AIP ADVANCES 8, 065301 (2018)
Ab-initio study of boron incorporation and compositional limits at GaN and AlN (0001) surfaces L. Lymperakisa Max-Planck-Institut f¨ur Eisenforschung GmbH, Max-Planck-Straße 1, 40237 D¨usseldorf, Germany (Received 13 March 2018; accepted 7 May 2018; published online 1 June 2018)
Density functional theory calculations are employed to investigate B incorporation at the GaN(0001) and AlN(0001) surfaces. It is found that under typical metal-organic chemical vapor deposition (MOCVD) and metal rich molecular beam epitaxy (MBE) conditions, the maximum B contents at the surfaces are in the order of 3% for GaN and 15% for AlN. Under MBE N-rich growth conditions the calculations reveal a rehybridization enhanced solubility mechanism that dominates at the surface. This mechanism offers a promising route to kinetically stabilize B contents above the bulk solubility limit and as high as 25%. © 2018 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5029339 The family of AlN, GaN, InN binary alloys as well as their solid solutions, nowadays dominate the optoelectronics industry with applications in light emitting devices (LED),1 laser diodes (LD)2 and power electronics.3 Boron containing GaN and AlN materials constitute emerging members of these alloys family and have been considered for a number of technological applications: Ultrathin BGaN layers implemented as a back barrier in AlGaN/GaN high electron mobility transistors have been found to improve the confinement of the 2-dimensional electron gas as well as the resistivity of the GaN buffer layer.4 The transparency of BAlN in the deep ultraviolet (UV) region as well as the high refractive index contrast between BAlN and AlN has been used to design BAlN/AlN deep-ultraviolet distributed Bragg reflectors.5 Furthermore, thanks to the considerably smaller lattice constant of wurtzite (wz) BN, which is ≈ 21% and 19% smaller than GaN and AlN, it has been proposed that co-alloying the former with the later or with InGaN could provide lattice matched layers to SiC or GaN.6–8 A major challenge towards exploiting the full potential of B containing III-Nitride alloys is the limited B incorporation. More specifically, employing the regular solution model it was predicted that the B solubility in GaN and AlN is 1.8% and 2.8% at 1000 ◦ C, respectively.9 First principles calculations of the zincblende phase of these alloys revealed a very large miscibility gap with the onset of spinodal decomposition of BGaN and BAlN at 2.8% and 3.7% B content, respectively, at the typical growth temperatures.10 Numerous previous experimental works have explored the compositional limits of BGa(Al)N alloys using MOCVD,6,11–19 reactive DC magnetron sputter deposition,20 and MBE.7,21–25 The vast majority of these reports employed the metal polar (0001) surface. In most of these reports the B composition in BGaN films is limited to less than ≈ 3.5%. Nevertheless, B contents as high as 7.4% have been reported for low pressure (in the order of 0.02 atm) and temperature (in the order of 1050-1200 K) metal-organic vapor phase epitaxy (MOVPE).11 However, structural characterization of the highest B content films indicated poor structural quality. In the case of BAlN and BGaAlN alloys several works reported B content in the range of ≈ 1–14%.14,16–19 The highest B content reported is 30% and corresponds to BAlN grown under plasma assisted MBE using electron beam (EB)-guns as group III sources.22 Moreover, secondary ion mass spectroscopy measurements on BGaN and BAlN alloys revealed an anticorrelation between Ga and B indicating that B preferentially incorporates in the cation sublattice.12,19 a
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Surfaces break the bulk translational symmetry, they provide different chemical environment, coordination and rehybridization than bulk, and allow for more efficient strain relaxation.26 These may result in an enhanced solute incorporation and in surface induced alloy ordering.27–29 Furthermore, in a recent study, the large impact of rehybridization on In solubility in InGaN alloys has been shown.30 Therefore, surface engineering offers a potential route to overcome the bulk solubility limits. The key idea is to employ surface structures with compositions higher than the bulk solubility limit which are kinetically stable and do not change their composition when overgrown. Therefore, understanding of the atomistic mechanisms at surfaces is a prerequisite to fully control the composition of the epitaxially grown alloys. However, detailed investigations of the atomistic mechanisms governing the growth of B containing III-Nitride alloys