J_{1}$, where $Z$ is the coordination number of the lattice.

Although the TIM is only microscopically justified for order-disorder ferroelectrics, it is often used as a tool to characterize ferroelectrics of all types, and variations of this model have been applied to ferroelectricity in perovskites, including BaTiO$_{3}$ [@zhang00] and SrTiO$_3$ (STO) [@hemberger96]. As a phenomenological model, the TIM is more complex than simple Landau-Ginzburg-Devonshire theories; however, it is also more versatile. The TIM, for example, is particularly well-suited to doped quantum paraelectrics, namely Sr$_{1-x}$M$_x$TiO$_3$ with M typically representing Ca or Ba [@kleemann00; @kleemann02; @kleemann98_di; @wu03; @tao04; @guo12]. In these materials, small dopant concentrations are sufficient to induce a ferroelectric transition. Several groups have successfully modeled these materials as binary alloys of SrTiO$_3$ and MTiO$_3$ with doping-independent model parameters [@kleemann02; @kleemann98_di; @wu03; @tao04; @guo12].

The current work is motivated by the application of the TIM to metallic LaAlO$_{3}$/SrTiO$_{3}$ (LAO/STO) interfaces. These, and other related perovskite interfaces, have been widely studied since the discovery in 2004 that a 2DEG appears spontaneously at the interface when the LAO film is more than four unit cells thick [@ohtomo04]. This system is rich with interesting properties, including coexisting ferromagnetism and superconductivity [@Brinkman:2007fk; @Reyren:2007gv; @Dikin:2011gl], nontrivial 