Direct Observation of the Proliferation of Ferroelectric Loop Domains and Vortex-Antivortex Pairs

We discovered stripe patterns of trimerization-ferroelectric domains in hexagonal ${\mathrm{REMnO}}_3$ ($\mathrm{RE}=\mathrm{Ho},\cdots ,\mathrm{Lu}$) crystals (grown below ferroelectric transition temperatures ($T_c$), reaching up to $1435°\mathrm{C}$), in contrast with the vortex patterns in ${\mathrm{YMnO}}_3$. These stripe patterns roughen with the appearance of numerous loop domains through thermal annealing just below $T_c$, but the stripe domain patterns turn to vortex-antivortex domain patterns through a freezing process when crystals cross $T_c$ even though the phase transition appears to not be Kosterlitz-Thouless-type. The experimental systematics are compared with the results of our six-state clock model simulation and also the Kibble-Zurek mechanism for trapped topological defects.

Figure 1

Two distinct domain patterns of ${\mathrm{REMnO}}_3$ crystals: stripe vs vortex patterns. (a) Optical microscope image of a chemically etched ${\mathrm{ErMnO}}_3$ crystal surface. The inset shows the atomic force microscope (AFM) image of a stripe domain pattern after chemical etching. These domain patterns for a long-range ordered phase are also observed in ${\mathrm{HoMnO}}_3$, ${\mathrm{TmMnO}}_3$, ${\mathrm{YbMnO}}_3$, ${\mathrm{LuMnO}}_3$ crystals, as shown in Fig. S1 in the Supplemental Material [17]. (b) Optical microscope image of a chemically etched ${\mathrm{YMnO}}_3$ crystal surface. The inset shows the AFM image of a ${\mathrm{YMnO}}_3$ surface after chemical etching, showing a vortex-antivortex domain pattern. Vortex and antivortex are distinguished by the arrangement of trimerization antiphase and ferroelectric domains with the opposite sense of rotation around a core. Note that the entire surfaces of all as-grown crystals ${\mathrm{REMnO}}_3$ ($\mathrm{RE}=\mathrm{Ho}$, Er, Tm, Yb, Lu) exhibit no hint of the presence of any vortices or antivortices.

Figure 2

The thermal evolution of stripe domain patterns in ${\mathrm{ErMnO}}_3$. (a),(b),(e), and (f) The AFM images of chemically etched ${\mathrm{ErMnO}}_3$ crystals annealed at $1120°\mathrm{C}$, $1140°\mathrm{C}$, $1170°\mathrm{C}$, and $1200°\mathrm{C}$, respectively. A stripe domain pattern with thermal roughening are evident in (b), but (e) and (f) display vortex-antivortex patterns. The inset displays $T_c$’s of ${\mathrm{REMnO}}_3$ estimated from the formation temperature of vortex-antivortex domain patterns. $T_c$ of ${\mathrm{YMnO}}_3$ is from Ref. 18. (c) and (d) TEM image of loop domains of ${\mathrm{ErMnO}}_3$ quenched from $1140°\mathrm{C}$ and a possible corresponding schematic, respectively. The loop domains observed in the TEM image are assigned with three antiphase domains ($\alpha +$, $\beta -$, $\gamma +$).

Figure 3

The evolution of vortex-antivortex domain patterns with varying cooling rate. The AFM images of chemically etched ${\mathrm{ErMnO}}_3$ crystals: (a) cooled from $1220°\mathrm{C}$ to $890°\mathrm{C}$ with a rate of $0.5°\mathrm{C}/\mathrm{h}$, followed by furnace cooling, (b) cooled from $1200°\mathrm{C}$ to room temperature with a rate of $5°\mathrm{C}/\mathrm{h}$, (c) from $1200°\mathrm{C}$ to $677°\mathrm{C}$ with a rate of $5°\mathrm{C}/\mathrm{h}$, followed by quenching, and (d) from $1200°\mathrm{C}$ to room temperature with a rate of $300°\mathrm{C}/\mathrm{h}$.

Figure 4

The pinning of vortex-antivortex domains. (a) and (b) TEM images of vortex-antivortex domain patterns of ${\mathrm{ErMnO}}_3$ at room temperature and $730°\mathrm{C}$, respectively. (c) The cooling rate dependence of the average distance of vortex-antivortex pairs, exhibiting a power law dependence with the power of 0.23. [The open triangle is from Fig. 3c.] (d) The optical microscope and AFM (inset) images of a chemically etched ${\mathrm{ErMnO}}_3$ crystal surface, suggesting the strong pinning of vortex-antivortex domains by surface defects.