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Theoretical study on the effect of pressure on the electronic structure of grey tin

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 احمد محمود عبد اللطيف الخفاجي 6/10/2011 5:04:40 PM
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 Introduction

 
  he electronic and structural properties of tin are very sensitive to pressure conditions [1]. The ground state, grey tin , has the diamond structure and is known to be a semiconductor with zero band gap (semi-metal). Tin also exists in the  structure (white tin) at atmospheric pressure above 13oC. The metallic  phase is a tetragonal distortion of diamond with two atoms per unit cell. This phase is stable up to 9.5 GPa at room temperature, where it transforms to a bct form, followed by a transformation to the cubic bcc structure [2]. These temperature and pressure-driven phase transformations have caused tin to be of considerable experimental and theoretical interest [2-11]. The electronic and structural properties of complex systems like tin require a fully quantum-mechanical description. Accurate full first-principles calculations such as Hartree-Fock with correlation correction [12] and the local density approximation [13,14] are extremely demanding from the computational point of view. The development of simpler yet reliable approximate methods of calculations is therefore crucial to progress in this field.Corresponding author: Ahmed M. Abdul-Lettif E-mail address: abdullettif@yahoo.com  An alternative to first principle methods, which has already provided numerous significant results, is the so called the large unit cell-intermediate neglect of differential overlap (LUC-INDO) [15-18]. The semi-empirical LUC-INDO method is capable of simulating real crystals because it takes into account the property of periodicity. Due to its semi-empirical character and a specific parameterization scheme, the computer program is not cumbersome and time consuming in the treatment of the electronic and spatial structure of complex systems.A lot of studies have been carried out on the properties of tin and its phase transitions [1-11]. However, there are comparatively fewer studies on the pressure and temperature dependence of these properties. The aim of the present work is to investigate the effect of pressure on the electronic and structural properties of  using the LUC-INDO formalism which will be outlined in the next section.      Calculation method   A quantum-chemical semi-empirical INDO method developed especially for crystals [19] is used in the present work. This quantum computational formalism has been used with great success especially exploiting the so-called LUC (Large Unit Cell) model [18]. Within the method each molecular orbital is constructed as a linear combination of atomic orbitals [19] in order to express the wave function of the system. Each energy value is calculated by the HF self-consistent field method and the total energy of the system is obtained. The basic idea of LUC is in computing the electronic structure of the unit cell extended in a special manner at k=0 in the reduced Brillouin zone. This equivalent to a band structure calculation at those k points; which transform to the Brillouin zone center on extending the unit cell [18]. The crystal wave function in the LUC-INDO formalism is written in the following form:where NL is the number of LUCs in the crystal, Cpk denotes the combination coefficients of the hybridized orbitals,  is the atomic wave function, k is the wave vector, and Tv denotes the lattice translation vector. The Hartree-Fock equation in this case ( Rothan-Hall equation ) can be written as [20]where the summation index p goes over all the atomic states of the LUC, Fpqk represents the Fock Hamiltonian which is given byand Spqk is the overlap integral defined bywhere HT in Eq.(3) represent the Hamiltonian operator of the total energy ET which is defined aswhere the first term of the right side of Eq.(5) represents the inter-nuclear potential energy, and P is the density matrix which has the following expression:The summation is over the occupied (occ) orbitals only.  refers to the matrix element of the Hamiltonian of a single electron in the field of the nuclei, and its operator representation iswhere ZA is the core charge, and the summation is over all nuclei. It should be pointed out that our calculations are carried out at k=0, so k value in eqs.(1,3,4,6) is set to zero. For more details of LUC-INDO formalism and the final form of the Fock matrix elements at k=0, see Refs [15-22] . The Roothan-Hall equation are solved by first assuming an initial set of the linear expansion coefficient (C ), generating the density matrix (P ), and computing the overlap integral and the first guess of the Fock matrix elements (F ). Then one can calculate the electronic energy (E ), and a new matrix of C  coefficients can be obtained. This procedure is continued until there is no significant variation between the calculated value of C  and E of the successive iterations. An initial guess of the wave function is predicted using the basis set and adopting Slater-type orbitals [19]. The initial guess of the wave function is important since an optimum guess reduces the number of iterations performed to obtain the converged electronic energy. A large number of iterations will result in an accumulation of the computational errors. The sp3 initial guess of the wave function is given as an expected linear combination of the atomic states of one cell. The tolerance of convergence of the total electronic energy adopted in our calculations is  eV. A large unit cell of 8 atoms, which is the conventional Bravais lattice of the diamond structure, has been used. Interactions of the atoms in 
  

  • وصف الــ Tags لهذا الموضوع
  • Electronic structure; Tin; Pressure; LUC-INDO; Semi-empirical

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