SUPPLEMENTARY INFORMATION

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1 SULEENTARY INFORATION doi: 1.138/nmat2469 Supplementary Information for Composite domain walls in a multiferroic perovskite ferrite Yusuke Tokunaga*, Nobuo Furukawa, Hideaki Sakai, Yasujiro Taguchi, Taka-hisa Arima & Yoshinori Tokura *To whom correspondence should be addressed. y-tokunaga@riken.jp Domain wall structures in O 3 Schematic of composite domain wall structures are depicted in Figs. S1a-c. While the actual domain walls should have finite thickness and the spin moments gradually rotate over this thickness, here we simplify such a feature and depict the domain wall as a kink so that one can grasp the main points of the change of spin configuration and electric polarization at the domain wall. As described in the main text, there exist three types of domain walls; the first one is ferromagnetic (df) domain wall across which the macroscopic quantity changes (Fig. S1a), the second one is ferroelectric (pf) domain wall across which the macroscopic quantity changes (Fig. S1b), and the third one is multiferroic (pd) domain walls across which and changes simultaneously. Domain wall motion in O 3 As an example, we schematically show the process of magnetic-field induced change of electric polarization (Figs. S2a-e). As an initial state, It is assumed that ferromagnetic (df) and ferroelectric (pf) domain walls exist (Fig. S2a). When the magnetic field is applied, the ferromagnetic domain wall will propagate so as to align ferromagnetic moment along the field direction (Figs. S2a and b). When the ferromagnetic domain wall collides with the ferroelectric domain wall, then two domain walls will merge to form the multiferroic (df) domain wall (Fig. S2c). Then, the multiferroic domain wall will propagate further via the coupling between weak ferromagnetic moment and magnetic field (Figs. S2c and d). However, if the pinning centre like the crystallographic defect exists, multiferroic (df) domain wall will be decomposed again into ferroelectric (pf) and ferromagnetic (df) domain walls and then only ferromagnetic domain wall will propagate again (Figs. S2d and f). Such a process gives rise to the change of the net electric polarization driven by the magnetic field. Electric field induced change of magnetization will also be explained in a similar manner, as depicted in Figs. S2f-j. As an intitial state, we also assume the existence of ferromagnetic (df) nature materials 1

2 supplementary information doi: 1.138/nmat2469 and ferroelectric (pf) domain walls (Fig. S2f). When the electric field is applied, the ferroelectric domain wall will propagate so as to align ferroelectric polarization along the electric field direction (Figs. S2f and g). When the ferroelectric domain wall collides with the ferromagnetic domain wall, then two domain walls will merged into a multiferroic (df) domain wall (Fig. S2g). Then, multiferroic domain wall will propagate further (Figs. S2h and i). However, if the pinning centre for the spins exists, again multiferroic (df) domain wall will be decomposed into the ferroelectric (pf) and the ferromagnetic (df) domain walls and then only the ferroelectric domain wall will propagate again (Figs. S2i and j). Such a process will give rise to the change of the net magnetization induced by the electric field. 2 nature ATERIALS

3 doi: 1.138/nmat2469 supplementary information a FDW(df) b (pf) c FDW(pd) Figure S1: Schema of composite domain walls in 3. a, ferromagnetic (df) domain wall, ferroelectric (pf) domain wall, and c, multiferroic (pd) domain wall. Dotted lines and circles denote the original lattice and of ions at the paraelectric state without ordering of spins. nature materials 3

4 supplementary information doi: 1.138/nmat2469 hase of order parameter FDW H-induced a E-induced b FDW g FDW c +++ FDW --+ h +++ FDW --+ FDW FDW d inning center i inning center hase of order parameter FDW f FDW FDW e j Figure S2: Schema of domain wall motion in 3. a-e, Schema of magnetic field induced change of electric polarization. In each panel, the phase of order parameter is plotted as a function of spatial. anels a-e represent the temporal evolution of the domain wall motion. Note that the of is different between the initial (a) and final (e) state of this magnetic-field driven process. f-, Schema of electric field induced change of magnetization. In each 4 nature ATERIALS

5 doi: 1.138/nmat2469 supplementary information panel, the phase of order parameter is plotted as a function of spatial. anels f-j represent the temporal evolution of the domain wall motion. Note that the of FDW is different between the initial (f) and final (j) state of this electric-field driven process. nature materials 5