phys. stat. sol. (b) 243, No. 12, 2858–2863 (2006) / DOI 10.1002/pssb.200642140

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Mechanical and optical properties of AIIBIVC2

V

and AIBIIIC2

VI semiconductors

A. S. Verma* and S. R. Bhardwaj

Department of Physics, B. S. A. College, 281004 Mathura, India

Received 25 March 2006, revised 28 May 2006, accepted 2 June 2006

Published online 10 August 2006

PACS 62.20.Dc, 77.22.Ch, 78.20.Ci

A simple method based on a plasma oscillations theory of solids is proposed for the calculation of me-

chanical and optical properties such as microhardness (H), bulk modulus (B), dielectric constant (ε), po-

larizability (α), electronic susceptibility (χ) of AIIBIVC2

V and AIBIIIC2

VI semiconductors. Our calculated val-

ues are in excellent agreement with the values reported by earlier researchers.

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The compounds of the type AIIBIVC2V and AIBIIIC2

VI have attracted considerable attention because of their interesting semiconducting, electrical, structural, mechanical and optical properties. Compared to their binary analogues these compounds have higher energy gaps and lower melting points because of which they are considered to be important in crystal growth studies and device applications. The ternary com-pounds are direct gap semiconductors with tetragonal chalcopyrite crystal structure. These families of material are relevant in many fields including non-linear optics, opto-electronic and photovoltaic devices. The chalcopyrite structure is common to compounds of chemical formula AIIBIVC2

V and AIBIIIC2VI. Struc-

turally these compounds are derived from that of the binary sphalerite structure (III–V and II–VI) with a slight distortion. Therefore, like binary compounds they have a high non-linear susceptibility. However, because of the presence of two types of bonds in chalcopyrites they become anisotropic. This anisotropy gives rise to high bifringence. High non-linear susceptibility coupled with high bifringence in these com-pounds makes them very useful for efficient second harmonic generation and phase matching. Apart from it, the other important technological applications of these materials are in light emitting diodes, infrared detectors, infrared oscillations, lasers etc. [1–7, 27, 28]. Frequent attempts have been made at understanding the mechanical and optical properties in AIIBIVC2

V and AIBIIIC2

VI semiconductors. Many researchers [5, 15–18, 20] have been developed various theories and calculated these properties for chalcopyrite semiconductors. Therefore, we thought it would be of interest to give an alternative explanation for the mechanical and optical properties in AIIBIVC2

V and AIBIIIC2

VI semiconductors. In this paper we propose a method based on a plasma oscillations theory of solids for the calculation of the microhardness (H), bulk modulus (B), dielectric constant (ε), polarizabil-ity (α), electronic susceptibility (χ) in AIIBIVC2

V and AIBIIIC2VI semiconductors. It is now well established

that the plasmon energy of a metal changes [8, 9], when it undergoes a chemical combination and forms a compound. A plasmon is a collective excitation of the conduction electrons in a metal with an energy, ħωp, which depends on the density of the conduction electrons. This is due to the fact that the plasmon energy depends on the density of the conduction electrons and effective number of valence electrons,

* Corresponding author: e-mail: ajay_phy@rediffmail.com, Phone: +91 565 2423417

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which changes when a metal forms a compound. We have calculated the mechanical and optical proper-ties of AIIBIVC

2

V and AIBIIIC2

VI semiconductors using this idea. Further it has been shown by Verma and Agarwal [10], that when the chemical composition of a solid is varied, the energy of the valence bands is shifted since the chemical bonds affect the lattice spacing.

2 Theory, results and discussion

The energy of a quantum of plasma oscillations of the valence electrons in both metal and compound is given by the relation [11]

p

28.8 ( / ) ,Z Wω σ� =

(1)

where Z is the number of electrons taking part in the plasma oscillations, σ the specific gravity and W the molecular weight. Equation (1) is valid for free electrons but it is also applicable for semiconductors and insulators, up to a first approximation. Raether [12], and Philipp and Ehrenreich [13] have shown that the plasmon energy for semiconductors and insulators is given by

