Frequency- and Electric Field-Dependent Physical Model of Ferroelectric Materials in the Tens of GHz

Ferroelectric materials are attractive for tunable components because their permittivity can be controlled by an applied electric field. The permittivity of these materials also depends on frequency, and can have a strongly nonlinear electric field dependence. A quantitative understanding of these behaviors is relevant for integration of tunable materials into devices. In this paper, we provide a simple closed-form expression for this dependence, which to our knowledge has never appeared in the literature. This expression is based on thermodynamic principles, and we expect it to be both widely applicable and generalizable. We test this model with measurements of transmission lines lithographically patterned on a ferroelectric thin film, and find that the relaxation timescales become shorter at higher bias fields. We attribute this faster relaxation to the steepening of the free energy gradient when a bias field is applied.