Introduction

Efficient manipulation of magnetic materials is essential for spintronic devices. While spin-orbit torques (SOTs)1,2 are well established to be an effective tool to manipulate metallic 3d ferromagnets (FMs), whether they can effectively control antiferromagnetically-ordered systems has remained elusive despite the recent blooming of interest in ferrimagnets (FIMs) and antiferromagnets (AFs)3,4,5,6,7,8,9. Experimentally, for reasons unclear, the SOTs exerted on nearly compensated FIMs5,6,7,10 are often measured to be considerably weaker than those on 3d FMs for a given spin-current generator (by up to >20 times, see below). More strikingly, it remains under debate whether perfectly compensated FIMs and collinear AFs (Ms = 0 emu/cm3) can be switched at all by SOTs11,12,13.

Microscopically, SOTs have been widely assumed as an interfacial transfer of spin (i.e., spin dephasing length λdp ≈ 0 nm for transverse spin current) that is independent of the bulk properties of the magnetic layer, such as in drift-diffusion analyses14,15,16. Under this assumption, spin current entering the magnet from an adjacent spin-generating layer is absorbed fully by the magnetization via dephasing to generate SOTs, and the damping-like SOT efficiency per current density (\({\xi }_{{DL}}^{j}\)) will depend only on the spin Hall ratio (θSH) of the spin-generating layer and the spin transparency (Tint) of the interface which determines what fraction of the spin current enters the magnet17,18, i.e.,

$${\xi }_{{DL}}^{j}={T}_{{{{{{\rm{int}}}}}}}{\theta }_{{{{{{\rm{SH}}}}}}}.$$
(1)

This picture is a reasonable approximation for sufficiently thick metallic FMs that have a short λdp (≤1 nm) due to strong exchange coupling19,20,21 and a long spin diffusion length (λs) associated with spin relaxation due to spin-orbit scattering22,23. However, in antiferromagnetically-ordered systems λdp can be quite long, as predicted more than a decade ago24,25,26,27, which, as discussed below, questions the widely accepted models of “interfacial torques”, particularly, in FIMs with strong spin-orbit scattering. So far, any roles of the bulk properties of the magnetic layer, e.g., the competing spin relaxation rates, in the determination of \({\xi }_{{DL}}^{j}\) have been overlooked in SOT analyses.

Here, we report measurements of SOTs acting on ferrimagnetic FexTb1-x layers with strong spin-orbit coupling (SOC)8 by tuning the FexTb1-x composition and temperature (T). We find that, in contrast to the prediction of Eq. (1), \({\xi }_{{DL}}^{j}\)varies strongly with the degree of magnetic compensation for a given Tint, due to changes in the fraction of spin current that relaxes directly to the lattice via SOC instead of being absorbed by the magnetization to apply SOTs. These results uncover the critical role of spin relaxation rates of the magnetic layer and provide a unified understanding for the diverse SOT phenomena in ferromagnetic and antiferromagnetically-ordered systems.

Results and discussion

Sample details

For this work, we sputter-deposited Pt0.75Ti0.25 (5.6 nm)/FexTb1-x (8 nm) bilayers with different Fe volumetric concentrations (x = 0.3–1). The Pt0.75Ti0.25 layer, a dirty-limit Pt alloy with strong intrinsic spin Hall effect17, sources spin current that exerts SOT on the FIM FexTb1-x. Each sample was deposited by co-sputtering on an oxidized Si substrate with a 1 nm Ta seed layer, and protected by a 2 nm MgO and a 1.5 nm Ta layer that was oxidized upon exposure to atmosphere. For electrical measurements, the samples were patterned into 5 × 60 µm2 Hall bars by photolithography and ion milling with a water-cooled stage. After processing, the magnetization hysteresis of the FexTb1-x measured from the anomalous Hall voltage (VAH) in patterned Hall bars shows fairly close coercivity (perpendicular depinning field) and squareness as the magnetization of unpatterned regions of the films measured by a superconducting quantum interference device (see Fig. 1a, b and Method). As shown in Fig. 1a–d, the FexTb1-x has strong bulk perpendicular magnetic anisotropy (PMA) for 0.3 ≤ x ≤ 0.62 and well-defined in-plane magnetic anisotropy for 0.75 ≤ x ≤ 1. All the PMA samples have large anisotropy fields (14.4–72.2 kOe, as estimated from the fits in Supplementary Fig. S2a) and square hysteresis loops for both the out-of-plane magnetization and anomalous Hall voltage.