1/2

pd p 0/(1 δ ) ,ω ω ε� �= -

(2)

where δε0 is a very small correction to the free-electron plasmon energy ħωp and can be neglected to a first approximation. Philipp and Ehrenreich [13] have shown that the calculated values of ħωp and ħωpd are in fair agreement with their observed values of plasmon energy in dielectrics. It has also been pointed out by Kittel [14] that the plasmon oscillations in dielectrics are physically the same as in metals. Presently, Kumar [18, 20] has shown that the following relations give electronic properties for chal-copyrites in terms of plasmon energy ħωp (wherein ħωp is given in eV): for the A–C bond in AIIBIVC2

V,

Eh,AC (eV) = 0.05118(ħωp,AC)ν , (3)

Ec,AC (eV) = 5.904bAC(ħωp, AC)µ exp [–5.971(ħωp,AC)µ/2] , (4)

dAC (eV) = 14.6337(ħωp,AC)–µ ; (5)

for the B–C bond in AIIBIVC2V,

Eh,BC (eV) = 0.04158(ħωp,BC)ν , (6)

Ec,BC (eV) = 1.81bBC(ħωp, BC)µ exp [–6.4930(ħωp,BC)µ/2] , (7)

dBC (eV) = 15.9124(ħωp, BC)–µ ; (8)

for the A–C bond in AIBIIIC2VI,

Eh,AC (eV) = 0.0246(ħωp,AC)ν , (9)

Ec,AC (eV) = 7.3208b(ħωp,AC)2/3 exp [–8.026(ħω ′p,AC)1/3 (ħωp,AC)2/3] , (10)

dAC (Å) = 19.67(ħωp,AC)–2/3 ; (11)

for the B–C bond in AIBIIIC2VI,

Eh,BC (eV) = 0.0416(ħωp,BC)ν , (12)

Ec,BC (eV) = 5.53b(ħωp,BC)2/3 exp [–6.5058(ħωp,BC)2/3] , (13)

dBC (Å) = 15.91(ħωp,BC)–2/3 . (14)

2860 A. S. Verma and S. R. Bhardwaj: Properties of AIIBIVC2

V and AIBIIIC2

VI semiconductors

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

Here Eh, Ec and d are homopolar gaps, ionic gaps and bond length, respectively. ν and µ are constants and b is prescreening factor. Recently, Kumar et al. [5], have shown that heat of formation for AIBIIIC2

VI and AIIBIVC2V semiconductors may be determined in terms of plasmon energy by following

form:

heat of formation = A(ħωp)B , (15)

where A and B are constants. Similarly, based on the above expressions, we are of the view that some parameters related to me-chanical and optical properties of AIBIIIC2

VI and AIIBIVC2V chalcopyrite semiconductors can be evaluated

using their plasmon energy by following relations: for AIIBIVC2

V

microhardness (H) = 0.001(ħωp)A ; (16)

for AIBIIIC2VI

microhardness (H) = 0.192(ħωp)A′

; (17)

for AIBIIIC2VI and AIIBIVC2

V

bulk modulus (B) = 0.005 (ħωp)S ; (18)

for AIBIIIC2VI and AIIBIVC2

V

dielectric constant (ε) = 500/(ħωp)V ; (19)

for AIBIIIC2VI and AIIBIVC2

V

polarizability (α) = 3500/(ħωp)D ; (20)

Table 1 In this table we have presented reported plasmon energy (ħωp in eV) [18] and calculated values of microhardness (H in kg/mm2) and bulk modulus (B in GPa) from relation (17) and (18) respectively.

solids ħωp*AC ħωp*BC

ħωp Hexp [24] HTheor [24] HThis work B[15, 16] BThis work

CuAlS2 25.110 18.295 21.703 250 251 94 93 CuAlSe2 23.231 17.041 20.136 210 226 210 80, 69 73 CuAlTe2 20.859 15.518 18.189 182 255 166 64, 45 53 CuGaS2 23.626 18.163 20.895 230 229 94 82 CuGaSe2 22.176 16.906 19.541 197 197 196 71, 68 66 CuGaTe2 20.703 15.195 17.949 180 240 161 44, 55 51 CuInS2 22.231 16.453 19.342 140 192 71, 81 64 CuInSe2 23.061 15.489 19.275 185 160 190 62 63 CuInTe2 20.167 14.332 17.250 152 166 147 36, 51 45 AgAlS2 21.536 18.708 20.122 210 73, 82 73 AgAlSe2 20.090 17.510 18.800 160 176 179 55, 70 59 AgAlTe2 18.681 15.408 17.045 149 167 143 54, 36 43 AgGaS2 21.201 18.598 19.900 175 205 67, 70 70 AgGaSe2 20.110 17.193 18.652 143 158 176 57 AgGaTe2 18.905 15.149 17.027 135 166 142 35, 49 43 AgInS2 21.979 16.313 19.146 187 56, 66 62 AgInSe2 20.513 15.291 17.902 127 102 160 50 50 AgInTe2 18.751 13.860 16.306 118 116 129 29, 44 37

