Piezoelectric properties of twinned ferroelectric perovskites with head-to-head and tail-to-tail domain walls

Longitudinal piezoelectric coefficient of a twinned ferroelectric perovskite material with an array of partially compensated head-to-head and tail-to-tail 90-degree domain walls has been studied by phase-field simulations in the framework of the Ginzburg-Landau-Devonshire model of BaTiO${}_3$ ferroelectric. In particular, it is shown that the magnitude of the build-in extrinsic charge at the domain wall and the nanoscale domain size can both promote rotation of the static polarization vector within the body of adjacent domains. This polarization rotation drives the domain closer to an orthorhombic state, and the proximity to this ferroelectric-ferroelectric phase transition is directly responsible for the enhancement of the properties. Our simulations and the theory also suggest that the same system with nominally overcompensated charged walls may show a negative effective longitudinal piezoelectric coefficient. The obtained results can be used for quantitative estimates of piezoelectric properties of domain-engineered crystals.

Figure 1

Schematic representation of initial conditions: (a) simple two-dimensional lamellar domain structure with charged ${90}^{\circ }$ domain walls assumed in present simulations and (b) the imposed piece-wise homogenous electric field ${\mathbf{E}}_{\sigma }$, associated with the compensating charge distribution defined by Eq. (7).

Figure 2

Effective longitudinal piezoelectric coefficient $d_{33}^{\left[100\right]}$ of the BaTiO${}_3$ ferroelectric material with a hypothetic domain structure sketched in Fig. 1a as a function of the compensation charge parameter ${\rho }^{*}$ (for 22.6-nm-thick domains). Values obtained from the present phase-field calculations (open circles) can be compared to the values of the phase-field calculations given in Table III of Ref. 13 (full circles). In both cases, the values were determined from the overall elongation of the material in the $\left[100\right]$ direction in response to a small probing electric field in the same direction (simulation of a converse piezoelectric response).

Figure 3

Effective longitudinal piezoelectric coefficients $d_{33}^{\left[100\right]}$ as a function of domain wall distance $w_{\mathrm{D}}$ for selected values of compensation charge parameter ${\rho }^{*}$. Open point symbols refer to this study, full circles are data from Table IV of Ref. 13. Full lines are guides for eye, the dotted line indicates the analytically calculated limiting value of the $d_{33}^{\left[100\right]}$ piezoelectric coefficient for ${\rho }^{*}=2$ case (see in Sec. 5a).

Figure 4

Local polarization increment induced by a small, $x$-axis tensile uniaxial stress (${\sigma }_{xx}={10}^4$ Pa), traced along the $y^{\prime }$ direction in the vicinity of a HH domain wall. The domain wall distance is fixed to 22.6 nm, various symbols refer to data calculated for various values of the charge parameter ${\rho }^{*}$, (a) $x$ component of the polarization increment and (b) $y$ component of the polarization increment.

Figure 5

Schema indicating the observed tilt of the equilibrium polarization P(A) and P(B) in the middle of the domain A and B, respectively. The equilibrium polarization is tilted towards the domain wall interface for ${\rho }^{*}<2$ as shown in (a) and away from it for ${\rho }^{*}>2$, as shown in (b). The role of the ${\alpha }_{\mathrm{A}}$ and ${\alpha }_{\mathrm{B}}$ angles is discussed in the main text.

Figure 6

Effective longitudinal piezoelectric coefficients $d_{33}^{\left[100\right]}$($w_{\mathrm{D}},{\rho }^{*}$) as a function of the deviation angle, joining the points with equal domain thicknesses. Dashed line was calculated from the Eqs. (15) and (22), derived in Sec. V C.

Figure 7

Phase diagram showing stability domains of distinct types of domain structures obtained in present phase-field simulations for a range of parameters $w_{\mathrm{D}}$ and ${\rho }^{*}$, defining the domain wall distance and the imposed compensating charge density, respectively. Point symbols are determined from the divergence of the piezoelectric response as a function of ${\rho }^{*}$ (for selected fixed values of $w_{\mathrm{D}}$), lines are just guides for eye. Regions II and III can be considered as the original domain structure of Fig. 1a, only with slight symmetric tilts of the polarization, as shown in Figs. 5a and 5b, respectively. Shaded areas I and IV correspond to a lower-symmetry domain structure (monoclinic one), with unequal polarization tilts, sketched in Figs. 8a and 8b. In the region V, the polarization is perpendicular to the imposed charge planes and the domain walls are transformed to 180-domain ones with HH and TT arrangements. See text for detailed explanation.

Figure 8

Schematic image indicating the observed tilt of the equilibrium polarization P(A) and P(B) in the middle of the domains A and B, respectively, as obtained in our phase-field simulations with input parameters falling (a) in the region I and (b) in the region IV of Fig. 7.