Switchable giant nonreciprocal frequency shift of propagating spin waves in synthetic antiferromagnets

We report the giant nonreciprocity and the electrical switching of propagating spin waves in synthetic antiferromagnets.


INTRODUCTION
Nonreciprocal spin wave propagation is of great interest in the emerging research field of magnonics (1)(2)(3). This specific property provides an advantage for the enhancement of logic circuits and communication devices (4). It is known that the amplitude nonreciprocity in magnetostatic surface waves is caused either by the local concentration of the spin waves at the upper and lower surfaces of the ferromagnetic films (5) or by the nonreciprocal coupling between microwave fields and spin waves (6)(7)(8). Another nonreciprocity is the frequency shift of the propagating spin waves due to the asymmetric spin wave dispersion. This effect can be evoked by an adjacent aluminum ground plane at one end of the ferromagnetic layer due to an additional boundary condition on the tangential electric field (9), the difference of the surface magnetic anisotropies at the two ferromagnetic surfaces (10,11), or electrical current flows in the ferromagnets (12,13). Recent reports have shown that the nonreciprocal frequency shifts in artificial structures are attributed to the presence of an interfacial Dzyaloshinsky-Moriya interaction (i-DMI) (14)(15)(16)(17)(18)(19). Although it is crucial to obtain high nonreciprocity for practical applications in spin wave logic devices, the nonreciprocal frequency shifts in the above demonstrations are limited to small values owing to interfacial effects. In addition to abovementioned interfacial effect, noncentrosymmetric magnets are one of the platforms to observe the nonreciprocal frequency shifts (20). Nevertheless, there are currently no solutions to satisfy practical requirements. In addition, switching of the spin wave nonreciprocity using electricity remains a challenging issue.
Here, we experimentally demonstrated a switchable giant nonreciprocal frequency shift of propagating spin waves in interlayer exchange-coupled synthetic antiferromagnets (SAFs) by spin wave spectroscopy using a vector network analyzer (VNA). The spin wave dispersion in SAFs has been first calculated and observed in the pioneering works by Grünberg et al. (21,22). Nonreciprocal frequency shift for thermally excited "incoherent" spin waves due to asymmetric spin wave dispersion have been experimentally observed by Brillouin light scattering techniques (22)(23)(24)(25)(26)(27). In the past few years, spin wave nonreciprocity in SAFs was micromagnetically calculated and experimentally observed (28)(29)(30)(31). However, despite the recent extensive studies on the research field of magnonics and antiferromagnetic spintronics, there have been few studies on antenna-excited "coherent" propagating spin waves in SAFs. Toward practical applications, it is necessary to investigate the characteristics of coherent propagating spin waves rather than incoherent spin waves. Moreover, switchable and highly nonreciprocal propagating spin waves will be useful for future spin wave-based applications.

