Electrodynamics of ferroelectric films with negative capacitance

We construct a theory of the electrodynamic response of ferroelectric thin films containing periodic domain textures (PDTs) with ${180}^{\circ }$ polarization-oriented domains. We demonstrate that a depolarization field induced by PDTs gives rise to the negative capacitance of the ferroelectric film. We derive frequency-dependent dielectric permittivity related to the PDT dynamics across the entire frequency range. We find the resonance mode of domain oscillations in the terahertz (THz) spectral band and the singular points in the phase of the reflected THz beam that are intimately related to the negative capacitance. Our findings provide a material platform for the THz negative-capacitance-based optics of ferroelectric films and for epsilon-near-zero plasmonic THz metamaterials.

Figure 1

Formation and dynamics of the periodic domain texture (PDT). (a) A single-domain state, the ferroelectric polarization and the depolarization field are shown by the blue and red arrows, respectively. The depolarization field traverses across the film. (b) The PDT with the alternating ${180}^{\circ }$ polarization domains. The depolarization field is confined to the surface of the film. (c) A sketch of the ferroelectric-paraelectric superlattice with PDT. (d) The response of the PDT structure to variations of the voltage $U$ applied to the ferroelectric-paraelectric system. The applied voltage induces the electric displacement $\mathbf{D}$ satisfying the continuity condition across the entire system. The average electric field $\mathbf{E}$ within the ferroelectric film is directed opposite the induced average polarization $\mathbf{P}$ and displacement $\mathbf{D}$. This manifests the effect of the negative capacitance. Stray fields are not shown in order to avoid an overcrowded picture.

Figure 2

Dynamical properties of the PDT. Real and imaginary parts of the frequency-dependent response are shown in red and blue, respectively. Dashed lines depict the behavior of the system in the absence of dissipation. (a)  Dielectric permittivity of the 30-nm-thick film of the PTO. (b) Response function, defined as ${\mathbf{P}}_{\mathrm{dw}\omega }=\gamma \left(\omega \right){\mathbf{D}}_{\omega }$, for the same film. (c) Dielectric permittivity for either the 30-/50-nm sandwich of the PTO/STO layers or the equivalent superlattice.

Figure 3

Resonance in the PDT. (a) Resonance behavior of the amplitude of the response function $\left|\gamma \right(\omega \left)\right|$ for different film thicknesses of the PTO. (b)  Dependence of the domain resonance frequency ${\omega }_{\mathrm{r}}$, of the damping frequency ${\omega }_{\mathrm{d}}$, and of the domain width $d$ upon the film thickness. The solid red line stands for ${\omega }_{\mathrm{r}}$ of the PTO; the dashed red line shows ${\omega }_{\mathrm{d}}$ of the PTO. The solid and dashed green lines show ${\omega }_{\mathrm{r}}$ and ${\omega }_{\mathrm{d}}$, respectively, for PZT. Green squares display the results of the ab initio simulations for PZT [23]. The blue solid line displays the behavior of the domain width $d$.

Figure 4

Reflection characteristics of the PDT. The $p$-polarized THz beam reflects from the $2a_{\mathrm{f}}\simeq 30\mathrm{nm}$ thick PTO film deposited on the STO dielectric substrate. (a) The dependence of the reflectance of the film on the substrate $R$ and the dependence of the net reflectance of the substrate $R_0$ on the incidence angle $\theta$ calculated at the PDT oscillation frequency ${\omega }_0/2\pi \simeq 0.75\mathrm{THz}$ are shown by the blue solid and dashed lines, respectively. The difference between the two is tiny, $\left|\mathrm{\Delta }R\right|=|R_0-R|\lesssim {10}^{-3}$, and, at the figure scale, is seen only under magnification. In contrast, the reflectivity $\mathrm{\Delta }R/R_0$ experiences a drastic variation of about unity and clearly displays a Lorentz-like spike at the Brewster angle ${\theta }_{\mathrm{B}}\simeq 86.7^{\circ }$, where $R_0$ almost vanishes. This spike is a fingerprint of the PDT resonance and can be used for its experimental identification. The inset shows the geometry of the experiment. (b) The 3D color plot of the reflectivity $\mathrm{\Delta }R/R_0$ as a function of the frequency and the incident angle in the immediate vicinity of the Brewster angle ${\theta }_{\mathrm{B}}$, marked by the thin blue line. The red line corresponds to the Lorentz-spike-like reflectivity behavior at the PDT oscillation frequency ${\omega }_0$. This spike smoothens upon deviation from ${\omega }_0$.

Figure 5

The complex Fresnel reflection coefficient $r_{123}^p$ of the PDT. (a) The 3D color plot of the phase and the amplitude of $r_{123}^p$ as a function of the frequency $\omega$ and the angle of incidence $\theta$ for the $2a_{\mathrm{f}}\simeq 30\mathrm{nm}$ thick PTO film deposited on the transparent substrate. The substrate permittivity ${\varepsilon }_{\mathrm{p}}=300$ is real, with a vanishingly small imaginary part. Two topologically protected phase singularity dark points, at which $|r_{123}^p|=0$, appear in the vicinity of the Brewster angle ${\theta }_{\mathrm{B}}$ at frequencies where ${\varepsilon }_{\mathrm{f}}=0$ and where ${\varepsilon }_{\mathrm{f}}\simeq -\sqrt{{\varepsilon }_{\mathrm{p}}{\varepsilon }_{\perp }}$. These singular points and the inversion of the phase rotation between them that appears in blue are the fingerprint of the negative permittivity in the low-frequency region at $\omega <{\omega }_0$. (b) The same as (a), but for the finite-transparency STO substrate with ${\varepsilon }_{\mathrm{p}}=300+10i$. The singular phase dark points merge and annihilate. The color legend refers to both panels. The color maps of the phase distribution are shifted down from the $|r_{123}^p|=0$ plain for convenience.

Figure 6

The geometry of the system.