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First-principles study on electronic and elastic properties of BN, AlN, and GaN Kazuhiro Shimada, Takayuki Sota, and Katsuo Suzuki Citation: J. Appl. Phys. 84, 4951 (1998); doi: 10.1063/1.368739 View online: http://dx.doi.org/10.1063/1.368739 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v84/i9 Published by the American Institute of Physics. Related Articles Note: Scale-free center-of-mass displacement correlations in polymer films without topological constraints and momentum conservation J. Chem. Phys. 135, 186101 (2011) A new Monte Carlo method for getting the density of states of atomic cluster systems J. Chem. Phys. 135, 144109 (2011) Ab initio molecular dynamics simulation of binary Cu64Zr36 bulk metallic glass: Validation of the cluster-plus- glue-atom model J. Appl. Phys. 109, 123520 (2011) Terahertz spectrum and normal-mode relaxation in pentaerythritol tetranitrate: Effect of changes in bond- stretching force-field terms J. Chem. Phys. 134, 244502 (2011) Structural and vibrational properties of amorphous GeO2 from first-principles Appl. Phys. Lett. 98, 202110 (2011) Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 21 Nov 2011 to 221.212.116.13. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions d S ne u cu l on e bl cu ti on of e aN © JOURNAL OF APPLIED PHYSICS VOLUME 84, NUMBER 9 1 NOVEMBER 1998 depending on a strain is indispensable. However there is little information on these properties, especially deformation potential constants, in spite of the fact that there have been many theoretical3–5 and experimental6–8 results on them. In recent work by Wright,9 all the elastic constants of AlN, GaN, and InN for both zinc-blende ~ZB! and wurtzite ~WZ! structures were obtained from first-principles calcula- tions based on the pseudopotential method. Kim et al.10,11 reported all the elastic constants of BN, AlN, GaN, and InN using the full potential linear muffin tin orbital ~FP-LMTO! method. Kato and Hama12 reported their work on elastic con- stants of WZ–AlN using the pseudopotential method. Majewski et al.4 and Kim et al.5 have obtained the de- formation potential constants for WZ crystal D3 and D4 in the Bir–Pikus notation by using the FP-LMTO and pseudo- potential methods within the local density approximation, re- spectively. Among recent calculations for deformation po- important role in InGaN quantum wells.13 However there are only a few reports on the WZ piezoelectric constants14,15 and those of ZB,16–19 and there is still no work which calculated many physical quantities of ZB and WZ BN, AlN, and GaN systematically, including piezoelectric constants with the same accuracy by using the pseudopotential-plane-wave method. It has been known that the physical parameters of them with the WZ structure sensitively depend on the cell shape and the atomic geometry. Thus in calculating the above men- tioned values of their various physical parameters, the opti- mization of the crystal structure is indispensable. In this article, we report on the systematic study of the electronic and elastic properties of BN, AlN, and GaN in both ZB and WZ structures. We have performed first- principles calculations using the molecular dynamics method with variable cell shapes20 to optimize their unit cell struc- tures. To find the equilibrium positions of atoms in a strained crystal, we have carried out first-principles molecular dy- namics calculation. Discussions on calculated results are given in comparison with the previous computational and a!Also at Material Research Laboratory for Bioscience and Photonics, Graduate School of Science and Engineering, Waseda University, Shin- juku, Tokyo 169-8555, Japan; electronic mail: sota@elec.waseda.ac.jp b!Also at: Kagami Memorial Laboratory for Material Science and Technol- ogy, Waseda University, Shinjuku, Tokyo 169-8555, Japan. First-principles study on electronic an of BN, AlN, and GaN Kazuhiro Shimada, Takayuki Sota,a) and Katsuo Department of Electrical, Electronics, and Computer Engi Shinjuku, Tokyo 169-8555, Japan ~Received 15 May 1998; accepted for publication 4 A We have carried out first-principles total energy cal properties of both zinc-blende and wurtzite BN, A parameters, elastic constants, deformation potential c effective charges, and piezoelectric constants. Lattic first-principles molecular dynamics method with varia crystal is also relaxed by the first-principles mole influences the elastic constants, the deformation poten effectively. We have calculated the wurtzite deformati internal strain correction. The piezoelectric constants been calculated using the Berry phase approach and w of BN have an inverse sign in contrast to AlN and G with results obtained herein with the previous ones. @S0021-8979~98!07921-3# I. INTRODUCTION AlN and GaN are promising materials for applications in optoelectronics in the short wavelength range, light emit- ting diode and laser diode, and BN is for the extreme hard- ness and high thermal conductivity electronics devices. Re- cently there has been much interest in III nitrides and their alloys because of the potential for fabrication of a light emitter.1,2 These materials grown on sapphire suffer a biaxial strain because of the lattice mismatch. In modeling these electronics devices, knowledge of their elastic constants, de- formation potential constants, and energy band information 4950021-8979/98/84(9)/4951/8/$15.00 Downloaded 21 Nov 2011 to 221.212.116.13. Redistribution subject to AIP elastic properties uzukib) ering, Waseda University, 3-4-1 Ohkubo, gust 1998! lations to investigate electronic and elastic N, and GaN. We have calculated lattice stants, phonon frequencies at G point, Born parameters are fully relaxed by using the e cell shape. The internal strain in a strained lar dynamics method. The internal strain al constants, and the piezoelectric constants potential constants D1 – D5 considering the wurtzite and also zinc-blende crystals have have found from first principles that those . Discussions will be given in comparison 1998 American Institute of Physics. tential constants, only Suzuki and Uenoyama3 have obtained all of the deformation potential constants D1 – D6 at the Bril- louin zone center based on the linearized augmented plane wave method together with a cubic approximation. However their calculation has not relaxed the internal strain which plays an important role in determining the deformation po- tential constants and the elastic constants. Their results are not satisfactory in comparison with experimental values by Shikanai et al.6 and by Yamaguchi et al.7,8 It has been pointed out that the piezoelectric field, aris- ing from the residual strain due to lattice mismatch, plays an 1 © 1998 American Institute of Physics license or copyright; see http://jap.aip.org/about/rights_and_permissions experimental results. This work is an extension of our previ- ous publication.21 II. COMPUTATIONAL METHOD We have performed first-principles total energy calcula- tions for BN, AlN, and GaN within the local density approxi- mation ~LDA! to the density functional theory22 ~DFT!. The pseudopotentials are generated through the Troullier– Martins scheme23 in the nonrelativistic limit and are cast into the Kleinman–Bylander24 separable form to save computa- tional time and memory. For each pseudopotential, the loga- rithmic derivatives are examined by using the criteria devel- oped by Gonze, Stumpf, and Scheffler25 to eliminate the appearance of unphysical states well known as the ‘‘ghost states.’’ For exchange–correlation potential we have used the Ceperley–Alder26 type parametrized by Perdew and Zunger.27 The Kohn–Sham equation is solved iteratively with the eigenvalue minimization scheme.28 The Brillouin zone k integration has been performed by the special-points method of Chadi and Cohen.29 We have used ten special k points for zinc blende-structure and six special k points for WZ structure in the irreducible wedge of the Brillouin zone. To treat the deep N 2p and Ga 3d potential, we have to take large energy cutoffs ~90 Ry for BN and GaN, 80 Ry for AlN! to eliminate computational uncertainties. These cutoff energies give good convergence of both total energy and pressure.30 III. RESULTS A. Structural optimization The calculations were first carried out assuming ideal ZB and WZ structures. For WZ structures, the axis ratio c/a 51.633 and the internal parameter u50.375 were used. The calculated total energies and pressures for several lattice con- stants were fitted with the empirical Murnaghan equation of state31 to obtain equilibrium lattice constants. For ZB struc- tures, the forces acting on atoms and the elements of the stress tensor have converged within 1025 and 1026 Ry/a.u.3, respectively. For WZ structures, the values of the various physical parameters depend sensitively on the shape of the unit cell and the atomic geometry, and, thus, structural opti- mization is needed as mentioned before. Starting from the resulting ideal atomic geometry, the crystal structure optimi- TABLE I. Lattice constants of ZB–BN, AlN, and GaN in units of Å. BN AlN GaN Present 3.596 4.376 4.537 Other calculations 3.56–3.77a 4.33–4.42b 4.30–4.56c Experimentd 3.615, 3.6157e 4.37, 4.38f 4.50,g 4.51h aReferences 10, 47–55. eReference 45. bReferences 10, 37, 56–59. fReference 66. cReferences 10, 37, 55, 58, 60–63. gReference 67. dReference 64. hReference 68. 4952 J. Appl. Phys., Vol. 84, No. 9, 1 November 1998 zation has been iterated using the first-principles molecular dynamics method with variable cell shape20 until the forces Downloaded 21 Nov 2011 to 221.212.116.13. Redistribution subject to AIP acting on atoms and the elements of the stress tensor have converged within 1024 and 1026 Ry/a.u.3, respectively. Tables I and II summarize the results of our calculations for lattice parameters. Because there are so many results for ZB structure, we only show the range of the results. For WZ structure, only fully relaxed calculations are listed in Table II. Our results agree very well with the experimental data. The lattice constants of GaN are overestimated by about 1%. This comes from the fact that we have considered Ga 3d electrons as valence states. If these electrons are considered as core states, lattice constants are underestimated by a few percent. B. Elastic constants The values of the elastic constants are obtained by cal- culating the full stress tensor for small strains within the harmonic approximation. The ZB elastic constants C11 and C12 are obtained with the relations C115s1 /e1 , C12 5s2 /e1 , where s is the stress tensor and e is the strain 32 TABLE II. Lattice constants of WZ–BN, AlN, and GaN. a and c are in units of Å. BN AlN GaN Present a 2.534 3.112 3.210 c 4.191 4.995 5.237 c/a 1.654 1.605 1.631 u 0.3738 0.3811 0.3762 Other calculationsa a 2.54 3.06 3.17 c 4.17 4.91 5.13 c/a 1.64 1.60 1.62 u 0.375 0.383 0.379 Other calculationsb a fl 3.084 3.162 c fl 4.948 5.142 c/a fl 1.604 1.626 u fl 0.3814 0.3770 Other calculationsc a 2.531 3.082 3.143 c/a 1.657 1.605 1.626 u 0.3751 0.3816 0.377 Other calculationsd a fl 3.10 3.22 c fl 4.97 5.26 u fl 0.381 0.371 Experimente a 2.558 fl fl c 4.228 fl fl c/a 1.656 fl fl Experimentf a fl 3.11 3.189 c fl 4.98 5.185 c/a fl 1.60 1.626 Experimentg a fl 3.110 3.190 c fl 4.980 5.189 c/a fl 1.601 1.627 u fl 0.3821 0.377 aFP-LMTO by Kim et al., Ref. 10. bPseudopotential LDA by Wright and Nelson, Ref. 59. cPseudopotential LDA by Karch and Bechstedt, Refs. 91, 92. dSIC-Pseudopotential by Vogel et al., Ref. 63. eSoma et al., Ref. 45. fReference 64. gSchulz and Thiemann, Ref. 46. Shimada, Sota, and Suzuki tensor. Voigt notation is used for the tensorial