Interconversion of multiferroic domains and domain walls

Systems with long-range order like ferromagnetism or ferroelectricity exhibit uniform, yet differently oriented three-dimensional regions called domains that are separated by two-dimensional topological defects termed domain walls. A change of the ordered state across a domain wall can lead to local non-bulk physical properties such as enhanced conductance or the promotion of unusual phases. Although highly desirable, controlled transfer of these properties between the bulk and the spatially confined walls is usually not possible. Here, we demonstrate this crossover by confining multiferroic Dy0.7Tb0.3FeO3 domains into multiferroic domain walls at an identified location within a non-multiferroic environment. This process is fully reversible; an applied magnetic or electric field controls the transformation. Aside from expanding the concept of multiferroic order, such interconversion can be key to addressing antiferromagnetic domain structures and topological singularities.

R ecently, the interests in ferroic materials with magnetic or electric order evolved from the spatially expanded bulk domains towards the spatially confined domain walls. As inherent inhomogeneity, domain walls are a source of specific phenomena that are forbidden in the uniform interior of the corresponding domains 1,2 . Examples are the occurrence of magnetization, polarization, magnetoelectric coupling, (super-) conductivity, memory effects, or a change of melting temperature [3][4][5][6][7][8][9][10][11][12][13][14][15] . Hence, domain walls may be regarded as additional state of a material, virtually a world in its own self and often discussed without reference to the surrounding bulk phase.
It would be very desirable if the dimensional limitation of these phenomena could be overcome. For example, one could consider the confinement of a multifunctional bulk state into domain walls where they could establish a form of rewritable electromagnetic circuits 5,16 . Reversely, the domain walls may seed the recovery of the original bulk state, acting as its memory 10,17 .
A class of materials in which the transition from bulk domains to domain walls could be virtually continuous are compounds with phase transitions, in which the symmetry of the domains and domain walls coincide crosswise on either sides of the phase boundary. In such materials, domain walls can in principle gradually transform into domains, and vice versa, across a first-order phase transition. This concept was discussed in theoretical physics [18][19][20] , and systems showing a behavior consistent with this concept were reported 21,22 . Conclusive observation of a smooth, deterministic transfer of a domain wall into a domain, however, and, in particular, the reversible interconversion between domains and domain walls have not been presented. Notably, the practical consequences of such interconversion for the physical properties and functionalities of materials were never debated.
We demonstrate this interconvertibility and apply it to tune a multiferroic state between three and two dimensions in a controlled way. Specifically, we confine a multiferroic bulk state into a multiferroic domain wall at an identified location within a nonmultiferroic environment. We act on this spatially confined magnetoelectric object with magnetic or electric fields and evidence the presence of a switchable magnetization and polarization in the wall. We furthermore employ the fields to transfer the multiferroic domain wall back into a multiferroic bulk state. We then discuss the general occurrence and benefits of an ordered state with controllable transfer in between spatial expansion and spatial confinement.

