Charge order in an interacting monolayer under transverse bias

A monolayer of molecules or quantum dots sandwiched between electrodes can be driven out of equilibrium by the application of a bias voltage between the electrodes. We study charge ordering, i.e., the spontaneous formation of a charge density wave, and the perpendicular current in such a system within a master-equation approach augmented by mean-field and classical Monte Carlo methods. Our approach is suitable for weak tunneling between the monolayer and the electrodes. For a square lattice with nearest-neighbor Coulomb repulsion, we present a comprehensive study of the zero-temperature phases controlled by the on-site energy, the bias voltage, and the degeneracy of the occupied single-site state. One of the most interesting results is the prediction of a conducting charge-density-wave phase that only occurs at a finite bias voltage. We also study the universality classes of the phase transitions towards charge-ordered states at zero and nonzero temperatures. While all transitions at $T>0$ and some at $T=0$ belong to the two-dimensional Ising universality class, we also find an absorbing-to-active phase transition in the $\mathbb{Z}_2$ symmetric directed percolation (DP2) class at $T=0$.

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