Spin wave dispersion in SAFs
Antiferromagnetically coupled ferromagnets exhibit two kinds of resonance precession modes: acoustic modes (in-phase precession) and optic modes (out-of-phase precession) (22,32). Considering the case of the canted magnetization state in SAFs, acoustic mode spin waves (A-SWs) can be excited when the microwave field is applied in a direction perpendicular to the bias magnetic field, namely, in a transverse pumping configuration (Fig. 1A), while optic mode spin waves (O-SWs) can be excited when the microwave field is applied along the bias magnetic field, namely, in a longitudinal pumping configuration (Fig. 1B) (33). The out-of-plane component of the microwave field effectively excites the A-SWs regardless of the bias magnetic field direction. Figure 1 (C and D) illustrates the sketches of A-SWs and O-SWs in canted magnetization states under the transverse and longitudinal pumping configurations. To describe the spin wave dispersion, it is important to consider the contribution to the spin wave energy from dipolar fields generated by the magnetization motion of spin waves (34,35). Since the dipolar fields from the two ferromagnetic layers across the nonmagnetic layer are in antiphase and are canceled out in the case of A-SWs (indicated by the squares in dashed line in Fig. 1C), spin wave dispersion of A-SWs in the transverse pumping configuration is symmetric with respect to the propagation direction. Conversely, that dipolar fields are in the same phase in the case of O-SWs (indicated by the squares in dashed line in Fig. 1D). They are antiparallel (parallel) to the local magnetic moments of the two ferromagnetic layers in the forward-(reverse-) propagation direction. Therefore, spin wave dispersion of O-SWs in the longitudinal pumping configuration is asymmetric with respect to the propagation direction.
with H 1 = (H ext cos  0 − H E cos 2 0 ) ± H E cos 2 0 and H 2 = (H ext cos  0 − H E cos 2 0 + M s ) ± H E . The upper (lower) sign of "±" is for the resonant frequency of A-SWs (O-SWs).  is the gyromagnetic ratio, H ext is the external magnetic field, and k is the wave number. The two magnetizations become the canted state with the angle  1 = − 2 =  0 = cos −1 (H ext /2H E ) in the low magnetic field region below the saturation field 2H E = −2J ex /(tM s ). According to Eq. 1, f T,A is more dispersive as  0 approaches zero and corresponds to the spin wave dispersion of the magnetostatic surface wave mode on a ferromagnetic film with a thickness of 2t above the saturation field. Conversely, according to Eq.

Sample description
To investigate the nonreciprocal frequency shift in SAFs, we fabricated multilayers, which consisted of Ta (3 nm)/Ru (3 nm)/FeCoB (15 nm)/Ru (0.6 nm) / FeCoB (15 nm)/Ru (3 nm) on thermally oxidized Si substrates by dc magnetron sputtering. From a magnetic hysteresis loop at room temperature, the canted magnetization states of the two ferromagnetic layers were confirmed in the low magnetic field region below the saturation field of approximately 100 mT (see section S1). The films were microfabricated into devices for spin wave spectroscopy measurements (36), as shown in Fig. 2A (see Materials and Methods for details). We measured the scattering parameters S 11 , S 12 , S 21 , and S 22 using a VNA at room temperature (see Materials and Methods for details). A static magnetic field was applied in the transverse direction H ext,T and in the longitudinal direction H ext,L . Figure 2 (B and C) shows the applied magnetic field dependence of the peak frequencies in measured Re[S 11 ] spectra (upper) and the calculated frequencies (lower) using analytical expressions of spin wave dispersion in SAFs (eqs. S8 and S9 with  = 1.89 × 10 11 rad/Ts, M s = 1.21 × 10 6 A/m, and J ex = −9.1 × 10 −4 J/ m 2 ), which yield a good agreement with the experimental data for the resonance for both the acoustic and optic modes. Note that we cannot distinguish the resonance peaks for +k and −k in the experimentally obtained Re[S 11 ] spectra due to broad linewidth, which originate from the intrinsic damping constant and wave number  This can be understood from the spin wave group velocity V g , which is given by the slope of spin wave dispersion (V g = 2df/dk). As expressed by Eqs. 1 and 2, V g increases (decreases) as increasing magnetic field in the |H ext | < 2H E limit for transverse (longitudinal) pumping configuration.

Propagating spin wave spectroscopy
To check the nonreciprocity of propagating spin waves, we extracted the Re[S 21 ] and Re [S 12 ] under the bias magnetic fields of ±20 mT, as indicated by the dashed lines in Fig. 2 (D and E). Regarding the propagation of A-SWs in the transverse pumping configuration (Fig. 3A), different amplitudes were observed owing to the nonreciprocal  Fig. 3 (A and B). The peak structures can be observed in the real part of transmitted signal when the magnetization precession between two antennas becomes in-phase. The phase difference of spin waves between f 1 and f 2 is 2.
In these measurements, we measured spin wave spectra by sweeping the magnetic field from positive to negative. For this sequence, the magnetization configuration always transforms from fig. S6A to  fig. S6B or from fig. S6C to fig. S6D (see section S4). Therefore, the sign of nonreciprocity is not changed when the applied magnetic field direction is reversed. The insets of Fig. 3 (C and D)      pumping configuration depending on the propagation direction, which decreases as H ext,L increases. This is consistent with the theoretical spin wave dispersion expressed by Eq. 2, whereby the nonreciprocal frequency shift is proportional to sin 0 . We have also confirmed that the linear increase of the nonreciprocal frequency shift depending on k (see section S5). Therefore, we conclude that the nonreciprocal frequency shift in our experiment originated from mutual dipolar interaction in SAFs.

