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Vegard's law deviation in lattice constant and band gap bowing parameter of zincblende InxGa1−xN
Optics Communications 237 (2004) 363–369 www.elsevier.com/locate/optcom
Vegard’s law deviation in lattice constant and band gap bowing parameter of zincblende InxGa1xN Yen-Kuang Kuo a
a,*
, Bo-Ting Liou b, Sheng-Horng Yen a, Han-Yi Chu
a
Department of Physics, National Changhua University of Education, 1 Jin-Der Road, Changhua 50058, Taiwan b Department of Mechanical Engineering, Hsiuping Institute of Technology, Taichung 41283, Taiwan Received 5 January 2004; received in revised form 2 April 2004; accepted 7 April 2004
Abstract Numerical simulation based on first-principles calculations is applied to study the structural characteristics and band-energy properties of the zincblende Inx Ga1x N. The deviation parameter of the lattice constant is determined to A band gap bowing parameter of 1.890 0.097 eV is obtained with the equilibrium lattice constant and be )0.004 A. 1.857 0.093 eV is obtained with the lattice constant derived from Vegard’s law. Ó 2004 Elsevier B.V. All rights reserved. PACS: 71.15.)m; 71.20.Nr; 81.05.Ea Keywords: Energy band structure; Numerical simulation; III–V semiconductors
1. Introduction The InGaN is important for the development of optoelectronic devices with quantum well active layers, since it is capable of being tuned from the near infrared across the ultraviolet spectral range. However, the physics related to the InGaN and InN materials were not well developed because of difficulties in growing high-quality crystal layers. Nevertheless, epitaxial technology has since progressed remarkably. The high-quality wurtzite InN
*
Corresponding author. Tel.: +886-4-724-8812; fax: +886-4721-1153. E-mail address: [email protected] (Y.-K. Kuo).
has recently become available and its direct band gap energy has been determined experimentally to be near 0.8 eV [1–4], which is much smaller than the 1.9 eV commonly accepted by the physics community in the past few years. On the other hand, the direct band gap energy of the zincblende InN has also been determined numerically to be near 0.8 eV [5,6]. In this paper, the structural characteristics and band-energy properties of the zincblende Inx Ga1x N are studied numerically based on first-principles calculations. Although most III-nitride semiconductors have been grown with the wurtzite structure, the zincblende structure still possesses distinct advantages over the wurtzite structure. For example, the zincblende structure is provided with larger optical
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.04.012
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Y.-K. Kuo et al. / Optics Communications 237 (2004) 363–369
gain and lower threshold current density because of its smaller effective mass, and has mirror facets compatible with substrates such as the GaAs [7]. Emitting wavelength is dependent on the band gap energy of the semiconductor, and the band gap bowing parameter, b, is important for calculating the band gap energy of the Inx Ga1x N material. Nevertheless, the band gap bowing parameter of the Inx Ga1x N is still under extensive research because high-quality Inx Ga1x N semiconductors with a high indium content were not available until recently. Several researchers are still devoted to obtaining the band gap bowing parameter of the Inx Ga1x N by experimental methods and numerical calculations. However, the results obtained by different researchers are quite diverging. For example, the band gap bowing parameter is 1.4 eV when the indium composition is between 0 and 0.2 [8], 1.5 [4] and 1.6 eV [9] when the indium composition is between 0 and 0.25, 1.02 eV when the indium composition is between 0 and 0.5 [10], 1.26 eV when the indium composition is between 0 and 0.75 [9], and 1.3 eV when the indium composition is between 0 and 1 [11]. Since the band gap energy of InN was confirmed to be near 0.8 eV, the In-rich Inx Ga1x N alloys have attracted extensive attention recently. Assuming that the lattice constants of the ternary compounds can be expressed as a linear combination of the lattice constants of the two forming binary compounds, the physical properties of the ternary compounds are usually investigated based on Vegard’s law. However, large Vegard’s law deviation in lattice constant and band gap bowing parameter is observed in some of the III-nitride alloys. For example, we have investigated the Vegard’s law deviation in lattice constant and band gap bowing parameter of the wurtzite Alx Ga1x N. The simulation results indicate that the lattice constants of the Alx Ga1x N obtained from the minimized equilibrium energy are larger than those obtained from Vegard’s law, which results in a noticeable difference between the band gap bowing parameter obtained with the lattice constants derived from Vegard’s law and that obtained with the lattice constants derived from the minimized equilibrium energy. Specifically, a band gap bowing parameter of 0.75 eV is
obtained with the equilibrium lattice constants and 0.35 eV is obtained with the lattice constants derived from Vegard’s law. The band gap bowing parameters of the wurtzite Alx Ga1x N obtained by other researchers are quite diverging [12–22]. In our study, the band gap bowing parameter obtained with the equilibrium lattice constants is in close agreement with that obtained experimentally by Steude et al. [13], and the band gap bowing parameter obtained with the lattice constants derived from Vegard’s law is in close agreement with that obtained numerically by Kuo and Lin [12]. In this work, by means of numerical simulation based on first-principles calculations, we first investigated the deviation of the zincblende Inx Ga1x N lattice constants from Vegard’s law. The bowing parameter obtained with the equilibrium lattice constants and that obtained with the lattice constants derived from Vegard’s law are also compared and discussed.
