Nonequilibrium states induced by an applied bias voltage (V) and the corresponding current-voltage characteristics of one-dimensional models describing band and Mott insulators are investigated theoretically by using nonequilibrium Greens functions. We attach the models to metallic electrodes whose effects are incorporated into the self-energy. Modulation of the electron density and the scalar potential coming from the additional long-range interaction are calculated self-consistently within the Hartree approximation. For both models of band and Mott insulators with length L_C, the bias voltage induces a breakdown of the insulating state, whose threshold shows a crossover depending on L_C. It is determined basically by the bias $V_{th}sim Delta$ for L_C smaller than the correlation length $xi=W/Delta$ where W denotes the bandwidth and $Delta$ the energy gap. For systems with $L_Cgg xi$, the threshold is governed by the electric field, $V_{th}/L_C$, which is consistent with a Landau-Zener-type breakdown, $V_{th}/L_Cpropto Delta^2/W$. We demonstrate that the spatial dependence of the scalar potential is crucially important for this crossover by showing the case without the scalar potential, where the breakdown occurs at $V_{th}sim Delta$ regardless of the length L_C.
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