Variable-range hopping is a model used to describe carrier transport in a disordered semiconductor or in
amorphous solid
In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid, glassy solid) is a solid that lacks the long-range order that is characteristic of a crystal.
Etymology
The term comes from the Greek ''a'' ("wit ...
by hopping in an extended temperature range. It has a characteristic temperature dependence of
:
where
is the conductivity and
is a parameter dependent on the model under consideration.
Mott variable-range hopping
The
Mott
Mott is both an English surname and given name. Notable people with the name include:
Surname B
*Basil Mott (1859–1938), British civil engineer
*Bitsy Mott (1918–2001), American baseball player C
* Charles James Mott (1880–1918), British bar ...
variable-range hopping describes low-temperature
conduction
Conductor or conduction may refer to:
Music
* Conductor (music), a person who leads a musical ensemble, such as an orchestra.
* Conductor (album), ''Conductor'' (album), an album by indie rock band The Comas
* Conduction, a type of structured f ...
in strongly disordered systems with
localized charge-carrier states and has a characteristic temperature dependence of
:
for three-dimensional conductance (with
= 1/4), and is generalized to ''d''-dimensions
:
.
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''
between two sites, which determines the probability of hopping between them.
Mott showed that the probability of hopping between two states of spatial separation
and energy separation ''W'' has the form:
: