Vortex Ferroelectric Domains, Large-loop Weak Ferromagnetic Domains, and Their Decoupling in Hexagonal (Lu, Sc)FeO3

The direct domain coupling of spontaneous ferroelectric polarization and net magnetic moment can result in giant magnetoelectric (ME) coupling, which is essential to achieve mutual control and practical applications of multiferroics. Recently, the possible bulk domain coupling, the mutual control of ferroelectricity (FE) and weak ferromagnetism (WFM) have been theoretically predicted in hexagonal LuFeO3. Here, we report the first successful growth of highly-cleavable Sc-stabilized hexagonal Lu0.6Sc0.4FeO3 (h-LSFO) single crystals, as well as the first visualization of their intrinsic cloverleaf pattern of vortex FE domains and large-loop WFM domains. The vortex FE domains are on the order of 0.1-1 {\mu}m in size. On the other hand, the loop WFM domains are ~100 {\mu}m in size, and there exists no interlocking of FE and WFM domain walls. These strongly manifest the decoupling between FE and WFM in h-LSFO. The domain decoupling can be explained as the consequence of the structure-mediated coupling between polarization and dominant in-plane antiferromagnetic spins according to the theoretical prediction, which reveals intriguing interplays between FE, WFM, and antiferromagnetic orders in h-LSFO. Our results also indicate that the magnetic topological charge tends to be identical to the structural topological charge. This could provide new insights into the induction of direct coupling between magnetism and ferroelectricity mediated by structural distortions, which will be useful for the future applications of multiferroics.


Introduction
Multiferroic materials, in which two or multiple ferroic orders coexist, have drawn a great deal of attentions due to their fundamental importance and potentials for the next generation devices [1][2][3] . However, in most multiferroics, the magnetoelectric (ME) coupling strength is quite weak, especially in linear-ME materials 4 , which limits their practical applications. Therefore, multiferroics with the direct domain coupling between spontaneous magnetization (M) and polarization (P) are highly sought after. The direct domain coupling can result in giant ME coupling, enabling the mutual control in the sense that flipping one of M or P can induce the flipping of the other. This direct domain coupling effect has been partially achieved, for example, at the hetero-interfaces of bilayer films. M in one film layer can be flipped by flipping P in the other film layer by electrical fields, however not the other way around 5 . The similar situation has also been discovered in single-phase Dy 0.7 Tb 0. 3  domain coupling between its ferroelectricity (FE) and weak ferromagnetism (WFM) 10 . Thus, it is of great importance to study experimentally the possibility of a direct domain coupling effect in h-LuFeO 3 , which is energy efficient and highly desirable for future applications.
Meanwhile, the recent ME coupling study on h-LuFeO 3 /LuFe 2 O 4 superlattices 11 demonstrates the capability of electrical field control of magnetism near the room temperature, which indicates h-LuFeO 3 and its related compounds are promising for future applications. However, with little knowledge of the intrinsic ME coupling of h-LuFeO 3 itself, a full understanding of the ME coupling in superlattices and other related materials is unrealistic.
For the possible giant ME coupling, h-LuFeO 3 systems in the P6 3 cm polar structure ( Figure   1a) has attracted a significant research attentions, since this metastable hexagonal phase of its orthorhombic bulk form was reported to be stabilized in thin film form by epitaxial strain 12,13 or in bulk by Sc or Mn doping [14][15][16]

Results
Despite all of these extensive investigations, the intrinsic FE domain structure in h-LuFeO 3 is still unexplored, and the direct experimental study of the ME coupling in h-LuFeO 3 is absent.
One of the main reasons is the instability of the h-LuFeO 3 phase at ambient synthesis conditions, which makes it challenging to synthesize bulk single crystals. Here, we report a successful growth of high-quality highly-cleavable h-Lu  Although most of our samples have FE domains with +P and -P domains in 50/50 ratio, 6 6 known as type-I domains, some cleaved samples closed to the surface of crystals do show type-II narrow domains ( Figure S2) due to the self-poling effect during the annealing process, which is also commonly observed in h-RMnO 3 compounds 24 .

A. Ferroelectricity
We plotted the density of topological defects (vortices and antivortices) in PFM images vs.
the cooling rate ( Figure 2e). The h-Lu 0.6 Sc 0.4 FeO 3 tends to show a much higher density of topological defects, compared with h-RMnO 3 (e.g. ErMnO 3 and TmMnO 3 , data obtained from ref. 25), even though it has a higher T C than, e.g., ErMnO 3 (T C ≈1129 °C ). Note that a higher-

B. Magnetism
The magnetic susceptibility of LSFO2 in magnetic fields along and perpendicular to the c axis should remain unchanged, we could confidently draw the conclusion that they are decoupled.

C. ME coupling
There are two and only two possible types of coupling among ferroelectric polarization, structural antiphases, and A 2 -type antiferromagnetic spins that have been theoretically  Figure   S6). Moreover, the similar magnetic susceptibility and PE loop can also be reproduced in different crystal pieces ( Figure S7) that are cleaved from the same batch, which is an indication of uniform ferroelectric and magnetic properties within the batch. Figure 4d illustrates the zoom-in view of the dotted area in Figure 4c, and the 3D spin configurations of trimerized Fe 3+ are drawn for each domain. Another evidence of this domain decoupling between FE and WFM is the absence of magnetoelectric current (or P switching) when M c is switched by magnetic fields (see Figure S8 and Supplementary Information note 3 for details).

Discussion
Our

Crystal growth
High-quality h-Lu 0.6 Sc 0.4 FeO 3 single crystals were grown using a floating zone method under

Data availability
All relevant data are available from the authors upon request.  The image of the whole crystal before cleaving is shown in the inset. Experimentally, it is also true that our h-Lu 0.6 Sc 0.4 FeO 3 has a slightly higher weak ferromagnetic transition temperature at 160 K-170 K, comparing to pure h-LuFeO 3   does not respond to the switching of magnetic fields and remains within the noise level. The sequence of the measurement was in the order of black, red, and green arrows.

Note 3: Estimation of the sensitivity in current measurement with magnetic fields.
We could estimate the minimum induced current (I min ) if a 180° polarization flipping happens when sweeping the field by having the following experimental parameters: electrode contact area (S=0.00021 cm 2 ), sweeping speed of magnetic fields (v=0.02 T/s), the saturated polarization according to PE loops (p=1 μC/cm 2 ). For the worst case, we assume the polarization flips during the whole field sweeping process (from 0 T to 9 T) which takes t= 9 T/ (0.02 T/s) = 450 s. Then I min = 2PS/t = 0.9 pA, which is at least 9 times of our current sensitivity (< 0.1 pA) in the measurement. Therefore, the absence of magnetoelectric current when sweeping magnetic fields is the strong evidence of the decoupling between WFM and FE domains.

Note 4: Structural topological charge and magnetic topological charge
A topological vortex concept is essential to understand the decoupling between FE and WFM domains. Here, structural distortions and spins of the identical sites by unit-cell translations (e.g. sites with darker grey triangle backgrounds in Figure 4a and 4b) need to be considered and compared. As the structural (e.g. oxygen) distortion directions (defined by the angle  in indicates that the magnetic topological charge (n m ) tends to be identical with the structural topological charge (n s ), which leads to the decoupling of FE and WFM domains. Therefore, we propose it is necessary to consider topological properties when ME coupling is studied in h-LuFeO 3 .