are lacking. In this work an extensive first-principles investigation of surface energetics and B incorporation at metal polar (0001) GaN and AlN surfaces is presented. The total energies and forces have been calculated employing density functional theory (DFT), the local density approximation (LDA), and the projector augmented-wave (PAW) method31,32 with the Ga 3d electrons treated as valence states. The surfaces are modeled using a slab geometry of 8 metal-N monolayers (MLs) separated consisting ¯ by a vacuum region of 16 Å. N atoms at the 0001 side of the slab are passivated with pseudohydrogen atoms of partial charge 0.75. The plane-wave energy cutoff was 450 eV and an equivalent of a 10 × 10 × 1 Monkhorst-Pack k-point mesh for the 1 × 1 surface unit cell was used to sample the Brillouin zone (BZ). Convergence checks were explicitly performed. They showed that the aforementioned settings provide surface energy differences with an accuracy better than 3 meV/Å2 . √ In total √ about 200 different GaN and AlN surfaces having n × n (n=1,2 and 4) and n 3 × n 3 R30o (n=1,2) reconstructions have been considered. These structures include B free surfaces as well as surfaces where B substitutes cations in the topmost cation-anion ML [layer L0 in Fig 1(c)] as well as in metallic adlayers [layers L1 and L2 in Fig 1(c)]. The surface energies are calculated with respect to the unreconstructed and relaxed bare 1 × 1 ideal surface. The relative surface energy ∆γ is expressed as:26 X tot tot ∆γ = Esurf − Eref − ∆ni µi , (1) tot and E tot are the total energies of the surface reconstruction and the reference surface where Esurf ref supercells, respectively. ∆ni and µi are the excess number of atoms of species i with respect to the reference structure and the corresponding chemical potential, respectively. Thermodynamic equilibrium with XN (X:Ga or Al) is assumed (i.e. µXN = µX + µN ). Under these conditions and in order to avoid the competing phases of bulk metal and nitrogen molecules the metal chemical potential is allowed to vary over the range: µX bulk + ∆HfXN ≤ µX ≤ µX bulk , where the upper (lower) limit
FIG. 1. Ball and stick models for selected BGaN(0001) surfaces in side view [(a)-(c))] √ and top √ view [(e)-(f)]. Morphologies correspond to (a) and (d) 2 × 2 NAd , (b) and (e) 2 × 2 NAd with 1 B×3 Ga , (c) 2 3 × 2 3 R30o LCB with 1 BGa and (f) 2 × 2 NAd with 1 B×4 Ga . In (c) the three top most surface layers (L2, L1, and L0) are indicated.
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corresponds to metal (nitrogen) rich conditions, respectively. µX bulk is the chemical potential of bulk metal and ∆HfXN is the formation enthalpy of XN as calculated within DFT. An additional constraint applied to the chemical potentials arise from the requirement that the competing phase of BN is thermodynamically unstable (i.e., µB + µN ≤ µwz−BN ), where µB is the chemical potential of B. The energetically most stable phase of BN is the sp2 bonded hexagonal structure. However, when BN is alloyed with wz GaN or AlN is expected to adopt the wz phase. Indeed, a number of previous experimental reports indicate that at least for low B contents the quaternary alloys inherit the wz crystal structure.14,17 Therefore, in the following thermodynamic analysis the calculated chemical potential of wz BN, µwz−BN , is employed. In Figs. 2(a) and (b) the phase diagrams of BGaN(0001) and BAlN(0001) surfaces, respectively, are shown as function of the metal and the hydrogen chemical potentials. The B chemical potential is fixed to the upper limit prior the formation of wz BN, i.e. µB + µN = µwz−BN . Both surfaces show qualitatively the same behavior and for typical MOCVD growth conditions only 2 × 2 surface reconstructions that obey electron counting rule (ECR) dominate. At the limit of the metal rich growth conditions the laterally contracted Ga bilayer (LCB) and Al monolayer (LCM) are the lowest energy reconstructions of the BGaN and BAlN surfaces, respectively. These findings are in full agreement with previous LDA calculations on GaN(0001) surfaces.33 Nevertheless, the energetics of ECR surfaces may be affected by the doping level.34 Recently, it has been shown that at high doping levels (in the order of 1019 − 1020 cm−3 ) and at low H chemical potential (∆µH < -1.0 eV) H desorbs from the ECR (2×1)-H ZnO(0001) ECR surface and the anion dangling bonds are passivated by free charge carriers introduced by the dopants.35 Therefore, at typical nitride growth conditions and doping levels in the order of 1018 − 1019 cm−3 , doping is not expected to affect both the phase diagrams in Fig. 2 as well as the following calculated B surface contents.
