This content has been downloaded from IOPscience. Please scroll down to see the full text.
Download details:
IP Address: 140.113.38.11
This content was downloaded on 28/04/2014 at 02:23
Please note that terms and conditions apply.
Cation substitution effects on structural, electronic and optical properties of nonlinear optical
AgGa(S
x Se1−x )2 crystals
View the table of contents for this issue, or go to the journal homepage for more 2003 J. Phys.: Condens. Matter 15 6043
(http://iopscience.iop.org/0953-8984/15/35/312)
J. Phys.: Condens. Matter 15 (2003) 6043–6055 PII: S0953-8984(03)62278-2
Cation substitution effects on structural, electronic
and optical properties of nonlinear optical
AgGa(S
x
Se
1
−x
)
2
crystals
L C Tang1, M H Lee2, C H Yang2, J Y Huang1and C S Chang1,3
1Institute of Electro-Optical Engineering, National Chiao Tung University Hsinchu 305,
Taiwan, Republic of China
2Department of Physics, Tamkang University Taipei 251, Taiwan, Republic of China
E-mail: [email protected]
Received 12 April 2003, in final form 29 July 2003 Published 22 August 2003
Online atstacks.iop.org/JPhysCM/15/6043
Abstract
The structural, electronic and optical properties of tetragonal nonlinear optical (NLO) crystals, AgGa(SxSe1−x)2 (x = 0.0, 0.25, 0.5, 0.75, and 1.0), were
investigated theoretically and experimentally. The results obtained indicated that the electronic bandgaps, optical properties and bulk moduli of these compounds were linearly dependent on the substitution concentration of cations. From partial density of state analysis, it was found that the electronic states near the band edges of AgGa(SxSe1−x)2 were a simple proportional
mixture of the atomic orbitals of sulfur and selenium. A cell-volume effect was proposed as the major cause of the linear dependence of material properties on the substitution concentration. It was calculated that the second-order NLO susceptibilities were scaled with the cubic power of bandgap, although a minor deviation existed. This deviation arose from the optical transition moment products.
1. Introduction
The most important technique to generate tunable coherent radiation in the mid-infrared spectrum is based on the second-order nonlinear optical (NLO) processes in a noncentrosymmetric crystal. These NLO processes include difference-frequency mixing, optical parametric generation and amplification.
The crystals used for mid-IR generation with frequency down conversion schemes should possess a wide optical transparency range from 1 µm to well beyond 5 µm. Currently those crystals are very limited and only a few of them have really become commercially available [1]. Among those available crystals, AgGaS2and AgGaSe2are well known for their 3 Author to whom any correspondence should be addressed.
Figure 1. The crystal structure of tetragonal chalcopyrite.
(This figure is in colour only in the electronic version)
wide transparency, large birefringence, high optical damage threshold and nonlinearity. To meet the demand from specific applications, the linear and NLO properties of AgGa(SxSe1−x)2
can be adjusted by varying the alloy composition [2–4]. Xue et al [5] proposed the similar concept that most of the properties of the materials were closely correlated to the micro-structure, which depends on the constituent atoms or ions or, more precisely, the average atomic number of the constituent atoms or ions. Xue et al [6] also applied this concept to some I–III–VI2chalcopyrites, e.g. AgGaS2and AgGaSe2, to analyse the linear and nonlinear
dielectric response of these semiconductors. Other than the work of Xue et al, there was little on the solid solution of these ternary and quaternary chalcogenides [4, 7–11].
In this paper, the first-principles calculation of electronic structures, and optical and bulk properties of tetragonal NLO crystals, AgGa(SxSe1−x)2(x = 0.0, 0.25, 0.5, 0.75, and 1.0), are
presented. Tang et al used the same approach to investigate the electronic structures, and optical and bulk properties of the rhombohedral ternary halides [12] and the orthorhombic ternary nitrides [13]. The analysis with band-by-band and atomic species projection techniques [12, 13] yielded useful information about material properties and provided deep insight into the fundamental understanding of the electronic structures and optical properties of tetragonal NLO crystals, AgGa(SxSe1−x)2.
The synthesis method of single crystals and measurements of the optical properties in each composition are reported. Nonlinear coefficients of AgGa(SxSe1−x)2were also carried
out to reveal the potential of these crystals in NLO applications. The calculations were then compared with the experimental data and values in the literature.