Quantitative comparison with i-DMI
Here, we discuss the magnitude of the nonreciprocal frequency shift of the propagating spin waves. The nonreciprocal frequency shift induced by the interfacial effect, such as i-DMI (17)(18)(19), is inversely proportional to the thickness of the ferromagnetic layer. Since V g is proportional to the thickness of the ferromagnetic layer for the spin waves in the magnetostatic limit, it is difficult to observe the nonreciprocal frequency shift in the propagating spin wave spectroscopy. Contrary to the interfacial effect, the nonreciprocal frequency shift in SAFs is proportional to the thickness of the ferromagnetic layer owing to dipolar contributions, as expressed by Eq. 2. Figure 4 shows the systematic comparison of the nonreciprocal frequency shift as a function of ferromagnetic layer thickness between the two contributions, where f/k is evaluated because the magnitude of both f SAF and f DMI is proportional to k in the magnetostatic limit. In the case of SAFs, we also plotted the thickness dependence of the nonreciprocal frequency shift obtained from numerical calculation with C z = t nm or C z = 3 nm, where C z is the cell size of z direction (see section S3 for details). It should be noted that the nonreciprocal frequency shifts were observed in both cases although those for C z = 3 nm are slightly smaller than those for C z = t nm because of the localization of spin wave in the thickness direction depending on the propagation direction (5). In addition, a discretization of C z results in the different magnetization state in the thickness direction, such as twisting of magnetization, which is not considered in the theoretically calculated spin wave dispersion and micromagnetic simulation with C z = t nm. Therefore, the nonreciprocal frequency shift calculated with a discretization of C z agrees well with the experiment. It is obvious that the dipolar contribution in SAFs reported in this study represents a larger nonreciprocal frequency shift compared to the i-DMI contribution in artificial structures. In addition, although the strength of the interlayer exchange coupling does not affect f SAF value in small magnetic field region (see section S6), SAFs with thicker ferromagnetic layers provide a further enhancement of nonreciprocal frequency shift as well as the spin wave group velocity, which will be useful for future spin wavebased applications.

Electrically switchable nonreciprocal devices
The sign of the nonreciprocal frequency shift depends on the relative magnetization angle between two ferromagnetic layers (see section S4). In the latter part of this study, we show an electrical switching of nonreciprocal frequency shift in SAFs to provide a previously unrealized manipulation technique of the relative magnetization configuration in two ferromagnetic layers (see Materials and Methods for sample details). The electrical switching of the two canted magnetization states was demonstrated as follows. First,  0 H ext,L = 10 mT was applied to obtain the canted magnetizations state in SAFs. We then applied the electric current pulse I pulse along the wire length direction with 100-s duration, which exerts a torque on the two magnetizations in opposite directions due to current-induced Oersted field. Then, propagating spin wave spectroscopy measurements were performed after applying the current pulse of various amplitudes. Depending on the current polarity, we achieve two different canted magnetization states (Fig. 5, A and B) and succeed to switch the sign of the nonreciprocal frequency shift, as shown in Fig. 5C. To investigate the threshold current, we measured nonreciprocal frequency shift, f SAF = f(S 12 ) − f(S 21 ), after applying the positive and negative current pulse of various amplitudes, as shown in Fig. 5D (see section S7 for the propagating spin wave spectra as a function of I pulse ). The switching of the nonreciprocal frequency shift was observed above ±18 mA corresponding to the current density of ±9.1 × 10 9 A/m 2 . Therefore, the electrical switching of nonreciprocal frequency shift without changing the bias magnetic field can be achieved by only applying the electrical current pulse to SAFs.

