Non-linear elastic characterization of hexagonal cadmium selenide Sindu Jones n , C.S. Menon School of Pure and Applied Physics, Mahatma Gandhi University, Kottayam 686650, India article info Article history: Received 11 November 2013 Received in revised form 23 December 2013 Accepted 23 December 2013 Available online 2 January 2014 Keywords: Elastic constants Cadmium selenide Strain energy Anharmonicity Pressure derivatives abstract The complete set of non-vanishing second- and third order elastic constants and the pressure derivatives of second order elastic constants of hexagonal wurtzite phase of cadmium selenide (CdSe) are obtained from the homogeneous deformation theory. The strain energy is derived using finite strain elasticity theory by considering interactions up to four nearest neighbors of each atom in the unit cell of CdSe. We then compare this strain energy with the strain dependent lattice energy density from the continuum model to obtain the expressions for the second- and third order elastic constants. The calculated third order elastic constants show that, the longitudinal wave velocities C 111 , C 222 and C 333 are greater than their shear mode, indicating an increase of vibrational frequencies under stress along these directions. Furthermore, the pressure derivatives of SOECs of the shear modes, C 44 and C 66 , are negative, which show the possibility of a structural transformation. Bulk modulus and its pressure derivative are calculated and comparison with experimental values is made. & 2013 Elsevier B.V. All rights reserved. 1. Introduction Cadmium selenide (CdSe) forms an important class of semi- conductor materials, which finds applications in gas sensors, light harvesting and image capturing devices [1–5]. CdSe is known to exist in three phases: wurtzite, zinc blend and rock- salt [6]. It is known that, by altering the epitaxial strain on substrates or buffer layers during the growth of the crystal, the crystal structure can be altered. This structural freedom pro- vides an opportunity for making more efficient and reliable devices by choosing the appropriate polytypism [7]. Thus, the understanding of the pressure dependence on elastic constants is decisive in device fabrication and the third order elastic constants (TOEC) provide a better understanding of the pressure dependence on elastic constants. The elastic constants of the materials are essential in predicting and understanding the material response, strength, mechanical stability and phase transitions [8]. In this paper we study systematically the elastic characteriza- tion of CdSe in wurtzite phase. Expressions for the second order elastic constants (SOEC) and TOEC of CdSe are derived from the lattice theory by the method of homogeneous deformation. The TOEC values are used to find the pressure derivatives of SOEC and the bulk modulus of CdSe. The results are compared with the available values and are in good agreement. 2. Theory The wurtzite structure can be regarded as two interpene- trating hexagonal close packed (hcp) lattices. One hcp lattice is made entirely of cadmium atoms at positions r ! 1 ¼ Dð0; 0; 0Þ; r ! 2 ¼ Dð 1 2 ffiffi 3 p ; 1 2 ; p=2Þ, and the other entirely of selenium atoms at positions r ! 3 ¼ Dð0; 0; uÞ; r ! 4 ¼ Dð 1 2 ffiffi 3 p ; 1 2 ; ðp þ uÞ=2Þ, with the two lattices displaced from each other by a distance uc along the c-axis, where u is the internal parameter and it is the ratio of nearest neighbor distance along the c-axis to c. Here, the basis vectors of the wurtzite lattice referred in the Cartesian system of axis are taken as a ! 1 ¼ Dð ffiffi 3 p 2 ; 1 2 ; 0Þ; a ! 2 ¼ Dð0; 1; 0Þ; a ! 3 ¼ Dð0; 0; pÞ, where D ¼ a, length of the edge of the hexagonal system perpendicular to the unique axis and p ¼ c/a. From the X-ray diffraction (XRD) measurements in CdSe, Sowa [9] has obtained the lattice constant, a ¼ 4.302 Å, the axial ratio, p ¼ c/a ¼ 1.63 and the conventional internal parameter, u ¼ 0.37. The potential energy is expanded in powers of changes in squares of the vector distances R(I) and R(J): ϕ ¼ ϕ 0 þ ∂ϕðrÞ ∂r 2 ∑ I ðΔR 2 ðI ÞÞþ ∑ J ðΔR 2 ðJ ÞÞ  ! þ ∂ 2 ϕðrÞ ð∂r 2 Þ 2 ∑ I ðΔR 2 ðI ÞÞ 2 þ ∑ J ðΔR 2 ðJ ÞÞ 2  ! þ ∂ 3 ϕðrÞ ð∂r 2 Þ 3 ∑ I ðΔR 2 ðI ÞÞ 3 þ ∑ J ðΔR 2 ðJ ÞÞ 3  ! ð1Þ Here I represents atoms of the same type and J represents the non- equivalent atoms which lie out of plane. The strained lattice Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/physb Physica B 0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.12.038 n Corresponding author. Tel.: +91 9446819214. E-mail address: jones@gmail.com (S. Jones). Physica B 437 (2014) 82–84