MAGNETISM ELECTRON TRANSPORT MAGNETORESISTIVE LANTHANUM CALCIUM MANGANITE

Electronic and Magnetic Measurements 35
polaron formation are more physically plausible and therefore the adiabatic
analysis is preferred.
3.1.3.2.3 Diffusive Conductivity
The same form of the conductivity derived for adiabatic small polaron
hopping is found more generally for diffusion limited conduction. If the
charge carrier must overcome an activation energy, E A , to hop to a
neighboring site, the probability for hopping will be proportional to
exp(-E A /k B T). From the theory of the random walk, the diffusion constant D
can be estimated using this hopping probability, the frequency, ω, with which
an attempt to hop is made (usually the frequency of the optical phonon which
provides the lowest barrier to hopping at some instant), and the site to site
distance, a: D = λωa 2 exp(-E A /k B T). λ is a geometrical factor approximately
equal to 1. The mobility is related to the diffusivity via the Nernst-Einstein
relation μ = eD/k B T. Thus the conductivity, σ = neμ, of a general, activated,
diffusive process is σ = ne 2 ωa 2 /k B T -1 exp(-E A /k B T). The transport of ions in a
crystal is diffusive, and therefore the ionic conductivity is often analyzed
assuming this form of the conductivity [66]. The transport of small polarons
is very similar and has also been described in this way [67-69].
3.1.3.2.4 Variable range Hopping
For a semiconductor or insulator at low enough temperatures, the
predominant conduction mechanism may no longer be by excitation of
carriers to the mobility edge or by thermally activated hopping to the nearest
neighbor but by variable range hopping. At low temperatures, the
mechanism with the lowest barrier energy will dominate. Due to any
randomness in the sample, the hopping site with the lowest barrier energy
will not in general be the nearest neighbor. The increased hopping distance
will of course reduce the probability that the carrier will tunnel to this