Variable-range hopping

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Variable-range hopping, or Mott variable-range hopping, is a model describing low-temperature conduction in strongly disordered systems with localized charge-carrier states.[1]

It has a characteristic temperature dependence of

\sigma= \sigma_0e^{-(T_0/T)^{1/4}}

for three-dimensional conductance, and in general for d-dimensions

\sigma= \sigma_0e^{-(T_0/T)^{1/(d+1)}}.

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.[2]

Derivation[edit]

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.[3] 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 \textstyle\mathcal{R} between two sites, which determines the probability of hopping between them.

Mott showed that the probability of hopping between two states of spatial separation \textstyle R and energy separation W has the form:

P\sim \exp \left[-2\alpha R-\frac{W}{kT}\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 \textstyle\mathcal{R} = 2\alpha R+W/kT, the range between two states, so \textstyle P\sim \exp (-\mathcal{R}). 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 \textstyle\mathcal{R}.

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

\sigma \sim \exp (-\overline{\mathcal{R}}_{nn})

where \textstyle\overline{\mathcal{R}}_{nn}is the average nearest-neighbour range. The problem is therefore to calculate this quantity.

The first step is to obtain \textstyle\mathcal{N}(\mathcal{R}), the total number of states within a range \textstyle\mathcal{R} of some initial state at the Fermi level. For d-dimensions, and under particular assumptions this turns out to be

\mathcal{N}(\mathcal{R}) = K \mathcal{R}^{d+1}

where \textstyle K = \frac{N\pi kT}{3\times 2^d \alpha^d}. The particular assumptions are simply that \textstyle\overline{\mathcal{R}}_{nn} is well less than the band-width and comfortably bigger than the interatomic spacing.

Then the probability that a state with range \textstyle\mathcal{R} is the nearest neighbour in the four-dimensional space (or in general the (d+1)-dimensional space) is

P_{nn}(\mathcal{R}) = \frac{\partial \mathcal{N}(\mathcal{R})}{\partial \mathcal{R}} \exp [-\mathcal{N}(\mathcal{R})]

the nearest-neighbour distribution.

For the d-dimensional case then

\overline{\mathcal{R}}_{nn} = \int_0^\infty (d+1)K\mathcal{R}^{d+1}\exp (-K\mathcal{R}^{d+1})d\mathcal{R}.

This can be evaluated by making a simple substitution of \textstyle t=K\mathcal{R}^{d+1} into the gamma function, \textstyle \Gamma(z) = \int_0^\infty  t^{z-1} e^{-t}\,\mathrm{d}t

After some algebra this gives

\overline{\mathcal{R}}_{nn} = \frac{\Gamma(\frac{d+2}{d+1})}{K^{\frac{1}{d+1}}}

and hence that

\sigma \propto \exp \left(-T^{-\frac{1}{d+1}}\right).

Non-constant density of states[edit]

When the density of states is not constant (odd power law N(E)), the Mott conductivity is also recovered, as shown in this article.

See also[edit]

Notes[edit]

  1. ^ Mott, N.F. (1969). Phil. Mag. 19: 835.  Missing or empty |title= (help)
  2. ^ P.V.E. McClintock, D.J. Meredith, J.K. Wigmore. Matter at Low Temperatures. Blackie. 1984 ISBN 0-216-91594-5.
  3. ^ Apsley, N. and Hughes, H.P. (1974). Phil. Mag. 30: 963.  Missing or empty |title= (help)