Studies of charging effects on resonant tunneling diodes in terms of extended Friedel sum rule

The Friedel sum rule is extended to express the electron number accumulated within an asymmetric potential well in terms of the phases of transmission and reflection amplitudes of a double-barrier potential, in the presence of a bias voltage and charging effects. It is applied to study non-equilibrium transport properties in asymmetric double-barrier systems. The current-voltage characteristics depends on the sign of the bias voltage, which is caused by the different accumulation of charges in the quantum well. Our self-consistent calculation shows the hysteresis in the current-voltage characteristics which agrees with the experimental observation.