Wannier function explained

The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937.^{[1] [2]} 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.

Definition

Although, like localized molecular orbitals, 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 _{k} ( r ) = e ^{ik ⋅ r} u _{k(r)} where u_k(r) has the same periodicity as the crystal. Then the Wannier functions are defined by\phi _{R} ( r ) = (1)/(\sqrt{N) } \sum_ e^ \psi_(\mathbf),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 _{k} \longrightarrow (N)/(\Omega) \int _{BZ} d ^{3k} 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 _{R} ( r ) = \phi _{R+R'} ( r + 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( r - R ) : = \phi _{R} ( r )

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

\psi _{k} ( r ) = (1)/(\sqrt{N) } \sum_ e^ \phi_(\mathbf),where the sum is over each lattice vector R in the crystal.

• The set of wavefunctions

\phi _{R} is an orthonormal basis for the band in question.\begin{align} \int _{crystal} \phi _{R} ( r ) ^{*} \phi _{R'} ( r ) d ^{3r} & = (1)/(N) \sum _{k,k'} \int _{crystal} e ^{ik ⋅ R} \psi _{k} ( r ) ^{*} e ^{-ik' ⋅ R'} \psi _{k'} ( r ) d ^{3r} \\ & = (1)/(N) \sum _{k,k'} e ^{ik ⋅ R} e ^{-ik' ⋅ R'} \delta _{k,k'} \\ & = (1)/(N) \sum _{k} e ^{ik ⋅ (R-R')} \\ & = \delta _{R,R'} \end{align}

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

Localization

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 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^[6] 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.

A Pipek-Mezey style localization scheme has also been recently proposed for obtaining Wannier functions.^[7] 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 results

The existence of exponentially localized Wannier functions in insulators was proved mathematically in 2006.^[8]

Modern theory of polarization

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,^[9] and Nakhmanson,^[10] and a power-point introduction by Vanderbilt.^[11] The polarization per unit cell in a solid can be defined as the dipole moment of the Wannier charge density:

p _{c} = - e \sum _{n} \int d ^{3} r r |W ^{2} _{n( r )|} , 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] [12]}

Wannier interpolation

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,^[13] anomalous Hall conductivity,^[14] orbital magnetization,^[15] thermoelectric and electronic transport properties,^[16] gyrotropic effects,^[17] shift current,^[18] spin Hall conductivity^{[19] [20]} and other effects.

See also

• Orbital magnetization

Further reading

• Book: Physics of Ferroelectrics: a Modern Perspective . Karin M Rabe . Karin M. Rabe . Jean-Marc Triscone . Charles H Ahn . 2 . Springer . 2007 . 978-3-540-34590-9.

External links

• 10.1103/PhysRev.52.191 . 52 . 3 . The Structure of Electronic Excitation Levels in Insulating Crystals . 1937 . Physical Review . 191–197 . Wannier Gregory H. 1937PhRv...52..191W .
• Wannier90 computer code that calculates maximally localized Wannier functions
• Wannier Transport code that calculates maximally localized Wannier functions fit for Quantum Transport applications
• WannierTools: An open-source software package for novel topological materials
• WannierBerri - a python code for Wannier interpolation and tight-binding calculations

