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Electronic structure and origin of ferromagnetism in Ga1−xMnxAs semiconductors
ARTICLE IN PRESS
Physica B 340-342 (2003) 874–877
Electronic structure and origin of ferromagnetism in Ga1xMnxAs semiconductors # Antonio J.R. da Silvaa,*, A. Fazzioa, Raimundo R. dos Santosb, Luiz E. Oliveirac b
a Instituto de F!ısica, Universidade de Sao * Paulo, CP 66318, 05315-970 Sao * Paulo-SP, Brazil Instituto de F!ısica, Universidade Fed. do Rio de Janeiro, CP 68528, 21945-970 Rio de Janeiro-RJ, Brazil c Instituto de F!ısica, Unicamp, CP 6165, 13083-970 Campinas-SP, Brazil
Abstract We perform a detailed theoretical study, within the density-functional theory, of the electronic structure and magnetic properties of Ga1xMnxAs diluted semiconductors. Ab initio total energy results provide evidence that the appearance of a ferromagnetic state in these materials is due to an exchange coupling between the localized m S ¼ 52 Mn spins mediated by quasi-localized k holes, with strong p-like character, surrounding the fully polarized Mn d5-electrons. From total energy differences, we find the effective Mn–Mn coupling always ferromagnetic, with the Mn–Mn interaction intermediated by an antiferromagnetic coupling of each Mn spin to the holes. r 2003 Elsevier B.V. All rights reserved. PACS: 42.65.Vh; 71.55.Eq; 73.20.Dx Keywords: Diluted magnetic semiconductors; (Ga,Mn)As; First-principles calculations
In the last decade or so, much interest has arisen in the understanding of the electronic, optical and transport properties of diluted magnetic semiconductors (DMS), due to the discovery of ferromagnetism in III–V materials alloyed with transition elements like Mn [1]. Fundamental work in ferromagnetic semiconductors, when combined with the capability of growing low-dimensional structures, opens up the possibility of controlling both spin and charge degrees of freedom, and leads to exciting new prospects in the production of spintronic devices, with potential applications
*Corresponding author. Tel.: +55-11-3818-6937; fax: +5511-3091-6831. E-mail address: [email protected] (A.J.R. da Silva).
such as non-volatile memory systems and quantum computing. Special attention has lately been focused on Ga1xMnxAs diluted semiconductors, which exhibit very interesting magnetic and transport properties [2,3]. It is well known that Mn forms acceptors when in substitutional Ga lattice sites (MnGa) in Ga1xMnxAs, and the generally accepted view of the origin of ferromagnetism in these DMS is a ferromagnetic effective exchange coupling between the localized Mn moments mediated by the holes [4,5]. If one aims to develop full-scale applications, however, it is of paramount importance to elucidate several issues in relation to these systems. One of the present crucial issues is that both the critical temperature Tc and hole concentration p as a function of Mn composition
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.09.210
ARTICLE IN PRESS A.J.R. da Silva et al. / Physica B 340-342 (2003) 874–877
in Ga1xMnxAs are very much dependent on the details of growth conditions, even in as-grown samples [2,3]. The highest critical temperature of 110 K, for instance, has only being reproduced by other groups after post-growth annealing. Of course, the existence of defects such as As occupying antisites, Mn interstitials (MnI), MnI– MnGa pairs, and the formation of MnAs complexes must play an important role in the magnetic, structural, and optoelectronic properties of Ga1xMnxAs, which, of course, are extremely sensitive to the actual molecular-beam epitaxy (MBE) growth conditions. It is clear, therefore, that a proper understanding of the physics of Ga1xMnxAs DMS should involve a detailed microscopic description of the effect of different defects on their electronic and magnetic properties. As a first step, however, it is fundamental to have an understanding of the basic interactions between the MnGa atoms in the system. With this goal in mind, we have performed first principles calculations, based on the density functional theory (DFT), for a set of configurations where the Mn–Mn distance is varied within the supercell. From total energy results for a ferromagnetic and an antiferromagnetic alignment of the Mn spins, one may extract the effective Mn–Mn exchange interaction. Our total energy results show that this effective coupling is always ferromagnetic, intermediated by an antiferromagnetic