Hopfions emerge in ferroelectrics

TL;DR

This paper demonstrates that Hopfions, knotted topological solitons characterized by the Hopf invariant, can naturally form in confined ferroelectric nanoparticles, revealing new functionalities in their electromagnetic responses.

Contribution

It introduces the emergence of Hopfions as fundamental polarization configurations in ferroelectric nanoparticles, linking topological physics with nanomaterial functionalities.

Findings

Hopfions are found in confined ferroelectric nanoparticles.

They influence the electromagnetic response of nanomaterials.

Potential for novel technological applications.

Abstract

Paradigmatic knotted solitons, Hopfions, that are characterized by topological Hopf invariant, are widely investigated in the diverse areas ranging from high energy physics, cosmology and astrophysics to biology, magneto- and hydrodynamics and condensed matter physics. Yet, while holding high promise for applications, they remain elusive and under-explored. Here we demonstrate that Hopfions emerge as a basic configuration of polarization field in confined ferroelectric nanoparticles. Our findings establish that Hopfions govern a wealth of novel functionalities in the electromagnetic response of composite nanomaterials opening route to unprecedented technological applications.

Figures

Equations

$$\mathcal{H}={\int }_{{\mathbb{M}}^3}(\mathbf{P}\cdot \mathbf{A})d\mathbf{r},$$

$$\displaystyle F=\int\Big{(}\left[a_{\rm\mathit{i}}(T)P_{\rm\mathit{i}}^{2}+a_{\rm\mathit{ij}}^{u}P_{\rm\mathit{i}}^{2}P_{\rm\mathit{j}}^{2}+a_{\rm\mathit{ijk}}P_{\rm\mathit{i}}^{2}P_{\rm\mathit{j}}^{2}P_{\rm\mathit{k}}^{2}\right]_{\rm\mathit{i}\leq\mathit{j}\leq\mathit{k}}$$

$$\displaystyle+\frac{1}{2}G_{\rm\mathit{ijkl}}(\partial_{\rm\mathit{i}}P_{\rm\mathit{j}})(\partial_{\rm\mathit{k}}P_{\rm\mathit{l}})+\frac{1}{2}\varepsilon_{\rm 0}\varepsilon_{\rm b}\left(\nabla\varphi\right)^{2}+\frac{1}{2}C_{\rm\mathit{ijkl}}u_{\rm\mathit{ij}}u_{\rm\mathit{kl}}\,\Big{)}\;d^{3}r\,,$$

$${\epsilon }_0{\epsilon }_{\mathrm{b}}{\nabla }^2\phi ={\partial }_i{P}_i,$$

$$C_{ijkl}{\partial }_j\left(u_{kl}-Q_{ijkl}{P}_k{P}_l\right)=0,$$

Full text

Hopfions emerge in ferroelectrics

I. Luk’yanchuk

  

1 Y. Tikhonov

  

1,2 A. Razumnaya

  

2 & V. M. Vinokur3∗

**Paradigmatic knotted solitons, Hopfions, that are characterized by topological Hopf invariant, are widely investigated in the diverse areas ranging from high energy physics, cosmology and astrophysics to biology, magneto- and hydrodynamics and condensed matter physics. Yet, while holding high promise for applications, they remain elusive and under-explored. Here we demonstrate that Hopfions emerge as a basic configuration of polarization field in confined ferroelectric nanoparticles. Our findings establish that Hopfions govern a wealth of novel functionalities in the electromagnetic response of composite nanomaterials opening route to unprecedented technological applications. **

Confinement of a ferroelectric material changes radically its electric properties. Termination of the of polarization at the surface leads to the depolarization charges $\rho=-\mathrm{div}\,\mathbf{P}$ that produce the depolarization field $\mathbf{E}$. In turn, the self-consistent interactions result in a nonuniform texture that minimizes electrostatic energy costs associated with these depolarization effects. The corresponding topologically nontrivial textures include regular patterns of Kittel domains  [1, 2, 3, 4, 5], which can be also viewed as the periodic array of vortex-antivortex pairs[6] and lattice of skyrmions [7] in the films and superlattices, and vortices and skyrmions in the nanords and nanodots  [8, 9, 10, 11, 12, 13, 14]. As we show below, a geometrical restriction of a feroelectric brings about yet another class of the topological formations, Hopfions which appear in a broad variety of nature phenomena [15, 16, 17, 18, 19, 20, 21, 22, 23, 24].