* Ref. [18]

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Table 2 In this table we have presented calculated values of dielectric constant (ε), polarizability (α in Å3) and susceptibility (χ) from relation (19)–(21) respectively.

solids ε[18] εThis work

αD.C. [17] αC.M. [17] αThis work χ[18] χThis work

CuAlS2 5.769 5.768 6.73 11.06 11.05 4.769 4.768 CuAlSe2 6.574 6.430 10.09 13.7 12.72 5.574 5.430 CuAlTe2 7.182 7.452 17.17 15.38 6.182 6.452 CuGaS2 5.866 6.094 7.25 12.04 11.87 4.866 5.094 CuGaSe2 6.705 6.716 10.91 14.24 13.45 5.705 5.716 CuGaTe2 7.628 7.597 19.2 15.77 6.628 6.597 CuInS2 6.680 6.816 8.42 13.1 13.71 5.680 5.816 CuInSe2 7.237 6.851 12.47 13.80 6.237 5.851 CuInTe2 9.026 8.047 20.86 16.98 8.026 7.047 AgAlS2 5.483 6.437 9.02 12.73 4.483 5.437 AgAlSe2 6.114 7.103 11.31 14.46 5.114 6.103 AgAlTe2 7.211 8.188 19.35 17.37 6.211 7.188 AgGaS2 5.773 6.541 8.22 12.47 13.00 4.773 5.541 AgGaSe2 6.602 7.185 12.13 14.79 14.67 5.602 6.185 AgGaTe2 7.626 8.200 20.79 17.40 6.626 7.200 AgInS2 6.324 6.918 9.04 13.97 5.324 5.918 AgInSe2 7.273 7.625 13.51 13.96 15.85 6.273 6.625 AgInTe2 8.494 8.731 23.23 18.87 7.494 7.731

D.C. = calculated by D. Chemla relation, C.M. = calculated by Clausius–Mossotti relation

for AIBIIIC2VI and AIIBIVC2

V

susceptibility (χ) = ε – 1 . (21)

Here A, A′, S, V and D are constants. These values are A = 4.89, A′ = 2.331, S = 3.193 and 3.54 for AIBIIIC2

VI and AIIBIVC2V chalcopyrite, respectively, V = 1.45 and 1.429 for AIBIIIC2

VI and AIIBIVC2V chal-

copyrite respectively and D = 1.871 and 1.962 for AIBIIIC2VI and AIIBIVC2

V chalcopyrites respectively. In these relations plasmon energy (ħωp) of AIBIIIC2

VI and AIIBIVC2V chalcopyrites can be calculated by

(ħωp,AC + ħωp,BC)/2. A detailed discussion of mechanical and optical properties for chalcopyrites has been given elsewhere [15–17, 19, 21, 24, 26, 28] and will not be presented here. Although the properties of the AIBIIIC2

VI and AIIBIVC2V chalcopyrite semiconductors have been exten-

sively investigated and some of these compounds have attracted attention for practical applications [2], the knowledge of their mechanical properties such as microhardness and bulk modulus are rather incom-plete. Experimental data are available for few compounds of chalcopyrite series AIBIIIC2

VI and AIIBIVC2V,

so there are many properties of the solid solution, which have not been investigated. In [2] an analysis of the dependence of their chemical composition is given. It should be noted that the experimental results differ widely for a number of reasons. First, the inconsistency of the results can be due to the experi-ments being carried out on poly crystalline samples, while AIBIIIC2

VI semiconductors are known to be anisotropic materials. Second, the shifts of the composition of the compounds from the stoichiometry affect greatly the values of microhardness. In the present work it is shown that analogous relations exists for the ternary chalcopyrite semiconductors, which can be successfully employed to estimate the me-chanical and optical properties from their plasmon energy (ħωp). Using Eqs. (16)–(21), the microhardness, bulk modulus, dielectric constant, polarizability and sus-ceptibility of AIBIIIC2

VI and AIIBIVC2V chalcopyrite semiconductors have been calculated and presented in

Tables 1–4.