2. Methods of simulation and experimental measurements
The unit cell exhibits I ¯42d symmetry (a= b = c, α = β = γ = 90◦) [14, 15], which can be considered as a superstructure of two zinc blende structures. There were four metal atoms Ag, Ga, and eight crystallographically equivalent S or Se atoms in a unit cell which occupied the symmetrical positions of [(x , y, z); (¯x, ¯y, ¯z); (y, ¯x, ¯z); ( ¯y, x, ¯z); ( ¯x + 1/2, y, ¯z + 3/4); (x + 1/2, ¯y, ¯z + 3/4); ( ¯y + 1/2, ¯x, z + 3/4); (y + 1/2, x, z + 3/4)]. In AgGa(SxSe1−x)2crystals,
metal atoms are tetrahedrally coordinated by S or Se and vice versa. 2.1. First-principles calculations
The first-principles calculations were performed using a plane-wave pseudopotential approach within the framework of density-functional theory (DFT) implemented in CASTEP software [16]. The supercell (SC) of AgGaS2 was assembled by enlarging the a-direction
and b-direction twice (2× 2 × 1), and then by substituting fractionally (x = 0.00, 0.25, 0.5, 0.75, and 1.0) atom by atom. The summation over the Brillouin zone (BZ) was carried out with a special k-point sampling using a Monkhorst-Pack grid [17]. A kinetic energy cut-off of 580 eV, 4-k special points and 96 bands were used to ensure convergence in the calculation of the optical properties.
In order to save computation time, the special k-point set was reduced to 8-k points (3× 3 × 3 mesh) for the calculation of the equilibrium lattice constants and mechanical properties. The smaller set of k-points was found to give less than 2% difference in the equilibrium lattice constants. For comparison, a non-SC model (1× 1 × 1) was also built up (hereafter named the single unit cell (SUC) model) with the same fractional substitution. It performed similar calculations to those with the SC models. In the ab initio calculation, both the 3-electron shallow-core pseudopotential in the SC model and the 13-electron shallow-core pseudopotential in SUC model for the gallium series were chosen to take into account the noticeable hybridization between the (3)d-shell and (4)p-shell electrons [18–20].
2.1.1. Calculating structural properties. The equilibrium lattice constants and fractional atomic coordinates were deduced from the total-energy minimization. Relaxation of the lattice parameters and atomic positions was carried out under the constraint of the unit-cell space-group symmetry. Energy–volume relations were obtained by varying unit-cell volume and the fitted results were obtained using the Murnaghan equation of state [21]
Etot= B0V B0 V0 V B0 1 B0− 1 + 1 + E0. (1)
From the fitted results, the estimate of the static bulk modulus B0at zero pressure, and
the first-order pressure derivative of the bulk modulus B0 were obtained. The chalcopyrite structure showed two types of distortion, namely, tetragonal (η = c/2a = 1) and tetrahedral (internal parameter u), with respect to the zinc blende cubic structure [22–24]. The parameter u can be used to yield a measure of bond length mismatch, through the relation
(u − 0.25)2· a2= d2 AC− d
2
BC (2)
where the subscripts A= Ag, B = Ga, and C = S or Se.
2.1.2. Electronic properties. The obtained lattice parameters were used to calculate the electronic properties of AgGa(SxSe1−x)2by proportionally substituting S and Se into the SC
and the SUC model. In order to understand the nature of optical transitions and other relevant effects on the calculated optical properties, an analysis of the local densities of states (LDOS) and other electronic properties of these AgGa(SxSe1−x)2crystals was made.
