Elastic properties of chalcopyrite structured solids

https://doi.org/10.1016/j.matchemphys.2011.11.047Get rights and content

Abstract

Elastic properties (i.e. six independent elastic stiffness constants, C11, C12, C13, C33, C44 and C66) of chalcopyrite structured solids were evaluated. Values of C11, C33, C44, C66, of AIBIIIC2VI and AIIBIVC2V chalcopyrite semiconductors exhibit a linear relationship when plotted against the kBTm/Ω (kB = Boltzmann's constant, Tm = melting temperature, Ω = atomic volume) normalization, but fall on two straight lines according to the product of ionic charges of the compounds. The calculated results are compared with available experimental data and previous calculations based on phenomenological models.

Highlights

► In this paper the authors have been evaluated six independent elastic stiffness constants, C11, C12, C13, C22, C33 and C44 for chalcopyrite (AIBIIIC2VI and AIIBIVC2V) structured solids with the help of ionic charge theory. ► The proposed relationship only the kBTm/Ω normalization and ionic charge are required as input, the computation of elastic constants itself is trivial, and the accuracy of the results compares well with experimental values. ► The method turns out to be widely applicable.

Introduction

Chalcogenide and pnictide semiconductors with the formula AIBIIIC2VI and AIIBIVC2V have been widely studied because of their possible technological applications as photo-voltaic detectors, solar cells, light emitting diodes, modulators, filters and their use in nonlinear optics [1], [2], [3], [4], [5]. These semi-conductors crystallize in the chalcopyrite structure, which is deduced from that of zinc blende by the replacement of the cationic sublattice by two different atomic species. This induces the doubling of the zinc blende unit cell and introduces a tetragonal distortion characterized [6] by the parameter η = c/2a, where a and c are the lattice parameters, and by the anion displacement u=0.25+(dAC2+dBC2)/a2 from its position in the cubic cell, where dA–C and dB–C are the cation–anion distances. Because of the added structural (η; u) and chemical (dA–C  dB–C) degrees of freedom relative to their binary analogue, the ternary semi-conductors exhibit a wide range of interesting physical and chemical properties [2], [3], [4]. Although different methods of material preparation for these chalcopyrite (AIBIIIC2VI and AIIBIVC2V) compounds have been suggested [6], the knowledge of many of the physico-chemical properties that are essential for designing appropriate conditions for the growth of bulk single crystals and of high-quality epitaxial layers is still inadequate. The elastic properties of these compounds are characterized by six independent elastic stiffness constants Cij: C11, C12, C13, C33, C44, and C66 [7], because of this anisotropy of the elastic properties of these compounds it is evident that the availability of sufficiently large, single phase, homogeneous and defect-free single crystals is an essential precondition for reliable experimental determinations of the elastic constants independent of the specific method used. Table 1 presents the six deformation adapted to the tetragonal I4¯2d(D2d12) space group (Laue class 4¯2m), that we have used to calculate the six elastic constants of chalcopyrite crystals. In general, the strain deformations reduce the symmetry of the cell, eventually increasing the number of degrees of freedom that have to be minimized. Deformations 4–6 in Table 1, for instance, have four internal coordinates of the atoms within the unit cell that are not fixed by symmetry and must be optimized for each deformed cell geometry.

Elastic constants of most of the chalcopyrite family of semiconductors have not been determined experimentally because of various difficulties in growing single crystals of these compounds [3], [8]. Experimental determinations of elastic stiffness constants, compressibilities and bulk moduli for chalcopyrite compounds have been reported in the literature but the results are often contradictory [9]. Attempts have been made to fill this gap in the knowledge of the elastic properties of the chalcopyrites by theoretical calculations using different approaches, but mostly the results obtained differ considerably, and in many cases no satisfactory agreement has been achieved with existing experimental data. On the other hand, the availability of reliable elastic constant data is an essential prerequisite for any calculation or analysis of the influence of pressure, stress and strain on the properties of crystals and thin epitaxial layers.

In the past few years [10], a number of theoretical calculations based on empirical relations have become an essential part of material research. In many cases empirical relations do not give highly accurate results for each specific material, but they still can be very useful. In particular, the simplicity of empirical relations allows a broader class of researchers to calculate useful properties, and often trends become more evident. Empirical concepts such as valence, empirical radii, ionicity and plasmon energy are then useful [11], [12], [13], [14], [15]. These concepts are directly associated with the character of the chemical bond and thus provide means for explaining and classifying many basic properties of molecules and solids.

