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.

Figure 1

(Color online) External dielectric susceptibility ${\chi }_{\gamma \gamma }^{\left(\mathrm{ext}\right)}$ of a $\mathrm{Pb}\left({\mathrm{Ti}}_{0.6}{\mathrm{Zr}}_{0.4}\right){\mathrm{O}}_3$ film (a), wire (b), and dot (c) versus the screening parameter $\beta$. Results obtained with fluctuation and direct approaches are shown in lines and symbols, respectively. Left and right insets show the projections of the dipole patterns in the structures under open-circuit (OC)-like and short-circuit (SC)-like boundary conditions, respectively. The vertical lines characterize the transition of the dipoles pattern from SC-like conditions to OC-like conditions.

Figure 2

(Color online) Internal dielectric susceptibility ${\chi }_{\gamma \gamma }^{\left(\mathrm{int}\right)}$ of a $\mathrm{Pb}\left({\mathrm{Ti}}_{0.6}{\mathrm{Zr}}_{0.4}\right){\mathrm{O}}_3$ film (a), wire (b), and dot (c) versus the screening parameter $\beta$. Results obtained with the direct approach are shown in symbols (open dots, open squares, and filled dots for the $xx$, $yy$, and $zz$ components, respectively). Results obtained with the fluctuation approach are shown in lines for the $\beta$’s different from 1, and via some filled squares for $\beta =1$. The vertical lines characterize the transition of the dipoles pattern from SC-like conditions to OC-like conditions. The insets emphasize that the ${\chi }_{\gamma \gamma }^{\left(\mathrm{int}\right)}$ coefficients are slightly negative for $\beta =0$, and become more negative as $\beta$ increases from 0 toward 1. Note that ${\chi }_{xx}^{\left(\mathrm{int}\right)}={\chi }_{xx}^{\left(\mathrm{ext}\right)}$ and ${\chi }_{yy}^{\left(\mathrm{int}\right)}={\chi }_{yy}^{\left(\mathrm{ext}\right)}$ in the film, while ${\chi }_{zz}^{\left(\mathrm{int}\right)}={\chi }_{zz}^{\left(\mathrm{ext}\right)}$ in the wire—since no depolarizing field exists along these directions. As a result, these latter coefficients are already shown in Fig. 1, and therefore are not shown here.