# Variable-range hopping

This article needs attention from an expert in Physics. (October 2008) |

**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

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

- .

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

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* between two sites, which determines the probability of hopping between them.

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

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 , the *range* between two states, so . 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 .

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

where is the average nearest-neighbour range. The problem is therefore to calculate this quantity.

The first step is to obtain , the total number of states within a range of some initial state at the Fermi level. For *d*-dimensions, and under particular assumptions this turns out to be

where . The particular assumptions are simply that is well less than the band-width and comfortably bigger than the interatomic spacing.

Then the probability that a state with range is the nearest neighbour in the four-dimensional space (or in general the (*d*+1)-dimensional space) is

the nearest-neighbour distribution.

For the *d*-dimensional case then

- .

This can be evaluated by making a simple substitution of into the gamma function,

After some algebra this gives

and hence that

- .

## Non-constant density of states

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