An exact time-interval distribution of single-electron-tunneling events in ultrasmall normal tunnel junctions is obtained under an arbitrary bias condition at temperatures much lower than e²/2CkB, where e is the electronic charge, C is the junction capacitance, and kB is Boltzmann's constant. Using the obtained formula we analytically describe the crossover from random to Coulomb-regulated single-electron tunneling in the time domain as the external source passes from a voltage source to a current source, and as the bias voltage decreases towards an optimum value above e/2C. Calculation procedure for this optimum value is given. In particular, a new analytic expression of the I-V characteristics is obtained for a microjunction operating under an arbitrary bias condition.
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Ueda et al. (1990) studied this question.