Fig. 1: Magnetization and anomalous Hall voltage hysteresis.
figure 1

a Magnetization vs out-of-plane field (Hz) and b Anomalous Hall voltage (VAH) vs Hz for Pt0.75Ti0.25 (5.6 nm)/Fe0.59Tb0.41 (8 nm) under a sinusoidal electric field E = 30 kV/m, indicating strong perpendicular magnetic anisotropy and a high coercivity of ≈1 kOe. VAH vs Hz for Pt0.75Ti0.25 (5.6 nm)/FexTb1-x (8 nm) with c perpendicular (x = 0.3, 0.43, and 0.61) and d in-plane magnetic anisotropy (x = 0.75, 0.85, and 1).

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Strong variation of spin-orbit torques with composition and temperature

We performed harmonic Hall voltage response (HHVR) measurements28,29 by carefully separating out any potential thermoelectric effects (see Supplementary Note 2 for details). We choose the HHVR technique with excitation of a sinusoidal electric field E (typically 30 kV/m) because it allows very accurate, consistent determination of SOTs for both IMA and PMA samples30,31,32 without introducing significant thermal heating (Fig. 1a, b and Supplementary Fig. S4). We calculate the SOT efficiency using \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) = (2e/\(\hslash\))Ms tFeTb HDL/jc18, where e is the elementary charge, \(\hslash\) the reduced Plank’s constant, tFeTb the FexTb1-x thickness, Ms the saturation magnetization of the FexTb1-x (Supplementary Note 1), and jc = E/ρxx is the sinusoidal current density in the Pt0.75Ti0.25 with resistivity ρxx (jc ≈ 2.2 × 106A/cm2 for E = 30 kV/m). HDL is the current-driven damping-like SOT field. The “planar Hall correction” is negligible for these PMA FexTb1-x samples (VPh/VAH < 0.1, Supplementary Fig. S3).

In Fig. 2a, b, we show the measured values of Ms and \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) at 300 K for the Pt0.75Ti0.25/FexTb1-x bilayers with different FexTb1-x compositions (we refer to this as the composition series). Ms decreases monotonically by a factor of 33, from 1560 emu/cm3 for x = 1 (pure Fe, 3d FM) to 47 emu/cm3 for x = 0.5 (nearly full compensation), and then increases slowly as x further decreases. More details about the composition dependent magnetic properties are shown in Supplementary Note 1. As x decreases in the Fe-dominated regime (x ≥ 0.5), \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) decreases by a factor of 7 at 300 K, first slowly from 0.38 ± 0.02 for x = 1 to 0.27 ± 0.01 for x = 0.61 and then more rapidly to 0.054 ± 0.002 for x = 0.5. \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) increases slowly with decreasing x in the Tb-dominated regime (x < 0.5). The field-like SOT from the same HHVR measurements is smaller than \({\xi }_{{DL}}^{j}\) for each x and also varies with x (Supplementary Fig. S7). We also measured Ms and \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) of Pt0.75Ti0.25/Fe0.59Tb0.41 as a function of temperature (we refer to this as the temperature series). Upon cooling from 350 K to 25 K, Ms and \({\xi }_{{DL}}^{j}\) for the Pt0.75Ti0.25/Fe0.59Tb0.41 sample are tuned by >2 times (Fig. 2c) and by >7.5 times (Fig. 2d), respectively. The dramatic tuning of \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) by the FexTb1-x composition and temperature is a striking observation because it suggests a strong dependence of SOTs on some bulk properties of the FexTb1-x, in contrast to \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) for heavy metal (HM)/3d FM samples, which is insensitive to the type of the FM33 and temperature34,35. Apparently, \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) for the Pt0.75Ti0.25/FexTb1-x correlates closely with net magnetization (Fig. 2a, b) but not in a proportional or monotonic manner (Fig. 2e, f), suggesting a rather critical role of the net magnetization as well as another bulk effect (which, as we discuss below, is spin-orbit scattering) in the determination of \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\).