We consider a spherical nanoparticle. A uniform mono-domain state is not energetically stable because of formation of the surface depolarization charges located at the termination points of polarization lines $\mathbf{P}(\mathbf{r})$, see Fig. 1a. To minimize the energy associated with depolarization field $\mathbf{E}$, the system transforms itself into a structure with the vanishing depolarization charges so that $\mathrm{div}\,\mathbf{P}=0$. Therefore, the divergenceless of the polarization field is the fundamental condition defining the physics of the spatially nonuniform ferroelectricity. The absence of the depolarization charges at the surface, implies that the polarization vector, $\mathbf{P}$ is tangent to the surface of the particle.

An instant configuration stemming from the above conditions is the vortex [10], see Fig. 1b. For the case of the isotropic spherical nanoparticle, such a solution [25] is stable just below the transition from the high-temperature paraelectric phase into the ferroelectric phase. However, in general, far from the transition, the system seeks for the configuration in which the amplitude of the polarization remains close to its equilibrium value everywhere, hence strives to eliminate singularities.

A singularity at the vortex core can be removed by the continuous deformation of the vector field $\mathbf{P}$ promoting its escape into the third dimension along the vortex axis [26], see Fig. 1c. Had this process been occurring in an unrestricted 3D space, it would have resulted in a uniform polarization. However, in the confined spherical geometry, this would have recovered the unfavorable mono-domain configuration shown in Fig. 1a. To avoid that, the polarization flow along the vortex axis spreads into a back-flow over the sphere’s surface, maintaining polarization tangent to the surface hence avoiding the onset of depolarization charges. The resulting $\mathbf{P}$-field configuration is a 3D knotted soliton, called ‘Hopfion,’ which is a set of interlinked circles or torus knots, see [21] and references therein. A simplest single polarization Hopfion is shown in Fig. 1d.

To unravel the nature of an emergent polarization structure in a nanoparticle, we observe that the lines of the divergenceless polarization field, $\mathbf{P}(\mathbf{r})$, have no intersections, are looped, tangent to the surface of the nanoparticle, and constitute a dense set in the sphere. Polarization lines are identical by their topological characteristics to the streamlines of an ideal incompressible liquid inside the restricted volume, which enables us to employ topological methods of hydrodynamics developed by Arnold [27]. According to Arnold’s theorem, the analytic stationary flows of the divergenceless vector field can belong in either of two classes: the field flows fibered into the nested cylindrical surfaces and those fibered into the set of the nested tori. The former configurations correspond to vortices and the latter configurations constitute Hopfion. We describe a layout of a Hopfion starting with filling the interior of the nanoparticle by the set of sequentially nested concentric tori, as shown in Fig. 2a. Then polarization field lines form the dense set of trajectories twining these tori, as shown by thin solid lines in Fig. 2b.

A topological structure that implements the streamlines of Hopfion is a map from the 3-sphere, $\mathbb{S}^{3}$, to the 2-sphere, $\mathbb{S}^{2}$, such that each point in a $\mathbb{S}^{2}$ corresponds to a distinct great circle at $\mathbb{S}^{3}$, called a fiber. An inverse map of $\mathbb{S}^{3}$ to a volume of a spherical nanoparticle, $\mathbb{M}^{3}$, projects fibers to a set of closed polarization field lines, and is characterized by its topological charge called Hopf invariant [27]

[TABLE]

where the gauge field $\mathbf{A}$ is defined by $\mathbf{P}=\mathrm{rot}\mathbf{A}$. The definitive property of the polarization lines in a Hopfion is that each field line links through others. This property illustrates an equivalent topological definition of the Hopf invariant ${\cal H}$ as the link index of two loops that are pre-images of two points in $\mathbb{S}^{2}$ [27].

