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Relationship between the thermal expansion coefficient, plasmon energy, and bond length of ternary chalcopyrite semiconductors
Journal of Physics and Chemistry of Solids 63 (2002) 107±112
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Relationship between the thermal expansion coef®cient, plasmon energy, and bond length of ternary chalcopyrite semiconductors V. Kumar*, B.S.R. Sastry Department of Electronics and Instrumentation, Indian School of Mines, Dhanbad 826 004, India Received 13 November 2000; accepted 9 March 2001
Abstract A simple relation between the average bond length and plasmon energy, that is, d 15.30 "vp 22=3 , has been proposed for A IIB IVC2V and A IB IIIC2VI chalcopyrite semiconductors. The average linear thermal expansion coef®cient (a L) of these semiconductors has been calculated using plasmon energy data. The linear expansion coef®cients (a a) and (a c) of the lattice parameters a and c, respectively, have also been calculated. The calculated values of d, a L, a a and a c have been compared with the experimental values and the values reported by different workers. A fairly good agreement has been obtained between them. q 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction In the recent past, increasing attention has been given towards the study of various electronic, electrical and optical properties of A IIB IVC2V and A IB IIIC2VI chalcopyrite semiconductors [1±10]. This is because of their potential applications in a variety of optoelectronic devices. Different workers [11±20] have studied thermodynamical properties such as melting point and thermal expansion coef®cient of these chalcopyrites. The study of thermal expansion coef®cients a L, a a and a c is important in the growth of single crystal [21,22], in the understanding of the temperature dependence of the electronic properties [9,10,23] and in the choice of substrate materials suited for epitaxial growth [7,24] of these semiconductors. Van Uitert et al. [13,15] have shown that average thermal expansion coef®cient (a L) depends upon melting point, orbital extension and site symmetry. They have further shown that the product of coef®cient of thermal expansion and melting point (a LTm) has a constant value for materials with ®xed crystal structure. The values of this constant for rectilinear, close-packed, rutite and tetrahedral structures have been obtained as 0.027, 0.016, 0.020 and 0.021, respec-
* Corresponding author. Tel.: 191-326-206247; fax: 191-326203042. E-mail address: [email protected] (V. Kumar).
tively [15]. The A IIB IVC2V and A IB IIIC2VI semiconductors crystallize in the tetragonal chalcopyrite structure and exhibit diamond-like or tetrahedral structure [7]. In this type of structure, two kinds of cations make an ordered sublattice which may result in a tetragonal distortion of the lattice de®ned as D ; 2 2 c/a, where c and a are the lattice parameters. Because of the axial symmetry of the crystals, the linear thermal expansion coef®cients a a and a c are anisotropic, which gives rise to a nonvanishing temperature dependence of the tetragonal distortion. In the studies of thermal expansion coef®cients a a and a c of A IIB IVC2V [21,22,25] and A IB IIIC2VI [22,23] semiconductors, it has been assumed that dD=dT increases monotonously with increasing D [26,27]. However, this supposition has been shown to be incorrect by Yamamoto et al. [23] and Kistaiah et al. [28±30]. Later, Neumann [16,17] gave a model for the temperature dependence of tetragonal distortion (D ) based on free energy consideration and explained the thermal expansion behavior of these semiconductors [16,19]. An X-ray bond method has also been used to explain the behavior of a a and a c in CuInTe2 compound in the temperature range of 30±300 K [18,20]. Recently, the ®rst author [9,10] has used the solid state theory of plasma oscillations to explain interatomic force constants of these ternary and some binary (A IIB IV and A IB V) semiconductors. The plasmon energy is related to the effective number of valence electrons in a compound. The bond length also depends on the effective number of
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108
V. Kumar, B.S.R. Sastry / Journal of Physics and Chemistry of Solids 63 (2002) 107±112
valence electrons. This shows that there must be a correlation between bond length and plasmon energy. Bond length is an important parameter for calculating the values of thermal expansion coef®cients. Earlier workers have used the experimental values of bond length while calculating the values of a L, a a and a c. The experimental values of bond length are still unknown for some semiconductors. Therefore, it has been of interest to develop a theoretical model for the calculation of bond length. In the present paper, we have extended the earlier work of the ®rst author [9,10] and proposed a simple relation between bond length and plasmon energy for the calculation of average bond length of A IIB IVC2V and A IB IIIC2VI semiconductors. The calculated values of the average bond length (d) have been further used to calculate the values of linear thermal expansion coef®cients a L, a a and a c. The calculated values of all four parameters are compared with the experimental values and the values reported by different workers [16±22]. 2. Calculation of bond length and plasmon energy The free-electron plasmon energy is given by the relation "vp 2
4pNe2 e2 ; m
1
also be written as [31] r Zs "vp 28:8 eV; W
6
where Z is the effective number of valence electrons taking part in plasma oscillations, s is the speci®c gravity and W the molecular weight. Using Eq. (6), plasmon energies of A IIB IVC2V and A IB IIIC2VI semiconductors have been calculated taking Z 16, and s and W data from a book of constants [32]. The calculated values of plasmon energies "v p are listed in Table 1, which can be written in terms of individual bonds as "vp
"vp;AC 1 "vp;BC ; 2
7
where "vp;AC and "vp;BC are the plasmon energies of the A± C and B±C bonds in A IIB IVC2V and A IB IIIC2VI semiconductors. In Table 1, the plasmon energies of ZnSnP2, CdSnP2 and CdSiAs2 are taken from our previous publications [33], since the values of speci®c gravity are not known [32]. However, from back calculation we can predict the values of s , which are equal to 4.48,4.02 and 5.15, respectively, for ZnSnP2, CdSnP2 and CdSiAs2 semiconductors. Several workers [1±8,34] have also used plasmon energy data and explained various electronic properties such as ionicity, energy gaps, dielectric constant and hardness of these semiconductors.
