Magnetoelectrical control of nonreciprocal microwave response in a multiferroic helimagnet

The control of physical properties by external fields is essential in many contemporary technologies. For example, conductance can be controlled by a gate electric field in a field effect transistor, which is a main component of integrated circuits. Optical phenomena induced by an electric field such as electroluminescence and electrochromism are useful for display and other technologies. Control of microwave propagation is also important for future wireless communication technology. Microwave properties in solids are dominated mostly by magnetic excitations, which cannot be easily controlled by an electric field. One solution to this problem is to use magnetically induced ferroelectrics (multiferroics). Here we show that microwave nonreciprocity, that is, different refractive indices for microwaves propagating in opposite directions, could be reversed by an external electric field in a multiferroic helimagnet Ba2Mg2Fe12O22. This approach offers an avenue for the electrical control of microwave properties.

M ultiferroics have been investigated extensively since the discovery of magnetically induced ferroelectrics in perovskite manganites 1 . This class of materials exhibits a giant magnetoelectric effect, which is an electric-fieldinduced change in magnetization, and its reciprocal effect, a magnetic-field-induced change in electric polarization. Magnetoelectric coupling is also valid during the course of magnetic oscillation. A spin wave excitation is coupled to an alternating electric field as well as an alternating magnetic field in multiferroics 2,3 . This dynamical magnetoelectric coupling gives rise to the unique nature of electromagnetic waves. The refractive indices of oppositely propagating electromagnetic waves are different from each other, an effect known as nonreciprocal directional dichroism. Nonreciprocal directional dichroism was first observed for the electronic transition of a non-centrosymmetric molecule in the visible region by Rikken et al. 4 Since then, optical nonreciprocity has been extensively investigated in non-centrosymmetric materials [5][6][7][8][9][10][11][12][13][14][15][16][17][18] . Similar nonreciprocity was also realized for magnon excitation in a transverse conical magnetic state [19][20][21] , which is one of the prototypical spin arrangements exhibiting multiferroic properties, as shown in Fig. 1a. This magnetic state has a net magnetization M in one direction. The magnetic moment components perpendicular to M rotate along the conical wave vector q 0 perpendicular to M. According to the spin current mechanism 22 , the rotating components give rise to static electric polarization P. Thus, P and M may oscillate in the spin excitations of the transverse conical state. Miyahara and Furukawa 23 have theoretically shown that the lowest magnon excitation simultaneously induces an alternating P (DP) and an alternating M (DM), which is denoted as toroidal magnon. Therefore, this mode is electrically and magnetically active and shows a strong magnetoelectric effect. The resulting large magnetoelectric coupling induces a term k Á (P Â M) in the refractive index for the electromagnetic wave, indicating a nonreciprocal directional dichroism, which can be controlled by a DC electric field as well as a DC magnetic field. Recently, controllable optical nonreciprocity has been observed for spin excitation in the terahertz region in high magnetic fields of 3rm 0 Hr7 T (ref. 19), which is ascribed to the toroidal magnon mode.
Here we report controllable nonreciprocity in ubiquitous frequency (10)(11)(12)(13)(14)(15) and magnetic field (160 mT) ranges in a multiferroic helimagnet Ba 2 Mg 2 Fe 12 O 22 with a longer period and smaller magnetic anisotropy. While microwave nonreciprocity was previously observed in a chiral magnet, it depended on the crystal chirality and could not be reversed by an electric field 13 . In Ba 2 Mg 2 Fe 12 O 22 , however, the electric polarization can be controlled by an electric field or a low magnetic field 24 . Related materials show a magnetoelectric response above room temperature 25 . The microwave functionality of this novel material may contribute to the development of interesting future technologies.

