Kinetics of 90° domain wall motions and high frequency mesoscopic dielectric response in strained ferroelectrics: A phase-field simulation

The dielectric and ferroelectric behaviors of a ferroelectric are substantially determined by its domain structure and domain wall dynamics at mesoscopic level. A relationship between the domain walls and high frequency mesoscopic dielectric response is highly appreciated for high frequency applications of ferroelectrics. In this work we investigate the low electric field driven motion of 90°-domain walls and the frequency-domain spectrum of dielectric permittivity in normally strained ferroelectric lattice using the phase-field simulations. It is revealed that, the high-frequency dielectric permittivity is spatially inhomogeneous and reaches the highest value on the 90°-domain walls. A tensile strain favors the parallel domains but suppresses the kinetics of the 90° domain wall motion driven by electric field, while the compressive strain results in the opposite behaviors. The physics underlying the wall motions and thus the dielectric response is associated with the long-range elastic energy. The major contribution to the dielectric response is from the polarization fluctuations on the 90°-domain walls, which are more mobile than those inside the domains. The relevance of the simulated results wth recent experiments is discussed.


Elastic strain energy of a domain structure
In the present framework, the 90 o -domain walls and their dynamics are associated with the total strain which includes the ferroelastic strain and externally imposed normal strain.
The ferroelastic strain associated with the FE polarization field is given by: where Q ij are the electrostrictive coefficients in the external stress free state.
The elastic strain e ij and the related elastic energy F e can be expressed as: where  ij is the total strain, c ijkl is the elastic stiffness tensor which has only three independent elastic constants C 11 , C 12 , and C 44 for a square lattice. Combining Eq.(S1) with Eq.(S2) gives term F e =F el +F es +F e , i.e. term F e can be decomposed into three terms F el , F es , and F e . Term F e is a function of polarization P(r) and thus only affects the Landau expression, i.e. A 11 and A 12 .
In our simulation the order parameter u(r) is reduced to a function of P(r) based on the assumption that the response time of elastic lattice strain is far shorter than the time for electric dipole relaxation. In that view, we can use a static condition for mechanical equilibrium to solve the displacement field u(x, y) as a function of the polarization field P(x, y). The mechanical equilibrium equation is expressed as  ij,j =0, and stress tensor  ij is related to strain e ij according to the Hooke's law via  ij =c ijkl e kl =c ijkl ( kl - 0 kl ). Therefore, the equilibrium equation can be rewritten as: which has a solution in the Fourier space, given a periodic boundary conditions: where g ij -1 =c ijkl ζ k ζ l and ζ is the coordinate in the Fourier space, ˆi u and 0 kl  are the Fourier transform of u i and  0 ij , respectively. We can use the inverse Fourier transformation to obtain u i in real space.

Domain wall motion under dc electric field
Figure S1 present two columns of snapshoted images of the domain structure at various times after a static electric field E ext =6A 1 P a is applied along the y-axis at t=0. The left column (a) refers to the case of external strain  0 =0 and the right one (b) to the case of  0 =0.4% (tensile strain) along the y-axis too (i.e.  0 //E ext ). The polarization in each domain is indicated by the black arrows and the a 1 -domain stands for P//x-axis and a 2 -domain for P//y-axis. Parameter l 0 and Δl are labeled for guide of eyes. By comparing the domain patterns in the two columns, it is clearly seen that the tensile strain along the y-axis not only broadens the domains with polarization parallel to the strain direction (a 2 -domains) but also remarkably delays the domain wall motion, i.e. the local mobility of the domain walls is reduced.
Although the a 2 -domains for  0 =0.4% are eventually wider than those for  0 =0, which is the strain induced effect, the Δl value, induced by E ext , is much smaller in this case. The domain wall motion will be eventually terminated, approaching another equilibrium state, as shown by the images at t=720. bottom rows refer to t=20 and t=720 after field E ext is applied to the lattice. E ext =6A 1 P a . Figure S2 evaluates the spatial contours of strain e xx and e yy in each step of Figure S1. The red surfaces stand for e xx and blue ones for e yy . Owing to the lattice volume conservation, for one domain, the compressive strain along one direction (e.g. the x-axis) is always accompanied with the tensile strain along the other direction (the y-axis), and verse vice. As shown in Fig.S2(a) at t=0, the a 1 -domain accommodates the tensile and compressive strains along the x-axis and y-axis respectively, while the compressive and tensile strains along the x-axis and y-axis are developed in the a 2 -domain.

Evolution of underlying strain distribution
First, we consider the  0 =0 case. Given a static electric field along the y-axis, the domain walls move into the a 1 -domain in compensation with the extension of the a 2 -domain width, and the electric field serves as the driving force of the wall motion. The continuous shrinking of the a 1 -domain is accompanied with the increasing magnitude of both e xx and e yy inside the a 1 -domain (e xx from 0.2% to 0.5% and e yy from -0.5% to -0.8%) with time. Correspondingly, both e xx and e yy inside the a 2 -domain have their magnitudes decreasing with time (e xx from -0.5% to -0.3% and e yy from 0.2% to 0).
When externally imposed strain, the calculated spatial distributions of e xx and e yy in the a 1 -domain and its neighbor a 2 -domain are shown in the Fig.S2(b). At t=0 with E ext =0, the whole strain field contour shifts upward with respect to the case of  0 =0. However, a careful observation shows that both the tensile strain e xx in the a 1 -domain and tensile strain e yy in the a 2 -domain are increased by ~0.3%, while the reductions of the compressive e yy in the a 1 -domain and compressive e xx in the a 2 -domain are much less (~0.1%). Therefore, additional shrinking of the a 1 -domain driven by the electric field would become even tougher, given that  0 =0.4% already makes the a 1 -domain narrow. lattice. The red surfaces stand for e xx and blue ones for e yy . The left column (a) refers to  0 =0 and the right one (b) refers to  0 =0.4%. The top row refers to E ext =0, and the middle and bottom rows refer to t=20 and t=720 given E ext (dc) along the x-axis. E ext =0.6A 1 P a .