2. Method In this study, the physical properties of the InGa x 1x N alloys are investigated using the Cambridge Serial Total Energy Package (CASTEP) simulation program [23]. To model the Inx Ga1x N zincblende alloys, we apply a 16-atom Inn Ga8n N8 supercell, which corresponds to a 2 1 1 supercell. In this simulation program, the calculation is performed using Kohn–Sham formation [24], which is based on the density functional theory (DFT). Local density approximation (LDA) is made for electronic exchange-correlation potential energy. The Coulomb potential energy caused by electron–ion interaction is described using ultrasoft scheme [25], in which the orbitals of In (4d10 5s2 5p1 ), Ga (3d10 4s2 4p1 ), and N (2s2 2p3 ) are treated as valence electrons. By the norm-conservation condition, the pseudo-wave function related to pseudopotential matches the plane-wave function expanded with Kohn–Sham formation beyond a cutoff energy. Using a high cutoff energy at the price of spending long computational time can make accurate results. For each ternary Inn Ga8n N8 supercell under study, once the ratio between the In and Ga atoms
Y.-K. Kuo et al. / Optics Communications 237 (2004) 363–369
is specified, the geometrical arrangement of In–Ga atoms is determined by the CASTEP simulation program. Geometry optimization is performed for the Inn Ga8n N8 supercell with symmetry P1. Atomic positions are relaxed and optimized with a density mixing scheme [26] using the conjugategradient (CG) method [27] for eigenvalues minimization. The iteration is repeated until the energy is less than 0.002 meV/atom and the root-meansquare (RMS) stress is less than 0.1 GPa. The calculation of total energy and charge density using DFT requires several integrals over the Brillouin zone. The integrals are approximated by numerical summation over a finite number of k points. The Monkhorst–Pack scheme [28] with uniform mesh points is applied in this work.
3. Results and discussion Bloch’s theorem states that the electronic planewave functions at each k point can be expanded in terms of a discrete plane-wave basis set. In principle, an infinite plane-wave basis set is required to expand the electronic wavefunctions. However, coefficients for plane waves with small kinetic energy are typically more important than those with large kinetic energy. Consequently, plane-wave basis sets can be truncated to include only plane waves that have kinetic energies less than some specified cutoff energy. To illustrate the reliability of the simulated results, the convergence behavior of the width of top valence band (arising from N 2p orbitals) at the C point for the zincblende InN and GaN is tested with various cutoff energies. The results, summarized in Table 1, indicate that the widths of top valence bands of the InN and GaN do not monotonically approach a certain value as the cutoff energy in-
365
creases; instead, they oscillate. A lower cutoff energy of 200 eV for the InN leads to an inadequate representation of the plane-wave function, which is reflected in a more non-monotonic behavior. The amplitude of oscillation is small when the cutoff energy is large. From Table 1 it is evident that, for the InN, the amplitude of fluctuation for the width of top valence band at the C point is within 0.002 eV when the cutoff energy ranges from 300 to 400 eV. On the other hand, for the GaN, the amplitude of fluctuation for the width of top valence band at the C point is also within 0.002 eV when the cutoff energy ranges from 300 to 350 eV. Many different schemes can be applied to study the energy band structures of semiconductors, for example, the BHS scheme [29], the Troullier–Martins scheme [30], the Lin scheme [31], which is modified from the RRKJ scheme [32] and has better convergence than the Troullier–Martins scheme, and the ultrasoft scheme, which is adopted in the present study. Different cutoff energies are usually used in different schemes. As a typical example, in a similar study, the band structures of the GaN calculations require a cutoff energy of 3200 eV (240 Ry) for the BHS scheme [33], 810 eV (60 Ry) for the Troullier–Martins scheme [34], and only 218 eV (16.2 Ry) with satisfactory results for the ultrasoft scheme [35]. Therefore, in the ternary compound Inx Ga1x N in this study, it is appropriate to use a cutoff energy of 300 eV, which corresponds to the grid mesh of 48 24 24. Note that a mesh of 2–8 k points is taken in the irreducible Brillouin zone for the convergence test. The first five decimals for the width of top valence band of InN were identical for all tested k points. Therefore, the expensive five k points in the irreducible Brillouin zone were used throughout this work. The LDA has been widely adopted to calculate the structural and electronic properties of III–V
Table 1 Width of top valence band at the C point for the zincblende InN and GaN with varying cutoff energies Cutoff energy (eV), InN
150
200
250
300
350
400
Width of top valence band (eV)
9.197
5.658
5.165
5.188
5.190
5.189
Cutoff energy (eV), GaN
100
150
200
250
300
350
Width of top valence band (eV)
6.093
6.935
6.375
6.338
6.324
6.326
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semiconductors. Many methods are proposed to improve the accuracy of the simulated results; for example, the plane-wave pseudopotential (PWPP) method, the full-potential linear muffin-tin orbital (FP-LMTO) method, and the full-potential linear augmented plane wave (FP-LAPW) method. The PWPP method in which the Ga 3d, In 4d, and N 2p electrons are treated explicitly is used in this study. The lattice constants of the InN and GaN obtained by our simulation and other researchers [35–41] are shown in Table 2. The results obtained in this work are in close agreement with those obtained experimentally and are better than results obtained using other methods. The element composition of the ternary InGaN compound plays an important role not only in the physical properties, but also in the epitaxial growth process. Assuming that Vegard’s law is valid, the lattice constants of the Inx Ga1x N are generally expressed as a linear relation to the indium composition x. In this study, the validity of Vegard’s law for the ternary InGaN compound with a zincblende structure in the ground state at 273 K is investigated. The numerical simulation is performed for the situation when the indium compositions are 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, and 1. The simulated results, listed in Table 3, show that the lattice constant increases with an increase in indium composition. This phenomenon occurs because the atom of indium is larger than that of gallium. To further analyze the degree of deviation from Vegard’s law, the lattice constant of the Inx Ga1x N as a function of the indium composition x can be approximated using the following formula: aðxÞ ¼ x aInN þ ð1 xÞ aGaN da x ð1 xÞ;