Letnk(r) be the self-consistent wavefunction of the crystal at the nth band and kth point
in the BZ, thennk(r) can be decomposed into a linear combination of the atomic orbitals
(i)lm(r) of each ith atom by
nk(r) = i∈(atoms) l m=+l m=−l Cnk(i),lmlm(i)(r) (3) where Cnk(i),lm =V 0nk(r)· (i)∗
lm (r) dV . The lth orbital of the α-species contributes to nk(r)
by a fraction of h(α)nk,l [25] h(α)nk,l = i∈(α) m=+l m=−l Cnk(i),lmC(i) ∗ nk,lm i∈(α,β,γ,...) l m=+l m=−l Cnk(i),lmC(i) ∗ nk,lm. (4)
By using the plane-wave basis set in a plane-wave pseudopotential scheme, the representation of natural atomic orbitals around the centres of atoms was neglected. However, this disadvantage could be overcome by projecting the solved Bloch states onto the atomic orbitals constructed using the radial pseudo-atomic wavefunctions of each angular momentum channel for each element (the angular parts were simply spherical harmonic functions) [26, 27]. This was useful for extracting the local (atomic) information of the materials. Those radial pseudo-atomic wavefunctions were the ones used to generate pseudopotentials and, therefore, had the best consistency with the solved Bloch states. The projected values were equivalent to the coefficients of a linear combination of atomic orbitals (LCAO) type expansion of the original Bloch states using pseudo-atomic orbitals as basis functions. The collection of those coefficients allows local information to be extracted from the system as a whole, whereas in the present work partial or projected density of states (PDOS) plots have been used as an analysis tool. This atomic projection concept was then employed for resolving interesting components from total density of states (TDOS)
LDOS(α, E) = n k l h(α)nk,lδ(E − Enk) (5) PDOS(α, l, E) = n k h(α)nk,lδ(E − Enk). (6)
The PDOS and LDOS could be used to provide a valuable insight into the formation of energy bandgaps and the nature of transitions from which the linear and NLO properties originated.
2.1.3. Nonlinear optical properties. For the second-order NLO response, the theoretical description was very complex [28]. However, at the zero frequency limits, the NLO susceptibility could be expressed as
χi j k(2)(0) = 1 V e¯h m 3 k vc c 1 EccEcv2 Ec2v (Di j k vcc+ D i j k cvc+ D i j k ccv) − v 1 EvvEcv2 Ec2v (Di j k vcv+ D i j k vvc+ D i j k cvv) (7) where Dnmli j k = Im[pnmi (p j mlp k ln+ p k mlp j
ln)]/2. Here pcvi (k, Å−1) denotes the momentum matrix
element (MME) from the conduction band c to the valence band v at the k-point of the BZ. These material properties were calculated using both the SC and the SUC model of AgGa(SxSe1−x)2.
Figure 2. X-ray powder diffraction of the NLO crystals AgGa(SxSe1−x)2. It can be seen that the
XRD peaks shift with S composition. A minimum peak height at [312] occurs with x= 0.5.
2.2. Single crystal growth and measurements of the characteristics of AgGa(S/xSe1−x)2
2.2.1. Single crystal growth and x-ray diffraction. The AgGa(SxSe1−x)2single crystals were
grown with the vertical Bridgman technique. All the starting reagents, Ag, Ga, S and/or Se, had a nominal purity of 99.999%. Particular attention was paid to the mole ratio among the starting reagents. Since vapour compositions and partial pressures of the gaseous species were unknown, the optimum conditions for growing high quality crystals were empirically discovered. X-ray diffraction (XRD) measurement was employed to determine the lattice parameters of all the crystals AgGa(SxSe1−x)2. In addition, the crystals of certain orientation
relative to crystallographic axes were cut into pellets and polished on both sides for optical measurements.
2.2.2. Optical transmission measurements. The polished sample plates were loaded into a liquid-nitrogen-cooled vacuum cryostat. Optical transmission spectra were recorded from visible to mid-IR in order to determine the optical absorption edges. Then, the photoluminescence (PL) spectra were measured. The measurements of both absorption edges and PL spectra gave information about the bandgap energy Eg. These measurement were
compared with the calculated results from section 2.1.2.
Fourier-transform infrared (FTIR) spectra of the AgGa(SxSe1−x)2crystals were measured
from 0.8 to 18µm to examine the transparency of these crystals.
3. Results and discussion
The calculated and measured lattice properties of five crystals are summarized in tables 1 and 2. The powder XRD patterns of AgGa(SxSe1−x)2crystals are shown in figure 2.
