Wannier function: Definition from Answers.com

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.

1 Simplest definition
- 1.1 Properties
2 Modern theory of polarization
3 References
4 Further reading
5 External links
6 See also

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

R is any lattice vector (i.e., there is one Wannier function for each Bravais lattice vector);
N is the number of primitive cells in the crystal;
The sum on k includes all the values of k in the Brillouin zone (or any other primitive cell of the reciprocal lattice) that are consistent with periodic boundary conditions on the crystal. This includes N different values of k, spread out uniformly through the Brillouin zone. Since N is usually very large, the sum can be written as an integral according to the replacement rule:

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

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:^[4]

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.

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

$\int_{crystal} \phi_{\mathbf{R}}(\mathbf{r})^* \phi_{\mathbf{R'}}(\mathbf{r}) d^3\mathbf{r} = \frac{1}{N} \sum_{\mathbf{k,k'}}\int_{crystal} e^{i\mathbf{k}\cdot\mathbf{R}} \psi_{\mathbf{k}}(\mathbf{r})^* e^{-i\mathbf{k'}\cdot\mathbf{R'}} \psi_{\mathbf{k'}}(\mathbf{r}) d^3\mathbf{r} = \frac{1}{N} \sum_{\mathbf{k,k'}} e^{i\mathbf{k}\cdot\mathbf{R}} e^{-i\mathbf{k'}\cdot\mathbf{R'}} \delta_{\mathbf{k,k'}} = \frac{1}{N} \sum_{\mathbf{k}} e^{i\mathbf{k}\cdot\mathbf{(R'-R)}}=\delta_{\mathbf{R,R'}}$

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]

Wannier functions have been extended to nearly periodic potentials as well.^[5]

Modern theory of polarization

Wannier functions have recently found application in describing the polarization in crystals, for example, ferroelectrics. See for example, Berghold,^[6] and Nakhmanson.^[7], and a power-point introduction by Vanderbilt^[8] The polarization per unit cell in a solid can be defined as the dipole moment of the Wannier charge density:

$\boldsymbol{p_c} = -e \sum_n \int\ d^3 r \,\, \boldsymbol r |W_n(\boldsymbol{r})|^2 \ ,$

where the summation is over the occupied bands, and W_n is the Wannier function localized in the cell for band n. The change in polarization during a continuous physical process is the time derivative of the polarization and also can be formulated in terms of the Berry phase of the occupied Bloch states. ^[4]^[9]

References

^ ^a ^b "The structure of electronic excitation levels in insulating crystals," G. H. Wannier, Phys. Rev. 52, 191 (1937)
^ ^a ^b Marzari et al.: Exponential localization of Wannier functions in insulators
^ Marzari et al.: An Introduction to Maximally-Localized Wannier Functions
^ ^a ^b A Bohm, A Mostafazadeh, H Koizumi, Q Niu and J Zqanziger (2003). The Geometric Phase in Quantum Systems. Springer. pp. §12.5, p. 292 ff. ISBN 3540000313. http://books.google.com/books?id=2XXWveovsRAC&pg=PA293&lpg=PA293&dq=berry++%22wannier+function%22&source=bl&ots=zbPBUW6K-g&sig=Jp-E3Tw6sheICB6QKW1amGC-OIg&hl=en&sa=X&oi=book_result&resnum=8&ct=result#PPA292,M1.
^ MP Geller and W Kohn Theory of generalized Wannier functions for nearly periodic potentials Physical Review B 48, 1993
^ Gerd Berghold et al. General and efficient algorithms for obtaining maximally localized Wannier functions
^ SM Nakhmanson et al. Spontaneous polarization and piezoelectricity in boron nitride nanotubes, 2008
^ D Vanderbilt Berry phases and Curvatures in Electronic Structure Theory.
^ C. Pisani (1994). Quantum-mechanical Ab-initio Calculation of the Properties of Crystalline Materials (Proceedings of the IV School of Computational Chemistry of the Italian Chemical Society ed.). Springer. p. 282. ISBN 3540616454. http://books.google.com/books?id=5ak5TwSLreAC&pg=PA282&dq=%22Berry+connection%22&lr=&as_brr=0.

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Wannier function

Contents

Simplest definition

Properties

Modern theory of polarization

References

Further reading

External links

See also