Spin homojunction with high interfacial transparency for efficient spin-charge conversion.

High interfacial transparency is vital to achieve efficient spin-charge conversion for ideal spintronic devices with low energy consumption. However, in traditional ferromagnetic/nonmagnetic heterojunctions, the interfacial Rashba spin-orbit coupling brings about spin memory loss (SML) and two-magnon scattering (TMS), quenching spin current crossing the heterointerfaces. To address the intrinsic deficiency of heterointerface, we design a ferromagnetic FeRh/antiferromagnetic FeRh spin homojunction for efficient spin-charge conversion, verified by a high interfacial transparency of 0.75 and a high spin torque efficiency of 0.34 from spin pumping measurements. First-principles calculations demonstrate that the interfacial electric field of homojunction is two orders of magnitude smaller than that of traditional heterojunction, producing negligible interfacial spin-orbit coupling to drastically reduce SML and TMS. Our spin homojunction exhibits potential and enlightenment for future energy-efficient spintronic devices.

INTRODUCTION

Pursing efficient spin-charge conversion lies at the heart of spintronics (, ). Numerous candidate materials for spin-charge conversion have emerged, such as heavy metals () and alloys (–), transition-metal dichalcogenides (, ), transition-metal oxides (), and antiferromagnets (, ). Notably, the spin transmission through the interface between the spin source layer and the spin sink layer is inevitably involved in all the spin-charge conversion process, such as the spin Hall effect and the inverse spin Hall effect (ISHE) (), spin-orbit torques (), and the spin Hall magnetoresistance (). Nevertheless, because of the existence of large interfacial spin-orbit coupling (–), spin memory loss (SML) (–) and two-magnon scattering (TMS) (, ) greatly reduce the interfacial transparency (–). On one hand, low interfacial transparency attenuates the spin-charge conversion efficiency drastically, although the spin source and the spin sink material may have strong spin-charge conversion capabilities. On the other hand, the ignorance of interfacial transparency results in an unphysical overestimation of effective spin mixing conductance (). This leads to inaccurate estimations of the spin Hall angle θSH and the spin diffusion length λs, which scatters in a large scale, such as θSH (0.004 to 0.64) and λs (0.5 to 10 nm) for the commonly used material Pt (, ). Therefore, improving the interfacial transparency is not only critical to efficient spin-charge conversion but also of great physical significance to the accurate determination of transport parameters and explanation of spin physics.

Interface engineering has been used to improve the interfacial transparency, including interfacial alloying () or inserting layers such as metal Hf (), Cu (, ), and insulator NiO (). However, these methods introduce new spin carriers (magnons) and new interfaces, complicating both film preparation and physical understanding. If the spin source and the spin sink are different materials to construct a “spin heterojunction” (Fig. 1A) with broken inversion symmetry, the interfacial spin-orbit coupling can be hardly eliminated. An ideal solution is an epitaxial “spin homojunction” with similar crystal structures and elemental composition for the spin source and the spin sink (Fig. 1B). It is reasonable to expect that the interfacial electric field can also be substantially weakened because of the conserved inversion symmetry, resulting in negligible interfacial Rashba spin-orbit coupling () and concomitant low SML and TMS. On the basis of this design, a spin homojunction with high interfacial transparency for efficient spin-charge conversion is promising but remains elusive.

Fig. 1.

Schematic of spin heterojunction and spin homojunction for spin transmission.

(A) Spin heterojunction. (B) Spin homojunction.