3.1. Crystal structure and mechanical properties
The calculated structural parameters shown in tables 1 and 2 were in good agreement with the experimental values of AgGaSe2as reported by Jaffe and Zunger [23, 24] and this work. The
Table 1. Structural parameters, bond lengths and bulk moduli of the NLO crystals AgGa(SxSe1−x)2. Values from first-principles calculations for the SC model compared with the
experimental values. x 0.00 0.25 0.50 0.75 1.00 a(Å) (exp.) 5.97 5.92 5.86 5.80 5.74 a(Å) (calc.) 5.87 5.89 5.77 5.66 5.63 c(Å) (exp.) 10.88 10.73 10.57 10.42 10.26 c(Å) (calc.) 10.64 10.60 10.24 10.09 10.02 Ga–S (Å) (calc.) 2.17 2.14 2.14 2.15 Ga–Se (Å) (calc.) 2.30 2.29 2.27 2.26 Ag–S (Å) (calc.) 2.64 2.60 2.59 2.59 Ag–Se (Å) (calc.) 2.70 2.68 2.67 2.65 η (exp.) 1.79 1.80 1.80 1.81 1.82 η (calc.) 1.81 1.87 1.84 1.85 1.84 u [S] (calc.) 0.32 0.32 0.32 0.32 u [Se] (calc.) 0.31 0.31 0.31 0.31 B0(GPa) (calc.) 7.31 7.71 8.15 8.64 9.11
Table 2. The calculated structural parameters and bond lengths of the NLO crystals AgGa(SxSe1−x)2using first-principles calculations for the SUC model.
x 0.00 0.25 0.50 0.75 1.00 a(Å) 5.87 5.89 5.81 5.73 5.63 c(Å) 10.78 10.32 10.25 10.15 10.17 Ga–S (Å) 2.34 2.30 2.29 2.31 Ga–Se (Å) 2.44 2.43 2.41 2.40 Ag–S (Å) 2.61 2.57 2.55 2.58 Ag–Se (Å) 2.67 2.65 2.63 2.65 η 1.81 1.79 1.78 1.75 1.84 u [S] 0.29 0.29 0.29 0.29 u [Se] 0.28 0.28 0.28 0.29
It was noted that the tetrahedral coordination of S and/or Se atoms could be distorted by the presence of two types of metal–S and/or metal–Se bonds. The average bond lengths were calculated to be Ga–S (2.17–2.15 Å), Ga–Se (2.30–2.26 Å), Ag–S (2.64–2.59 Å), and Ag–Se (2.70–2.65 Å). The calculated bond lengths of Xue et al were Ga–S (2.235 Å) and Ag–S (2.605 Å) in AgGaS2, then Ga–Se (2.416 Å) and Ag–Se (2.601 Å) in AgGaSe2. Our
research presented a shorter bond length for Ga–X (X= S, Se) than the results calculated by Xue et al. On the other hand, we have a longer bond length for Ag–X(X = S, Se) than the results calculated by Xue et al.
The calculated structures of AgGa(SxSe1−x)2 appeared to be more distorted than the
observed structure, which was reflected in a larger value for the bond length mismatch parameter u (table 1). The calculated values of AgGa(SxSe1−x)2 crystals were about 0.3
SUC models (about 0.29) and the experimental values of AgGa(SxSe1−x)2, reported by Jaffe
and Zunger (about 0.27).
From the DFT calculations, the equation-of-state parameters such as the bulk modulus and its pressure derivative were deduced, and the results are also summarized in table 1. The bulk modulus for AgGaS2 was approximately 9.11 GPa. It decreased to 8.15 GPa for
AgGa(SxSe1−x)2and 7.31 GPa for AgGaSe2.
XRD patterns from figure 2 showed that the substitution-related diffraction peaks, e.g. at the [112] plane of 2θ ≈ 26◦ or at the [312] plane of 2θ ≈ 52◦, shifted gradually with substitution composition. From the crystal models used in this study, the S and Se atoms were located between these atomic planes. The substitution of S with Se atoms caused the change of interplanar spacing. The peaks from the [312] plane were found to have a minimum peak height when S and Se are 50% occupied. The S and Se atoms randomly occupied the crystallographic sites between the [312] planes, which resulted in a reduction in this structural factor.
The lattice parameters of AgGa(SxSe1−x)2 crystals varied linearly with x : a =
b ∼= (5.99 − 0.55x) Å, c ∼= (10.87 − 0.57x) Å. This also generated a linear dependence of the cell volume with x : V ∼= (387.88 − 54.50x) Å3.
