|
The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier.〔("The structure of electronic excitation levels in insulating crystals," G. H. Wannier, Phys. Rev. 52, 191 (1937) )〕〔("Dynamics of Band Electrons in Electric and Magnetic Fields", G. H. Wannier, Rev. Mod. Phys. 34, 645 (1962) )〕 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.〔(Marzari ''et al.'': Exponential localization of Wannier functions in insulators )〕 Specifically, these functions are also used in the analysis of excitons and condensed Rydberg matter. ==Definition== Although Wannier functions can be chosen in many different ways,〔(Marzari ''et al.'': An Introduction to Maximally-Localized Wannier Functions )〕 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 : where ''u''k(r) has the same periodicity as the crystal. Then the Wannier functions are defined by :, 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: : where "BZ" denotes the Brillouin zone, which has volume Ω. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wannier function」の詳細全文を読む スポンサード リンク
|