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.
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
has the same periodicity as the crystal. Then the Wannier functions are defined by
where "BZ" denotes the Brillouin zone, which has volume Ω
On the basis of this definition, the following properties can be proven to hold:
- For any lattice vector R' ,
In other words, a Wannier function only depends on the quantity (r
). As a result, these functions are often written in the alternative notation
- The Bloch functions can be written in terms of Wannier functions as follows:
where the sum is over each lattice vector R
in the crystal.
- The set of wavefunctions is an orthonormal basis for the band in question.
It is generally assumed that the function 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.