3.2. Electronic structures, density of states and optical spectrum
The calculated bandgap values for the AgGa(SxSe1−x)2 crystals are now presented. The
calculated bandgap which varied from 1.76 to 2.43 eV was found to be proportional to the composition of S, but inversely proportional to the cell volume. The results for the bond length in section 3.1 agreed with the concept that stronger bonding between (Ag, Ga) and (S, Se) atoms leads to a shorter bond length in AgGa(SxSe1−x)2crystals with increasing S (higher
(x)) composition. This observed tendency of bulk modulus in section 3.1 was consistent with the trend of the weaker bonding strength (larger unit-cell volume) and the narrower bandgap. Grown crystals up to 10 mm in diameter and 5 cm in length could be prepared routinely. All grown samples were single crystals, which exhibit various colours from yellow (AgGaS2)
through red AgGa(SxSe1−x)2 to black (AgGaSe2). For comparison with the calculated
electronic bandgap, the results were obtained by both absorption edges and PL spectra which gave information about the bandgap energy Eg. Figure 3 shows the absorption edge of the
resulting crystals.
In figure 3, the absorption spectra measured using a single crystal AgGa(SxSe1−x)2in the
visible light range are shown. The absorption edge is found to increase from 1.7 to 2.64 eV with the composition of sulfur. PL spectra (figure 4) reveal a similar trend of x -dependence. From PL measurements at room temperature, the bandgap values of AgGa(SxSe1−x)2crystals
can be fitted to EPL(x) ∼= (1.76 + 0.65x) eV. The experimental data taken from Matsushita
et al [7] and this work place the calculated bandgap between the values for the SUC model and the SC model. The resulting bandgap of AgGa(SxSe1−x)2can be tuned linearly in a wide
spectral range by adjusting the substitution composition. Furthermore, these crystals are all direct bandgap (see figure 5) with potential applications as efficient absorber or photon emitter. To gain further insight into the calculated optical properties, the PDOS and LDOS spectra of these crystals were calculated (see figures 6(a)–(e)).
We found that the valence band maximums (VBM) of AgGa(SxSe1−x)2 were mainly
contributed by the (3)p-orbital of S and (4)p-orbital of Se atoms. However the roles played by S or Se atoms at the band edges cannot be distinguished. Their contribution to the TDOS is a simple weighted summation over the S and Se compositions (see figure 6(f)). The substitution effect of the S or Se in AgGa(SxSe1−x)2crystals appeared from the influences of cell volume
Figure 3. Absorption edge measurements of the NLO crystals AgGa(SxSe1−x)2.
Figure 4. The first-principles calculated bandgaps and PL measurements of the NLO crystals AgGa(SxSe1−x)2. (*): [7].
change. The performance of first-principles calculations on a series of artificial structures was prepared by expanding the cell volume of AgGaS2and by compressing that of AgGaSe2.
The calculated results showed clearly a similar effect on the bandgap and the related optical properties. This finding agreed with an empirical rule proposed by Vegard, which stated that the change of material properties caused by substitution could be calculated by simply taking a composition-weighted average of the pure substances, here AgGaS2and AgGaSe2[29–31].
FTIR measurements showed that the long wavelength limit of the transparent range of the crystals exhibited a similar dependence on substitution composition. Crystal AgGaS2had an
(a) (b)
(d) (c)
(e)
Figure 5. Band structures of the NLO crystals AgGa(SxSe1−x)2.
infrared cut-off wavelength at approximately 12µm, which was shorter than the cut-off value of AgGaSe2(approximately 18µm). The infrared absorption edge of AgGa(SxSe1−x)2with
x = 0.25, 0.5, 0.75 lay approximately from 13 to 15 µm. This result was within acceptable range in view of the fact that the Se atom is heavier than S and the infrared absorption edge was dominated by the stretching modes of the S– and Se–metal bonds.
3.3. Second-order nonlinear optical susceptibilities
The second-order NLO susceptibility was known to be much more sensitive to crystal structure than linear optical susceptibility. The variation of the calculated second-order nonlinear susceptibilities was compared with the substitution-induced distortion. The results are summarized in the table 3. Complications from dispersion and resonant effects can be avoided by using equation (7) to calculate the static second-harmonic susceptibility (the zero frequency limit) of the AgGa(SxSe1−x)2 crystals. For chalcopyrites, only two independent
Figure 6. Analysis of the PDOS of s- and p-orbital contributions via S and Se in the NLO crystals AgGa(SxSe1−x)2.
components, the 123 and 312 components, ofχ(2)exist, where 1, 2 and 3 refer to the x , y and z directions along the cubic axes respectively [32]. However, the input photon frequency being below the material absorption edge, these two components were essentially equal according to the Kleinman symmetry.