An associated feature that arises in the Hopfion state is the chirality which is the asymmetry with respect to mirror-reflection. The corresponding symmetry group is $C_{\rm\it\infty}$, and is eventually reduced by the crystal anisotropy. Hopfions can be the “left” and “right” ones, hence spontaneous chiral symmetry breaking upon the formation of the Hopfion state. Chirality marks the Hopfion off from the vortex, endowed with the group $C_{\rm\it\infty h}$ that includes the reflection in the plane, $\sigma_{\rm\it h}$, perpendicular to the vortex axis. We thus use the chirality parameter, $\chi=\mathbf{P}\cdot\mathrm{rot}\mathbf{P}$,[29] to characterize the Hopfion state. This parameter compliments the toroidal moment $\mathbf{T}=\mathrm{rot}\,\mathbf{P}$ that is ordinarily used for the description of the state containing topological excitations in ferroelectric nanoparticles [8], since $\mathbf{T}$ cannot expose the difference between the vortex state and the chiral Hopfion state. Note, that the spontaneously arising chirality opens an unprecedented opportunity of manipulating the ferroelectric nanoparticles by circular-polarized laser tweezers, inducing and tuning the optical activity of the media.

To investigate the Hopfions arising in ferroelectric nanoparticle, we perform the relaxation minimization of the Ginzburg-Landau (GL) functional coupled with the electrostatic and elastic equations, see Methods. An insight into the Hopfion emergence is gained by the purposeful initial using of the isotropic model functional, simplifying the anisotropy and elasticity effects. We select, however, parameters that are close to those in the realistic oxide materials and that partially account for the elastic interaction. Shown in Fig. 2c-k are the results of computations. Panels (c) and (d) display the Hopf fibration in the isotropic nanoparticle with radius $R=25$ nm at room temperature realizing the self-linking spiral-like structure of polarization lines. The dense set of lines forming the knots at a single torus is shown in panel (c), whereas the panel (d) exhibits the compactification of entwined tori in the bulk of the nanoparticle. Figure 2e demonstrates the pairwise intra- and inter-tori linking of the polarization lines belonging to the tori shown in panel (d). The nontrivial knotting of the field lines leads to the peculiar spatial distributions of the polarization characteristics of the system. The tendency for the polarization vectors to escape in the third dimension results in the up-stream of the polarization lines near the Hopfion core and to their down-stream at the periphery, as reflected in Fig. 2f showing the $P_{\rm z}$ component. At the same time, the distribution of amplitudes of polarization vectors, becomes nearly homogeneous, (Fig. 2g), and the residual singularities settle as whorles of the polarization at the points of the termination of the Hopfion axis at the poles of the sphere as shown in Fig. 2h. These residual singularities are essentially non-removable and manifest the Poincaré hairy ball theorem stating that there is no non-vanishing continuous tangent vector field on two-dimensional sphere [30]. Figure 2i demonstrates the distribution of the chirality, $\chi$, inside the particle that concentrates mostly along the Hopfion core.

