are incompletely screened by the electrodes. There is thus a residual depolarizing field in the STO film that reduces the overall polarization of the film.

To make the dependence of $\kappa$ on the TIM parameters explicit, we substitute values for the LGD parameters from equations - into equation  in the limit $T\rightarrow 0$. For spin-1 we find $$\kappa = \sqrt{-\frac{2(\Omega - J_{0})}{J_{1}a^{2}} + \frac{\mu^{2} \eta}{\epsilon_{0} J_{1} a^{2}}}.$$ For fixed $\Omega$ and $J_{0}$ (i.e. for a fixed value of the bulk $T_\mathrm{c}$), $\kappa^{-1}$ increases as $\sqrt{J_{1}}$. Because the difference between the polarizations at the film surface and interior depends on $\kappa^{-1}$, the depolarizing field also grows with $J_1$; it then follows immediately that $P_\mathrm{ave}$ decreases as $J_1$ increases. This suppression is illustrated in (a), which shows the dependence of both the average polarization and $\kappa^{-1}$ on $J_1$. The polarization equals its bulk value when $J_1=0$ and drops as $J_1$ increases. Notably, there is a critical value of $J_1$ (which depends on the number of layers, $N_L$, in the film) above which ferroelectricity is completely suppressed. For the 50-layer film modelled here, this value is approximately 17 meV.

Alternatively, one can fix $J_1$ and consider how $P_\mathrm{ave}$ depends on film thickness, as 