PHYSICAL REVIEW B 81, 155301 共2010兲 Continuously tunable band gap in GaN/AlN (0001) superlattices via built-in electric field X. Y. Cui,1 D. J. Carter,1,2 M. Fuchs,3 B. Delley,4 S. H. Wei,5 A. J. Freeman,6 and C. Stampfl1 1School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia Research Institute, Curtin University of Technology, Perth, Western Australia 6845, Australia 3 Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany 4 Paul Scherrer Institut, WHGA/123, CH-5232 Villigen PSI, Switzerland 5 National Renewable Energy Laboratory, 1617 Cole Boulevard, Golden, Colorado 80401, USA 6 Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208-3112, USA 共Received 24 September 2009; revised manuscript received 9 March 2010; published 1 April 2010兲 2Nanochemistry Based on all-electron density-functional theory calculations using the generalized gradient approximation, we demonstrate the continuous tunability of the band gap and strength of the built-in electric field in GaN/AlN 共0001兲 superlattices by control of the thickness of both the well 共GaN兲 and barrier 共AlN兲 regions. The effects of strain for these quantities are also studied. Calculations taking into account the self-interaction correction exhibit the same dependence on thickness. The calculated electric field strength values are in good agreement with recent experiments. Spontaneous polarization dominates the contribution to the electric field and the strain-induced piezoelectric polarization is estimated to contribute only about 5–10%. DOI: 10.1103/PhysRevB.81.155301 PACS number共s兲: 68.65.Cd, 71.15.Mb, 71.20.⫺b I. INTRODUCTION The polarization properties of 共0001兲-grown GaN-based heterostructures have been an intense research subject due to their crucial influence on, and thus affording a tailoring of, the electronic and optical performance of technologically important nitride devices.1–8 Macroscopic polarization, both of an intrinsic and piezoelectric nature, originates from polarization discontinuities at heterointerfaces, and manifests itself as built-in electrostatic fields across the heterostructure. The electric field, acting via the quantum-confined Stark effect 共QCSE兲,9 changes the energies 共spectra shift兲 and intensity 共oscillator strength兲 共Ref. 10兲 of the emitted light. It can be engineered, for example, to influence favorably transistor characteristics11 and spin injection.12 The intensity of the built-in electric fields and the effect on the resulting properties depend on the composition, structural geometry, and strain of the samples. Compared with using 共Ga,Al兲N, using pure AlN as the barrier material leads to a stronger macroscopic polarization 共both spontaneous and piezoelectric兲 and larger built-in electric fields.2,13 This offers a wider range of effective band-gap 共optical emission兲 tunability of both interband transitions14 and improved intersubband transitions.15–17 In particular, earlier experimental results for using 共Ga,Al兲N as a barrier have shown that the band gap is set primarily by the thickness of the well layer,16 as well as by the barrier width.5 However, a systematic investigation, particularly from the theoretical side, of the dependence of the band gap of GaN/AlN 共0001兲 multiple quantum wells 共MQWs兲 共with a few dozen periods兲, on the thickness of both well and barrier is still missing. Regarding the strength of the electric field, a key parameter governing the optical properties of GaN/AlN heterostructures, and often used as an input parameter for nonselfconsistent simulations,4,18 most of the reported values show a wide disparity. Tight-binding simulations reported that the spontaneous and piezoelectric components are comparable, i.e., 1.14 MV/cm and 1.12 MV/cm, respectively, for a 1098-0121/2010/81共15兲/155301共5兲 GaN/ Al0.2Ga0.8N superlattice.3,4 However, Park and Chuang reported that for GaN/ Al0.2Ga0.8N QW structures, there exist strong spontaneous polarizations 共about 1.1 MV/cm兲 even when the piezoelectric field in the well is zero.5 By fitting the emission energy to the well width, the electric field was deduced to be 10⫾ 1 MV/ cm 共Ref. 7兲 in GaN/AlN 共0001兲 MQWs, and 8–10 MV/cm 共Refs. 