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. It has a characteristic temperature dependence of
where formula_2 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
for three-dimensional conductance (with formula_2 = 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.
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. 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" formula_6 between two sites, which determines the probability of hopping between them.
Mott showed that the probability of hopping between two states of spatial separation formula_7 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 formula_9, the "range" between two states, so formula_10. 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 formula_6.
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 formula_13is the average nearest-neighbour range. The problem is therefore to calculate this quantity.
The first step is to obtain formula_14, the total number of states within a range formula_6 of some initial state at the Fermi level. For "d"-dimensions, and under particular assumptions this turns out to be
where formula_17.
The particular assumptions are simply that formula_13 is well less than the band-width and comfortably bigger than the interatomic spacing.
Then the probability that a state with range formula_6 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 formula_22 into the gamma function, formula_23
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
for all dimensions (i.e. formula_2 = 1/2).