Resonant Tunneling with an Electron-Phonon Interaction

Using nonequilibrium quantum-statistical mechanics, we study both the equilibrium behavior and the nonlinear transport in a one-dimensional model of resonant tunneling with an electron-phonon interaction at zero temperature. There are several unique features of our work: (1) We demonstrate the importance of the Hartree-like electron-phonon self-energy diagram in this problem lacking translational invariance. (2) We identify a new perturbation parameter in the study of the equilibrium polaron shift. (3) We provide a thorough understanding of when the noninteracting current is a poor predictor of the interacting current-voltage characteristic, which we calculate and explain in detail. Our calculation treats a weak electron-phonon interaction at zero temperature perturbatively in the dimensionless coupling constant, g, to lowest self-consistent order. Because the system is not translational invariant, we must retain both a Fock-like and a Hartree-like diagram. Depending of the filling condition of the resonant level, the model can be in two qualitatively different regimes: when the resonant level is empty (occupied), the polaron shift of the level with the interaction is −gΩ (−3gΩ), where Ω is the frequency of the phonon. The crossover occurs on a scale set by the noninteracting escape rate, Γ, and we find that gΩ/Γ is an additional perturbation parameter in the study of the equilibrium polaron shift. We furthermore evaluate typical nonlinear current-voltage characteristics. When the Fermi seas of the leads are much thicker than Ω (and Γ), the interaction affects the total current only near the onset of the large resonant current and in the valley region. However, the interaction modifies the I-V characteristics at all biases when the Fermi seas are shallow. To interpret our results we separate the current densities per unit energy for the left and right leads into elastic and inelastic contributions. We finally show that the qualitative behavior of the inelastic current contributions does not depend on the filling condition of the resonant level for a system with thick Fermi seas at a typical bias where a large current flows.