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Ab initio calculations of the electronic, linear and nonlinear optical properties of zinc chalcogenides
ARTICLE IN PRESS
Physica B 388 (2007) 34–42 www.elsevier.com/locate/physb
Review
Ab initio calculations of the electronic, linear and nonlinear optical properties of zinc chalcogenides Ali Hussain Reshak, Sushil Auluck Physics Department, Indian Institute of Technology, Roorkee (Uttaranchal) 247667, India Received 18 January 2006; received in revised form 9 March 2006; accepted 3 May 2006
Abstract We report calculations of the electronic, linear and nonlinear optical properties of ZnX (X ¼ S, Se, Te) compounds using the full potential linear augmented plane wave (FP-LAPW) method. We present results for the band structure, density of states, and imaginary part of the frequency-dependent linear and nonlinear optical response. Our calculations show that the energy band gap of these compounds decreases when S is replaced by Se and Se by Te. This can be attributed to the increase in the bandwidth of the conduction bands. Our calculated e2(o) shows good agreement with the experimental data. We find that the intra- and inter-band contributions of the second harmonic generation increase when moving from S to Se to Te. The smaller energy band gap compounds have larger values of wð2Þ 123 ð0Þ in agreement with the experimental measurements and other theoretical calculations. r 2006 Elsevier B.V. All rights reserved. PACS: 70; 71.15.Ap; 71.1 Keywords: Electronic structure; Optical properties (linear and nonlinear); DFT; LDA; FPLAPW; DOS; Band structure
Contents 1. 2. 3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Band structure and density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Linear optical response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Nonlinear response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34 35 35 35 36 38 41 42 42 42
1. Introduction
Corresponding author. Tel.: +420 777 729583; fax: +420 386 361231.
E-mail address: [email protected] (A.H. Reshak). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.05.003
The semiconducting binary ZnX (X ¼ S, Se, Te) compounds have attracted much attention because they possess direct energy band gaps and are efficient light emitters even at room temperature. These compounds
ARTICLE IN PRESS A.H. Reshak, S. Auluck / Physica B 388 (2007) 34–42
appear to be promising candidates for many technological applications [1], such as blue laser fabrication techniques, optical switching devices, fabrication of visible lightemitting devices and modulated hetero-structures, optical wave guides, optical computing, and cascaded laser operation chip-sized platforms [2]. There have been many empirical and ab initio calculations of the electronic properties [3–8] for ZnX compounds. Sapra et al. [4] have analyzed the electronic structures of ZnX compounds within linear muffin-tin orbital (LMTO) approach in order to arrive at a parametrized tight-binding model which can provide an accurate description of both valence and conduction bands. Rossler [5] used the Green’s function Korringa–Kohn–Rostoker (KKR) method for calculating the band structure of ZnS. Walter et al. [6] used the empirical pseudopotential method to calculate the band structure of ZnSe and ZnTe. Huang and Ching [3] used the first principle-orthogonalized linear combination of atomic orbitals (OLCAO) method to calculate the band structure and density of states for ZnX compounds. Ghahramani et al. [7] have reported band structure calculations of ZnSe and ZnTe using the linear combination of Gaussian orbitals technique in conjunction with the Xa method. Tsuchiya et al. [8] have calculated the energy band structure and the density of states for ZnS using the empirical pseudopotential method. All the above calculations show that the valence band maximum (VBM) and the conduction band minimum (CBM) are located at G. Chelikowsky et al. [9] used the empirical nonlocal pesudopotential method to calculate the reflectivity of ZnSe. Huang and Ching [3] calculated the linear and nonlinear optical properties of ZnX compounds. Erbarut [2] has used the localized orbital method (LOM) to compute the dielectric response functions of ZnX compounds. The reflectivity spectra of ZnSe and ZnTe has been studied experimentally and theoretically by Walter et al. [6]. Ghahramani et al. [7] have calculated the linear and nonlinear optical properties of ZnSe and ZnTe. Aspnes [10] has used energy band theory to calculate the zero frequency limit of the second-order optical susceptibility of ZnX compounds. Bang et al. [11] used spectroscopic ellipsometry to measure the real and imaginary parts of the pseudo dielectric function for ZnX and Zn1xTex. Freeouf [12] has used synchrotron radiation and standard light source to measure the polarization dependent optical properties of ZnX. Ronnow et al. [13] measured the piezo-optical coefficients of ZnSe and ZnTe above the fundamental gap using reflectance difference spectroscopy. The measured spectra of these compounds show good Kramer– Kronig consistency between their real and imaginary parts. Even though there exist a number of calculations for the electronic band structure and optical properties using different methods, no full-potential calculation exist for these compounds. The calculated energy gaps vary from 2.3 to 3.8 eV for ZnS, 1.6 to 2.9 eV for ZnSe and 1.4 to 2.4 eV for ZnTe. The measured energy gaps [3] are 3.8 eV
35
