Variable-range hopping

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

It has a characteristic temperature dependence of

$\sigma= \sigma_0e^{-(T_0/T)^{1/4}}$

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

$\sigma= \sigma_0e^{-(T_0/T)^{1/(d+1)}}$.

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 $\textstyle\mathcal{R}$ between two sites, which determines the probability of hopping between them.

Mott showed that the probability of hopping between two states of spatial separation $\textstyle R$ and energy separation W has the form:

$P\sim \exp \left[-2\alpha R-\frac{W}{kT}\right]$

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 $\textstyle\mathcal{R} = 2\alpha R+W/kT$, the range between two states, so $\textstyle P\sim \exp (-\mathcal{R})$. 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 $\textstyle\mathcal{R}$.

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

$\sigma \sim \exp (-\overline{\mathcal{R}}_{nn})$

where $\textstyle\overline{\mathcal{R}}_{nn}$is the average nearest-neighbour range. The problem is therefore to calculate this quantity.

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

$\mathcal{N}(\mathcal{R}) = K \mathcal{R}^{d+1}$

where $\textstyle K = \frac{N\pi kT}{3\times 2^d \alpha^d}$. The particular assumptions are simply that $\textstyle\overline{\mathcal{R}}_{nn}$ is well less than the band-width and comfortably bigger than the interatomic spacing.

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

$P_{nn}(\mathcal{R}) = \frac{\partial \mathcal{N}(\mathcal{R})}{\partial \mathcal{R}} \exp [-\mathcal{N}(\mathcal{R})]$

the nearest-neighbour distribution.

For the d-dimensional case then

$\overline{\mathcal{R}}_{nn} = \int_0^\infty (d+1)K\mathcal{R}^{d+1}\exp (-K\mathcal{R}^{d+1})d\mathcal{R}$.

This can be evaluated by making a simple substitution of $\textstyle t=K\mathcal{R}^{d+1}$ into the gamma function, $\textstyle \Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,\mathrm{d}t$

After some algebra this gives

$\overline{\mathcal{R}}_{nn} = \frac{\Gamma(\frac{d+2}{d+1})}{K^{\frac{1}{d+1}}}$

and hence that

$\sigma \propto \exp \left(-T^{-\frac{1}{d+1}}\right)$.

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.

See also

• Coulomb gap
• Mobility edge
• Bloch wave