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

Charge order in an interacting monolayer under transverse bias