for ZnS, 2.8 eV for ZnSe and 2.4 eV for ZnTe. Thus there is a large variation in the energy gaps, suggesting that the energy band gap depends on the method of the band structure calculation. Also some of the calculated energy gaps are equal to the measured energy gaps which is not expected from calculations based on the local density approximation. We therefore thought it worthwhile to perform calculations using the full potential linear augmented plane wave [14] (FP-LAPW) method. We report calculations of the linear and nonlinear optical properties. Our calculations will highlight the effect of replacing S by Se and Se by Te on the electronic and optical properties in ZnX compounds. We compare our calculations with the experimentally measured frequency-dependent dielectric function and previous theoretical calculations [3,7]. Such comparisons are missing in previous works. In Section 2, we give details of our calculations. The band structure, densities of states, the linear and nonlinear optical susceptibilities are presented and discussed in Section 3. In Section 4, we summarize our conclusions. 2. Methodology We have performed ab initio calculations for the noncentro-symmetric cubic semiconducting ZnX compounds. The Zn atom is located at the origin and the X atom is located at (14 14 14). This is the zinc blende structure. The space ¯ group is F43m. We have used the experimental lattice constants [15–17] Table 1). In our calculations we used the FP-LAPW [14] method based on the density functional theory (DFT). Calculations are performed with local density approximation (LDA) and generalized gradient approximation (GGA). Self-consistency is obtained using 200 k-points in the irreducible Brillouin zone (IBZ). The linear optical properties are calculated using 500 k-points and the nonlinear optical properties using 1300 k-points in the IBZ to obtain converged results. 3. Results and discussion 3.1. Band structure and density of states The band structure, total density of states (TDOS) along with the X–s, X–p and Zn–d partial densities of states (PDOS) for ZnX (X ¼ S, Se, Te) are shown in Fig. 1. From the PDOS we are able to identify the angular momentum character of the various structures. The lowest energy group has mainly X–s states. The second group between 7.0 and 6.0 eV is composed of Zn–d and X–p states. The third group from 5.0 eV up to the Fermi energy (EF) is composed of Zn–d and X–p states. The last group from 1 eV and above has contributions from X–p and Zn–spd states. The trends in the band structures (as we move from S to Se to Te) can be summarized as follows: (1) The first group in ZnSe is shifted towards lower energies by around 0.5 eV in comparison with ZnS, while in ZnTe it is shifted
ARTICLE IN PRESS A.H. Reshak, S. Auluck / Physica B 388 (2007) 34–42
36 Table 1 Lattice parameters, energy gaps, and e1(0)
a (A˚) E exp g ðeVÞ E Theory ðeVÞ g
ZnS
ZnSe
ZnTe
5.409a 3.80a 2.34a, 3.2b, 3.8c, 2.4d
5.668a 2.82a 1.65a, 2.9e, 2.8f, 1.6d
6.089a 2.39a 2.24a, 2.4e, 1.4b, 2.2f, 1.6d
E LDA ðeVÞ g
2.1
1.3
1.1
E GGA ðeVÞ g
2.3
1.7
1.4
e1(0) exp. e1(0) Theory
5.2g 5.63a, 5.4d, 6.5j, 6k
5.9h 5.56a, 6.7d, 7.5j, 7k
7.3i 5.24a, 8.1d 9.5j, 9k
a
Ref. [3]. Ref. [4]. c Ref. [5,8]. d Ref. [19]. e Ref. [7]. f Ref. [6]. g Ref. [10,26]. h Ref. [10,27]. i Ref. [10,28]. j This work (LDA). k This work (GGA). b
towards higher energies by around 1 eV. (2) The bandwidth of the second group is reduced. This group is shifted towards lower energies by around 0.5 eV causing to increase the gap between the second and third groups. (3) The bandwidth of the conduction band increases slightly by around 0.5 eV on going from S to Se to Te. From the PDOS, we note a strong hybridization between Zn–d and X–p states. Following Yamasaki et al. [18] we can defined degree of hybridization by the ratio of Zn–d states and X–p states within the muffin tin sphere. Based on this we can say that the hybridization between Zn–d and chalcogen-p states becomes stronger when moving from S to Se to Te. The VBM and the CBM are located at G resulting in a direct gap in agreement with experiment and previous theoretical work [3–8,19]. We have performed calculations using LDA and GGA and find that both underestimate the energy gaps. GGA yields slightly larger energy gaps. In Table 1 we list the values of the energy gaps calculated by LDA and GGA along with the experimental values [3] and previous theoretical work [3–8]. We note that the energy gap reduces when S replaced by Se and Se by Te in agreement with the experimental data. This trend in the energy gaps is not present in previous calculations [3,19]. The reduction in the energy gaps can be attributed to the increase in bandwidth of the conduction bands. This reduction in the gap is consistent with an overall weakening of the bonds and therefore with a smaller bondingantibonding splitting. In a previous calculation [3] the second and third groups are merged for ZnS and ZnSe but not for ZnTe. 3.2. Linear optical response Since ZnX compounds have cubic symmetry, we need to calculate only one dielectric tensor component to com-
pletely characterize the linear optical properties. This component is e2(o) the imaginary part of the frequency dependent dielectric function and is given by [20] Z 1 h e i2 X 2 ðoÞ ¼ f Pji Pij dðE ji _oÞ dk, p mo i;j BZ ij where E ¼ _o, E ji ¼ E j E i and f il ¼ f i f l . fi is the Fermi occupation factor of the single-particle state i. The Pij are the momentum matrix elements. Here i and j indicates the conduction and valence bands. Fig. 2 shows the calculated e2(o) for the ZnX compounds. Broadening is taken to be 0.2 eV. The linear optical properties are scissors corrected [21] by the difference between the calculated and measured energy gaps (Table 1). We note that the e2(o) shows a large peak (located at 7.5 eV for ZnS, 6.5 eV for ZnSe, and 5.5 eV for ZnTe) between two small peaks/humps. All the structures in e2(o) are shifted towards the lower energies with increase in the peak heights when S is replaced by Se and Se by Te, in agreement with the experimental data [12]. This is attributed to the reduction in the band gaps. We compare our calculated e2(o) with the most recent calculations of Huang and Ching [3], Ghahramani et al. [7] and the experimental data [12]. In order to make a meaningful comparison, we have shifted the calculated e2(o) [3,7] to match the large peak in the experimental data. Previous calculations [3,7] underestimate the magnitude of e2(o) in the low-energy regime. This could be due to an inaccurate representation of the wave functions. However our calculated e2(o) shows very good agreement with the experimental data [12] in the matter of the peaks position and peak heights. The calculated e1(0) is compared with the experimental values in Table 1. We note that a smaller energy gap yields
ARTICLE IN PRESS A.H. Reshak, S. Auluck / Physica B 388 (2007) 34–42
37
Fig. 1. Band structure and total density of states (states /eV unit cell), along with X–p (———), Zn–d (yy), and X–s (———) partial densities of states.
