  With the recent experimental verification that ferroelectric lattice
distortions survive in the metallic phase of some materials, there is a desire
to create devices that are both switchable and take advantage of the novel
functionalities afforded by polar interfaces. In this work, we explore a simple
model for such an interface and demonstrate a mechanism by which a metallic
ferroelectric substrate may be switched by a bias voltage. This finding is in
contrast to the reasonable expectation that hysteresis is prevented by
screening of external fields in ferroelectric metals. Instead, the electron gas
binds to polarization gradients to form a compensated state. Uncompensated
electrons, which may screen external fields, are generated either when the
electron density exceeds the ferroelectric polarization or when the bias
voltage exceeds a "spillover" threshold. We propose that switchable thin films
may be optimized by choosing an electron density that is slightly less than the
lattice polarization. In addition to the high-polarization states, we find that
thin metallic films also have a low-polarization state with average
polarization near zero. Unlike in insulating films, where the polarization is
small everywhere in this state, the low-polarization state in the metallic
films consists of two head-to-head domains of opposite polarization. This
domain formation is enabled by the screening of depolarizing fields by the
electron gas.