Results and discussion
For the reasons given below, we choose the rare-earth orthoferrite Dy 0.7 Tb 0.3 FeO 3 (see elsewhere 23 about sample preparation) as our model system. Multiferroicity occurs at 2.65 K, caused by simultaneous antiferromagnetic ordering of the rare-earth and iron spin systems, coupled by exchange striction 23 . An electric polarization P s = 0.12 μC cm −2 coexists with a Dzyaloshinskii-Moriya-type weak magnetization M s = 0.15 μ B per formula unit, both oriented along the c-axis. Subsequently, at T C ≃ 2.3 K a first-order spin reorientation of the iron sublattice from the multiferroic to an antiferromagnetic phase with M s , P s = 0 occurs. The simplest possible form to describe such a transition in between two magnetic phases M1 and M2 is the phenomenological effective Hamiltonian 19,20 H ¼ Aj∇θj 2 þ K 1 sin 2 θ þ K 2 sin 4 θ: As depicted in Fig. 1, the angle θ distinguishes the domain states on the two sides of the M1 ↔ M2 phase transition. K 1 and K 2 are magnetocrystalline anisotropy parameters. K 1 = 0 and K 1 = −2K 2 set the limits of the phase-coexistence region of M1 and M2. The gradient term with A as exchange constant further determines the order-parameter rotation across the domain walls. In Dy 0.7 Tb 0.3 FeO 3 , M1 and M2 are associated with the multiferroic (weakly ferromagnetic) and non-multiferroic (purely antiferromagnetic) phases, respectively, with θ ±M1 = +π/2, −π/2 and θ ±M2 = 0, π. Figure 1b depicts the evolution according to Eq. (1) of a configuration starting with two well-defined domains outside the coexistence region and separated by a single domain wall 18 . By entering the M1/M2 coexistence region, the domain wall widens and transforms into a domain of the other phase. Conversely, the initial domain shrinks and eventually transforms into a domain wall separating the domains on the opposite side of the coexistence region. Hence, the first-order phase transition between the multiferroic M1 and the non-multiferroic M2 phase in Dy 0.7 Tb 0.3 FeO 3 should in principle allow us to reversibly convert a multiferroic domain into a multiferroic domain wall in a non-multiferroic environment.
We first verify the theoretically predicted transformation of a domain wall into a domain 18 experimentally. In Dy 0.7 Tb 0.3 FeO 3 , we can exploit the magnetization of the multiferroic phase to distinguish the M1 and M2 domain states by spatially resolved real-time imaging experiments, using the magneto-optical Faraday effect as magnetization probe. Furthermore, Dy 0.7 Tb 0.3 FeO 3 exhibits a relatively broad region of phase coexistence. This gives us ample time to image the phase coexistence while tuning the balance between the competing multiferroic and non-multiferroic phases. Figure 1c-e shows sequential Faraday images and the associated order-parameter distribution of a c-oriented Dy 0.7 Tb 0.3 FeO 3 sample cooled across the M1 ↔ M2 transition. We refrain from quantifying K 1 and K 2 since their absolute values are not relevant for our discussion. In Fig. 1c we see a ± M1 domain pair at T > T C . Because of the opposite direction of magnetization and, hence, Faraday rotation, the domains appear as bright and dark regions. Figure 1d, e show the same region about 4 and 8 min after cooling the sample to T ≲ T C . A gray stripe centers at the domain-wall position of Fig. 1c that widens with time. Its zero Faraday rotation identifies it as antiferromagnetic region. Below T C , this is the dominating phase and, starting from the original +M1/−M1 domain wall, we perceive the homogeneous expansion of this wall into an M2 domain engulfing the sample as time progresses. Note that this expansion of the non-multiferroic phase occurs in a deterministic way, starting uniformly from the center of the domain wall of the multiferroic phase. This deterministic behavior is in contrast to the needle-like growth of domains of the emergent phase at domain walls of the original phase 21 or to the nucleation on random defects as in most first-order phase transitions 24 . Defects do not play a role in our case as we experimentally verify in Supplementary Note 3. Furthermore, the deterministic expansion requires an order-to-order transition, whereas order-todisorder transitions into phase coexistence 25 are not covered by Eq. (1) and must have a different origin 18 . Now we employ this deterministic aspect for confining a multiferroic domain into a multiferroic domain wall at an identified location within a non-multiferroic environment. We begin at T > T C with a −M1 domain sandwiched between +M1 domains (Fig. 2a). In a magnetic field we enlarge the +M1 domains (Fig. 2b) until they meet (Fig. 2c). After cooling the sample to T ≲ T C , the antiferromagnetic state emerges from the spatially confined object at the meeting point of the +M1 domains (Fig. 2d) and grows with time until it has filled the entire field of view (Fig. 2e).
As we see in Fig. 3, there are two types of meetings between +M1 domains on a −M1 background. As graphically illustrated in the Supplementary Note 4, they reflect the two types of domain walls a multiferroic domain is expected to exhibit, namely +M1 → −M1 walls with either clockwise or counterclockwise rotation of the order parameter across the wall 26 . When domain walls of the opposite type meet, the respective rotations cancel and the walls annihilate so that the meeting domains coalesce. Alternatively, for a meeting of the same type of domain walls, a 360 ∘ spin rotation occurs as sketched in Fig. 3b and Supplementary Note 4. This topologically protected object prevents the coalescence of the domains and is visible as brightness dip in Fig. 3c. Instead, it can seed a +M2 and a −M2 domain as sketched beneath Fig. 2d, e. Note that a magnetic field three times the value of the coercive field is required to destroy the 360 ∘ wall.