Panel (j) presents the $T-R$ phase diagram for the spherical particles with the radius $R<30$ nm. Notably, the Hopfion state occupies its major part. The transition temperature $T_{\rm c}$ from the high-temperature paraelectric state to the low-temperature ferroelectric one, lies only slightly below the bulk temperature $T_{\rm 0}$ in large particles with $R>20$ nm. In small particles with $R<20$ nm, $T_{\rm c}$ is noticeably suppressed by the size-driven confinement. The polarization texture of the ferroelectric state, which forms just below the transition, has the vortex-like structure. In general, the dependence $T_{\rm c}(R)$ is well fitted by the formula, following from the dimensional analysis of GL equations, $(T_{\rm 0}-T_{\rm c})/T_{\rm 0}\simeq_{\rm c}(\mu\xi_{\rm 0}/R)^{2}$, where $\xi_{\rm 0}\simeq 0.7$ nm is the coherence lengths and $\mu_{\rm c}\simeq 2.0$. Vortices start expelling their core singularities into the third dimension below the critical temperature $T_{\rm h}$, which also scales as $R^{-2}$, with the coefficient $\mu_{\rm h}\simeq 2.8$. The temperature interval of the vortex phase existence is negligibly small for $R>10-15$ nm and further cooling drives the system into a Hopfion state. The vortex state becomes noticeable only in small enough nanoparticles, where the geometry restriction stabilizes vortices.

The temperature dependence of the principal ferroelectric characteristics, the mean-squared polarization, $(\overline{{\mathbf{P}}^{2}})^{1/2}$, the absolute value of the mean toroidal moment, $|\overline{\mathbf{T}}|$, directed along the Hopfion/vortex axis, and the mean chirality, $\overline{\chi}$, chosen negative for concreteness, are shown in the panel (j) for three characteristic sizes of the nanoparticles: $R=5,10$ and $25$ nm. The mean squared polarization vanishes as a square root on approach to the ferroelectric transition temperature, $T_{\rm c}$, similar to the uniform bulk case. The toroidal moment also vanishes at $T_{\rm c}$, whereas the chirality disappears below $T_{\rm c}$ at the vortex-Hopfion transition and is used to determine $T_{\rm h}$. Note that for the 5 nm nanoparticles $\overline{\chi}=0$ since the system remains in the vortex state.

Now we turn to concrete embodiments of Hopfions in specific ferroelectric materials. We focus on the technologically impactful quasi-isotropic Pb0.4Zr0.6TiO3 (PZT) compound, which is close to the so-called morphotropic phase boundary. This distincts favorably the PZT from other ferroelectrics, like PbTiO3 and BaTiO3, where strong anisotropy may result in breaking a ferroelectric nanoparticle into domains [29, 31]. Numerical simulations of the PZT nanoparticle of $R=25$ nm are based on the relaxation of the full GL functional with the elastic terms, see Methods. We find that the Hopfion state spans almost the entire ferroelectric phase, so that the resulting phase diagram resembles the phase diagram for the isotropic system with the similar parameters, Fig. 2j. The field-polarization characteristics of the nanoparticle of $R=25$ nm at room temperature are shown in Fig. 3. The distinct polarization topological configurations are marked as states (i)-(vii). The set-up used in the simulations of the nanoparticle under the applied field is sketched in the insert in the plot. The voltage $U$ is applied to the plates of the capacitor embracing the nanoparticle, which is oriented to have its [111] crystallographic axis perpendicular to the capacitor’s plates. Accordingly, $U$, induces average polarization, ${\bar{P}}$, and average internal field, ${\bar{E}}$, along the [111] direction. The relation between these quantities completely describes the dielectric properties of the system, and is the constitutive relation $\bar{P}(\bar{E})$. We show that $\bar{P}(\bar{E})$ dependence is the S-shape curve with slight hysteresis that we describe below.

The state (i) in Fig. 3 corresponds to the Hopf fibration arising after the zero-field quench of the nanoparticle from the paraelectric state. The polarization lines maintain the torus-winding structure as in the isotropic case. However, the crystal anisotropy drives the polarization towards [111] and [110] (or equivalent) crystallographic directions fixing the orientation of the Hopfion axis along [111] direction. In addition, the pole singularities extend slightly into the bulk, to form two respective vortices coexisting with the bulk Hopfion. Upon the application of the external field (in the negative direction), the virgin curve $\bar{P}(\bar{E})$ jumps first to the left, across the singular internal field, ${\varepsilon_{\rm 0}\bar{E}}\simeq 1.4\times 10^{-5}$ C m*-2* (where $\varepsilon_{\rm 0}=8.85\times 10^{-12}$ C V*-1m-1*), related to the topological piercing of the Hopfion by helical polarization lines that will be described further. Then it descends along the left branch of the hysteresis loop forming finally the down-oriented directional helical structure, (state (ii) in Fig. 3). In the emerging polarization structure the open polarization lines thrust the nanoparticle, so that the mean polarization flux gets aligned with the applied field.