FIG. 2. Phase diagrams for the (a) BGaN(0001) and (b) BAlN(0001) surfaces as function of the Ga and Al, respectively and the hydrogen chemical potentials. ∆µH is the H chemical potential with respect to H2 molecules at T=0 K. The B chemical potential is fixed to the thermodynamically allowed maximum value when GaN and AlN are stable against the competing phase of wz-BN. The dashed lines indicate constant B surface concentrations at T=1300 K. The H2 partial pressures, pH2 , on the right are calculated from ∆µ H assuming an ideal gas of H√2 molecules at T=1300 K. All reconstructions have a 2 × 2 √ periodicity, except the LCB in (a) and LCM in (b) which have a 3 × 3R30o periodicity.
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An important result from the above mentioned diagrams is that all B containing GaN (AlN) reconstructions are energetically unfavorable at T=1300 K for H2 partial pressures larger than ≈ 5 × 10−4 ( ≈ 5 × 10−3 ) atm, respectively. In order to estimate the B solubility at the surface, the lowest energy reconstruction at each ∆µX and ∆µH is treated as a host matrix and all the other reconstructions as defects with formation energy ∆E i . The surface B content cB (∆µX , ∆µH ) is then calculated as: ! ∆Ei 1X i gi cB exp − , (2) cB (∆µX , ∆µH ) = Z i kB T where gi and cBi are the degeneracy and B surface content of reconstruction i, respectively and Z is the partition function. k B and T are the Boltzmann constant and temperature, respectively. This approach is valid because all the low energy reconstructions have the 2 × 2 or 4 × 4 symmetry. Hence, they are commensurate and the energy of the interface between different reconstructions is very small compared to k B T.36 The vast majority of reported MOCVD grown alloys employed low pressure conditions with the total gas pressure in the order of 0.1 atm. Under these conditions and at T=1300 K the H2 partial pressure is in the order of 10−2 − 10−1 atm.37 As can be seen in Fig. 2, the calculated maximum B surface solubility in this region of H2 partial pressures at GaN and AlN (0001) surfaces is in the order of 0.1-1% and 1-10%, respectively. Nevertheless, in the aforementioned analysis, the calculated surface energies do not include vibrational entropic contributions. These were estimated to be in the order of 0.1 eV per H atom at the T=1300 K.33 Furthermore, errors in the same order of magnitude have to be considered for the LDA calculated potentials of the chemical reservoirs (e.g. 0.03 and -0.2 eV for ∆µGa and ∆µH , respectively).38 Therefore, considering these, the calculated surface B contents should be estimated in the order of 0.1-3% for GaN and 1-15% for AlN. These values are in good agreement with the experimentally reported B solubilities. An interesting outcome from Fig. 2 is that at low values of the H chemical potential surface reconstructions with B content as high as 25% are energetically favorable. Arbitrary low values of H chemical potential, ∆µH , and hence H2 partial pressures correspond to MBE growth conditions. In Figs. 3(a) and (b) the phase diagrams of BGaN(0001) and BAlN(0001) under MBE conditions as function of the metal and B chemical potentials are shown, respectively. The surfaces of both BGaN and BAlN show qualitatively similar behavior: In both cases, B incorporation is energetically favorable for growth under N rich conditions. Growth under metal rich conditions is preferentially employed in MBE of III-Nitrides to achieve smooth surface morphologies.39,40 However, the calculations indicate that under these conditions B incorporations is highly unfavorable. The maximum B content under N rich conditions is 25% and B incorporates at the low coordinated cation site of the 2 × 2 N adatom (NAd ) reconstruction. Moreover, in both cases the B layer content can be tuned from 0% to 25% as we √ move√from B poor to B rich conditions. Furthermore, all the symmetry inequivalent 4 × 4 and 2 3 × 2 3R30o NAd reconstructions with B substituting the low coordinated cation sites have been explicitly calculated. For 25% content these are symmetry equivalent and degenerate. However, √ B√ for lower B content the 2 3 × 2 3 R30o reconstructions are energetically more favorable and the B content changes abruptly from 0 to 1/12, 2/12 and 3/12. The favorable incorporation of B atom at the low coordinated cation site of the 2 × 2NAd reconstruction can be explained as follows: The 2 × 2NAd reconstruction obeys electron counting rule (ECR) and the low coordinated cation adopts a planar sp2 configuration [see Figs. 1(a) and (e)].41 The sp2 configuration shifts the bonding states of the × 3 coordinated