Lecture note supplement for PHYS 7635: tight-binding model (09/14/12 TBM lecture notes(only) will also be available online at my site http://www.ocf.berkeley.edu/~mwg/7635/)

The tight bind model (TBM) works well when the potential force is large enough that electrons spend most the time bound to their ionic cores. This enables us to begin with the valence electron wavefunction (ϕk(r)), and treat the potential imparted by neighboring ions using perturbation theory. The conditions assumed for the TBM limit are essentially opposite to the nearly free electron approach.

For a 1D array of N atoms we derived that ground state atomic levels of the individual atomic sites have a k-dependent total energy from nearest neighbor interactions given by: )cos(2 11 kaVEk −=∆ ε

Figure S1: Schematic of TBM (credit: Chap. 5, M. Omar, Elementary Solid State Physics). (a) atomic ion crystal potential (b) ion valence atomic orbitals (c) schematic of TBM Bloch functions. In the TBM, the probability for an electron to leak or tunnel to adjacent sites decreases, and is approximately zero beyond next-nearest neighbor interactions.

Electron interactions with the nearest neighbor sights lead to band broadening that scales inversely with the inter-ion spacing:

Figure S2: Schematic of band broadening under the TBM (credit: Fig. 10.4 of AS&M). (left) Discrete nondegenerate electronic levels in an atomic potential. (right) Energy broadening of levels for N ions as a function the inverse inter-ion spacing. When ions are far apart (small overlap integral) the bands are nearly degenerate, but

broaden into bands as they move closer (with bandwidth of roughly 4V where nnnm HV ϕϕ '= for m=n±1,

nearest neighbor (nn) interactions).