Reversing the change in the field, we follow now the right hand side branch of the hysteresis loop from the bottom to the top. The evolution of the system upon the monotonic variation of the field from the negative to the positive value, occurs through first, compression, and then, stretching the helical polarization lines with the change of the mean polarization flux direction from the negative to the positive one. We observe that the system passes through the sequence of topological phases that follow the Arnold’s partitioning of the nanoparticle space into cells where the field lines are entwined around the nested sets of either cylindrical or toroidal surfaces. In the initial helical state (ii) the polarization lines are entwined around the cylindrical surfaces. Upon the decreasing field, the helical structure compresses while broadening its central part (state (iii)) and, finally, a spherical cell containing the toroidal Hopf fibration nucleates at the center of the nanoparticle (state (iv)). The emerging Hopfion grows further ousting the helical states towards the nanoparticle periphery, which bypass Hopfion outside and carry the mean polarization of the nanoparticle along the nanoparticle surface. When the applied field vanishes, the Hopfion fills up the entire nanoparticle asymptotically approaching to the state (i) with $\bar{P}=0$.

The change of the sign of the external field from negative to positive leads to the topological phase transition in the course of which the system “turns inside out” and a hyperbolic cell filled up with the nested cylinders sets in along the axis of the nanoparticle. This cell ruptures the Hopfion sphere into a torus (state (v)) and hosts the open polarization helical lines carrying now the mean polarization inside the nanoparticle along its axis. This transition occurs abruptly at some threshold applied field and manifests as a disruption of the smooth behaviour of the internal field $\bar{E}$ which makes a singular turn in polarization curve. Upon the further field increase, the Hopfion torus retires towards the equator and eventually disappears there draining out of the nanoparticle, which thus falls into the helical state. Just after the transition the polarization helices maintains the fitted structure (state (vi)), as a legacy of the vanished Hopfion tori, and only at higher applied fields it crosses over to the helical state (vii) equivalent, up to sign ${\mathbf{P}(r)}$ reversal, to the state (ii). Upon the sequential field reversal and decreasing the field back, the system does not pass through the reverse sequence of the states, but repeats the (ii)$\to$(vii) scenario with the replacement ${\mathbf{P}(r)}\to-{\mathbf{P}(r)}$, demonstrating thus the weakly hysteretic behaviour (left descending branch in Fig. 3). Again, the mean polarization flows, first, along the surface of the nanoparticle and then along the nanoparticle axis.

Complexity of the intertwined topological states encoded in the ${\bar{P}}$-${\bar{E}}$ characteristic of Fig. 3 stems from the interplay of confinement and depolarizing effects. Most importantly, the system is highly responsive to even weak internal fields, owing to the utmost softness of the helical springs of the polarization lines. In other words, the ease at which the open lines reconnect, ensures an unobstructed redistribution of the field-induced depolarization charge at the points of their termination, guaranteeing the almost perfect screening of the applied field. This behaviour is similar to that of the ferroelectric with domains, where the easy domain wall motion results in the similar softness [32]. Because of that, the absolute value of the effective dielectric permittivity of the nanoparticle, $\bar{\varepsilon}=\bar{P}/\varepsilon_{\rm 0}\bar{E}$, can reach giant values of order $10^{4}$ and even more. Moreover, in the close resemblance of the nanoparticles with domains, the S-shape ${\bar{P}}$-${\bar{E}}$ characteristic demonstrate segments having the negative slops, hence negative capacitance effect, which is explained by the advancing reaction of the polarization texture to the applied field [33].