Results
Variation of magnetic structure in the magnetic field. Figure 2a shows the crystal structure of Ba 2 Mg 2 Fe 12 O 22 . The crystal structure is composed of two types of alternately stacked blocks denoted as S and L blocks. Within each block, the magnetic moments are ferrimagnetically ordered. As a result, the L and S blocks have large and small net magnetic moments, respectively. Figure 2b shows the magnetization curve at 6 K for Ba 2 Mg 2-Fe 12 O 22 . At low temperature, the magnetic moments of the S and L blocks showed conical magnetic orderings where the wave vector of the spin structure was q 0 ¼ (0, 0, 0.59) 26,27 . Therefore, there was a finite spontaneous magnetization. When the magnetic field was increased, the magnetization curve showed several kinks reflecting magnetostructural transitions, as indicated by inverted triangles in Fig. 2b. While the helical plane was perpendicular to the [001] axis at zero magnetic field, it was inclined and a spontaneous electric polarization was induced parallel to the [1 10] axis in a small magnetic field along the [110] axis while maintaining q 0 ¼ (0, 0, 0.59). When the magnetic field was further increased, the helical plane became perpendicular to the magnetic structure in the q 0 ¼ (0, 0, 3/4) state is shown in Fig. 2a.
The phase transitions between helimagnetic structures were firstorder transitions, and phase coexistence was observed around the phase boundaries. In particular, the q 0 ¼ (0, 0, 3/2) state extended down to around zero magnetic field. The q 0 ¼ (0, 0, 3/2) state showed a subtle spin structural change at B200 mT (ref. 27). Above 4.5 T, a collinear ferrimagnetic state appeared, and the ferroelectricity was quenched.
Microwave absorption. Figure 2c shows DS 12 spectra at various magnetic fields at 6 K, measured using the experimental set-up shown in Fig. 1c. The spatial distribution of electric and magnetic fields of microwaves is shown in Fig. 1d. The DS 12 spectra reflect the absorption of microwaves owing to magnetic resonance during the course of propagation from port 2 to port 1. For details of the experimental set-up and a precise definition of DS 12 , see Methods. While the microwave absorption was almost absent at zero field, a small and broad peak appeared at B12 GHz when the magnetic field was applied along the [110] direction. When the magnetic field was increased above the phase boundary between the q 0 ¼ (0, 0, 0.59) and q 0 ¼ (0, 0, 3/4) states (E60 mT), the frequency was slightly increased, and the intensity also increased (for a detailed comparison of frequency, see also Supplementary Fig. 1a,b). The peak at B12 GHz was suppressed, and the high-frequency peak at B19 GHz evolved instead near the phase boundary between the q 0 ¼ (0, 0, 3/4) and q 0 ¼ (0, 0, 3/2) states (E200 mT). Two peaks were simultaneously observed around the phase boundary because of the phase coexistence. The higher peak frequency increased with magnetic field and disappeared from the measured frequency range above 500 mT. The high-frequency peak was suppressed below 200 m. However, it did not vanish but coexisted with the lower frequency peak even around zero field, which was consistent with the presence of the q 0 ¼ (0, 0, 3/2) state around zero field. We demonstrated the control of nonreciprocity mainly for the low-frequency peak in the q 0 ¼ (0, 0, 3/4) state because the measurement sensitivity in the frequency range of 9-14 GHz was better than that in the higher-frequency region in our experimental set-up.
Magnetoelectrical control of nonreciprocity. Figure 3 demonstrates the magnetoelectrical control of nonreciprocal microwave absorption. DS 21 is a microwave absorption spectrum similar to DS 12 , but with the opposite microwave propagation direction. We performed a poling procedure using an external electric field in order to fix the spin helicity (for the details, see Methods). After the poling procedure, we turned off the external electric field and changed the magnetic field to a certain value and then measured DS 12 and DS 21 . Figure 3a,c shows the spectra measured after the poling procedure with an electric field E of þ 0.5 MV m À 1 , and Fig. 3b,d shows those measured with E ¼ À 0.5 MV m À 1 . The measured magnetic field H was þ 160 mT for Fig. 3a,b and À 160 mT for Fig. 3c,d. For all cases, there were clear differences between DS 12 and DS 21 , indicating the microwave nonreciprocity. The nonreciprocity was reversed by the inversion of either E or H, but it was unchanged by the simultaneous inversion of E and H.
To further study the effect of an external field, we investigated the poling electric field dependence of microwave nonreciprocity, DS 12 À DS 21 , at m 0 H ¼ 160 mT and the magnetic field dependence at E ¼ ± 0.5 MV m À 1 . The results are shown in Fig. 4a,b, respectively. The sign of the nonreciprocity reflected that of E, and the magnitude monotonically increased with E. The frequency of the nonreciprocity increased with the magnetic field, corresponding to the increase of absorption peak frequency (see also Supplementary Fig. 2). In Fig. 4c, we show the integrated intensity of nonreciprocity I 12 between 9 and 14.4 GHz at m 0 H ¼ ±160 mT as a function of E. The sign of I 12 depended on those of E and H, and the magnitude of I 12 gradually increased in the low-E region and tended to be saturated at B0.5 MV m À 1 . A similar E dependence was discerned in the polarization results. Therefore, these field dependences were dominated by the ferroelectric domain population.

