Wannier function

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An example of WF in Barium Titanate.
An example of WF in Barium Titanate.

The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier.[1]

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[2]. Specifically, these functions are also used in the analysis of excitons and condensed Rydberg matter.

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[edit] Simplest definition

Although Wannier functions can be chosen in many different ways,[3] the original,[1] 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_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_{\mathbf{k}}(\mathbf{r})

where \, u_{\mathbf{k}}(\mathbf{r}) has the same periodicity as the crystal. Then the Wannier functions are defined by

\phi_{\mathbf{R}}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{k}} e^{-i\mathbf{k}\cdot\mathbf{R}} \psi_{\mathbf{k}}(\mathbf{r}),

where

\sum_{\mathbf{k}} \longrightarrow \frac{N}{\Omega} \int_{BZ} d^3\mathbf{k}

where "BZ" denotes the Brillouin zone, which has volume Ω.

[edit] Properties

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

  • For any lattice vector R' ,
\phi_{\mathbf{R}}(\mathbf{r}) = \phi_{\mathbf{R}+\mathbf{R}'}(\mathbf{r}+\mathbf{R}')

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(\mathbf{r}-\mathbf{R}) := \phi_{\mathbf{R}}(\mathbf{r})
  • The Bloch functions can be written in terms of Wannier functions as follows:
\psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}} \phi_{\mathbf{R}}(\mathbf{r}),

where the sum is over each lattice vector R in the crystal.

It is generally assumed that the function \phi_{\mathbf{R}} 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.[2]

[edit] References

  1. ^ a b "The structure of electronic excitation levels in insulating crystals," G. H. Wannier, Phys. Rev. 52, 191 (1937)
  2. ^ a b http://arxiv.org/pdf/cond-mat/0606726
  3. ^ http://quasiamore.mit.edu/wannier/papers/MSVpsik.pdf

[edit] External links

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