Realistic tight-binding calculations

Tight binding (TB) and linear combination of atomic orbitals (LCAO) methods represent the more chemical approach to the problem of surface state calculations. They are basically fitting techniques, but, given a reasonable choice of parameters, they can add considerable detail to the basis provided by self-consistent calculations. The method, as applied to surfaces, was initiated by Hirabayashi [73] and developed into a useable form by Pandey and Phillips [74, 75]. 

The major problem with tight-binding calculations is the change of parameters from bulk values when the surface relaxes, causing first and second nearest neighbour bond distances to change. This is resolved, however, by the availability of self-consistent calculations for relaxed surfaces. The method as developed by Pandey and Phillips assumes that the wave functions of a thin slab can be written as 

A plethora of electron diffraction results amply testifies to the fact that clean surfaces of semiconductors undergo relaxation or reconstruction  

In elemental semiconductors and the polar faces of compound semiconductors, an odd number of electrons is formed per surface atom by the creation of a surface. The solid therefore undergoes a metal—insulator phase transition [82] to produce an even number of electrons per surface unit cell, thus reducing its symmetry in the plane of the surface. For non-polar faces of compound semiconductors, the simple truncated bulk geometry is already insulating in character because anionic and cationic species are electronically inequivalent. No distortions which reduce the symmetry are therefore necessary to provide stability, but the unbalanced ionic forces and unsaturated covalencies can produce quite large ( 0.5 A) atomic movements ( surface relaxation ). 

Thus to define the atomic geometry of a clean semiconductor surface, it is necessary to determine (1) the depth of the reconstructed layer, (2) its structure, and (3) its registry with respect to the underlying substrate. 

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The methods used to calculate surface states need not concern us here in any detail, but it will be instructive to give a brief indication of the two approaches currently employed (self-consistent calculations of the electronic energy and surface potential and realistic tight binding models), since this will provide some insight into semiconductor surface bonds and hence into chemisorption. 

The naive structural assumptions of crystalline models have been eliminated with the development of methods to calculate the electronic states of realistic liquid structures. An early approach of this type was the multiband tight-binding model that Yonezawa and Martino (1976) applied to an assumed hard-core liquid structure. Later, it was recognized that solution of the many-body statistical mechanics to calculate the liquid structure automatically solves the corresponding quantum mechanical problem required for the electronic states (Logan and Winn, 1988 Xu... 

In summary, using tight-binding molecular dynamics simulations, we have demonstrated qu ilitative differences in the physical properties of carbon nanotubes and graphitic carbon. Furthermore, we have presented an efficient Green s function formalism for calculating the quantum conductance of SWCNs. Our work reveals that use of full orbital basis set is necessary for realistic ceilculations of quantum conductance of carbon nanotubes. Rirthermore, our approach allows us to use the same Hamiltonian to ceilculate quantum conductivity as well as to perform structural relaxation. 

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