cation with the N atoms as well as the fully occupied state of the N adatom dangling bond to lower energies and thus lowers the surface energy.30 B adopts the sp2 bonded hexagonal structure when alloyed with N under ambient conditions. Thus, when it is incorporated at the × 3 coordinated site [see Figs. 1(b) and (e)] its bonding states with the N atoms as well as the fully occupied state of the N adatom dangling bond are shifted to even lower energies. Therefore, as the calculations confirm, the sp2 hybridized low coordinated cation site is highly preferable for B incorporation. This hybridization enhanced solubility mechanism allows to incorporate at the surface B contents well above the bulk solubility limit. This mechanism operates in the opposite direction to the recently identified elastically frustrated rehybridization mechanism which results in preferential incorporation of In at the × 4 coordinated cation sites of the 2 × 2NAd GaN(0001) surfaces.30 Nevertheless, when B atom is placed at a four fold coordinated site, i.e. bound to the NAd , it forms a planar configuration with the NAd and two surface anions. In order to
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FIG. 3. Surface phase diagrams for the (a) BGaN and (b) BAlN(0001) surfaces under MBE conditions as function of the Ga and Al, respectively and B chemical potentials. In (a) and (b) thermodynamic equilibrium with GaN √ and√AlN, respectively, is assumed. The √shaded√triangles denote instability against the competing phase of wz BN. 2 3 × 2 3NAd nB, n=1,2, and 3, indicate 2 3 × 2 3R30o NAd reconstructions with one, two and three B atoms incorporated at the low coordinated surface sites, respectively.
achieve this the two surface anions shift towards the B atom and considerably deform the surface [see Fig. 1(f)]. To increase the B content above 25%, B should incorporate at ×4 coordinated sites. However, exchanging a B atom from the ×3 to the ×4 coordinated cation surface site costs more than 1.5 eV. This introduces a strict upper limit of 25% to the thermodynamically allowed surface B content. Furthermore, the in basal plane lattice constant of BN is ≈ 21 % and ≈ 19 % smaller than that of GaN and AlN, respectively. Thus, substitutional B introduces considerable tensile strain into the host matrix. At the 2 × 2NAd reconstruction the bonds of the low coordinated cation with the N atoms are ≈ 5% contracted. Therefore, these cation site readily offer more efficient strain accommodation than in bulk. Moreover, solute-solute stain interaction has been shown to result in ordering phenomena in InGaN alloys.42 In order to investigate the strain interactions among B substitutionals, the B-B interactions in the basal plane of the GaN matrix are calculated following the methodology described in Ref. 42. The calculations indicate that first and third nearest neighbors B-B interactions are strongly repulsive. The calculated interaction energies of 0.25 and 0.14 eV are larger than the corresponding 42 In-In interaction energies in GaN ( ≈ 50 meV D ). EHowever, the second nearest neighbor interactions ¯ where the B atoms are aligned along the 1100 direction are only weakly repulsive (interaction energy of 5 meV). Inspection of the atomic displacements introduced by the BGa atoms reveals that the preferable alignment of BGa as second nearest neighbors is, as in InGaN alloys, due to an efficient local√strain mechanism.42 The later results in preferential arrangement of B atoms √ accommodating o in a √ 3 × 3R30 pattern. The highest symmetry surface reconstruction that is commensurate √ √ √ to the 3 × 3R30o pattern with the symmetry of the 2 × 2NAd reconstruction is the 2 3 × 2 3 R30o reconstruction. This explains the above mentioned finding that these reconstructions are energetically
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more favorable than all the B containing 4 × 4 reconstructions. It also indicates that B contents below √ √ 1/4 should be arranged in the basal plane in a 2 3 × 2 3R30o pattern. The above mentioned results provide valuable insights into the growth of B containing III-Nitrides. First of all they show that the maximum B content that can be achieved at the surfaces under typical MOCVD growth conditions does not exceed the bulk solubility limit for GaN and 15% for AlN. To further increase B composition the B chemical potential should be raised into a regime that the competing phase of BN is favorable. Thus, increased B content will be accompanied by the formation of the BN phase at the surfaces and severe degradation of the materials quality. On the other hand, MBE growth offers