Variable-range hopping

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

${\displaystyle \sigma =\sigma _{0}e^{-(T_{0}/T)^{1/4}}}$

for three-dimensional conductance (with ${\displaystyle \beta }$ = 1/4), and is generalized to d-dimensions

${\displaystyle \sigma =\sigma _{0}e^{-(T_{0}/T)^{1/(d+1)}}}$.

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.^[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). Later it was shown that this assumption was unnecessary, and this proof is followed here.^[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 ${\displaystyle \textstyle {\mathcal {R}}}$ between two sites, which determines the probability of hopping between them.

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

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

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

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

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

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

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

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

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

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

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

the nearest-neighbour distribution.

For the d-dimensional case then

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

This can be evaluated by making a simple substitution of ${\displaystyle \textstyle t=K{\mathcal {R}}^{d+1}}$ into the gamma function, ${\displaystyle \textstyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,\mathrm {d} t}$

After some algebra this gives

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

and hence that

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

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

${\displaystyle \sigma =\sigma _{0}e^{-(T_{0}/T)^{1/2}}}$

for all dimensions (i.e. ${\displaystyle \beta }$ = 1/2).^[6][7]

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