Iron-rhodium alloy (FeRh) is an intriguing spintronic material having first-order phase transition between antiferromagnetic (AFM) and ferromagnetic (FM) when sweeping temperature, potentially used in heat-assisted magnetic recording and magnetic refrigerant (–). Recently, the spin pumping effect was verified in FM-FeRh/Pt and NiFe/AFM-FeRh films with a high θSHλs value of ~0.2 to 0.8 for FeRh (). The high θSHλs value indicates a superior intrinsic spin-charge conversion capability of FeRh alloy (, , , ). Meanwhile, the FM and the AFM phase of the same FeRh material can separately play the role of the spin source layer and the spin sink layer, which brings about possibilities for a FeRh-based homojunction. Here, we design an FM-FeRh/AFM-FeRh spin homojunction for efficient spin-charge conversion, demonstrated by spin pumping measurements. The interfacial transparency T reaches 0.75 and the spin torque efficiency is up to 0.34, which are higher than traditional heavy metal/FM spin heterojunctions based on the spin Hall effect, such as Pt/Py (, ). First-principles calculations reveal that the interfacial Rashba spin-orbit coupling of this FM-FeRh/AFM-FeRh spin homojunction is negligibly small, highly suppressing the SML and the TMS. Novel spin homojunction brings about a new dimension to the pursuit of efficient spin-charge conversion.

RESULTS

Growth and characterizations of the spin homojunction

We prepared 15-nm FeRh films on MgO(001) substrates by magnetron sputtering under different Ar pressures at 300°C, followed by annealing at 750°C for 1 hour. Figure 2A shows the temperature-dependent four-point resistivity (ρ-T) curves of the FeRh film grown at the Ar pressure of 0.4 and 0.7 Pa, with an in-plane (IP) field of 5 T. Abrupt decrease/increase of resistivity is observed when the temperature increases/decreases, corresponding to the AFM-FM and FM-AFM phase transition, respectively (, ). As the Ar pressure increases from 0.4 to 0.7 Pa, the composition of FeRh changes from Fe0.52Rh0.48 to Fe0.50Rh0.50, making the room-temperature (RT) phase of FeRh change from FM to AFM, denoted as FM-FeRh and AFM-FeRh, respectively. Figure 2B presents the x-ray diffraction (XRD) spectra of these FM-FeRh and AFM-FeRh films. They exhibit strong FeRh(001) texture with diffraction peaks located at similar diffraction angles, indicating highly matched lattice constants. The average roughness (Ra) of 15-nm-thick FM-FeRh and AFM-FeRh films is 0.15 and 0.25 nm, respectively, showing a flat surface (fig. S1). As a result, different magnetic phases at RT, highly matched lattice and composition, as well as flat surfaces of FM-FeRh and AFM-FeRh single layers provide perquisite for a high-equality spin homojunction of FM-FeRh/AFM-FeRh.

Fig. 2.

Characterization of the FM-FeRh/AFM-FeRh spin homojunction.

(A) ρ-T curves of individual FM-FeRh and AFM-FeRh thin films under an IP magnetic field of 5 T. (B) XRD spectra of the FM-FeRh and the AFM-FeRh thin films. a.u., arbitrary units. (C) ρ-T curve of the FM-FeRh/AFM-FeRh homojunction under an IP magnetic field of 5 T. (D) XRD spectra of the FM-FeRh, the AFM-FeRh, and the FM-FeRh/AFM-FeRh homojunction at high diffraction angle. (E) HAADF-STEM image of the FM-FeRh/AFM-FeRh homojunction. The red dash line indicates the interface between the FM-FeRh layer and the AFM-FeRh layer. Schematics of crystalline layout are included, where the purple and green atoms represent Fe and Rh atoms, and the vectors indicate the direction of spin. (F) High-resolution HAADF-STEM image and corresponding (G) EELS spectrum of the FM-FeRh/AFM-FeRh homojunction at the interface. Fe L2–3 edge fine structures are extracted from 1′ to 6′ Fe atomic planes.

