Diffusion model and the anisotropic relaxation time in metals: Application to potassium

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Abstract
The Boltzmann equation for the anisotropic relaxation time leads to a generalization of the diffusion model in which umklapp electron-phonon scattering is treated explicitly, and does not require Fermi-surface distortion or intersection with zone boundaries. The diffusion equation contains an anisotropic function which may be interpreted as the zeroth-order relaxation time in the iteration sequence of the Boltzmann equation, and which has been used commonly as an approximation to the true relaxation time. The solution of the diffusion equation tends toward this zeroth-order relaxation time at higher temperatures, where the latter is known to become a good approximation. The importance of treating umklapp explicitly is illustrated by calculating deviations from Matthiessen's rule in the electrical resistivity of potassium, for which previous diffusion models were of course not intended. Comparison with data and with previous numerical calculations by others lends credence to the model.