Recently, Verma and co-authors [15], [16], [17], [18] have been evaluated the structural, electronic, mechanical and ground state properties of binary and ternary crystals with the help of valence electron theory of solids. In this paper, we explore the applicability of the kBTm/Ω (kB = Boltzmann's constant, Tm = melting temperature, Ω = atomic volume) normalization for the elastic constants (i.e. six independent elastic stiffness constants, C11, C12, C13, C22, C33, C44,) of chalcopyrite (AIBIIIC2VI and AIIBIVC2V) semiconductors. We note that Yonenaga and Suzuki have already used the parameter /kB (G = shear modulus) to scale temperatures in a study of the elevated temperature mechanical properties of compound semiconductors. [19]. The proposed empirical relationship only the kBTm/Ω normalization and ionic charge are required as input, the computation of mechanical properties itself is trivial, and the accuracy of the results compares well with experimental values. The method turns out to be widely applicable.

Section snippets

Physical concepts

Several attempts have been made to estimate the elastic constants of chalcopyrite compounds using phenomenological models. In the first application of the rigid ion model to chalcopyrites [20], it has been shown that the frequencies of the infrared active zone-centre optical phonon modes can be well reproduced using an approximation for the inter-atomic forces which takes into account only interaction between nearest neighbours and two effective charge parameters. The Keating model, one of the

Concept of ionic charge theory

A chemical bond is formed when the atoms with incomplete valence shells combine. There are following main types of bonds:

  • 1.

    Ionic or electrovalent bond.

  • 2.

    Covalent bond.

  • 3.

    Coordinate bond.

  • 4.

    Mettalic bond.

The valence electrons refer to the electrons that take part in chemical bonding. These electrons reside in the outer most electron shell of the atom. The participation of valence shell electrons in chemical bonding may be explained on the basis of following grounds:

  • (i)

    The outermost-shell electrons are

Verification of ionic charge theory from the graphs and proposed expression for elastic properties with the comparison of calculated and reported values

As an example to verification of ionic charge theory, we have plotted a curve between experimental C11 and kBTm/Ω for AIBIIIC2VI and AIIBIVC2V semiconductors and presented in the following Fig. 1. we observe that in the plot of B and kBTm/Ω normalization, the group AIIBIVC2V semiconductors lie on line nearly parallel to the group AIBIIIC2VI semiconductors. This effect induced by the ionic charges of the compounds in the case of AIBIIIC2VI and AIIBIVC2V semiconductors. This condition (as we see

Summary and conclusions

There are several methods in determining elastic properties in semiconductors, but due to the small changes of the unit cell dimensions, the accuracy of determining these parameters always have been unpredictable. Furthermore, we found that in the compounds investigated here, the elastic stiffness constants exhibit a linear relationship when plotted against the kBTm/Ω, but fall on different straight lines according to the product of ionic charge of the compounds, which are presented in Fig. 1,

Acknowledgements

One of the authors (Dr. Ajay Singh Verma, PH/08/0049) thankful to University Grant Commission, New Delhi, India, for supporting this research under the scheme of U.G.C. Dr. D.S. Kothari Post Doctoral Fellowship.

References (28)

  • A.S. Verma et al.

    Mater. Chem. Phys.

    (2011)
  • M.S. Omar

    Mat. Res. Bull.

    (2007)
  • A.S. Verma

    Solid State Commun.

    (2009)
  • V. Kumar et al.

    Solid State Commun.

    (2009)
  • A.S. Verma et al.

    J. Alloys Compd.

    (2009)
  • A.S. Verma et al.

    J. Alloys Compd.

    (2009)
  • A.M. Brown et al.

    Acta Metall.

    (1980)
  • J. Lazÿewski et al.

    J. Appl. Phys.

    (2003)
  • T. Gurel et al.

    J. Phys. Condens. Matter

    (2006)
  • A.V. Kosobutsky et al.

    Phys. Stat. Sol. B

    (2009)
  • J.L. Shay et al.

    Ternary Chalcopyrite Semi-conductors: Growth Electronic Properties and Applications

    (1975)
  • A. Kelly et al.

    Crystallography and Crystal Defects

    (2000)
  • H. Neumann

    Cryst. Res. Technol.

    (2004)
  • F.W. Ohrendorf et al.

    Cryst. Res. Technol.

    (1999)
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