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Magnetization-polarization cross-control near room temperature in hexaferrite single crystals


Mutual control of the electricity and magnetism in terms of magnetic (H) and electric (E) fields, the magnetoelectric (ME) effect, offers versatile low power consumption alternatives to current data storage, logic gate, and spintronic devices. Despite its importance, E-field control over magnetization (M) with significant magnitude was observed only at low temperatures. Here we have successfully stabilized a simultaneously ferrimagnetic and ferroelectric phase in a Y-type hexaferrite single crystal up to 450 K, and demonstrated the reversal of large non-volatile M by E field close to room temperature. Manipulation of the magnetic domains by E field is directly visualized at room temperature by using magnetic force microscopy. The present achievement provides an important step towards the application of ME multiferroics.


Multiferroic materials, endowed with both orders of polarization (P) and M, exhibit various intriguing phenomena due to the interplay of magnetic and electric degrees of freedom, such as H-induced P flop1,2, E-control of magnetic helicity3,4, and optical nonreciprocal directional dichroism5,6. The cross-coupling phenomena can greatly expand the functions of materials, and hence the multiferroic materials are anticipated to be applied to technological devices. In particular, nonvolatile, E-driven reversal of M without significant dissipation will lead to magnetic memory devices with ultra-low power-consumption7,8,9,10,11,12,13.

To accomplish this goal, strong coupling between P and M is necessary. In general, depending on the microscopic mechanism of P generation, the strength of the cross-coupling is different: In type-I multiferroics where P emerges potentially at a high temperature, but independently of the magnetic ordering, the coupling between P and M is weak, while in type-II multiferroics where P is induced by magnetic ordering, the coupling between P and M is strong14. Thus far, the E-induced M reversal has been investigated for both type-I and type-II mutiferroics. Heterostructures based on BiFeO3 belonging to the type-I category have been demonstrated to be promising12,13,15,16, while H-induced P reversal is difficult17. In the type-II category, good performance has been reported for hexaferrite materials10,11,18,19,20,21,22 with various structural types, including those at room temperature23,24,25. However, in previous studies23,24, ME effects with symmetric PH and ME field dependence were observed at room temperature, which unfortunately cannot achieve E-field-controlled magnetic memory function, or in another study25 the spontaneous P was found to diminish slightly below room temperature without any ME control. Among the various compounds, the largest M switching (~3 μB per f.u.) by E was obtained in Y-type hexaferrites10,11 at cryogenic temperatures, which is attributed to the simultaneous reversal of the ferroelectric and ferrimagnetic order parameters in a particular multiferroic phase, termed FE3 phase. Moreover, this FE3 phase was found to emerge as a metastable state even at room temperature22,26.

Here we demonstrate that by choosing appropriate chemical composition and performing high-pressure oxygen annealing, the FE3 phase can be partially stabilized up to above room temperature. This enables us to observe reversal of M with considerable magnitude by E field as well as the nearly full reversal of P by H field in a single-component material near room temperature. By using magnetic force microscopy (MFM) technique, magnetic domain switching by E field is visualized and the fundamental insight into the PM clamping is provided, which is expected to promote further advances in the magnetization-polarization cross-control.


Structural and magnetic properties

Figure 1a shows the structural unit cell of the Y-type hexaferrite studied in the present work, Ba0.8Sr1.2Co2Fe12−xAlxO22 with x = 0.9 (BSCFAO), which is composed of Fe3+/Co2+ and Fe3+/Al3+ ions in tetrahedral and octahedral oxygen coordinations, respectively, similarly to the other members of the material family. It has been known21 that the magnetic structure in the hexaferrites is well described by ferrimagnetically ordered spin-blocks with large (\({\mathbf{S}}_i^{\mathrm{L}}\)) and small (\({\mathbf{S}}_i^{\mathrm{S}}\)) net magnetizations alternately stacked along the c-axis. As a result of complex magnetic interactions among the adjacent magnetic blocks, various magnetic structures have been identified20,22. These structures, such as commensurate phases FE3 and FE2′ (ref. 20), alternating longitudinal conical (ALC)27, proper screw (PS), and collinear ferrimagnetic (FiM) phases (schematically illustrated in Fig. 1a, b), are also observed in the present material. The magnetic ground state reached via zero-field cooling was reported to be ALC for a hexaferrite with a similar composition28,29. The FE3 phase is induced by H field applied within the magnetic easy-plane, but preserved as a metastable state even after the field is removed22,26. In the FE3 phase the magnetic moments of the SL and SS blocks form a double fan structure22, lying in the ab plane and a plane containing c-axis, respectively. Spin-driven P emerges within the ab plane and perpendicular to the net M, due to the inverse Dzyaloshinskii−Moriya mechanism19,20.