DISCUSSION
We have experimentally demonstrated the switchable giant nonreciprocal frequency shift in interlayer exchange-coupled SAFs. The magnitude of the nonreciprocal frequency shift is proportional to the thickness of the ferromagnetic layer owing to dipolar contributions between two ferromagnetic layers. It should also be noted that the wavelength symmetry of the propagating spin waves can be changed depending on the propagation direction with respect to the external magnetic field (Fig. 1, E and F), which cannot be achieved in the magnetostatic surface wave using a single ferromagnetic layer. Last, we demonstrate an electrical way to control the sign of the nonreciprocal frequency shift. Our results offer a new key for switchable and highly nonreciprocal spin wave-based applications.

Sample preparation
Ta (3 nm)/Ru (3 nm)/Fe 60 Co 20 B 20 (15 nm)/Ru (0.6 nm) / Fe 60 Co 20 B 20 (15 nm)/Ru (3 nm) were deposited on Si/SiO 2 substrates by dc magnetron sputtering. The two in-plane magnetized FeCoB layers separated by a Ru layer with a thickness of 0.6 nm were antiferromagnetically coupled via interlayer exchange coupling. The films were patterned into 50 m by 100 m wires by electron beam (EB) lithography and Ar ion milling. Subsequently, an insulating SiO 2 layer with a thickness of 80 nm was deposited by radio frequency magnetron sputtering. Two coplanar waveguides (CPWs), which consisted of Cr (5 nm)/ Au (100 nm), were then fabricated at the distance of 10 m with the use of EB lithography and an evaporator. The designed widths of a center strip and the two side strips were 2 and 1 m, respectively. From the calculation of the spatial distribution of the microwave current in CPWs (36), the spin wave with the wave number k of 1.2 m −1 was efficiently excited in our devices. Samples for electrically switchable nonreciprocity devices consist of Ta (3 nm)/Ru (3 nm)/Fe 40 Co 40 B 20 (15 nm)/Ru (0.5 nm) / Fe 40 Co 40 B 20 (15 nm)/Ru (3 nm) deposited on a Si/SiO 2 substrate by dc magnetron sputtering. The films were patterned into 30 m by 120 m wires, and two CPWs and dc pads connecting to the wire to apply a dc current were fabricated by the same process as above.

VNA measurement
Spin wave spectroscopy was performed using a VNA. The microwave probes were connected to the VNA via coaxial cables. After setting the parameters in VNA, such as the frequency range (0.01 to 20 GHz with a step of 0.01 GHz), the microwave power (−20 dBm), and bandwidth (1 kHz), the microwave apparatus was calibrated using a calibration substrate, which include short-open-load-through coplanar standards. The scattering parameters S 11, S 21 , S 12 , and S 22 were measured by VNA in transverse and longitudinal pumping configurations at room temperature. Each spectrum was subtracted by a reference spectrum in the cases at which the resonant peaks were shifted away from the relevant frequency regime. On the basis of the self-scattering parameters S 11 and S 22 , we can extract the local spin wave resonance under CPWs. On the basis of the mutual-scattering parameters S 12 and S 21 , we can extract the propagation characteristics of the spin waves between the two CPWs.

Amplitude nonreciprocity
Because S parameters has the following relation, S ∝ i2fL, where f is the excitation frequency and L is a complex inductance induced by the magnetization precession, the amplitude nonreciprocity for positive field  + and negative field  − can be defined as where S ij (f ij ) (i,j = 1,2) is the maximum spin wave amplitude at spin wave resonance frequency f ij .