Tunneling theory without the transfer-Hamiltonian formalism. II. Resonant and inelastic tunneling across a junction of finite width

Abstract

A novel theory of electronic tunneling between two semi-infinite systems separated by an insulating barrier of finite width is developed. The theory based on Keldysh's perturbation theory for nonequilibrium processes does not invoke the transfer-Hamiltonian formalism. The analysis is a reasonably straightforward extension of the theory developed in the first paper of this series, dealing with the abrupt junction (of zero width). The results for the abrupt junction are shown to correspond to those of the finite junction in the zero-thickness limit. The general formalism is applied to the study of effects of impurities, incorporated into the barrier, on the tunneling-current energy distribution. It is shown that tunneling resonances may reflect "interfacial" as well as "atomic" impurity states localized within the barrier. The general discussion is illustrated by somewhat more detailed analysis of two common models for such impurities: the Anderson model for the localized resonant (impurity) state, and the elementary $\delta$-function pseudopotential. The entire formalism is developed so as to allow for explicitly time-dependent potentials in the barrier region. This feature of the formalism is being applied to a phenomenological analysis of inelastic tunneling associated with localized vibrational excitations in the barrier region, i.e., the vibrating-impurity problem. This work will be reported separately. The theory allows a clear separation between the normal tunneling and the so-called resonant and inelastic channels. In the normal channel the energy density of tunneling current displays the expected dependence on the product of the local densities of states in the right and left electrodes evaluated at the interfaces. The other channels have a characteristically different dependence on these quantities, and are inherently capable of producing a left-right asymmetry in the tunneling current. Besides opening the new channels the (time-dependent) impurity potential also affects the normal channel by modifying the elastic barrier transmissivity. An extension of the one-dimensional theory to three dimensions is being developed.