Charge order in an interacting monolayer under transverse bias

Abstract

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 $T=0$.Comment: 16 pages, 16 figures; minor change

Electronic structure, transport, and collective effects in molecular layered systems

“…F 16 CoPc/MnPc has a twofold spin degenerate HOMO so that a model with a single orbital per site with interactions should be reasonable. The main difference from the previously studied case [ 15 ] is the asymmetry of the molecule. The asymmetry can be modeled by assuming different tunneling probabilities between the molecular orbital and the two electrodes.…”

“…An advanced theoretical description extending the mean-field-type description within the DFT-NEGF approach requires a method that can deal with strong interactions in systems far from equilibrium, and the prime candidate is again the ME approach discussed in Section ’DFT combined with the master equation’, ideally using parameters from DFT calculations. The non-local interaction adds another level of complication [ 15 , 79 ]. In the sequential-tunneling and diagonal approximation described above, this interaction can be treated essentially exactly using Monte Carlo simulations [ 15 ].…”

“…The non-local interaction adds another level of complication [ 15 , 79 ]. In the sequential-tunneling and diagonal approximation described above, this interaction can be treated essentially exactly using Monte Carlo simulations [ 15 ]. The main idea is to use the sequential-tunneling rates, which are analogous to Eq.…”

“…A simple model system consisting of a square lattice with a single spinful orbital per site and with very strong intraorbital and arbitrary nearest-neighbor Coulomb interactions has recently been studied by two of us [ 15 ]. There, the molecules have been assumed to be symmetric, which would for example be appropriate for a CoPc layer.…”

“…Approaches beyond mean-field-type approximations are required for the treatment of correlation effects such as Coulomb blockade and the Kondo effect [ 14 ]. Such interactions not only occur within a single molecule but also between neighboring molecules in a film [ 15 ], where they can lead to ordering phenomena.…”