energy is decomposed into elastic and chemical energy contributions, the alloys with more than 0.2 In composition have a chemical tendency of mixing, while the phase separation is driven by the large elastic energy contribution. & 2010 Elsevier B.V. All rights reserved.
Keywords: A1. Phase equilibria A1. Solubility B2. Magnetic materials B2. Semiconducting III–V materials
1. Introduction Understanding the structural, electronic, and magnetic properties of magnetic semiconductors, such as Mn-doped GaAs, is of crucial importance in realizing novel spintronic applications [1,2]. So far, many efforts have been performed to synthesis the magnetic semiconductors with high Mn composition in order to increase the Curie temperature, and the solubility of the Mn has been achieved up to about 10% by means of low-temperature molecular beam epitaxy (MBE) [3]. Although the MBE growth introduces high Mn interstitial and As antisite defects, a low-temperature annealing (2002300 1C) can eliminate most of the defects and so the Mn atoms preferably occupy substitutional cation sites [4,5]. From a theoretical viewpoint, we previously [6] investigated the stability of the pseudo-binary (Ga, Mn)As alloy with a zincblende structure by means of the full-potential linearized augmented plane-wave (FLAPW) method [7] within the local density approximation (LDA) [8], generalized gradient approximation (GGA) [9], and LDA+ U [10]. In contrast to LDA and GGA predictions, importantly, the calculated lattice constant in the LDA+U is found to increase when the Mn composition increases, due to the correlation correction of the pd hybridization strength between the Mn 3d bands and the As 4p valence bands, which agrees with experimental findings [11]. In addition, we found that the system has a tendency to segregate into constituent GaAs and MnAs. The thermodynamic equilibrium Mn solubility into GaAs is very low at low-temperatures, namely, 1% at 300 1C, and the spinodal decomposition occurs for more than about 10% at the low-temperatures. Moreover, the phase separation is found to be mainly driven by a positive chemical energy contribution and the system favors clustering
[12]. In the present work, we extend our FLAPW calculations to the pseudo-ternary (Ga, In, Mn)As alloy and discuss the phase stability.
2. Method We considered 15 ordered zincblende structures, Gam Inn Mn4mn As (m and n ¼0–4), with no tetragonal distortion, in which the Ga, In, and Mn atoms substitutionally occupy cation sites (a, b, g, and d in Fig. 1). Assuming a ferromagnetic state, total energy, E(a), as a function of the lattice constant (a) is calculated by using the FLAPW method within the LDA+ U, in which the core states are treated fully relativistically and the valence states are treated semi-relativistically, i.e., without spin–orbit coupling. Local atomic relaxations of anions (As) that surround the Ga, In, or Mn atoms are introduced in accordance with atomic force calculations [13]. The literature values of Coulomb (U) and exchange (J) parameters used in the LDA+U for the Mn 3d states are 4 and 0.7 eV, respectively [14]. The LAPW basis with a cut-off of jk þGjp 3:9 a:u:1 and muffin-tin sphere radii of 2.3 a.u. for the Ga, In, and Mn and 2.0 a.u. for the As are used. Lattice harmonics with angular momentum up to ‘ ¼ 8 are employed to expand the charge density, potential and wavefunctions.
3. Results and discussion Table 1 summarizes the calculated formation energy, Emn, of the 15 ordered Gam Inn Mn4mn As at the equilibrium volume (the lattice constant of amn ) that minimizes the total energy, referred to the segregation limit as Emn ¼ Eðamn Þ
Corresponding author. Tel.: +81 59 231 5361; fax: +81 59 231 9726.
E-mail address: [email protected] (K. Nakamura). 0022-0248/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2010.11.023
m n 4mn EGaAs EInAs EMnAs , 4 4 4
where EGaAs (InAs, MnAs) is the total energy of the constituent GaAs (InAs, MnAs), and the Emn(a) is fitted by the Murnaghan equation of
K. Nakamura et al. / Journal of Crystal Growth 318 (2011) 360–362
361
InAs 20
γ δ
40
β
α
60 Fig. 1. Zincblende structure; a, b, g, and d indicate sites on a fcc sublattice, which are occupied by Ga, In, and Mn atoms.
Table 1 Calculated formation energy, Emn in meV/f.u., at the equilibrium volume, and the ˚ for the 15 ordered zincblende Gam Inn Mn4mn As. a, b, g, lattice constant, amn in A, and d indicate sites of a tetrahedron in Fig. 1.
