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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 Variable-range hopping ： ウィキペディア英語版 Variable-range hopping Variable-range hopping, or Mott variable-range hopping, is a model describing low-temperature conduction in strongly disordered systems with localized charge-carrier states. It has a characteristic temperature dependence of :$\backslash sigma=\; \backslash sigma\_0e^\}$. Hopping conduction at low temperatures is of great interest because of the savings the semiconductor industry could achieve if they were able to replace single-crystal devices with glass layers.〔P.V.E. McClintock, D.J. Meredith, J.K. Wigmore. ''Matter at Low Temperatures''. Blackie. 1984 ISBN 0-216-91594-5.〕 ==Derivation== The original Mott paper introduced a simplifying assumption that the hopping energy depends inversely on the cube of the hopping distance (in the three-dimensional case). Later it was shown that this assumption was unnecessary, and this proof is followed here. In the original paper, the hopping probability at a given temperature was seen to depend on two parameters, ''R'' the spatial separation of the sites, and ''W'', their energy separation. Apsley and Hughes noted that in a truly amorphous system, these variables are random and independent and so can be combined into a single parameter, the ''range'' $\backslash textstyle\backslash mathcal$ between two sites, which determines the probability of hopping between them. Mott showed that the probability of hopping between two states of spatial separation $\backslash textstyle\; R$ and energy separation ''W'' has the form: :$P\backslash sim\; \backslash exp\; \backslash left(R-\backslash frac\backslash right)$ where α−1 is the attenuation length for a hydrogen-like localised wave-function. This assumes that hopping to a state with a higher energy is the rate limiting process. We now define $\backslash textstyle\backslash mathcal\; =\; 2\backslash alpha\; R+W/kT$, the ''range'' between two states, so $\backslash textstyle\; P\backslash sim\; \backslash exp\; (-\backslash mathcal)$. The states may be regarded as points in a four-dimensional random array (three spatial coordinates and one energy coordinate), with the `distance' between them given by the range $\backslash textstyle\backslash mathcal$. Conduction is the result of many series of hops through this four-dimensional array and as short-range hops are favoured, it is the average nearest-neighbour `distance' between states which determines the overall conductivity. Thus the conductivity has the form :$\backslash sigma\; \backslash sim\; \backslash exp\; (-\backslash overline)$ where $\backslash textstyle\backslash overline$is the average nearest-neighbour range. The problem is therefore to calculate this quantity. The first step is to obtain $\backslash textstyle\backslash mathcal(\backslash mathcal)$, the total number of states within a range $\backslash textstyle\backslash mathcal$ of some initial state at the Fermi level. For ''d''-dimensions, and under particular assumptions this turns out to be :$\backslash mathcal(\backslash mathcal)\; =\; K\; \backslash mathcal^$ where $\backslash textstyle\; K\; =\; \backslash frac$. The particular assumptions are simply that $\backslash textstyle\backslash overline$ is well less than the band-width and comfortably bigger than the interatomic spacing. Then the probability that a state with range $\backslash textstyle\backslash mathcal$ is the nearest neighbour in the four-dimensional space (or in general the (''d''+1)-dimensional space) is :$P\_(\backslash mathcal)\; =\; \backslash frac)\}(\backslash mathcal)"\; TITLE="-\backslash mathcal(\backslash mathcal)">)$ the nearest-neighbour distribution. For the ''d''-dimensional case then :$\backslash overline\; =\; \backslash int\_0^\backslash infty\; (d+1)K\backslash mathcal^\backslash exp\; (-K\backslash mathcal^)d\backslash mathcal$. This can be evaluated by making a simple substitution of $\backslash textstyle\; t=K\backslash mathcal^$ into the gamma function, $\backslash textstyle\; \backslash Gamma(z)\; =\; \backslash int\_0^\backslash infty\; t^\; e^\backslash ,\backslash mathrmt$ After some algebra this gives :$\backslash overline\; =\; \backslash frac)\}\}\}$ and hence that :$\backslash sigma\; \backslash propto\; \backslash exp\; \backslash left(-T^\}\backslash right)$. 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「Variable-range hopping」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.