Next, the FM-FeRh (15 nm)/AFM-FeRh (12 nm) homojunction is prepared by in situ tuning of Ar pressure from 0.4 to 0.7 Pa during sputtering of the FM-FeRh layer, followed by the AFM-FeRh layer. As shown in Fig. 2C, there are two hysteresis windows in the ρ-T loop of the homojunction, originated from two phase transitions of FM-FeRh and AFM-FeRh layers. XRD spectra at high diffraction angles are displayed in Fig. 2D, corresponding to the diffraction peaks of FeRh(300) for the FM-FeRh, the AFM-FeRh, and the FM-FeRh/AFM-FeRh homojunction. The diffraction angle of AFM-FeRh is slightly larger than that of FM-FeRh, consistent with the compressive stress in AFM-FeRh to stabilize the AFM phase (). Notably, the XRD spectrum of FM-FeRh/AFM-FeRh exhibits two diffraction peaks, as a combination of both layers. To make in-depth characterization of the homointerface of our homojunction, we performed the cross-sectional high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM). The interface between the FM-FeRh and the AFM-FeRh can be observed in Fig. 2E, as denoted by a red dash line. The contrast of the interface is low, indicating similar structure and composition of the FM-FeRh bottom layer and the AFM-FeRh upper layer. Figure 2F displays a high-resolution HAADF-STEM cross-sectional image of the FM-FeRh/AFM-FeRh interface with few interfacial disorders, where the electron energy-loss spectroscopy (EELS) spectra in Fig. 2G are acquired. Brighter Rh atoms are located at the body center of darker Fe atoms, indicating bcc α′-FeRh phase for both FM-FeRh and AFM-FeRh, which is able to exhibit the phase transition characteristic (). Concomitant schematics of crystalline layout are displayed in Fig. 2E. The full epitaxial growth relationship with substrate is determined to be MgO(001)[010]//α′-FM-FeRh(001)[110]//α′-AFM-FeRh(001)[110] by φ-scan XRD measurements (fig. S2), highly consistent with the HAADF-STEM image. As indicated by the Fe L2–3 edge fine structures extracted from 1′ to 6′ atomic planes of Fe (Fig. 2G), the binding energy of Fe does not change when crossing the interface of FM-FeRh and AFM-FeRh, demonstrating analogous valence states of Fe atoms in the FM-FeRh layer and the AFM-FeRh layer. Therefore, the spin homojunction is composed of FM-FeRh and AFM-FeRh with distinct FM and AFM magnetic phases while with similar crystal structure and chemical environment for the spin source and the spin sink.

Spin pumping measurements for the spin homojunction

Spin pumping measurements were carried out to investigate the spin-charge conversion efficiency of the FM-FeRh/AFM-FeRh homojunction. As shown in Fig. 3A, spin pumping process involves spin current generated by microwave-excited FM resonance (FMR) of the FM-FeRh layer, transmitted through the homointerface of FM-FeRh/AFM-FeRh and converted to charge current via the ISHE in the AFM-FeRh layer (). Details for device configurations and measurement principles are discussed in Materials and Methods. Figure 3B presents a typical voltage signal of the stripe along FM-FeRh/AFM-FeRh[110] when sweeping the magnetic field along the direction with the out-of-plane (OOP) polar angle θH of 90° and the IP azimuthal angle φH of 45°, under the excitation of 17-GHz, 19-dBm microwave. The resonance curve can be divided into symmetric and antisymmetric Lorentz line shapes as V = VasymLasym + VsymLsym, drawn by yellow and blue lines. Lasym = − 4ΔH(H − HR)/[4(H − HR)2 + ΔH2] and Lsym = ΔH2/[4(H − HR)2 + ΔH2], where ΔH and HR are the linewidth and the resonance field, respectively. anisotropic magnetoresistance (AMR) contributes to both symmetric and antisymmetric line shapes, while ISHE only contributes to symmetric line shape. The AMR component is zero at φH = 0°, 90°, 180°, and 270°, while the ISHE signal theoretically disappears only at φH = 0° and 180° (, ). Hence, as shown in Fig. 3C, spin pumping measurements at φH = 90° and 270° reveal opposite pure ISHE signals with symmetric line shape, well matching the ISHE effect. For the stripe along FM-FeRh/AFM-FeRh[100], the spin pumping signal is slightly smaller (fig. S3). We focus on the stripes along FM-FeRh/AFM-FeRh[110] in the following measurements. Note that the actual microwave power on the stripe is measured to be 7.5 dBm because of microwave loss by a thermal-equivalent method (fig. S4). A clear linear dependence of signals with microwave power confirms that we are still in the linear regime of excitation (fig. S5). Thermal effects usually have negligible contribution to spin pumping signals, especially under such a low microwave power (–).

Fig. 3.