coupling of each Mn spin to the holes. We have performed DFT total energy calculations with a generalized-gradient approximation (GGA) for the exchange-correlation potential. The electron–ion interactions are described using ultrasoft pseudopotentials and a plane wave expansion up to 230 eV as implemented in the VASP code [6]. We used a 128-atom and 250-atom FCC supercell and L-points for the Brillouin zone sampling. The positions of all atoms in the supercell were relaxed until all the forces components were smaller than ( Spin-orbit effects were checked through 0.02 eV/A. the projector augmented-wave (PAW) method and we found that they may be safely neglected. Let us first consider the case of a single isolated MnGa acceptor. Calculations indicate that the ground state of the MnGa defect is consistent with the picture of a k hole interacting antiferromag-
875
netically with the m S ¼ 52 spin of the d5-configuration at the Mn site. We found (for the 250 supercell) the excited ferromagnetic configuration (with an m hole) lying 0.25 eV above the groundstate antiferromagnetic level. Fig. 1 shows isosurfaces for the net local magnetization m(r) = rm(r)rk(r) for both the neutral (MnGa)0 and the negative (MnGa) defects. We note, in Fig. 1(a),
Fig. 1. Isosurfaces for the net local magnetization mðrÞ ¼ rm ðrÞ rk ðrÞ for the (a) neutral (MnGa)0, and (b) negative (MnGa) defects. The grey surface corresponds to net m spins at ( 3 and the black surface to a value of rm ðrÞ rk ðrÞ ¼ þ0:04e=A ( 3 : For comparison, note k spins at rm ðrÞ rk ðrÞ ¼ 0:004e=A that a uniform charge density of 1 electron/unit cell in GaAs ( 3, where e is the electron charge. The corresponds to 0.022 e/A small black spheres denote Ga atoms, whereas the larger grey ones represent the As. The Mn atoms are at the center of the positively valued isosurfaces.
ARTICLE IN PRESS 876
A.J.R. da Silva et al. / Physica B 340-342 (2003) 874–877
that near the MnGa acceptor, the local magnetization has a strong m d-like character, due essentially to the valence-band resonant d5 electrons, whereas as one approaches its As neighbors, the character changes to k p-like. The signature of the abovementioned antiferromagnetic interaction consists of a sign change in mðrÞ as one moves from the Mn site to any of its neighboring As. In the case of the negative (MnGa) defect (cf. Fig. 1(b)), one notes a quenching of the local magnetization near the As sites, although a remaining k p-like polarization survives due to the highly localized spin at the Mn site. From the total energy calculations for the neutral (MnGa)0 and the ionized negative (MnGa) defects, with 250-atom in the supercell, we obtain a localized acceptor level lying at 0.1 eV above the top of the valence band, in good agreement with the 0.11 eV experimental value [7]. The nature of ferromagnetism in diluted Ga1xMnxAs semiconductors may be elucidated by focusing on interacting MnGa substitutional defects in a 128-atom supercell, considering both a ferromagnetic as well as an antiferromagnetic alignment between the Mn spins. We have performed calculations for two Mn substitutional atoms in configurations corresponding to all nonequivalent positions within the supercell, i.e., Mn– ( Mn distances varying from 4.06 up to 11.48 A. Calculated results yield an unambiguous Mn–Mn ferromagnetic ground state in all cases. Fig. 2 shows the net magnetization m(r) isosurfaces for two Mn defects in nearest-neighbor positions (in the Ga sublattice), in the ferromagnetic configuration. In Fig. 2, the Mn atoms are at the center of the spherical-like regions of m(r), whereas the plike regions are always centered on As atoms. Note that the antiferromagnetic coupling between the Mn and hole spins is maintained. Another important feature illustrated in Fig. 2 is the fact that, again, the Mn–Mn defect is quite localized, with the magnetization density clearly spreading out from one MnGa site to the other. The picture that emerges is that of a cloud of k holes surrounding the substitutional Mn, with the distribution of quasi-localized holes giving rise to a nearly dispersionless impurity band, so that a description via the effective-mass approximation would be inappropriate [8,9]. From the difference
Fig. 2. Isosurfaces for the net local magnetization m(r)= rm(r)rk(r) in the case of two nearest-neighbor MnGa defects, with parallel S ¼ 52 spins. The color code and isosurface values are the same as in Fig. 1.
