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Electronic properties and elastic constants of wurtzite, zinc-blende and rocksalt AlN
ARTICLE IN PRESS
Journal of Physics and Chemistry of Solids 67 (2006) 1888–1892 www.elsevier.com/locate/jpcs
Electronic properties and elastic constants of wurtzite, zinc-blende and rocksalt AlN S. Saiba, N. Bouarissab, a
Physics Department, Faculty of Science and Engineering, University of M‘sila, 28000 M’sila, Algeria Department of Physics, Faculty of Science, King Khalid University, Abha, P.O. Box 9004, Saudi Arabia
b
Received 25 January 2006; received in revised form 23 April 2006; accepted 8 May 2006
Abstract Electronic properties and elastic constants of AlN in the wurtzite, zinc-blende and rocksalt structures are investigated using an ab initio pseudopotential method based on the density-functional theory with both the local-density approximation and the generalized gradient approximation for the exchange-correlation functional. The numerically calculated results compare well with the existing experimental data. For elastic constants of rocksalt AlN our results are predictions. r 2006 Elsevier Ltd. All rights reserved. PACS: 71.15.Mb; 71.20.b; 71.20.Nr; 71.15.Dx Keywords: C. High-pressure; D. Electronic structure; D. Elastic properties
1. Introduction Group-III nitrides have received extensive attention in recent years due to their superior properties with respect to conventional III–V semiconductors (the nitrides have larger band gaps and stronger bonds). Because of their wide band gaps and strong bond strength, group-III nitrides can be used for violet, blue, and green light emitting devices and for high temperature transistors [1–5]. Among them, AlN has a high melting point, a high thermal conductivity, and a large bulk modulus [6]. It has also the largest band gap (6.2 eV) and thus, is the best material for constructing devices for the violet region [7]. In the bulk form AlN stabilizes in the wurtzite structure. However, it has been reported that AlN can be grown in zinc-blende phase [5,8]. The optical gap of the wurtzite phase is direct and that of the zinc-blende phase is indirect. This can be useful in constructing different kinds of quantum wells or superlattices [7]. The very basis for these important applications is the specific electronic properties
Corresponding author.
E-mail address: [email protected] (N. Bouarissa). 0022-3697/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2006.05.007
as well as the mechanical response that is related to the elastic constants. Experimental and computational research on the physical properties of group-III nitrides at high pressure has seen much activity in recent years. Theoretical studies are most valuable so as to help understand and control the materials and device properties. In the present work, we perform density-functional-theory calculation for AlN, using the pseudopotential plane-wave method, where we employed both the local density approximation (LDA) and the generalized gradient approximation (GGA) for the exchange-correlation functional. We report electronic properties, namely the band-gap pressure coefficients of the three main transitions and the band-gap volume deformation potentials as well as elastic constants for AlN in the wurtzite, zinc-blende, and rocksalt structures. The paper is organized as follows. After a brief introduction in Section 1, the theoretical framework within which all the calculations have been performed is outlined in Section 2. In Section 3, we present and discuss the results of our study regarding the electronic properties and elastic constants of the material of interest in different structures being considered here. We compare our results with available experimental data as well as with previous
ARTICLE IN PRESS S. Saib, N. Bouarissa / Journal of Physics and Chemistry of Solids 67 (2006) 1888–1892
theoretical published values. A conclusion of the present study is given in Section 4. 2. Computational method
1889
Table 1a Values of the parameters a and b obtained by fitting the dependence of E GG , L M H K EA A ; E L ; E M ; E H and E K band-gap energies of wurtzite AlN on pressure (p) to EðpÞ ¼ Eð0Þ þ ap þ bp2 Wurtzite
A first principles pseudopotential plane-wave method based on the density functional theory (DFT) [9] is performed employing the LDA with the Ceperley–Alder [10] form of the exchange-correlation energy as parameterized by Perdew and Zunger [11], and the GGA with the Perdew et al. [12] form of the exchange-correlation potential. The interaction of the valence electrons with the ionic cores was represented with a separable, normconserving non-local Troullier–Martins [13] pseudopotentials. An energy cutoff of 160 Ry was used. Reciprocal space integration was performed by k-point sampling with sets of special points obtained by using the standard special k-points technique of the Monkhorst and Pack (MP) [14]. In this paper, a 4 4 4, 8 8 4, and 6 6 6 MP meshes were used for the semi-conducting zincblende, wurtzite and rocksalt AlN, respectively. In zincblende AlN the Al and N atoms are at the positions: Al(0, 0, 0) and N(1/4, 1/4, 1/4). For wurtzite AlN, there are four atoms per hexagonal unit cell where the positions of the atoms Al and N are: Al(0, 0, 0), (1/3, 2/3, 1/2) and N(0, 0, u), (1/3, 2/3, u+1/2), where u is the dimensionless internal parameter that represents the distance between Al plane and its nearest neighbor N plane, in the unit of c. For rocksalt AlN the Al and N atom positions are Al(0, 0, 0) and N(1/2, 1/2, 1/2).
G-G
A-A
L-L
M-M
H-H
K-K
a (meV/GPa)
41.14a 43.42b 40c 44d
44.06a 46.79b
20.33a 21.52b
16.22a 18.15b
17.00a 17.93b
11.28a 12.06b
b (meV/GPa2)
0.33a 0.30b 0.32c
0.26a 0.32b
0.15a 0.17b
0.11a 0.19b
0.13a 0.14b
0.08a 0.11b
a
LDA calculation. GGA calculation. c Ref. [15]: Theoretical calculations using the band-structure scheme LMTO. d Ref. [16]: Theoretical calculations using the full-potential LMTO method. b
Table 1b Values of the parameters a and b obtained by fitting the dependence of E GG , X L EX G , E X and E L band-gap energies of zinc-blende AlN on pressure (p) to EðpÞ ¼ Eð0Þ þ ap þ bp2 Zinc-blende G-G
G-X
X-X
L-L
a (meV/GPa)
43.14a 45.44b 42c
2.16a 2.38b 1.7c 1.9d
7.44a 7.93b
43.95a 46.42b
b (meV/GPa2)
0.27a 0.33b 0.34c
0.34a 0.01b 0.03c
0.06a 0.07b
0.30a 0.33b
3. Results 3.1. Electronic properties Table 1 shows the LDA and GGA-obtained band gap pressure coefficients of the transitions that are related to the expression: EðpÞ ¼ Eð0Þ þ ap þ bp2 .
(1)
For comparison, available theoretical calculations reported by Christensen and Gorczyca using the band structure scheme ‘linear muffin-tin-orbitals (LMTO)’ [15] and those of Kim et al. [16] using the full-potential LMTO method are also included. Note that our GGA results regarding the linear pressure coefficient (a) are slightly larger than those of Ref. [15]. The LDA values of a for all structures being studied here are smaller than those from GGA. The situation seems to be different for the coefficient b, which has generally larger values reported in Ref. [15] as compared to our both LDA and GGA results. As compared to the results of Kim et al. [16], our GGA value for a is in good agreement with that reported in Ref. [16] for wurtzite G–G gap. However, their value of a for the zinc-blende G-X gap seems to be smaller than our obtained values from both LDA and GGA, although it is closer to our results than the value reported in Ref. [15].
a
LDA calculation. GGA calculation. c Ref. [15]: Theoretical calculations using the band-structure scheme LMTO. d Ref. [16]: Theoretical calculations using the full-potential LMTO method. b
Table 2 lists the LDA and GGA-numerically calculated band-gap deformation potentials of the transitions G-G, X-X, L-L and G-X for zinc-blende and rocksalt AlN and the transitions G-G, A-A, L-L, M-M, H-H and K-K for wurtzite AlN. For comparison, results reported in Refs. [15–17] are also presented. Note that the agreement between our results and those reported in Refs. [15–17] is reasonable except may be for aGX in the rocksalt V structure. The reasonable agreement between our results as derived from our scheme and those reported in Refs. [15–17] clearly shows that it is possible to establish, a relation between the first-principles theories and the parameters entering the more empirically based models.
ARTICLE IN PRESS S. Saib, N. Bouarissa / Journal of Physics and Chemistry of Solids 67 (2006) 1888–1892
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Table 1c Values of the parameters a and b obtained by fitting the dependence of E GG , X L EX G , E X and E L band-gap energies of rocksalt AlN on pressure (p) to EðpÞ ¼ Eð0Þ þ ap þ bp2 Rocksalt
a (meV/GPa)
G-G
G-X
X-X
L-L
52.75a 54.95b
34.42a 36.09b
43.92b 45.96b
30.23a 31.67b
Table 2c Band-gap deformation potentials of the transitions for rocksalt AlN Rocksalt ðeVÞ aGG V aGX ðeVÞ V X aX ðeVÞ V aLL ðeVÞ V
15.14a; 14.99b; 14.9c 9.85a; 9.82b; 2.3c 12.56a; 12.49b 8.63a; 8.60b
a
LDA calculation. GGA calculation. c Ref. [15]: Theoretical calculations using the band-structure scheme LMTO. b
b (meV/GPa2) a
0.25a 0.27b
0.17a 0.18b
0.22a 0.24b
0.16b 0.17b
LDA calculation. GGA calculation.
