GroupTheory`
GroupTheory`

GTTbHamiltonianElement

GTTbHamiltonianElement[l1,m1,l2,m2,shell,shell vectors]

constructs the k-dependent contribution of shell characterized by the shell vectors to the tight-binding matrix element between functions of symmetry and .

Details and Options

  • The tables in the seminal paper of Slater and Koster and a series of other papers in the field contain the k-dependent matrix elements of a tight-binding Hamiltonian including usually (*), and functions.
  • The tables are given for certain crystal structures. GTTbHamiltonianElement generates the analytical expressions automatically and therefore avoids mistakes. k-vectors are represented in units of 2π/a. The components of the k-vector are , , .
  • The following option can be given:
  • GOTbBasis 0Supresses superscripts with element names
  • See: J. C. Slater, G. F. Koster,Simplified LCAO method for the periodic potential problem, Phys. Ref. 94, 1498-1524 (1954)
  • W. Hergert, M. Geilhufe, Group Theory in Solid State Physics and Photonics. Problem Solving with Mathematica, chapter 9.4.

Examples

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Basic Examples  (1)

First load the package.

We assume a cubic lattice. The vectors of the nearest neighbor shell are:

The Hamilton matrix element between to s function is given by

This is exactly the form given by Slater and Koster., , are the components of the k-vector in units 2π/a.

Other Examples:

In some considerations of semiconductors an excited orbital * is included in the basis. This can be simulated using .

Options  (1)

GOTbBasis  (1)

If two different types of atoms are involved, this has to be indicated by a corresponding superscript. The atom names in the superscript are given in alphabetic order.