Fig. 2: Spin-orbit toque and spin-mixing conductance.
figure 2

a, b Saturation magnetization (Ms) and Damping-like SOT efficiency per unit current density (\({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\)) for Pt0.75Ti0.25/FexTb1-x with different Fe concentration (x) at 300 K. c, d Ms and \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) for the Pt0.75Ti0.25/Fe0.59Tb0.41 at different temperatures. e \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) vs Ms for the Pt0.75Ti0.25/FexTb1-x with different Fe concentration (x) at 300 K. f \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) vs Ms for the Pt0.75Ti0.25/Fe0.59Tb0.41 at different temperatures. g Frequency dependence of ferromagnetic resonance linewidth (∆H) of the FeCoB layer in FeCoB (5.2 nm)/Ti (1 nm), FeCoB (5.2 nm)/Ti (1 nm)/Fe0.5Tb0.5 (8 nm), and FeCoB (5.2 nm)/Ti (1 nm)/Fe0.61Tb0.39 (8 nm) samples. The solid lines represent linear fits, the slopes of which yield the damping. In (a–e) some error bars are smaller than the data points. h \({G}_{{{{{{\rm{eff}}}}}}}^{\uparrow \downarrow }\) of the FeCoB/Ti/FexTb1-x interfaces measured from spin pumping into the FexTb1-x. The blue circles are for the composition series (300 K) and the red dots for the temperature series (x = 0.59). The blue dashed line represents \({G}_{{{{{{\rm{eff}}}}}}}^{\uparrow \downarrow }\) = 0.31 × 1015 Ω−1 m−2 previously reported for typical Pt/3d FM interfaces33. Error bars represent fitting uncertainty.

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Robustness of the spin Hall ratio and the effective spin-mixing conductance

These strong variations cannot be explained by changes in either θSH or Tint (Eq. (1)). First, θSH is a property of the Pt0.75Ti0.25 layer, not the FexTb1-x layer. The Pt0.75Ti0.25 layer is made identically for all of the samples, and has a sufficiently large resistivity (ρxx = 135 µΩ cm) such that its properties can hardly be affected significantly by either a neighboring layer or temperature. We have verified that \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) for a ferromagnetic Pt0.75Ti0.25 (5.6 nm)/FePt (8 nm) bilayer only has very weak temperature dependence (Supplementary Fig. S13), in good consistence with previous reports on other HM/3d FM samples34,35. We have also measured negligible SOT signal from the 8 nm FexTb1-x layers in control samples without a Pt0.75Ti0.25 layer (Supplementary Note 6), indicating that changes in our signals are not due to SOT arising from the FexTb1-x bulk. Note that a bulk torque of a magnetic layer is strongly thickness dependent36,37 and vanishes at small thicknesses of a few nm36.

As for the possibility of changes in Tint, if we employ a drift-diffusion analysis14,15,16, the effect on Tint of spin backflow at the Pt0.75Ti0.25/FexTb1-x interface should be proportional to the effective spin-mixing conductance (\({G}_{{{{{{\rm{eff}}}}}}}^{\uparrow \downarrow }\)) of the interface, i.e., Tint ≈ 2\({G}_{{{{{{\rm{eff}}}}}}}^{\uparrow \downarrow }\)/GPtTi38, with GPtTi = 1/λsρxx ≈ 1.3 × 105 Ω−1 m−1 being the spin conductance18,39 of the Pt0.75Ti0.25. To quantify \({G}_{{{{{{\rm{eff}}}}}}}^{\uparrow \downarrow }\), we measure the change in the damping (α) of a precessing Fe60Co20B20 (= FeCoB) layer due to the absorption of the FeCoB-emitted spin current at the FexTb1-x interfaces (Fig. 2 g and Supplementary Fig. S8). The samples used here had the structure FeCoB (5.2 nm)/Ti (1 nm) and FeCoB (5.2 nm)/Ti (1 nm)/FexTb1-x (8 nm). Each of these samples is sputter-deposited on a 1 nm Ta seed layer and protected by capping with MgO (2 nm)/Ta (1 nm). The value of α for the FeCoB layers is determined from the linear dependence of the ferromagnetic resonance linewidth (∆H, half width at half maximum) on the frequency (f) using the relation ΔH = ΔH0 + 2παf/γ, where ΔH0 is the inhomogeneous broadening of the linewidth and γ the gyromagnetic ratio. The damping enhancement of the FeCoB layer induced by spin pumping into the 8 nm FexTb1-x layers, ∆α = αFeCoB/Ti/FeTb - αFeCoB/Ti, can be related to \({G}_{{{{{{\rm{eff}}}}}}}^{\uparrow \downarrow }\) as refs. 40,41,42

