Wannier function

Wannier functions of triple- and single-bonded nitrogen dimers in palladium nitride.

The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937.<ref name=Wannier1937>Template:Cite journal</ref><ref name=Wannier1962>Template:Cite journal</ref> Wannier functions are the localized molecular orbitals of crystalline systems.

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.

DefinitionEdit

File:WanF-BaTiO3.png

Example of a localized Wannier function of titanium in barium titanate (BaTiO3)

Although, like localized molecular orbitals, Wannier functions can be chosen in many different ways,<ref>Marzari et al.: An Introduction to Maximally-Localized Wannier Functions</ref> the original,<ref name=Wannier1937/> 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

<math>\psi_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_\mathbf{k}(\mathbf{r})</math>

where u_k(r) has the same periodicity as the crystal. Then the Wannier functions are defined by

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

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:

<math>\sum_{\mathbf{k}} \longrightarrow \frac{\sqrt{N}}{\Omega} \int_\text{BZ} d^3\mathbf{k}</math>

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

PropertiesEdit

On the basis of this definition, the following properties can be proven to hold:<ref name=Bohm>Template:Cite book</ref>

For any lattice vector R' ,

<math>\phi_{\mathbf{R}}(\mathbf{r}) = \phi_{\mathbf{R}+\mathbf{R}'}(\mathbf{r}+\mathbf{R}')</math>

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

<math>\phi(\mathbf{r}-\mathbf{R}) := \phi_{\mathbf{R}}(\mathbf{r})</math>

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

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

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

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

<math>\begin{align}

\int_\text{crystal} \phi_{\mathbf{R}}(\mathbf{r})^* \phi_{\mathbf{R'}}(\mathbf{r}) d^3\mathbf{r} & = \frac{1}{N} \sum_{\mathbf{k,k'}}\int_\text{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'}} \end{align} </math>

Wannier functions have been extended to nearly periodic potentials as well.<ref name=Kohn0>MP Geller and W Kohn Theory of generalized Wannier functions for nearly periodic potentials Physical Review B 48, 1993</ref>

LocalizationEdit

The Bloch states ψ_k(r) are defined as the eigenfunctions of a particular Hamiltonian, and are therefore defined only up to an overall phase. By applying a phase transformation e^iθ(k) to the functions ψ_k(r), for any (real) function θ(k), one arrives at an equally valid choice. While the change has no consequences for the properties of the Bloch states, the corresponding Wannier functions are significantly changed by this transformation.

One therefore uses the freedom to choose the phases of the Bloch states in order to give the most convenient set of Wannier functions. In practice, this is usually the maximally-localized set, in which the Wannier function Template:Math is localized around the point R and rapidly goes to zero away from R. For the one-dimensional case, it has been proved by Kohn<ref name=Kohn1>Template:Cite journal</ref> that there is always a unique choice that gives these properties (subject to certain symmetries). This consequently applies to any separable potential in higher dimensions; the general conditions are not established, and are the subject of ongoing research.<ref name=Arxiv-Localization/>

A Pipek-Mezey style localization scheme has also been recently proposed for obtaining Wannier functions.<ref name=Jonsson2016>Template:Cite journal</ref> Contrary to the maximally localized Wannier functions (which are an application of the Foster-Boys scheme to crystalline systems), the Pipek-Mezey Wannier functions do not mix σ and π orbitals.

Rigorous resultsEdit

The existence of exponentially localized Wannier functions in insulators was proved mathematically in 2006.<ref name=Arxiv-Localization>Template:Cite journal</ref>

Modern theory of polarizationEdit

Wannier functions have recently found application in describing the polarization in crystals, for example, ferroelectrics. The modern theory of polarization is pioneered by Raffaele Resta and David Vanderbilt. See for example, Berghold,<ref name=Berghold>Template:Cite journal</ref> and Nakhmanson,<ref name=Nakhmanson>Template:Cite journal</ref> and a power-point introduction by Vanderbilt.<ref name=Vanderbilt>D Vanderbilt Berry phases and Curvatures in Electronic Structure Theory.</ref> The polarization per unit cell in a solid can be defined as the dipole moment of the Wannier charge density:

<math>\mathbf{p_c} = -e \sum_n \int\ d^3 r \,\, \mathbf{r} |W_n(\mathbf{r})|^2 \ , </math>

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.<ref name=Bohm/><ref name=Resta>Template:Cite book</ref>

Wannier interpolationEdit

Wannier functions are often used to interpolate bandstructures calculated ab initio on a coarse grid of k-points to any arbitrary k-point. This is particularly useful for evaluation of Brillouin-zone integrals on dense grids and searching of Weyl points, and also taking derivatives in the k-space. This approach is similar in spirit to the tight binding approximation, but in contrast allows for an exact description of bands in a certain energy range. Wannier interpolation schemes have been derived for spectral properties,<ref name="Yates Wang Vanderbilt Souza p. ">Template:Cite journal</ref> anomalous Hall conductivity,<ref name="Wang Yates Souza Vanderbilt p. ">Template:Cite journal</ref> orbital magnetization,<ref name="Lopez Vanderbilt Thonhauser Souza p. ">Template:Cite journal</ref> thermoelectric and electronic transport properties,<ref name="Computer Physics Communications 2014 pp. 422–429">Template:Cite journal</ref> gyrotropic effects,<ref name="Tsirkin Puente Souza p. ">Template:Cite journal</ref> shift current,<ref name="Ibañez-Azpiroz Tsirkin Souza p. ">Template:Cite journal</ref> spin Hall conductivity <ref name="Qiao Zhou Yuan Zhao p. ">Template:Cite journal</ref> <ref name="Ryoo Park Souza p. ">Template:Cite journal</ref> and other effects.

ReferencesEdit

Template:Reflist

Wannier function

Contents