b
Table 2a Band-gap deformation potentials of the direct transitions for wurtzite AlN Wurtzite 8.80a; 9.68a; 4.37a; 3.52a; 3.65a; 2.44a;
ðeVÞ aGG V aAA ðeVÞ V aLL ðeVÞ V aMM ðeVÞ V aHH ðeVÞ V aKK ðeVÞ V
9.00b; 8.8c; 9.0d 10.15b 4.61b 3.76b 3.84b 2.54b
a
LDA calculation. GGA calculation. c Ref. [15]: Theoretical calculations using the band-structure scheme LMTO. d Ref. [16]: Theoretical calculations using the full-potential LMTO method. b
the elastic constants of the material under study are of particular interest and has been calculated at zero pressure by computing the components of the stress tensor d for small strains, using the method reported in Ref. [18]. It is well known that a cubic crystal such as zinc-blende and rocksalt structures has only three independent elastic constants, namely C11, C12 and C44. Thus, a set of three equations is required to determine the constants. The first equation involves calculating the bulk modulus (B), which is related to the elastic constants by [19] 1 B ¼ ðC 11 þ 2C 12 Þ. (2) 3 Our obtained B’s for zinc-blende structure are 208.27 and 197.92 GPa, from LDA and GGA calculations, respectively. For rocksalt structure we obtained 274.87 GPa from LDA and 260.6 GPa from GGA. The second one involves performing volume-conservative tetragonal strain tensor d. ! 1 d ¼ d; d; 1; 0; 0; 0 . (3) 1 þ d2
Table 2b Band-gap deformation potentials of the transitions for zinc-blende AlN
Application of this strain changes the total energy from its unstrained value as follows:
Zinc-blende
EðdÞ ¼ ðC 11 C 12 Þ6V 0 d2 ,
ðeVÞ aGG V aGX ðeVÞ V X aX ðeVÞ V aLL ðeVÞ V
9.45a; 0.48a; 1.60a; 9.57a;
9.43b; 9.0c 0.49b; 0.37c; 0.38d; 0.39e 1.61b 9.62b
a
LDA calculation. b GGA calculation. c Ref. [15]: Theoretical calculations using the band-structure scheme LMTO. d Ref. [16]: Theoretical calculations using the full-potential LMTO method. e Ref. [17]: Theoretical calculations using the empirical pseudopotential method (EPM).
3.2. Elastic constants The effect of strain on electronic properties requires knowledge of the material’s mechanical properties, and specifically the elastic constants, which describe the response to an applied macroscopic stress. To this purpose
(4)
where V0 is the volume of the unit cell. Finally, for the last type of deformation, we used the volume-conserving rhombohedral strain tensor given by 1 d ¼ ðd; d; d; 2d; 2d; 2dÞ, 3 which transform the total energy to
!
(5)
V0 ðC 11 þ 2C 12 þ 4C 44 Þd2 (6) 3 These three equations form the set of equations required for the determination of the full elastic tensor. The elastic constants of the wurtzite structure are obtained directly using a similar procedure as that used by Wright [20]. In fact, there are five elastic constants for the wurtzite structure denoted C 11 ; C 12 ; C 13 ; C 33 and C 44 . These quantities were determined from energy calculations for five different strain configurations. EðdÞ ¼
! d1 ¼ ðd; d; 0; 0; 0; 0Þ ! E 1 ðdÞ ¼ ðC 11 þ C 12 ÞV 0 d2 ,
(7)
ARTICLE IN PRESS S. Saib, N. Bouarissa / Journal of Physics and Chemistry of Solids 67 (2006) 1888–1892
! d2 ¼ ðd; d; 2d; 0; 0; 0Þ ! E 2 ðdÞ ¼ ðC 11 þ C 12 4C 13 þ 2C 33 ÞV 0 d2 ,
ð8Þ
! 1 d3 ¼ ð0; 0; d; 0; 0; 0Þ ! E 3 ðdÞ ¼ C 33 V 0 d2 , 2
(9)
! 1 d4 ¼ ð0; 0; 0; 0; 0; dÞ ! E 4 ðdÞ ¼ ðC 11 C 12 ÞV 0 d2 , 4
(10)
! d5 ¼ ð0; 0; 0; d; d; 0Þ ! E 5 ðdÞ ¼ C 44 V 0 d2 .