Hence, Fig. 2e shows a +M2 and a −M2 domain; according to Fig. 1a, they are separated by a multiferroic domain wall into which the multiferroic −M1 bulk state of Fig. 2a has been confined. This wall has been placed at the meeting point of the two former +M1 domains in Fig. 2c. For confirmation, we re-heat the sample to T ≳ T C ; Fig. 2f shows that this reconverts the suspected multiferroic domain wall into a multiferroic −M1 domain. Note that even though the optical resolution is not sufficient to reveal the microscopic structure of the domain wall, we still can estimate its width to be about 1 μm through deconvolution of the optical signal as described in Supplementary Note 1. In complementary experiments on the closely related compound DyFeO 3 , we verify the value directly by magnetic force microscopy measurements in Supplementary Note 2.
These measurements allow us to show the clear difference between the domain wall that we probe and the alternative case of a narrow domain.
To describe the wall as multiferroic, we furthermore have to show that we can act on it and affect its magnetization and electric polarization by an applied magnetic or electric field. This is also important for its technological utility. This is done in Fig. 4 which shows the response of a multiferroic domain wall in a nonmultiferroic environment, generated as in Fig. 2e, to static coriented magnetic or electric fields. We see that both fields initiate the transfer of the wall back into a domain of the multiferroic phase, even though temperature-wise the material still favors the non-multiferroic phase. This is only possible because the wall has a magnetization M s and a polarization P s the respective applied field can act on to initiate the transfer. Furthermore, the transfer is energetically beneficial only if M s or P s points in the direction of the applied field. Since the field has triggered the transfer in all our experiments, the wall magnetization and polarization itself must be switchable, and hence set the direction of M s or P s of the expanding domain. Determination of the sign of M s by the direction of the magnetic field is evident from Fig. 4a, b, whereas Fig. 4c shows that the electric field acts on the magnetization via magnetoelectric coupling. (Note that because of the involvement of three order parameters in this coupling 27 , the sign of M s is not determined by the sign of P s (E) but rather by the history of the sample, as comparison of Fig. 4c, d shows.) We thus see that despite the spatial confinement and the non-multiferroic environment, a wall as in Fig. 2e retains the magnetoelectric manipulability of the multiferroic bulk phase.
To conclude, we have experimentally demonstrated the conversion of a multiferroic bulk phase into a multiferroic domain wall at an identified location inside a non-multiferroic environment. The initial multiferroic hallmark properties, namely a magnetization and a polarization that are coupled and switched by an applied field, are retained by the wall. Magnetic or electric fields act on the wall and convert it back into a bulk multiferroic domain with a predetermined direction of magnetization or polarization.
The concept of interconversion can be expanded beyond multiferroics. First, there are plenty of first-order magnetic phase transitions fulfilling the rather basic conditions of Eq. (1) with CaFe 2 O 4 as a very recent example to catch scientific attention 28 . Note that even if the region of phase coexistence may be narrower than in our case, possibly even too narrow for convenient experimental probing during the transition, a system will still be subject to the same dynamics as found in Dy 0.7 Tb 0.3 FeO 3 . Even non-magnetic types of order can display the spatial interconversion. For example, a (static) expansion of domain walls into domains with phase coexistence based on a chemical gradient rather than the first-order nature of a phase transition was observed in the ferroelectric-to-nonferroelectric transition in InMnO 3 29 . Even organic materials may show a transition between bulk phases with neutral and ionic building blocks emerging out of an expanding singularity by progressive charge transfer 30,31 .
Second, our work expands the possibilities for functionalizing domain walls. Unlike in previous examples 14,15 , the magnetoelectric properties are not confined to the domain walls but can be expanded into the bulk when needed. The magnetoelectric properties of the walls may be employed to visualize or even tailor domain patterns in materials whose order is normally difficult to access, such as 180°a ntiferromagnets. Finally, our work provides a rare opportunity of deterministic nucleation in a first-order phase transition since the nucleating phase is seeded by the order and symmetry in domain walls rather than occurring on random defects.

Methods
Single-crystals of Dy 0.7 Tb 0.3 FeO 3 were grown by the floating-zone method under O 2 atmosphere. Using back-reflection X-ray Laue diffractometry, thin c-oriented plates of 2.5 × 2.5 mm 2 were cut, lapped, and polished with a silica slurry down tõ 60-80 μm with optically flat surfaces on both sides. Temperature and magnetic field treatment was done in a liquid-helium-cooled magnetic-field cryostat (Oxford Spectromag 4000-10T). For Faraday imaging, a light-emitting diode with a central wavelength of 660 nm (Thorlabs M660L4) was used along with a monochromatic digital camera (The Imaging Source DMK 22BUC03) for image acquisition. Image processing is limited to adjustments of brightness, contrast and levels and homogeneously applied to the entire image, unless noted. The images of each figure were adjusted equally. For electric field measurements, 3-nm platinum electrodes were deposited on both surfaces of the specimen.
Reporting summary. Further information on research design is available in the Nature Research Reporting Summary linked to this article.

Data availability
The data that support the findings of this study are available from the corresponding author on request.