Definitions

Wannier function

The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier.

The Wannier functions for different lattice sites in a crystal are orthogonal, allowing a convenient basis for the expansion of electron states in certain regimes. Wannier functions have found widespread use, for example, in the analysis of binding forces acting on electrons; they have proven to be in general localized, at least for insulators, in 2006. Specifically, these functions are also used in the analysis of excitons and condensed Rydberg matter.

Simplest definition

Although Wannier functions can be chosen in many different ways, the original, simplest, and most common definition in solid-state physics is as follows. Choose a single band in a perfect crystal, and denote its Bloch states by

$psi_\left\{mathbf\left\{k\right\}\right\}\left(mathbf\left\{r\right\}\right) = e^\left\{imathbf\left\{k\right\}cdotmathbf\left\{r\right\}\right\}u_\left\{mathbf\left\{k\right\}\right\}\left(mathbf\left\{r\right\}\right)$
where $, u_\left\{mathbf\left\{k\right\}\right\}\left(mathbf\left\{r\right\}\right)$ has the same periodicity as the crystal. Then the Wannier functions are defined by
$phi_\left\{mathbf\left\{R\right\}\right\}\left(mathbf\left\{r\right\}\right) = frac\left\{1\right\}\left\{sqrt\left\{N\right\}\right\} sum_\left\{mathbf\left\{k\right\}\right\} e^\left\{-imathbf\left\{k\right\}cdotmathbf\left\{R\right\}\right\} psi_\left\{mathbf\left\{k\right\}\right\}\left(mathbf\left\{r\right\}\right)$,
where

$sum_\left\{mathbf\left\{k\right\}\right\} longrightarrow frac\left\{N\right\}\left\{Omega\right\} int_\left\{BZ\right\} d^3mathbf\left\{k\right\}$
where "BZ" denotes the Brillouin zone, which has volume Ω.

Properties

On the basis of this definition, the following properties can be proven to hold:

• For any lattice vector R' ,

$phi_\left\{mathbf\left\{R\right\}\right\}\left(mathbf\left\{r\right\}\right) = phi_\left\{mathbf\left\{R\right\}+mathbf\left\{R\right\}\text{'}\right\}\left(mathbf\left\{r\right\}+mathbf\left\{R\right\}\text{'}\right)$
In other words, a Wannier function only depends on the quantity (r-R). As a result, these functions are often written in the alternative notation
$phi\left(mathbf\left\{r\right\}-mathbf\left\{R\right\}\right) := phi_\left\{mathbf\left\{R\right\}\right\}\left(mathbf\left\{r\right\}\right)$

• The Bloch functions can be written in terms of Wannier functions as follows:

$psi_\left\{mathbf\left\{k\right\}\right\}\left(mathbf\left\{r\right\}\right) = frac\left\{1\right\}\left\{sqrt\left\{N\right\}\right\} sum_\left\{mathbf\left\{R\right\}\right\} e^\left\{imathbf\left\{k\right\}cdotmathbf\left\{R\right\}\right\} phi_\left\{mathbf\left\{R\right\}\right\}\left(mathbf\left\{r\right\}\right)$,
where the sum is over each lattice vector R in the crystal.

• The set of wavefunctions $phi_\left\{mathbf\left\{R\right\}\right\}$ is an orthonormal basis for the band in question.

It is generally assumed that the function $phi_\left\{mathbf\left\{R\right\}\right\}$ is localized around the point R, and rapidly goes to zero away from that point. However, quantifying and proving this assertion can be difficult, and is the subject of ongoing research.