from which we have Ne2
m "vp 2 ; 4pe2
2
where Ne is the effective number of free electrons taking part in the plasma oscillations, e is the charge and m the mass of electron. Ne can be written in terms of individual bond properties as ZA Z 1 B NCA NCB ; 3 Ne vb where ZA and ZB are the numbers of valence electrons of the atoms A and B, respectively, in an AB compound. NCA and NCB are the coordination numbers of the atoms A and B, and vb is the bond volume. For tetrahedral p crystals, NCA NCB 4, ZA 1 ZB 8 and vb 4d 3/3 3. Substituting these values in Eq. (3), we get p 3 3 Ne : 4 2d 3 From Eqs. (2) and (4), we get the following equation for the average bond length of A IIB IVC2V and A IB IIIC2VI semiconductors d 15:30 "vp 22=3
d in A and "vp in eV:
5
For a compound, the plasmon energy shown in Eq. (1) can
3. Average linear thermal expansion coef®cient The melting point Tm can be taken as a direct measurement of binding energy and an inverse measure of the volume expansion coef®cient a v or average linear thermal expansion coef®cient, that is, a L (1/3)a v. Van Uitert et al. [13±15] have shown that the product a LTm is a constant for materials with ®xed ionicity under these conditions. In reality, the spatial extension of the orbital around the atoms involved may signi®cantly affect anharmonicity and, therefore, thermal expansion. This means that besides the bond ionicity, the bond length must be taken into account in considering the average linear thermal expansion coef®cient of a given material and we have developed Eq. (5) for the calculation of bond length. Based on thermal expansion coef®cient data, Neumann [16] has proposed the following expression for average thermal expansion coef®cient for binary tetrahedral semiconductors
aL
A 2 B d 2 d0 3 ; Tm
8
where A is a constant, Tm is the melting temperature in Kelvin and d the bond length. The value of A 0.021 for all tetrahedrally coordinated compounds (A IV, A IIIB V, A IIB VI) as estimated from a hard sphere model based on the diamond structure. The value of d0 should be equal to
V. Kumar, B.S.R. Sastry / Journal of Physics and Chemistry of Solids 63 (2002) 107±112
109
Table 1 Average bond length (d) and thermal expansion coef®cient (a L) of ternary chalcopyrites Compd.
A IIB IVC2V ZnSiP2 CdSiP2 ZnGeP2 CdGeP2 ZnSnP2 CdSnP2 ZnSiAs2 CdSiAs2 ZnGeAs2 CdGeAs2 ZnSnAs2 CdSnAs2 A IB IIIC2VI CuAlS2 CuAlSe2 CuAlTe2 CuGaS2 CuGaSe2 CuGaTe2 CuInS2 CuInSe2 CuInTe2 AgAlSe2 AgAlTe2 AgGaS2 AgGaSe2 AgGaTe2 AgInS2 AgInSe2 AgInTe2 CuTlSe2 a
s
3.39 4.00 4.17 4.48
W
"v p
Ê) Cal. `d' (A
5.32 5.60 5.53 5.72
155.40 202.43 199.90 246.94 246.00 243.03 242.20 290.34 287.80 334.83 333.90 380.92
17.02 16.19 16.64 15.52 15.55 a 14.82 a 16.05 15.35 a 15.66 14.90 14.82 14.12
2.312 2.390 2.347 2.459 2.456 2.536 2.404 2.477 2.445 2.527 2.536 2.619
3.47 4.70 5.50 4.35 5.56 5.99 4.75 5.77 6.10 5.07 6.18 4.72 5.84 6.05 5.00 5.81 6.12 7.11
154.65 248.45 354.73 197.39 291.19 388.47 242.49 336.29 433.57 292.77 390.05 241.71 355.51 432.79 286.87 380.61 477.89 425.85
17.25 15.86 14.35 17.10 15.92 14.30 16.12 15.09 13.66 15.16 14.50 16.10 14.76 13.63 15.21 14.23 13.04 14.88
2.292 2.424 2.590 2.305 2.417 2.597 2.397 2.505 2.677 2.498 2.573 2.399 2.542 2.681 2.492 2.605 2.761 2.529
4.70
Expt. `d' (Angstrom unit) [16]
Tm (K) [16,34]
2.313 2.402 2.355 2.441
2.530 2.620
2.300 2.416 2.602 2.509 2.676
2.525
2.773
a L (10 26 K 21) Cal. Eq. (9)
Rep. [16]
1643 1393 1298 1073 1203 843 1369 1120 1148 938 1048 871
6.28 6.29 8.71 8.37 6.37 10.53 6.10 6.85 7.61 8.41 5.66 5.68
± ± ± ± 7.0 4.9 5.3 5.0 6.7
1570 1273 1163 1553 1313 1143 1300 1263 1053 1223 1002 1313 1123 987 1145 1053 965 900
9.64 10.99 9.65 9.63 10.59 9.83 11.65 9.81 9.68 10.47 12.89 10.86 11.22 10.92 11.75 11.24 9.45 16.09
9.5 8.9 9.8
4.5
10.9
10.5 10.6 11.3
Expt. [16,32] 6.3 7.2 6.7 6.1 3.2
1.0 6.0 2.3 4.7
8.9 10.5, 5.4 11.2, 6.9 10.6, 6.66 9.5, 7.1
7.1 4.2 1.9 7.3, 9.4, 0.69
Taken from Ref. [33].
