# Variable-range hopping

**Variable-range hopping** is a model used to describe carrier transport in a disordered semiconductor or in amorphous solid by hopping in an extended temperature range.[1] It has a characteristic temperature dependence of

where is a parameter dependent on the model under consideration.

## Mott variable-range hopping

The **Mott variable-range hopping** describes low-temperature conduction in strongly disordered systems with localized charge-carrier states[2] and has a characteristic temperature dependence of

for three-dimensional conductance (with = 1/4), and is generalized to *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.[3]

### 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.[4] 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.

## Efros–Shklovskii variable-range hopping

The **Efros–Shklovskii (ES) variable-range hopping** is a conduction model which accounts for the Coulomb gap, a small jump in the density of states near the Fermi level due to interactions between localized electrons.[5] It was named after Alexei L. Efros and Boris Shklovskii who proposed it in 1975.[5]

The consideration of the Coulomb gap changes the temperature dependence to

## See also

## Notes

- Hill, R. M. (1976-04-16). "Variable-range hopping".
*Physica Status Solidi A*.**34**(2): 601–613. doi:10.1002/pssa.2210340223. ISSN 0031-8965. - Mott, N. F. (1969). "Conduction in non-crystalline materials".
*Philosophical Magazine*. Informa UK Limited.**19**(160): 835–852. doi:10.1080/14786436908216338. ISSN 0031-8086. - P.V.E. McClintock, D.J. Meredith, J.K. Wigmore.
*Matter at Low Temperatures*. Blackie. 1984 ISBN 0-216-91594-5. - Apsley, N.; Hughes, H. P. (1974). "Temperature-and field-dependence of hopping conduction in disordered systems".
*Philosophical Magazine*. Informa UK Limited.**30**(5): 963–972. doi:10.1080/14786437408207250. ISSN 0031-8086. - Efros, A. L.; Shklovskii, B. I. (1975). "Coulomb gap and low temperature conductivity of disordered systems".
*Journal of Physics C: Solid State Physics*.**8**(4): L49. doi:10.1088/0022-3719/8/4/003. ISSN 0022-3719. - Li, Zhaoguo (2017). et. al. "Transition between Efros–Shklovskii and Mott variable-range hopping conduction in polycrystalline germanium thin films".
*Semiconductor Science and Technology*.**32**(3): 035010. doi:10.1088/1361-6641/aa5390. - Rosenbaum, Ralph (1991). "Crossover from Mott to Efros-Shklovskii variable-range-hopping conductivity in InxOy films".
*Physical Review B*.**44**(8): 3599–3603. doi:10.1103/physrevb.44.3599. ISSN 0163-1829.