Fig. 1
figure 1

Structural and magnetic properties. a, b Schematic of the structural and magnetic unit cells of Ba0.8Sr1.2Co2Fe12−xAlxO22 (x = 0.9, BSCFAO), the latter of which is composed of alternately stacked spin-blocks with large (SL) and small (SS) magnetic moments. Alternate longitudinal conical (ALC), proper screw (PS), ferrimagnetic (FiM) orders as well as the multiferroic FE3 and FE2′ phases are illustrated in terms of the spin-blocks. The FE3 phase can be viewed as a double fan structure, where the SL and SS spins are staggered in the ab plane and the plane including c-axis, respectively, and the net magnetization (M) and net polarization (P) are perpendicular to each other and also to the c-axis. Respective magnetic phases are identified by the characteristic magnetic reflections indicated below each of the magnetic structure. c Real-space magnetic force microscopy (MFM) image of a BSCFAO sample with an ac surface after zero-field cooling. The striped and dark regions correspond to the magnetoelectric FE3/FE2′ and the incommensurate ALC/PS phases with typical dimensions of 5–30 μm with high and low contrast of MFM phase signals, respectively, indicating the phase separation. The magnetic domain within the FE3/FE2′ region has 200–300 nm in thickness along the c- axis and 10–20 μm in length along the ab plane. d Zero-field-cooled magnetic phase diagram presented as a chart diagram, showing coexistence of the magnetic phases. Horizontal axis is common for panels df. e Temperature dependence of the field-cooled M in H = 100 Oe for H || c and Hc. f The integrated intensities of selected neutron diffraction peaks representing the different magnetic phases in the zero-field-cooled measurements are plotted against temperature

The magnetic phases in BSCFAO have been investigated by the zero-field-cooled (ZFC) MFM, low-field-cooled magnetization, and neutron diffraction measurements as shown in Fig. 1c–f. The neutron diffraction measurements revealed a complex magnetic phase diagram with several coexisting magnetic orders (Fig. 1d and see Supplementary Notes 3 and 4 and Supplementary Figs. 4 and 5), which was determined by taking also the previous results into account20,22,30. Below TC1 = 450 K, a magnetic peak with a commensurate wavevector of q = 3/2 appears together with the onset of M for Hc, which indicates the coexistence of the FE3 and the collinear FiM phases. At TC2 = 400 K, the magnetic peaks with commensurate q = 3/4 and incommensurate qIC wavenumbers emerge, while M for Hc decreases, indicating that the FiM phase is turned into the PS and FE2′ phases. Finally at TC3 = 300 K, the PS order changes to the ALC phase as M for H || c shows a slight decrease. Real-space MFM image of an ac surface at room temperature indicates the phase separation between the strongly magnetic (large averaged-magnetization hosting) FE3/FE2′ and weakly magnetic (little averaged-magnetization hosting) ALC/PS phases as shown in Fig. 1c (for details see Supplementary Fig. 5). A prominent feature of the present-composition compound, being distinct from the previous report22 on a similar Y-type hexaferrite, is the presence of stable FE3 phase among the ZFC states.

Magnetic state under H applied within the ab plane was investigated by magnetization and neutron-diffraction experiments. In Fig. 2, M and the neutron diffraction intensities corresponding to each of the co-existing phases are separably plotted. Prior to the application of H, three phases coexist in the ZFC initial state in agreement with the temperature-dependent measurements. At 100 K (Fig. 2a), the ALC and FE2′ phases disappear at H = 2 kOe and H = 4 kOe, respectively, while the FE3 phase takes over their places. Once the single-phase state of FE3 is attained, it is fully preserved even when the H field is removed or reversed. On the contrary, at 250 K (Fig. 2b) and 295 K (Fig. 2c), both the ALC and FE2′ phases reappear upon the reversal of the H field. At relatively high temperatures, thermal agitation is large enough to overcome the energy barriers between the competing phases with almost degenerated free energies, while not at low temperatures. It is noted that magnetic anisotropy within the ab plane is negligible at room temperature (Supplementary Fig. 6), and hence the MH curve as well as the diffraction intensity are least affected by the anisotropy.