14 and 19兲 in isolated QWs. The highest value of the electric field reported, by considering piezoelectric polarization alone, is 4.7 MV/cm.20 On the other hand, recent experiments reported that for well widths of 2.3 and 1.4 nm 共and barrier width 1.9 nm兲 the intrinsic electric field strengths in the wells were 5.04 MV/cm and 6.07 MV/cm,16 respectively. In this paper, we report a systematic first-principles investigation of the dependence of the band gap and built-in electric field as a function of the thickness of both well and barrier in GaN/AlN共0001兲 superlattices. The effect of strain on these two quantities is also studied. We find that the energy band gap and the built-in electric field of GaN/AlN 共0001兲 superlattices can be tuned continuously by varying the thickness of both well and barrier. The calculated electric field strength values are in good agreement with recent experiments. Investigation into the effect of strain reveals that piezoelectric polarization only contributes about 5–10% of the total macroscopic polarization. II. COMPUTATIONAL DETAILS MQWs, which consist of a large number of alternating well 共GaN兲 and barrier 共AlN兲 layers, are simulated using periodic superlattices. We perform all-electron densityfunctional theory 共DFT兲 calculations using the generalized gradient approximation 共GGA兲 共Ref. 21兲 for the exchangecorrelation functional as implemented in the DMOL3 code.22 The wave functions are expanded in terms of a doublenumerical quality localized basis set with a real-space cutoff of 9 bohr. We construct various sized 共GaN兲m / 共AlN兲n 共0001兲 superlattices, where m and n are the number of double layers 155301-1 ©2010 The American Physical Society 1 [0001] 5 AlN region A FIG. 1. 共Color online兲 Geometry of the 共GaN兲5 / 共AlN兲5共0001兲 heterostructure. The center 共middle dashed line兲 represents the type B interface while the two edges correspond to the type A interface. Small 共green兲 spheres indicate N atoms and large spheres, light 共pink兲 and dark 共blue兲, indicate Ga and Al atoms, respectively. of GaN and AlN in the well and barrier region, respectively. For short, we label it the “m + n” superlattice. As an example, Fig. 1 shows the 5 + 5 superlattice. It involves two different interfaces, “type A” and “type B.” The Brillouin-zone integrations are performed using a 10⫻ 10⫻ 4 k-point grid for the 6 + 6 structure. For other superlattices, the same/similar sampling of reciprocal space is used. We impose the assumption of pseudomorphic growth, i.e., the condition that the in-plane lattice constants of GaN region and AlN region are equal. Three approaches regarding the geometry relaxation are employed; 共i兲 full relaxation, including lattice constants 共a and c兲 and internal parameters, representing “freestanding” 共strain-free兲 superlattices; then, in order to investigate the effect of strain, we fix the in-plane lattice constant of the superlattices at that of either 共ii兲 GaN or 共iii兲 AlN, relaxing the c lattice constant and internal parameters. As expected, DFT-GGA leads to an underestimation of the band-gap values for semiconductors. For example, our calculated direct band-gap values are 4.21 eV and 1.81 eV, compared to the corresponding experimental values 6.12 eV 共Ref. 23兲 and 3.51 eV,24 for AlN and GaN, respectively. To obtain a more accurate description of the