Influence of elastic strain gradient on the upper limit of flexocoupling strength, spatially modulated phases, and soft phonon dispersion in ferroics

Using the Landau-Ginzburg-Devonshire theory, we established the role of the flexoelectric coupling between the gradients of elastic strain and polarization in the stability of spatially modulated phases in ferroics, such as incipient and proper ferroelectrics with commensurate and incommensurate long-range-ordered phases. We included the square of elastic strain gradient in the Landau-Ginzburg-Devonshire functional because this term provides the functional stability for all values of the strain gradient. Analytical expressions for polarization, strain, dielectric susceptibility, and stability threshold were derived for a one-dimensional case. The expressions show that the maximal possible values of the static flexoelectric effect coefficients (upper limits) established by Yudin, Ahluwalia, and Tagantsev without the square of elastic strain gradient and other higher order gradients terms lose their direct meaning. Considering the gradients, the temperature dependent condition for the flexocoupling magnitude exists instead of the upper limits. Also, we established that spatially modulated phases appear and become stable in commensurate ferroelectrics if the flexocoupling constant exceeds a critical value. The critical value depends on the electrostriction and elastic constants, temperature, and gradient coefficients in the Landau-Ginzburg-Devonshire functional. We calculated soft phonon dispersion in commensurate and incommensurate long-range-ordered phases of ferroelectrics with the square of elastic strain gradient, static, and dynamic flexocoupling. It appeared that the dispersion of the optical mode is slightly sensitive to the flexocoupling, and the dispersion of acoustic mode strongly depends on the coupling magnitude. Obtained results demonstrate that nontrivial differences in the dispersion of optical and acoustic modes occur with the change of flexocoupling constant. Therefore, experimental determination of soft phonon dispersion might be very informative to study the influence of strain gradients and flexocoupling on the spatially modulated phase in ferroelectrics with commensurate and incommensurate long-range order.

Figure 1

(a) Diagrams of the spatially modulated phase (SMP) and homogeneous ferroelectric phase (HFP) stability in ferroics with $g>0$ (right side) and $g<0$ (left side) plotted in dimensionless coordinates—flexoconstant $F^{*}$ and reduced temperature ${\alpha }_v^{*}$. Parameters $Q^{*}=0.5,F^{*}\le 0,{a_v}^{*}\le 0$, and $w_{}^{*}=-2$ for the case $g<0;F^{*}\ge 0,{{\alpha }_v}^{*}\ge 0$, and $w_{}^{*}=0$ for $g>0$. (b) Temperature dependence of the flexoconstant critical value $F_{cr}^{*}$. (c), (d) Dependence of the flexoelectric coefficient critical value $f_{cr}^{*}$ on the strain gradient constant $v$ (c) and temperature $T$ (d). The curves in (b)–(d) are calculated for dimension and dimensionless parameters of incipient ferroelectric $\mathrm{SrTi}{\mathrm{O}}_3$ and proper ferroelectrics $\mathrm{PbTi}{\mathrm{O}}_3, \mathrm{S}{\mathrm{n}}_2{\mathrm{P}}_2{\mathrm{S}}_6$, and $\mathrm{S}{\mathrm{n}}_2{\mathrm{P}}_2\mathrm{S}{\mathrm{e}}_6$ from Tables 1 and 2, with the exception of $f$ values, which change in (b) and (d), and $v$ values, which change in (c). Points in (c) correspond to the values of $f$ and $v$, determined from the phonon spectra of ferroelectrics shown in Fig. 4.

Figure 2

Dependences of the wave vectors $k_{+}^{*}$ (solid curves) and $k_{-}^{*}$ (dashed curves) on the flexoconstant $F^{*}$ (a) and reduced temperature ${\alpha }_v^{*}$ (b). Different curves are calculated for several values of ${\alpha }_v^{*}=0.001,0.1,0.5,1$ for $g>0$ and ${\alpha }_v^{*}=-0.001,-0.1,-0.5,-1$ for $g<0$ [curves 1–4 in plot (a)]; $F_{}^{*}=1.5,2,2.5,5$ for $g>0$; and $F_{}^{*}=-0.001,-1,-2.5,-5$ for $g<0$ [curves 1–4 in plot (b)]. Parameters $Q^{*}=0.5, a^{*}=0.1$, and $w_{}^{*}=-2$ for ferroelectrics with $g>0$, and $w_{}^{*}=0$ for $g<0$.

Figure 3

Dependences of the dimensionless phonon frequency ${\omega }^{*}$ on the wave vector $k_{}^{*}$ calculated for ferroelectrics with $g>0$ (a), (b) and $g<0$ (c), (d). Curves 1–3 in plots (a), (c) are calculated for ${\alpha }_v^{*}=1$ and several flexoconstants $F_{}^{*}=1,3,5$ for ferroelectrics with $g>0$ [plot (a)], and ${\alpha }_v^{*}=-1$ and $F_{}^{*}=-1,-3,-5$ for ferroelectrics with $g<0$ [plot (c)]. Curves 1–3 in plots (b), (d) are calculated for several values of reduced temperature ${\alpha }_v^{*}=1,2.5,5$ and $F_{}^{*}=1$ for ferroelectrics with $g>0$ [plot (b)], ${\alpha }_v^{*}=-1,-2.5,-5$ and $F_{}^{*}=-1$ for ferroelectrics with $g<0$ [plot (d)], as indicated by legends on the plots. Parameters $Q^{*}=0.5, M^{*}=0.05, a^{*}=0.1, w_{}^{*}=-2$, and ${\mu }^{*}=-1$ for ferroelectrics with $g<0$; $w_{}^{*}=0$ and ${\mu }^{*}=1$ for ferroelectrics with $g>0$.

Figure 4

Dependences of the phonon frequency ω on the wave vector $k$ measured experimentally (symbols) and calculated theoretically (solid and dashed curves) for the lowest transverse optical and acoustic modes in $\mathrm{SrTi}{\mathrm{O}}_3$ (a), in the paraelectric phase of $\mathrm{PbTi}{\mathrm{O}}_3$ (b), in the paraelectric (c) and ferroelectric (d) phases of $\mathrm{S}{\mathrm{n}}_2{\mathrm{P}}_2{\mathrm{S}}_6$, and in the paraelectric (e) and ferroelectric (f) phases $\mathrm{S}{\mathrm{n}}_2{\mathrm{P}}_2\mathrm{S}{\mathrm{e}}_6$. Experimental data are taken from Refs. [43, 44, 61, 62]. Temperatures are specified in plots (a)–(f). Parameters used in our calculations are listed in Table 1. Solid and dashed curves were calculated with and without lattice discreteness, respectively.