Fig. 2
figure 2

Magnetization and magnetic phases in magnetic field. Magnetic-field (H) dependence of magnetization and integrated intensity of neutron diffraction peaks for relevant phases at a 100 K, b 250 K and c 295 K. The measurements were started from a zero-field-cooled state, then H field was applied perpendicular to the c-axis. The FE3, FE2′, and ALC/PS phases are represented by the neutron diffraction peaks (0, 0, 9 − q) with q = 3/2, q = 3/4 commensurate, and qIC incommensurate wavenumbers, respectively. In all the panels, red symbols indicate the initial H-increasing process, while blue and green symbols denote field-decreasing and second increasing runs, respectively. The ALC phase can be identified in the MH measurements as exhibiting low M and low magnetic susceptibility (not shown) appearing at the low H region near the origin, while the FE3 phase has high M, showing distinction from the ALC phase. At T = 100 K (a), when magnetic field of 5 kOe is applied, the FE2′ and ALC phases disappear, leaving FE3 the only phase. Once the FE3 phase is stabilized, it is preserved throughout the subsequent reversal processes of the H field. At T = 250 K (b), the FE2′ and ALC phases re-emerge when the H field is reversed. The fraction of the re-appearing phases becomes largest not at zero field, but at ±500 Oe. Finally at T = 295 K (c), ratio of the re-appearing phases becomes larger, but the FE3 phase is partially preserved

Magnetoelectric properties

H-induced P and E-controlled M are shown in Fig. 3. Prior to the measurements, the single-domain ME state was attained by the application of (+E0, +H0) poling fields in a crossed configuration (EH; E, Hc). Below T = 250 K both P and M show antisymmetric dependence on H and E fields, respectively, indicating that the PM coupling is conserved throughout the reversal of the fields. Magnitude of the saturation value of the spin-driven polarization (Psat) is significantly larger than the earlier observations in other Y-type hexaferrites10,19,22,31, while comparable to TbMnO33 and the spin-driven component of BiFeO3 which can be controlled by the field of more than H = 100 kOe 17. Correspondingly, the magnetization change between ±Emax fields, ΔME = 5.5 μB per f.u., at T = 100 K is larger than in any former experiments performed at lower temperatures9,10,11. Even at 250 K, a significant portion of M can be reversed (ΔME = 4.1 μB per f.u.) by the E field. Near room temperature, symmetry of the PH and ME loops begins to change to a symmetric butterfly shape, indicating that the PM clamping is not fully preserved during the reversal. Moreover, PH loops show a secondary hysteresis (indicated with black triangles), which is attributed to the re-emergence and disappearance of the PS and FE2′ phases as shown in Fig. 2c.

Fig. 3
figure 3

Cross-control of polarization and magnetization. Prior to the measurements, a single-domain ME state was prepared by (+E0, +H0) poling fields applied in EH; E, Hc-axis configuration. The PH and ME measurements were performed in the absence of E and H fields, respectively. At low temperatures, both PH and ME curves exhibit antisymmetric shape with respect to the fields, indicating that the P × M is conserved due to the strong PM coupling. Around room temperature, components with symmetric field-dependence are mixed in the PH and ME loops, implying that the P × M is not fully preserved anymore. The region where the ALC and PS phases re-emerge is highlighted by light shading in the PH data at 295 K, while the secondary hysteresis of the PH loops is indicated by black triangles

Importantly, the remanent M of BSCFAO can be switched in a nonvolatile manner between positive and negative values by E field even at 250 K, which is favorable for ME memory and spintronic applications. Changes in the remanent M for the first two ME loops are as large as ΔM1 = 3.9 μB per f.u. and ΔM2 = 3.0 μB per f.u. at 100 K and ΔM1 = 2.5 μB per f.u. and ΔM2 = 1.5 μB per f.u. at 250 K. Correspondingly, the remanent P is also switched between positive and negative values with Prem = 95 μCm−2 at 250 K. As for the retention, PH loop exhibits good characteristics for the repeated reversal processes even at 295 K (see Supplementary Fig. 7). However, ME loops are subject to deterioration at higher temperatures than 250 K. This decrease in the magnitude of the reversible M is attributed to the weakened PM coupling as well as insufficient magnitude of the applicable E field.