Ê . With the the bond length of diamond, that is, d0 1.545 A available experimental data [13±16] of a L and d, Neumann [16] has obtained the values of B and d0 on least square ®ts of Eq. (8). The values of B are equal to 17.0, 3.3, 10.0, 16.1 Ê 23), respectively, for A IV, A IIB VI, and 4.2 (10 26 K 21 A A IIIB V, A IIB IVC2V and A IB IIIC2VI semiconductors and d0 are, Ê [16]. respectively, 1.549, 1.382, 1.561, 1.573, and 1.330 A The ternary chalcopyrites of general composition A IIB IVC2V and A IB IIIC2VI can be considered as similar to those of A IIIB V and A IIB VI semiconductors. Thus, Eq. (8) can also be reasonably used to describe thermal expansion coef®cients of the ternary chalcopyrites. In Eq. (8), the value of d has taken the mean value of bond length, that is, d dA±C 1 dB±C =2 where dA±C and dB±C are the individual bond lengths of the A±C and B±C bonds in an ABC compound. The values of d have been calculated from Eq. (5).
From Eqs. (5) and (8), we get following relation between a L and "vp
aL
0:021 2 B15:30 "vp 22=3 2 d0 3 : Tm
9
Using Eq. (9), we have calculated the values of a L. The calculated values a L are listed in Table 1 along with the experimental values and the values reported by different workers. 4. Anisotropy of the thermal expansion coef®cients The average linear thermal expansion coef®cient a L depends on a a and a c, and is de®ned as
aL ;
1 2aa 1 ac : 3
10
110
Table 2 Linear expansion coef®cients a a and a c of lattice parameters a and c of ternary chalcopyrites Compd. (Z 16) Lattice parameters (D )cal.
CdSiAs2 ZnGeAs2 CdGeAs2 ZnSnAs2 CdSnAs2 A IB IIIC2VI CuAlS2 CuAlSe2 CuAlTe2 CuGaS2 CuGaSe2 CuGaTe2 CuInS2 CuInSe2 CuInTe2 AgAlSe2 AgAlTe2 AgGaS2 AgGaSe2 AgGaTe2 AgInS2 AgInSe2 AgInTe2 CuTlSe2
Ê) c (A
5.400 5.678 5.465 5.741 5.900 5.610
10.441 10.431 10.771 10.775 11.518 10.880
5.884 5.672 5.9427 5.8515 6.0944
10.882 0.151 11.153 0.033 11.2172 0.112 11.704 20.00017 11.9182 0.044
5.323 5.617 5.976 5.360 5.618 6.013 5.528 5.785 6.179 5.968 6.309 5.753 5.985 6.301 5.828 6.102 6.42 5.844
10.44 10.92 11.0 10.49 11.01 11.92 11.08 11.56 12.365 10.77 11.85 10.28 10.90 11.96 11.19 11.69 12.59 11.65
0.066 0.163 0.029 0.123 0.048 0.061
0.0387 0.056 0.0254 0.043 0.040 0.016 20.00434 0.00173 20.0013 0.195 0.122 0.214 0.17878 0.102 0.0799 0.084 0.0389 20.0065
0.067 0.164 0.029 0.123
0.112
0.041 0.037 20.01
dD /dT (10 26 K 21)
a a (10 26 K 21)
Authors Cal. Eq. (13)
Expt. [16]
Authors. Cal. Eq. (16)
9.52 20.58 5.306 16.02 7.47 8.95
9.09 20.21 5.52 16.81
7.921 10.02 9.64 11.21 11.80 7.64
7.8 8.8 8.8 9.7 6.2 6.9
10.31 8.58 11.02 5.99 6.87
8.5 7.7 9.5 4.8 5.1
12.45 14.49 11.93 12.61 13.45 11.74 12.78 11.17 10.93 20.05 19.16 21.33 20.04 16.33 16.23 15.89 12.27 17.30
12.5 11.8 12.1 12.5 13.0 11.5 11.7 10.6 10.9
19.21 5.76 14.77 1.98 7.016 16.546 20.456 13.540 17.52 16.84 11.416 6.82 8.19 7.50 51.87 35.37 56.16 48.20 30.85 25.85 26.78 16.59 9.27
14.47
13.89 14.07 5.59
a c (10 26 K 21) Reported [16] Expt. [16,20]
20.8 20.5 16.0 15.8 13.7
7.9 7.8 8.9 7.6, 9.6, 7.2, 6.9 11.0 8.5
11.8 13.1 11.4, 9.9
Authors Cal. Eq. (17)
Reported [16] Expt. [16,20]
2.99 21.18 6.91 2.68 7.97 3.02
2.9 22.4 5.7 1.2 2.3 2.3
10.07 5.65 3.19 5.00 3.28
21.9 4.8 1.7 3.8 1.5
4.01 3.97 5.07 3.66 4.86 5.99 9.38 7.07 7.18 28.69 0.33 210.09 26.42 1 0.08 2.77 1.92 3.81 13.65
3.7 3.1 5.3 3.9 4.6 4.6 9.6 7.9 7.0 210.1 26.6 20.3 2.3 0.6
3.2 5.0 0.37 3.3, 2.8, 3.6, 2.3 0.0 11.1
4.1 5.2 8.6, 8.4
V. Kumar, B.S.R. Sastry / Journal of Physics and Chemistry of Solids 63 (2002) 107±112
A IIB IVC2V ZnSiP2 CdSiP2 ZnGeP2 CdGeP2 CdSnP2 ZnSiAs2
Ê) a (A
(D )expt
V. Kumar, B.S.R. Sastry / Journal of Physics and Chemistry of Solids 63 (2002) 107±112
More information regarding a a and a c can be obtained from temperature dependence of the tetragonal distribution (D ) [35]. Expanding the free energy F of a crystal with chalcopyrite structure as a function of the tetragonal distortion D , we have F a0 1 a1 D 1 a2 D2 1 ¼:
11