Table 1 Total energy differences (DEAFF) between the Mn–Mn antiferromagnetic and ferromagnetic spins alignment, for the non-equivalent Mn–Mn distances (dMnMn) in a 128-atom supercell. To obtain JMnMn from these values, the Mn atoms in neighboring supercell must be taken into account ( dMnMn (A) DEAFF (eV)
4.06 0.29
5.74 0.13
7.03 0.17
8.12 0.14
9.08 0.066
9.95 0.22
11.48 0.001
between total energies of the ferro- and antiferromagnetic configurations of the S ¼ 12 hole and the S ¼ 52 Mn spins, and assuming an interaction of the type Nb s:S; one may calculate the strength of the Kondo-like antiferromagnetic exchange coupling Nb E +0.1 eV. This result contrasts with the previous estimates of Nb E 1.5–3.0 eV. We note that these larger values are open to question, as they are obtained via fittings, to experimental data, of results based on the effective-mass approximation with a hole effective mass m=0.5me. We have also estimated, from total
ARTICLE IN PRESS A.J.R. da Silva et al. / Physica B 340-342 (2003) 874–877
energy calculations (see Table 1), the effective exchange coupling between pairs of S ¼ 52 Mn spins, J Mn–Mn, as a function of the Mn–Mn distance, for all non-equivalent pair positions within the supercell. Results clearly show that the coupling between the Mn spins is always ferromagnetic, irrespective of their relative distance. As it is well known that the bare coupling between two Mn spins should be antiferromagnetic, one concludes that the resulting Mn–Mn ferromagnetic effective coupling, in Ga1xMnxAs, is essentially intermediated by the antiferromagnetic coupling of each Mn spin to the quasi-localized holes. Here we note that this result contrasts with the RKKY coupling obtained by Zhao et al. [10] in the case of MnxGe1x.
Acknowledgements Partial financial support by the Brazilian Agencies CNPq, CENAPAD-Campinas, Rede Nacional de Materiais Nanoestruturados/CNPq, FAPESP, FAPERJ, and Millenium Institute for Nanosciences/MCT is gratefully acknowledged.
877
References [1] H. Ohno, et al., Phys. Rev. Lett. 68 (1992) 2664; H. Ohno, et al., Appl. Phys. Lett. 69 (1996) 363. [2] F. Matsukura, et al., Phys. Rev. B 57 (1998) R2037; H. Ohno, J. Magn. Magn. Mater. 200 (1999) 110; K.W. Edmonds, et al., Appl. Phys. Lett. 81 (2002) 3010; M.J. Seong, et al., Phys. Rev. B 66 (2002) 033202; S.J. Potashnik, et al., Appl. Phys. Lett. 79 (2001) 1495; A. Van Esch, et al., Phys. Rev. B 56 (1997) 13103; H. Asklund, et al., Phys. Rev. B 66 (2002) 115319; K.M. Yu, et al., Phys. Rev. B 65 (2002) R201303; K.M. Yu, et al., Appl. Phys. Lett. 81 (2002) 844; R. Moriya, H. Munekata, J. Appl. Phys. 93 (2003) 4603; S.J. Potashnik, et al., Phys. Rev. B 66 (2002) 012408. [3] T. Hayashi, et al., Appl. Phys. Lett. 78 (2001) 1691. [4] T. Dietl, et al., Phys. Rev. B 55 (1997) R3347. [5] H. Ohno, F. Matsukura, Solid State Commun. 117 (2001) 179. [6] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892; G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) R558; G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169; G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. [7] R.A. Chapman, W.G. Hutchinson, Phys. Rev. Lett. 18 (1967) 443; J. Schneider, et al., Phys. Rev. Lett. 59 (1987) 240; M. Linnarsson, et al., Phys. Rev. B 55 (1997) 6938. [8] E.J. Singley, et al., Phys. Rev. Lett. 89 (2002) 097203. [9] T. Dietl, et al., Phys. Rev. B 63 (2001) 195205; M. Abolfath, et al., Phys. Rev. B 63 (2001) 054418. [10] Y-J. Zhao, et al., Phys. Rev. Lett. 90 (2003) 047204.