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Fig. 2. Calculated e2(o) (solid curve) along with the experimental data [12] (dashed curve) and theoretical calculation (light curve) of Huang and Ching [3] for ZnS and of Ghahramani et al. [7] for ZnSe and ZnTe.
a larger e1(0) value. This could be explained on the basis of the Penn model [22]. 3.3. Nonlinear response The complex second-order nonlinear optical susceptibility tensor wð2Þ ijk ð2o; o; oÞ can be generally written as the sum of three physically different contributions in the form [7,14,23].
wð2Þ ijk ð2o; o; oÞ
ac0 Z i h e i3 cX dk vc cc0 c0 v ¼ fP P P g 2 mo vcc0 BZ 4p3 1 ðE cv 2_oÞðE c0 v _oÞ 1 þ ðE c0 v þ 2_oÞðE cv þ _oÞ 1 þ , ðE cv þ _oÞðE c0 v _oÞ
ARTICLE IN PRESS A.H. Reshak, S. Auluck / Physica B 388 (2007) 34–42
where v stands for a valence band state and c, c0 two different conduction band states, E cv ðkÞ ¼ E c ðkÞ E v ðkÞ is the difference in the R band energies between two states c and v, and Pcv ¼ i_ cc ðk; rÞrcv ðk; rÞ dr is the corresponding momentum matrix element. It has been demonstrated by Aspnes [10] that only one virtual-electron transitions (transitions between one valence band state and two conduction band states) give a significant contribution to the second-order tensor. Here we ignored the virtual-hole contribution (transitions between two valence band states
39
and one conduction band state) because it was shown to be negative and more than an order of magnitude smaller than the virtual-electron contribution for ZnX compounds. For ð2Þ simplicity we call wð2Þ ijk ð2o; o; oÞ as wijk ðoÞ. The subscripts i, j, and k are Cartesian indices. It has 18 elements. For non-centro-symmetric compounds with cubic group symmetry, there is only one nonzero element [3] wð2Þ 123 ðoÞ. We have calculated the imaginary and real parts of the second harmonic generation (SHG) susceptibility wð2Þ 123 ðoÞ. The calculated imaginary part is shown in Fig. 3. It is calculated
ð2Þ Fig. 3. Calculated Im w123 ðoÞ for ZnX compounds without scissors correction (dark curve) and with scissors correction (light curve).
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with and without the scissors correction. We see that the scissors correction [21] has a profound effect on magnitude and sign of wð2Þ 123 ðoÞ. That is attributed to fact that LDA calculations underestimate the energy gaps by 30–40%. A very simple way to overcome this drawback is to use the scissor correction, which merely makes the calculated energy gap equal to the experimental gap. This is a very crude method and empirical in nature. Moreover, it is clear from the formulae that, wð2Þ ijk ð2o; o; oÞ depends on the energy gap. Hence when we use the scissor correction it gives a considerable effect on wð2Þ ijk ð2o; o; oÞ. We note that all the structures in Im wð2Þ ðoÞ are shifted towards lower 123 energies when S replaced by Se and Se by Te. It is well known that nonlinear optical properties are more sensitive to small changes in the band structure than the linear optical properties. There are two reasons for that sensitivity. For one, the second harmonic response wð2Þ ijk ðoÞ [7,14,23] involves more resonances than the linear one. In addition to the usual o resonance contribution there is a 2o resonance contribution. Both the o and 2o resonances can be further separated into inter-band and intra-band contributions. Secondly, the products of matrix elements, which control the strength of a given resonance in Im wð2Þ 123 ðoÞ, can be positive or negative. In contrast, for the linear response the corresponding factors involve only the square of matrix elements, which ensures, that e2(o) is positive. As a result, the structure in Im wð2Þ 123 ðoÞ is more pronounced than in the e2(o). In Fig. 4 we present the inter-band and intra-band contributions to the o and 2o resonances for ZnS. We find that the o resonance is smaller than the 2o resonance. As can be seen the total SHG susceptibility is zero below half the band gap. The 2o terms start contributing at energies (1/2)Eg and the o terms for energy values above Eg. In the low energy regime (p3 eV) the SHG optical spectra is dominated by the 2o contributions. Beyond 3 eV the major contribution comes from the o terms.
The structures in Im wð2Þ 123 ðoÞ can be understood from the structures in e2(o). Unlike the linear optical spectra, the features in the SHG susceptibility are very difficult to identify from the band structure because of the presence of 2o and o terms. But we can make use of the linear optical spectra to identify the different resonance leading to various features in the SHG spectra. The first structure in Im wð2Þ 123 ðoÞ between 1.0 and 3.5 eV for ZnS, 0.5 and 3.0 eV for ZnSe, and 0.5 and 2.0 eV for ZnTe is associated with interference between a o resonance and 2o resonance and arises from the first hump in e2(o). The second structure between 3.5 and 4.5 eV for ZnS, 3.0 and 4.0 eV for ZnSe, and 2.0 and 3.5 eV for ZnTe is due mainly to o resonance and associated with high peak in e2(o). The last structure from 4.5 and 5.5 for ZnS, 4.0 and 5.5 eV for ZnSe, and 3.5 and 5.5 eV for ZnTe is manly due to o resonance and associated with the second hump in e2(o). In Fig. 5 we show the 2o inter-band and intra-band contributions for ZnX compounds. We note the opposite signs of the two contributions throughout the frequency range. Both the contributions increases on moving from S to Se to Te. This could be attributed to the decrease in the energy gaps on moving from S to Se to Te. In Table 2 we present the values of wð2Þ 123 ð0Þ. These values clearly increase on going from S to Se to Te in agreement with experiment and theoretical calculations (Table 2). We notice that the smaller band gap compounds gives higher values of wð2Þ 123 ð0Þ in agreement with the experiment [24,25] and theory [3,7,10,17,26–29]. Table 2 shows that there is a large variation in the theoretical values of wð2Þ 123 ð0Þ. That is attributed to the large variation in the calculated energy gaps (Table 1) because of using different methods of calculations. As we mentioned above that wð2Þ ijk ð2o; o; oÞ depends on the energy gap, so it is a natural result that one can get different values of wð2Þ 123 ð0Þ for different energy gaps. We compare our calculated wð2Þ 123 ð0Þ with the experimental data [24,25]; it shows reasonable agreement. The lack of
Fig. 4. Calculated Im wð2Þ 123 ðoÞ along with the intra (2o) and (o)-inter (2o) and (o)-band contributions for ZnS.