(11)
In Table 3, we show our results regarding the elastic constants for AlN in the wurtzite, zinc-blende and rocksalt structures at zero pressure. For comparison, the available experimental and other theoretical data are also presented. Of note is that our results for wurtzite structure are in good agreement with the experimental values reported in Ref. [21] within the experimental uncertainties. Fair but slightly less good agreement is also obtained with calculations using Martin’s transformation approach [23,24]. One may note also that our LDA values are somewhat different from our GGA obtained ones. For the zinc-blende structure, the
Table 3 Elastic constants for AlN in the (a) wurtzite structure, (b) zincblende structure, (c) rocksalt structure at normal pressure (a) Wurtzite structure Elastic constants (GPa) C11
C12
C13
C33
C44
Reference
376a 358b 345 411710 388 398 396
130a 121b 125 149710 154 140 137
122a 126b 120 9974 84 127 108
411a 391b 395 389710 458 382 373
122a 120b 118 12575 99 96 116
This work [21] [22] [23] [24] [20]
c c d d
1891
numerically calculated elastic constants are in reasonable agreement (except may be for C44) with those reported in Refs. [20,23,24]. Our LDA C’s are somewhat larger than those of GGA ones. In the absence of both theoretical and experimental results regarding the C’s of AlN in the rocksalt structure, to our knowledge, our elastic constants of rocksalt AlN are predictions and may serve as a reference. It is worth noting that the same trend as that in zinc-blende phase could be seen for the results of LDA as compared to those of GGA. One may then conclude that the C’s calculated by LDA are somewhat larger than those obtained by GGA for both zinc-blende and rocksalt phase of AlN, while in the wurtzite structure only the elastic constants C11 and C33 still observe the same trends. The shear constants show a less clear trend as is to be expected. In fact, LDA underestimates lattice constants, overestimates bonding energies, and thus also leads to overestimates of stiffness or stretch elastic constants but not necessarily for shear moduli, which correspond to more subtle deformations. 4. Conclusion In conclusion, using a first-principle pseudopotential method based on the density-functional theory with both the LDA and GGA for the exchange-correlation functional, the electronic properties and elastic constants of wurtzite, zinc-blende and rocksalt AlN have been calculated. The numerically calculated values showed generally reasonable agreement with the available experimental data. The LDA obtained values for C’s are found to be somewhat larger than those calculated by GGA for both zinc-blende and rocksalt structures. The same trend still hold for the elastic constants C11 and C33 for wurtzite structure, which was not the case for the shear constants which showed a less clear trend as is to be expected. The elastic constants of rocksalt AlN are predictions.
d
References (b) Zincblende structure Elastic constants (GPa) C12
C11 a
302 288b 304 304 298
C44 a
161 153b 152 160 164
Reference a
160 155b 199 193 187
This work [24] [20] [23]
d d d
(c) Rocksalt structure Elastic constants (GPa) C11
C12
C44
Reference
397a 379b
214a 201b
202a 196b
This work
a
LDA calculation. GGA calculation. c Experiment. d Theoretical calculations. b
[1] F.A. Ponce, D.P. Bour, Nature (London) 386 (1997) 351. [2] J.W. Orton, C.T. Foxon, Rep. Prog. Phys. 61 (1998) 1 and references cited therein. [3] S.C. Jain, M. Willander, J. Narayan, R. van Overstraeten, J. Appl. Phys. 87 (2000) 965 and references cited therein. [4] I. Vurgaftman, J.R. Meyer, L.-R. Ram-Mohan, J. Appl. Phys. 89 (2001) 5815 and references cited therein. [5] I. Vurgaftman, J.R. Meyer, J. Appl. Phys. 94 (2003) 3675 and references cited therein. [6] Properties of Group-III Nitrides, in: J.H. Edgar (Ed.), EMIS Datareviews Series, IEE, London, 1994. [7] P. Jonnard, N. Capron, F. Semond, J. Massies, E. MartinezGuerrero, H. Mariette, Eur. Phys. J. B 42 (2004) 351. [8] M.P. Thompson, G.W. Auner, T.S. Zheleva, K.A. Jones, S.J. Simko, J.N. Hilfiker, J. Appl. Phys. 89 (2001) 3331. [9] 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, D.C. Allan, Comput. Mater. Sci. 25 (2002) 478. [10] D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. [11] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048.