ð1Þ
Table 2 Lattice constants of the InN and GaN obtained by our simulation and other researchers Method
InN, a (A)
GaN, a (A)
This work FP-LMTO [35] PWPP [36] PWPP [37] FP-LAPW [38] FP-LAPW [39] Experiment
4.974 4.92 5.004 4.964 5.03 4.94 4.986 [40]
4.469 4.46 4.518 4.446 4.552 4.46 4.52 [41]
Table 3 Lattice constants of the zincblende Inx Ga1–x N Material
a (A)
InN In0:875 Ga0:125 N In0:75 Ga0:25 N In0:625 Ga0:375 N In0:50 Ga0:50 N In0:375 Ga0:625 N In0:25 Ga0:75 N In0:125 Ga0:875 N GaN
4.974 4.906 4.854 4.788 4.717 4.660 4.597 4.533 4.469
where aðxÞ is the lattice constant of the Inx Ga1x N, aInN and aGaN are the lattice constants of the InN and GaN, respectively, and da is the deviation parameter for the lattice constant. If we best fit the results shown in Table 3 with Eq. (1), the result is obtained. We have done similar da ¼ 0:004 A study for the wurtzite AlGaN. The simulation results indicate that, for the wurtzite AlGaN, the for a lattice condeviation parameter is )0.040 A for c lattice constant. Hence, stant and )0.125 A compared to the wurtzite AlGaN, the lattice constant of the zincblende InGaN has small deviation from Vegard’s law. The energy band structure of the Inx Ga1x N is next investigated. Fig. 1 shows a typical example of the calculated energy band structure of the In0:5 Ga0:5 N. Underestimation of the band gap energy but accurate estimation of the valence band is a well-known consequence of using LDA pseudopotential calculations. To amend these band gap energies, the results are calculated using scissors operation with a rigid upward shift of the conduction band with respect to the valence band from known band gap energies of the InN (0.78 eV [6]) and GaN (3.3 eV). The difference between the corrected and LDA energies of the ternary compounds is obtained from the linear combination of the differences of the InN and GaN. The band gap energy at the C point shown in Fig. 1 is 1.60 eV. As indicated in Fig. 1, the valence bands consist of two major sub-bands. The top valence band between 0 and )6.85 eV originates predominantly from the N 2p orbitals. The deep band between )10.59 and )13.10 eV originates predominantly from the N 2s orbitals. The highly overlapped band structures around )12.90 eV are compounds
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Fig. 1. Energy band diagram of the zincblende In0:5 Ga0:5 N.
of the Ga 3d and In 4d orbitals. It is obvious that there is considerable hybridization among the Ga 3d, In 4d, and N 2s states. Using the lattice constants of the Inx Ga1x N obtained from the minimized equilibrium energy (labeled ‘‘Non-linear’’) and those obtained from Vegard’s law (labeled ‘‘Linear’’), Table 4 shows the widths of top valence band and band gap energies at the C point for the Inx Ga1x N with different indium compositions. It is obvious that the width of top valence band at the C point of the Inx Ga1x N decreases with an increase in indium composition. The width of top valence band at the C point is 6.324 eV for the GaN and 5.188 eV for the InN, which are in close agreement with the results of 6.2–6.5 eV for the GaN and 5.1–5.3 eV for the InN obtained by other researchers [11].
Table 4 Widths of top valence band and band gap energies at the C point for the zincblende Inx Ga1x N with different indium compositions Material
InN In0:875 Ga0:125 N In0:75 Ga0:25 N In0:625 Ga0:375 N In0:50 Ga0:50 N In0:375 Ga0:625 N In0:25 Ga0:75 N In0:125 Ga0:875 N GaN
Width of top valence band (eV)
Direct band gap energy (eV)
Linear
Non-linear
Linear
Non-linear
5.188 5.484 5.698 5.802 5.915 6.048 6.123 6.233 6.324
5.188 5.499 5.683 5.795 5.931 6.042 6.118 6.230 6.324
0.78 0.860 0.956 1.270 1.607 1.951 2.381 2.798 3.3
0.78 0.860 0.942 1.262 1.603 1.945 2.373 2.794 3.3
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minimized equilibrium energy has a negative de . For the zincblende viation parameter of )0.004 A Inx Ga1x N, a band gap bowing parameter of 1.890 0.097 eV is obtained with the equilibrium lattice constant, and 1.857 0.093 eV is obtained with the lattice constant derived from Vegard’s law.
Band gap energy (eV)
3.5 3.0 2.5 2.0 1.5 1.0 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Indium composition x Fig. 2. Band gap energies of the zincblende Inx Ga1x N obtained with the equilibrium lattice constants.
The band gap energies of the zincblende Inx Ga1x N obtained with the equilibrium lattice constants are plotted in Fig. 2 for further analysis. The band gap energy of the Inx Ga1x N can be depicted as a function of the indium composition x and expressed using the following formula [42]: Eg ðxÞ ¼ x Eg;InN þ ð1 xÞ Eg;GaN b x ð1 xÞ;
ð2Þ
where Eg ðxÞ is the band gap energy of the Inx Ga1x N, Eg;InN is the band gap energy of the InN, Eg;GaN is the band gap energy of the GaN, and b is the band gap bowing parameter of the Inx Ga1x N. If we best fit the results shown in Fig. 2 with Eq. (2), a band gap bowing parameter of 1.890 0.097 eV is obtained. Similarly, a band gap bowing parameter of 1.857 0.093 eV is obtained with the lattice constants derived from Vegard’s law. The band gap bowing parameters of the zincblende Inx Ga1x N obtained in this study are close to that obtained experimentally by Goldhahn et al. [8], and those obtained numerically by other researchers [5,9,11].
4. Conclusion Numerical simulation based on first-principles calculations is applied to study the structural characteristics and band-energy properties of the zincblende Inx Ga1x N. From the simulation results, it is found that the lattice constant of the Inx Ga1x N in the ground state obtained from the
Acknowledgements The authors acknowledge the National Center for High-Performance Computing of the Republic of China, Taiwan, for the use of CASTEP simulation program. This work is supported by the National Science Council of the Republic of China, Taiwan, under grant No. NSC-92-2112-M018-008.
