Effect of scattering in electrodes on double-barrier resonant tunneling

Assuming that electrons in quantum-well and electrode regions are well separated from each other, resonant tunneling is analyzed as a quantum transition from one electrode to the other, in which the effects of scattering in the electrodes are introduced through damping constants. Explicit formulas for tunneling currents are obtained for long- and short-electrode limits. It is found, for the long-electrode case, that a coherent tunneling process predominates over an incoherent process when the damping constant ${\gamma }_{12}$ for the current operator is smaller than the electron transfer rate through a barrier, while the incoherent process dominates when ${\gamma }_{12}$ is larger. It is also found that there is a quantum-transition region near the barrier inside the electrode, where the carrier density and the local current are not homogeneous. The coherent interaction length, defined as the size of the quantum-transition region, is a monotonic decreasing function with respect to the damping constant. In the short-electrode limit too, there is a quantum-transition region. The tunneling current in this region is linearly dependent on the electrode size.