We investigate now the Hopfion-governed physics in a composite material comprising the high-$\varepsilon$ ferroelectric nanoparticles embedded in the low-$\varepsilon$ dielectric matrix. Note, that according to the Maxwell Garnett mixing rule [34] the dilute high-$\varepsilon$ nanoparticles, see Fig. 4a, only weakly renormalize the properties of the low-$\varepsilon$ hosting matrix. This was confirmed by numerical simulation in [35]. We thus focus on the composite of sintered nanoparticles, where polarization lines can pierce the entire system passing from one grain to another, and model a sintered composite as a rectangular array of contacting nanoparticles as shown in Fig. 4b.

Figure 4c displays the configuration that forms under the conditions of the moderate densification of PZT nanoparticles with $R=25$ nm, contacted along the [111] direction. The degree of densification is quantified by the thickness of the contacting neck, $2h=0.4$ nm (see inset), so that the area of the mutual contact of the adjacent particles serves as an aperture for polarization lines. Only a fraction of polarization helical lines passes through the interfacial aperture. Another part of the lines is confined into the Hopfions and does not interact with the applied electric field. They are invisible to the entire dielectric response of the system and can be viewed as an electric realization of the dark matter in the universe. In the case of strongly-densified nanoparticles with $2h=2$ nm all the field lines form helices flowing through the area of the contact (Fig. 4d).

Figure 4e displays the polarization characteristics of the nanocomposite, comprising the sintered 25 nm nanoparticles, as functions of the applied voltage. The chain of the connected particles with the repeating polarization pattern is modeled by a single particle with cut off the skullcaps of the height $h$, the electrodes covering its top and bottom cuts respectively. The model setup is shown in the inset. As we already mentioned, the particles with the vanishing contact area ($h\simeq 0$ nm) gives almost no contribution to the dielectric properties of the system. The dielectric response grows with the degree of the densification. For the moderate contact $h=0.2$ nm the switching between the up- to down-polarized helices, occurs along the gently-sloping hysteresis loop. At each branch, the system quasistatically passes through the sequence of the topological states, similar to those, shown in Fig. 3. For the high degree of the densification, $h=1$ nm, the sharp hysteresis loop with the polarization jumps is observed. The switching follows through the same sequence of the states, but the system passes through the Hopfion states via dynamic instability, as illustrated in Supplementary Video.

A far reaching implication of our findings is that an array of sintered nanoparticles hosts Hopfion fibrations imprinting in the polarization lines configurations. The emerging topological frustrations are an expected origin of yet mysterious relaxor behaviour of disordered systems. As a marked analogy to a dark matter, the Hopfions store an energy in composite ferroelectrics which opens new routes for design of the ferroelectric-based energy storage devices.

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Acknowledgements. This work was supported by H2020 RISE-MELON and ITN-MANIC actions (I.L. and Y.T.), by the Southern Federal University, Russia (A.R. and Y.T.) and by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (V.M.V and partially I.L.).

METHODS

Phase-field simulations. The phase-field simulations were performed using the FERRET package [29], designed for the multi-physics simulation environment MOOSE [2]. We solved time-dependent relaxation equation $-\gamma\,\partial P_{\rm\mathit{i}}/\partial t=\delta F/\delta P_{\rm\mathit{i}}$ where $F$ is the total free energy functional that also includes electrostatic and elastic effects, the relaxation parameter $\gamma$ was taken as unity since the time scale is irrelevant to the problem. The details of simulations are similar to those, described in [29]. The geometry and conceptual setup of simulated system are shown in the insets to the Fig. 3 and Fig. 4e of the article. The internal field was controlled by the voltage $U$, applied to the electrodes. The paraelectric state with the small randomly-distributed polarization was used as the initial condition. To ensure the stability of solution we used the quench from the different initial paraelectric states and the temperature annealing procedure as well as the different finite-element meshes. In order to calculate the $P(E)$ characteristics in the ascending/descending field we used the quasi-stating poling when the polarization distribution at previous stage was used as the initial condition.