References [1] J. Wu, W. Walukiewicz, K.M. Yu, J.W. Ager III, E.E. Haller, H. Lu, W.J. Schaff, Appl. Phys. Lett. 80 (2002) 4741. [2] J. Wu, W. Walukiewicz, K.M. Yu, J.W. Ager III, E.E. Haller, H. Lu, W.J. Schaff, Y. Saito, Y. Nanishi, Appl. Phys. Lett. 80 (2002) 3967. [3] T. Matsuoka, H. Okamoto, M. Nakao, H. Harima, E. Kurimoto, Appl. Phys. Lett. 81 (2002) 1246. [4] T. Miyajima, Y. Kudo, K.-L. Liu, T. Uruga, T. Honma, Y. Saito, M. Hori, Y. Nanishi, T. Kobayashi, S. Hirata, Phys. Status Solidi B 234 (2002) 801. [5] F. Bechstedt, J. Furthm€ uller, M. Ferhat, L.K. Teles, L.M.R. Scolfaro, J.R. Leite, V.Yu. Davydov, O. Ambacher, R. Goldhahn, Phys. Status Solidi A 195 (2003) 628. [6] I. Vurgaftman, J.R. Meyer, J. Appl. Phys. 94 (2003) 3675. [7] S.H. Park, S.L. Chuang, J. Appl. Phys. 87 (2000) 353. [8] R. Goldhahn, J. Scheiner, S. Shokhovets, T. Frey, U. K€ ohler, D.J. As, K. Lischka, Appl. Phys. Lett. 76 (2000) 291. [9] M. Ferhat, J. Furthm€ uller, F. Bechstedt, Appl. Phys. Lett. 80 (2002) 1394. [10] A.F. Wright, J.S. Nelson, Appl. Phys. Lett. 66 (1995) 3051. [11] M. van Schilfgaarde, A. Sher, A.-B. Chen, J. Cryst. Growth 178 (1997) 8. [12] Y.K. Kuo, W.W. Lin, Jpn. J. Appl. Phys. 41 (2002) 73. [13] G. Steude, B.K. Meyer, A. G€ oldner, A. Hoffmann, F. Bertram, J. Christen, H. Amano, I. Akasaki, Appl. Phys. Lett. 74 (1999) 2456. [14] S.R. Lee, A.F. Wright, M.H. Crawford, G.A. Petersen, J. Han, R.M. Biefeld, Appl. Phys. Lett. 74 (1999) 3344.
Y.-K. Kuo et al. / Optics Communications 237 (2004) 363–369 [15] Y. Koide, H. Itoh, M.R.H. Khan, K. Hiramatu, N. Sawaki, I. Akasaki, J. Appl. Phys. 61 (1987) 4540. [16] D. Brunner, H. Angerer, E. Bustarret, F. Freudenberg, R. H€ opler, R. Dimitrov, O. Ambacher, M. Stutzmann, J. Appl. Phys. 82 (1997) 5090. [17] S. Yoshida, S. Misawa, S. Gonda, J. Appl. Phys. 53 (1982) 6844. [18] D.K. Wickenden, C.B. Bargeron, W.A. Bryden, J. Miragliotta, T.J. Kistenmacher, Appl. Phys. Lett. 65 (1994) 2024. [19] T. Takeuchi, H. Takeuchi, S. Sota, H. Sakai, H. Amano, I. Akasaki, Jpn. J. Appl. Phys. 36 (1997) L177. [20] L.-C. Duda, C.B. Stagarescu, J. Downes, K.E. Smith, D. Korakakis, T.D. Moustakas, J. Guo, J. Nordgren, Phys. Rev. B 58 (1998) 1928. [21] S.K. Pugh, D.J. Dugdale, S. Brand, R.A. Abram, J. Appl. Phys. 86 (1999) 3768. [22] T.J. Ochalski, B. Gil, P. Lefebvre, N. Grandjean, M. Leroux, J. Massies, S. Nakamura, H. Morkoc, Appl. Phys. Lett. 74 (1999) 3353. [23] CASTEP 4.6 program developed by Accelrys Inc., 2001. [24] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [25] J. Furthm€ uller, P. Kackell, F. Bechstedt, Phys. Rev. B 61 (2000) 4576. [26] G. Kresse, J. Furthm€ uller, Phys. Rev. B 54 (1996) 11169. [27] M.C. Payne, M.P. Teter, D.C. Allen, T.A. Arias, J.D. Joannopolous, Rev. Mod. Phys. 64 (1992) 1045. [28] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188.
369
[29] G.B. Bachelet, D.R. Hamann, M. Schluter, Phys. Rev. B 26 (1982) 4199. [30] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [31] J.S. Lin, A. Qteish, M.C. Payne, V. Heine, Phys. Rev. B 47 (1993) 4174. [32] A.M. Rappe, K.M. Rabe, E. Kaxiras, J.K. Joannopoulos, Phys. Rev. B 41 (1990) 1227. [33] A.F. Wright, J.S. Nelson, Phys. Rev. B 50 (1994) 2159. [34] J. Neugebauer, C.G. Van de Walle, Phys. Rev. B 50 (1994) 8067. [35] K. Kim, W.R.L. Lambrecht, B. Segall, Phys. Rev. B 53 (1996) 16310. [36] C. Stampfl, C.G. Van de Walle, Phys. Rev. B 59 (1999) 5521. [37] A. Zoroddu, F. Bernardini, P. Ruggerone, V. Fiorentini, Phys. Rev. B 64 (2001) 045208. [38] L.E. Ramos, L.K. Teles, L.M.R. Scolfaro, J.L.P. Castineira, A.L. Rosa, J.R. Leite, Phys. Rev. B 63 (2001) 165210. [39] M. Fuchs, J.L.F. Da Silva, C. Stampfl, J. Neugebauer, M. Scheffler, Phys. Rev. B 65 (2002) 165210. [40] V. Cimalla, J. Pezoldt, G. Ecke, R. Kosiba, O. Ambacher, L. Spieß, G. Teichert, H. Lu, W.J. Schaff, Appl. Phys. Lett. 83 (2003) 3468. [41] T. Lei, T.D. Moustakas, R.J. Graham, Y. He, S.J. Berkowitz, J. Appl. Phys. 71 (1992) 4933. [42] K. Osamura, S. Naka, Y Murakami, J. Appl. Phys. 46 (1975) 3432.
