Variable-range hopping

The Mott variable-range hopping describes low-temperature conduction in strongly disordered systems with localized charge-carrier states{{cite journal | last=Mott | first=N. F. | title=Conduction in non-crystalline materials | journal=Philosophical Magazine | publisher=Informa UK Limited | volume=19 | issue=160 | year=1969 | issn=0031-8086 | doi=10.1080/14786436908216338 | pages=835–852| bibcode=1969PMag...19..835M }} and has a characteristic temperature dependence of

:$\backslash sigma=\; \backslash sigma\_0e^\{-(T\_0/T)^\{1/4\}\}$

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

:$\backslash sigma=\; \backslash sigma\_0e^\{-(T\_0/T)^\{1/(d+1)\}\}$.

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. 1984 {{ISBN|0-216-91594-5}}.

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.{{cite journal | last1=Apsley | first1=N. | last2=Hughes | first2=H. P. | title=Temperature-and field-dependence of hopping conduction in disordered systems | journal=Philosophical Magazine | publisher=Informa UK Limited | volume=30 | issue=5 | year=1974 | issn=0031-8086 | doi=10.1080/14786437408207250 | pages=963–972| bibcode=1974PMag...30..963A }} 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 $\backslash textstyle\backslash mathcal\{R\}$ between two sites, which determines the probability of hopping between them.

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

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

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

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

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

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

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

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

where $\backslash textstyle\; K\; =\; \backslash frac\{N\backslash pi\; kT\}\{3\backslash times\; 2^d\; \backslash alpha^d\}$.

The particular assumptions are simply that $\backslash textstyle\backslash overline\{\backslash mathcal\{R\}\}\_\{nn\}$ is well less than the band-width and comfortably bigger than the interatomic spacing.

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

:$P\_\{nn\}(\backslash mathcal\{R\})\; =\; \backslash frac\{\backslash partial\; \backslash mathcal\{N\}(\backslash mathcal\{R\})\}\{\backslash partial\; \backslash mathcal\{R\}\}\; \backslash exp\; [-\backslash mathcal\{N\}(\backslash mathcal\{R\})]$

the nearest-neighbour distribution.

For the d-dimensional case then

:$\backslash overline\{\backslash mathcal\{R\}\}\_\{nn\}\; =\; \backslash int\_0^\backslash infty\; (d+1)K\backslash mathcal\{R\}^\{d+1\}\backslash exp\; (-K\backslash mathcal\{R\}^\{d+1\})d\backslash mathcal\{R\}$.

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

After some algebra this gives

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

and hence that

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

When the density of states is not constant (odd power law N(E)), the Mott conductivity is also recovered, as shown in [http://hal.archives-ouvertes.fr/ccsd-00004661 this article].

{{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.{{Cite journal|last1=Efros|first1=A. L.|last2=Shklovskii|first2=B. I.|date=1975|title=Coulomb gap and low temperature conductivity of disordered systems|url=http://stacks.iop.org/0022-3719/8/i=4/a=003|journal=Journal of Physics C: Solid State Physics|language=en|volume=8|issue=4|pages=L49|doi=10.1088/0022-3719/8/4/003|bibcode=1975JPhC....8L..49E |issn=0022-3719|url-access=subscription}} It was named after Alexei L. Efros and Boris Shklovskii who proposed it in 1975.

The consideration of the Coulomb gap changes the temperature dependence to

:$\backslash sigma=\; \backslash sigma\_0e^\{-(T\_0/T)^\{1/2\}\}$

for all dimensions (i.e. $\backslash beta$ = 1/2).{{Cite journal|last=Li|first=Zhaoguo|date=2017|others=et. al|title=Transition between Efros–Shklovskii and Mott variable-range hopping conduction in polycrystalline germanium thin films|journal=Semiconductor Science and Technology|volume=32|issue=3|pages=035010|doi=10.1088/1361-6641/aa5390|bibcode=2017SeScT..32c5010L |s2cid=99091706 }}{{Cite journal|last=Rosenbaum|first=Ralph|date=1991|title=Crossover from Mott to Efros-Shklovskii variable-range-hopping conductivity in InxOy films|journal=Physical Review B|volume=44|issue=8|pages=3599–3603|doi=10.1103/physrevb.44.3599|pmid=9999988 |bibcode=1991PhRvB..44.3599R |issn=0163-1829}}

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