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| {{Expert-subject|Physics|date=October 2008}}
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| '''Variable-range hopping''', or [[Nevill Francis Mott|Mott]] variable-range hopping, is a model describing low-temperature [[Electrical conduction|conduction]] in strongly disordered systems with [[Anderson localization|localized]] charge-carrier states.<ref>{{cite journal|author=Mott, N.F.|journal=Phil. Mag.|volume=19|page=835|year=1969}}</ref>
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| It has a characteristic temperature dependence of
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| :<math>\sigma= \sigma_0e^{-(T_0/T)^{1/4}}</math>
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| for three-dimensional conductance, and in general for ''d''-dimensions
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| :<math>\sigma= \sigma_0e^{-(T_0/T)^{1/(d+1)}}</math>.
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| 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.<ref>P.V.E. McClintock, D.J. Meredith, J.K. Wigmore. ''Matter at Low Temperatures''. Blackie. 1984 ISBN 0-216-91594-5.</ref>
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| ==Derivation==
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| 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.<ref>{{cite journal|author=Apsley, N. and Hughes, H.P.|journal=Phil. Mag.|volume=30|page=963|year=1974}}</ref> 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'' <math>\textstyle\mathcal{R}</math> between two sites, which determines the probability of hopping between them.
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| Mott showed that the probability of hopping between two states of spatial separation <math>\textstyle R</math> and energy separation ''W'' has the form:
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| :<math>P\sim \exp \left[-2\alpha R-\frac{W}{kT}\right]</math>
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| where α<sup>−1</sup> 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.
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| We now define <math>\textstyle\mathcal{R} = 2\alpha R+W/kT</math>, the ''range'' between two states, so <math>\textstyle P\sim \exp (-\mathcal{R})</math>. 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 <math>\textstyle\mathcal{R}</math>.
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| 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
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| :<math>\sigma \sim \exp (-\overline{\mathcal{R}}_{nn})</math> | |
| where <math>\textstyle\overline{\mathcal{R}}_{nn}</math>is the average nearest-neighbour range. The problem is therefore to calculate this quantity.
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| The first step is to obtain <math>\textstyle\mathcal{N}(\mathcal{R})</math>, the total number of states within a range <math>\textstyle\mathcal{R}</math> of some initial state at the Fermi level. For ''d''-dimensions, and under particular assumptions this turns out to be
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| :<math>\mathcal{N}(\mathcal{R}) = K \mathcal{R}^{d+1}</math>
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| where <math>\textstyle K = \frac{N\pi kT}{3\times 2^d \alpha^d}</math>.
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| The particular assumptions are simply that <math>\textstyle\overline{\mathcal{R}}_{nn}</math> is well less than the band-width and comfortably bigger than the interatomic spacing.
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| Then the probability that a state with range <math>\textstyle\mathcal{R}</math> is the nearest neighbour in the four-dimensional space (or in general the (''d''+1)-dimensional space) is
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| :<math>P_{nn}(\mathcal{R}) = \frac{\partial \mathcal{N}(\mathcal{R})}{\partial \mathcal{R}} \exp [-\mathcal{N}(\mathcal{R})]</math>
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| the nearest-neighbour distribution.
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| For the ''d''-dimensional case then
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| :<math>\overline{\mathcal{R}}_{nn} = \int_0^\infty (d+1)K\mathcal{R}^{d+1}\exp (-K\mathcal{R}^{d+1})d\mathcal{R}</math>.
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| This can be evaluated by making a simple substitution of <math>\textstyle t=K\mathcal{R}^{d+1}</math> into the [[gamma function]], <math>\textstyle \Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,\mathrm{d}t</math>
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| After some algebra this gives
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| :<math>\overline{\mathcal{R}}_{nn} = \frac{\Gamma(\frac{d+2}{d+1})}{K^{\frac{1}{d+1}}}</math>
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| and hence that
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| :<math>\sigma \propto \exp \left(-T^{-\frac{1}{d+1}}\right)</math>.
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| ==Non-constant density of states==
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| When the [[density of states]] is not constant (odd power law N(E)), the Mott conductivity is also recovered, as shown in [http://hal.archives-ouvertes.fr/ccsd-00004661 this article].
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| ==See also==
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| {{Portal|Physics}}
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| * [[Coulomb gap]]
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| * [[Mobility edge]]
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| * [[Bloch wave]]
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| == Notes ==
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| {{reflist}}
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| <!-- '''Sadly''', it is still not widely recognised that the "simplifying assumption" (as it is described above) in the Mott derivation is actually a serious fundamental error, in that it simultaneously employs a distance R to represent two very different parameters - the '''''actual''''' distance hopped and the radius of the sphere '''''within which''''' hopping occurs! Simple averaging of the hopping distance to yield 3/4 of the sphere radius (N.F.Mott and E.A.Davis, ''Electronic Processes in Non-Crystalline Materials'', 2nd Edition (Clarendon Press, Oxford, 1979)) is also inappropriate, since it fails to apply the necessary weightings to tunnelling over various distances within the sphere.
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| The above derivation, albeit in a more complex 4-dimensional formulation, contains a corresponding error in respect of the phrase "the total number of states '''''within''''' a range R of some initial state at the Fermi level". A more appropriate approach would be to consider the states within a '''''surface''''' element (dR) of a sphere of radius R, (i.e. 4.pi.R<sup>2</sup>.dR) and then (possibly?) include their resulting average energy spacing in maximizing the hopping probability. Of course, (a) this would not lead to a T<sup>-1/4</sup> dependence, due to the R<sup>2</sup> rather than R<sup>3</sup> term, and (b) the resulting value would be subject to some arbitrary assumption regarding an appropriate value of dR.
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| As a consequence of these errors, values of the density of hopping states (DOS) close to the Fermi level, and of other associated parameters (as calculated using the parameter ''T<sub>0</sub>'' above) have often been totally physically unreasonable (DOS values in excess of 10<sup>39</sup> /cm<sup>3</sup>/eV have been obtained, and values of 10<sup>28</sup> are not untypical! - D.K.Paul and S.S.Mitra, Phys. Rev. Lett., '''31''', 1000 (1973) is a good example). (Actual DOS values are not expected to exceed 10<sup>21</sup> /cm<sup>3</sup>/eV without metallic conduction becoming dominant (N.F.Mott and E.A.Davis, as referenced above, p.359)). It has also frequently been observed that a T<sup>-1/4</sup> relationship is by no means the best fit to experimental data.
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| This does not mean that the underlying '''''qualitative''''' concept of "variable-range hopping" (i.e. the obvious tendency to favour a more spatially distant site with a smaller hopping-energy requirement at sufficiently low temperatures) is invalid. It simply means that the ('''''unfortunately still''''') unquestioning application of the Mott model to infer the associated material properties is entirely inappropriate.
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| J.M.Marshall and C.Main, J. Physics: Condensed Matter '''20''', 285210 (2008) provides extensive further detail and examples, and also advances an alternative model which eliminates these massive discrepancies.-->
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| [[Category:Electrical phenomena]]
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| [[Category:Condensed matter physics]]
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