Spin pumping measurements for the FM-FeRh/AFM-FeRh spin homojunction.

(A) Schematic of the spin pumping experimental setup and principles. (B) Typical resonance curve measured at θH = 90° and φH = 45° excited by the microwave of 17 GHz, 19 dBm. (C) Symmetric resonance signals from pure ISHE at φH = 90° and φH = 270°, equivalent to reversing the polarity of the magnetic field.

Angle-dependent measurements were performed to demonstrate the spin pumping process with the exclusion of AMR signals more specifically. The magnetic field is swept at different φH, and corresponding symmetric and antisymmetric parts of signals at each φH are extracted to draw Fig. 4 (A and B, respectively). Note that inductive currents in the AFM-FeRh layer can also give a y-oriented Oersted field IP to excite the adjacent FM-FeRh, in addition to the OOP excitation field from the microwave in the coplanar waveguide (CPW). Thus, AMR and ISHE signals are originated from both IP and OOP excitation source. We fit the angle-dependent curve by the following expressions (detailed derivations can be found in Materials and Methods)where φM is the azimuthal angle from x axis to the magnetization direction, which is approximately equal to φH here because the excitation frequency and the resultant resonance field are quite high. From the fitting, we obtain C = −3.9 μV, which is consistent with the magnitude of the spin pumping voltage observed at 90° and 270° (Fig. 3C). No exchange bias is found in our homojunction (fig. S6), judged by the M-H curve and the angle-dependent resonance field data (). Consequently, angle-dependent experiments further verify the spin pumping process from FM-FeRh to AFM-FeRh in the FM-FeRh/AFM-FeRh spin homojunction.

Fig. 4.

Frequency-dependent and angle-dependent spin pumping measurements for the FM-FeRh/AFM-FeRh spin homojunction.

(A) Symmetric and (B) asymmetric angle-dependent line shapes measured at 17 GHz, 19 dBm. (C) The resonance frequency f as a function of the resonance magnetic field HR at φH = 45°. (D) The relationship between the resonance linewidth ΔH and the resonance frequency f at φH = 45°. (E) Comparison of the interfacial transparency T and the spin torque efficiency of our spin homojunction with representative works. Several topological materials and transition-metal dichalcogenides are also candidate spin sources but not included, because the former is beyond the scope of the spin Hall effect and the latter’s interfacial transparency is seldom reported. The hollow points denote the results that are not directly presented in corresponding papers but are calculated by Eq. 5 according to relevant data given in corresponding papers. Solid points are given directly in corresponding papers. CFB denotes CoFeB and Py represents NiFe alloy.

Frequency-dependent measurements were carried out to investigate the spin transmission properties of the homogenous interface between FM-FeRh and AFM-FeRh, quantified by the effective spin mixing conductance . Figure 4C shows the dispersion relation of the resonance frequency f versus the resonance field μ0HR at φH = 45°. We use the simplified Kittel formula [considering φM = φH due to the negligible anisotropy of FM-FeRh (fig. S6)] to fit and get the effective magnetization μ0Meff = 863.59 mT, along with the gyromagnetic ratio γ = 176.22 GHz T−1. The saturation magnetization μ0Ms is measured to be 926.29 mT. Figure 4D presents the linear growth of resonance linewidth ΔH with f, which can be described by ()where μ0ΔH0 is the inhomogeneous term, which is independent with f. αFM/AFM is the effective damping of the FM-FeRh/AFM-FeRh homojunction. αFM/AFM equals 0.0125 as fitted by Eq. 2. Then, of the interface of FeRh(AFM)/FeRh(FM) can be quantified by ()where μB is the Bohr magnetron, g is the g factor, ℏ is the reduced Planck constant, e is the electron charge, and dFM is the thickness of FM-FeRh, respectively. Δα = αFM/AFM − αFM. αFM represents the damping of the FM-FeRh single layer, which is determined to be 0.0115 by Eq. 2 with frequency-dependent measurements under the same condition as FM-FeRh/AFM-FeRh (fig. S7). No observable resonant signals are found for an AFM-FeRh single layer (fig. S8). The is then calculated to be 0.259 × 1015 ohm−1 m−2, corresponding to a of 6.69 × 1018 m−2.