$$\triangle \alpha=\gamma {{{\hslash }}}^{2} {G}_{{{{{{\rm{eff}}}}}}}^{\uparrow \downarrow }/2{e}^{2}{M}_{{{{{{\rm{FeCoB}}}}}}}{t}_{{{{{{\rm{FeCoB}}}}}}}$$
(2)

where tFeCoB = 5.2 nm and MFeCoB = 1255 emu/cm3 is the saturation magnetization of the FeCoB layer as measured by SQUID. The value of α = 0.0053 for the bare FeCoB/Ti sample with no FexTb1-x coincides closely with the intrinsic damping of FeCoB (≈0.00633), indicating that the damping in this system does not contain any significant contributions from interfacial two-magnon scattering or spin memory loss. As shown in Fig. 2h, \({G}_{{{{{{\rm{eff}}}}}}}^{\uparrow \downarrow }\) of the FeCoB/Ti/FexTb1-x interfaces is insensitive to temperature and the FexTb1-x composition within experimental uncertainty, and has a value as high as that of typical 3d FM/Pt interfaces (≈0.31 × 1015 Ω−1 m−2) 33. This indicates that compensated FexTb1-x alloys act as spin sinks that are just as good as 3d FMs and Pt, and that there is no enhancement in the amount of spin reflection and backflow due to magnetic compensation. In principle, changes in Tint for SOT measurements could also arise from spin memory loss induced by interfacial SOC29,43,44, but this should be a minor effect for Tint of our un-annealed Pt0.75Ti0.25/FexTb1-x just as is the case of un-annealed Pt/Co45. As noted above, we also do not observe any enhancement in damping due to spin memory loss in the spin-pumping measurements. Note that the large and robust \({G}_{{{{{{\rm{eff}}}}}}}^{\uparrow \downarrow }\) for electron-mediated spin transport at the metallic FexTb1-x interface is in sharp contrast to that of ferrimagnetic insulator interfaces46,47 where thermal magnons mediate the spin transport such that a very low magnetic moment density reduces \({G}_{{{{{{\rm{eff}}}}}}}^{\uparrow \downarrow }\).

Variation of SOT with the relative spin relaxation rates

Since we have ruled out any significant change in θSH or Tint as contributing to the large variations we measure in \({\xi }_{{DL}}^{j}\) as a function of composition and temperature, these large variations must be due to physics that is not captured in the simple Eq. (1). We show below that spin relaxation induced by SOC in the bulk of the FexTb1-x layer is the most likely physics that is neglected in Eq. (1). As schematically shown in Fig. 3a, a spin current, in general, can be relaxed in a magnetic layer through two competing mechanisms: exchange-interaction-induced angular momentum transfer from spin current to magnetization (with a relaxation rate τM−1 and a length scale of λdp) and bulk spin-orbit-scattering-induced loss of spin angular momentum to the lattice (with a relaxation rate τso−1 and a length scale of λs). Theory24,25,26,27 and experiments48,49 have suggested that, in fully or partially compensated magnetic systems, the rate of spin angular momentum transfer via exchange interaction can be greatly decreased because of cancellations between exchange fields of antiferromagnetically-aligned magnetic sub-lattices, resulting in long λdp and low τM−1. Spin-orbit scattering is well known to result in spin relaxation in both magnetic and non-magnetic materials22,23. While relatively weak in light, highly conductive 3d FMs (e.g., λs was measured to be 5–8 nm for Fe, Co, and CoFe at room temperature22,23), spin-orbit scattering becomes very strong in strong-SOC, resistive materials (e.g., dirty heavy metals18 and rare-earth FIMs) and substantially reduces λs and enhances τso−1. This makes it possible for spin currents in FIMs to relax partially or even primarily via spin-orbit scattering to the lattice, instead of applying a spin-transfer torque to the magnetization.

Fig. 3: Variation of SOT with the relative spin relaxation rates.
figure 3

a Schematic of the spin relaxation processes that can influence the SOT, highlighting the competition between exchange interaction (with relaxation rate τM−1 ∝ Ms) and spin-orbit scattering (τso−1 ∝ ζsoτe−1). Only the spin current relaxed by exchange interaction contributes to SOTs. b Momentum scattering time (τe), c Estimated SOC strength (ζso), d ζsoτe−1, and e \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) of Pt0.75Ti0.25/FexTb1-x vs Ms/ζsoτe−1 for the composition series (x = 0.3–1, T = 300 K, black circles) and for the temperature series (x = 0.59, T = 25-300 K, red circles). The solid curve in (e) represents the fit of the data to Eq. (3). Error bars represent fitting uncertainty.