2862 A. S. Verma and S. R. Bhardwaj: Properties of AIIBIVC2

V and AIBIIIC2

VI semiconductors

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Table 3 In this table we have presented reported plasmon energy (ħωp in eV) [20] and calculated values of microhardness (H in kg/mm2) and bulk modulus (B in Gpa) from relation (16) and (18) respectively.

solids ħωp*AC ħωp*BC

ħωp Hexp [24] HTheor [24] HThis work B[19] BThis work

ZnSiP2 15.294 18.757 17.026 1100 900 1048 120 114 ZnGeP2 15.198 17.944 16.571 980 635 918 108 104 ZnSnP2 14.907 16.203 15.555 650 530 673 84 83 ZnSiAs2 14.473 17.620 16.047 920 820 784 93 93 ZnGeAs2 14.316 16.997 15.657 680 630 695 86 85 ZnSnAs2 14.094 15.524 14.809 455 430 530 67 70 CdSiP2 13.659 18.845 16.252 730 834 97 97 CdGeP2 13.675 17.905 15.790 565, 410 470 725 86 87 CdSnP2 13.446 16.194 14.820 255 531 67 70 CdSiAs2 13.050 17.642 15.346 615 630 77 79 CdGeAs2 13.035 16.85 14.943 470 470 553 70 72 CdSnAs2 12.831 15.388 14.110 350 310 418 55 59

* Ref. [20]

In Tables 1 and 2 we have been presented mechanical and optical properties of AIBIIIC2

VI chalcopyrite semiconductors and in Tables 3 and 4 have been presented mechanical and optical properties of AIIBIVC2

V chalcopyrite semiconductors.

3 Conclusion

From the above results and discussions obtained by using the proposed approach, it is quite obvious that the parameters such as microhardness (H), bulk modulus (B), dielectric constant (ε), polarizability (α) and electronic susceptibility (χ) reflecting the mechanical and optical properties respectively, can be expressed in terms of plasmon energies of these materials, which is definitely a surprising phenomenon and need further investigations of the reason. The calculated values are presented in Tables 1–4. We note that the evaluated values of these parameters by our proposed relations are in close agreement with experimental data as compared to the values reported by previous researchers. The various parameters

Table 4 In this table we have presented calculated values of dielectric constant (ε), polarizability (α in Å3) and susceptibility (χ) from relation (19)–(21) respectively.

solids ε[20] ε[26]

εThis work αD.C. [17] αC.M. [17] αThis work χThis work

ZnSiP2 8.709 8.625 8.704 12.45 13.4 13.45 7.704 ZnGeP2 9.010 9.195 9.047 14.24 14.27 14.18 8.047 ZnSnP2 9.789 9.747 9.904 16.36 13.95 16.06 8.904 ZnSiAs2 9.285 9.875 9.473 18.12 15.2 15.10 8.473 ZnGeAs2 9.568 11.12 9.811 20.52 15.85 8.811 ZnSnAs2 10.30 13.02 10.62 23.84 17.68 9.62 CdSiP2 9.057 8.70 9.302 14.34 14.84 14.73 8.302 CdGeP2 9.377 9.27 9.694 15.95 17.4 15.59 8.694 CdSnP2 10.14 9.82 10.61 18.4 17.7 17.67 9.61 CdSiAs2 9.591 10.01 10.10 20.71 16.49 9.10 CdGeAs2 9.966 11.26 10.49 23.16 18.4 17.37 9.49 CdSnAs2 10.72 13.15 11.38 26.68 21.36 19.44 10.38

D.C. = calculated by D. Chemla relation, C.M. = calculated by Clausius–Mossotti relation

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evaluated in this work hardly deviates 0% to 10% from experimental data. The values evaluated show a systematic trend and are consistent with the available data reported so far, which proves the validity of the approach. It is also noteworthy that the proposed empirical relations are simpler and widely applica-ble. Since we have been reasonably successful in calculating these parameters using the plasmon fre-quency of the materials for chalcopyrite crystals. It is natural to say that this model can easily be ex-tended to rock salt, CsCl and zinc blende crystals for which the work is in progress and will be appearing in forthcoming papers. Hence it is possible to predict the order of mechanical and optical properties of semi-conducting compounds from their plasmon energies.

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