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Measurements of the dark conductivity and thermoelectric power in hydrogenated amorphous silicon–germanium alloys (a-Si 1- x Ge x :H) reveal that charge transport is not well described by an Arrhenius expression. For alloys with concentrations of Ge below 20%, anomalous hopping conductivity is observed with a power-law exponent of 3/4, while the temperature dependence of the conductivity of alloys with higher Ge concentrations is best fit by a combination of anomalous hopping and a power-law temperature dependence. The latter has been attributed to charge transport via multi-phonon hopping. Corresponding measurements of the Seebeck coefficient reveal that the thermopower is n-type for the purely a-Si:H and a-Ge:H samples but that it exhibits a transition from negative to positive values as a function of the Ge content and temperature. These findings are interpreted in terms of conduction via hopping through either exponential band tail states or dangling bond defects, suggesting that the concept of a mobility edge, accepted for over five decades, may not be necessary to account for charge transport in amorphous semiconductors.