Functional. The free functional, $F$, is written as follows

[TABLE]

were the summation over the repeated indices $i,j,...=1,2,3$ (or $x,y,z$) is performed. The first square brackets term of (2) stands for the Ginzburg-Landau energy written in the form given in [3]. Importantly, the 4th-order coefficients $a_{\rm 11}^{u}$ and $a_{\rm 12}^{u}$ (and their cubic-symmetry homologs) are taken at zero strain and are calculated by the Legendre transformation from the stress-free coefficients $a_{\rm 11}^{\sigma},a_{\rm 12}^{\sigma}$ [4], see also [5]. The second term of (2) with coefficients $G_{\rm\mathit{ijkl}}$ corresponds to the gradient of energy. The two last terms are the electrostatic and elastic energies, written in terms of the electrostatic potential $\varphi$ and strain tensor $u_{\rm\mathit{ij}},$ respectively. Here $\varepsilon_{\rm 0}=8.85\times 10^{-12}$ C V*-1m-1* is the vacuum permittivity, $\varepsilon_{\rm b}\simeq 10$ is the background dielectric constant of the non-polar ions, typical for PbTiO3 [29] and $C_{\rm\mathit{ijkl}}$ is the elastic stiffness tensor.

The polarization-induced distribution of the electrostatic potential $\varphi$ and elastic strains $u_{\rm\mathit{ij}}$ in functional (2) are found at each relaxation step as solutions of equations

[TABLE]

[TABLE]

implemented in the MOOSE-FERRET package. Here $Q_{\rm\mathit{ijkl}}$ is the electrostriction tensor.

Material parameters. For the Pb0.4Zr0.6TiO3 the coefficients for the uniform part of the functional (2) are as follows, $a_{\rm 1}=2.3\,(T-364^{\circ}C)\times 10^{-13}$ C*-2m2N, $a_{\rm 11}^{u}=0.44\times 10^{-9}$ C-4m6N, $a_{\rm 12}^{u}=0.074\times 10^{-9}$ C-4m6N, $a_{\rm 111}=0.27\times 10^{-9}$ C-6m10N, $a_{\rm 112}=1.21\times 10^{-9}$ C-6m10N, and $a_{\rm 123}=-5.69\times 10^{-9}$ C-6* m10N. The electrostriction coefficients are $Q_{\rm 1111}=0.073$ C*-2m4* , $Q_{\rm 1122}=-0.027$ C*-2m4*, and $Q_{\rm 1212}=0.064$ C*-2m4* (with cubic symmetry permutations) were calculated on the base of the expression, given in [4] after transformation from Voigt to tensor notations [5]. The gradient energy coefficients $G_{\rm 1111}=2.77\times 10^{-10}$ C*-2m4N, $G_{\rm 1122}=0$, and $G_{\rm 1212}=1.38\times 10^{-10}$ C-2m4N selected to be the same as for PbTiO3 [6]. The elements $C_{\rm 1111}=1.68\times 10^{11}$ m-2N, $C_{\rm 1122}=0.82\times 10^{11}$ m-2N, and $C_{\rm 1212}=0.41\times 10^{11}$ m-2*N of the stiffness tensor $C_{\rm\mathit{ijkl}}$ were obtained by inversion of the compliance tensor $s_{\rm\textit{ijkl}}$, which elements are given in [4].

To explore the isotropic model we dropped out the elastic part of the functional (2), neglected the 6th-order polarization terms, and imposed $a_{\rm 12}^{u}=2a_{\rm 11}^{u}=0.27\times 10^{-9}$ C*-4* m 6N, so that the uniform part of the functional acquired the isotropic form $a_{\rm 1}(T)P^{2}+a_{\rm 11}^{u}P^{4}$. Note that the gradient energy with selected coefficients $G_{\rm\mathit{ijkl}}$ is already invariant with respect to rotation.

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