Tight Binding Model: Reference Book for Band Structure

[paweld]

Can anyone give a reference to book in which the model of tight binding is well explained.
This model is used to find band structure of metals and semiconductors. I'm interested
in a book which states all needed assumptions and gives logically consistent mathematical
reasoning explaing why energy bands exist.
Thanks.

 

[mavipranav]

Quantum theory of solids, Kittel
Solid state physics, Ashcroft and Mermin

Mavi

 

[cgk]

Note that the band structure has nothing to do with the tight binding model. Tight binding is simply one of the many approximations one can invoke to construct matrix elements for electronic structure theories (like Kohn-Sham DFT).

The bands originate from a unitary transformation from the local orbital basis (Wannier basis) to the crystal orbital basis (Bloch orbital basis). In the Wannier basis, there is one set of more-or-less atomic orbitals for each atom in the lattice, and these are orthonormal and degenerate (i.e., equal at each site, apart from position and phase of course). However, they do not diagonalize any kind of Fock matrix. After the CO transformation you don't have atomic sites anymore, but inverse lattice sites; and the matrix element computation is simplified because different k vectors belong to different irreducible representations of the spatial symmetry group, and for most matrix elements there is a theorem that the product representation of all its indices must contain the totally symmetric irrep in order to not vanish.

 

[mavipranav]

What cgk meant by the "totally symmetric representation" is actually the trivial representation. Landau & Lifgarbagez:Volume 3 explains in simple terms why this wonderful theorem is true (sort of like the Kraemers degeneracy theorem).

Mavi

 

[paweld]

cgk could you tell me where I can find some details about what you've written above.
Thanks for all answers.