In the tight binding model, it is assumed that the full Hamiltonian <math>H</math> of the system may be approximated by the Hamiltonian of an isolated atom centred at each lattice point. The atomic orbitals <math>\psi_n</math>, which are eigenfunctions of the single atom Hamiltonian <math>H_{at}</math>, are assumed to be very small at distances exceeding the lattice constant. This is what is meant by tight-binding. It is further assumed that any corrections to the atomic potential <math>\Delta U</math>, which are required to obtain the full Hamiltonian <math>H</math> of the system, are appreciable only when the atomic orbitals are small. A solution to the time-independent single electron Schrödinger equation <math>\Phi</math> is then assumed to be a linear combination of atomic orbitals <math>\ {\psi}_n</math>
- <math>\Phi(\vec{r}) = \sum_{n,\vec{R}} b_{n, \vec{R}}\ \psi_n(\vec{r}-\vec{R})</math>,
where n refers to the n-th atomic energy level and <math>^{{\vec{R}}}</math> is an atomic site in the crystal lattice. Using this approximate form for the wavefunction, and assuming only the m-th atomic energy level is important for the m-th energy band, the Bloch energies <math>\varepsilon_m</math> are of the form
- <math>\varepsilon_m(\vec{k}) = E_m - {\beta_m + \sum_{\vec{R}\neq 0} \gamma_m(\vec{R}) e^{i \vec{k} \cdot \vec{R}}\over {b_{m,\vec{R}}} + \sum_{\vec{R}\neq 0} \alpha_m(\vec{R}) e^{i \vec{k} \cdot \vec{R}}}</math>,
where <math>E_m</math> is the energy of the <math>m</math>th atomic level,
- <math> \beta_m = -\int \psi_m^*(\vec{r})\Delta U(\vec{r}) \Phi(\vec{r}) d\vec{r}</math>,
- <math> \alpha_m(\vec{R}) = \int \psi_m^*(\vec{r}) \Phi(\vec{r}-\vec{R}) d\vec{r}</math>,
and
- <math> \gamma_m(\vec{R}) = -\int \psi_m^*(\vec{r}) \Delta U(\vec{r}) \Phi(\vec{r}-\vec{R}) d\vec{r}</math>
are the overlap integrals.
The tight-binding model is typically used for calculations of electronic band structure and energy gaps in the static regime. However, in combination with other methods such as the random phase approximation (RPA) model, the dynamic response of systems may also be studied.
Further reading
- J.C. Slater and G.F. Koster, Phys. Rev. 94, 1498 (1954).
- C.M. Goringe, D.R. Bowler and E. Hernández, Rep. Prog. Phys. 60, 1447 (1997).
- N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, Toronto, 1976).