The equilibrium value of D corresponding to the minimum of the free energy, that is, dF=dD 0 is given by a D2 1 : 12 2a2 From Eq. (12), the change in D with temperature T can be written as dD 1 da1 1 da2 2 2 D C0 1 C1 D 2a2 dT a2 dT dT
13
111
four parameters, that is, d, a L, a a and a c are compared with the experimental values and the values reported by different workers. Fairly good agreement has been obtained between them. The main advantage of the present model is the simplicity of the formulae, which do not require any experimental data except the plasmon energy of the ternary chalcopyrites. In contrast, the previous models require the experimental values of bond length of these semiconductors. Thus, it is possible to calculate the values of d, a L, a a and a c of the A IIB IVC2V and A IB IIIC2VI semiconductors from their plasmon energy data even if the experimental values of bond lengths are unknown. The values of d and a L of A IIB VI and A IIIB V groups of semiconductors can also be described with this model. Hence, one can predict the values of these parameters for unknown semiconductors belonging to these groups of semiconductors.
where C0 2
1 da1 1 da2 and C1 2 : 2a2 dT a2 dT
14
By extrapolating the curve with the known experimental values of a a, a c and D [7], Neumann [16] has estimated the constants C0 and C1 to be 2.0 £ 10 26 and 1.14 £ 10 24 K 21 for the A IIB IVC2V and 7.8 £ 10 26 and 2.26 £ 10 24 K 21 for A IB IIIC2VI compounds. The tetragonal distortion can also be de®ned as D ; 2 2 c=a from which we get dD c 1 da 1 dc c 15 2 aa 2 ac : dT a a dT c dT a Solving Eqs. (10) and (15), we get the following equations for a a and a c
aa aL 1
1 a dD 3 c dT
16
ac aL 2
2 a dD 3 c dT
17
where dD=dT is given by Eq. (13). The values of a a and a c have been calculated from Eqs. (16) and (17) using values of a L calculated from Eq. (9). The calculated values of a a and a c are listed in Table 2 along with the experimental values and the values reported by different workers. 5. Conclusion The values of average bond length (d) and plasmon energy ("vp ) of A IIB IVC2V and A IB IIIC2VI chalcopyrites have been calculated using Eqs. (5) and (6), respectively. The calculated values of d have been used further to calculate the average linear expansion coef®cient (a L) of these semiconductors. The linear expansion coef®cients a a and a c have also been calculated using Eqs. (16) and (17). The calculated values of "vp , d and a L are listed in Table 1 and a a and a c in Table 2. The calculated values of all
Acknowledgements The authors are grateful to Professor B.B. Bhattacharya, Director, and Professor D. Chandra, Head, Department of Electronics and Instrumentation, Indian School of Mines, Dhanbad, for their continuous inspiration and encouragement in conducting this part of the work.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
J.A. Van Vechten, Phys. Rev. 182 (1969) 891. J.A. Van Vechten, Phys. Rev. B1 (1970) 3351. J.C. Phillips, Rev. Mod. Phys. 42 (1970) 317. D.S. Chemla, Phys. Rev. 23 (1971) 1441. B.F. Levine. Phys. B7 (1973) 2591, 2600. B.F. Levine, J. Chem. Phys. 59 (1973) 1463. J.L. Shay, J.H. Wernik, Ternary Chalcopyrite Semiconductors: Growth, Electronic Properties, and Applications Pergamon Press, Oxford, 1975. H. Matsusita, S. Endo, T. Irje, Jpn. J. Appl. Phys. 30 (1991) 1181. V. Kumar, J. Phys. Chem. Solids 61 (2000) 91. V. Kumar, D. Chandra, Phys. Stat. Sol. (b) 212 (1999) 37. C. Rincon, J. Gonzalez, Phys. Rev. B40 (1992) 282. T.L. Makarova, G.A. Medvedkin, Semiconductor Sci. Technol. 7 (1992) 278. L.G. Van Uitert, H.M. O'Bryan, M.E. Lines, H.J. Guggenheim, G. Zydzik, Mat. Res. Bull. 12 (1977) 261. L.G. Van Uitert, H.M. O'Bryan, H.J. Guggenheim, R.L. Barns, G. Zydzik, Mat. Res. Bull. 12 (1977) 307. L.G. Van Uitert, Mat. Res. Bull. 12 (1977) 315. H. Neumann, Kristall. Technik. 15 (1980) 849. H. Neumann, Cryst. Res. Technol. 18 (1983) 659. H. Neumann, P. Deus, R.D. Tomlinson, G. Kuhn, B. Hintze, Phys. Stat. Sol. (a) 84 (1984) 87. P. Deus, H. Neumann, G. Kuhn, B. Minze, Phys. Stat. Sol. (a) 80 (1983) 205. P. Deus, U. Voland, H. Neumann, Phys. Stat. Sol. (a) 108 (1988) 225.