The interfacial transparency and the spin torque efficiency of the spin homojunction

The interfacial transparency T is related to the spin mixing conductance G↑↓ of the FM-FeRh/AFM-FeRh interface according to the drift-diffusion model ()where λs and dAFM are the spin diffusion length and the thickness of the AFM-FeRh. GAFM = 1/λsρAFM denotes the spin conductance of the AFM-FeRh (), and ρAFM is the resistivity of the AFM-FeRh, measured to be 144 microhm·cm. λs is measured to be around 1 nm (fig. S9), which is close to former works by FMR measurements (λs < 1.15 nm) (). Considering dAFM >> λs, , and then T can be simplified as ()

To further investigate the spin-charge conversion efficiency of the FM-FeRh/AFM-FeRh homojunction, we calculate the spin Hall angle θSH of AFM-FeRh by (), where VISHE is the inverse spin Hall voltage generated by the spin pumping process. R and w are the resistance and the width of stripe, equaling 1989 ohms and 6 μm, respectively. Js is the magnitude of spin current at the interface. We carefully calibrate the microwave current by the thermal-equivalent method and calculate OOP via the electromagnetic field simulation with COMSOL Multiphysics (fig. S4), and then Js can be determined. As a result, θSH is calculated as 0.46. Detailed deviations are included in Materials and Methods. After that,, which represents the damping-like spin Hall torque efficiency per unit current density (), can be determined as

We also carried out spin-torque FMR for independent verification, which is widely used to extract (). turns out to be 0.28, consistent with the spin pumping measurements within experimental uncertainty (fig. S9 and table S1). Besides, the spin polarization direction of spin current from AFM-FeRh is found to be dependent on Néel vector (fig. S10). of a reference FM-FeRh(15 nm)/Pt(12 nm) heterojunction is measured to be 0.18, indicating the high spin-torque efficiency of our homojunction (fig. S11).

As presented in Fig. 4E, we compare T and of our FM-FeRh/AFM-FeRh homojunction with that of available and representative two-layer heterojunctions for spin-charge conversion, including heavy metals and alloys, transition-metal oxides, antiferromagnets, etc. (, , , , , , –). As a result, our FM-FeRh/AFM-FeRh spin homojunction has a high interfacial transparency T and a high , demonstrating its unique advantage for spin-charge conversion.

DISCUSSION

It should be noted that the damping enhancement Δα is usually not simply originated from the spin pumping effect (, , , ). Specifically, considering SML and TMS, αFM/AFM can be expressed aswhere αGil, αSP, αSML, and αTMS parameter the intrinsic Gilbert damping, the damping from SML, TMS, and the spin pumping effect. As a result

It means that the ignorance of additional damping due to SML and TMS gives rise to an unphysically giant and resultant overestimations of and (). Principally, the true has two upper limits (): (i) should be at least less than 2GAFM; otherwise, a negative G↑↓ will be obtained and T will excess 1 according to Eqs. 4 and 5. (ii) G↑↓ has a limitation of G↑↓ < Gsh, where Gsh is the Sharvin conductance of the AFM-FeRh. Hence, . The violation of the above two limitations indicate non-negligible SML or TMS at the interface (–), and these reports are not included in Fig. 4E.