ARTICLE IN PRESS A.H. Reshak, S. Auluck / Physica B 388 (2007) 34–42
41
Fig. 5. Calculated Im wð2Þ 123 ðoÞ along with the intra (2o) and inter (2o)-band contributions for ZnX compounds.
experimental data prevents any conclusive comparison with experiment over a large energy range. 4. Conclusions We have performed calculations of the band structure, DOS, linear and nonlinear optical response for the ZnX compounds, using the FP-LAPW method. Our results for band structure and DOS show that these compounds have similar structures and the energy gap decreases when S is
replaced by Se and Te. This is attributed to the fact that the bandwidth of the conduction bands increases on going from S to Se to Te. All the structures in the imaginary part of the dielectric function e2(o) are shifted towards lower energies when S is replaced by Se and Te. We compare our calculated e2(o) with the experimentally measured frequency-dependent dielectric function and previous theoretical calculations [3,7]. Our e2(o) shows better agreement with experiment data than the previous calculations. We find that the values of e1(0) increases with decreasing
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Table 2 Experimental, calculated total and intra-inter- band of the zero frequency of the real part of the wð2Þ 123 ðoÞ ZnS 123
w (0) w123(0) w123(0) w123(0) w123(0)
theory exp. total inter intra
a
ZnSe b
c
d
e
f
g
16 , 43 , 17 , 19 , 35 , 10 , 48 17i, 14.6j 20k 80k 100k
a
ZnTe b
c
f
h
d
e
g
19 , 63 , 30 , 15 , 44 27 , 53 , 38 22i, 37.4j 50k 150k 200k
53a, 112b, 73c, 51d, 90e, 146g, 29f, 73h 73i, 44j 90k 310k 400k
a
Ref. [17]. Ref. [26]. c Ref. [27]. d Ref. [28]. e Ref. [29]. f Ref. [10]. g Ref. [24]. h Ref. [7]. i Ref. [3]. j Ref. [25]. k This work (LDA). b
energy gap, in agreement with the Penn model. Our calculations of the SHG susceptibility show that the intra-band and inter-band contributions are significantly increased when S is replaced by Se and Se by Te. All the structures in wð2Þ 123 ðoÞ are shifted towards lower energies when moving from S to Se to Te. Our calculations show that the smaller band gap materials have higher wð2Þ 123 ð0Þ values. Since wð2Þ 123 ð0Þ is roughly inversely correlated to the band gap, one might think that this would lead to a possible route to further enhancement of Im wð2Þ 123 ðoÞ. The enhancement of the SHG when one substitutes S by Se and Te is considerable. Acknowledgments We would like to thank the Institute Computer Center and Physics Department for providing the computational facilities. Also to thank Prof. Claudia Ambrosch-Draxl and Dr. Sangeeta Sharma, Institute of Physics, University Graz, Universita¨tsplatz 5, A-8010 Graz, Austria for using the nonlinear code. References [1] J.L. Bredin, Phys. Today 47 (1994) 5; J.L. Bredin, Science 263 (1994) 487. [2] E. Erbarut, Solid State Commun. 127 (2003) 515. [3] M.-Z. Huang, W.Y. Ching, Phys. Rev. B 47 (1993) 9449; M.-Z. Huang, W.Y. Ching, Phys. Rev. B 47 (1993) 9464. [4] S. Sapra, N. Shanthi, D.D. Sarma, arxiv: Cond-Matt, 0308048, v1, 4 August 2003. [5] U. Rossler, Phys. Rev. 184 (1969) 733. [6] J.P. Walter, M.L. Cohen, Y. Petroff, M. Balkanski, Phys. Rev. B 1 (1970) 2661. [7] E. Ghahramani, D.J. Moss, J.E. Sipe, Phys. Rev. B 43 (1991) 9700.
[8] T. Tsuchiya, S. Ozaki, S. Adachi, J. Phys.: Condens. Matter 15 (2003) 3717. [9] J.R. Chelikowsky, M.L. Chen, Phys. Rev. B 14 (1976) 556. [10] D.E. Aspnes, Phys. Rev. B 6 (1972) 4648. [11] C.Y. Bang, M.S. Lee, T.J. Kim, Y.D. Kim, A.E. Aspnes, Y.M. Yu, B. Sung O, Y.D. Choi, J. Korean Phys. Soc. 39 (2001) 462. [12] J.L. Freeouf, Phys. Rev. B 7 (1973) 3810. [13] D. Ronnow, M. Cardona, L.F. Lastras-Martinez, Phys. Rev. B. 59 (1999) 5581. [14] S. Sharma, C. Ambrosch-Draxl, Phys. Scr. T 109 (2004) 128. [15] S.J. Czyzak, W.M. Barker, R.C. Crane, J.B. Howe, J. Opt. Soc. Am. 47 (1957) 240. [16] A. Manabe, A. Mitsuishi, H. Yoshinaga, Jpn. J. Appl. Phys. 6 (1967) 593. [17] D.T.F. Marple, J. Appl. Phys. 35 (1964) 539. [18] T. Yamasaki, N. Suzuki, K. Motizuki, J. Phys. C 20 (1987) 395. [19] A. Dal Corso, F. Mauri, A. Rubio, Phys. Rev. B 53 (1996) 15638. [20] D.J. Moss, E. Ghahramani, J.E. Sipe, H.M. van Driel, Phys. Rev. B 41 (1990) 1542. [21] S.N. Rashkeev, W.R.L. Lambrecht, Phys. Rev. B 63 (2001) 165212. [22] D.R. Penn, Phys. Rev. 128 (1962) 2093. [23] D.J. Moss, J.E. Sipe, H.M. van Driel, Phys. Rev. B 36 (1987) 9708. [24] R.A. Soref, H.W. Moos, J. Appl. Phys. 35 (1964) 2152. [25] C.K.N. Patel, Phys Rev. Lett. 16 (1966) 613. [26] D.A. Kleinman, Phys. Rev. B 2 (1970) 2139. [27] C.L. Tang, C. Flytzanis, Phys. Rev. B 4 (1971) 2520. [28] M.I. Bell, Phys Rev. B 6 (1972) 516. [29] C.L. Tang, IEEE J. Quantum Electron. QE-9 (1973) 755.
