Dimensionality-Induced Change in Topological Order in Multiferroic Oxide Superlattices

We construct ferroelectric ${{(\mathrm{LuFeO}}_3)}_m/{(\mathrm{LuFe}}_2{\mathrm{O}}_4)$ superlattices with varying index $m$ to study the effect of confinement on topological defects. We observe a thickness-dependent transition from neutral to charged domain walls and the emergence of fractional vortices. In thin ${\mathrm{LuFeO}}_3$ layers, the volume fraction of domain walls grows, lowering the symmetry from $P{6}_{3}cm$ to $P3c1$ before reaching the nonpolar $P{6}_3/mmc$ state, analogous to the group-subgroup sequence observed at the high-temperature ferroelectric to paraelectric transition. Our study shows how dimensional confinement stabilizes textures beyond those in bulk ferroelectric systems.

Figure 1

(a) HAADF STEM image of an $m=9$ ${{(\mathrm{LuFeO}}_3)}_m{/(\mathrm{LuFe}}_2{\mathrm{O}}_4)$ superlattice, where bright (dark) contrast indicates lutetium (iron) atomic columns. The lower panels show a cartoon of the crystal structure. A sinusoidal curve fits the lutetium displacements, giving the amplitude $Q$. The phase $\mathrm{\Phi }$, reflecting the in-plane apical oxygen rotations, can also be retrieved from the phase of the atomic displacements, as previously done in Ref. [12]. (b),(c) Polarization color overlay, with cyan indicating polarization down and red indicating polarization up, for (b) $m=1$, 2, and 4, and (c) $m=9$, where head-to-head walls fall in the middle of the ${\mathrm{LuFeO}}_3$ block. (d) EELS spectra of the $\mathrm{Fe}\text{−}{L}_{2,3}$ edge of the double iron layers and single iron layers (the ${\mathrm{LuFeO}}_3$ block) in an $m=7$ superlattice compared to that of ${\mathrm{LuFe}}_2{\mathrm{O}}_4$. (e) The fractional occurrence of the tail-to-tail configuration across the double iron layer, per in-plane distance along the double iron layer. (f) Fractional occurrence of neutral walls (angles $>60°$ from horizontal), charged walls (angles $<30°$), and diagonal walls (30°60°) within the ${\mathrm{LuFeO}}_3$ block, per in-plane distance along the block.

Figure 2

$\mathrm{\Phi }$ overlay of STEM images showing the domain structure in ${{(\mathrm{LuFeO}}_3)}_m{/(\mathrm{LuFe}}_2{\mathrm{O}}_4)$ superlattices for large $m$ ($m=7$, 9). (a) Cartoon of the projection of lutetium positions for different $\mathrm{\Phi }$, with the color corresponding to the color overlays. (b) A stable charged domain wall, with black arrows showing polarization directions. (c),(d) At the end of a domain in plane, the phase rotates clockwise or anticlockwise (white arrows), forming half vortices and antivortices to maintain the charged domain wall configuration. (e) For short in-plane domains, the phase can first wrap on one neighboring double iron layer and then the other. (f) Rarely, a vortex with five domains can be observed.

Figure 3

The order parameter $\mathrm{\Phi }$ as a color overlay on the HAADF STEM images for small $m$ ($m=4$, 2) where the neutral domain walls are observed with more equal occurrence. (a) for $m=4$, we see a fairly equal coexistence of charged ferroelectric domain walls (left), and $\mathrm{neutral}+\mathrm{diagonal}$ domain walls forming stripe patterns (right). We also see irregular numbers of domains coming together to a point in the middle image. (b) In the $m=2$ case, we see smaller areas of charged domain walls (left) and more neutral walls forming in a stripe pattern (right). We also observe unusual numbers of domains intersecting (middle).

Figure 4

(a) Histograms of the order parameter for different $m$ in ${{(\mathrm{LuFeO}}_3)}_m{/(\mathrm{LuFe}}_2{\mathrm{O}}_4)$ superlattices, with the logarithm of the occurrences plotted. $m=1$ shows a parabolic well, consistent with a paraelectric order parameter distribution. For $m=2$ and 3, $Q$ is slightly larger and the values of $\mathrm{\Phi }$ are widely spread, indicating a combination of ferroelectric domain and intermediate $\mathrm{\Phi }$ states. For $m=4$, the six ferroelectric domains are visible while maintaining a large fraction of intermediate states. For $m=7$ and 9, the six ferroelectric domains comprise the majority of the observed states, and the $Q$ value is higher. (b) Relative amount of states in ferroelectric (FE) wells vs in the intermediate states (between wells) vs the paraelectric state. The difference between the states in the well and in the intermediate states tracks the relative amount of $Z_6$ symmetry.