Next, we theoretically investigate the influence of αSML and αTMS in our spin homojunction. (i) First, both SML and TMS have been demonstrated to be intrinsically determined by the magnitude of the interfacial Rashba spin-orbit coupling (, , –). Here, αR denotes the Rashba coefficient, which is proportional to the magnitude of the interfacial electric field (). The interfacial electric field is calculated for both our FM-FeRh/AFM-FeRh homojunction and a common Fe/Pt heterojunction for comparison. Atomic spin-orbit coupling is included to consider the influence of magnetic symmetry breaking on the interfacial electric field. As shown in Fig. 5A, the evolution feature of charge density along the OOP z direction is quite similar before and after stepping crosshomogeneous interface (indicated by yellow shadow), and thus, the averaged planar potential in Fig. 5B only varies 0.04 eV. In contrast, the charge density changes severely between the heterogeneous interface of Pt and Fe, as presented in Fig. 5C. Hence, the averaged planar potential experiences a sharp increase of 7.71 eV from Pt to Fe (Fig. 5D), two orders of magnitude larger than that of the homojunction, indicating a much stronger interfacial electric field than that of the homojunction. (ii) Second, both SML and TMS have extrinsic contributions from interfacial disorder (including defects, roughness, polycrystalline structure, etc.) (, ). Compared with the common polycrystalline heterojunctions such as Fe/Pt, interfacial disorders in our single crystalline and epitaxial homojunction are negligible, which can also be verified by the HAADF-STEM image (Fig. 2F). Besides, SML originated from the bulk spin-orbit coupling () is little for our homojunction with conserved inversion symmetry and epitaxial interface. The conserved inversion symmetry and negligible interfacial spin-orbit coupling also exclude self-torques () from FM-FeRh in our FM-FeRh/AFM-FeRh homojunction. As a result, SML and TMS are negligible in our homojunction.

Fig. 5.

First-principles calculations for spin homojunction and spin heterojunction.

(A) Charge density and (B) planar potential for the FM-FeRh/AFM-FeRh homojunction. The interfacial potential difference is calculated to be 0.04 eV. (C) Charge density and (D) planar potential for the Pt/Fe heterojunction. The interfacial potential difference is calculated to be 7.71 eV.

It should be noted that T can never reach 1, although the SML and TMS are ruled out for our homojunction. That is because the spin backflow should be taken into account in the definition of T () in bilayers, even in our ideal homojunction. By inserting a NiO layer to suppress the spin backflow by thermal magnon excitation, trilayer Pt/NiO/CoFeB junction realized T = 1 (). Nevertheless, this design involves excitations of other spin carriers, indicating it is not a simple spin transmission process at the interface. The transparency of our spin homojunction has almost reached its physical limit based on the above two limits of (see a detailed explanation in Materials and Methods).

We report an FM-FeRh/AFM-FeRh spin homojunction for efficient spin-charge conversion, which has both practical and physical advances. On one hand, spin homojunction makes use of the analogous crystal structure and chemical environment, high single-crystallinity, and high flatness of FM-FeRh and AFM-FeRh, inherently reducing the interfacial spin-orbit coupling to eliminate SML and TMS at the interface. It leads to a crucially large T and of 0.75 and 0.34, demonstrating efficient spin-charge conversion in our spin homojunction, which gives fundamental enlightenment to energy-efficient spintronic devices. On the other hand, besides our FM-FeRh/AFM-FeRh spin homojunction, CrBr3 and Mn3GaN have the potential to construct FM-CrBr3/AFM-CrBr3 () and FM-Mn3GaN/AFM-Mn3GaN () spin homojunctions. These spin homojunctions provide simple and clean platforms to investigate spin-charge conversion phenomena (spin pumping, spin Hall effect, spin-orbit torques, spin Hall magnetoresistance, etc.) without the complicated influence of SML and TMS on interfacial spin transport.

MATERIALS AND METHODS

Sample preparations and measurement setups

The FM-FeRh/AFM-FeRh spin homojunction was grown on a single-crystal MgO(001) substrate by magnetron sputtering. The bottom 15-nm FM-FeRh and the upper 12-nm AFM-FeRh were deposited at 300°C at the Ar atmosphere of 0.4 and 0.7 Pa, respectively, followed by annealing at 750°C for 1 hour. All the samples were kept in a glove box with O2 and H2O < 0.01 parts per million to prevent degradation or oxidation. Four-terminal transport measurements were carried out in Physical Property Measurement System to collect ρ-T curves for characterizing the phase transition behavior of FeRh films. θ-2θ and φ-scan XRDs were carried out to obtain the crystal quality and grown modes of the homojunction. Atomic force microscopy was used to determine the roughness of FeRh films.