Investigation of the PM coupling

To further clarify the behavior of the PM coupling, M and P reversal was investigated simultaneously in pulsed E-field experiments (Fig. 4). Similarly to the quasi-static measurements, the single-domain FE3 state was initially prepared with (+E0, +H0) poling fields (EH; E, Hc). After the removal of the poling fields, triangular-shaped E-field pulse pairs were applied antiparallel, then parallel with respect to the E0 poling field (see Fig. 4a). M was measured before and after the pulses, while P was measured during the same period as the E-field pulses were applied (see Methods). The ΔPE curve of magnetic origin at T = 250 K is displayed in Fig. 4d, where the partial reversal of the ferroelectric P is attained by the pulsed E field.

Fig. 4
figure 4

Magnetization reversal by pulsed electric field. a Schematic of the experimental procedure. Measurements were started from a single-domain ME state achieved by (+E0, +H0) poling in a [EH; E, Hc] configuration. Triangular-shaped E-field pulses were applied to the sample. Magnetization was measured before and after the pulses, while polarization at the same time. Duration (τ) of the pulses was 50 ms and 1 ms for T < 280 K and \(T\geqslant 280\,{\mathrm K}\), respectively. b Schematic illustration of the definitions for the ΔP1 and ΔP2 polarization changes. A minor loop without saturation of P is represented by red line. c Schematic ME loop showing the definitions for M0, ΔM1, ΔM2 and ΔME. A minor loop is illustrated by a green curve, M0 is the initial remanent magnetization after poling, and ΔM1, ΔM2 are the magnetization changes after −E and +E pulses. The ΔME is defined only for quasi-static experiments (Fig. S8) as the magnetization difference between +E0 and −E0 fields. d The polarization change of magnetic origin determined from the pulsed E-field experiments at 250 K after (+E0, +H0) poling. e Changes in the remanent M and P induced by the E-field pulses at 250 K. Blue downward and red upward arrows indicate the sign of the E pulses. Although both M and ΔPE = 0 decrease due to the insufficient E-field strength for complete reversal, their parallel change demonstrates their strong coupling. Temperature dependence of f ΔP1 and ΔP2, and g M0, ΔM1 and ΔM2 switched by the first and second E-field pulses. Up to 260 K, the E-field-induced ΔP and ΔM exhibit similar temperature variation. Upon approaching T = 295 K, temperature dependence of ΔP and ΔM shows clear departure, indicating that the PM coupling starts to get weaker. From a technological viewpoint, the magnetization changes are shown also in emu cm−3 unit

Figure 4e shows simultaneous reversal of the remanent P and M by four pairs of E-field pulses at 250 K. Upon the first negative E-field pulse, both the remanent P and M change from positive to negative, causing the magnetization change ΔM1 = 2.3 μB per f.u. The M is almost completely reversed at this point, and there is only a small change in M for the second negative E-field pulse. For the subsequent two positive E-field pulses, P and M were again reversed from negative to positive, with the change of ΔM2 = 1.8 μB per f.u. Although the magnitudes of both ΔP and ΔM decrease as further pulses are applied, similarly to the quasi-static experiments, their parallel reduction demonstrates the strong PM clamping in this temperature range (T ~ 250 K).

Temperature dependence of the P and M switched by the first negative and positive E-field pulses, as defined as ΔP1, ΔP2, ΔM1, and ΔM2 (Fig. 4b, c) respectively, are shown in Fig. 4f, g. Irrespective of the strength of the PM coupling, P can be reversed by the E field. Magnitude of the reversed ΔP1 and ΔP2 slightly decreases as the temperature is increased, but remains finite at 300 K, since the multiferroic FE3 phase is present in the whole temperature region shown here. The initial value M0 and the switched magnetizations ΔM1 and ΔM2 show similar temperature dependence with the ΔP up to 260 K. In contrast to the ΔP, however, the ΔM exhibits more rapid decrease above 260 K, and almost vanishes at 300 K. Therefore, the E-control over the M is lost due to the weakened PM coupling rather than to the reduced volume fraction of the FE3 phase.