band gap, and to compare the trend with varying well and barrier widths, we also perform self-interaction corrected local-density approximation 共SIC-LDA兲 pseudopotential calculations25,26 for selected superlattices using the ABINIT code.27 The SIC-LDA calculated band-gap values are 6.96 eV and 3.93 eV for AlN and GaN, respectively. Self-interaction corrections were included for the Ga 3d and the N 2s and N 2p states in the nonlocal part of the respective pseudopotentials.28 In the SIC-LDA band-structure calculations we used the same relaxed structures as obtained in our DFT-GGA calculations. III. RESULTS AND DISCUSSIONS We start with n = m superlattices. The calculated effective band-gap values, for layer thickness varying from 1 to 20 double layers 共DLs兲 with a 1 DL increment, obtained using DFT-GGA, as well as several selected values by SIC-LDA, are displayed in Fig. 2共a兲. Both methods predict that the band gap decreases monotonically with increasing well and barrier width. In the interband transition regime, laser diodes generate light via the process of electron-hole recombination, and the energy of the photon and hence the emission wavelength is therefore determined by the band gap. Our results thus show that the emission energy can be tuned continuously by changing the quantum well and barrier thickness over a large range. The DFT-GGA calculations predict that metallization 4 (a) SIC-LDA GGA 3 2 1 0 0 n+n 2 4 6 8 5.8 (b) 5.6 well barrier 5.4 5.2 5 n+n 4.8 2 10 12 14 16 18 20 4 8 6 10 12 14 18 16 Layer number of barrier and well (n) Layer number of barrier and well (n) 4 8 3.5 (c) 3 SIC-LDA GGA 2.5 2 8+n 1.5 1 2 5 4 6 8 10 12 Barrier layer number (n) SIC-LDA GGA 3 m +8 2 1 2 4 6 8 10 12 Well layer number (m) 14 (d) well barrier 7 6 5 8+n 4 3 2 14 (e) 4 Electric field (MV/cm) B 1 Electric field (MV/cm) 1 Bandgap (eV) GaN region 1 Bandgap (eV) A 1 Bandgap (eV) 1 Electric field (MV/cm) PHYSICAL REVIEW B 81, 155301 共2010兲 CUI et al. 4 8 6 10 12 Barrier layer number (n) 14 8 (f) 7 6 well barrier 5 4 3 2 m+8 4 6 8 10 12 Well layer number (m) 14 FIG. 2. Calculated band-gap values 共left兲 and the absolute values for the strength of the built-in electric field 共right兲 for various free-standing superlattices. occurs at around n = m = 20. As expected, the SIC-LDA bandgap values are systematically larger than those obtained from GGA by around 2 eV, and the predicted 共by linearly extrapolating兲 metallization occurs at around n = m = 28. The bandgap evolution as a function of thickness changes from the quantum confinement 共blueshift兲 regime to the QCSE 共redshift兲 regime. The bulk well 共GaN兲 band-gap values 关GGA: 1.82 eV, SIC-LDA: 4.01 eV, and expt.: 3.475 eV 共Ref. 16兲兴 separate these two regions, which for both DFT-GGA and SIC-LDA results occurs around the 7 + 7 superlattice. Note in the small-thickness region, the electric field induced redshift concedes with the strong confinement blueshift, which accounts for the more dramatic increase in band gap against decreasing thickness. The presence of a built-in electric field can be demonstrated by using the core levels as reference energies to determine the relative alignment of the valence-band edges.29,30 Here we use the N 1s orbital binding energy, ECL b , in different layers, as shown in Fig. 3共a兲 for the various superlattices. There is a different slope of the curves in the well GaN 共positive兲 and barrier AlN 共negative兲 regions, showing V-shaped profiles and the electric fields have opposite signs at the two sides of the interface. The slopes of the linear regions correspond to the absolute values of the electric CL CL fields30 共兩E兩 = ⌬ECL b / ⌬z兲, where ⌬Eb is the difference, Eb , of the N atoms with a distance of ⌬z along the 关0001兴 