Unfolding of Vortices into Topological Stripes in a Multiferroic Material

Multiferroic hexagonal $R{\mathrm{MnO}}_3$ ($R=\text{rare}\text{earths}$) crystals exhibit dense networks of vortex lines at which six domain walls merge. While the domain walls can be readily moved with an applied electric field, the vortex cores so far have been impossible to control. Our experiments demonstrate that shear strain induces a Magnus-type force pulling vortices and antivortices in opposite directions and unfolding them into a topological stripe domain state. We discuss the analogy between this effect and the current-driven dynamics of vortices in superconductors and superfluids.

Figure 1

Effect of the crystal shape for annealing under strain (triangle vs rectangle). (a) and (b) are OM images of chemically etched ${\mathrm{ErMnO}}_3$ crystals, which indicate two distinct domain patterns: stripes and vortices. (c) Collaged optical microscope image of EMO-$A$ after chemical etching. (d) Enlarged OM image of the green-box area in Fig. 1 showing the vortex-to-stripe transformation. (e) OM images of triangular EMO-$B$. (f) Enlarged image of the green-boxed area in Fig. 1 shows only vortices. (g) Schematics of in-plane strain on the top surface, average (or middle region), and bottom surface. Blue, red, and black colors indicate compressive, tensile, and no strain, respectively.

Figure 2

(a) Large-range AFM image of the area in Fig. 1 showing vortex-to-stripe transformation. (b) Fine-scan AFM image of the green-boxed area in Fig. 3. (c) Expanded AFM image of the purple-box area in Fig. 3 with the self-consistently assigned trimerization and ferroelectric phases of all vortex and stripe domains. An excess of vortices (red circles, the clockwise ($\alpha -$, $\beta +$, $\gamma -$, $\alpha +$, $\beta -$, $\gamma +$) sequence) at the vortex-to-stripe transformation boundary results from predominant expulsion of antivortices [dashed blue circles, the anticlockwise ($\alpha -$, $\beta +$, $\gamma -$, $\alpha +$, $\beta -$, $\gamma +$) sequence]. All domain walls in stripes exhibit the same chirality corresponding to the sequence ($\alpha -$, $\beta +$, $\gamma -$, $\alpha +$, $\beta -$, $\gamma +$) from right to left in the whole stripe.

Figure 3

(a) A schematic of Fig. 2. (b) A topologically deformed cartoon of the boundary between the vortex-antivortex domains (upper part) and the topological stripe domains (lower part). Vortices (antivortices) are marked with red (blue) circles. (c) A vortex-antivortex pair under the shear $u_{xx}-u_{yy}$ strain inducing vertical forces (light-blue vertical arrows) on the vortex and antivortex. Purple-curved arrows indicate the direction of the phase gradient. (d) Vertical stripe domains which are about to form, as the Magnus-type force pulls the vortex and antivortex apart.

Figure 4

STEM-HAADF analysis of different types of domain walls. (a) SEM image of EMO-$D$ $ab$-plane surface after chemical etching. Mountain plateaus (valleys) are ferroelectric domains with downward (upward) polarization. The green box indicates the area cut for STEM-HAADF experiments using FIB. (b) Dark-field TEM image of a STEM-HAADF specimen with the ($00\overline{4}$) reflection near the [100] zone axis. Dark lines are narrow ferroelectric domains with upward polarization. Also shown are crystallographic axes and an enlarged image of the red box area. (c) False-colored STEM-HAADF images of 1–4 domain walls in the inset of Fig. 4. DW1 and DW3 domain walls are of $C$ type (with distorted Er ions at walls), and DW2 and DW4 domain walls are of $A$ type (with undistorted Er ions at walls). Each STEM-HAADF image is followed by schematics of Er-ion columns. $C$-type domain walls are slightly slanted (by a unit cell), so near the wall centers two upward and downward Er columns overlap, giving rise to elongated Er column images.