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.
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] 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).
[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.
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.
is well less than the band-width and comfortably bigger than the interatomic spacing.
For the d-dimensional case then This can be evaluated by making a simple substitution of
After some algebra this gives and hence that When the density of states is not constant (odd power law N(E)), the Mott conductivity is also recovered, as shown in this article.
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 for all dimensions (i.e.