Identification of defect distribution at ferroelectric domain walls from evolution of nonlinear dielectric response during the aging process

The motion of ferroelectric domain walls greatly contributes to the macroscopic dielectric and piezoelectric response of ferroelectric materials. The domain-wall motion through the ferroelectric material is, however, hindered by pinning on crystal defects, which substantially reduces these contributions. Here, using thermodynamic models based on the Landau-Ginzburg-Devonshire theory, we find a relation between the microscopic reversible motion of nonferroelastic ${180}^{\circ }$ domain walls interacting with a periodic array of pinning centers and the nonlinear macroscopic permittivity. We show that the reversible motion of domain walls can be split into two basic modes: first, the bending of a domain wall between pinning centers, and, second, the uniform movement of the domain-wall plane. We show that their respective contributions may change when the distribution of pinning centers is rearranged during the material aging. We demonstrate that it is possible to indicate which mechanism of the domain-wall motion is affected during material aging. This allows one to judge whether the defects only homogeneously accumulate at domain walls or prefer to align in certain directions inside the domain-wall plane. We suggest that this information can be obtained using simple macroscopic dielectric measurements and a proper analysis of the nonlinear response. Our results may therefore serve as a simple and useful tool to obtain details on domain-wall pinning in an aging process.

Figure 1

Reversible movements of ${180}^{\circ }$ ferroelectric domain walls pinned by crystal-lattice defects. Domain walls in the absence of the external electric field (position A) are pinned by pinning centers indicated by red spheres. When the external electric field $E$ is applied to the system, the equilibrium position of the domain wall (position C) is given by the superposition of two mechanisms: uniform displacement ${\delta }_{\mathrm{pln}}$ of the wall from the pinning center (position B) and the spatially dependent displacement ${\delta }_{\mathrm{ben}}(y,z)$ due to domain-wall bending.

Figure 2

Result of the phase-field model simulation of the ${180}^{\circ }$ domain wall interacting with a pinning center in $\mathrm{BaTiO}{}_3$ for $r_0=4\mathrm{nm}$ and $r=19\mathrm{nm}$ during aging. Two aging mechanisms were modeled: (a) bending mode (BM) aging, where the dielectric response is controlled by bending movements of ${180}^{\circ }$ domain walls, and (b) planar mode (PM) aging, where the planar ${180}^{\circ }$ domain walls oscillate around the pinning centers. The magnitude of the external field is $E=180\mathrm{kV}/\mathrm{m}$. Color on the surface indicates the value of the $P_3$ component of the polarization vector. Blue and red colors indicate the values of $P_3$ equal to $-0.221 \text{C}/\mathrm{m}{}^2$ and 0.221 $\text{C}/\mathrm{m}{}^2$, respectively.

Figure 3

Evolution of the electric field dependence of extrinsic permittivity computed from numerical phase-field model simulations of (a),(c) BM aging and (b),(d) PM aging. (a),(b) Numerical simulations computed for $\mathrm{PbTiO}{}_3$. (c),(d) Results for $\mathrm{BaTiO}{}_3$.

Figure 4

Dielectric nonlinearity constant ${\gamma }_w$ vs small-signal extrinsic permittivity ${\varepsilon }_L$ during PM (filled markers) and BM (open markers) aging. Numerical computations were carried out for (a) $\mathrm{PbTiO}{}_3$ and (b) $\mathrm{BaTiO}{}_3$. Dashed and solid lines indicate fitting to the model of PM domain-wall motion [see Eq. (28)] and BM domain-wall motion [see Eq. (29)], respectively.

Figure 5

Parameter $A$ computed according to Eq. (32) as a function of the inverse macroscopic small-signal permittivity ${\varepsilon }_F$ normalized to its minimal value measured during the aging experiment for the PM (filled markers) and BM (open markers) aging mechanisms. It is seen that values of the parameter $A$ can be used for the identification of the aging mechanism. Numerical computations were carried out for (a) $\mathrm{PbTiO}{}_3$ and (b) $\mathrm{BaTiO}{}_3$.

Figure 6

Geometry of the representative volume element (RVE) in the periodic structure of the ferroelectric polydomain system with a single ${180}^{\circ }$ domain wall (green rectangle), which interacts with a pinning center (red sphere) in the applied external field.

Figure 7

Comparison of the numerically computed values [see Eq. (D2)] vs predictions of analytical formulas [see Eqs. (B12a) and (C32a)] of small-signal permittivity controlled by (a) bending movements and (b) uniform planar movement of ${180}^{\circ }$ domain walls, respectively.

Figure 8

Comparison of the numerically computed values [see Eq. (D4)] vs predictions of analytical formulas [see Eqs. (B12b) and (C32b)] of the nonlinearity constant of extrinsic permittivity controlled by (a) bending movements and (b) planar movement of ${180}^{\circ }$ domain walls, respectively.