Mean-field theory for underdamped Josephson-junction arrays with an offset voltage

We study a model Hamiltonian for superconductivity in underdamped Josephson-junction arrays in the presence of an offset voltage between the array and the substrate. We develop an approximate zero-temperature (T=0) phase diagram as a function of Josephson coupling, charging energy, and offset voltage, using a simple Hartree-type mean-field approximation. With diagonal charging energy, the calculated phase diagram is periodic in offset voltage, in agreement with previous results. At a special value of this voltage such that states with n and n+1 Cooper pairs per grain are degenerate, only an infinitesimal Josephson coupling is needed to establish long-range phase coherence in this approximation. With both diagonal and nearest-neighbor charging energies, the T=0 phase diagram has two types of insulating lobes with different kinds of charge order, and two types of superconducting regions. One of these is a ``supersolid'' in which long-range phase coherence coexists with a frozen charge-density wave. We briefly discuss connections to previous calculations, and possible relevance to experiments.