Vegard’s law deviation in lattice constant and band gap bowing parameter of zincblende InxGa1xN Yen-Kuang Kuo a
a,*
, Bo-Ting Liou b, Sheng-Horng Yen a, Han-Yi Chu
a
Department of Physics, National Changhua University of Education, 1 Jin-Der Road, Changhua 50058, Taiwan b Department of Mechanical Engineering, Hsiuping Institute of Technology, Taichung 41283, Taiwan Received 5 January 2004; received in revised form 2 April 2004; accepted 7 April 2004
Abstract Numerical simulation based on first-principles calculations is applied to study the structural characteristics and band-energy properties of the zincblende Inx Ga1x N. The deviation parameter of the lattice constant is determined to A band gap bowing parameter of 1.890 0.097 eV is obtained with the equilibrium lattice constant and be )0.004 A. 1.857 0.093 eV is obtained with the lattice constant derived from Vegard’s law. Ó 2004 Elsevier B.V. All rights reserved. PACS: 71.15.)m; 71.20.Nr; 81.05.Ea Keywords: Energy band structure; Numerical simulation; III–V semiconductors
1. Introduction The InGaN is important for the development of optoelectronic devices with quantum well active layers, since it is capable of being tuned from the near infrared across the ultraviolet spectral range. However, the physics related to the InGaN and InN materials were not well developed because of difficulties in growing high-quality crystal layers. Nevertheless, epitaxial technology has since progressed remarkably. The high-quality wurtzite InN
*
Corresponding author. Tel.: +886-4-724-8812; fax: +886-4721-1153. E-mail address: [email protected] (Y.-K. Kuo).
has recently become available and its direct band gap energy has been determined experimentally to be near 0.8 eV [1–4], which is much smaller than the 1.9 eV commonly accepted by the physics community in the past few years. On the other hand, the direct band gap energy of the zincblende InN has also been determined numerically to be near 0.8 eV [5,6]. In this paper, the structural characteristics and band-energy properties of the zincblende Inx Ga1x N are studied numerically based on first-principles calculations. Although most III-nitride semiconductors have been grown with the wurtzite structure, the zincblende structure still possesses distinct advantages over the wurtzite structure. For example, the zincblende structure is provided with larger optical
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.04.012
364
Y.-K. Kuo et al. / Optics Communications 237 (2004) 363–369
gain and lower threshold current density because of its smaller effective mass, and has mirror facets compatible with substrates such as the GaAs [7]. Emitting wavelength is dependent on the band gap energy of the semiconductor, and the band gap bowing parameter, b, is important for calculating the band gap energy of the Inx Ga1x N material. Nevertheless, the band gap bowing parameter of the Inx Ga1x N is still under extensive research because high-quality Inx Ga1x N semiconductors with a high indium content were not available until recently. Several researchers are still devoted to obtaining the band gap bowing parameter of the Inx Ga1x N by experimental methods and numerical calculations. However, the results obtained by different researchers are quite diverging. For example, the band gap bowing parameter is 1.4 eV when the indium composition is between 0 and 0.2 [8], 1.5 [4] and 1.6 eV [9] when the indium composition is between 0 and 0.25, 1.02 eV when the indium composition is between 0 and 0.5 [10], 1.26 eV when the indium composition is between 0 and 0.75 [9], and 1.3 eV when the indium composition is between 0 and 1 [11]. Since the band gap energy of InN was confirmed to be near 0.8 eV, the In-rich Inx Ga1x N alloys have attracted extensive attention recently. Assuming that the lattice constants of the ternary compounds can be expressed as a linear combination of the lattice constants of the two forming binary compounds, the physical properties of the ternary compounds are usually investigated based on Vegard’s law. However, large Vegard’s law deviation in lattice constant and band gap bowing parameter is observed in some of the III-nitride alloys. For example, we have investigated the Vegard’s law deviation in lattice constant and band gap bowing parameter of the wurtzite Alx Ga1x N. The simulation results indicate that the lattice constants of the Alx Ga1x N obtained from the minimized equilibrium energy are larger than those obtained from Vegard’s law, which results in a noticeable difference between the band gap bowing parameter obtained with the lattice constants derived from Vegard’s law and that obtained with the lattice constants derived from the minimized equilibrium energy. Specifically, a band gap bowing parameter of 0.75 eV is
obtained with the equilibrium lattice constants and 0.35 eV is obtained with the lattice constants derived from Vegard’s law. The band gap bowing parameters of the wurtzite Alx Ga1x N obtained by other researchers are quite diverging [12–22]. In our study, the band gap bowing parameter obtained with the equilibrium lattice constants is in close agreement with that obtained experimentally by Steude et al. [13], and the band gap bowing parameter obtained with the lattice constants derived from Vegard’s law is in close agreement with that obtained numerically by Kuo and Lin [12]. In this work, by means of numerical simulation based on first-principles calculations, we first investigated the deviation of the zincblende Inx Ga1x N lattice constants from Vegard’s law. The bowing parameter obtained with the equilibrium lattice constants and that obtained with the lattice constants derived from Vegard’s law are also compared and discussed.