ME switching revealed by MFM at room temperature

Using the real-space MFM imaging, we have investigated E-field-induced motion of the magnetic domain walls (DW) at room temperature (Fig. 5). The measurement was started from an initial state (0th in Fig. 5a), where poling E and H fields were once applied and then removed, and the evolution of the magnetic domain pattern of the same region was followed after several applications of the E field (1st–2nd in Fig. 5a and 1st–4th Supplementary Fig. 10). As displayed in Fig. 5a, magnetic domain pattern clearly shows changes in response to the applied E fields with different sign, which demonstrates that these are composite PM domain walls32. The most typical cases of domain dynamics are observed in regions R1 and R2.

Fig. 5
figure 5

Real-space magnetic force microscopy (MFM) images. The MFM images were taken on the same 10 × 10 μm2 region of a BSCFAO crystal with an ac face (see Supplementary Figs. 3, 9 and 10) at room temperature. Prior to the MFM measurements, the sample was poled to a single-domain ME state using (+E0, +H0) poling fields in a EH; E, Hc configuration. Panel a shows the changes in the magnetic domain pattern caused by two successive applications of the E field with different signs (the initial state is labeled as the 0th). The images include small regions, R1 and R2, where two representative cases of DW motion are observed. Around R1, the negatively magnetized domain (denoted with blue color, MFM phase shift Δφ < 0) expands and shrinks along the c-axis upon the first and second applications of E-field, respectively. On the other hand, around R2, a positively magnetized domain (denoted with red color, Δφ > 0) is pushed into the view area from the upper side along the ab plane. These two cases are further displayed as line profiles of the MFM phase shift (Δφ) data along the b A−A′ and c B−B′ lines. Panels d, e show the schematic illustration of these two cases of domain wall motions for the second E-field switch, respectively

At region R1 in Fig. 5a, the negatively magnetized region expands and shrinks due to the successive applications of E field with alternate sign, which corresponds to DW propagation along the c-axis. Figure 5b shows the MFM signals taken along the A−A′ line, clearly demonstrating the DW motion along c-axis. The process of M switch is schematically illustrated in Fig. 5d, where one of the M domains expands along the c-axis, so that the ME domain with P parallel to E expands. The small magnetic anisotropy within the ab plane (see Supplementary Fig. 6) suggests that in this boundary between the oppositely magnetized regions, local net M is likely to rotate around the c-axis.

The region R2 in Fig. 5a, c exemplifies a different process, where a positively magnetized domain is pushed in the image area from the upper side. This behavior is clearly illustrated in the line profile of Fig. 5c. The process of M switch (shown in Fig. 5e) is similar to the previous case; however, in this case the domains are separated by a DW, where local M appears to form a cycloidal structure. Apart from these successful examples, change in the ratio between the majority and minority magnetic domains is relatively small, pointing to the decreased PM coupling at a relatively high temperature, e.g. room temperature.


By choosing appropriate chemical composition and applying high-pressure oxygen annealing procedure, a multiferroic FE3 phase was stabilized up to 450 K in a single crystal of Y-type hexaferrite Ba0.8Sr1.2Co2Fe12−xAlxO22 with x = 0.9 (BSCFAO). At low temperatures, P and M are tightly clamped and reversed simultaneously, while at high temperatures, the PM coupling becomes weaker, and the domain walls are deconfined although the multiferroic phase still survives. MFM measurements at room temperature demonstrated that the M switching by E field in BSCFAO is realized via the propagation of magnetic domain walls throughout the material. To further improve the E-field-induced M reversal in Y-type hexaferrites, the confinement−deconfinement crossover of the domain walls should be pushed to higher temperature, while the coexisting ALC, PS, and FE2′ phases have to be suppressed.


Single crystal growth and oxygen annealing procedures

Single crystals of Y-type hexaferrite, Ba0.8Sr1.2Co2Fe12−xAlxO22 with x = 0.9, were grown by the laser floating zone (LFZ) technique in 10 atm oxygen atmosphere. First, SrCO3, BaCO3, Co3O4, Fe2O3, and Al2O3 were mixed in stochiometric amount and sintered in air at 1150 °C for 24 h. Then the resulting product was pressed into rods and re-sintered for 14 h in the same conditions. Y-type hexaferrite single crystals from earlier growths were used as seeds for the LFZ growth. The single crystal rods were oriented with a back-scattering Laue camera and cut into discs with the surfaces containing c-axis. To increase the resistivity of the samples for the ME as well as neutron diffraction measurements, the cut pieces were annealed in 10 atm O2 at 1000 °C for 100 h in sealed quartz tubes, by adopting the technique described in ref. 33 (see Supplementary Notes 1 and 2 and Supplementary Figs 13).