The cross-sectional HAADF-STEM and the EELS were carried out on an FEI Titan 80-300 electron microscopy equipped with a monochromator unit, a probe spherical aberration corrector, a postcolumn energy filter system (Gatan Tridiem 865 ER) and a Gatan 2k slow-scan charge-coupled device system, operating at 300 kV (), combining an energy resolution of ~0.6 eV and a dispersion of 0.2 eV per channel with a spatial resolution of ~0.08 nm.

Spin homojunctions were patterned to 6 μm by 300 μm stripes using optimal lithography combined with Ar ion milling for spin pumping measurements. Then, Ti(10 nm)/Au(140 nm) ground-signal-ground CPW was thermally evaporated to conduct microwaves with ~50-ohm impedance match. Microwave of 19 dBm was introduced into the CPW from the microwave generator to excite FMR of the stripes. The actual microwave power on the stripe was measured to be 7.5 dBm by a thermal-equivalent method (fig. S4). The voltage across the stripe was captured via a nanovoltmeter when sweeping the magnetic field, which is composed of the ISHE signal from spin pumping and the AMR rectification signal. These stripes were located in the gap between the ground line and the signal line, leading to a oop excitation configuration for feasible distinguishment between the ISHE signal and the AMR signal via angle-dependent measurements ().

COMSOL simulations

The microwave magnetic field intensity distribution in CPW was simulated with a radio frequency module (electromagnetic waves and frequency domain) in COMSOL Multiphysics. Propagation region was bound by scattering boundary conditions. To simplify the model, Ti layer was ignored, and Au layer was approximated by transition boundary conditions with a thickness of 140 nm. We used lumped exports, the known impedance of 50 ohms, and microwave power of 7.5 dBm to excite the CPW. Because of the relatively small thickness of Au, the meshing was chosen finer around the stripe. Then, magnetic field intensity distribution around the CPW could be obtained (fig. S4). The micromagnetic magnetic field μ0h0 around the stripe was calculated to be 0.052 mT.

Density functional theory calculations

All the first-principles calculations were performed in Vienna ab initio simulation package, with the Perdew-Burke-Ernzerhof functional. The calculations were based on a cutoff energy of 500 eV. The γ-centered k-point mesh of 24 × 24 × 1 was used in the calculations of both FeRh homojunction and Pt/Fe heterostructure. A vacuum layer larger than 15 Å was adopted in all calculations of thin films, which is large enough for the structures.

Principles of the spin pumping measurements and determination of

The spin pumping process in FM-FeRh/AFM-FeRh involves the FMR of FM-FeRh, pumping spin current into the adjacent AFM-FeRh, which is then converted to charge current by ISHE. The Landau-Lifshitz-Gilbert (LLG) equation is modified by the spin pumping process as ()where is the magnetic moment and eff is the localized effective magnetic field on . γ is the gyromagnetic ratio, αGil is the intrinsic Gilbert damping, and Ms and V are the saturation magnetization and the volume of FM-FeRh. s is the total spin current, which reads , considering the spin backflow . By substituting αGil by the effective damping constant of the FM-FeRh/AFM-FeRh homojunction αFM/AFM, the last term in Eq. 9 can be combined with the second term, which results in the same form as the original LLG equation ()where is the magnetization vector and d/dt is its time derivative. eff is the effective magnetic field and μ0 is the vacuum permeability. We define two Cartesian coordinate systems as shown in fig. S12. x′, y′, and z′ axes are based on the direction of magnetization x′, while x, y, and z axes are related to the direction of microwave current x. The microwave current (t) is along the stripe, generating inductive current FM(t) and AFM(t) in FM-FeRh and AFM-FeRh in the same direction, respectively. φM/φH denotes the angle between the magnetization/magnetic field and x axis. Two main excitations are involved, the OOP microwave magnetic field OOP from CPW and the IP Oersted field IP due to AFM(t). In the coordinate of (x′, y′, z′), (M, me, m), OOP = e(0, 0, h0), IP = e(hOesinφM, hOecosφM, 0), where ω is the angle frequency of microwave with ω = 2πf. Considering the external constant magnetic field H and the perpendicular anisotropic effective field, the total effective field is ()

