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. It has a characteristic temperature dependence of

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 and has a characteristic temperature dependence of

= 1/4), and is generalized to d-dimensions

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.

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., 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.

has the form:

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.

, 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

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

-dimensions, and under particular assumptions this turns out to be

is well less than the band-width and comfortably bigger than the interatomic spacing.

+1)-dimensional space) is

the nearest-neighbour distribution.

-dimensional case then

1. .

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. 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

= 1/2).

See also

• Mobility edge

Notes