component. The calculation of the elastic constants C44 is not as simple license or copyright; see http://jap.aip.org/about/rights_and_permissions as that of C11 and C12 . For a dilation along the @111# axis, the atomic positions in the unit cell are no longer determined by symmetry, and one has to find the atomic positions where the forces acting on atoms vanish. Following Nielsen and Martin,30 C44 is determined by an independent calculation of the forces and the stresses. Bulk moduli are obtained by us- ing the relation B5(C1112C12)/3. Table III also lists the values of B in square brackets calculated by Murnaghan’s equation of state. Experimental data of the elastic constants for the ZB structure are available only for BN and our results are in good agreement with the experimental data to within a few %. The other calculations predict nearly the same value within a few percent compared with ours except for Sokolovskii’s results.33 Our value of internal strain param- eter of ZB–BN is larger than the others by about 50%. For ZB–AlN and ZB–GaN, there are no experimental data and there are several other calculations. Our calculations are all in good agreement with the others except for Ruiz’s results.34 In a WZ crystal, any strain makes the forces acting on atoms finite. To delete these forces, the optimization of the atomic position was performed by using the first-principles TABLE III. Elastic constants, internal strain parameters, and bulk moduli of ZB–BN, AlN, and GaN. Elastic constants and bulk moduli are in units of GPa. The values in parentheses are calculated without internal strain correc- tion. The values in square brackets are obtained from Murnaghan’s equation of state. C11 C12 C44 z B BN Present 819 195 475 0.16 403 fl fl ~483! fl @401# Expt.a 820 190 480 fl 400 Calc.b 837 182 493 0.1 400 fl fl ~495! fl @400# Calc.c 830 420 450 fl 556 Calc.d 844 190 483 0.11 410 fl fl ~486! fl @386# Calc.e 812 182 464 0.07 397 fl fl ~466! fl fl AlN Present 313 168 192 0.56 216 fl fl ~236! fl @216# Calc.b 304 152 199 0.6 203 fl fl ~230! fl @203# Calc.f 348 168 135 fl 228 Calc.g 304 160 193 0.55 208 fl fl ~237! fl fl Calc.e 294 160 189 0.57 214 fl fl ~233! fl fl GaN Present 285 161 149 0.67 202 fl fl ~202! fl @203# Calc.b 296 154 206 0.5 201 fl fl ~225! fl @201# Calc.g 293 159 155 0.61 203 fl fl ~200! fl fl aGrimsditch et al., Ref. 69. bFP-LMTO LDA by Kim et al., Ref. 10. cSemiempirical by Sokolovskii, Ref. 33. dPseudopotential LDA by Rodriguez-Herna´ndez et al., Ref. 54. ePseudopotential LDA by Karch and Bechstedt, Ref. 91. fHartree–Fock by Ruiz et al., Ref. 34. gPseudopotential LDA by Wright, Ref. 9. J. Appl. Phys., Vol. 84, No. 9, 1 November 1998 molecular dynamics method. After this optimization, the forces are converged within 1024 Ry/a.u.. Induced strains in Downloaded 21 Nov 2011 to 221.212.116.13. Redistribution subject to AIP calculating the values of elastic constants for WZ crystals are e5$e100000% for C11 , C12 , and C13 , e5$00e3000% for C13 and C33 , and e5$000e4e5e6% with e45e55e6 for C44 and C66 . The results are listed in Table IV. The values in paren- theses are obtained without internal strain correction. It is shown that the internal strain corrects the values of the elas- tic constants explicitly. The bulk moduli of ZB crystals are determined by the Murnaghan equation fitting for total ener- gies and pressures of several lattice constants, and those of WZ are calculated using the following relation: B5 C33~C111C12!22~C13!2 C111C1212C3324C13 . ~1! For WZ structure, there is only one result calculated from first principles for all elastic constants.9 Kim et al.10,11 have calculated only C33 directly from first principles, and other elastic constants are obtained by using Martin’s trans- formation method.35 For BN and AlN, our results indicate TABLE IV. Elastic constants of WZ–BN, AlN, and GaN in units of GPa. The values in parentheses