Frequency dependent dielectric response of ferroelectric-dielectric junction with negative electric capacitance

We calculated the frequency dependent dielectric response (electric susceptibility) of layered ferroelectric-dielectric junction, biased by the time-dependent harmonic voltage with single frequency $\omega$. Working point is stabilized, by the charge boundary condition between the layers, in the region with negative electric capacitance. The static susceptibility $\chi_0$ is negative and relative dielectric constant $\epsilon_r$ smaller than one, clearly indicating the opposite direction of electric field and polarization in the ferroelectric layer due to the negative electric capacitance. At finite frequencies this sign is preserved in real part of susceptibility which gains the frequency dependence. Also, frequency dependent imaginary part arises due to the phase shift between electric field and polarization. The type of that frequency dependence in linear regime is so-called relaxation (Debye) response, i.e. $\chi'(\omega)=\chi_0/(1+(\tau \omega)^2)$ and $\chi"(\omega)=\chi_0\tau \omega/(1+(\tau \omega)^2)$, where $\tau$ is polarization switching time characteristic to ferroelectric material. In particular, we modeled the junction of ferroelectric BaTiO$_3$ and dielectric Al$_2$O$_3$, taking the experimental values of material parameters, and addressed the role of nonlinearity with respect to result of the linear response theory.