Physica B 340-342 (2003) 874–877
Electronic structure and origin of ferromagnetism in Ga1xMnxAs semiconductors # Antonio J.R. da Silvaa,*, A. Fazzioa, Raimundo R. dos Santosb, Luiz E. Oliveirac b
a Instituto de F!ısica, Universidade de Sao * Paulo, CP 66318, 05315-970 Sao * Paulo-SP, Brazil Instituto de F!ısica, Universidade Fed. do Rio de Janeiro, CP 68528, 21945-970 Rio de Janeiro-RJ, Brazil c Instituto de F!ısica, Unicamp, CP 6165, 13083-970 Campinas-SP, Brazil
Abstract We perform a detailed theoretical study, within the density-functional theory, of the electronic structure and magnetic properties of Ga1xMnxAs diluted semiconductors. Ab initio total energy results provide evidence that the appearance of a ferromagnetic state in these materials is due to an exchange coupling between the localized m S ¼ 52 Mn spins mediated by quasi-localized k holes, with strong p-like character, surrounding the fully polarized Mn d5-electrons. From total energy differences, we find the effective Mn–Mn coupling always ferromagnetic, with the Mn–Mn interaction intermediated by an antiferromagnetic coupling of each Mn spin to the holes. r 2003 Elsevier B.V. All rights reserved. PACS: 42.65.Vh; 71.55.Eq; 73.20.Dx Keywords: Diluted magnetic semiconductors; (Ga,Mn)As; First-principles calculations
In the last decade or so, much interest has arisen in the understanding of the electronic, optical and transport properties of diluted magnetic semiconductors (DMS), due to the discovery of ferromagnetism in III–V materials alloyed with transition elements like Mn [1]. Fundamental work in ferromagnetic semiconductors, when combined with the capability of growing low-dimensional structures, opens up the possibility of controlling both spin and charge degrees of freedom, and leads to exciting new prospects in the production of spintronic devices, with potential applications
*Corresponding author. Tel.: +55-11-3818-6937; fax: +5511-3091-6831. E-mail address: [email protected] (A.J.R. da Silva).
such as non-volatile memory systems and quantum computing. Special attention has lately been focused on Ga1xMnxAs diluted semiconductors, which exhibit very interesting magnetic and transport properties [2,3]. It is well known that Mn forms acceptors when in substitutional Ga lattice sites (MnGa) in Ga1xMnxAs, and the generally accepted view of the origin of ferromagnetism in these DMS is a ferromagnetic effective exchange coupling between the localized Mn moments mediated by the holes [4,5]. If one aims to develop full-scale applications, however, it is of paramount importance to elucidate several issues in relation to these systems. One of the present crucial issues is that both the critical temperature Tc and hole concentration p as a function of Mn composition
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.09.210
ARTICLE IN PRESS A.J.R. da Silva et al. / Physica B 340-342 (2003) 874–877
in Ga1xMnxAs are very much dependent on the details of growth conditions, even in as-grown samples [2,3]. The highest critical temperature of 110 K, for instance, has only being reproduced by other groups after post-growth annealing. Of course, the existence of defects such as As occupying antisites, Mn interstitials (MnI), MnI– MnGa pairs, and the formation of MnAs complexes must play an important role in the magnetic, structural, and optoelectronic properties of Ga1xMnxAs, which, of course, are extremely sensitive to the actual molecular-beam epitaxy (MBE) growth conditions. It is clear, therefore, that a proper understanding of the physics of Ga1xMnxAs DMS should involve a detailed microscopic description of the effect of different defects on their electronic and magnetic properties. As a first step, however, it is fundamental to have an understanding of the basic interactions between the MnGa atoms in the system. With this goal in mind, we have performed first principles calculations, based on the density functional theory (DFT), for a set of configurations where the Mn–Mn distance is varied within the supercell. From total energy results for a ferromagnetic and an antiferromagnetic alignment of the Mn spins, one may extract the effective Mn–Mn exchange interaction. Our total energy results show that this effective coupling is always ferromagnetic, intermediated by an antiferromagnetic coupling of each Mn spin to the holes. We have performed DFT total