a limitation of the TIM. Nonetheless, this calculation shows that $J_{1}$ is orders of magnitude larger than the value $J_1 = J_0/Z$ that is implicit in the conventional TIM.

This large discrepancy between $J_1$ and $J_0$ is a key feature of STO, and that there is more than an order of magnitude difference between their values can be related to their different physical origins. Further, from equation  it follows that $J_\mathrm{an}$ is not small; rather, it is negative and nearly cancels $ZJ_1$. $J_\mathrm{an}$ would however play less of a role in a material with a high transition temperature, where $J_1$ and $J_0$ would be closer in value.

Ferroelectric Thin Films {#sec:FEfilm}
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We first model the polarization in ferroelectric thin films as a simple application of the modified TIM. A ferroelectric’s properties can vary drastically between the bulk and thin-film forms, and the origins and applications of these differences have been increasingly studied in recent years [@setter06]. Ferroelectric thin films provide significant advantages in electronic devices such as increased efficiency in photovoltaic cells [@liu16; @zenkevich14; @kutes14] and decreased power usage in non-volatile memory storage [@muller11].

We focus on weakly ferroelectric materials, like those obtained by doping STO with $^{18}$O, Ca, 