are calculated without internal strain correction. C11 C12 C13 C33 C44 C66 B BN Present 982 134 74 1077 388 424 401 ~998! ~127! ~73! ~1076! ~397! ~436! ~402! Calc.a 987 143 70 1020 369 422 395 AlN Present 398 142 112 383 127 128 212 ~474! ~106! ~68! ~478! ~147! ~184! ~212! Expt.b 345 125 120 395 118 110 fl Expt.c 411 149 99 389 125 131 fl Calc.a 398 140 127 382 96 129 218 Calc.d 380 114 127 382 109 133 207 Calc.e 396 137 108 373 116 fl 207 GaN Present 350 140 104 376 101 115 197 ~420! ~110! ~66! ~448! ~119! ~156! ~197! Expt.f 365 135 114 381 109 115 fl Expt.g 377 160 114 209 81.4 109 fl Expt.h 390 145 106 398 105 123 fl Calc.a 396 144 100 392 91 126 207 Calc.e 367 135 103 405 95 fl 202 aFP-LMTO LDA by Kim et al., Ref. 10. bTsubouchi and Mikoshiba, Ref. 15. cMcNeil et al., Ref. 75. dPseudopotentail LDA by Kato and Hama, Ref. 12. ePseudopotential LDA by Wright, Ref. 9. fBrillouin scattering by Yamaguchi et al., Ref. 79. gResonance ultrasound by Schwarz et al., Ref. 80. hBrillouin scattering by Polian et al., Ref. 81. TABLE V. Deformation potential constants b and d of ZB–BN, AlN, and GaN in units of eV. The Bir–Pikus notation is used. The values in paren- theses are calculated without internal strain correction. BN AlN GaN Present Calc.a Present Calc.a Present Calc.a b 23.41 23.4 21.44 21.4 21.62 21.6 d 23.75 23.3 25.02 25.3 24.03 23.7 (22.19) (22.3) (23.04) (23.3) (22.12) (22.4) 4953Shimada, Sota, and Suzuki aFP-LMTO LDA by Kim et al., Ref. 10. license or copyright; see http://jap.aip.org/about/rights_and_permissions good agreement w tions. For GaN, o percent compared C. Deformation The values o obtained in a wa constants. We ha pute the values o spin orbit interact dilation along the DE52b~e32 Under a dilation DE5 ) de4 . ~3! lN A 6 fl 2,j , R f WZ–BN, AlN, and The values in paren- . D4 D5 23.32 24.30 (22.04) (23.01) 23.92 23.36 (21.94) (22.49) (22.04) (22.59) 24.8 fl 24.10 fl fl fl 23.25 22.85 (21.61) (22.04) 24.41 fl 23.4 23.3 23.6 fl Expt. 22 fl fl fl fl fl Expt.i 11 fl fl fl fl fl 4954 J. Appl. P , Sota, and Suzuki 6 Here b and d are the deformation potential constants of ZB crystals. For WZ structure, assuming that D25D350 and e45e55e650, a 636 strain Hamiltonian leads to three doubly degenerated states: E15D1e31D2~e11e2!, ~4! E2,35Dcr1~D11D3!e31~D21D4!~e11e2! 6D5~e12e2!. ~5! Here D2 and D3 denote the spin orbit splittings, Dcr is the crystal field splitting, and D1 – D5 are the deformation poten- tial constants of WZ crystals. From these equations, energy shifts under a strain are as follows: E1!E11D1e31D2~e11e2!, ~6! Dcr!Dcr1D3e31D4~e11e2!. ~7! From Eqs. ~5!, ~6!, and ~7!, D1 – D5 are obtained directly under uniaxial strain. The results are listed in Tables V and VI. Experimental data are only available for WZ–GaN to our knowledge. The values in parentheses are obtained without internal strain correction. It can be found that the internal strain plays a TABLE VII. Optical phonon frequencies of ZB–BN, A of cm21. BN Present 1082 Experiment 1056a,b Other calculations 1000,f 1070g,h 648,i 65 aReference 64. bRaman data by Alvarenga et al., Ref. 65. cRaman data by Murugkar et al., Ref. 86. dRaman data by Tabata et al., Ref. 87. eRaman data by Siegle et al., Ref. 88. fPseudopotential LDA by Lam et al., Ref. 51. gFP-LMTO LDA by Kim et al., Ref. 10. hPseudopotential LDA by Rodriguez-Herna´ndez et al. iMixed basis LDA by Miwa and Fikumoto, Ref. 37. jFP-LMTO LDA by Gorczyca et al., Ref. 38. kFP-LMTO LDA by Fiorentini et al., Ref. 89. lPseudopotential LDA by Karch et al., Ref. 90. mPseudopotential LDA by Wright, Ref. 9. Downloaded 21 Nov 2011 to 221.212.116.13. Redistribution subject to AIP significant role in calculating the deformation potential con- stants. Especially for the ZB shear deformation potential constant d , the WZ deformation potential constants D3 , and D4 , the values are nearly twice as large as the values calcu- lated without internal