Self-Consistent Tight-Binding Approximation Including Polarisable Ions
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ABSTRACT Until recently, tight-binding has been applied to either covalent or metallic solid state systems, or charge transfer treated in a simple point charge framework. We present a selfconsistent tight-binding model which, for the first time, includes electrostatic ion polarisability and crystal field splitting. The tight-binding eigenvectors are used to construct multipole moments of the ionic charges which are used to obtain angular momentum components of the electrostatic potential in structure constant expansions. Our first test of the model is to study the phase stability in zirconia; in particular the instability of the fluorite phase to a spontaneous symmetry breaking, and its interpretation in terms of band effects and ion polarisability. This new formalism opens up the way to apply the tight-binding approximation to problems in which polarisationof atomic charges is important, for example oxides and other ceramic materials and surfaces of metals. INTRODUCTION Although the tight-binding approximation was originally conceived to deal with homopolar covalently bonded solids, an extension to include charge transfer in alloys and heteropolar semiconductors was proposed by Falicov and by Harrison some 15-20 years ago [1,2]. This self-consistent solution was extended and used very effectively by Majewski and Vogl to describe the energetics of ionically sp-bonded compounds [3]. A self-consistent scheme for metals has also been used [4], which in its simplest form assumes local charge neutrality [5] and leads to the tight-binding bond model [6]. Recently some questions have arisen that have motivated us to extend the self-consistent tight-binding approximation with charge transfer to include the effects of polarisation of the atomic charges. Such observations include, 1. The electronic structure of a low coverage of Nb on the (0001) a-A12 0 3 surface has been observed to have the character of a single non degenerate d-orbital. We found it impossible to describe this with conventional orthogonal or non orthogonal tight-binding, in which the on site hamiltonian matrix elements are constrained to be the same for each angular momentum (ie, crystal field splitting is ignored). 2. There is a question whether surface dipole barriers and hence work functions could be calculated in the tight-binding approximation. 3. It is thought that the spontaneous distortion of cubic zirconia into the tetragonal structure is driven by the quadrupole polarisability of the oxygen atoms [7]. 4. One would like to include crystal field terms as a natural consequence of the non spherical Madelung potential, rather than by direct parameterisation as has been done previously [8]. 265 Mat. Res. Soc. Symp. Proc. Vol. 491 ©1998 Materials Research Society
THEORY Traditional self-consistent tight-binding In order to make our new approach clear we begin with a brief reminder of Harrison's scheme which will also serve to establish our notation. In non self-consistent tight-binding, one supposes the existence of a hamiltonian 7W0whose matrix eleme
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