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.
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