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[21] G.W. Iseler, N. Kildal, N. Menyuk, J. Electron. Mater. 7 (1978) 737. [22] G.W. Iseler, J. Cryst. Growth 41 (1977) 146. [23] N. Yamamoto, H. Horinaka, T. Miyauchi, Jpn. J. Appl. Phys. 18 (1997) 225. [24] H.G. Bruhle, H. Neumann, G. Kuhn, Solid State Commun. 34 (1980) 225. [25] T. Makam, H. Nimura, Y. Ozaki, S. Kamada, T. Hismatsn, Jpn. J. Appl. Phys. 17 (1975) 1677. [26] G.C. Bhar, L.K. Samanta, Phys. Rev. B26 (1982) 2275. [27] J. Moolamdi, Phys. Rev. B 10 (1974) 2490. [28] P. Kistaiah, Y.C. Venudhar, K.S. Murthy, L. Jyengar, K.V.K. Rao, J. Phys. D14 (1981) 457, 1131.
[29] P. Kistaiah, Y.C. Venudhar, K.S. Murthy, L. Jyengar, K.V.K. Rao, J. Mater. Sci. 16 (1981) 1417. [30] P. Kistaiah, Y.C. Venudhar, K.S. Murthy, L. Jyengar, K.V.K. Rao, J. Mater. Sci. 16 (1981) 2911. [31] L. Martin, L.B. Leder, H. Mendlowitz, in: L. Marton (Ed.), Advances in Electronics and Electron Physics, vol. 7, Academic Press, New York, 1955, p. 225. [32] D.R. Lide (Ed.), Handbook of Chemistry and Physics, 72nd, CRC Press, 1991±1992. [33] V. Kumar, G.M. Prasad, D. Chandra, Cryst. Res. Technol. 31 (1996) 501. [34] L. Garbato, A. Rucci, Phil. Mag. 35 (1977) 1685. [35] D. Weaire, J. Modandi, J. Phys. Suppl. 36 (1975) C3±27.
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Relationship between the thermal expansion coef®cient, plasmon energy, and bond length of ternary chalcopyrite semiconductors V. Kumar*, B.S.R. Sastry Department of Electronics and Instrumentation, Indian School of Mines, Dhanbad 826 004, India Received 13 November 2000; accepted 9 March 2001
Abstract A simple relation between the average bond length and plasmon energy, that is, d 15.30 "vp 22=3 , has been proposed for A IIB IVC2V and A IB IIIC2VI chalcopyrite semiconductors. The average linear thermal expansion coef®cient (a L) of these semiconductors has been calculated using plasmon energy data. The linear expansion coef®cients (a a) and (a c) of the lattice parameters a and c, respectively, have also been calculated. The calculated values of d, a L, a a and a c have been compared with the experimental values and the values reported by different workers. A fairly good agreement has been obtained between them. q 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction In the recent past, increasing attention has been given towards the study of various electronic, electrical and optical properties of A IIB IVC2V and A IB IIIC2VI chalcopyrite semiconductors [1±10]. This is because of their potential applications in a variety of optoelectronic devices. Different workers [11±20] have studied thermodynamical properties such as melting point and thermal expansion coef®cient of these chalcopyrites. The study of thermal expansion coef®cients a L, a a and a c is important in the growth of single crystal [21,22], in the understanding of the temperature dependence of the electronic properties [9,10,23] and in the choice of substrate materials suited for epitaxial growth [7,24] of these semiconductors. Van Uitert et al. [13,15] have shown that average thermal expansion coef®cient (a L) depends upon melting point, orbital extension and site symmetry. They have further shown that the product of coef®cient of thermal expansion and melting point (a LTm) has a constant value for materials with ®xed crystal structure. The values of this constant for rectilinear, close-packed, rutite and tetrahedral structures have been obtained as 0.027, 0.016, 0.020 and 0.021, respec-
* Corresponding author. Tel.: 191-326-206247; fax: 191-326203042. E-mail address: [email protected] (V. Kumar).