See also

• Bloch's theorem
• Hannay angle
• Geometric phase

Notes and References

1. 10.1103/PhysRev.52.191 . 52 . 3 . The Structure of Electronic Excitation Levels in Insulating Crystals . 1937 . Physical Review . 191–197 . Wannier Gregory H. 1937PhRv...52..191W .
2. Wannier . Gregory H. . Dynamics of Band Electrons in Electric and Magnetic Fields . Reviews of Modern Physics . American Physical Society (APS) . 34 . 4 . 1 September 1962 . 0034-6861 . 10.1103/revmodphys.34.645 . 645–655 . 1962RvMP...34..645W.
3. https://cfm.ehu.es/ivo/publications/marzari-psik03.pdf Marzari et al.: An Introduction to Maximally-Localized Wannier Functions
4. Book: The Geometric Phase in Quantum Systems . A Bohm, A Mostafazadeh, H Koizumi, Q Niu and J Zqanziger . 978-3-540-00031-0 . Springer . 2003 . §12.5, p. 292 ff. 10.1007/978-3-662-10333-3 .
5. http://www.physast.uga.edu/~mgeller/4.pdf MP Geller and W Kohn
6. 10.1103/PhysRev.115.809 . 115 . 4 . Analytic Properties of Bloch Waves and Wannier Functions . 1959. Physical Review . 809–821 . W. Kohn. 1959PhRv..115..809K.
7. 10.1021/acs.jctc.6b00809 . 28099002 . 13 . 2 . Theory and Applications of Generalized Pipek–Mezey Wannier Functions . 2017 . Journal of Chemical Theory and Computation . 460–474 . Jónsson Elvar Ö., Lehtola Susi, Puska Martti, Jónsson Hannes. 1608.06396 . 206612913 .
8. Brouder . Christian . Panati . Gianluca . Calandra . Matteo . Mourougane . Christophe . Marzari . Nicola . Exponential Localization of Wannier Functions in Insulators . Physical Review Letters . American Physical Society (APS) . 98 . 4 . 25 January 2007 . 0031-9007 . 10.1103/physrevlett.98.046402 . 046402. 17358792 . cond-mat/0606726. 2007PhRvL..98d6402B . 32812449 .
9. Berghold . Gerd . Mundy . Christopher J. . Romero . Aldo H. . Hutter . Jürg . Parrinello . Michele . General and efficient algorithms for obtaining maximally localized Wannier functions . Physical Review B . American Physical Society (APS) . 61 . 15 . 15 April 2000 . 0163-1829 . 10.1103/physrevb.61.10040 . 10040–10048. 2000PhRvB..6110040B .
10. Nakhmanson . S. M. . Calzolari . A. . Meunier . V. . Bernholc . J. . Buongiorno Nardelli . M. . Spontaneous polarization and piezoelectricity in boron nitride nanotubes . Physical Review B . 67 . 23 . 10 June 2003 . 0163-1829 . 10.1103/physrevb.67.235406 . 235406. cond-mat/0305329v1. 2003PhRvB..67w5406N . 119345964 .
11. http://www.physics.rutgers.edu/~dhv/talks/rahman.pdf D Vanderbilt
12. Book: C. Pisani . Quantum-mechanical Ab-initio Calculation of the Properties of Crystalline Materials . 978-3-540-61645-0 . 1994 . Springer . Proceedings of the IV School of Computational Chemistry of the Italian Chemical Society . 282 .
13. Yates . Jonathan R. . Wang . Xinjie . Vanderbilt . David . Souza . Ivo . Spectral and Fermi surface properties from Wannier interpolation . Physical Review B . American Physical Society (APS) . 75 . 19 . 2007-05-21 . 195121 . 1098-0121 . 10.1103/physrevb.75.195121 . cond-mat/0702554. 2007PhRvB..75s5121Y . 31224663 .
14. Wang . Xinjie . Yates . Jonathan R. . Souza . Ivo . Vanderbilt . David . Ab initiocalculation of the anomalous Hall conductivity by Wannier interpolation . Physical Review B . 74 . 19 . 2006-11-21 . 195118 . cond-mat/0608257. 1098-0121 . 10.1103/physrevb.74.195118 . 2006PhRvB..74s5118W . 30427871 .
15. Lopez . M. G. . Vanderbilt . David . Thonhauser . T. . Souza . Ivo . Wannier-based calculation of the orbital magnetization in crystals . Physical Review B . 85 . 1 . 2012-01-31 . 014435 . 1098-0121 . 10.1103/physrevb.85.014435 . 1112.1938 . 2012PhRvB..85a4435L . 44056938 .
16. BoltzWann: A code for the evaluation of thermoelectric and electronic transport properties with a maximally-localized Wannier functions basis . Computer Physics Communications . 185 . 1 . 2014-01-01 . 0010-4655 . 10.1016/j.cpc.2013.09.015 . 422–429 . 1305.1587 . 2020-07-13. Pizzi . Giovanni . Volja . Dmitri . Kozinsky . Boris . Fornari . Marco . Marzari . Nicola . 2014CoPhC.185..422P . 6140858 .
17. Tsirkin . Stepan S. . Puente . Pablo Aguado . Souza . Ivo . Gyrotropic effects in trigonal tellurium studied from first principles . Physical Review B . 97 . 3 . 2018-01-29 . 035158 . 2469-9950 . 10.1103/physrevb.97.035158 . 1710.03204. 2018PhRvB..97c5158T . 55517213 .
18. Ibañez-Azpiroz . Julen . Tsirkin . Stepan S. . Souza . Ivo . Ab initio calculation of the shift photocurrent by Wannier interpolation . Physical Review B . 97 . 24 . 2018-06-26 . 245143 . 2469-9950 . 10.1103/physrevb.97.245143 . 1804.04030. 2018PhRvB..97x5143I . 67751414 .
19. Qiao . Junfeng . Zhou . Jiaqi . Yuan . Zhe . Zhao . Weisheng . Calculation of intrinsic spin Hall conductivity by Wannier interpolation . Physical Review B . 98 . 21 . 2018-12-03 . 214402 . 1810.07637. 2469-9950 . 10.1103/physrevb.98.214402 . 2018PhRvB..98u4402Q . 119223848 .
20. Ryoo . Ji Hoon . Park . Cheol-Hwan . Souza . Ivo . Computation of intrinsic spin Hall conductivities from first principles using maximally localized Wannier functions . Physical Review B . 99 . 23 . 2019-06-07 . 235113 . 1906.07139. 2469-9950 . 10.1103/physrevb.99.235113 . 2019PhRvB..99w5113R . 189928182 .