direction. The calculated absolute values in the well and barrier regions are about 5 ⫾ 0.1 MV/ cm and 5.6⫾ 0.1 MV/ cm, respectively, for n = m superlattices 关Fig. 2共b兲兴. The periodic condition of superlattices imply that Eb / Ew = −Lw / Lb,5 where superscripts “w” and “b” represent the well and barrier, and E and L are the strength of electric field and layer thickness, 155301-2 PHYSICAL REVIEW B 81, 155301 共2010兲 N-1s binding energy (eV) CONTINUOUSLY TUNABLE BAND GAP IN GaN/AlN… -375 (a) GaN AlN -376 4+4 6+6 8+8 10+10 12+12 14+14 16+16 -377 N-1s binding energy (eV) -378 -50 -40 -30 -20 -10 -375 (b) GaN 0 10 20 30 40 AlN -376 8+4 8+6 8+8 8+10 8+12 8+14 -377 N-1s binding energy (eV) -30 -375 -20 -10 0 10 20 30 40 (c) GaN AlN -376 4+8 6+8 8+8 10+8 12+8 14+8 -377 -40 -30 -20 -10 0 10 o Distance in the [0001] direction (A) 20 FIG. 3. 共Color online兲 Calculated variation in the N 1s corelevel binding energy 共electron volt兲 along the 关0001兴 direction for 共a兲 n + n; 共b兲 n + 8, and 共c兲 8 + n free-standing superlattices. respectively. For the cases of n = m, the larger magnitude in the barrier region is due to the slightly shorter AlN layer 共2.49 Å兲 than the GaN layer 共2.64 Å兲. Such strong electric fields correspond to a dramatic charge accumulation and large monopole around the interfaces,12,31 and have significant effects on the electronic structure of the superlattices. The total band structure and layer-resolved 关each layer containing one metal 共Ga or Al兲 and one N atom兴 density of states 共DOS兲 are demonstrated, as an example, in Fig. 4 for the 16+ 16 superlattice. First, the existence of the electric field leads to a bending of the conduction- and valence-band profiles across the superlattice 4 2 Energy [eV] Energy [eV] 4 0 0 -2 -2 -4 -4 K Γ MH A L . 2 A GaN region B AlN region A FIG. 4. 共Color online兲 Total band-structure 共left兲 and layerresolved DOS 共right兲 for the free-standing 16+ 16 superlattice. Dark 共black兲 represents the s orbital, light 共green兲 the p orbital, and gray the d orbital. The dashed line indicates the type B interface. and the related localization of electron and hole gases on the two opposite interfaces of the quantum well. The resulted valence-band maximum 共VBM兲 around interface A and the conduction-band minimum 共CBM兲 at interface B determine the overall band gap of superlattices. With increasing well and barrier thickness, the strong built-in electric field in the AlN/GaN superlattices gives rise to a marked reduction in the effective band gap. Second, the progressive separation of the VBM and CBM with increasing thickness, particularly the well thickness, will diminish the electron-hole recombination process 共emission strength兲, and thus the matrix element of the optical transition due to the decrease in the electron and hole wave functions overlap.4 The latter adverse effect has lead to the conclusion that for optimal optical devices 共such as QW lasers兲, the maximum well thickness should be around 3 nm 共12 DL GaN兲.13,14 Note that in Ref. 14, the large linewidth broadening indicated severe thickness fluctuations and interface roughness, which may contribute to a prevalence of nonradiative recombination. A large oscillator strength was achieved for a 4 nm well thickness in GaN/ Al0.15Ga0.85N samples4 and up to 8 nm in GaN/ Al0.17Ga0.83N.2 The feasibility of tuning the band gap of MQW structures is further boosted by changing only either well or barrier thickness while keeping the thickness of the other one unchanged. To demonstrate this, we report a well-thickness and barrier-thickness study by using m + 8 共m = 4, 6, 8, 10, 12, and 14兲 and 8 + n 共n = 4, 6, 8, 10, 12, and 14兲 superlattices. The calculated band-gap values and the strength of the electric field in the well