Elastic-scattering effects on resonant tunneling in double-barrier quantum-well structures

We investigate, using a microscopic perturbation theory, the effects of elastic scattering on resonant tunneling in double-barrier quantum-well structures. Using a perturbation expansion in the scattering strength, we calculate corrections to the average transmission matrix for low impurity densities. For two- and three-dimensional structures in the absence of a magnetic field, the Lorentzian line shape associated with tunneling events that conserve transverse momentum is broadened asymmetrically by impurity scattering. There are also contributions to the tunneling probability that do not conserve transverse momentum; for incident energy {ε}>E₀, where E₀ is the resonant energy, this contribution is strongly peaked when the energy of motion perpendicular to the well closely matches E₀. This leads to a ``focusing'' of electrons into a particular set of final states. In the absence of a magnetic field, a striking spatial distribution in the tunneling probability results. For the total tunneling current, our microscopic theory gives results consistent with a Breit-Wigner-type formula for the impurity scattering induced broadening. For the case in which a magnetic field is applied perpendicular to the well region, we find that the ``focused'' electrons tend to scatter into Landau levels n such that {ε}-(n+1/2)ω₀ is as close as possible to E₀. We also find a broadening of the usual Lorentzian contribution to the tunneling probability in the magnetic field case and an upward shift in the resonant energy that varies logarithmically with magnetic field. Finally, we test our predictions numerically by calculating the exact transmission matrix for a finite two-dimensional system; the results are in good qualitative agreement with the perturbation theory analysis. Our numerical work also shows that for stronger disorder scattering, the conductance is a direct measure of the density of states inside the well.