Neutron diffraction measurements

Neutron diffraction measurements were carried out at the triple-axis neutron spectrometer (PTAX) in the High Flux Isotope Reactor of Oak Ridge National Laboratory. Sliced and O2-annealed single crystal of BSCFAO (approximately 25 mm3) were placed in a cryomagnet with H applied along the [010] axis, while (h, 0, l) plane was set to be the scattering planes.

PH, ME and PE measurements

For each type of experiments, single crystals with the surface containing c-axis were coated with Au/Pt as electrodes; thus E field was applied in the ab plane, while H field was perpendicular to both the E field and the c-axis (EH; E, Hc). H field dependence of the polarization was measured in a PPMS (Quantum Design) with an electrometer (Keithley 6517A) by monitoring the displacement current as the H field was swept with 100 Oe/s continuously between ±5 kOe for 11–21 cycles, depending on the signal to noise ratio. Current peaks around 0 Oe did not show degradation; therefore, the PH curves were obtained by integrating the current after averaging. Magnetization measurement under E field was carried out in an MPMS-XL (Quantum Design), while the electrometer (Keithley 6517A) was used as a voltage source. The thickness, surface area, and mass of the sample were 70 μm, 1.64 mm2, and 0.71 mg, respectively. Pulsed E-field measurements were performed with a ferroelectric tester (Radiant Inc., Precision Premiere II) equipped with 500 V option. The ferroelectric polarization of magnetic origin was measured by the Positive-Up-Negative-Down (PUND) technique. At low temperatures, triangular-shaped E-field pulses with 7 MV/m in amplitude and 50 ms in duration were applied. Above 280 K, however, the pulse duration was reduced to 1 ms due to the lower resistivity (for further details see Supplementary Notes 58 and Supplementary Figs. 68).

Magnetic force microscopy measurements

MFM measurements were carried out with a commercially available scanning probe microscope (MFP-3D, Asylum Research) using Co-coated cantilever (MFMR-10, Nano World). We used the two-pass technique (NAP mode in our MFM instrument of Asylum Research). Throughout the whole measurements, operation condition related to the cantilever was unchanged, which indicates no deterioration of it. For the analysis the linear flattening procedure was used. For E-field-dependent measurements, samples were poled to a single-domain ME state using +3 MV/m and +4 kOe poling fields in EH; E, Hc configuration in a PPMS (for sample preparation see Supplementary Notes 5 and 9 and Figs. 5, 9 and 10). Static E field (+3 MV/m or −3 MV/m) was applied to manipulate magnetic domains using a Keithley 6517A electrometer, and the MFM images were taken after the E field was switched off. The sign and magnitude of the MFM phase shift, Δφ, roughly correspond to those of the magnetization perpendicular to the plane.

Data availability

The data presented in the current study are available from the corresponding authors on reasonable request.


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This research used resources at the High Flux Isotope Reactor, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory. This research was supported in part by the U.S.−Japan Cooperative Program on Neutron Scattering. Structural unit cell of the Y-type hexaferrite crystal was illustrated using the software VESTA34. The authors are grateful for the technical assistance provided by Maximilian A. Hirschberger.

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V.K., T.N., M.M., K.K., T.A. performed the measurements and analyzed the data; V.K., Y.K., A.K., Y. Tokunaga prepared the single crystal samples; J.T. investigated the O2 annealing; V.K. and F.K. took the MFM images; V.K., Y. Taguchi wrote the manuscript; Y. Taguchi and Y. Tokura conceived the project; all the authors contributed to the discussion of the results.

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Correspondence to V. Kocsis or Y. Taguchi.

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Journal peer review information: Nature Communications thanks Chan-Ho Yang, and other anonymous reviewer(s) for their contribution to the peer review of this work.

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Kocsis, V., Nakajima, T., Matsuda, M. et al. Magnetization-polarization cross-control near room temperature in hexaferrite single crystals. Nat Commun 10, 1247 (2019).

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