a promising route to increase the layer B content. Rehybridization under N rich conditions and strain accommodation at the surface allows to efficiently incorporate B up to maximum surface composition of 25%. This can be achieved under B rich conditions, i.e. just before the onset of the BN phase. Thus, careful tuning of the growth conditions is a necessary √ prerequisite. √ An interesting outcome of this approach is that ordering in the form of 2 × 2 [ 2 3 × 2 3 R30o ] in the basal plane pattern for the maximum (lower) B contents, respectively, is introduced at the surface. Nevertheless, a major drawback of this growth strategy is the use of N rich conditions. It is well established that due to poor adatom surface kinetics, these conditions result in rough surface morphologies and low material quality.39 A possible strategy to address this challenge is to focus on the growth of high B content BGaN/GaN or BAlN/AlN superlattices (SLs) and apply modulated MBE, where N rich and metal conditions alternate. In summary, using DFT calculations, B incorporation at the GaN(0001) and AlN(0001) surfaces has been investigated. It is found that the surface B compositional limits under typical MOCVD and metal rich MBE growth conditions do not exceed the bulk solubility limit for GaN and 15% for AlN. However, a hybridization enhanced B incorporation mechanism dominates under MBE N rich growth conditions and offers a promising route to kinetically stabilize B contents well above the bulk solubility limit and as high as 25%. This growth regime has not been used so far to grow ternary III-Nitride alloys with B. However, as the aforementioned results indicate, it worth to be considered and investigated by experiment. Financial support from the Electronic Component Systems for European Leadership Joint Undertaking under grant agreement No 662133, Powerbase project is acknowledged. This Joint Undertaking receives support from the European Union’s Horizon 2020 research and innovation programme and Austria, Belgium, Germany, Italy, Netherlands, Norway, Slovakia, Spain, United Kingdom. The author would also like to acknowledge fruitful discussions with Prof. J¨org Neugebauer. 1 S.
Nakamura and M. R. Krames, Proceedings of the IEEE 101, 2211 (2013). Nakamura, S. Pearton, and G. Fasol, The Blue Laser Diode (Springer-Verlag Berlin Heidelberg, 2000). 3 S. Chowdhury, B. L. Swenson, M. H. Wong, and U. K. Mishra, Semiconductor Science and Technology 28, 074014 (2013). 4 V. Ravindran, M. Boucherit, A. Soltani, S. Gautier, T. Moudakir, J. Dickerson, P. L. Voss, M. A. di Forte-Poisson, J. C. de Jaeger, and A. Ougazzaden, Appl. Phys. Lett. 100, 243503 (2012). 5 M. Abid, T. Moudakir, G. Orsal, S. Gautier, A. E. Naciri, Z. Djebbour, J. H. Ryou, G. Patriarche, L. Largeau, H. J. Kim, Z. Lochner, K. Pantzas, D. Alamarguy, F. Jomard, R. D. Dupuis, J. P. Salvestrini, P. L. Voss, and A. Ougazzaden, Appl. Phys. Lett. 100, 051101 (2012). 6 A. Y. Polyakov, M. Shin, W. Qian, M. Skowronski, D. W. Greve, and R. G. Wilson, J. Appl. Phys. 81, 1715 (1997). 7 V. Vincent, Y. Satoshi, S. Hiroyuki, and U. Satoshi, Jpn. J. Appl. Phys. 36, L1483 (1997). 8 L. Williams and E. Kioupakis, Appl. Phys. Lett. 111, 211107 (2017). 9 C. H. Wei and J. H. Edgar, J. Cryst. Growth 208, 179 (2000). 10 L. K. Teles, J. Furthm¨ uller, L. M. R. Scolfaro, A. Tabata, J. R. Leite, F. Bechstedt, T. Frey, D. J. As, and K. Lischka, Physica E: Low-Dimensional Systems and Nanostructures 13, 1086 (2002). 11 B. P. Gunning, M. W. Moseley, D. D. Koleske, A. A. Allerman, and S. R. Lee, J. Cryst. Growth 464, 190 (2017). 12 A. Ougazzaden, S. Gautier, C. Sartel, N. Maloufi, J. Martin, and F. Jomard, J. Cryst. Growth 298, 316 (2007). 13 K. Ueyama, H. Mimura, Y. Inoue, T. Aoki, and T. Nakano, Jpn. J. Appl. Phys. 55 (2016). 14 S. Wang, X. H. Li, A. M. Fischer, T. Detchprohm, R. D. Dupuis, and F. A. Ponce, J. Cryst. Growth 475, 334 (2017). 15 T. Akasaka and T. Makimoto, Appl. Phys. Lett. 88 (2006). 16 X. Li, S. Sundaram, Y. El Gmili, F. Genty, S. Bouchoule, G. Patriache, P. Disseix, F. Reveret, J. Leymarie, J. P. Salvestrini, R. D. Dupuis, P. L. Voss, and A. Ougazzaden, J. Cryst. Growth 414, 119 (2015). 17 X. H. Li, S. Wang, H. X. Liu, F. A. Ponce, T. Detchprohm, and R. D. Dupuis, Physica Status Solidi B 254 (2017). 18 T. Takano, M. Kurimoto, J. Yamamoto, and H. Kawanishi, J. Cryst. Growth 237-239, 972 (2002). 19 X. Li, S. Sundaram, Y. El Gmili, T. Moudakir, F. Genty, S. Bouchoule, G. Patriarche, R. D. Dupuis, P. L. Voss, J. P. Salvestrini, and A. Ougazzaden, Physica Status Solidi A 212, 745 (2015). 20 L. Liljeholm and J. Olsson, Vacuum 86, 466 (2011). 2 S.
