Current And Rate-Equation For Resonant Tunneling

We have used nonequilibrium quantum statistical mechanics to derive an exact expression for the current and a rate equation for a simple model of a resonant-tunneling diode including scattering within the resonant state. The rate equation is identical to the classical rate equation for resonant tunneling provided that two conditions are met. First, the driving frequency must be slow compared with the Fermi energy of incoming electrons, the lifetime of the resonance does not set a scale. Second, the resonance must be narrow compared with the range of incoming energies and on a scale set by the variation in energy of the tunneling rates of the individual barriers. Our derivation shows that the rate equation holds both in the coherent limit, in which transport can be described by elementary wave mechanics alone, and in the limit where scattering within the resonant state makes a classical ''sequential'' picture more appropriate; all information about coherence is lost in the averaging required to deduce the rate equation. The current through the resonant state is independent of scattering provided that the rate equation holds. We also examine the current when the rate equation does not hold, and show how strongly inelastic processes enter the quantum kinetic description through scattering-out and scattering-in processes at different energies.