Variable range hopping

Introduction

Variable range hopping or Mott variable range hopping, is a model describing low temperature conduction in strongly disordered systems with localized states. [Mott, N.F., "Phil. Mag.",19,835,1969]

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. [P.V.E. McClintock,D.J. Meredith,J.K. Wigmore. "Matter at Low Temperatures". Blackie. ISBN 0-216-91594-5.]

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. [Apsley, N. and Huges, H.P., "Phil. Mag.",30,963,1974] 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::$Psim\; exp\; [-2alpha\; R-frac\{W\}\{kT\}]$where $alpha^\{-1\}$ 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 $mathcal\{R\}\; =\; 2alpha\; R+W/kT$, the "range" between two states, so $Psim\; exp\; (-mathcal\{R\})$. 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 $K\; =\; frac\{Npi\; kT\}\{3\; imes\; 2^d\; alpha^d\}$.The particular assumptions are simply that $overline\{mathcal\{R\_\{nn\}$ is well less that 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:$overline\{mathcal\{R\_\{nn\}\; =\; int\_0^infty\; (d+1)Kmathcal\{R\}^\{d+1\}exp\; (-Kmathcal\{R\}^\{d+1\})dmathcal\{R\}$.

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

After some algebra this gives:$overline\{mathcal\{R\_\{nn\}\; =\; frac\{Gamma(frac\{d+2\}\{d+1\})\}\{K^\{frac\{1\}\{d+1\}$and hence that:$sigma\; propto\; exp\; (T^\{-frac\{1\}\{d+1)$

Notes

Sadly, it is still not widely recognised that the Mott derivation contains a fundamental error, in that it simultaneously employs a distance R in relation to two very different parameters - the "actual" distance hopped and the radius of the sphere "within which" hopping occurs! Simple averaging of the hopping distance to yield 3/4 of the sphere radius (Mott & Davis 1979) is also inappropriate, since it fails to apply the necessary weightings to tunnelling over various distances within the sphere.

As a consequence of these errors, calculated values of the density of states (DOS) and other associated parameters are often totally physically unreasonable (DOS values in excess of 10³⁹ /cm³/eV have been obtained, and values of 10²⁸ are not untypical! - D.K.Paul & S.S.Mitra, Phys. Rev. Lett., Vol. 31, 1000, 1973 is a good example)See J. Optoelectronics and Advanced Materials Vol. 9, 84, 2007 provides some initial details.