Relation between dielectric responses and polarization fluctuations in ferroelectric nanostructures

Analytical derivations and first-principles-based computations are combined to establish the link between dielectric responses and thermal fluctuations of polarization in low-dimensional systems under any possible electrical boundary condition. It is proven that the analytical expression (in terms of polarization fluctuations) of the external dielectric susceptibility of a nanostructure, which is defined as the polarization's response to an external field, is identical to that of the dielectric susceptibility in the bulk material. On the other hand, such expression has to be modified for the internal dielectric susceptibility of a nanostructure that characterizes the response of the polarization to the total internal field. Such modification originates from the existence of the depolarizing field, and involves the depolarizing coefficients, as well as the degree of screening of the polarization-induced surface charges.