tively [15]. The A IIB IVC2V and A IB IIIC2VI semiconductors crystallize in the tetragonal chalcopyrite structure and exhibit diamond-like or tetrahedral structure [7]. In this type of structure, two kinds of cations make an ordered sublattice which may result in a tetragonal distortion of the lattice de®ned as D ; 2 2 c/a, where c and a are the lattice parameters. Because of the axial symmetry of the crystals, the linear thermal expansion coef®cients a a and a c are anisotropic, which gives rise to a nonvanishing temperature dependence of the tetragonal distortion. In the studies of thermal expansion coef®cients a a and a c of A IIB IVC2V [21,22,25] and A IB IIIC2VI [22,23] semiconductors, it has been assumed that dD=dT increases monotonously with increasing D [26,27]. However, this supposition has been shown to be incorrect by Yamamoto et al. [23] and Kistaiah et al. [28±30]. Later, Neumann [16,17] gave a model for the temperature dependence of tetragonal distortion (D ) based on free energy consideration and explained the thermal expansion behavior of these semiconductors [16,19]. An X-ray bond method has also been used to explain the behavior of a a and a c in CuInTe2 compound in the temperature range of 30±300 K [18,20]. Recently, the ®rst author [9,10] has used the solid state theory of plasma oscillations to explain interatomic force constants of these ternary and some binary (A IIB IV and A IB V) semiconductors. The plasmon energy is related to the effective number of valence electrons in a compound. The bond length also depends on the effective number of
0022-3697/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(01)00085-3
108
V. Kumar, B.S.R. Sastry / Journal of Physics and Chemistry of Solids 63 (2002) 107±112
valence electrons. This shows that there must be a correlation between bond length and plasmon energy. Bond length is an important parameter for calculating the values of thermal expansion coef®cients. Earlier workers have used the experimental values of bond length while calculating the values of a L, a a and a c. The experimental values of bond length are still unknown for some semiconductors. Therefore, it has been of interest to develop a theoretical model for the calculation of bond length. In the present paper, we have extended the earlier work of the ®rst author [9,10] and proposed a simple relation between bond length and plasmon energy for the calculation of average bond length of A IIB IVC2V and A IB IIIC2VI semiconductors. The calculated values of the average bond length (d) have been further used to calculate the values of linear thermal expansion coef®cients a L, a a and a c. The calculated values of all four parameters are compared with the experimental values and the values reported by different workers [16±22]. 2. Calculation of bond length and plasmon energy The free-electron plasmon energy is given by the relation "vp 2
4pNe2 e2 ; m
1
also be written as [31] r Zs "vp 28:8 eV; W
6
where Z is the effective number of valence electrons taking part in plasma oscillations, s is the speci®c gravity and W the molecular weight. Using Eq. (6), plasmon energies of A IIB IVC2V and A IB IIIC2VI semiconductors have been calculated taking Z 16, and s and W data from a book of constants [32]. The calculated values of plasmon energies "v p are listed in Table 1, which can be written in terms of individual bonds as "vp
"vp;AC 1 "vp;BC ; 2
7
where "vp;AC and "vp;BC are the plasmon energies of the A± C and B±C bonds in A IIB IVC2V and A IB IIIC2VI semiconductors. In Table 1, the plasmon energies of ZnSnP2, CdSnP2 and CdSiAs2 are taken from our previous publications [33], since the values of speci®c gravity are not known [32]. However, from back calculation we can predict the values of s , which are equal to 4.48,4.02 and 5.15, respectively, for ZnSnP2, CdSnP2 and CdSiAs2 semiconductors. Several workers [1±8,34] have also used plasmon energy data and explained various electronic properties such as ionicity, energy gaps, dielectric constant and hardness of these semiconductors.
from which we have Ne2
m "vp 2 ; 4pe2
2
where Ne is the effective number of free electrons taking part in the plasma oscillations, e is the charge and m the mass of electron. Ne can be written in terms of individual bond properties as ZA Z 1 B NCA NCB ; 3 Ne vb where ZA and ZB are the numbers of valence electrons of the atoms A and B, respectively, in an AB compound. NCA and NCB are the coordination numbers of the atoms A and B, and vb is the bond volume. For tetrahedral p crystals, NCA NCB 4, ZA 1 ZB 8 and vb 4d 3/3 3. Substituting these values in Eq. (3), we get p 3 3 Ne : 4 2d 3 From Eqs. (2) and (4), we get the following equation for the average bond length of A IIB IVC2V and A IB IIIC2VI semiconductors d 15:30 "vp 22=3
d in A and "vp in eV:
5
For a compound, the plasmon energy shown in Eq. (1) can
3. Average linear thermal expansion coef®cient The melting point Tm can be taken as a direct measurement of binding energy and an inverse measure of the volume expansion coef®cient a v or average linear thermal expansion coef®cient, that is, a L (1/3)a v. Van Uitert et al. [13±15] have shown that the product a LTm is a constant for materials with ®xed ionicity under these conditions. In reality, the spatial extension of the orbital around the atoms involved may signi®cantly affect anharmonicity and, therefore, thermal expansion. This means that besides the bond ionicity, the bond length must be taken into account in considering the average linear thermal expansion coef®cient of a given material and we have developed Eq. (5) for the calculation of bond length. Based on thermal expansion coef®cient data, Neumann [16] has proposed the following expression for average thermal expansion coef®cient for binary tetrahedral semiconductors
aL
A 2 B d 2 d0 3 ; Tm
8
where A is a constant, Tm is the melting temperature in Kelvin and d the bond length. The value of A 0.021 for all tetrahedrally coordinated compounds (A IV, A IIIB V, A IIB VI) as estimated from a hard sphere model based on the diamond structure. The value of d0 should be equal to
V. Kumar, B.S.R. Sastry / Journal of Physics and Chemistry of Solids 63 (2002) 107±112
109
Table 1 Average bond length (d) and thermal expansion coef®cient (a L) of ternary chalcopyrites Compd.
