Emergent domain topology in the multiferroic hexagonal manganites

Abstract

Emergent topological phenomena in multiferroic materials arise from the intricate coupling between structural, electric, and magnetic order parameters. Hexagonal manganites provide a paradigmatic platform for such studies. These compounds exhibit a strongly coupled distortive-improper ferroelectric order, arising from trimerizing lattice distortions, and a 120° noncollinear antiferromagnetic spin structure. While their two-dimensional domain topology has been extensively studied, the full three-dimensional multiferroic domain architecture has remained largely unexplored, mainly due to the experimental challenges of probing bulk structures beyond surfaces. Here, we employ a Landau free-energy framework combined with large-scale phase-field simulations to reveal the intricate three-dimensional multiferroic domain network of hexagonal manganites. We demonstrate that the coupling between the structural and antiferromagnetic order parameters gives rise to a rich variety of three-dimensional topological features. In particular, these features give rise to an attraction between different types of domain walls. Moreover, we identify bifurcations of vortex-like lines at domain-wall intersections, a phenomenon that can exist only in three dimensions and fundamentally alters the topology of the domain network. Our results provide a comprehensive theoretical basis for understanding three-dimensional domain interactions in multiferroics and highlight the essential role of dimensionality in coupling improper ferroelectricity, magnetism, and topology in hexagonal manganites.

Figures (7)

• Figure 1: $\mid$Visualization of a vortex line. A vortex line corresponds to a 1D line in 3D space. Following a path that leads around this line, the angular coordinate or complex phase of the order parameter performs a full 360° rotation or multiples thereof.
• Figure 2: $\mid$Structural and magnetic order of hexagonal manganites.a Structural order of hexagonal manganites, corresponding to the zone-boundary mode $K_3$. The mode consists of a MnO$_5$ bipyramid tilt and an up-down order of R-atoms. b Visualization of the bipyramid tilt and the shift of the R atoms. Where all three bipyramids tilt towards a common center, the R-atoms shift away from the bipyramids, while the other R-atoms move in the opposite direction. c The structural order parameter components $Q$ and $\Phi$ correspond to the amplitude and azimuthal angle of the bipyramid tilt. d Top view of the structural order with arrows illustrating bipyramid tilts with respect to the associated Mn$^{3+}$ ions. e Top view of the antiferromagnetic order of the system on the example of ErMnO$_3$, with arrows corresponding to the spins of the Mn$^{3+}$ ions. In d and e, the reference Mn$^{3+}$ ion is ion marked with a green circle. Different shades of Mn$^{3+}$ ions and arrows correspond to atoms in adjacent Mn-O layers. f Definition of $\Phi$ and $\Psi$ in reference to d and e, respectively. g 2D $z$-cut of a structural domain pattern, with 60° structural domain walls in $\Phi$ and six $\Phi$-domains meeting at a vortex. The secondary ferroelectric polarization follows an alternating pattern, depicted with $\odot$ and $\otimes$ symbols.
• Figure 3: $\mid$Illustration of structural and magnetic domain patterns.a Schematic domain pattern of the structural order with 60° domain walls in $\Phi$. b Illustrations of structural and magnetic six-fold vortex lines. In minimization of Eq. \ref{['eq:Condition']}, they show a one-to-one correspondence. c Ising-like magnetostructural $\Phi - \Psi$ domain pattern. The two domain states with $\Phi - \Psi = \pm 90$° are separated by a magnetostructural 180° domain wall. d Magnetic domain pattern. Note that the correlation between the domain pattern in a and the $\Phi - \Psi$ domain pattern in c results in three types of magnetic domain walls with a change of $\Psi$ by 60°, 180° or $-$120°. e Legend of magnetic and structural domain states. Pairs of $\Phi$ and $\Psi$ in the same row fulfill the condition $\Phi - \Psi = \pm 90$°.
• Figure 4: $\mid$Pseudo-vortex lines and bifurcation.a Four-fold pseudo-vortex line (blue dashed line, marked with '4') as an intersection of a 180° magnetic domain wall (purple) and a 60° magnetic domain wall (turquoise). b Three-fold pseudo-vortex line (red dashed line, marked with '3') as a junction of a 180° and a 60° magnetic domain wall, forming a $-$120° magnetic domain wall (ocher) in the process. c Bifurcation of two three-fold pseudo-vortex lines merging at the point marked 'B' to form a four-fold pseudo-vortex line.
• Figure 5: $\mid$Three-dimensional visualization of the simulated magnetic domain pattern. Both figures visualize the same data. a Overview of the simulated data, with 2D cuts perpendicular to the $y$- and $z$-directions, and 3D visualization elsewhere. b Visualization of six-fold vortex lines (marked with '6') in the magnetic order along with an exemplary $z$-cut. Vortex lines always form loops which in this visualization are interrupted at the periodic boundaries of the system.