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We can consider how the SOT should scale as a function of the ratio τM−1/τso−1. Quantitative measurements of these rates (e.g., from the dependence on layer thicknesses of spin valve or spin-pumping experiments) are quite challenging because the bulk properties of FexTb1-x50 and other ferrimagnetic alloys48,49 vary sensitively with the layer thicknesses51,52,53 (e.g., the magnetic compensation, the bulk PMA, the orientation of the magnetic easy axis, and resistivity all change with thickness). Nonetheless, it is reasonable to expect τM−1 ∝ Ms for such ferrimagnetic alloys considering the canceling effects of the exchange fields from the antiferromagnetically-aligned magnetic sub-lattices. For the spin-orbit scattering rate, the Elliot-Yafet mechanism51,52,53 predicts τso−1 ∝ ζsoτe−1, where ζso is the bulk SOC strength and τe−1 is the momentum scattering rate. One can thus expect τM−1/τso−1 = kMs/ζsoτe−1, with k being a constant. Since only the spin current relaxed through exchange interaction with magnetization contributes to SOTs, we propose that

$${\xi }_{{{{{{\rm{DL}}}}}}}^{j}\approx {\xi }_{{{{{{\rm{DL}}}}}},0}^{j}{{\tau }_{{{{{{\rm{M}}}}}}}}^{-1}/\left({{\tau }_{{{{{{\rm{M}}}}}}}}^{-1}+{{\tau }_{{{{{{\rm{so}}}}}}}}^{-1}\right)={\xi }_{{{{{{\rm{DL}}}}}},0}^{j}{\left(1+{\left({{kM}}_{{{{{{\rm{s}}}}}}}/{\zeta }_{{{{{{\rm{so}}}}}}}{{\tau }_{{{{{{\rm{e}}}}}}}}^{-1}\right)}^{-1}\right)}^{-1}$$
(3)

where \({\xi }_{{{{{{\rm{DL}}}}}},0}^{j}\) is the value of \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) in the limit τM−1/τso−1 ≫ 1.

Figure 3b–d shows the estimated values of τe, ζso, and ζsoτe−1 for both our composition series (x = 0.3-1, T = 300 K, black symbols) and our temperature series (x = 0.59, T = 25–350 K, red symbols). Here, the value of τe is a rough estimate from the resistivity of the FexTb1-x following Drude model ρFeTb = m*/ne2τe, where m* is the effective mass of the carriers and n is the carrier density measured from the ordinary Hall coefficient (ROH = 1/ne for a single-band model54,55, see Supplementary Note 5 for more details). τe of the FexTb1-x varies by a factor of ≈6 by composition and a factor of ≈7 by temperature, suggesting a significant tuning of the Fermi surface properties. The increase of τe in the FexTb1-x metal with raising temperature likely originates from electron-electron interaction56 (ρFeTb \(\propto\) T 1/2, see Supplementary Fig. S14a) or magnetic Brillouin zone scattering (the periodic potentials due to antiferromagnetic alignment of the magnetic sublattices can produce an additional magnetic Brillouin zone, of smaller volume in k-space than the ordinary lattice potential, whose planes further incise and contort the Fermi surface57). Both electron-electron interaction and magnetic Brillouin zone lead to additional electron scattering manifesting as a resistivity upturn upon cooling.56,57 The average SOC strength of the FexTb1-x is estimated as ζso ≈ xζso,Fe + (1-x)ζso,Tb following the linear dependence on alloy composition of the bulk SOC58 and the theoretical values59 of ζso,Fe = 0.069 eV for Fe and ζso,Tb = 0.283 eV for Tb (while the actual values of ζso,Fe and ζso,Tb within the amorphous FexTb1-x, which are not trivial to obtain, may be slightly different from the theoretical ones, this estimation should, at least, provide a reasonable functional approximation for the expected dramatic variation of ζso as a function of the FexTb1-x composition, from light Fe to Tb-rich Fe0.3Tb0.7). We estimate that ζsoτe−1 decreases by a factor of >16 as x varies between 0.3 to 1 and by a factor of >7 as temperature increases from 25 K to 350 K (Fig. 3d). In Fig. 3e, we plot \({\xi }_{{DL}}^{j}\) as a function of Ms/ζsoτe−1 for both the composition series and the temperature series. As Ms/ζsoτe−1, or equivalently τM−1/τso−1, decreases, we find that \({\xi }_{{DL}}^{j}\) decreases first slowly and then rapidly towards a vanishing value. The variation of \({\xi }_{{DL}}^{j}\) with Ms/ζsoτe−1 can be fit very well by Eq. (3) with \({\xi }_{{{{{{\rm{DL}}}}}},0}^{j}\) = 0.395 ± 0.022 and k = (1.66 ± 0.23) × 1011 s−1 emu−1 cm3 eV. Note that the applicability of the Drude model, the approximated value of m*, and the single-band model for the ordinary Hall effect for estimating τe−1 is not essential for our conclusion of the variation of \({\xi }_{{DL}}^{j}\)with relative spin relaxation rates, since similar scaling in Fig. 3e is present even when simply plotting \({\xi }_{{DL}}^{j}\) as a function of Ms/ζsoρxx (Supplementary Fig. S11). In the above discussions, we ignored any effect of local distribution of magnetizations (also known as sperimagnetism50) because it, if present, may only indirectly affect the average spin relaxation rates of spin-magnetization exchange interaction and spin-orbit scattering via reducing the net magnetization and strengthening SOC-related momentum scattering of spin carriers, respectively.

