  Charge carriers that execute multi-phonon hopping generally interact strongly
enough with phonons to form polarons. A polarons sluggish motion is linked to
slowly shifting atomic displacements that severely reduce the intrinsic width
of its transport band. Here a means to estimate hopping polarons bandwidths
from Seebeck-coefficient measurements is described. The magnitudes of
semiconductors Seebeck coefficients are usually quite large (greater than 86
microvolts/K) near room temperature. However, in accord with the third law of
thermodynamics, Seebeck coefficients must vanish at absolute zero. Here the
transition of the Seebeck coefficient of hopping polarons to its
low-temperature regime is investigated. The temperature and sharpness of this
transition depends on the concentration of carriers and on the width of their
transport band. This feature provides a means of estimating the width of a
polarons transport band. Since the intrinsic broadening of polaron bands is
very small, less than the characteristic phonon energy, the net widths of
polaron transport bands in disordered semiconductors approach the energetic
disorder experienced by their hopping carriers, their disorder energy.