and barrier regions are shown in Fig. 2, and the plots of core level N 1s binding energies are shown in Fig. 3. Increasing either well or barrier thickness will lead to a decrease in the band gap, as is predicted by both the GGA and SIC-LDA calculations. As expected, changing the well thickness is more effective because as a type I heterostructure system, i.e., the VBM 共CBM兲 of GaN is higher 共lower兲 than that of AlN, the band gap of the GaN/AlN共0001兲 superlattice is mainly determined by the well GaN region. Increasing the thickness of either the barrier or well will lead to a decrease in the magnitude of the electric field in the same region and an increase in the other region. More importantly, compared to the modest change in the electric field for n = m superlattices, such a change for the n ⫽ m cases is much more efficient. Thus, our results demonstrate that the values of the band gap and the built-in electric field exhibit a significant dependence on both the well and the barrier widths, which offers the possibility to tune the emission energy over a wide range. In addition, by varying the thickness of both well and barrier, one can realize multitarget band-gap design.32 For example, 9 + 9, 12+ 8, and 8 + 14 superlattices give a target band gap of 1.65 eV 共DFT-GGA兲. So far, we have only considered free-standing, i.e., strainfree superlattices. Biaxial strain perpendicular to the 关0001兴 direction will induce a piezoelectric field along the 关0001兴 direction. Fixing the lateral lattice constant of the superlattices at that of GaN 共fix-GaN兲, and of AlN 共fix-AlN兲, represents two “extremes” of the compressive and tensile-strained superlattices. For this purpose, we consider 6 + 8, 8 + 8, and 155301-3 PHYSICAL REVIEW B 81, 155301 共2010兲 CUI et al. TABLE I. Calculated lattice constant c in the 关0001兴 direction 共angstrom兲 and the built-in electric field in the well 共GaN兲 region, Ew, and in the barrier 共AlN兲 region, Eb, 共megavolt per centimeter兲. c 共Å兲 Strain type Fix AlN Fix GaN Free standing 6+8 36.07 35.47 35.74 8+8 41.41 40.71 40.97 Eb 共MV/cm兲 Ew 共MV/cm兲 10+ 8 46.76 45.94 46.20 6+8 5.19 6.09 5.41 10+ 8 superlattices. The calculated in-plane lattice parameters of pure AlN 共3.134 Å兲 and GaN 共3.228 Å兲 differ by 2.91%, compared to the experimental value of 2.41%.24 The calculated in-plane lattice constant for the free-standing 6 + 8, 8 + 8, and 10+ 8 superlattices are 3.162 Å, 3.168 Å, and 3.172 Å, respectively. For these superlattices, the calculated c lattice constants and strength of the electric fields are listed in Table I, and the band-gap values in Table II. In the GaN/AlN共0001兲 heterostructure, the direction of spontaneous polarization is from the nitrogen 共anion兲 atom to the nearest-neighbor metal 共cation兲 atom along the c axis, i.e., the 关0001̄兴 direction.1,33 The alignment of the piezoelectric and spontaneous polarization is parallel in the case of tensile strain and antiparallel for compressive strain.33 This is in agreement with our calculated values shown in Table I, where the calculated strength of the electric field values in the well and barrier regions for fix-AlN 共fix-GaN兲 are smaller 共larger兲 than the corresponding free-standing ones. Our calculated values are in good agreement with recent experimental results,16 namely, 6.07 and 5.04 MV/cm for the 20 period 2.3 nm/1.9 nm and 1.4 nm/1.9 nm 关corresponding to 共8 – 10兲 + 8 and 共5 – 6兲 + 8兴 GaN/AlN MQWs. Moreover, in all the superlattices considered, the magnitudes of the electric fields in the well region are around 4–6 MV/cm, which are significantly lower than the 10⫾ 1 MV/ cm 共Ref. 7兲 deduced by fitting spectra. It is known that dielectric screening and geometrical factors also affect the polarization fields in III-V nitride heterostructures.2,3 Consequently, the electric field of single QWs is expected to be larger than that in MQWs, as obtained in Refs. 14 and 19. Note, our study shows that the field strength is a function of supercell thickness, not a constant as assumed in Ref. 14. Moreover, we confirm that the significant built-in electric field is dominantly due to spontaneous polarization. Comparison of the electric field for the three strain conditions reveals that piezoelectric polarization plays only a minor role, estimated to be 5–10%. 8+8 4.72 5.19 4.89 10+ 8 4.24 4.76 4.52 6+8 4.36 5.08 4.55 8+8 4.77 5.19 4.89 Another approach to estimate the electric filed is based on 兩E兩 = ⌬Egap / ⌬d, where ⌬Egap is the difference of the band gap with the changes of the superlattice thickness, ⌬d.3 Assuming the band edge changes linearly with the field, this method gives an averaged electric field of 3.2 MV/cm by DFT-GGA and 3.9 MV/cm by LDA-SIC for the n + n superlattices in Fig. 2共a兲. The underestimated values are partial due to the fact that such a method neglects the contribution of the quantum confinement, which also plays a significant role in affecting the band-gap values, particularly in short superlattices. Indeed, for shorter than 7 + 7 superlattices, the estimated electric field is about 4.6 MV/cm by DFT-GGA. Strain effects also influence the band gap, both the DFTGGA and SIC-LDA calculations demonstrate that compressive 共tensile兲 strain leads to larger 共smaller兲 band-gap values compared to the free-standing superlattices with the exception of 6 + 8 by SIC-LDA. Such a small deviation may be related to the fact that we have used the relaxed DFT-GGA structures for the SIC-LDA calculations. Thus, strain offers another degree of freedom for tuning the band gap and electric field. IV. CONCLUSION To summarize, our first-principles DFT calculations systematically demonstrate the continuous tunability of the band gap and the strength of the built-in electric fields in GaN/ AlN 共0001兲 superlattices by adjustment of the thickness of both the well 共GaN兲 and barrier 共AlN兲 regions leaving much room for future improvement and tailoring of various nitride devices. The effects of strain in affecting the band gap and electric field are also established. We confirm that spontaneous polarization plays a dominant role for the large built-in electric fields, typically 4–6 MV/cm, while strain-induced piezoelectric polarizations contribute only about 5–10%. TABLE II. Calculated band-gap values for strained and free-standing GaN/AlN superlattices from GGA and SIC-LDA. Units are in electron volt. GGA Strain type Fix AlN Fix GaN Free standing 6+8 2.13 2.00 2.08 8+8 1.84 1.75 1.80 10+ 8 6.01 6.81 6.30 SIC-LDA 10+ 8 1.67 1.56 1.60 155301-4 6+8 4.12 3.98 4.21 8+8 3.78 3.63 3.76 10+ 8 3.57 3.35 3.49 PHYSICAL REVIEW B 81, 155301 共2010兲 CONTINUOUSLY TUNABLE BAND GAP IN GaN/AlN… ACKNOWLEDGMENTS We acknowledge the computing resources provided by the NCI National Facility in Canberra, Australia, 1 F. Bernardini, V. Fiorentini, and D. Vanderbilt, Phys. Rev. B 56, R10024 共1997兲. 2 N. Grandjean, B. Damilano, S. Dalmasso, M. Leroux, M. Laugt, and J. Massies, J. Appl. Phys. 86, 3714 共1999兲. 3 V. Fiorentini, F. Bernardini, F. Della Sala, A. Di Carlo, and P. Lugli, Phys. Rev. B 60, 8849 共1999兲. 4 R. Cingolani, A. Botchkarev, H. Tang, H. Morkoc, G. Traetta, G. Coli, M. Lomascolo, A. Di Carlo, F. Della Sala, and P. Lugli, Phys. Rev. B 61, 2711 共2000兲. 5 S.-H. Park and S.