Quantum-Transport Theories and Multiple Scattering in Doped Semiconductors. I. Formal Theory

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Abstract
Previous quantum-transport theories relevant to the calculation of multiple-scattering contributions to the electron mobility of a doped semiconductor are re-examined. The Green's-function evaluations of the Kubo formula are extended to include contributions of second order in the impurity concentration in addition to the first-order terms included previously. The simple rigorous results obtained are used to justify and simplify the Kohn-Luttinger transport equations. Apparent difficulties with the impurity averaging are clarified, and the second-order Kohn-Luttinger equation is reduced to a Boltzmann form. For an isotropic solid a simple closed formula for the mobility is derived, which contains such quantum corrections to the Brooks-Herring formula as coherent scattering from pairs of impurities, higher Born approximations for the single scattering, and dressing effects of the impurities on the electron energy levels and wave functions. It is noted that a variational principle of the Kohler type exists, which should be of use in estimating multiple-scattering contributions in anisotropic solids.