Shear-strain gradient induced polarization reversal in ferroelectric ${\mathbf{BaTiO}}_3$ thin films: A first-principles total-energy study

Based on the first-principles total-energy calculation, we have studied the shear-strain gradient effect on the polarization reversal of ferroelectric ${\mathrm{BaTiO}}_3$ thin films. By calculating the energies of double-domain supercells for different electric polarization, shear-strain gradients, and domain-wall displacement, we extracted, in addition to the domain-wall energy, the polarization energy, elastic energy, and flexoelectric coefficient of a single domain. The constructed Landau-Devonshire phenomenological theory yields a critical shear-strain gradient of $9.091\times {10}^7/\text{m}$ (or a curvature radius ($R$) of 110 Å) for reversing the ${180}^{\circ }$ domain at room temperature, which is on the same order of the experimentally estimated value of $3.333\times {10}^7/\text{m}$ ($R=300\AA$). In contrast to the commonly used linear response theory, the flexoelectric coefficient derived from fitting the total energy to a Landau-Devonshire energy functional does not depend on the specific pseudopotential. Thus, our method offers an alternative numerical approach to study the flexoelectric effect.

Figure 1

Sketches of double-domain supercells. The supercell is composed of $2Na\times b\times c$ tetragonal ${\mathrm{BaTiO}}_3$ unit cells with an up-domain ($p$) and a down-domain ($-p$) of $N$ unit cells each along $a$ axis. (a) Free-shear-strain case, and (b) with sinusoidal shear strain and strain gradient. Ba, Ti, and O atoms are represented as big green, middle light blue, and small red spheres, respectively.

Figure 2

The polarization-dependent energy density of single-domain ${\mathrm{BaTiO}}_3$. The saturated polarization $P_0=0.251{\mathrm{C}/\mathrm{m}}^2$ is set to that of the ferroelectric ground state. The circles are the calculated points, while the solid line represents the best fit by Eqs. (4) and (5).

Figure 3

The domain-wall energy densities as functions of electric polarization for different $\delta$. The symbols are the calculated points while solid lines represent the best fit by Eqs. (6) and (7). $\delta \left(p\right)={\delta }_0+{\delta }_1{p}^2$ with ${\delta }_0=0.0142$ and ${\delta }_1=-0.0016$.

Figure 4

The effect of shear-strain gradient on the polarization-dependent energy density of ${\mathrm{BaTiO}}_3$ at $T=300\mathrm{K}$ ($T_C\approx 388\mathrm{K}$). The values of shear-strain gradient $\eta =1/R$ are indicated by different line types.