For these materials, the calculatedχ123(2) was negative with a magnitude lying between −22.07 and −51.21 pm V−1for AgGa(S
xSe1−x)2with the SC model. The bandgap correction
scheme was not utilized. Theseχ123(2) values agreed very well with the experimental results for AgGaS2 and AgGaSe2 [2, 6, 33–35]. Using the SUC model, the calculatedχ123(2) ranged
from−47.07 to −140.411 pm V−1 with different x (see table 3). Theseχ(2) values were approximately twice as large as the values calculated using the SC model. The nonlinearity of material is inversely proportional to the cubic power of the bandgap [36]. This simple scaling rule was used to remove bandgap influence on the second-order nonlinearity. These results are presented in figure 7. AgGaS2was used as the reference and the cubic power scaling rule was
employed to predict the optical nonlinearity of other crystals. The rescaled calculatedχ123(2) using the SC model were labelled as the scaled values (filled symbols) and were compared to
Figure 7. The first-principles calculations of the second-harmonic susceptibilityχ(2)and the scaled values ofχ(2)of the NLO crystals AgGa(SxSe1−x)2. The scaledχ(2)values were estimated from
that of the AgGaS2with a cubic power of Eg.
Table 3. The calculatedχ123(2) (pm V−1) of NLO crystals AgGa(SxSe1−x)2using first-principles
calculations with the SC model and the SUC model.
x
0.00 0.25 0.50 0.75 1.00 Condition
This work −51.21 −43.31 −32.14 −26.08 −22.07 SC model
Reference [6] −42.50 — — — −16.50 Bond charge model
Reference [2] 65.50 — — — 25.90 —
Reference [33] 64.00 — — — 23.00 Measured
Reference [34] 68.00 — — — 18.00 Measured
Reference [35] 66.00 — — — 22.00 Measured
This work −140.41 −107.03 −67.90 −54.22 −47.07 SUC model
the first-principles calculations (open symbols) with the SUC model. As shown in figure 7, the deviation can be as large as 20% at x= 0.25. The difference was likely to originate from the transition moment products which were known to be sensitive to the detailed arrangement of the unit cell.
Based on these findings, the linearly increasing x in AgGa(SxSe1−x)2 caused a linear
compression in cell volume, a linear increase in bandgap values and a decrease in second-order NLO susceptibility,χ(2)in the cubic power of bandgap.
4. Conclusions
The structural, electronic and optical properties of tetragonal NLO crystals, AgGa(SxSe1−x)2
(x = 0.0, 0.25, 0.5, 0.75, and 1.0), have been investigated theoretically and experimentally to reveal the cation substitution effect. Based on the results, the linearly increasing x in AgGa(SxSe1−x)2 caused a linear compression in cell volume, a linear increase in bandgap
values and a decrease second-order NLO susceptibility. Xue et al discussed the linear and nonlinear dielectric responses relevant to the constituent effects. This research revealed that the NLO properties were sensitive to the electronic structures which were influenced by the unit cell structures and the constituent atoms.
The bond lengths of Ag–S, Ag–Se, Ga–S and Ga–Se were found to vary with the substitution composition, but the bond length mismatching parameter u did not change. It was found that the major influence of cation substitution was simply introducing a change of cell volume. PDOS analysis showed that the VBM of AgGa(SxSe1−x)2were mainly contributed
by the (3)p-orbital of S and (4)p-orbital of Se atoms. Their contribution to the TDOS was a simple weighted summation over S and Se compositions. The CBM and VBM were relatively shifted with the substitution concentration as well as the fundamental bandgap and, therefore, were reduced accordingly. The second-harmonic generation coefficientsχ123(2)were also found to change with substitution. The substitution dependence could be ascribed to the bandgap change and a minor contribution from the transition moment products. Based on theoretical and experimental results obtained, these materials could be useful for both linear and NLO applications.
Acknowledgments
The authors are indebted to the financial support from the National Science Council of the Republic of China under grant NSC 91-2112-M-009-036. M H Lee acknowledges with gratitude the funding support from NSC 91-2112-M-032-014. Address any correspondence to C S Chang.