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[12] J. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. [13] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [14] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [15] N.E. Christensen, I. Gorczyca, Phys. Rev. B 50 (1994) 4397. [16] K. Kim, W.R.L. Lambrecht, B. Segall, Phys. Rev. B 53 (1996) 16310. [17] N. Bouarissa, Phys. Stat. Sol. B 231 (2002) 391. [18] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN 2K, An augmented plane wave+local orbitals program for calculating crystal properties, Karlheinz Schwarz, Techn. Universitat, Wien, Austria, 2001, ISBN: 3-9501031-1-2
[19] E. Schreiber, O.L. Anderson, N. Soga, Elastic Constants and their Measurement, Mc Graw-Hill, New York, 1973. [20] A.F. Wright, J. Appl. Phys. 82 (1997) 2833. [21] K. Tsubouchi, N. Mikoshiba, IEEE. Trans. Sonics Ultrason. Su-32 (1985) 634. [22] K. Kim, W.R.L. Lambrecht, B. Segall, Phys. Rev. B 56 (1997) 7018 (erratum). [23] M. van Schilfgaarde, A. Sher, A.-B. Chen, J. Cryst. Growth 178 (1997) 8. [24] L.E. McNeil, M. Grimsditch, R.H. French, J. Am. Ceram. Soc. 76 (1993) 1132.
Journal of Physics and Chemistry of Solids 67 (2006) 1888–1892 www.elsevier.com/locate/jpcs
Electronic properties and elastic constants of wurtzite, zinc-blende and rocksalt AlN S. Saiba, N. Bouarissab, a
Physics Department, Faculty of Science and Engineering, University of M‘sila, 28000 M’sila, Algeria Department of Physics, Faculty of Science, King Khalid University, Abha, P.O. Box 9004, Saudi Arabia
b
Received 25 January 2006; received in revised form 23 April 2006; accepted 8 May 2006
Abstract Electronic properties and elastic constants of AlN in the wurtzite, zinc-blende and rocksalt structures are investigated using an ab initio pseudopotential method based on the density-functional theory with both the local-density approximation and the generalized gradient approximation for the exchange-correlation functional. The numerically calculated results compare well with the existing experimental data. For elastic constants of rocksalt AlN our results are predictions. r 2006 Elsevier Ltd. All rights reserved. PACS: 71.15.Mb; 71.20.b; 71.20.Nr; 71.15.Dx Keywords: C. High-pressure; D. Electronic structure; D. Elastic properties
1. Introduction Group-III nitrides have received extensive attention in recent years due to their superior properties with respect to conventional III–V semiconductors (the nitrides have larger band gaps and stronger bonds). Because of their wide band gaps and strong bond strength, group-III nitrides can be used for violet, blue, and green light emitting devices and for high temperature transistors [1–5]. Among them, AlN has a high melting point, a high thermal conductivity, and a large bulk modulus [6]. It has also the largest band gap (6.2 eV) and thus, is the best material for constructing devices for the violet region [7]. In the bulk form AlN stabilizes in the wurtzite structure. However, it has been reported that AlN can be grown in zinc-blende phase [5,8]. The optical gap of the wurtzite phase is direct and that of the zinc-blende phase is indirect. This can be useful in constructing different kinds of quantum wells or superlattices [7]. The very basis for these important applications is the specific electronic properties
Corresponding author.
