Tight Binding - Example: One-dimensional S-band

Example: One-dimensional S-band

Here the tight binding model is illustrated with a s-band model for a string of atoms with a single s-orbital in a straight line with spacing a and σ bonds between atomic sites.

To find approximate eigenstates of the Hamiltonian, we can use a linear combination of the atomic orbitals

where N = total number of sites and is a real parameter with . (This wave function is normalized to unity by the leading factor 1/√N provided overlap of atomic wave functions is ignored.) Assuming only nearest neighbor overlap, the only non-zero matrix elements of the Hamiltonian can be expressed as

The energy E_i is the ionization energy corresponding to the chosen atomic orbital and U is the energy shift of the orbital as a result of the potential of neighboring atoms. The elements, which are the Slater and Koster interatomic matrix elements, are the bond energies . In this one dimensional s-band model we only have -bonds between the s-orbitals with bond energy . The overlap between states on neighboring atoms is S. We can derive the energy of the state using the above equation:

where, for example,

and

Thus the energy of this state can be represented in the familiar form of the energy dispersion:

.

• For the energy is and the state consists of a sum of all atomic orbitals. This state can be viewed as a chain of bonding orbitals.
• For the energy is and the state consists of a sum of atomic orbitals which are a factor out of phase. This state can be viewed as a chain of non-bonding orbitals.
• Finally for the energy is and the state consists of an alternating sum of atomic orbitals. This state can be viewed as a chain of anti-bonding orbitals.

This example is readily extended to three dimensions, for example, to a body-centered cubic or face-centered cubic lattice by introducing the nearest neighbor vector locations in place of simply n a. Likewise, the method can be extended to multiple bands using multiple different atomic orbitals at each site. The general formulation above shows how these extensions can be accomplished.