References
[1] Petrov V, Rotermond F and Noack F 2001 J. Opt. A: Pure Appl. Opt. 3 R1 [2] Rashkeev S N and Lambrecht W R L 2001 Phys. Rev. B 63 165212 [3] Roberts D A 1992 IEEE J. Quantum Electron. 28 2057
[4] Mikkelson J C Jr and Kidal H 1978 J. Appl. Phys. 49 426
[5] Xue D, Betzler K, Hesse H and Lammers D 1999 Relationship between dielectric responses and constituent atoms in crystal materials Phys. Status Solidi b 216 R7
[6] Xue D, Betzler K and Hesse H 2000 Dielectric properties of I–III–VI2-type chalcopyrite semiconductors Phys.
Rev. B 62 13546
[7] Matsushita H, Shiono O, Endo S and Irie T 1995 Japan. J. Appl. Phys. 34 5546
[8] Yamamoto N, Mouri A, Seyoyama M and Horinaka H 1989 Japan. J. Appl. Phys. 28 2513 [9] Balakrishnan K, Vengatesan B and Ramasamy P 1994 J. Mater. Sci. 29 1879
[10] Robbins M and Lambrecht V 1973 MRS Bull. 8 703
[11] Jackson A, Ohmer M and LeClair S 1997 Infrared Phys. Technol. 38 233
[12] Tang L-C, Chang C-S and Huang J Y 2000 Electronic structure and optical properties of rhombohedral CsGeI3
crystal J. Phys.: Condens. Matter 12 9129
[13] Huang J Y, Tang L C and Lee M H 2001 Ab initio study of the structural and optical properties of orthorhombic ternary nitride crystals J. Phys.: Condens. Matter 13 10417
[14] Abrahams S and Berstein J 1973 J. Chem. Phys. 59 1625 [15] Tell B and Kasper H 1971 Phys. Rev. B 4 4455
[16] Payne M C, Teter M P, Allan D C, Arias T A and DJoannopoulos J 1992 Rev. Mod. Phys. 64 1045 [17] Monkhorst H J and Pack J D 1976 Phys. Rev. B 13 5188
[18] Fiorentini V, Methfessel M and Scheffler M 1993 Phys. Rev. B 47 13353 [19] Wright A F and Nelson J S 1994 Phys. Rev. B 50 2159
[20] Karch K, Bechstedt F and Pletl T 1997 Phys. Rev. B 56 3560 [21] Milman V, Lee M H and Payne M C 1994 Phys. Rev. B 49 16300
[22] Camassel J, Artus L and Pascual J 1990 Lattice dynamics of AgGaSe2 i experiment Phys. Rev. B 41 5717 [23] Jaffe J and Zunger A 1984 Phys. Rev. B 28 5822
[24] Jaffe J and Zunger A 1984 Phys. Rev. B 29 1882
[25] Sanchez-Portal D, Artacho E and Soler J M 1996 J. Phys.: Condens. Matter 8 3859
[26] Segall M D, Pickard C J, Shah R and Payne M C 1996 Population analysis in plane wave electronic structure calculations Mol. Phys. 89 571–7
[27] Segall M D, Shah R, Pickard C J and Payne M C 1996 Population analysis of plane-wave electronic structure calculations of bulk materials Phys. Rev. B 54 16317–20
[28] Rashkeev S N, Lambrecht W R L and Segall B 1998 Phys. Rev. B 57 3905 [29] Fong C and Phillips J 1976 Phys. Rev. B 14 5387
[30] Wooly J, Williardson R and Goering H 1962 Compound Semiconductors (New York: Reinhold) [31] Wooly J 1962 Compound Semiconductors (New York: Reinhold)
[32] Wyckoff R W G 1963 Crystal Structures (New York: Interscience)
[33] Shay J L and Wernick J H 1975 Ternary Chalcopyrite Semiconductors: Growth, Electronic Properties, and
Applications (Oxford: Pergamon)
[34] Kurtz S K, Jerphagnon J and Choy M M 1979 Numerical Data and Functional Relationships in Science and
Technology (Oxford: Pergamon) p 671
[35] Roberts D A 1992 IEEE J. Quantum Electron. 27 142