E-mail address: [email protected] (N. Bouarissa). 0022-3697/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2006.05.007
as well as the mechanical response that is related to the elastic constants. Experimental and computational research on the physical properties of group-III nitrides at high pressure has seen much activity in recent years. Theoretical studies are most valuable so as to help understand and control the materials and device properties. In the present work, we perform density-functional-theory calculation for AlN, using the pseudopotential plane-wave method, where we employed both the local density approximation (LDA) and the generalized gradient approximation (GGA) for the exchange-correlation functional. We report electronic properties, namely the band-gap pressure coefficients of the three main transitions and the band-gap volume deformation potentials as well as elastic constants for AlN in the wurtzite, zinc-blende, and rocksalt structures. The paper is organized as follows. After a brief introduction in Section 1, the theoretical framework within which all the calculations have been performed is outlined in Section 2. In Section 3, we present and discuss the results of our study regarding the electronic properties and elastic constants of the material of interest in different structures being considered here. We compare our results with available experimental data as well as with previous
ARTICLE IN PRESS S. Saib, N. Bouarissa / Journal of Physics and Chemistry of Solids 67 (2006) 1888–1892
theoretical published values. A conclusion of the present study is given in Section 4. 2. Computational method
1889
Table 1a Values of the parameters a and b obtained by fitting the dependence of E GG , L M H K EA A ; E L ; E M ; E H and E K band-gap energies of wurtzite AlN on pressure (p) to EðpÞ ¼ Eð0Þ þ ap þ bp2 Wurtzite
A first principles pseudopotential plane-wave method based on the density functional theory (DFT) [9] is performed employing the LDA with the Ceperley–Alder [10] form of the exchange-correlation energy as parameterized by Perdew and Zunger [11], and the GGA with the Perdew et al. [12] form of the exchange-correlation potential. The interaction of the valence electrons with the ionic cores was represented with a separable, normconserving non-local Troullier–Martins [13] pseudopotentials. An energy cutoff of 160 Ry was used. Reciprocal space integration was performed by k-point sampling with sets of special points obtained by using the standard special k-points technique of the Monkhorst and Pack (MP) [14]. In this paper, a 4 4 4, 8 8 4, and 6 6 6 MP meshes were used for the semi-conducting zincblende, wurtzite and rocksalt AlN, respectively. In zincblende AlN the Al and N atoms are at the positions: Al(0, 0, 0) and N(1/4, 1/4, 1/4). For wurtzite AlN, there are four atoms per hexagonal unit cell where the positions of the atoms Al and N are: Al(0, 0, 0), (1/3, 2/3, 1/2) and N(0, 0, u), (1/3, 2/3, u+1/2), where u is the dimensionless internal parameter that represents the distance between Al plane and its nearest neighbor N plane, in the unit of c. For rocksalt AlN the Al and N atom positions are Al(0, 0, 0) and N(1/2, 1/2, 1/2).
G-G
A-A
L-L
M-M
H-H
K-K
a (meV/GPa)
41.14a 43.42b 40c 44d
44.06a 46.79b
20.33a 21.52b
16.22a 18.15b
17.00a 17.93b
11.28a 12.06b
b (meV/GPa2)
0.33a 0.30b 0.32c
0.26a 0.32b
0.15a 0.17b
0.11a 0.19b
0.13a 0.14b
0.08a 0.11b
a
LDA calculation. GGA calculation. c Ref. [15]: Theoretical calculations using the band-structure scheme LMTO. d Ref. [16]: Theoretical calculations using the full-potential LMTO method. b
Table 1b Values of the parameters a and b obtained by fitting the dependence of E GG , X L EX G , E X and E L band-gap energies of zinc-blende AlN on pressure (p) to EðpÞ ¼ Eð0Þ þ ap þ bp2 Zinc-blende G-G
G-X
X-X
L-L
a (meV/GPa)
43.14a 45.44b 42c
2.16a 2.38b 1.7c 1.9d
7.44a 7.93b
43.95a 46.42b
b (meV/GPa2)
0.27a 0.33b 0.34c
0.34a 0.01b 0.03c
0.06a 0.07b
0.30a 0.33b
3. Results 3.1. Electronic properties Table 1 shows the LDA and GGA-obtained band gap pressure coefficients of the transitions that are related to the expression: EðpÞ ¼ Eð0Þ þ ap þ bp2 .
(1)
For comparison, available theoretical calculations reported by Christensen and Gorczyca using the band structure scheme ‘linear muffin-tin-orbitals (LMTO)’ [15] and those of Kim et al. [16] using the full-potential LMTO method are also included. Note that our GGA results regarding the linear pressure coefficient (a) are slightly larger than those of Ref. [15]. The LDA values of a for all structures being studied here are smaller than those from GGA. The situation seems to be different for the coefficient b, which has generally larger values reported in Ref. [15] as compared to our both LDA and GGA results. As compared to the results of Kim et al. [16], our GGA value for a is in good agreement with that reported in Ref. [16] for wurtzite G–G gap. However, their value of a for the zinc-blende G-X gap seems to be smaller than our obtained values from both LDA and GGA, although it is closer to our results than the value reported in Ref. [15].
a
LDA calculation. GGA calculation. c Ref. [15]: Theoretical calculations using the band-structure scheme LMTO. d Ref. [16]: Theoretical calculations using the full-potential LMTO method. b
Table 2 lists the LDA and GGA-numerically calculated band-gap deformation potentials of the transitions G-G, X-X, L-L and G-X for zinc-blende and rocksalt AlN and the transitions G-G, A-A, L-L, M-M, H-H and K-K for wurtzite AlN. For comparison, results reported in Refs. [15–17] are also presented. Note that the agreement between our results and those reported in Refs. [15–17] is reasonable except may be for aGX in the rocksalt V structure. The reasonable agreement between our results as derived from our scheme and those reported in Refs. [15–17] clearly shows that it is possible to establish, a relation between the first-principles theories and the parameters entering the more empirically based models.
ARTICLE IN PRESS S. Saib, N. Bouarissa / Journal of Physics and Chemistry of Solids 67 (2006) 1888–1892
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Table 1c Values of the parameters a and b obtained by fitting the dependence of E GG , X L EX G , E X and E L band-gap energies of rocksalt AlN on pressure (p) to EðpÞ ¼ Eð0Þ þ ap þ bp2 Rocksalt
a (meV/GPa)
G-G
G-X
X-X
L-L
52.75a 54.95b
34.42a 36.09b
43.92b 45.96b
30.23a 31.67b
Table 2c Band-gap deformation potentials of the transitions for rocksalt AlN Rocksalt ðeVÞ aGG V aGX ðeVÞ V X aX ðeVÞ V aLL ðeVÞ V
15.14a; 14.99b; 14.9c 9.85a; 9.82b; 2.3c 12.56a; 12.49b 8.63a; 8.60b
a
LDA calculation. GGA calculation. c Ref. [15]: Theoretical calculations using the band-structure scheme LMTO. b
b (meV/GPa2) a
0.25a 0.27b
0.17a 0.18b
0.22a 0.24b
0.16b 0.17b
LDA calculation. GGA calculation.
b
Table 2a Band-gap deformation potentials of the direct transitions for wurtzite AlN Wurtzite 8.80a; 9.68a; 4.37a; 3.52a; 3.65a; 2.44a;
ðeVÞ aGG V aAA ðeVÞ V aLL ðeVÞ V aMM ðeVÞ V aHH ðeVÞ V aKK ðeVÞ V
9.00b; 8.8c; 9.0d 10.15b 4.61b 3.76b 3.84b 2.54b
a
LDA calculation. GGA calculation. c Ref. [15]: Theoretical calculations using the band-structure scheme LMTO. d Ref. [16]: Theoretical calculations using the full-potential LMTO method. b
the elastic constants of the material under study are of particular interest and has been calculated at zero pressure by computing the components of the stress tensor d for small strains, using the method reported in Ref. [18]. It is well known that a cubic crystal such as zinc-blende and rocksalt structures has only three independent elastic constants, namely C11, C12 and C44. Thus, a set of three equations is required to determine the constants. The first equation involves calculating the bulk modulus (B), which is related to the elastic constants by [19] 1 B ¼ ðC 11 þ 2C 12 Þ. (2) 3 Our obtained B’s for zinc-blende structure are 208.27 and 197.92 GPa, from LDA and GGA calculations, respectively. For rocksalt structure we obtained 274.87 GPa from LDA and 260.6 GPa from GGA. The second one involves performing volume-conservative tetragonal strain tensor d. ! 1 d ¼ d; d; 1; 0; 0; 0 . (3) 1 þ d2
Table 2b Band-gap deformation potentials of the transitions for zinc-blende AlN
Application of this strain changes the total energy from its unstrained value as follows:
Zinc-blende
EðdÞ ¼ ðC 11 C 12 Þ6V 0 d2 ,
ðeVÞ aGG V aGX ðeVÞ V X aX ðeVÞ V aLL ðeVÞ V
9.45a; 0.48a; 1.60a; 9.57a;
9.43b; 9.0c 0.49b; 0.37c; 0.38d; 0.39e 1.61b 9.62b
a
LDA calculation. b GGA calculation. c Ref. [15]: Theoretical calculations using the band-structure scheme LMTO. d Ref. [16]: Theoretical calculations using the full-potential LMTO method. e Ref. [17]: Theoretical calculations using the empirical pseudopotential method (EPM).
3.2. Elastic constants The effect of strain on electronic properties requires knowledge of the material’s mechanical properties, and specifically the elastic constants, which describe the response to an applied macroscopic stress. To this purpose
(4)
where V0 is the volume of the unit cell. Finally, for the last type of deformation, we used the volume-conserving rhombohedral strain tensor given by 1 d ¼ ðd; d; d; 2d; 2d; 2dÞ, 3 which transform the total energy to
!