A IIB IVC2V ZnSiP2 CdSiP2 ZnGeP2 CdGeP2 ZnSnP2 CdSnP2 ZnSiAs2 CdSiAs2 ZnGeAs2 CdGeAs2 ZnSnAs2 CdSnAs2 A IB IIIC2VI CuAlS2 CuAlSe2 CuAlTe2 CuGaS2 CuGaSe2 CuGaTe2 CuInS2 CuInSe2 CuInTe2 AgAlSe2 AgAlTe2 AgGaS2 AgGaSe2 AgGaTe2 AgInS2 AgInSe2 AgInTe2 CuTlSe2 a
s
3.39 4.00 4.17 4.48
W
"v p
Ê) Cal. `d' (A
5.32 5.60 5.53 5.72
155.40 202.43 199.90 246.94 246.00 243.03 242.20 290.34 287.80 334.83 333.90 380.92
17.02 16.19 16.64 15.52 15.55 a 14.82 a 16.05 15.35 a 15.66 14.90 14.82 14.12
2.312 2.390 2.347 2.459 2.456 2.536 2.404 2.477 2.445 2.527 2.536 2.619
3.47 4.70 5.50 4.35 5.56 5.99 4.75 5.77 6.10 5.07 6.18 4.72 5.84 6.05 5.00 5.81 6.12 7.11
154.65 248.45 354.73 197.39 291.19 388.47 242.49 336.29 433.57 292.77 390.05 241.71 355.51 432.79 286.87 380.61 477.89 425.85
17.25 15.86 14.35 17.10 15.92 14.30 16.12 15.09 13.66 15.16 14.50 16.10 14.76 13.63 15.21 14.23 13.04 14.88
2.292 2.424 2.590 2.305 2.417 2.597 2.397 2.505 2.677 2.498 2.573 2.399 2.542 2.681 2.492 2.605 2.761 2.529
4.70
Expt. `d' (Angstrom unit) [16]
Tm (K) [16,34]
2.313 2.402 2.355 2.441
2.530 2.620
2.300 2.416 2.602 2.509 2.676
2.525
2.773
a L (10 26 K 21) Cal. Eq. (9)
Rep. [16]
1643 1393 1298 1073 1203 843 1369 1120 1148 938 1048 871
6.28 6.29 8.71 8.37 6.37 10.53 6.10 6.85 7.61 8.41 5.66 5.68
± ± ± ± 7.0 4.9 5.3 5.0 6.7
1570 1273 1163 1553 1313 1143 1300 1263 1053 1223 1002 1313 1123 987 1145 1053 965 900
9.64 10.99 9.65 9.63 10.59 9.83 11.65 9.81 9.68 10.47 12.89 10.86 11.22 10.92 11.75 11.24 9.45 16.09
9.5 8.9 9.8
4.5
10.9
10.5 10.6 11.3
Expt. [16,32] 6.3 7.2 6.7 6.1 3.2
1.0 6.0 2.3 4.7
8.9 10.5, 5.4 11.2, 6.9 10.6, 6.66 9.5, 7.1
7.1 4.2 1.9 7.3, 9.4, 0.69
Taken from Ref. [33].
Ê . With the the bond length of diamond, that is, d0 1.545 A available experimental data [13±16] of a L and d, Neumann [16] has obtained the values of B and d0 on least square ®ts of Eq. (8). The values of B are equal to 17.0, 3.3, 10.0, 16.1 Ê 23), respectively, for A IV, A IIB VI, and 4.2 (10 26 K 21 A A IIIB V, A IIB IVC2V and A IB IIIC2VI semiconductors and d0 are, Ê [16]. respectively, 1.549, 1.382, 1.561, 1.573, and 1.330 A The ternary chalcopyrites of general composition A IIB IVC2V and A IB IIIC2VI can be considered as similar to those of A IIIB V and A IIB VI semiconductors. Thus, Eq. (8) can also be reasonably used to describe thermal expansion coef®cients of the ternary chalcopyrites. In Eq. (8), the value of d has taken the mean value of bond length, that is, d dA±C 1 dB±C =2 where dA±C and dB±C are the individual bond lengths of the A±C and B±C bonds in an ABC compound. The values of d have been calculated from Eq. (5).
From Eqs. (5) and (8), we get following relation between a L and "vp
aL
0:021 2 B15:30 "vp 22=3 2 d0 3 : Tm
9
Using Eq. (9), we have calculated the values of a L. The calculated values a L are listed in Table 1 along with the experimental values and the values reported by different workers. 4. Anisotropy of the thermal expansion coef®cients The average linear thermal expansion coef®cient a L depends on a a and a c, and is de®ned as
aL ;
1 2aa 1 ac : 3
10
110
Table 2 Linear expansion coef®cients a a and a c of lattice parameters a and c of ternary chalcopyrites Compd. (Z 16) Lattice parameters (D )cal.