40
40
a
b
g
d
amn
Emn
Ga In Mn Ga Ga Ga In In In Ga Ga Ga Ga Ga Ga
Ga In Mn Ga Ga Mn In In Mn Ga Ga In Ga In In
Ga In Mn Ga Mn Mn In Mn Mn Ga In In In In Mn
Ga In Mn Mn Mn Mn Mn Mn Mn In In In Mn Mn Mn
5.656 6.057 5.720 5.679 5.693 5.708 5.976 5.890 5.802 5.757 5.858 5.956 5.777 5.875 5.791
0.0 0.0 0.0 59.0 55.0 39.1 39.2 44.8 31.3 32.4 37.4 24.7 59.8 50.4 47.7
20
20
MnAs
GaAs InAs
6.00 5.95
5.90 5.85
state [15] from the total energy calculated by the FLAPW method. The results indicate that all ordered structures considered are higher in energy and so these structures are unstable with respect to the phase separation. Next, we estimated the energy of a disordered state (random solid solution), in which energy in a system is assumed to given by XXXX EðaÞ ¼ eijkl ðaÞwijkl , i
j
k
l
where the wijkl is a probability of finding an i–j–k–l atomic configuration in the tetrahedron (Fig. 1). In the disordered state, since the atoms align randomly, the mean-field approximation of wijkl ¼ xi xj xj xl can be applied where the xi is a mole fraction of the i-th atom. The effective tetrahedron interactions, eijkl ðaÞ, as a function of the lattice constant are obtained by mapping on the atomic configurations of the ordered structures considered (Table 1). Results for the formation energy, ExGa xIn , of the disordered state with respect to the segregation limit and the lattice constant, axGa xIn , at the equilibrium volume are shown in Fig. 2. The calculated ExGa xIn has positive values over the whole compositions, indicating that the disordered phase is less favorable with respect to the phase separation. In the case of the pseudo-binary (Ga, Mn)As alloy, as demonstrated previously [6], the maximum in the ExGa ðxIn ¼ 0Þ (46 meV/f.u.) appears at xGa ¼0.56, which corresponds to the critical temperatures of 1130 K. Then, when In atoms are incorporated and the In composition increases, the ExGa xIn achieves up to 65 meV/f.u. at xGa ¼0.38 and xIn ¼0.36. The calculated axGa xIn is also found to roughly follow Vegard’s law as seen in Fig. 2(b). In the pseudo-binary (Ga, Mn)As alloy [6], the lattice constant slightly increases when the Mn composition increases, since the difference in the lattice constant between
5.80
5.75
MnAs
GaAs
Fig. 2. Calculated formation energy, ExGa xIn in meV/f.u., with respect to the ˚ for the random disordered state segregation limit and lattice constant, axGa xIn in A, at the equilibrium volume for the pseudo-ternary (Ga,In,Mn)As alloy.
the GaAs and MnAs is only about 1%. When the In composition increases, however, the lattice constant largely increases due to the large lattice constant of the InAs compared to those of the GaAs and MnAs. In order to discuss an origin of the phase separation, we decomposed the ExGa xIn of the disordered state into elastic and chemical energy contributions as [16,17] Eel xGa xIn ¼ xGa EGaAs ðaxGa xIn Þ þ xIn EInAs ðaxGa xIn Þ þ ð1xGa xIn ÞEMnAs ðaxGa xIn Þ and el Echem xGa xIn ¼ ExGa xIn ðaxGa xIn ÞExGa xIn ðaxGa xIn Þ:
The Eel xGa xIn is the energy required to expand or contract the constituent GaAs, InAs, and MnAs to the equilibrium lattice constant of the disordered state, while the Echem xGa xIn is the energy required to form the random atomic alignment (i.e., mixing) from the separated constituent phases on the fixed lattice.
362
K. Nakamura et al. / Journal of Crystal Growth 318 (2011) 360–362
Echem xGa xIn turns to be negative values at about 0.2 In composition. Thus, the systems with more than 0.2 In composition have inherently a chemical tendency of mixing. Of course, the large positive value of chem the Eel xGa xIn exceeds the magnitude of ExGa xIn , which causes the positive ExGa xIn in Fig. 2. When the elastic energy contribution is relaxed, e.g., by imposing an elastic constraint from substrates, the alloys with higher Mn composition may be realized.