2. Method In this study, the physical properties of the InGa x 1x N alloys are investigated using the Cambridge Serial Total Energy Package (CASTEP) simulation program [23]. To model the Inx Ga1x N zincblende alloys, we apply a 16-atom Inn Ga8n N8 supercell, which corresponds to a 2 1 1 supercell. In this simulation program, the calculation is performed using Kohn–Sham formation [24], which is based on the density functional theory (DFT). Local density approximation (LDA) is made for electronic exchange-correlation potential energy. The Coulomb potential energy caused by electron–ion interaction is described using ultrasoft scheme [25], in which the orbitals of In (4d10 5s2 5p1 ), Ga (3d10 4s2 4p1 ), and N (2s2 2p3 ) are treated as valence electrons. By the norm-conservation condition, the pseudo-wave function related to pseudopotential matches the plane-wave function expanded with Kohn–Sham formation beyond a cutoff energy. Using a high cutoff energy at the price of spending long computational time can make accurate results. For each ternary Inn Ga8n N8 supercell under study, once the ratio between the In and Ga atoms
Y.-K. Kuo et al. / Optics Communications 237 (2004) 363–369
is specified, the geometrical arrangement of In–Ga atoms is determined by the CASTEP simulation program. Geometry optimization is performed for the Inn Ga8n N8 supercell with symmetry P1. Atomic positions are relaxed and optimized with a density mixing scheme [26] using the conjugategradient (CG) method [27] for eigenvalues minimization. The iteration is repeated until the energy is less than 0.002 meV/atom and the root-meansquare (RMS) stress is less than 0.1 GPa. The calculation of total energy and charge density using DFT requires several integrals over the Brillouin zone. The integrals are approximated by numerical summation over a finite number of k points. The Monkhorst–Pack scheme [28] with uniform mesh points is applied in this work.
3. Results and discussion Bloch’s theorem states that the electronic planewave functions at each k point can be expanded in terms of a discrete plane-wave basis set. In principle, an infinite plane-wave basis set is required to expand the electronic wavefunctions. However, coefficients for plane waves with small kinetic energy are typically more important than those with large kinetic energy. Consequently, plane-wave basis sets can be truncated to include only plane waves that have kinetic energies less than some specified cutoff energy. To illustrate the reliability of the simulated results, the convergence behavior of the width of top valence band (arising from N 2p orbitals) at the C point for the zincblende InN and GaN is tested with various cutoff energies. The results, summarized in Table 1, indicate that the widths of top valence bands of the InN and GaN do not monotonically approach a certain value as the cutoff energy in-
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creases; instead, they oscillate. A lower cutoff energy of 200 eV for the InN leads to an inadequate representation of the plane-wave function, which is reflected in a more non-monotonic behavior. The amplitude of oscillation is small when the cutoff energy is large. From Table 1 it is evident that, for the InN, the amplitude of fluctuation for the width of top valence band at the C point is within 0.002 eV when the cutoff energy ranges from 300 to 400 eV. On the other hand, for the GaN, the amplitude of fluctuation for the width of top valence band at the C point is also within 0.002 eV when the cutoff energy ranges from 300 to 350 eV. Many different schemes can be applied to study the energy band structures of semiconductors, for example, the BHS scheme [29], the Troullier–Martins scheme [30], the Lin scheme [31], which is modified from the RRKJ scheme [32] and has better convergence than the Troullier–Martins scheme, and the ultrasoft scheme, which is adopted in the present study. Different cutoff energies are usually used in different schemes. As a typical example, in a similar study, the band structures of the GaN calculations require a cutoff energy of 3200 eV (240 Ry) for the BHS scheme [33], 810 eV (60 Ry) for the Troullier–Martins scheme [34], and only 218 eV (16.2 Ry) with satisfactory results for the ultrasoft scheme [35]. Therefore, in the ternary compound Inx Ga1x N in this study, it is appropriate to use a cutoff energy of 300 eV, which corresponds to the grid mesh of 48 24 24. Note that a mesh of 2–8 k points is taken in the irreducible Brillouin zone for the convergence test. The first five decimals for the width of top valence band of InN were identical for all tested k points. Therefore, the expensive five k points in the irreducible Brillouin zone were used throughout this work. The LDA has been widely adopted to calculate the structural and electronic properties of III–V
Table 1 Width of top valence band at the C point for the zincblende InN and GaN with varying cutoff energies Cutoff energy (eV), InN
150
200
250
300
350
400
Width of top valence band (eV)
9.197
5.658
5.165
5.188
5.190
5.189
Cutoff energy (eV), GaN
100
150
200
250
300
350
Width of top valence band (eV)
6.093
6.935
6.375
6.338
6.324
6.326
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semiconductors. Many methods are proposed to improve the accuracy of the simulated results; for example, the plane-wave pseudopotential (PWPP) method, the full-potential linear muffin-tin orbital (FP-LMTO) method, and the full-potential linear augmented plane wave (FP-LAPW) method. The PWPP method in which the Ga 3d, In 4d, and N 2p electrons are treated explicitly is used in this study. The lattice constants of the InN and GaN obtained by our simulation and other researchers [35–41] are shown in Table 2. The results obtained in this work are in close agreement with those obtained experimentally and are better than results obtained using other methods. The element composition of the ternary InGaN compound plays an important role not only in the physical properties, but also in the epitaxial growth process. Assuming that Vegard’s law is valid, the lattice constants of the Inx Ga1x N are generally expressed as a linear relation to the indium composition x. In this study, the validity of Vegard’s law for the ternary InGaN compound with a zincblende structure in the ground state at 273 K is investigated. The numerical simulation is performed for the situation when the indium compositions are 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, and 1. The simulated results, listed in Table 3, show that the lattice constant increases with an increase in indium composition. This phenomenon occurs because the atom of indium is larger than that of gallium. To further analyze the degree of deviation from Vegard’s law, the lattice constant of the Inx Ga1x N as a function of the indium composition x can be approximated using the following formula: aðxÞ ¼ x aInN þ ð1 xÞ aGaN da x ð1 xÞ;
