Self-consistent theory of nanodomain formation on nonpolar surfaces of ferroelectrics

We propose a self-consistent theoretical approach capable of describing the features of the anisotropic nanodomain formation induced by a strongly inhomogeneous electric field of a charged scanning probe microscopy tip on nonpolar cuts of ferroelectrics. We obtained that a threshold field, previously regarded as an isotropic parameter, is an anisotropic function that is specified from the polar properties and lattice pinning anisotropy of a given ferroelectric in a self-consistent way. The proposed method for the calculation of the anisotropic threshold field is not material specific, thus the field should be anisotropic in all ferroelectrics with the spontaneous polarization anisotropy along the main crystallographic directions. The most evident examples are uniaxial ferroelectrics, layered ferroelectric perovskites, and low-symmetry incommensurate ferroelectrics. Obtained results quantitatively describe the differences at several times in the nanodomain length experimentally observed on $X$ and $Y$ cuts of $\mathrm{LiNb}{\mathrm{O}}_3$ and can give insight into the anisotropic dynamics of nanoscale polarization reversal in strongly inhomogeneous electric fields.

Figure 1

(a) Sketch of the domain shape on the $XZ$ plane $\left(Y=0\right)$ induced by an antiferromagnetic probe on CLN nonpolar cuts. (b) and (c) Experimentally observed domain shape from Alikin et al. [37] on (b) $Y$ and (c) $X$ cuts of 20-μm-thick CLN.

Figure 2

Schematics of the tip-induced polarization switching on the nonpolar $Y$ cut of a uniaxial ferroelectric.

Figure 3

Atomic structure of (a) $\mathrm{LiNb}{\mathrm{O}}_3 Z$ cut ($XY$ plane), (b) $X$ cut ($ZY$ plane), and (c) $Y$ cut ($ZX$ plane). Big blue balls are Nb atoms, smaller green balls are Li atoms, and the smallest red balls are O atoms.

Figure 4

(a) Threshold fields dependence on the domain-wall half-width $w$ calculated within the Suzuki-Ishibashi model for $\mathrm{LiNb}{\mathrm{O}}_3$ parameters $\alpha =-1.95\times {10}^9\mathrm{m}/\mathrm{F}$ and $P_S=0.735\mathrm{C}/{\mathrm{m}}^2$. (b) Activation fields dependence on the voltage $U$ applied to the probe calculated within the Burtsev-Chervonobrodov model for the parameters $R=100\mathrm{nm},{\sigma }_{min}=0.160\mathrm{J}/{\mathrm{m}}^2$, and $\delta \sigma =0.150\mathrm{J}/{\mathrm{m}}^2$ [33]. (c) The threshold fields' ratio versus the domain-wall half-width $w$. (d) Activation fields' ratio versus applied voltage $U$.

Figure 5

Domain shape (top view) calculated for (a) the $Y$ cut and (b) the $X$ cut. Colored domain cross sections with solid borders are calculated in a self-consistent way for different threshold fields $E_{th}=50\mathrm{kV}/\mathrm{mm}$ for the $Y$ cut and $E_{th}=550\mathrm{kV}/\mathrm{mm}$ for the $X$ cut. Domain cross sections shown by dotted ellipselike curves are calculated under the assumptions that the threshold field is the same for the $X$ and $Y$ cuts, the depolarization field is the same as created by the semiellipsoidal domains, and domain walls are infinitely thin.

Figure 6

Temporal dependencies of the (a) domain length and (b) width on the ferroelectric surface calculated for different threshold fields $E_{th}=(21,50,100,200,550)\mathrm{kV}/\mathrm{mm}$. Points correspond to the numerical results simulated in comsol. Solid curves 1–5 correspond to the interpolation functions: (a) Eq. (5) plotted for parameters $C_k,B_k,t_{0k}$, and $t_{ck}$ shown in the plot (c). (b) Equation (6) plotted for parameters $C_k,B_k$, and $t_{ck}$ shown in plot (d). The scales for $t_{ck}$ and $t_{0k}$ are ${10}^2$. (c) and (d) Dependence of the fitting constants $C_k,B_k$, and $t_{ck}$ on the threshold field. As anticipated, the critical times $t_{c1}=0.065,t_{c2}=0.06,t_{c3}=0.05,t_{c4}=0.03$, and $t_{c5}=0.02$ are the same for plots (a) and (b).

Figure 7

(a) Temporal dependence of the domain apex angle θ calculated for different threshold fields $E_{th}=\left(21,50,100,200,550\right)\mathrm{kV}/\mathrm{mm}$. Points correspond to the numerical results simulated in comsol. Solid curves 1–5 are the interpolation functions (7) plotted for parameters $C_1=4$; $C_2=6$; $C_3=8$; $C_4=14$; $C_5=19$; $B_1=30$; $B_2=38$; $B_3=40$; $B_4=42$; $B_5=42$; $t_{01}=0.3$; $t_{02}=0.33$; $t_{03}=0.43$; $t_{04}=0.60$; $t_{05}=0.8$. (b) Dependence of the constants $C_k$ and $B_k$ on the threshold field.

Figure 8

Dependencies of (a) domain length and (b) domain width versus the switching pulse duration on the $X$ and $Y$ cuts of $\mathrm{LiNb}{\mathrm{O}}_3$. Symbols with error bars are experimental data [37] for the $X$ and $Y$ cuts of 20-μm-thick CLN placed in dry nitrogen, and the solid curves are our fitting using the interpolation functions (5) and (6). The function for the domain length is Eq. (5) with parameters $C_X=90,B_X=80,t_{cX}=0.1\mathrm{ms}$, and $t_{0X}=1\mathrm{ms}$ for the $X$ cut and $C_Y=205,B_Y=150,t_{cY}=0.15\mathrm{ms}$, and $t_{0Y}=1\mathrm{ms}$ for the $Y$ cut. The function for the domain width is Eq. (6) with parameters $C_Y=23,B_Y=138$, and $t_{cY}=0.15\mathrm{ms}$ for the $Y$ cut, $C_X=15,B_X=82.5$, and $t_{cX}=0.1\mathrm{ms}$ for the $X$ cut. (c) and (d) The ratio $B/C$ extracted from numerical modeling.