InAs 20 40
60 4. Summary
100
80
60 40 20 MnAs
GaAs InAs
We investigated the phase stability of the pseudo-ternary (Ga, In, Mn)As alloy with the zincblende structure by using the FLAPW method within the LDA+ U. The calculated results demonstrated that the ordered and disordered states are energetically less favorable over the phase separation and so the system has a tendency to segregate. When the formation energy is decomposed into the elastic and chemical energy contributions, we found that the alloys with more than 0.2 In composition have a chemical tendency of mixing. This is in contrast to the case in the pseudobinary (Ga, Mn)As alloy, where the chemical energy contribution has a positive value that favors clustering.
Acknowledgments Work at Mie University was supported by a Grant-in-Aid for Scientific Research (No. 20540334) from the Japan Society for the Promotion of Science, and computations were partially performed at ISSP, University of Tokyo.
-10 -40
-20
-30
References
-20 0 10 MnAs
20
-10
30 GaAs
Fig. 3. Calculated elastic energy contribution, Eel xGa xIn , and chemical energy contribution, Echem xGa xIn (in meV/f.u.) for the random disordered state at the equilibrium volume for the pseudo-ternary (Ga,In,Mn)As alloy.
The results are shown in Fig. 3. In the pseudo-binary (Ga, Mn)As alloy [6], since the lattice mismatch between the GaAs and MnAs is chem very small, almost no Eel xGa ðxIn ¼ 0Þ is observed. However, the ExGa ðxIn ¼ 0Þ has a positive value, which favors clustering and drives the phase separation. In contrast, when the In composition increases, the
[1] H. Ohno, J. Magn. Magn. Mater. 200 (1999) 110. [2] H. Katayama-Yoshida, K. Sato, T. Fukushima, M. Toyoda, H. Kizaki, V.A. Dinh, P.H. Dederichs, Phys. Status Solidi A 204 (15). [3] D. Chiba, Y. Nishitani, F. Matsukura, H. Ohno, Appl. Phys. Lett. 90 (2007) 22503. [4] S.J. Potashnik, K.C. Ku, S.H. Chun, J.J. Berry, N. Samarth, P. Schiffer, Appl. Phys. Lett. 79 (2001) 1495. [5] K.M. Yu, W. Walukiewicz, T. Wojtowicz, I. Kuryliszyn, X. Liu, Y. Sasaki, J.K. Furdyna, Phys. Rev. B 65 (2002) 201303(R). [6] K. Nakamura, K. Hatano, T. Akiyama, T. Ito, A.J. Freeman, Phys. Rev. B 75 (2007) 205205. [7] E. Wimmer, H. Krakauer, M. Weinert, A.J. Freeman, Phys. Rev. B 24 (1981) 864; M. Weinert, E. Wimmer, A.J. Freeman, Phys. Rev. B 26 (1982) 4571. [8] U. von Barth, L. Hedin, J. Phys. C 5 (1972) 1629. [9] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [10] A.B. Shick, A.I. Liechtenstein, W.E. Pickett, Phys. Rev. B 60 (1999) 10763. [11] L.X. Zhao, C.R. Staddon, K.Y. Wang, K.W. Edmonds, R.P. Campion, B.L. Gallagher, C.T. Foxon, Appl. Phys. Lett. 86 (071902). [12] K. Hatano, K. Nakamura, T. Akiyama, T. Ito, J. Cryst. Growth 301–302 (2007) 631. [13] R. Yu, D. Singh, H. Krakauer, Phys. Rev. B 43 (1991) 6411. [14] J.H. Park, S.K. Kwon, B.I. Min, Physica B 281, 282 (2000) 703. [15] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244. [16] T. Mohri, K. Koyanagi, T. Ito, K. Watanabe, Jpn. J. Appl. Phys. 28 (1989) 1312. [17] T. Mohri, K. Nakamura, T. Ito, J. Appl. Phys. 70 (1991) 1320.
