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Variable range hopping

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Variable-range hopping, or Mott variable-range hopping, is a model describing low-temperature conduction in strongly disordered systems with localized charge-carrier states.

Contents

It has a characteristic temperature dependence of

σ = σ 0 e − ( T 0 / T ) 1 / 4

for three-dimensional conductance, and in general for d-dimensions

σ = σ 0 e − ( T 0 / T ) 1 / ( d + 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. 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 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 R and energy separation W has the form:

P ∼ exp ⁡ [ − 2 α R − W k T ]

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 R = 2 α R + W / k T , the range between two states, so P ∼ exp ⁡ ( − 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 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

σ ∼ exp ⁡ ( − R ¯ n n )

where R ¯ n n is the average nearest-neighbour range. The problem is therefore to calculate this quantity.

The first step is to obtain N ( R ) , the total number of states within a range R of some initial state at the Fermi level. For d-dimensions, and under particular assumptions this turns out to be

N ( R ) = K R d + 1

where K = N π k T 3 × 2 d α d . The particular assumptions are simply that R ¯ n n is well less than the band-width and comfortably bigger than the interatomic spacing.

Then the probability that a state with range R is the nearest neighbour in the four-dimensional space (or in general the (d+1)-dimensional space) is

P n n ( R ) = ∂ N ( R ) ∂ R exp ⁡ [ − N ( R ) ]

the nearest-neighbour distribution.

For the d-dimensional case then

R ¯ n n = ∫ 0 ∞ ( d + 1 ) K R d + 1 exp ⁡ ( − K R d + 1 ) d R .

This can be evaluated by making a simple substitution of t = K R d + 1 into the gamma function, Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t

After some algebra this gives

R ¯ n n = Γ ( d + 2 d + 1 ) K 1 d + 1

and hence that

σ ∝ exp ⁡ ( − T − 1 d + 1 ) .

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.

References

Variable-range hopping Wikipedia
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