CdSiAs2 ZnGeAs2 CdGeAs2 ZnSnAs2 CdSnAs2 A IB IIIC2VI CuAlS2 CuAlSe2 CuAlTe2 CuGaS2 CuGaSe2 CuGaTe2 CuInS2 CuInSe2 CuInTe2 AgAlSe2 AgAlTe2 AgGaS2 AgGaSe2 AgGaTe2 AgInS2 AgInSe2 AgInTe2 CuTlSe2
Ê) c (A
5.400 5.678 5.465 5.741 5.900 5.610
10.441 10.431 10.771 10.775 11.518 10.880
5.884 5.672 5.9427 5.8515 6.0944
10.882 0.151 11.153 0.033 11.2172 0.112 11.704 20.00017 11.9182 0.044
5.323 5.617 5.976 5.360 5.618 6.013 5.528 5.785 6.179 5.968 6.309 5.753 5.985 6.301 5.828 6.102 6.42 5.844
10.44 10.92 11.0 10.49 11.01 11.92 11.08 11.56 12.365 10.77 11.85 10.28 10.90 11.96 11.19 11.69 12.59 11.65
0.066 0.163 0.029 0.123 0.048 0.061
0.0387 0.056 0.0254 0.043 0.040 0.016 20.00434 0.00173 20.0013 0.195 0.122 0.214 0.17878 0.102 0.0799 0.084 0.0389 20.0065
0.067 0.164 0.029 0.123
0.112
0.041 0.037 20.01
dD /dT (10 26 K 21)
a a (10 26 K 21)
Authors Cal. Eq. (13)
Expt. [16]
Authors. Cal. Eq. (16)
9.52 20.58 5.306 16.02 7.47 8.95
9.09 20.21 5.52 16.81
7.921 10.02 9.64 11.21 11.80 7.64
7.8 8.8 8.8 9.7 6.2 6.9
10.31 8.58 11.02 5.99 6.87
8.5 7.7 9.5 4.8 5.1
12.45 14.49 11.93 12.61 13.45 11.74 12.78 11.17 10.93 20.05 19.16 21.33 20.04 16.33 16.23 15.89 12.27 17.30
12.5 11.8 12.1 12.5 13.0 11.5 11.7 10.6 10.9
19.21 5.76 14.77 1.98 7.016 16.546 20.456 13.540 17.52 16.84 11.416 6.82 8.19 7.50 51.87 35.37 56.16 48.20 30.85 25.85 26.78 16.59 9.27
14.47
13.89 14.07 5.59
a c (10 26 K 21) Reported [16] Expt. [16,20]
20.8 20.5 16.0 15.8 13.7
7.9 7.8 8.9 7.6, 9.6, 7.2, 6.9 11.0 8.5
11.8 13.1 11.4, 9.9
Authors Cal. Eq. (17)
Reported [16] Expt. [16,20]
2.99 21.18 6.91 2.68 7.97 3.02
2.9 22.4 5.7 1.2 2.3 2.3
10.07 5.65 3.19 5.00 3.28
21.9 4.8 1.7 3.8 1.5
4.01 3.97 5.07 3.66 4.86 5.99 9.38 7.07 7.18 28.69 0.33 210.09 26.42 1 0.08 2.77 1.92 3.81 13.65
3.7 3.1 5.3 3.9 4.6 4.6 9.6 7.9 7.0 210.1 26.6 20.3 2.3 0.6
3.2 5.0 0.37 3.3, 2.8, 3.6, 2.3 0.0 11.1
4.1 5.2 8.6, 8.4
V. Kumar, B.S.R. Sastry / Journal of Physics and Chemistry of Solids 63 (2002) 107±112
A IIB IVC2V ZnSiP2 CdSiP2 ZnGeP2 CdGeP2 CdSnP2 ZnSiAs2
Ê) a (A
(D )expt
V. Kumar, B.S.R. Sastry / Journal of Physics and Chemistry of Solids 63 (2002) 107±112
More information regarding a a and a c can be obtained from temperature dependence of the tetragonal distribution (D ) [35]. Expanding the free energy F of a crystal with chalcopyrite structure as a function of the tetragonal distortion D , we have F a0 1 a1 D 1 a2 D2 1 ¼:
11
The equilibrium value of D corresponding to the minimum of the free energy, that is, dF=dD 0 is given by a D2 1 : 12 2a2 From Eq. (12), the change in D with temperature T can be written as dD 1 da1 1 da2 2 2 D C0 1 C1 D 2a2 dT a2 dT dT
13
111
four parameters, that is, d, a L, a a and a c are compared with the experimental values and the values reported by different workers. Fairly good agreement has been obtained between them. The main advantage of the present model is the simplicity of the formulae, which do not require any experimental data except the plasmon energy of the ternary chalcopyrites. In contrast, the previous models require the experimental values of bond length of these semiconductors. Thus, it is possible to calculate the values of d, a L, a a and a c of the A IIB IVC2V and A IB IIIC2VI semiconductors from their plasmon energy data even if the experimental values of bond lengths are unknown. The values of d and a L of A IIB VI and A IIIB V groups of semiconductors can also be described with this model. Hence, one can predict the values of these parameters for unknown semiconductors belonging to these groups of semiconductors.
where C0 2
1 da1 1 da2 and C1 2 : 2a2 dT a2 dT
14
By extrapolating the curve with the known experimental values of a a, a c and D [7], Neumann [16] has estimated the constants C0 and C1 to be 2.0 £ 10 26 and 1.14 £ 10 24 K 21 for the A IIB IVC2V and 7.8 £ 10 26 and 2.26 £ 10 24 K 21 for A IB IIIC2VI compounds. The tetragonal distortion can also be de®ned as D ; 2 2 c=a from which we get dD c 1 da 1 dc c 15 2 aa 2 ac : dT a a dT c dT a Solving Eqs. (10) and (15), we get the following equations for a a and a c
aa aL 1
1 a dD 3 c dT
16
ac aL 2
2 a dD 3 c dT
17
where dD=dT is given by Eq. (13). The values of a a and a c have been calculated from Eqs. (16) and (17) using values of a L calculated from Eq. (9). The calculated values of a a and a c are listed in Table 2 along with the experimental values and the values reported by different workers. 5. Conclusion The values of average bond length (d) and plasmon energy ("vp ) of A IIB IVC2V and A IB IIIC2VI chalcopyrites have been calculated using Eqs. (5) and (6), respectively. The calculated values of d have been used further to calculate the average linear expansion coef®cient (a L) of these semiconductors. The linear expansion coef®cients a a and a c have also been calculated using Eqs. (16) and (17). The calculated values of "vp , d and a L are listed in Table 1 and a a and a c in Table 2. The calculated values of all
Acknowledgements The authors are grateful to Professor B.B. Bhattacharya, Director, and Professor D. Chandra, Head, Department of Electronics and Instrumentation, Indian School of Mines, Dhanbad, for their continuous inspiration and encouragement in conducting this part of the work.
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