Since the strong variation of \({\xi }_{{DL}}^{j}\)with relative spin relaxation rates we propose here is unlikely to be specific just to the HM/FexTb1-x system (as indicated by the widespread presence of spin-orbit scattering in various materials22,23 and by the general fact that the SOT provided by a given spin-current generator is significantly weaker on FIMs than on FMs, see Table 1), we generalize Eq. (1) as

$${\xi }_{{DL}}^{j}={T}_{{{{{{\rm{int}}}}}}}{\theta }_{{{{{{\rm{SH}}}}}}}{{\tau }_{{{{{{\rm{M}}}}}}}}^{-1}/\left({{\tau }_{{{{{{\rm{M}}}}}}}}^{-1}+{{\tau }_{{{{{{\rm{so}}}}}}}}^{-1}\right).$$
(4)
Table 1 Comparison of \({\xi }_{{{{{{\rm{DL}}}}}}}^{{{{{{\rm{j}}}}}}}\) for FIMs and 3d FMs in contact with spin current sources that have similar resistivities, thicknesses, and thus similar values of θSH and Tint
Full size table

This generalized equation should apply to FMs, FIMs, and AFs that are metals or insulators. In magnetic insulators, an incident spin current carried by magnons transfers angular momentum to the magnetization via exchange interaction (with spin relaxation rate τM−1) and also to the lattice via spin-orbit scattering of spin carriers (with spin relaxation rate τso−1).

We note that our conclusions are contrary to some previous experiments, which reported \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) to remain constant60,61,62 or even diverge63 near the magnetic compensation point of HM/CoTb or HM/CoFeGd bilayers. While it might be possible that τM−1 /τso−1 is different in CoTb and CoFeGd compared to FexTb1-x (e.g., Gd has zero atomic orbital angular momentum8 and thus considerably weaker SOC than Tb), we also question these previous conclusions for a variety of technical experimental reasons. In three of the previous experiments, the PMA of the FIM layer was weak and showed gradual magnetization hysteresis62, non-parabolic first-harmonic signal in HHVR measurements60,62,63, and/or non-linear second-harmonic signal in HHVR measurements60,62 as a function of a small in-plane applied magnetic field. This indicates magnetization behavior outside of the simple macrospin model assumed in the HHVR analysis. The HHVR results in refs. 62,63 also applied “planar Hall correction”, the latter, if significant, causes erroneous estimates of \({\xi }_{{DL}}^{j}\) (see refs. 30,64,65,66 for more discussions). References 60,61 reported substantial changes of sample properties before and after device patterning, resulting in large uncertainties in the estimation of Ms and \({\xi }_{{DL}}^{j}\)for those samples. The loop-shift measurements in ref. 61 also applied large dc current densities of ~107 A/cm2, close to the switching current density, in the resistive Ta/CoxTb1-x samples, leading to considerable Joule heating that had significantly altered the temperature, Ms, and ferrimagnetic compensation points of those samples. The latter resulted in additional uncertainties in those loop-shift results of \({\xi }_{{DL}}^{j}\) and ultimately prevented resolving the variation of \({\xi }_{{DL}}^{j}\) with the bulk properties of FIMs (i.e., τM−1 /τso−1) in that work.