-L. Chuang, Appl. Phys. Lett. 76, 1981 共2000兲. 6 K. Omae, Y. Kawakami, S. Fujita, M. Yamada, Y. Narukawa, and T. Mukai, Phys. Rev. B 65, 073308 共2002兲. 7 M. Tchernycheva, L. Nevou, L. Doyennette, F. H. Julien, E. Warde, F. Guillot, E. Monroy, E. Bellet-Amalric, T. Remmele, and M. Albrecht, Phys. Rev. B 73, 125347 共2006兲. 8 E. P. Pokatilov, D. L. Nika, and A. A. Balandin, Appl. Phys. Lett. 89, 113508 共2006兲. 9 M. Leroux, N. Grandjean, M. Laugt, J. Massies, B. Gil, P. Lefebvre, and P. Bigenwald, Phys. Rev. B 58, R13371 共1998兲. 10 J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, Phys. Rev. B 57, R9435 共1998兲. 11 J. A. Majewski, S. Hachenbuchner, G. Zandler, and P. Vogl, Comput. Mater. Sci. 30, 81 共2004兲. 12 X. Y. Cui, J. E. Medvedeva, B. Delley, A. J. Freeman, and C. Stampfl, Phys. Rev. B 78, 245317 共2008兲. 13 N. Grandjean, J. Massies, and M. Leroux, Appl. Phys. Lett. 74, 2361 共1999兲. 14 C. Adelmann, E. Sarigiannidou, D. Jalabert, Y. Hori, J.-L. Rouviere, B. Daudin, S. Fanget, C. Bru-Chevallier, T. Shibata, and M. Tanaka, Appl. Phys. Lett. 82, 4154 共2003兲. 15 E. Baumann, F. R. Giorgetta, D. Hofstetter, S. Golka, W. Schrenk, G. Strasser, L. Kirste, S. Nicolay, E. Feltin, J. F. Carlin, and N. Grandjean, Appl. Phys. Lett. 89, 041106 共2006兲. 16 C. Buchheim, R. Goldhahn, A. T. Winzer, G. Gobsch, U. Rossow, D. Fuhrmann, A. Hangleiter, F. Furtmayr, and M. Eickhoff, Appl. Phys. Lett. 90, 241906 共2007兲. 17 C. Bayram, N. Pere-laperne, R. McClintock, B. Fain, and M. Razeghi, Appl. Phys. Lett. 94, 121902 共2009兲. 18 N. Iizuka, K. Kaneko, and N. Suzuki, Appl. Phys. Lett. 81, 1803 共2002兲. 19 A. Helman, M. Tchernycheva, A. Lusson, E. Warde, F. H. Julien, Kh. Moumanis, G. Fishman, E. Monroy, B. Daudin, D. Le Si Dang, E. Bellet-Amalric, and D. Jalabert, Appl. Phys. Lett. 83, which is supported by the Australian Commonwealth Government, and from the NSF 共U.S.兲 共through its MRSEC program at the Northwestern Materials Research Center兲. 5196 共2003兲. G. Martin, A. Botchkarev, A. Rockett, and H. Morkoc, Appl. Phys. Lett. 68, 2541 共1996兲. 21 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 共1996兲. 22 B. Delley, J. Chem. Phys. 113, 7756 共2000兲; 92, 508 共1990兲. 23 J. Li, K. B. Nam, M. L. Nakarmi, J. Y. Lin, H. X. Jiang, P. Carrier, and S.-H. Wei, Appl. Phys. Lett. 83, 5163 共2003兲. 24 I. Vurgaftman and J. R. Meyer, J. Appl. Phys. 94, 3675 共2003兲. 25 D. Vogel, P. Kruger, and J. Pollmann, Phys. Rev. B 55, 12836 共1997兲; Specifically, we use the Troullier-Martins scheme 关N. Troullier and J. L. Martins, ibid. 43, 1993 共1991兲兴; as implemented in the FHIPP code 关M. Fuchs and M. Scheffler, Comput. Phys. Commun. 119, 67 共1999兲兴 to generate the SIC pseudopotentials. 26 A. Qteish, A. I. Al-Sharif, M. Fuchs, M. Scheffler, S. Boeck, and J. Neugebauer, Phys. Rev. B 72, 155317 共2005兲. 27 X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.-M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, P. Ghosez, J.-Y. Raty, and D. C. Allan, Comput. Mater. Sci. 25, 478 共2002兲. 28 Troullier-Martins pseudopotentials were constructed in atomic LDA calculations for Al共3s2 , 3p3 , 3d0兲, Ga共3d10 , 4s2 , 4p1兲, and N共2s2 , 2p3 , 3d0兲 with pseudoization radii 共rs , r p , rd兲 of 共1.5, 1.5, 1.5兲 bohr, 共2.2, 2.4, 2.1兲 bohr, and 共1.8, 2.0, 2.1兲 bohr, respectively. SIC potentials were then added for the Ga 3d and N 2s and N 2p states, constructed with a cutoff radius of 7.3 bohr. The potentials were used in Kleinman-Bylander form 关L. Kleinman and D. M. Bylander. Phys. Rev. Lett. 48, 1425 共1982兲.兴 with the d-共Al and N兲 and s-components 共Ga兲 as local potentials and checked to be free of ghost states. 29 S. Massidda, B. I. Min, and A. J. Freeman, Phys. Rev. B 35, 9871 共1987兲. 30 S. Picozzi, A. Continenza, and A. J. Freeman, Phys. Rev. B 55, 13080 共1997兲. 31 F. Bernardini and V. Fiorentini, Phys. Rev. B 57, R9427 共1998兲. 32 P. Piquini, P. A. Graf, and A. Zunger, Phys. Rev. Lett. 100, 186403 共2008兲. 33 O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, J. Appl. Phys. 85, 3222 共1999兲. 20 155301-5