Solving Eqs. 10 to 13, one can obtain the resonance condition, namely, the Kittel formulation , where f and HR are the frequency of microwave and corresponding resonance field. Note that the IP anisotropy is negligible in our FM-FeRh layer compared with Meff and HR at a relative high microwave frequency of 17 GHz (fig. S6), and thus is left out for simplicity. In this case, φM ≈ φH. Assuming αFM/AFM << 1, Mx ≈ Ms, we relate with OOP and IP by the dynamical magnetic complex susceptibility χ ()

Here, is the complex diagonal (off-diagonal) dynamics magnetic susceptibility due to h0. is the complex diagonal (off-diagonal) dynamics magnetic susceptibility due to hOe. Φ1 (Φ2) represents the phase difference between magnetization dynamics and hOe (h0). Eq. 14 results inwhere χ can be decomposed into real parts and imaginary parts and reads ()

The ISHE voltage VISHE can be expressed as the following with negligible SML ()

For systems with substantial SML, please refer to the work by Krishnia et al. with T and SML included (), R is the resistance of the FM-FeRh/AFM-FeRh homojunction, Ic is the ISHE-induced charge current, and Js is the magnitude of the spin current at the interface. e and ℏ are the electron charge and the reduced Planck constant. w is the width of the stripe. dAFM, θSH, and λs are the thickness, the spin Hall angle, and the spin diffusion length of the AFM-FeRh layer, respectively. ρAFM is the resistivity of the AFM-FeRh. Considering dAFM >> λs, . Js is determined by ()

As a result, for φM-dependent spin pumping measurements

Merging similar items containing hOe as and h0 as , we have , and the angle dependence of and can be expressed as

Considering that Re(χ) and Im(χ) have antisymmetric and symmetric line shapes, respectively, we conclude that ISHE only contributes to the symmetric line shape.

For AMR rectification volage, it can be expressed as ()

Here, Δρ is the resistivity difference between the magnetization parrel to the current and the magnetization perpendicular to the current. L is the length of the stripe, equaling 300 μm. Similarly, we have , and

It is clear that AMR rectification contributes to both the symmetric and the antisymmetric line shapes. Consequently, we can separate AMR and ISHE signals from angle-dependent spin pumping measurements by V = VasymLasym + VsymLsym, where Lsym = − 4ΔH(H − H)/[4(H − HR)2 + ΔH2] and Lasym = ΔH2/[4(H − HR)2 + ΔH2] are the symmetric and the antisymmetric Lorentz line shapes

For φM = 90°, only contributes to the voltage V. On the basis of Eq. 21, we substitute the by data shown in Fig. 3C. R = 1989 ohms, w = 6 μm, h = 6.626 × 10−34 J·s, f = 17 GHz, ρAFM = 144 microhm·cm, e = 1.6 × 1019 C, μ0Ms = 926.29 mT, and μ0h0 = 0.052 mT. Last, θSH turns out to be 0.46. After that, we determine as

Estimation on the upper limitation of the interfacial transparency based on the Sharvin conductance

Sharvin conductance means the contact conductance dominated by the Sharvin mechanism, when the size of microcontacts is less than the mean free path λ of electron (). For a microcontact of area ai, local Sharvin conductance , where (). ρ1 and ρ2 are the specific resistivities of contacting surfaces. The resistivity of our FM-FeRh and AFM-FeRh are 66 and 144 microhm·cm, respectively; hence, . For an interface of area A, the Sharvin conductance reads

Here, ∑a/A is influenced by the roughness of the interface. We estimate ∑a/A to be close to 1, considering that the roughness of our FM-FeRh/AFM-FeRh homojunction is negligibly small, verified by both the HAADF-STEM image and atomic force microscope measurements. Note that the spin diffusion length λs is always larger than the mean free path λ of the electron (). By simply considering λs = (1 − 1.5)λ, we can roughly estimate that Gsh ~ (1.3 to 1.8) GAFM. On the basis of , we have the upper limit of , which is close to our . As a result, the of our spin homojunction and concomitant with the interfacial transparency has almost reached its physical limit.