Further reading [30] M. Cardona, G. Harbeke, Phys. Rev. 137 (1965) A1467. [31] C.S. Wang, B.M. Klein, Phys. Rev. B 24 (1981) 3417. [32] D.J. Moss, E. Ghahramani, J.E. Sipe, H.M. van Driel, Phys. Rev. B 34 (1986) 8758. [33] S. Adachi, Phys. Rev. B 35 (1987) 7454. [34] J.C. Phillips, J.A. Van Vechten, Phys. Rev. 183 (1969) 709.
Physica B 388 (2007) 34–42 www.elsevier.com/locate/physb
Review
Ab initio calculations of the electronic, linear and nonlinear optical properties of zinc chalcogenides Ali Hussain Reshak, Sushil Auluck Physics Department, Indian Institute of Technology, Roorkee (Uttaranchal) 247667, India Received 18 January 2006; received in revised form 9 March 2006; accepted 3 May 2006
Abstract We report calculations of the electronic, linear and nonlinear optical properties of ZnX (X ¼ S, Se, Te) compounds using the full potential linear augmented plane wave (FP-LAPW) method. We present results for the band structure, density of states, and imaginary part of the frequency-dependent linear and nonlinear optical response. Our calculations show that the energy band gap of these compounds decreases when S is replaced by Se and Se by Te. This can be attributed to the increase in the bandwidth of the conduction bands. Our calculated e2(o) shows good agreement with the experimental data. We find that the intra- and inter-band contributions of the second harmonic generation increase when moving from S to Se to Te. The smaller energy band gap compounds have larger values of wð2Þ 123 ð0Þ in agreement with the experimental measurements and other theoretical calculations. r 2006 Elsevier B.V. All rights reserved. PACS: 70; 71.15.Ap; 71.1 Keywords: Electronic structure; Optical properties (linear and nonlinear); DFT; LDA; FPLAPW; DOS; Band structure
Contents 1. 2. 3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Band structure and density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Linear optical response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Nonlinear response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34 35 35 35 36 38 41 42 42 42
1. Introduction
Corresponding author. Tel.: +420 777 729583; fax: +420 386 361231.
E-mail address: [email protected] (A.H. Reshak). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.05.003
The semiconducting binary ZnX (X ¼ S, Se, Te) compounds have attracted much attention because they possess direct energy band gaps and are efficient light emitters even at room temperature. These compounds
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appear to be promising candidates for many technological applications [1], such as blue laser fabrication techniques, optical switching devices, fabrication of visible lightemitting devices and modulated hetero-structures, optical wave guides, optical computing, and cascaded laser operation chip-sized platforms [2]. There have been many empirical and ab initio calculations of the electronic properties [3–8] for ZnX compounds. Sapra et al. [4] have analyzed the electronic structures of ZnX compounds within linear muffin-tin orbital (LMTO) approach in order to arrive at a parametrized tight-binding model which can provide an accurate description of both valence and conduction bands. Rossler [5] used the Green’s function Korringa–Kohn–Rostoker (KKR) method for calculating the band structure of ZnS. Walter et al. [6] used the empirical pseudopotential method to calculate the band structure of ZnSe and ZnTe. Huang and Ching [3] used the first principle-orthogonalized linear combination of atomic orbitals (OLCAO) method to calculate the band structure and density of states for ZnX compounds. Ghahramani et al. [7] have reported band structure calculations of ZnSe and ZnTe using the linear combination of Gaussian orbitals technique in conjunction with the Xa method. Tsuchiya et al. [8] have calculated the energy band structure and the density of states for ZnS using the empirical pseudopotential method. All the above calculations show that the valence band maximum (VBM) and the conduction band minimum (CBM) are located at G. Chelikowsky et al. [9] used the empirical nonlocal pesudopotential method to calculate the reflectivity of ZnSe. Huang and Ching [3] calculated the linear and nonlinear optical properties of ZnX compounds. Erbarut [2] has used the localized orbital method (LOM) to compute the dielectric response functions of ZnX compounds. The reflectivity spectra of ZnSe and ZnTe has been studied experimentally and theoretically by Walter et al. [6]. Ghahramani et al. [7] have calculated the linear and nonlinear optical properties of ZnSe and ZnTe. Aspnes [10] has used energy band theory to calculate the zero frequency limit of the second-order optical susceptibility of ZnX compounds. Bang et al. [11] used spectroscopic ellipsometry to measure the real and imaginary parts of the pseudo dielectric function for ZnX and Zn1xTex. Freeouf [12] has used synchrotron radiation and standard light source to measure the polarization dependent optical properties of ZnX. Ronnow et al. [13] measured the piezo-optical coefficients of ZnSe and ZnTe above the fundamental gap using reflectance difference spectroscopy. The measured spectra of these compounds show good Kramer– Kronig consistency between their real and imaginary parts. Even though there exist a number of calculations for the electronic band structure and optical properties using different methods, no full-potential calculation exist for these compounds. The calculated energy gaps vary from 2.3 to 3.8 eV for ZnS, 1.6 to 2.9 eV for ZnSe and 1.4 to 2.4 eV for ZnTe. The measured energy gaps [3] are 3.8 eV
35
