Electric field driven evolution of topological domain structure in hexagonal manganites

Controlling and manipulating the topological state represents an important topic in condensed matters for both fundamental researches and applications. In this work, we focus on the evolution of a real-space topological domain structure in hexagonal manganites driven by electric field, using the analytical and numerical calculations based on the Ginzburg-Landau theory. It is revealed that the electric field drives a transition of the topological domain structure from the type-I pattern to the type-II one. In particular, it is identified that a high electric field can enforce the two antiphase-plus-ferroelectric ($\mathrm{AP}+\mathrm{FE}$) domain walls with $\mathrm{\Delta }\mathrm{\Phi }=\pi /3$ to approach each other and to merge into one domain wall with $\mathrm{\Delta }\mathrm{\Phi }=2\pi /3$ eventually if the electric field is sufficiently high, where $\mathrm{\Delta }\mathrm{\Phi }$ is the difference in the trimerization phase between two neighboring domains. Our simulations also reveal that the vortex cores of the topological structure can be disabled at a sufficiently high critical electric field by suppressing the structural trimerization therein, beyond which the vortex core region is replaced by a single ferroelectric domain without structural trimerization ($Q=0$). Our results provide a stimulating reference for understanding the manipulation of real-space topological domain structure in hexagonal manganites.

Figure 1

A schematic drawing of real-space topological structure in hexagonal manganites $R\mathrm{Mn}{\mathrm{O}}_3$, projected on the $ab$ plane: (a) type-I pattern and (b) type-II pattern. The structural trimerization leads to the threefold antiphase domain structure (α, β, γ), i.e., the $Z_3$ symmetry. The ferroelectric polarization $P$ has two degenerate moments $P_{\uparrow }$ and $P_{\downarrow }$ oriented respectively along the $\pm c$ axis, with the $Z_2$ symmetry. The vortex-antivortex pairs are properly labeled.

Figure 2

The plotted ground-state solutions of parameters $Q$ (a), $P$ (b), and $F$ (c) as a function of $E$ respectively, as calculated from Eq. (6). Here the gradient energy $F_G$ is not included. The dashed lines are the unstable solutions.

Figure 3

The calculated ferroelectric ($P\text{−}E$) hysteresis loops at different frequencies ω of the ac electric field $E$. The values of ω are inserted numerically.

Figure 4

The simulated three-dimensional domain patterns at different $E$. The magnitude of $P$ is scaled by the color bar. The varying sequence of electric field $E$ is (a) $0.00\to$ (b) $0.16\to$ (c) $0.32\to$ (d) $0.66\to$ (e) 0.00. A small rectangle area covering a domain wall is marked for reference in Fig. 5.

Figure 5

The stationary $P$ contours of the rectangle area as marked in Fig. 4, as obtained upon implication of different electric field $E=0.00$ (a), 0.16 (b), and 0.32 (c). The spatial line profiles of parameters $P$($x$), $Q$($x$), Φ($x$), and $F$($x$) across the domain wall, as marked in (a)–(c) are presented in (d)–(g) respectively. The domain-wall motion speed $v$ ($h$) and energy density difference ($F_{\uparrow }-F_{\downarrow }$) (i) as a function of electric field $E$ respectively.

Figure 6

The stationary $P$ contours across two ferroelectric domain walls, as obtained upon sequential implication of different electric fields $E=0.00$ (a), 0.16 (b), 0.32 (c), 0.66 (d), and 0.00 (e). The spatial line profiles of parameters $P$($x$), $Q$($x$), Φ($x$), and $F$($x$) across the domain walls, as marked in (a)–(e) are presented in (f)–(i) respectively.

Figure 7

The lattice configurations for two types of AP-only domain walls with $\mathrm{\Delta }\mathrm{\Phi }=2\pi /3$. The atomic structures of type-A (a) and type-B (b) domain walls seen along the [001] and [100] axes. The “up” and “down” indicate the directions of the Y ions’ displacements from the paraelectric $P{6}_3/mmc$ to the ferroelectric $P{6}_{3}cm$ structure. The arrow in the in-plane lattice configurations indicates the trimerization phase orientations. The coarse open arrow indicates the polarization which changes its sign from negative value to positive value with increasing electric field.

Figure 8

The spatial contours of parameters $P$, $Q$, Φ, and $F$ over a vortex core region, given different electric fields (a) $E=0$, (b) $E=0.20$, and (c) $E=0.66$, respectively.

Figure 9

The calculated polarization $P$ at the vortex core tip, energy density $F$ at the vortex core tip, and difference ($\mathrm{\Delta }F$) in energy density between the vortex core tip and outer domain, as a function of electric field $E$ respectively.

Figure 10

The spatial contours of parameters $P$, $Q$, and $F$ over a vortex core region, given different electric fields (a) $E=1.48$ after cycling for $6\times {10}^5\mathrm{ISs}$, (b) $E=1.49$ after cycling for $5\times {10}^4\mathrm{ISs}$, and (c) $E=1.49$ after cycling for $8\times {10}^4\mathrm{ISs}$, respectively.

Figure 11

The side view of the paraelectric (PE, $P{6}_3/mmc$) (a) and ferroelectric (FE, $P{6}_{3}cm$) (b) crystal structures of hexagonal $\mathrm{YMn}{\mathrm{O}}_3$. The “up” and “down” indicate the directions of the Y ions's displacements from the PE to the FE structure. Distinct $\mathrm{Y}\text{−}{\mathrm{O}}_{\mathrm{ep}}$ bonds resulting from the inhomogeneous Y ions distortion are highlighted with green and red lines, respectively.

Figure 12

The plotted ground-state solutions of parameters $Q$ (a), $P$ (b), and $F$ (c) as a function of $E$ respectively when the fourth-order term in $P$ is added to the Landau energy. Along the violet arrow, the curves successively correspond to the results at $b_P=0.0,0.8,1.6,2.4$, and 10. The red arrow marks the shifting of the intersection point of the two $F$($E$) curves for the $P_{\uparrow }$ state and the $Q=0$ state as $b_P$ increases.