ð1Þ
Table 2 Lattice constants of the InN and GaN obtained by our simulation and other researchers Method
InN, a (A)
GaN, a (A)
This work FP-LMTO [35] PWPP [36] PWPP [37] FP-LAPW [38] FP-LAPW [39] Experiment
4.974 4.92 5.004 4.964 5.03 4.94 4.986 [40]
4.469 4.46 4.518 4.446 4.552 4.46 4.52 [41]
Table 3 Lattice constants of the zincblende Inx Ga1–x N Material
a (A)
InN In0:875 Ga0:125 N In0:75 Ga0:25 N In0:625 Ga0:375 N In0:50 Ga0:50 N In0:375 Ga0:625 N In0:25 Ga0:75 N In0:125 Ga0:875 N GaN
4.974 4.906 4.854 4.788 4.717 4.660 4.597 4.533 4.469
where aðxÞ is the lattice constant of the Inx Ga1x N, aInN and aGaN are the lattice constants of the InN and GaN, respectively, and da is the deviation parameter for the lattice constant. If we best fit the results shown in Table 3 with Eq. (1), the result is obtained. We have done similar da ¼ 0:004 A study for the wurtzite AlGaN. The simulation results indicate that, for the wurtzite AlGaN, the for a lattice condeviation parameter is )0.040 A for c lattice constant. Hence, stant and )0.125 A compared to the wurtzite AlGaN, the lattice constant of the zincblende InGaN has small deviation from Vegard’s law. The energy band structure of the Inx Ga1x N is next investigated. Fig. 1 shows a typical example of the calculated energy band structure of the In0:5 Ga0:5 N. Underestimation of the band gap energy but accurate estimation of the valence band is a well-known consequence of using LDA pseudopotential calculations. To amend these band gap energies, the results are calculated using scissors operation with a rigid upward shift of the conduction band with respect to the valence band from known band gap energies of the InN (0.78 eV [6]) and GaN (3.3 eV). The difference between the corrected and LDA energies of the ternary compounds is obtained from the linear combination of the differences of the InN and GaN. The band gap energy at the C point shown in Fig. 1 is 1.60 eV. As indicated in Fig. 1, the valence bands consist of two major sub-bands. The top valence band between 0 and )6.85 eV originates predominantly from the N 2p orbitals. The deep band between )10.59 and )13.10 eV originates predominantly from the N 2s orbitals. The highly overlapped band structures around )12.90 eV are compounds
Y.-K. Kuo et al. / Optics Communications 237 (2004) 363–369
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Fig. 1. Energy band diagram of the zincblende In0:5 Ga0:5 N.
of the Ga 3d and In 4d orbitals. It is obvious that there is considerable hybridization among the Ga 3d, In 4d, and N 2s states. Using the lattice constants of the Inx Ga1x N obtained from the minimized equilibrium energy (labeled ‘‘Non-linear’’) and those obtained from Vegard’s law (labeled ‘‘Linear’’), Table 4 shows the widths of top valence band and band gap energies at the C point for the Inx Ga1x N with different indium compositions. It is obvious that the width of top valence band at the C point of the Inx Ga1x N decreases with an increase in indium composition. The width of top valence band at the C point is 6.324 eV for the GaN and 5.188 eV for the InN, which are in close agreement with the results of 6.2–6.5 eV for the GaN and 5.1–5.3 eV for the InN obtained by other researchers [11].
Table 4 Widths of top valence band and band gap energies at the C point for the zincblende Inx Ga1x N with different indium compositions Material
InN In0:875 Ga0:125 N In0:75 Ga0:25 N In0:625 Ga0:375 N In0:50 Ga0:50 N In0:375 Ga0:625 N In0:25 Ga0:75 N In0:125 Ga0:875 N GaN
Width of top valence band (eV)
Direct band gap energy (eV)
Linear
Non-linear
Linear
Non-linear
5.188 5.484 5.698 5.802 5.915 6.048 6.123 6.233 6.324
5.188 5.499 5.683 5.795 5.931 6.042 6.118 6.230 6.324
0.78 0.860 0.956 1.270 1.607 1.951 2.381 2.798 3.3
0.78 0.860 0.942 1.262 1.603 1.945 2.373 2.794 3.3
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minimized equilibrium energy has a negative de . For the zincblende viation parameter of )0.004 A Inx Ga1x N, a band gap bowing parameter of 1.890 0.097 eV is obtained with the equilibrium lattice constant, and 1.857 0.093 eV is obtained with the lattice constant derived from Vegard’s law.
Band gap energy (eV)
3.5 3.0 2.5 2.0 1.5 1.0 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Indium composition x Fig. 2. Band gap energies of the zincblende Inx Ga1x N obtained with the equilibrium lattice constants.
The band gap energies of the zincblende Inx Ga1x N obtained with the equilibrium lattice constants are plotted in Fig. 2 for further analysis. The band gap energy of the Inx Ga1x N can be depicted as a function of the indium composition x and expressed using the following formula [42]: Eg ðxÞ ¼ x Eg;InN þ ð1 xÞ Eg;GaN b x ð1 xÞ;
ð2Þ
where Eg ðxÞ is the band gap energy of the Inx Ga1x N, Eg;InN is the band gap energy of the InN, Eg;GaN is the band gap energy of the GaN, and b is the band gap bowing parameter of the Inx Ga1x N. If we best fit the results shown in Fig. 2 with Eq. (2), a band gap bowing parameter of 1.890 0.097 eV is obtained. Similarly, a band gap bowing parameter of 1.857 0.093 eV is obtained with the lattice constants derived from Vegard’s law. The band gap bowing parameters of the zincblende Inx Ga1x N obtained in this study are close to that obtained experimentally by Goldhahn et al. [8], and those obtained numerically by other researchers [5,9,11].
4. Conclusion Numerical simulation based on first-principles calculations is applied to study the structural characteristics and band-energy properties of the zincblende Inx Ga1x N. From the simulation results, it is found that the lattice constant of the Inx Ga1x N in the ground state obtained from the
Acknowledgements The authors acknowledge the National Center for High-Performance Computing of the Republic of China, Taiwan, for the use of CASTEP simulation program. This work is supported by the National Science Council of the Republic of China, Taiwan, under grant No. NSC-92-2112-M018-008.
