A study of umklapp scattering and transport coefficients in simple metals
Two-plane-wave models for electron-electron and electron-phonon umklapp scattering, and their effect upon the transport properties of simple metals, are discussed. For the electron-electron scattering problem, a variational approach to the Boltzmann equation is used. According to the model the electron-electron scattering contribution to resistivity is as large in alkali metals as in the simple polyvalent metals that were studied, in apparent contradiction with tentative conclusions drawn from experiment. If further experiments confirm these conclusions, then electron-phonon umklapp scattering is suggested as a possible explanation. It is shown that its contribution to resistivity does not die exponentially at low temperatures in polyvalent metals, as it does in alkali metals.
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1st zone 2-plane-wave A(Ak alkali metals angular average angular integration anisotropic Appendix approximation assume Asymtotic values band gap Boltzmann equation Bragg planes calculate chemical potential component contribution to resistivity correction term defined discussed effect electrical resistivity electron-electron scattering electron-phonon scattering energy estimate exact solution expression Fermi surface distortion Figure form factor formula free-electron geometry indium interband intraband limiting low-temperature low-temperature behavior lower bounds magnitude matrix element mixing coefficients momentum transfer N. W. Ashcroft Nu/N pair of Bragg parameter phonon Phys plane-waves planes G polyvalent metals pseudopotential reciprocal lattice vectors relative error remapped surface restricted umklapp result scattering event scattering parameter scattering phase space shown on Fig simple metals single-plane-wave small-q spherical Fermi surface surface integral temperatures thermal conductivities transition probability transport coefficients transverse modes umklapp contribution umklapp processes umklapp region umklapp scattering phase upper bound velocity Vg/Ep wavefunctions Wilkins Ziman