for ZnS, 2.8 eV for ZnSe and 2.4 eV for ZnTe. Thus there is a large variation in the energy gaps, suggesting that the energy band gap depends on the method of the band structure calculation. Also some of the calculated energy gaps are equal to the measured energy gaps which is not expected from calculations based on the local density approximation. We therefore thought it worthwhile to perform calculations using the full potential linear augmented plane wave [14] (FP-LAPW) method. We report calculations of the linear and nonlinear optical properties. Our calculations will highlight the effect of replacing S by Se and Se by Te on the electronic and optical properties in ZnX compounds. We compare our calculations with the experimentally measured frequency-dependent dielectric function and previous theoretical calculations [3,7]. Such comparisons are missing in previous works. In Section 2, we give details of our calculations. The band structure, densities of states, the linear and nonlinear optical susceptibilities are presented and discussed in Section 3. In Section 4, we summarize our conclusions. 2. Methodology We have performed ab initio calculations for the noncentro-symmetric cubic semiconducting ZnX compounds. The Zn atom is located at the origin and the X atom is located at (14 14 14). This is the zinc blende structure. The space ¯ group is F43m. We have used the experimental lattice constants [15–17] Table 1). In our calculations we used the FP-LAPW [14] method based on the density functional theory (DFT). Calculations are performed with local density approximation (LDA) and generalized gradient approximation (GGA). Self-consistency is obtained using 200 k-points in the irreducible Brillouin zone (IBZ). The linear optical properties are calculated using 500 k-points and the nonlinear optical properties using 1300 k-points in the IBZ to obtain converged results. 3. Results and discussion 3.1. Band structure and density of states The band structure, total density of states (TDOS) along with the X–s, X–p and Zn–d partial densities of states (PDOS) for ZnX (X ¼ S, Se, Te) are shown in Fig. 1. From the PDOS we are able to identify the angular momentum character of the various structures. The lowest energy group has mainly X–s states. The second group between 7.0 and 6.0 eV is composed of Zn–d and X–p states. The third group from 5.0 eV up to the Fermi energy (EF) is composed of Zn–d and X–p states. The last group from 1 eV and above has contributions from X–p and Zn–spd states. The trends in the band structures (as we move from S to Se to Te) can be summarized as follows: (1) The first group in ZnSe is shifted towards lower energies by around 0.5 eV in comparison with ZnS, while in ZnTe it is shifted
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36 Table 1 Lattice parameters, energy gaps, and e1(0)
a (A˚) E exp g ðeVÞ E Theory ðeVÞ g
ZnS
ZnSe
ZnTe
5.409a 3.80a 2.34a, 3.2b, 3.8c, 2.4d
5.668a 2.82a 1.65a, 2.9e, 2.8f, 1.6d
6.089a 2.39a 2.24a, 2.4e, 1.4b, 2.2f, 1.6d
E LDA ðeVÞ g
2.1
1.3
1.1
E GGA ðeVÞ g
2.3
1.7
1.4
e1(0) exp. e1(0) Theory
5.2g 5.63a, 5.4d, 6.5j, 6k
5.9h 5.56a, 6.7d, 7.5j, 7k
7.3i 5.24a, 8.1d 9.5j, 9k
a
Ref. [3]. Ref. [4]. c Ref. [5,8]. d Ref. [19]. e Ref. [7]. f Ref. [6]. g Ref. [10,26]. h Ref. [10,27]. i Ref. [10,28]. j This work (LDA). k This work (GGA). b
towards higher energies by around 1 eV. (2) The bandwidth of the second group is reduced. This group is shifted towards lower energies by around 0.5 eV causing to increase the gap between the second and third groups. (3) The bandwidth of the conduction band increases slightly by around 0.5 eV on going from S to Se to Te. From the PDOS, we note a strong hybridization between Zn–d and X–p states. Following Yamasaki et al. [18] we can defined degree of hybridization by the ratio of Zn–d states and X–p states within the muffin tin sphere. Based on this we can say that the hybridization between Zn–d and chalcogen-p states becomes stronger when moving from S to Se to Te. The VBM and the CBM are located at G resulting in a direct gap in agreement with experiment and previous theoretical work [3–8,19]. We have performed calculations using LDA and GGA and find that both underestimate the energy gaps. GGA yields slightly larger energy gaps. In Table 1 we list the values of the energy gaps calculated by LDA and GGA along with the experimental values [3] and previous theoretical work [3–8]. We note that the energy gap reduces when S replaced by Se and Se by Te in agreement with the experimental data. This trend in the energy gaps is not present in previous calculations [3,19]. The reduction in the energy gaps can be attributed to the increase in bandwidth of the conduction bands. This reduction in the gap is consistent with an overall weakening of the bonds and therefore with a smaller bondingantibonding splitting. In a previous calculation [3] the second and third groups are merged for ZnS and ZnSe but not for ZnTe. 3.2. Linear optical response Since ZnX compounds have cubic symmetry, we need to calculate only one dielectric tensor component to com-
pletely characterize the linear optical properties. This component is e2(o) the imaginary part of the frequency dependent dielectric function and is given by [20] Z 1 h e i2 X 2 ðoÞ ¼ f Pji Pij dðE ji _oÞ dk, p mo i;j BZ ij where E ¼ _o, E ji ¼ E j E i and f il ¼ f i f l . fi is the Fermi occupation factor of the single-particle state i. The Pij are the momentum matrix elements. Here i and j indicates the conduction and valence bands. Fig. 2 shows the calculated e2(o) for the ZnX compounds. Broadening is taken to be 0.2 eV. The linear optical properties are scissors corrected [21] by the difference between the calculated and measured energy gaps (Table 1). We note that the e2(o) shows a large peak (located at 7.5 eV for ZnS, 6.5 eV for ZnSe, and 5.5 eV for ZnTe) between two small peaks/humps. All the structures in e2(o) are shifted towards the lower energies with increase in the peak heights when S is replaced by Se and Se by Te, in agreement with the experimental data [12]. This is attributed to the reduction in the band gaps. We compare our calculated e2(o) with the most recent calculations of Huang and Ching [3], Ghahramani et al. [7] and the experimental data [12]. In order to make a meaningful comparison, we have shifted the calculated e2(o) [3,7] to match the large peak in the experimental data. Previous calculations [3,7] underestimate the magnitude of e2(o) in the low-energy regime. This could be due to an inaccurate representation of the wave functions. However our calculated e2(o) shows very good agreement with the experimental data [12] in the matter of the peaks position and peak heights. The calculated e1(0) is compared with the experimental values in Table 1. We note that a smaller energy gap yields
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37
Fig. 1. Band structure and total density of states (states /eV unit cell), along with X–p (———), Zn–d (yy), and X–s (———) partial densities of states.