Scientific implications

Our finding of the critical role of the relative rates of spin-orbit-induced relaxation to the lattice versus spin transfer to the magnetization has important implications for this field and resolves outstanding puzzles in previous experiments. We first schematically demonstrate in Fig.4a the effect of spin-orbit scattering on SOTs suggested by Eq. (4). Only in absence of spin-orbit scattering (τso−1 = 0), should the simple form of Eq. (1) apply such that \({\xi }_{{DL}}^{j}\) is independent of τM−1 and thus Ms for a sufficiently thick magnetic layer with nonzero Ms. This is a good approximation only for magnetic materials with τM−1 /τso−1 ≫ 1, e.g., 3d FMs that have high Ms, low resistivity, and weak SOC. However, in the presence of non-negligible spin-orbit scattering (τso−1 > 0), \({\xi }_{{DL}}^{j}\)decreases more and more rapidly with reducing τM−1/τso−1 (with τM−1 ∝ Ms). This is generally the case of uncompensated “AF” domains (Ms > 0 emu/cm3) and FIMs with strong SOC and large resistivities (e.g., FexTb1-x and CoxTb1-x). However, \({\xi }_{{DL}}^{j}\) always diminishes at zero τM−1 (Ms = 0 emu/cm3), e.g., in perfectly compensated FIMs and AFMs and non-magnetic materials.

Fig. 4: Schematic depicts of the implications.
figure 4

a Dependence on τM−1 (∝ Ms) of the efficiency of the damping-like spin-orbit torque (\({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\)) exerted by a given spin current on a magnetic layer with zero spin-orbit scattering (τso−1 = 0, red) and non-negligible spin-orbit scattering (τso−1 ≠ 0, blue), highlighting the critical role of spin-orbit scattering. “Potential” spin current switching of local atomic magnetizations by local spin-orbit torque in b perfectly compensated single-layer antiferromagnet, c perfectly compensated synthetic antiferromagnet, and d disordered ferrimagnet, with negligible barrier against switching (magnetic anisotropy, pinning, damping, etc.). In the case of perfectly compensated antiferromagnets, a spin current generates zero net spin-orbit torque and zero net magnetization change.

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Therefore, SOTs will be reduced in any experiments which use magnetic free layers for which τM−1 is not much greater than τso−1 or not sufficiently large to allow complete spin relaxation within the layer thickness. The condition τM−1/ τso−1 ≪ 1 resolves the puzzles why the measured \({\xi }_{{DL}}^{j}\)values for SOT acting on nearly-compensated FIMs are often several to over 20 times smaller than for corresponding measurements using 3d FMs (see Table 1 for a few representative examples with spin current sources that have similar resistivities, thicknesses, and thus similar values of θSH and Tint)5,6,7,10,35,67,68,69. We have also verified from study of a large number of samples that τM−1/(τM−1 + τso−1) is 0.58 for 5 nm Co0.65Tb0.35 layers at room temperature such that \({\xi }_{{{{{{\rm{DL}}}}}}}^{j}\) for Pt-X/Co0.65Tb0.35 is only 58% of that of Pt-X/Co for given Pt-X (Pt-X being Pt-based alloys and multilayers)70. Spin-orbit scattering within the magnetic layer should, therefore, be minimized and the average exchange coupling maximized for efficient SOT devices.

Taking into account the relative spin relaxation rates also clarifies the ongoing debate as to whether AFs can be switched at all by SOTs11,12,13. The previous observation of current-driven switching of uncompensated magnetic domains or magnetizations embedded within AF hosts using transport or imaging methods71,72,73,74,75 (which is not real switching of AF Néel vector) is naturally explained by the small but sizable SOTs in nearly but not fully-compensated systems, while the absence of macroscopic transport evidence of current switching of some more uniform HM/AF11,12,13 is well consistent with the diminishment of SOTs in fully-compensated systems (Fig. 4a).

