Variable-range hopping

Variable-range hopping is a model used to describe carrier transport in a disordered semiconductor or in amorphous solid by hopping in an extended temperature range. ^[1] It has a characteristic temperature dependence of

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

where ${\displaystyle \sigma }$ is the conductivity and ${\displaystyle \beta }$ is a parameter dependent on the model under consideration.

Mott 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]

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. ^[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

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

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.

Efros–Shklovskii variable-range hopping

See also: Coulomb gap

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]}

See also

• Physics portal

• Mobility edge

Notes

1. Hill, R. M. (1976-04-16). "Variable-range hopping". Physica Status Solidi A. 34 (2): 601–613. Bibcode:1976PSSAR..34..601H. doi:10.1002/pssa.2210340223. ISSN   0031-8965.
2. Mott, N. F. (1969). "Conduction in non-crystalline materials". Philosophical Magazine. 19 (160). Informa UK Limited: 835–852. Bibcode:1969PMag...19..835M. doi:10.1080/14786436908216338. ISSN   0031-8086.
3. P.V.E. McClintock, D.J. Meredith, J.K. Wigmore. Matter at Low Temperatures. Blackie. 1984 ISBN   0-216-91594-5.
4. Apsley, N.; Hughes, H. P. (1974). "Temperature-and field-dependence of hopping conduction in disordered systems". Philosophical Magazine. 30 (5). Informa UK Limited: 963–972. Bibcode:1974PMag...30..963A. doi:10.1080/14786437408207250. ISSN   0031-8086.
5. Efros, A. L.; Shklovskii, B. I. (1975). "Coulomb gap and low temperature conductivity of disordered systems". Journal of Physics C: Solid State Physics. 8 (4): L49. Bibcode:1975JPhC....8L..49E. doi:10.1088/0022-3719/8/4/003. ISSN   0022-3719.
6. Li, Zhaoguo (2017). "Transition between Efros–Shklovskii and Mott variable-range hopping conduction in polycrystalline germanium thin films". Semiconductor Science and Technology. 32 (3). et. al: 035010. Bibcode:2017SeScT..32c5010L. doi:10.1088/1361-6641/aa5390. S2CID   99091706.
7. Rosenbaum, Ralph (1991). "Crossover from Mott to Efros-Shklovskii variable-range-hopping conductivity in InxOy films". Physical Review B. 44 (8): 3599–3603. Bibcode:1991PhRvB..44.3599R. doi:10.1103/physrevb.44.3599. ISSN   0163-1829. PMID   9999988.