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Fig. 2. Calculated e2(o) (solid curve) along with the experimental data [12] (dashed curve) and theoretical calculation (light curve) of Huang and Ching [3] for ZnS and of Ghahramani et al. [7] for ZnSe and ZnTe.
a larger e1(0) value. This could be explained on the basis of the Penn model [22]. 3.3. Nonlinear response The complex second-order nonlinear optical susceptibility tensor wð2Þ ijk ð2o; o; oÞ can be generally written as the sum of three physically different contributions in the form [7,14,23].
wð2Þ ijk ð2o; o; oÞ
ac0 Z i h e i3 cX dk vc cc0 c0 v ¼ fP P P g 2 mo vcc0 BZ 4p3 1 ðE cv 2_oÞðE c0 v _oÞ 1 þ ðE c0 v þ 2_oÞðE cv þ _oÞ 1 þ , ðE cv þ _oÞðE c0 v _oÞ
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where v stands for a valence band state and c, c0 two different conduction band states, E cv ðkÞ ¼ E c ðkÞ E v ðkÞ is the difference in the R band energies between two states c and v, and Pcv ¼ i_ cc ðk; rÞrcv ðk; rÞ dr is the corresponding momentum matrix element. It has been demonstrated by Aspnes [10] that only one virtual-electron transitions (transitions between one valence band state and two conduction band states) give a significant contribution to the second-order tensor. Here we ignored the virtual-hole contribution (transitions between two valence band states
39
and one conduction band state) because it was shown to be negative and more than an order of magnitude smaller than the virtual-electron contribution for ZnX compounds. For ð2Þ simplicity we call wð2Þ ijk ð2o; o; oÞ as wijk ðoÞ. The subscripts i, j, and k are Cartesian indices. It has 18 elements. For non-centro-symmetric compounds with cubic group symmetry, there is only one nonzero element [3] wð2Þ 123 ðoÞ. We have calculated the imaginary and real parts of the second harmonic generation (SHG) susceptibility wð2Þ 123 ðoÞ. The calculated imaginary part is shown in Fig. 3. It is calculated
ð2Þ Fig. 3. Calculated Im w123 ðoÞ for ZnX compounds without scissors correction (dark curve) and with scissors correction (light curve).
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with and without the scissors correction. We see that the scissors correction [21] has a profound effect on magnitude and sign of wð2Þ 123 ðoÞ. That is attributed to fact that LDA calculations underestimate the energy gaps by 30–40%. A very simple way to overcome this drawback is to use the scissor correction, which merely makes the calculated energy gap equal to the experimental gap. This is a very crude method and empirical in nature. Moreover, it is clear from the formulae that, wð2Þ ijk ð2o; o; oÞ depends on the energy gap. Hence when we use the scissor correction it gives a considerable effect on wð2Þ ijk ð2o; o; oÞ. We note that all the structures in Im wð2Þ ðoÞ are shifted towards lower 123 energies when S replaced by Se and Se by Te. It is well known that nonlinear optical properties are more sensitive to small changes in the band structure than the linear optical properties. There are two reasons for that sensitivity. For one, the second harmonic response wð2Þ ijk ðoÞ [7,14,23] involves more resonances than the linear one. In addition to the usual o resonance contribution there is a 2o resonance contribution. Both the o and 2o resonances can be further separated into inter-band and intra-band contributions. Secondly, the products of matrix elements, which control the strength of a given resonance in Im wð2Þ 123 ðoÞ, can be positive or negative. In contrast, for the linear response the corresponding factors involve only the square of matrix elements, which ensures, that e2(o) is positive. As a result, the structure in Im wð2Þ 123 ðoÞ is more pronounced than in the e2(o). In Fig. 4 we present the inter-band and intra-band contributions to the o and 2o resonances for ZnS. We find that the o resonance is smaller than the 2o resonance. As can be seen the total SHG susceptibility is zero below half the band gap. The 2o terms start contributing at energies (1/2)Eg and the o terms for energy values above Eg. In the low energy regime (p3 eV) the SHG optical spectra is dominated by the 2o contributions. Beyond 3 eV the major contribution comes from the o terms.
The structures in Im wð2Þ 123 ðoÞ can be understood from the structures in e2(o). Unlike the linear optical spectra, the features in the SHG susceptibility are very difficult to identify from the band structure because of the presence of 2o and o terms. But we can make use of the linear optical spectra to identify the different resonance leading to various features in the SHG spectra. The first structure in Im wð2Þ 123 ðoÞ between 1.0 and 3.5 eV for ZnS, 0.5 and 3.0 eV for ZnSe, and 0.5 and 2.0 eV for ZnTe is associated with interference between a o resonance and 2o resonance and arises from the first hump in e2(o). The second structure between 3.5 and 4.5 eV for ZnS, 3.0 and 4.0 eV for ZnSe, and 2.0 and 3.5 eV for ZnTe is due mainly to o resonance and associated with high peak in e2(o). The last structure from 4.5 and 5.5 for ZnS, 4.0 and 5.5 eV for ZnSe, and 3.5 and 5.5 eV for ZnTe is manly due to o resonance and associated with the second hump in e2(o). In Fig. 5 we show the 2o inter-band and intra-band contributions for ZnX compounds. We note the opposite signs of the two contributions throughout the frequency range. Both the contributions increases on moving from S to Se to Te. This could be attributed to the decrease in the energy gaps on moving from S to Se to Te. In Table 2 we present the values of wð2Þ 123 ð0Þ. These values clearly increase on going from S to Se to Te in agreement with experiment and theoretical calculations (Table 2). We notice that the smaller band gap compounds gives higher values of wð2Þ 123 ð0Þ in agreement with the experiment [24,25] and theory [3,7,10,17,26–29]. Table 2 shows that there is a large variation in the theoretical values of wð2Þ 123 ð0Þ. That is attributed to the large variation in the calculated energy gaps (Table 1) because of using different methods of calculations. As we mentioned above that wð2Þ ijk ð2o; o; oÞ depends on the energy gap, so it is a natural result that one can get different values of wð2Þ 123 ð0Þ for different energy gaps. We compare our calculated wð2Þ 123 ð0Þ with the experimental data [24,25]; it shows reasonable agreement. The lack of
Fig. 4. Calculated Im wð2Þ 123 ðoÞ along with the intra (2o) and (o)-inter (2o) and (o)-band contributions for ZnS.
