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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Wannier equation ： ウィキペディア英語版 Wannier equation The Wannier equation describes a quantum mechanical eigenvalue problem in solids where an electron in a conduction band and an electronic vacancy (i.e. hole) within a valence band attract each other via the Coulomb interaction. For one electron and one hole, this problem is analogous to the Schrödinger equation of the hydrogen atom; and the bound-state solutions are called excitons. When an exciton's radius extends over several unit cells, it is referred to as a Wannier exciton in contrast to Frenkel excitons whose size is comparable with the unit cell. An excited solid typically contains many electrons and holes; this modifies the Wannier equation considerably. The resulting generalized Wannier equation can be determined from the homogeneous part of the semiconductor Bloch equations or the semiconductor luminescence equations. ==Background== Since an electron and a hole have opposite charges their mutual Coulomb interaction is attractive. The corresponding Schrödinger equation, in relative coordinate $\backslash mathbf$, has the same form as the hydrogen atom: $$ -\left(\frac + V(\mathbf) \right ) \phi_\lambda (\mathbf) = E_\lambda \phi_\lambda (\mathbf)\,, with the potential given by $$ V(\mathbf) = \frac\,. Here, $\backslash hbar$ is the reduced Planck constant, $\backslash nabla$ is the nabla operator, $\backslash mu$ is the reduced mass, $-|e|$ ($+|e|$) is the elementary charge related to an electron (hole), $\backslash varepsilon\_r$ is the relative permittivity, and $\backslash varepsilon\_0$ is the vacuum permittivity. The solutions of the Hydrogen atom are described by eigenfunction $\backslash phi\_\backslash lambda\; (\backslash mathbf)$ and eigenenergy $E\_\backslash lambda$ where $\backslash lambda$ is a quantum number labeling the different states. In a solid, the scaling of $E\_\backslash lambda$ and the wavefunction size are orders of magnitude different from the hydrogen problem because the relative permittivity $\backslash varepsilon\_r$ is roughly ten and the reduced mass in a solid is much smaller than the electron rest mass $m\_e$, i.e., $\backslash mu\; \backslash ll\; m\_e$. As a result, the exciton radius can be large while the exciton binding energy is small, typically few to hundreds of meV, depending on material, compared to eV for the Hydrogen problem.〔Haug, H.; Koch, S. W. (2009). ''Quantum Theory of the Optical and Electronic Properties of Semiconductors'' (5th ed.). World Scientific. p. 216. ISBN 9812838848.〕〔Klingshirn, C. F. (2006). ''Semiconductor Optics''. Springer. ISBN 978-3540383451.〕 The Fourier transformed version of the presented Hamiltonian can be written as $$ E_) - \sum_-\mathbf} \phi_\lambda (\mathbf) = E_\lambda \phi_\lambda (\mathbf)\,, where $\backslash mathbf$ is the electronic wave vector, $E\_\}$, $\backslash phi\_\backslash lambda\; (\backslash mathbf)$ are the Fourier transforms of $V(\backslash mathbf)$, $\backslash phi\_\backslash lambda\; (\backslash mathbf)$, respectively. The Coulomb sums follows from the convolution theorem and the $\backslash mathbf$-representation is useful when introducing the generalized Wannier equation. 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 equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.