Discussion
We observed controllable nonreciprocity for the lowest-energy magnetic resonance modes in the transverse conical state. The observed nonreciprocity cannot be ascribed to the effect of magnetic dipole interaction because it should not be changed by the poling electric field in the case of magnetic dipolar nonreciprocity. As explained in the Introduction, the lowest-energy magnetic resonance in the transverse conical state is the toroidal magnon mode, which is expected to show a large nonreciprocity along P Â M. This is quite consistent with the present observations. While similar magnon modes were previously observed at large magnetic fields in the THz region for perovskite RMnO 3 (refs 19,20), we have observed it here at a low magnetic field in the GHz region. While RMnO 3 and Ba 2 Mg 2 Fe 12 O 22 are both helimagnets, the period and magnetic anisotropy are different. For TbMnO 3 , which is a typical material of perovskite helimagnets, the period is 2 nm, and the magnetic anisotropy constant K 1 is 6 Â 10 7 erg cm À 3 , whereas for Ba 2 Mg 2 Fe 12 O 22 , the period is 23 nm and the magnetic anisotropy constant K 1 þ 2K 2 is À 6 Â 10 5 erg cm À 3 (refs 28-31). The frequency difference reflects the differences in the magnetic anisotropy 32 and helical period. The magnitude of nonreciprocity in this study was as large as 6-8%, which is smaller than that observed in the THz region. One of the reasons for this is that the intensity of pure magnetic excitation becomes relatively large and, therefore, the relative nonreciprocity becomes small in the case of the spontaneous conical state. One of the advantages of microwave nonreciprocity is compatibility with other microwave technologies. For example, the nonreciprocity can be adequately enhanced by utilizing a high-Q resonator. While the electrical control of microwave properties has been extensively investigated for multiferroic heterostructures mainly with the use of mechanical strainmediated magnetoelectric coupling 33 , controllable nonreciprocity in the transverse conical state seems more useful. Thus, the new microwave functionality of controllable nonreciprocity demonstrated here has a large potential for practical applications.

Methods
Sample preparation and magnetization measurement. A single crystal of Ba 2 Mg 2 Fe 12 O 22 was grown by the flux method 26 . The crystal orientations were determined by X-ray diffraction. The magnetization curve was measured using a superconducting quantum interference device magnetometer (Magnetic Property Measurement System, Quantum Design) at 6 K in magnetic fields parallel to the [110] direction. Polarization measurement and poling procedure. The spontaneous electric polarization along the [1 10] direction at 6 K was obtained by the integration of measured displacement currents. Before the current measurement, we performed the following poling procedure to fix the spin helicity in a manner similar to a previous study 24 . After cooling the sample to 50 K without external fields, an electric field was applied in the [1 10] direction, and then a magnetic field as large as 5 T was applied parallel to the [110] direction. Next, the magnetic field was decreased to 1 T, followed by cooling to 6 K. Finally, the electric field was removed. The displacement current was measured while sweeping the magnetic field.
Microwave spectroscopy. We fabricated a microwave device composed of a Ba 2 Mg 2 Fe 12 O 22 sample and a microwave coplanar waveguide shown in Fig. 1c in order to measure the microwave response after the electric poling procedure. The dimensions of the sample were 1.1 mm Â 1.1 mm Â 0.3 mm. The largest plane was perpendicular to the [1 10] direction, and the two longer sides were parallel to the [110] and [001] directions. The aluminium electrodes were attached to the largest sample planes for applying the electric field. The coplanar waveguide was designed so that the characteristic impedance matched 50 O. The width of the strip line was 0.2 mm, and the gap between the strip line and the ground plane was 0.05 mm. The sample was placed on the coplanar waveguide so that the [001] direction was parallel to the wave guide. A Teflon sheet with a thickness of 20 mm was inserted between the sample and the waveguide for insulation. The microwave response was measured in a superconducting magnet using a vector network analyser (E5071C, Agilent). Before the microwave measurement, we performed a poling procedure similar to that used previously for the polarization measurement. For the measurements in negative magnetic fields, the magnetic field during the poling was also negative. The microwave absorption spectrum DS 12 at m 0 H is defined as |S 12 (5 T)| À |S 12 (m 0 H)|, where S 12 is the microwave transmittance from port 2 to port 1. Because the magnetic resonance frequency was high enough in the entire measurement range at 5 T, the microwave absorption owing to magnetic resonance can be obtained using this formula. The microwave absorption with the opposite wave vector DS 21 was also obtained by a similar procedure.
Data availability. All relevant data are available from the corresponding authors upon request.