While in principle a spin current still has the potential to switch the local nonzero atomic magnetizations within a perfectly compensated FIMs, single-layer AFs, and synthetic AFs (e.g., FM/Ru/FM) by local SOT on each magnetic site (Fig. 4b–d), such switching is unfortunately very inefficient in overcoming the switching barriers of the samples (e.g., magnetic anisotropy, pinning field, damping, etc.). It is rather typical that the effective magnetic anisotropy fields and pinning fields (i.e., the coercivities in the magnetization hysteresis) of hard rare-earth transition-metal FIMs50,60,61,62 and synthetic AFs11 turn to diverge at the magnetization compensation point, making them unswitchable for a realistic spin current. Collective 180o reversal of the Néel vector of collinear AFs also requires net SOT to overcome the barriers against switching (e.g., magnetic anisotropy, pinning, damping, etc.), which is challenging to achieve, particularly so in present of spin-orbit scattering. Therefore, fully-compensated FIMs and AFs themselves are most likely not an option as “Néel-vector-type” free layer of magnetic memory in terms of electrical writability and readability. We also note that field-like effective SOT field is too weak to reach the spin-flop field required for 90o rotation of Néel vector (typically of the order of 1−103 kOe)76.

In summary, we have shown that the strength of SOTs depends critically on the ratio of rate of spin-orbit-induced spin relaxation within a magnetic layer relative to the rate of exchange-induced spin transfer to the magnetization. We find experimentally that SOT efficiencies decrease strongly upon approaching the magnetic compensation point in ferrimagnetic FexTb1-x due to a decrease in the rate of exchange-induced spin transfer on account of partial cancellation between the oppositely-directed exchange interactions from the magnetic sub-lattices. Near the compensation point, spin-orbit-induced spin relaxation dominates over spin transfer to the magnetization so that the measured SOT goes to zero. These results suggest the breakdown of the “interfacial torques” concept in FIMs and AFs. We find no indication of any dependence of the spin transparency of FexTb1-x interfaces on the degree of compensation. Our finding suggests that it will be essential to modify spin transport models that assumed an infinite τM−1 to include spin decoherence by spin-orbit scatting on an equal footing with dephasing by the exchange interaction. This work provides not only a unified understanding of the very different efficiencies of SOTs that have been reported in the literature for FMs, FIMs, and AFs, but also insight about how the different sources of spin relaxation should be optimized in the design of FIMs and AFs for spintronic technologies8,9.

Methods

Sample fabrication

Samples for this study includes Pt0.75Ti0.25 (5.6 nm)/FexTb1-x (8 nm) bilayers, FeCoB (5.2 nm)/Ti (1 nm)/FexTb1-x (8 nm), FeCoB (5.2 nm)/Ti (1 nm), Pt0.75Ti0.25 (5.6 nm)/FePt (8 nm), and FexTb1-x (8 nm), with x being the Fe volumetric concentration (x = 0.3−1). Each sample was sputter-deposited on an oxidized Si substrate with a 1-nm Ta seed layer, and protected by a 2 nm MgO and a 1.5 nm Ta layer that was oxidized upon exposure to atmosphere. The FeCoB (5.2 nm)/Ti (1 nm)/FexTb1-x (8 nm) and FeCoB (5.2 nm)/Ti (1 nm) samples were diced into pieces for ferromagnetic resonance measurements. The other samples (2 × 2 cm in area) were partly patterned into 5 × 60 µm2 Hall bars by photolithography and ion milling with a water-cooled stage. After processing, the samples were separated into pieces by a dicing saw (Fig. 5). The patterned pieces were used for harmonic Hall voltage response (HHVR) measurements, and un-patterned regions of the films for magnetization characterizations using a superconducting quantum interference device (SQUID). The “SQUID” pieces underwent the same processing as the Hall bars during the device fabrications, providing good consistency between the electrical and magnetic properties of the samples in our analyses.

Fig. 5: Photo of a Pt0.75Ti0.25/FexTb1-x sample separated into pieces by a dicing saw.
figure 5

The patterned pieces are used for HHVR measurements and un-patterned regions for SQUID measurements. The “SQUID” pieces underwent the same processing as the Hall bars during the device fabrication, providing good consistency between the electrical and magnetic properties of the samples in our analyses.

Full size image

Measurements

The saturation magnetization of each sample was measured by a Quantum Design SQUID as a function of magnetic field and temperature. The SOTs were measured using harmonic Hall voltage response (HHVR) measurements, during which a Signal Recovery DSP Lock-in Amplifier (Model 7625) or a Stanford Research System Lock-in Amplifier (Model SR830) was used to source a sinusoidal electric field E (typically 30 kV/m) onto the Hall bars and to detect the first and second harmonic Hall voltage responses. The magnetic damping was measured using flip-chip ferromagnetic resonance with a radio frequency signal generator, two Stanford Research System Lock-in Amplifiers (Model SR830), and a sweeping in-plane magnetic field.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.