Friedlander–Iwaniec theorem

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

$\sigma ={\sigma }_0{e}^{-(T_0/T{)}^{1/4}}$

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

$\sigma ={\sigma }_0{e}^{-(T_0/T{)}^{1/(d+1)}}$.

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

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

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

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

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

The first step is to obtain $\mathcal{𝒩}(\mathcal{ℛ})$, the total number of states within a range $\mathcal{ℛ}$ 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 $K=\frac{N\pi kT}{3\times 2^d{\alpha }^d}$. The particular assumptions are simply that ${\overline{\mathcal{ℛ}}}_{nn}$ is well less than the band-width and comfortably bigger than the interatomic spacing.

Then the probability that a state with range $\mathcal{ℛ}$ 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

${\overline{\mathcal{ℛ}}}_{nn}=\underset{0}{\overset{\mathrm{\infty }}{\int }}(d+1)K{\mathcal{ℛ}}^{d+1}\exp (-K{\mathcal{ℛ}}^{d+1})d\mathcal{ℛ}$.

This can be evaluated by making a simple substitution of $t=K{\mathcal{ℛ}}^{d+1}$ into the gamma function, $\mathrm{\Gamma }(z)={\int }_0^{\mathrm{\infty }}{t}^{z-1}{e}^{-t}\mathrm{d}t$

After some algebra this gives

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

and hence that

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

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.

See also

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• Coulomb gap
• Mobility edge
• Bloch wave

Notes

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