energy calculations with a generalized-gradient approximation (GGA) for the exchange-correlation potential. The electron–ion interactions are described using ultrasoft pseudopotentials and a plane wave expansion up to 230 eV as implemented in the VASP code [6]. We used a 128-atom and 250-atom FCC supercell and L-points for the Brillouin zone sampling. The positions of all atoms in the supercell were relaxed until all the forces components were smaller than ( Spin-orbit effects were checked through 0.02 eV/A. the projector augmented-wave (PAW) method and we found that they may be safely neglected. Let us first consider the case of a single isolated MnGa acceptor. Calculations indicate that the ground state of the MnGa defect is consistent with the picture of a k hole interacting antiferromag-
875
netically with the m S ¼ 52 spin of the d5-configuration at the Mn site. We found (for the 250 supercell) the excited ferromagnetic configuration (with an m hole) lying 0.25 eV above the groundstate antiferromagnetic level. Fig. 1 shows isosurfaces for the net local magnetization m(r) = rm(r)rk(r) for both the neutral (MnGa)0 and the negative (MnGa) defects. We note, in Fig. 1(a),
Fig. 1. Isosurfaces for the net local magnetization mðrÞ ¼ rm ðrÞ rk ðrÞ for the (a) neutral (MnGa)0, and (b) negative (MnGa) defects. The grey surface corresponds to net m spins at ( 3 and the black surface to a value of rm ðrÞ rk ðrÞ ¼ þ0:04e=A ( 3 : For comparison, note k spins at rm ðrÞ rk ðrÞ ¼ 0:004e=A that a uniform charge density of 1 electron/unit cell in GaAs ( 3, where e is the electron charge. The corresponds to 0.022 e/A small black spheres denote Ga atoms, whereas the larger grey ones represent the As. The Mn atoms are at the center of the positively valued isosurfaces.
ARTICLE IN PRESS 876
A.J.R. da Silva et al. / Physica B 340-342 (2003) 874–877
that near the MnGa acceptor, the local magnetization has a strong m d-like character, due essentially to the valence-band resonant d5 electrons, whereas as one approaches its As neighbors, the character changes to k p-like. The signature of the abovementioned antiferromagnetic interaction consists of a sign change in mðrÞ as one moves from the Mn site to any of its neighboring As. In the case of the negative (MnGa) defect (cf. Fig. 1(b)), one notes a quenching of the local magnetization near the As sites, although a remaining k p-like polarization survives due to the highly localized spin at the Mn site. From the total energy calculations for the neutral (MnGa)0 and the ionized negative (MnGa) defects, with 250-atom in the supercell, we obtain a localized acceptor level lying at 0.1 eV above the top of the valence band, in good agreement with the 0.11 eV experimental value [7]. The nature of ferromagnetism in diluted Ga1xMnxAs semiconductors may be elucidated by focusing on interacting MnGa substitutional defects in a 128-atom supercell, considering both a ferromagnetic as well as an antiferromagnetic alignment between the Mn spins. We have performed calculations for two Mn substitutional atoms in configurations corresponding to all nonequivalent positions within the supercell, i.e., Mn– ( Mn distances varying from 4.06 up to 11.48 A. Calculated results yield an unambiguous Mn–Mn ferromagnetic ground state in all cases. Fig. 2 shows the net magnetization m(r) isosurfaces for two Mn defects in nearest-neighbor positions (in the Ga sublattice), in the ferromagnetic configuration. In Fig. 2, the Mn atoms are at the center of the spherical-like regions of m(r), whereas the plike regions are always centered on As atoms. Note that the antiferromagnetic coupling between the Mn and hole spins is maintained. Another important feature illustrated in Fig. 2 is the fact that, again, the Mn–Mn defect is quite localized, with the magnetization density clearly spreading out from one MnGa site to the other. The picture that emerges is that of a cloud of k holes surrounding the substitutional Mn, with the distribution of quasi-localized holes giving rise to a nearly dispersionless impurity band, so that a description via the effective-mass approximation would be inappropriate [8,9]. From the difference
Fig. 2. Isosurfaces for the net local magnetization m(r)= rm(r)rk(r) in the case of two nearest-neighbor MnGa defects, with parallel S ¼ 52 spins. The color code and isosurface values are the same as in Fig. 1.