(5)
V0 ðC 11 þ 2C 12 þ 4C 44 Þd2 (6) 3 These three equations form the set of equations required for the determination of the full elastic tensor. The elastic constants of the wurtzite structure are obtained directly using a similar procedure as that used by Wright [20]. In fact, there are five elastic constants for the wurtzite structure denoted C 11 ; C 12 ; C 13 ; C 33 and C 44 . These quantities were determined from energy calculations for five different strain configurations. EðdÞ ¼
! d1 ¼ ðd; d; 0; 0; 0; 0Þ ! E 1 ðdÞ ¼ ðC 11 þ C 12 ÞV 0 d2 ,
(7)
ARTICLE IN PRESS S. Saib, N. Bouarissa / Journal of Physics and Chemistry of Solids 67 (2006) 1888–1892
! d2 ¼ ðd; d; 2d; 0; 0; 0Þ ! E 2 ðdÞ ¼ ðC 11 þ C 12 4C 13 þ 2C 33 ÞV 0 d2 ,
ð8Þ
! 1 d3 ¼ ð0; 0; d; 0; 0; 0Þ ! E 3 ðdÞ ¼ C 33 V 0 d2 , 2
(9)
! 1 d4 ¼ ð0; 0; 0; 0; 0; dÞ ! E 4 ðdÞ ¼ ðC 11 C 12 ÞV 0 d2 , 4
(10)
! d5 ¼ ð0; 0; 0; d; d; 0Þ ! E 5 ðdÞ ¼ C 44 V 0 d2 .
(11)
In Table 3, we show our results regarding the elastic constants for AlN in the wurtzite, zinc-blende and rocksalt structures at zero pressure. For comparison, the available experimental and other theoretical data are also presented. Of note is that our results for wurtzite structure are in good agreement with the experimental values reported in Ref. [21] within the experimental uncertainties. Fair but slightly less good agreement is also obtained with calculations using Martin’s transformation approach [23,24]. One may note also that our LDA values are somewhat different from our GGA obtained ones. For the zinc-blende structure, the
Table 3 Elastic constants for AlN in the (a) wurtzite structure, (b) zincblende structure, (c) rocksalt structure at normal pressure (a) Wurtzite structure Elastic constants (GPa) C11
C12
C13
C33
C44
Reference
376a 358b 345 411710 388 398 396
130a 121b 125 149710 154 140 137
122a 126b 120 9974 84 127 108
411a 391b 395 389710 458 382 373
122a 120b 118 12575 99 96 116
This work [21] [22] [23] [24] [20]
c c d d
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numerically calculated elastic constants are in reasonable agreement (except may be for C44) with those reported in Refs. [20,23,24]. Our LDA C’s are somewhat larger than those of GGA ones. In the absence of both theoretical and experimental results regarding the C’s of AlN in the rocksalt structure, to our knowledge, our elastic constants of rocksalt AlN are predictions and may serve as a reference. It is worth noting that the same trend as that in zinc-blende phase could be seen for the results of LDA as compared to those of GGA. One may then conclude that the C’s calculated by LDA are somewhat larger than those obtained by GGA for both zinc-blende and rocksalt phase of AlN, while in the wurtzite structure only the elastic constants C11 and C33 still observe the same trends. The shear constants show a less clear trend as is to be expected. In fact, LDA underestimates lattice constants, overestimates bonding energies, and thus also leads to overestimates of stiffness or stretch elastic constants but not necessarily for shear moduli, which correspond to more subtle deformations. 4. Conclusion In conclusion, using a first-principle pseudopotential method based on the density-functional theory with both the LDA and GGA for the exchange-correlation functional, the electronic properties and elastic constants of wurtzite, zinc-blende and rocksalt AlN have been calculated. The numerically calculated values showed generally reasonable agreement with the available experimental data. The LDA obtained values for C’s are found to be somewhat larger than those calculated by GGA for both zinc-blende and rocksalt structures. The same trend still hold for the elastic constants C11 and C33 for wurtzite structure, which was not the case for the shear constants which showed a less clear trend as is to be expected. The elastic constants of rocksalt AlN are predictions.
d
References (b) Zincblende structure Elastic constants (GPa) C12
C11 a
302 288b 304 304 298
C44 a
161 153b 152 160 164
Reference a
160 155b 199 193 187
This work [24] [20] [23]
d d d
(c) Rocksalt structure Elastic constants (GPa) C11
C12
C44
Reference
397a 379b
214a 201b
202a 196b
This work
a
LDA calculation. GGA calculation. c Experiment. d Theoretical calculations. b
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