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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 Coulomb gap ： ウィキペディア英語版 Coulomb gap First introduced by M. Pollak,〔M. Pollak. Discuss. Faraday Soc. 50 (1970), p. 13 〕 the Coulomb gap is a soft gap in the Single-Particle Density of States (DOS) of a system of interacting localized electrons. Due to the long-range Coulomb interactions, the single-particle DOS vanishes at the chemical potential, at low enough temperatures, such that thermal excitations do not wash out the gap. == Theory == At zero temperature, a classical treatment of a system gives an upper bound for the DOS near the Fermi-energy, first suggested by Efros and Shklovskii.〔A L Efros and B I Shklovskii , J. Phys. C8, L49 (1975)〕 The argument is as follows: Let us look at the ground state configuration of the system. Defining $E\_i$ as the energy of an electron at site $i$, due to the disorder and the Coulomb interaction with all other electrons (we define this both for occupied and unoccupied sites), it is easy to see that the energy needed to move an electron from an occupied site $i$ to an unoccupied site $j$ is given by the expression: :$\backslash Delta\; E=E\_j-E\_i-e^2/r\_$. The subtraction of the last term accounts for the fact that $E\_j$ contains a term due to the interaction with the electron present at site $i$, but after moving the electron this term should not be considered. It is easy to see from this that there exists an energy $E\_f$ such that all sites with energies above it are empty, and below it are full (this is the Fermi energy, but since we are dealing with a system with interactions it is not obvious a-priori that it is still well-defined). Assume we have a finite single-particle DOS at the Fermi energy, $g(E\_f)$. For every possible transfer of an electron from an occupied site i to an unoccupied site j, the energy invested should be positive, since we are assuming we are in the ground state of the system, i.e., $\backslash Delta\; E>=0$. Assuming we have a large system, let us consider all the sites with energies in the interval $(E\_f+\backslash epsilon).$ The number of these, by assumption, is $N=\; 2\; \backslash epsilon\; g(E\_f).$ As explained, $N/2$ of these would be occupied, and the others unoccupied. Of all pairs of occupied and unoccupied sites, let us choose the one where the two are closest to each other. If we assume the sites are randomly distributed in space, we find that the distance between these two sites is of order: $R\; \backslash sim\; (N/V)^$, where d is the dimension of space. Plugging the expression for $N$ into the previous equation, we obtain the inequality: $E\_j-E\_i-C\; e^2\; (\backslash epsilon\; g(E\_f)/V)^\; >0$ where $C$ is a coefficient of order unity. Since , this inequality will necessarily be violated for small enough $\backslash epsilon$. Hence, assuming a finite DOS at $E\_f$ led to a contradiction. Repeating the above calculation under the assumption that the DOS near $E\_f$ is proportional to $(E-E\_f)^\backslash alpha$ shows that $\backslash alpha>=d-1$. This is an upper bound for the Coulomb gap. Efros 〔A. L. Efros, J. Phys. C: Solid State Phys 9, 2021 (1976)〕 considered single electron excitations, and obtained an integro-differential equation for the DOS, showing the Coulomb gap in fact follows the above equation (i.e., the upper bound is a tight bound). Other treatments of the problem include a mean-field numerical approach,〔M. Grunewald, B. Pohlmann, L. Schweitzer, and D.Wurtz,J. Phys. C: Solid State Phys., 15, L1153 (1982)〕 as well as more recent treatments such as,〔M. Muller and S. Pankov, Phys. Rev. B. 75, 144201 (2007)〕 also verifying the upper bound suggested above is a tight bound. Many Monte-Carlo simulations were also performed,〔J. H. Davies, P. A. Lee, and T. M. Rice, Phys. Rev. Lett. 49, 758 - 761 (1982) 〕〔A. Mobius, M. Richter, and B. Drittler, Phys. Rev. B 45, 11568 (1992)〕 some of them in disagreement with the result quoted above. Few works deal with the quantum aspect of the problem.〔G. Vignale, Phys. Rev. B 36, 8192(1987) 〕 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「Coulomb gap」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.