Table 1 Total energy differences (DEAFF) between the Mn–Mn antiferromagnetic and ferromagnetic spins alignment, for the non-equivalent Mn–Mn distances (dMnMn) in a 128-atom supercell. To obtain JMnMn from these values, the Mn atoms in neighboring supercell must be taken into account ( dMnMn (A) DEAFF (eV)
4.06 0.29
5.74 0.13
7.03 0.17
8.12 0.14
9.08 0.066
9.95 0.22
11.48 0.001
between total energies of the ferro- and antiferromagnetic configurations of the S ¼ 12 hole and the S ¼ 52 Mn spins, and assuming an interaction of the type Nb s:S; one may calculate the strength of the Kondo-like antiferromagnetic exchange coupling Nb E +0.1 eV. This result contrasts with the previous estimates of Nb E 1.5–3.0 eV. We note that these larger values are open to question, as they are obtained via fittings, to experimental data, of results based on the effective-mass approximation with a hole effective mass m=0.5me. We have also estimated, from total
ARTICLE IN PRESS A.J.R. da Silva et al. / Physica B 340-342 (2003) 874–877
energy calculations (see Table 1), the effective exchange coupling between pairs of S ¼ 52 Mn spins, J Mn–Mn, as a function of the Mn–Mn distance, for all non-equivalent pair positions within the supercell. Results clearly show that the coupling between the Mn spins is always ferromagnetic, irrespective of their relative distance. As it is well known that the bare coupling between two Mn spins should be antiferromagnetic, one concludes that the resulting Mn–Mn ferromagnetic effective coupling, in Ga1xMnxAs, is essentially intermediated by the antiferromagnetic coupling of each Mn spin to the quasi-localized holes. Here we note that this result contrasts with the RKKY coupling obtained by Zhao et al. [10] in the case of MnxGe1x.
Acknowledgements Partial financial support by the Brazilian Agencies CNPq, CENAPAD-Campinas, Rede Nacional de Materiais Nanoestruturados/CNPq, FAPESP, FAPERJ, and Millenium Institute for Nanosciences/MCT is gratefully acknowledged.
877
References [1] H. Ohno, et al., Phys. Rev. Lett. 68 (1992) 2664; H. Ohno, et al., Appl. Phys. Lett. 69 (1996) 363. [2] F. Matsukura, et al., Phys. Rev. B 57 (1998) R2037; H. Ohno, J. Magn. Magn. Mater. 200 (1999) 110; K.W. Edmonds, et al., Appl. Phys. Lett. 81 (2002) 3010; M.J. Seong, et al., Phys. Rev. B 66 (2002) 033202; S.J. Potashnik, et al., Appl. Phys. Lett. 79 (2001) 1495; A. Van Esch, et al., Phys. Rev. B 56 (1997) 13103; H. Asklund, et al., Phys. Rev. B 66 (2002) 115319; K.M. Yu, et al., Phys. Rev. B 65 (2002) R201303; K.M. Yu, et al., Appl. Phys. Lett. 81 (2002) 844; R. Moriya, H. Munekata, J. Appl. Phys. 93 (2003) 4603; S.J. Potashnik, et al., Phys. Rev. B 66 (2002) 012408. [3] T. Hayashi, et al., Appl. Phys. Lett. 78 (2001) 1691. [4] T. Dietl, et al., Phys. Rev. B 55 (1997) R3347. [5] H. Ohno, F. Matsukura, Solid State Commun. 117 (2001) 179. [6] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892; G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) R558; G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169; G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. [7] R.A. Chapman, W.G. Hutchinson, Phys. Rev. Lett. 18 (1967) 443; J. Schneider, et al., Phys. Rev. Lett. 59 (1987) 240; M. Linnarsson, et al., Phys. Rev. B 55 (1997) 6938. [8] E.J. Singley, et al., Phys. Rev. Lett. 89 (2002) 097203. [9] T. Dietl, et al., Phys. Rev. B 63 (2001) 195205; M. Abolfath, et al., Phys. Rev. B 63 (2001) 054418. [10] Y-J. Zhao, et al., Phys. Rev. Lett. 90 (2003) 047204.