Resistance of random Kronig-Penney potentials and the effects of phase-disrupting collisions

We consider the resistance of a Kronig-Penney potential with a finite number of barriers. The average resistance and its variance are computed for random Kronig-Penney potentials as a function of the number of barriers. Various types of randomizations are considered, including variations of the barrier width, distance between barriers, positions of the discontinuities of the potential, and barrier height. We find striking differences in the average resistance depending on the type of randomization considered. The resulting resistance is exponentially dependent on the resistor length and the variance grows with length faster than the resistance. We develop a procedure for incorporating the effects of phase-disrupting collisions, such as electron-phonon collisions, into random potential calculations of the resistance. The problematic exponential length dependence and growth of the variance are thereby eliminated. The resulting resistivity is determined by both the inelastic collision length and the quantum resistance as manifested by the Landauer formula. For sample lengths large compared with the inelastic collision length, Ohms law is derived and the variance of the resistance is shown to vary as the square root of the sample length. The rate of convergence to Ohms-law behavior with increasing sample length, and its dependence upon the localization length and inelastic scattering length, is discussed.