Linear and Nonlinear Electronic Transport in Electron-Phonon Systems: Self-Consistent Approach within the Path-Integral Formalism

Abstract

The expectation value of the steady-state velocity acquired by an electron interacting with the phonons of a crystal in finite electric and magnetic fields is analyzed quantum mechanically for arbitrary coupling strength, field strengths, and temperature. The rate of loss of momentum by an electron drifting through the crystal in the applied fields is expressed in a form in which the lattice coordinates (the phonons) have been eliminated exactly by path-integral methods. The quadratic influence functional used to simulate the electron-lattice interaction is shown to be derivable from a self-consistent relation for the impedance tensor of the electron in its drifting frame of reference. This eliminates the ambiguity of which influence functional to use in path-integral treatments of electron transport. For zero applied electric and magnetic fields, it is shown that the self-consistency is equivalent to minimizing the free energy of the electron-phonon system at finite temperatures. The Feynman one-oscillator model is discussed in the light of the self-consistency relation. Several important changes are indicated by an approximate self-consistent solution for low temperature. The applied magnetic and electric field problem is briefly discussed. For optical-phonon scattering, it is shown that the cyclotron mass is the same as the dynamic effective mass in the absence of a magnetic field. For the Fr\"ohlich polaron model, it is shown that, because of the influence of the magnetic field on the scattering rate, a longitudinal magnetoresistivity should exist equal to exactly one-half of the transverse value. Finally, two limitations of this general approach are noted and discussed briefly.