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41
Fig. 5. Calculated Im wð2Þ 123 ðoÞ along with the intra (2o) and inter (2o)-band contributions for ZnX compounds.
experimental data prevents any conclusive comparison with experiment over a large energy range. 4. Conclusions We have performed calculations of the band structure, DOS, linear and nonlinear optical response for the ZnX compounds, using the FP-LAPW method. Our results for band structure and DOS show that these compounds have similar structures and the energy gap decreases when S is
replaced by Se and Te. This is attributed to the fact that the bandwidth of the conduction bands increases on going from S to Se to Te. All the structures in the imaginary part of the dielectric function e2(o) are shifted towards lower energies when S is replaced by Se and Te. We compare our calculated e2(o) with the experimentally measured frequency-dependent dielectric function and previous theoretical calculations [3,7]. Our e2(o) shows better agreement with experiment data than the previous calculations. We find that the values of e1(0) increases with decreasing
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Table 2 Experimental, calculated total and intra-inter- band of the zero frequency of the real part of the wð2Þ 123 ðoÞ ZnS 123
w (0) w123(0) w123(0) w123(0) w123(0)
theory exp. total inter intra
a
ZnSe b
c
d
e
f
g
16 , 43 , 17 , 19 , 35 , 10 , 48 17i, 14.6j 20k 80k 100k
a
ZnTe b
c
f
h
d
e
g
19 , 63 , 30 , 15 , 44 27 , 53 , 38 22i, 37.4j 50k 150k 200k
53a, 112b, 73c, 51d, 90e, 146g, 29f, 73h 73i, 44j 90k 310k 400k
a
Ref. [17]. Ref. [26]. c Ref. [27]. d Ref. [28]. e Ref. [29]. f Ref. [10]. g Ref. [24]. h Ref. [7]. i Ref. [3]. j Ref. [25]. k This work (LDA). b
energy gap, in agreement with the Penn model. Our calculations of the SHG susceptibility show that the intra-band and inter-band contributions are significantly increased when S is replaced by Se and Se by Te. All the structures in wð2Þ 123 ðoÞ are shifted towards lower energies when moving from S to Se to Te. Our calculations show that the smaller band gap materials have higher wð2Þ 123 ð0Þ values. Since wð2Þ 123 ð0Þ is roughly inversely correlated to the band gap, one might think that this would lead to a possible route to further enhancement of Im wð2Þ 123 ðoÞ. The enhancement of the SHG when one substitutes S by Se and Te is considerable. Acknowledgments We would like to thank the Institute Computer Center and Physics Department for providing the computational facilities. Also to thank Prof. Claudia Ambrosch-Draxl and Dr. Sangeeta Sharma, Institute of Physics, University Graz, Universita¨tsplatz 5, A-8010 Graz, Austria for using the nonlinear code. References [1] J.L. Bredin, Phys. Today 47 (1994) 5; J.L. Bredin, Science 263 (1994) 487. [2] E. Erbarut, Solid State Commun. 127 (2003) 515. [3] M.-Z. Huang, W.Y. Ching, Phys. Rev. B 47 (1993) 9449; M.-Z. Huang, W.Y. Ching, Phys. Rev. B 47 (1993) 9464. [4] S. Sapra, N. Shanthi, D.D. Sarma, arxiv: Cond-Matt, 0308048, v1, 4 August 2003. [5] U. Rossler, Phys. Rev. 184 (1969) 733. [6] J.P. Walter, M.L. Cohen, Y. Petroff, M. Balkanski, Phys. Rev. B 1 (1970) 2661. [7] E. Ghahramani, D.J. Moss, J.E. Sipe, Phys. Rev. B 43 (1991) 9700.
[8] T. Tsuchiya, S. Ozaki, S. Adachi, J. Phys.: Condens. Matter 15 (2003) 3717. [9] J.R. Chelikowsky, M.L. Chen, Phys. Rev. B 14 (1976) 556. [10] D.E. Aspnes, Phys. Rev. B 6 (1972) 4648. [11] C.Y. Bang, M.S. Lee, T.J. Kim, Y.D. Kim, A.E. Aspnes, Y.M. Yu, B. Sung O, Y.D. Choi, J. Korean Phys. Soc. 39 (2001) 462. [12] J.L. Freeouf, Phys. Rev. B 7 (1973) 3810. [13] D. Ronnow, M. Cardona, L.F. Lastras-Martinez, Phys. Rev. B. 59 (1999) 5581. [14] S. Sharma, C. Ambrosch-Draxl, Phys. Scr. T 109 (2004) 128. [15] S.J. Czyzak, W.M. Barker, R.C. Crane, J.B. Howe, J. Opt. Soc. Am. 47 (1957) 240. [16] A. Manabe, A. Mitsuishi, H. Yoshinaga, Jpn. J. Appl. Phys. 6 (1967) 593. [17] D.T.F. Marple, J. Appl. Phys. 35 (1964) 539. [18] T. Yamasaki, N. Suzuki, K. Motizuki, J. Phys. C 20 (1987) 395. [19] A. Dal Corso, F. Mauri, A. Rubio, Phys. Rev. B 53 (1996) 15638. [20] D.J. Moss, E. Ghahramani, J.E. Sipe, H.M. van Driel, Phys. Rev. B 41 (1990) 1542. [21] S.N. Rashkeev, W.R.L. Lambrecht, Phys. Rev. B 63 (2001) 165212. [22] D.R. Penn, Phys. Rev. 128 (1962) 2093. [23] D.J. Moss, J.E. Sipe, H.M. van Driel, Phys. Rev. B 36 (1987) 9708. [24] R.A. Soref, H.W. Moos, J. Appl. Phys. 35 (1964) 2152. [25] C.K.N. Patel, Phys Rev. Lett. 16 (1966) 613. [26] D.A. Kleinman, Phys. Rev. B 2 (1970) 2139. [27] C.L. Tang, C. Flytzanis, Phys. Rev. B 4 (1971) 2520. [28] M.I. Bell, Phys Rev. B 6 (1972) 516. [29] C.L. Tang, IEEE J. Quantum Electron. QE-9 (1973) 755.
Further reading [30] M. Cardona, G. Harbeke, Phys. Rev. 137 (1965) A1467. [31] C.S. Wang, B.M. Klein, Phys. Rev. B 24 (1981) 3417. [32] D.J. Moss, E. Ghahramani, J.E. Sipe, H.M. van Driel, Phys. Rev. B 34 (1986) 8758. [33] S. Adachi, Phys. Rev. B 35 (1987) 7454. [34] J.C. Phillips, J.A. Van Vechten, Phys. Rev. 183 (1969) 709.