Hidden symmetry of flexoelectric coupling

Considering the importance of flexoelectric coupling for the physical understanding of the gradient-driven couplings in mesoscale and nanoscale solids, one has to determine its full symmetry and numerical values. The totality of available experimental and theoretical information about the flexocoupling tensor symmetry (specifically the amount of measurable independent components) and numerical values is contradictory. However, the discrepancy between the theory and experiment can be eliminated by consideration of all possible inner symmetries of the flexocoupling tensor and physical limits on its component values. Specifically, this paper reveals the inner “hidden” symmetry of the static flexoelectric tensor that allows minimizing the number of its independent components. Revealed hidden symmetry leads to nontrivial physical sequences, namely, it affects the upper limits of the static flexocoupling constants. Also we analyze the dynamic flexoelectric coupling symmetry and establish the upper limits for the numerical values of its components. These results can help us to understand and quantify the fundamentals of the gradient-type couplings in different solids.

Figure 1

Effective flexoelectric coefficient $f_{1111}^{\prime }\left(\phi \right)=f_{1111}-\mu {sin}^2\left(2\phi \right)$ (a) and $f_{1212}^{\prime }\left(\phi \right)=f_{1212}+\mu {sin}^2\left(2\phi \right)$ (b) as a function of the sample rotation angle $\phi$ calculated for several values of the ratio $f^{*}=f_{1212}/\phantom{f_{1212}{f}_{1111}}{f}_{1111}$ equal to 1 (dashed blue curves 1), 1/3 (dotted red curves 2), and 0.1 (solid magenta curves 3).

Figure 2

Temperature dependence of the soft phonon TO frequency ${\omega }_{}^{\mathrm{opt}}$ (a) and the ratio (b) $\eta =\mu {\left({\omega }_{}^{\mathrm{opt}}\right)}^2/\phantom{\mu {\left({\omega }_{}^{\mathrm{opt}}\right)}^{2}2{\alpha }_T\left(T-T_C\right)}2{\alpha }_T\left(T-T_C\right)$ calculated from Eq. (8) at $M_{ij}^{}=0$ (dashed black curve) and $|M_{ii}|=(2,5,8)\times {10}^{-8}\mathrm{V}{\mathrm{s}}^2/{\mathrm{m}}^2$ (solid red, magenta, and blue curves), ${\alpha }_T=3.765\times {10}^5\mathrm{F}/\mathrm{m}, T_C=752\mathrm{K}, \mu =1.59\times {10}^{-18}{\mathrm{s}}^2\mathrm{Jm}, \rho =7.986\times {10}^3\mathrm{kg}/{\mathrm{m}}^3$, and ${\beta }_T=3.2\times {10}^{-5}{\mathrm{K}}^{-1}$ corresponding to ${\mathrm{PbTiO}}_3$ in a paraelectric phase.