References [1] J. Wu, W. Walukiewicz, K.M. Yu, J.W. Ager III, E.E. Haller, H. Lu, W.J. Schaff, Appl. Phys. Lett. 80 (2002) 4741. [2] J. Wu, W. Walukiewicz, K.M. Yu, J.W. Ager III, E.E. Haller, H. Lu, W.J. Schaff, Y. Saito, Y. Nanishi, Appl. Phys. Lett. 80 (2002) 3967. [3] T. Matsuoka, H. Okamoto, M. Nakao, H. Harima, E. Kurimoto, Appl. Phys. Lett. 81 (2002) 1246. [4] T. Miyajima, Y. Kudo, K.-L. Liu, T. Uruga, T. Honma, Y. Saito, M. Hori, Y. Nanishi, T. Kobayashi, S. Hirata, Phys. Status Solidi B 234 (2002) 801. [5] F. Bechstedt, J. Furthm€ uller, M. Ferhat, L.K. Teles, L.M.R. Scolfaro, J.R. Leite, V.Yu. Davydov, O. Ambacher, R. Goldhahn, Phys. Status Solidi A 195 (2003) 628. [6] I. Vurgaftman, J.R. Meyer, J. Appl. Phys. 94 (2003) 3675. [7] S.H. Park, S.L. Chuang, J. Appl. Phys. 87 (2000) 353. [8] R. Goldhahn, J. Scheiner, S. Shokhovets, T. Frey, U. K€ ohler, D.J. As, K. Lischka, Appl. Phys. Lett. 76 (2000) 291. [9] M. Ferhat, J. Furthm€ uller, F. Bechstedt, Appl. Phys. Lett. 80 (2002) 1394. [10] A.F. Wright, J.S. Nelson, Appl. Phys. Lett. 66 (1995) 3051. [11] M. van Schilfgaarde, A. Sher, A.-B. Chen, J. Cryst. Growth 178 (1997) 8. [12] Y.K. Kuo, W.W. Lin, Jpn. J. Appl. Phys. 41 (2002) 73. [13] G. Steude, B.K. Meyer, A. G€ oldner, A. Hoffmann, F. Bertram, J. Christen, H. Amano, I. Akasaki, Appl. Phys. Lett. 74 (1999) 2456. [14] S.R. Lee, A.F. Wright, M.H. Crawford, G.A. Petersen, J. Han, R.M. Biefeld, Appl. Phys. Lett. 74 (1999) 3344.
Y.-K. Kuo et al. / Optics Communications 237 (2004) 363–369 [15] Y. Koide, H. Itoh, M.R.H. Khan, K. Hiramatu, N. Sawaki, I. Akasaki, J. Appl. Phys. 61 (1987) 4540. [16] D. Brunner, H. Angerer, E. Bustarret, F. Freudenberg, R. H€ opler, R. Dimitrov, O. Ambacher, M. Stutzmann, J. Appl. Phys. 82 (1997) 5090. [17] S. Yoshida, S. Misawa, S. Gonda, J. Appl. Phys. 53 (1982) 6844. [18] D.K. Wickenden, C.B. Bargeron, W.A. Bryden, J. Miragliotta, T.J. Kistenmacher, Appl. Phys. Lett. 65 (1994) 2024. [19] T. Takeuchi, H. Takeuchi, S. Sota, H. Sakai, H. Amano, I. Akasaki, Jpn. J. Appl. Phys. 36 (1997) L177. [20] L.-C. Duda, C.B. Stagarescu, J. Downes, K.E. Smith, D. Korakakis, T.D. Moustakas, J. Guo, J. Nordgren, Phys. Rev. B 58 (1998) 1928. [21] S.K. Pugh, D.J. Dugdale, S. Brand, R.A. Abram, J. Appl. Phys. 86 (1999) 3768. [22] T.J. Ochalski, B. Gil, P. Lefebvre, N. Grandjean, M. Leroux, J. Massies, S. Nakamura, H. Morkoc, Appl. Phys. Lett. 74 (1999) 3353. [23] CASTEP 4.6 program developed by Accelrys Inc., 2001. [24] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [25] J. Furthm€ uller, P. Kackell, F. Bechstedt, Phys. Rev. B 61 (2000) 4576. [26] G. Kresse, J. Furthm€ uller, Phys. Rev. B 54 (1996) 11169. [27] M.C. Payne, M.P. Teter, D.C. Allen, T.A. Arias, J.D. Joannopolous, Rev. Mod. Phys. 64 (1992) 1045. [28] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188.
369
[29] G.B. Bachelet, D.R. Hamann, M. Schluter, Phys. Rev. B 26 (1982) 4199. [30] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [31] J.S. Lin, A. Qteish, M.C. Payne, V. Heine, Phys. Rev. B 47 (1993) 4174. [32] A.M. Rappe, K.M. Rabe, E. Kaxiras, J.K. Joannopoulos, Phys. Rev. B 41 (1990) 1227. [33] A.F. Wright, J.S. Nelson, Phys. Rev. B 50 (1994) 2159. [34] J. Neugebauer, C.G. Van de Walle, Phys. Rev. B 50 (1994) 8067. [35] K. Kim, W.R.L. Lambrecht, B. Segall, Phys. Rev. B 53 (1996) 16310. [36] C. Stampfl, C.G. Van de Walle, Phys. Rev. B 59 (1999) 5521. [37] A. Zoroddu, F. Bernardini, P. Ruggerone, V. Fiorentini, Phys. Rev. B 64 (2001) 045208. [38] L.E. Ramos, L.K. Teles, L.M.R. Scolfaro, J.L.P. Castineira, A.L. Rosa, J.R. Leite, Phys. Rev. B 63 (2001) 165210. [39] M. Fuchs, J.L.F. Da Silva, C. Stampfl, J. Neugebauer, M. Scheffler, Phys. Rev. B 65 (2002) 165210. [40] V. Cimalla, J. Pezoldt, G. Ecke, R. Kosiba, O. Ambacher, L. Spieß, G. Teichert, H. Lu, W.J. Schaff, Appl. Phys. Lett. 83 (2003) 3468. [41] T. Lei, T.D. Moustakas, R.J. Graham, Y. He, S.J. Berkowitz, J. Appl. Phys. 71 (1992) 4933. [42] K. Osamura, S. Naka, Y Murakami, J. Appl. Phys. 46 (1975) 3432.