Variable-range hopping explained

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

\sigma = \sigma ^{- (T ^{\beta} _{0 /T)}} _{0 e}

where

\sigma is the conductivity and \beta 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

\sigma = \sigma ^{- (T ^{1 / 4} _{0 /T)}} _{0 e}

for three-dimensional conductance (with

\beta = 1/4), and is generalized to d-dimensions\sigma = \sigma ^{- (T ^{1 /(d +1 )} _{0 /T)}} _{0 e} .

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

style l{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

style R and energy separation W has the form:P\sim \exp \left[ -2 \alpha R - (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

style l{R} = 2 \alpha R + W/kT , the range between two states, so style P\sim \exp ( - l{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 style l{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{ l{R}} _{nn} ) where style\overline{ l{R}} _{nn} is the average nearest-neighbour range. The problem is therefore to calculate this quantity.

The first step is to obtain

style l{N}( l{R}) , the total number of states within a range style l{R} of some initial state at the Fermi level. For d-dimensions, and under particular assumptions this turns out to bel{N}( l{R}) = K l{R} ^{d+1} where style K = (N\pi kT)/(3 x 2 ^{d} \alpha ^{d}) .The particular assumptions are simply that style\overline{ l{R}} _{nn} is well less than the band-width and comfortably bigger than the interatomic spacing.

Then the probability that a state with range

style l{R} is the nearest neighbour in the four-dimensional space (or in general the (d+1)-dimensional space) isP _{nn} ( l{R}) = (\partial l{N)/(( l{R})}{\partial l{R}} \exp) [ - l{N}( l{R})] the nearest-neighbour distribution.

For the d-dimensional case then

\overline{ l{R}} _{nn} = \int ^{inf ty} _{0} (d +1 )K l{R} ^{d+1} \exp ( - K l{R} ^{d+1} )d l{R} .

This can be evaluated by making a simple substitution of

style t = K l{R} ^{d+1} into the gamma function, style \Gamma(z) = \int ^{inf ty} _{0} t ^{z-1} e ^{-t} d t

After some algebra this gives

\overline{ l{R}} _{nn} = (\Gamma( (d +2)/() d +1))/(K ^{(1)/(d +1)}) and hence that\sigma \propto \exp \left( - T ^{- (1)/(d +1)} \right) .

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

See also: Coulomb gap. 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.

The consideration of the Coulomb gap changes the temperature dependence to

\sigma = \sigma ^{- (T ^{1 / 2} _{0 /T)}} _{0 e}

for all dimensions (i.e.

\beta = 1/2).^{[6] [7]}

See also

• Mobility edge

Notes and References

1. Hill. R. M.. 1976-04-16. Variable-range hopping. Physica Status Solidi A. en. 34. 2. 601–613. 10.1002/pssa.2210340223. 1976PSSAR..34..601H . 0031-8965.
2. Mott . N. F. . Conduction in non-crystalline materials . Philosophical Magazine . Informa UK Limited . 19 . 160 . 1969 . 0031-8086 . 10.1080/14786436908216338 . 835–852. 1969PMag...19..835M .
3. P.V.E. McClintock, D.J. Meredith, J.K. Wigmore. Matter at Low Temperatures. Blackie. 1984 .
4. Apsley . N. . Hughes . H. P. . Temperature-and field-dependence of hopping conduction in disordered systems . Philosophical Magazine . Informa UK Limited . 30 . 5 . 1974 . 0031-8086 . 10.1080/14786437408207250 . 963–972. 1974PMag...30..963A .
5. Efros. A. L.. Shklovskii. B. I.. 1975. Coulomb gap and low temperature conductivity of disordered systems. Journal of Physics C: Solid State Physics. en. 8. 4. L49. 10.1088/0022-3719/8/4/003. 1975JPhC....8L..49E . 0022-3719.
6. 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. 10.1088/1361-6641/aa5390. 2017SeScT..32c5010L . 99091706 .
7. Rosenbaum. Ralph. 1991. Crossover from Mott to Efros-Shklovskii variable-range-hopping conductivity in InxOy films. Physical Review B. 44. 8. 3599–3603. 10.1103/physrevb.44.3599. 9999988 . 1991PhRvB..44.3599R . 0163-1829.