Universal Properties of Ferroelectric Domains

Based on the Ginzburg-Landau approach, we generalize the Kittel theory and derive the interpolation formula for the temperature evolution of a multidomain polarization profile $\mathbf{P}(x,z)$. We resolve the long-standing problem of the near-surface polarization behavior in ferroelectric domains and demonstrate polarization vanishing instead of the usually assumed fractal domain branching. We propose an effective scaling approach to compare the properties of different domain-containing ferroelectric plates and films.

Figure 1

(a) Multidomain texture of ferroelectric polarization in uniaxial ferroelectric film, sandwiched by two paraelectric (dead) layers. The emerging depolarization electric field is provided by alternating polarization-induced surface charges and confined in the near-surface layer of thickness, comparable with domain width $d$. (b) Elliptical functions $y=\mathrm{sn}(x,m)$ for different parameters $m$ that we use to model the domain profile at different $t$. (c) Phase diagram of domain states as function of sample thickness $2a_f$ and reduced critical temperature $t=T/T_{c0}-1$. Polarization profiles of hard and soft domains were obtained by numerical solution of Eqs. (6, 7, 8, 9). We assume that ${\varkappa }_{\parallel }\simeq 500$, ${\epsilon }_{\perp }\simeq 100$, ${\epsilon }_p\ll {\epsilon }_{\perp }$, ${\varkappa }_{\parallel }$, ${\xi }_{0x}\simeq 1\mathrm{nm}$, and $a_p\simeq 30\mathrm{nm}$.

Figure 2

Temperature dependencies of parameters of Eq. (19): (a) elliptic arguments $m_1$ and $m_2$, elliptic integrals $K_1$ and $K_2$, (b) domain amplitude $A$ and domain lattice period $d^{\prime }$. All the variables are dimensionless.

Figure 3

Polarization of Kittel domain. (a) Numerical solution of complete CT Eqs. (6, 7, 8, 9). (b) Numerical solution of simplified Eqs. (14, 15, 16). (c) Interpolation formula (19).