MAGNETISM ELECTRON TRANSPORT MAGNETORESISTIVE LANTHANUM CALCIUM MANGANITE

36 Chapter 3
position; however, this is always offset by the lower barrier energy at
sufficiently low temperatures.
The simplest quantitative derivation of the form of variable range
hopping is the following. For a given site, the number of states within a
range R per unit energy is (4π/3)R 3 N(E F ), where N(E F ) is the density of
localized states. Thus the smallest energy difference for a site within a radius
R is on average the reciprocal of this ΔE = 1/[(4π/3)R 3 N(E F )]. Thus, the further
the carrier hops, the lower the activation energy.
The carrier has an electronic wave function exponentially localized on a
particular site with a decay or localization length of ξ. The tunneling
probability that the electron will hop to a site a distance R away will contain a
factor exp(-2R/ξ). The further the distance, the lower the tunneling
probability.
Since the hopping favors large R while the tunneling favors small R,
there will be an optimum hopping distance R for which the hopping
probability proportional to exp(-2R/ξ) exp(ΔE/k B T) is a maximum. This will
occur when 1/R 4 = 8πN(E F )k B T/ξ. Substituting this value for R, the hopping
probability and thus the conductivity is proportional to exp(-(T 0 /T) 1/4 ) where
T 0 = Cξ 3 /k B N(E F ). C is a constant which in this derivation is 24/π ≈ 7.6, but
other values of C are obtained from more sophisticated analyses. C ≈ 21 is
recommended by Shklovskii and Efros.
The conductivity for variable range hopping is usually given the form σ 0
exp(-(T 0 /T) ν ), ν = 1/4. The exact form of σ 0 depends on the model and may
have a power law temperature dependence of its own. For example T 0.33 has
been found [70].
Other values of ν can be obtained theoretically using different
assumptions. The above derivation assumes a three dimensional system. In