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Phase stability of Mn-doped GaInAs alloys Kohji Nakamura , Masahiro Miyake, Toru Akiyama, Tomonori Ito Department of Physics Engineering, Mie University, Tsu, Mie 514-8507, Japan
a r t i c l e i n f o
a b s t r a c t
Available online 18 November 2010
Phase stability of pseudo-ternary (Ga,In,Mn)As alloys with a zincblende structure is investigated by means of the full-potential linearized augmented plane-wave method. Calculated results predict that the ordered and disordered states are energetically less favorable with respect to the phase separation. When the formation energy is decomposed into elastic and chemical energy contributions, the alloys with more than 0.2 In composition have a chemical tendency of mixing, while the phase separation is driven by the large elastic energy contribution. & 2010 Elsevier B.V. All rights reserved.
Keywords: A1. Phase equilibria A1. Solubility B2. Magnetic materials B2. Semiconducting III–V materials
1. Introduction Understanding the structural, electronic, and magnetic properties of magnetic semiconductors, such as Mn-doped GaAs, is of crucial importance in realizing novel spintronic applications [1,2]. So far, many efforts have been performed to synthesis the magnetic semiconductors with high Mn composition in order to increase the Curie temperature, and the solubility of the Mn has been achieved up to about 10% by means of low-temperature molecular beam epitaxy (MBE) [3]. Although the MBE growth introduces high Mn interstitial and As antisite defects, a low-temperature annealing (2002300 1C) can eliminate most of the defects and so the Mn atoms preferably occupy substitutional cation sites [4,5]. From a theoretical viewpoint, we previously [6] investigated the stability of the pseudo-binary (Ga, Mn)As alloy with a zincblende structure by means of the full-potential linearized augmented plane-wave (FLAPW) method [7] within the local density approximation (LDA) [8], generalized gradient approximation (GGA) [9], and LDA+ U [10]. In contrast to LDA and GGA predictions, importantly, the calculated lattice constant in the LDA+U is found to increase when the Mn composition increases, due to the correlation correction of the pd hybridization strength between the Mn 3d bands and the As 4p valence bands, which agrees with experimental findings [11]. In addition, we found that the system has a tendency to segregate into constituent GaAs and MnAs. The thermodynamic equilibrium Mn solubility into GaAs is very low at low-temperatures, namely, 1% at 300 1C, and the spinodal decomposition occurs for more than about 10% at the low-temperatures. Moreover, the phase separation is found to be mainly driven by a positive chemical energy contribution and the system favors clustering
[12]. In the present work, we extend our FLAPW calculations to the pseudo-ternary (Ga, In, Mn)As alloy and discuss the phase stability.
2. Method We considered 15 ordered zincblende structures, Gam Inn Mn4mn As (m and n ¼0–4), with no tetragonal distortion, in which the Ga, In, and Mn atoms substitutionally occupy cation sites (a, b, g, and d in Fig. 1). Assuming a ferromagnetic state, total energy, E(a), as a function of the lattice constant (a) is calculated by using the FLAPW method within the LDA+ U, in which the core states are treated fully relativistically and the valence states are treated semi-relativistically, i.e., without spin–orbit coupling. Local atomic relaxations of anions (As) that surround the Ga, In, or Mn atoms are introduced in accordance with atomic force calculations [13]. The literature values of Coulomb (U) and exchange (J) parameters used in the LDA+U for the Mn 3d states are 4 and 0.7 eV, respectively [14]. The LAPW basis with a cut-off of jk þGjp 3:9 a:u:1 and muffin-tin sphere radii of 2.3 a.u. for the Ga, In, and Mn and 2.0 a.u. for the As are used. Lattice harmonics with angular momentum up to ‘ ¼ 8 are employed to expand the charge density, potential and wavefunctions.
3. Results and discussion Table 1 summarizes the calculated formation energy, Emn, of the 15 ordered Gam Inn Mn4mn As at the equilibrium volume (the lattice constant of amn ) that minimizes the total energy, referred to the segregation limit as Emn ¼ Eðamn Þ
Corresponding author. Tel.: +81 59 231 5361; fax: +81 59 231 9726.
E-mail address: [email protected] (K. Nakamura). 0022-0248/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2010.11.023
m n 4mn EGaAs EInAs EMnAs , 4 4 4
where EGaAs (InAs, MnAs) is the total energy of the constituent GaAs (InAs, MnAs), and the Emn(a) is fitted by the Murnaghan equation of
K. Nakamura et al. / Journal of Crystal Growth 318 (2011) 360–362
361
InAs 20
γ δ
40
β
α
60 Fig. 1. Zincblende structure; a, b, g, and d indicate sites on a fcc sublattice, which are occupied by Ga, In, and Mn atoms.
Table 1 Calculated formation energy, Emn in meV/f.u., at the equilibrium volume, and the ˚ for the 15 ordered zincblende Gam Inn Mn4mn As. a, b, g, lattice constant, amn in A, and d indicate sites of a tetrahedron in Fig. 1.
40
40
a
b
g
d
amn
Emn
Ga In Mn Ga Ga Ga In In In Ga Ga Ga Ga Ga Ga
Ga In Mn Ga Ga Mn In In Mn Ga Ga In Ga In In
Ga In Mn Ga Mn Mn In Mn Mn Ga In In In In Mn
Ga In Mn Mn Mn Mn Mn Mn Mn In In In Mn Mn Mn
5.656 6.057 5.720 5.679 5.693 5.708 5.976 5.890 5.802 5.757 5.858 5.956 5.777 5.875 5.791
0.0 0.0 0.0 59.0 55.0 39.1 39.2 44.8 31.3 32.4 37.4 24.7 59.8 50.4 47.7
20
20
MnAs
GaAs InAs
6.00 5.95
5.90 5.85
state [15] from the total energy calculated by the FLAPW method. The results indicate that all ordered structures considered are higher in energy and so these structures are unstable with respect to the phase separation. Next, we estimated the energy of a disordered state (random solid solution), in which energy in a system is assumed to given by XXXX EðaÞ ¼ eijkl ðaÞwijkl , i
j
k
l
where the wijkl is a probability of finding an i–j–k–l atomic configuration in the tetrahedron (Fig. 1). In the disordered state, since the atoms align randomly, the mean-field approximation of wijkl ¼ xi xj xj xl can be applied where the xi is a mole fraction of the i-th atom. The effective tetrahedron interactions, eijkl ðaÞ, as a function of the lattice constant are obtained by mapping on the atomic configurations of the ordered structures considered (Table 1). Results for the formation energy, ExGa xIn , of the disordered state with respect to the segregation limit and the lattice constant, axGa xIn , at the equilibrium volume are shown in Fig. 2. The calculated ExGa xIn has positive values over the whole compositions, indicating that the disordered phase is less favorable with respect to the phase separation. In the case of the pseudo-binary (Ga, Mn)As alloy, as demonstrated previously [6], the maximum in the ExGa ðxIn ¼ 0Þ (46 meV/f.u.) appears at xGa ¼0.56, which corresponds to the critical temperatures of 1130 K. Then, when In atoms are incorporated and the In composition increases, the ExGa xIn achieves up to 65 meV/f.u. at xGa ¼0.38 and xIn ¼0.36. The calculated axGa xIn is also found to roughly follow Vegard’s law as seen in Fig. 2(b). In the pseudo-binary (Ga, Mn)As alloy [6], the lattice constant slightly increases when the Mn composition increases, since the difference in the lattice constant between
5.80
5.75
MnAs
GaAs
Fig. 2. Calculated formation energy, ExGa xIn in meV/f.u., with respect to the ˚ for the random disordered state segregation limit and lattice constant, axGa xIn in A, at the equilibrium volume for the pseudo-ternary (Ga,In,Mn)As alloy.
the GaAs and MnAs is only about 1%. When the In composition increases, however, the lattice constant largely increases due to the large lattice constant of the InAs compared to those of the GaAs and MnAs. In order to discuss an origin of the phase separation, we decomposed the ExGa xIn of the disordered state into elastic and chemical energy contributions as [16,17] Eel xGa xIn ¼ xGa EGaAs ðaxGa xIn Þ þ xIn EInAs ðaxGa xIn Þ þ ð1xGa xIn ÞEMnAs ðaxGa xIn Þ and el Echem xGa xIn ¼ ExGa xIn ðaxGa xIn ÞExGa xIn ðaxGa xIn Þ:
The Eel xGa xIn is the energy required to expand or contract the constituent GaAs, InAs, and MnAs to the equilibrium lattice constant of the disordered state, while the Echem xGa xIn is the energy required to form the random atomic alignment (i.e., mixing) from the separated constituent phases on the fixed lattice.
