the correlation length is $\xi=\sqrt{-C/2A}$ (in terms of LGD parameters), which is proportional to $\sqrt{J_1}$. The conventional TIM with $J_\mathrm{an}=0$ has $J_1=0.65$ meV, which corresponds to a correlation length of $\xi=2.7$ Å. Consistent with this, (a) shows that for the conventional TIM, surface effects are confined to narrow regions near the edges of the film. Conversely, the modified TIM with a more realistic value of $J_1=100$ meV gives the correlation length $\xi=3.3$ nm, which is comparable to the film thickness. In this case, the polarization is inhomogeneous throughout the film. In contrast to both of these cases, the polarization is nearly constant across the film when electric fields are included \[(b)\]; the polarization decreases with increasing $J_1$, and is suppressed completely for $J_1=100$ meV.

The apparent uniformity of the polarization across the film in (b) is because the correlation length is replaced by a shorter length scale $\kappa^{-1}$ when electric fields are included, with [@kretschmer79] $$\label{kappa}
\kappa = \sqrt{\xi^{-2} + \frac{\mu^{2} \eta}{\epsilon_{0} C}}.$$ In STO, this length scale is less than a unit cell, and the polarization is therefore nearly constant, with only a small reduction in the surface layer. This slight reduction is, nonetheless, enough that the depolarizing fields 