§5.6 §5.6 Tight-binding Tight-binding is first proposed in 1929 by Bloch. The primary idea is to use a linear combination of atomic orbitals as a set of basis functions, thereby, we can solve the solid Schrodinger equation. This method is based on such physical image, there is little difference between the electronic states in solids and the free atoms which are composed by them. Tight-binding is very successful in the study of the band structure of the insulator. Since the atomic orbitals are locate on different grids, basis function composed of them are generally non-orthogonal. So, inevitably, we would encounter the computing problems of multi-center integrals, and the form of eigenequation is not easy.

Potential fieldis a periodic function of lattice, which can be From the Bloch theorem ， in the wave vector space, bloch wave function is a periodic function of reciprocal lattice vectors. Similar to the potential function, the Bloch wave function in the wave vector space is expanded to a Fourier series: In the above formula, is called Wannier function, α is the band number. developed into the real space Fourier series:

The above equation was multiplied by All wave vector within the brillouin zone are summed up: Combined with (5.15): Get

The above formula shows that Wannier function of the different energy bands or the same band with different grid points are orthogonal. Known from the translational symmetry of the Bloch wave function: When the spacing of atoms in the crystal is large, the probability of electrons are trapped by nearby atoms is much larger than it move away from atoms, the behavior of electrons in the vicinity of a grid is similar to the behavior of electrons in the isolated atoms; when the electrons obviously deviated from the grids, the wave function is a small quantity. Isolated atom wave functions can be used to describe the wave function under tight-binding conditions. Take

Wannier function can be translated into: Using the orthogonality of the Wannier function, we can get:

We can get: The formula is called the Bloch wave function ， it is a linear combination of atomic orbital wave function, so the tight-binding method is often called linear combination of atomic orbital method. The above equation is substituted into the Schrodinger equation, and is rewritten as ： Among them, is the potential field formed by atoms with the grid point.

We discuss the s-state of those non-degenerated electrons. When the principal quantum number is certain, the s-state wavefunction will be more localized ， more suitable to the tight-binding. Using the following relationship ： The above equation is multiplied by integrated over the crystal volume, we can get: then,

Using the tight-binding model, we ignore the quadratic terms, and only reserve the term then, the first part of the above equation is equal to ： W hen, the integral term of the second part of the above formula is written as The integral term is negative obtained from the above figure.

When,since the overlap is small for the isolated atom wave function of the two adjacent grids, so only the overlap integral of the adjacent grids are considered. S-state is spherical symmetry, so the integral values of the nearest grids are same. So, the second part can be simplified as : In summary, the band of the s-state of the tight-binding electron is is the nearest lattice vector.

For example, for a simple cubic crystal, there are six of the nearest neighbor atoms. Substitute coordinates of the six atoms into the above formula, we can get: The maximum of the Energy is: The width of the band: The minimum of the Energy is:

The width of the band is determined by the size and the coefficient of J s, and J s depends on the overlap integral, the coefficient depends on the number of the nearest neighbor grid points, i.e., the coordination number of crystal. We can expect, the larger overlap degree of the wave function, and the more coordination numbers of the crystal, the wider band, on the contrary, the narrower band. An energy level of an electron in the isolated atom becomes a band in solid.