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K. Gupta, C. C. Wamsley, M. W. Koch, and G. W. Wicks, J. Vac. Sci. Technol. B 17, 1246 (1999). Yamashita, Y. Ishikawa, H. Ohsato, and N. Shibata, Key Engineering Materials 301, 95 (2005). 23 T. L. Williamson, N. R. Weisse-Bernstein, and M. A. Hoffbauer, Phys. Status Solidi C 11, 462 (2014). 24 R. C. Cramer, B. Bonef, J. English, C. E. Dreyer, C. G. Van de Walle, and J. S. Speck, J. Vac. Sci. Technol. A 35, 041509 (2017). 25 A. Nakajima, Y. Furukawa, H. Yokoya, and H. Yonezu, J. Cryst. Growth 278, 437 (2005). 26 J. Z´ un˜ iga-P´erez, V. Consonni, L. Lymperakis, X. Kong, A. Trampert, S. Fern´andez-Garrido, O. Brandt, H. Renevier, S. Keller, K. Hestroffer, M. R. Wagner, J. S. Reparaz, F. Akyol, S. Rajan, S. Rennesson, T. Palacios, and G. Feuillet, Applied Physics Reviews 3, 041303 (2016). 27 D. M. Wood and A. Zunger, Phys. Rev. Lett. 61, 1501 (1988). 28 J. E. Bernard, S. Froyen, and A. Zunger, Phys. Rev. B 44, 11178 (1991). 29 J. Tersoff, Phys. Rev. Lett. 74, 434 (1995). 30 L. Lymperakis, T. Schulz, C. Freysoldt, M. Anikeeva, Z. Chen, X. Zheng, B. Shen, C. Ch` eze, M. Siekacz, X. Q. Wang, M. Albrecht, and J. Neugebauer, Phys. Rev. Materials 2, 011601 (2018). 31 G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993). 32 G. Kresse and J. Furthm¨ uller, Phys. Rev. B 54, 11169 (1996). 33 C. G. Van de Walle and J. Neugebauer, Phys. Rev. Lett. 88, 066103 (2002). 34 O. Sinai, O. T. Hofmann, P. Rinke, M. Scheffler, G. Heimel, and L. Kronik, Phys. Rev. B 91, 075311 (2015). 35 S. Erker, P. Rinke, N. Moll, and O. T. Hofmann, New J. Phys. 19, 083012 (2017). 36 H. Abu-Farsakh and J. Neugebauer, Phys. Rev. B 79, 155311 (2009). 37 C. H. Wei and J. H. Edgar, Journal of Crystal Growth 217, 109 (2000). 38 C. Freysoldt, B. Lange, J. Neugebauer, Q. Yan, J. L. Lyons, A. Janotti, and C. G. Van de Walle, Phys. Rev. B 93, 165206 (2016). 39 C. Adelmann, J. Brault, G. Mula, B. Daudin, L. Lymperakis, and J. Neugebauer, Phys. Rev. B 67, 165419 (2003). 40 J. Neugebauer, T. K. Zywietz, M. Scheffler, J. E. Northrup, H. Chen, and R. M. Feenstra, Phys. Rev. Lett. 90, 056101 (2003). 41 M. Himmerlich, L. Lymperakis, R. Gutt, P. Lorenz, J. Neugebauer, and S. Krischok, Physical Review B 88, 125304 (2013). 42 S. Lee, C. Freysoldt, and J. Neugebauer, Phys. Rev. B 90, 245301 (2014). 22 M.