362
K. Nakamura et al. / Journal of Crystal Growth 318 (2011) 360–362
Echem xGa xIn turns to be negative values at about 0.2 In composition. Thus, the systems with more than 0.2 In composition have inherently a chemical tendency of mixing. Of course, the large positive value of chem the Eel xGa xIn exceeds the magnitude of ExGa xIn , which causes the positive ExGa xIn in Fig. 2. When the elastic energy contribution is relaxed, e.g., by imposing an elastic constraint from substrates, the alloys with higher Mn composition may be realized.
InAs 20 40
60 4. Summary
100
80
60 40 20 MnAs
GaAs InAs
We investigated the phase stability of the pseudo-ternary (Ga, In, Mn)As alloy with the zincblende structure by using the FLAPW method within the LDA+ U. The calculated results demonstrated that the ordered and disordered states are energetically less favorable over the phase separation and so the system has a tendency to segregate. When the formation energy is decomposed into the elastic and chemical energy contributions, we found that the alloys with more than 0.2 In composition have a chemical tendency of mixing. This is in contrast to the case in the pseudobinary (Ga, Mn)As alloy, where the chemical energy contribution has a positive value that favors clustering.
Acknowledgments Work at Mie University was supported by a Grant-in-Aid for Scientific Research (No. 20540334) from the Japan Society for the Promotion of Science, and computations were partially performed at ISSP, University of Tokyo.
-10 -40
-20
-30
References
-20 0 10 MnAs
20
-10
30 GaAs
Fig. 3. Calculated elastic energy contribution, Eel xGa xIn , and chemical energy contribution, Echem xGa xIn (in meV/f.u.) for the random disordered state at the equilibrium volume for the pseudo-ternary (Ga,In,Mn)As alloy.
The results are shown in Fig. 3. In the pseudo-binary (Ga, Mn)As alloy [6], since the lattice mismatch between the GaAs and MnAs is chem very small, almost no Eel xGa ðxIn ¼ 0Þ is observed. However, the ExGa ðxIn ¼ 0Þ has a positive value, which favors clustering and drives the phase separation. In contrast, when the In composition increases, the
[1] H. Ohno, J. Magn. Magn. Mater. 200 (1999) 110. [2] H. Katayama-Yoshida, K. Sato, T. Fukushima, M. Toyoda, H. Kizaki, V.A. Dinh, P.H. Dederichs, Phys. Status Solidi A 204 (15). [3] D. Chiba, Y. Nishitani, F. Matsukura, H. Ohno, Appl. Phys. Lett. 90 (2007) 22503. [4] S.J. Potashnik, K.C. Ku, S.H. Chun, J.J. Berry, N. Samarth, P. Schiffer, Appl. Phys. Lett. 79 (2001) 1495. [5] K.M. Yu, W. Walukiewicz, T. Wojtowicz, I. Kuryliszyn, X. Liu, Y. Sasaki, J.K. Furdyna, Phys. Rev. B 65 (2002) 201303(R). [6] K. Nakamura, K. Hatano, T. Akiyama, T. Ito, A.J. Freeman, Phys. Rev. B 75 (2007) 205205. [7] E. Wimmer, H. Krakauer, M. Weinert, A.J. Freeman, Phys. Rev. B 24 (1981) 864; M. Weinert, E. Wimmer, A.J. Freeman, Phys. Rev. B 26 (1982) 4571. [8] U. von Barth, L. Hedin, J. Phys. C 5 (1972) 1629. [9] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [10] A.B. Shick, A.I. Liechtenstein, W.E. Pickett, Phys. Rev. B 60 (1999) 10763. [11] L.X. Zhao, C.R. Staddon, K.Y. Wang, K.W. Edmonds, R.P. Campion, B.L. Gallagher, C.T. Foxon, Appl. Phys. Lett. 86 (071902). [12] K. Hatano, K. Nakamura, T. Akiyama, T. Ito, J. Cryst. Growth 301–302 (2007) 631. [13] R. Yu, D. Singh, H. Krakauer, Phys. Rev. B 43 (1991) 6411. [14] J.H. Park, S.K. Kwon, B.I. Min, Physica B 281, 282 (2000) 703. [15] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244. [16] T. Mohri, K. Koyanagi, T. Ito, K. Watanabe, Jpn. J. Appl. Phys. 28 (1989) 1312. [17] T. Mohri, K. Nakamura, T. Ito, J. Appl. Phys. 70 (1991) 1320.