In ferromagnetic metals, transverse spin currents are thought to be absorbed via dephasing—i.e., destructive interference of spins precessing about the strong exchange field. Yet, due to the ultrashort coherence length of ≈1 nm in typical ferromagnetic thin films, it is difficult to distinguish dephasing in the bulk from spin-flip scattering at the interface. Here, to assess which mechanism dominates, we examine transverse spin-current absorption in ferromagnetic NiCu alloy films with reduced exchange fields. We observe that the coherence length increases with decreasing Curie temperature, as weaker dephasing in the film bulk slows down spin absorption. Moreover, nonmagnetic Cu impurities do not diminish the efficiency of spin-transfer torque from the absorbed spin current. Our findings affirm that the transverse spin current is predominantly absorbed by dephasing inside the nanometer-thick ferromagnetic metals, even with high impurity contents.

Spin currents underpin a variety of fundamental condensed-matter phenomena and technological applications,1–3 especially those based on magnetic materials. Of particular interest is coherent transverse spin current, where the flowing spins are uniformly polarized transverse to the magnetization. This spin current generates a spin-transfer torque that can switch a nanomagnetic memory or drive a GHz-range oscillator.4–6 While spins may be carried by magnons7 and phonons,8 they are often primarily carried by electrons in practical metallic multilayers incorporating ferromagnetic thin films. Therefore, it is crucial to understand the nanoscale transport of the electron-mediated transverse spin current in ferromagnetic metals.

A spin current in any material ultimately becomes absorbed (loses coherence) within a finite length scale.1 In ferromagnetic metals, transverse spin-current absorption can occur via dephasing,9–11 i.e., destructive interference of coherent spins that precess about the magnetic exchange field. The dephasing mechanism is illustrated in Fig. 1: the transverse electronic spins enter the ferromagnetic metal with a wide distribution of incident wavevectors; these spins traverse and precess about the magnetic exchange field at different rates, thereby averaging out the net transverse polarization (destroying the phase coherence) of the spin current within a finite length scale. Another possible mechanism of spin-current absorption is diffusive spin-flip scattering:12 when electrons carrying the spin current are scattered, e.g., by impurities or an interface, the orientation of the propagating spins may be flipped to various orientations.

FIG. 1.

Dephasing of a transverse spin current generated by FMR in the ferromagnetic (FM) spin source. The propagating spins are coherent in the normal metal (NM) spacer—as illustrated by the aligned black arrows–but they enter the spin sink with different incident wavevectors. In the FM spin sink, the spins precess about the ferromagnetic exchange field (red vertical arrows) by different amounts, thereby losing phase coherence.

FIG. 1.

Dephasing of a transverse spin current generated by FMR in the ferromagnetic (FM) spin source. The propagating spins are coherent in the normal metal (NM) spacer—as illustrated by the aligned black arrows–but they enter the spin sink with different incident wavevectors. In the FM spin sink, the spins precess about the ferromagnetic exchange field (red vertical arrows) by different amounts, thereby losing phase coherence.

Close modal

Prior experiments13 have quantified the absorption length scale—i.e., coherence length λc—of transverse spin current through ferromagnetic resonance (FMR) spin pumping.14 These experiments indicate λc1 nm from the ferromagnetic film thickness, where the measured spin absorption saturates. This ultrashort λc is presumably due to rapid dephasing9–11 from the strong ferromagnetic exchange field of ≫ 100 T.15 Hence, the conventional wisdom is that transverse spin current is absorbed via dephasing, rather than spin-flip scattering. However, λc1 nm corresponds to a nominal film thickness of a few lattice parameters, likely just at the threshold of forming a continuous film layer. Spin-flip scattering at the “interface” could be significant for such ultrathin ferromagnets. Thus, a plausible alternative explanation for λc1 nm is that interfacial spin-flip scattering saturates at the ferromagnetic thickness of ≈1 nm. Spin-flip scattering by impurities in the ferromagnet bulk may also contribute to the short λc. Therefore, it generally remains a challenge to distinguish spin-flip scattering from spin dephasing.

In this Letter, we experimentally address the following fundamental question: which mechanism—spin dephasing or spin-flip scattering—is responsible for the ultrashort coherence length λc of the transverse spin current in ferromagnetic metals? By employing the FMR spin pumping protocol similar to Ref. 13, we quantify λc for ferromagnetic Ni films alloyed with nonmagnetic Cu that reduces the ferromagnetic exchange strength. Our hypothesis is that λc must increase with increasing nonmagnetic Cu impurity content, if dephasing in the bulk is dominant. On the other hand, if spin-flip scattering at the interface is dominant, λc is expected to remain mostly unchanged or become shorter as the Cu impurities may enhance interfacial scattering. Similarly, λc should shorten if spin-flip scattering by the impurities in the bulk dominates. Thus, testing the above hypothesis permits us to confirm—or refute—the long-held notion that dephasing in the ferromagnet's bulk drives transverse spin-current absorption. It is also timely to examine basic spin transport in NiCu alloys, which have attracted attention for their reportedly sizable spin–orbit effects16–18 that may hold promise for spintronic devices.

Ni and Cu readily form homogeneous solid solutions, permitting continuous tuning of ferromagnetic exchange while maintaining the same face-centered cubic structure in NiCu alloys. Figure 2 summarizes the Curie temperatures TC (the metric for the ferromagnetic exchange strength) and electrical resistivities ρ (the metric for the electronic scattering rate) of 10-nm-thick Ni, Ni80Cu20, and Ni60Cu40 films. We limit the maximum Cu content to 40 at. % to attain ferromagnetism close to room temperature, where our FMR spin pumping measurements were performed. The monotonic drop in TC seen in Fig. 2(a) is consistent with prior reports19,20 and verifies that the Cu impurities dilute the ferromagnetic exchange. The monotonic increase in ρ [Fig. 2(b)] confirms enhanced electronic scattering by the Cu impurities in the film bulk.

FIG. 2.

Compositional dependence of (a) the Curie temperature TC and (b) the electrical resistivity ρ of 10-nm-thick Ni(Cu) films.

FIG. 2.

Compositional dependence of (a) the Curie temperature TC and (b) the electrical resistivity ρ of 10-nm-thick Ni(Cu) films.

Close modal

To derive λc, we conducted FMR spin pumping measurements on film stacks Si-SiO2(substrate)/Ti(3)/Cu(3)/Ni80Fe20(10)/Ag(5)/Ni(Cu)(0–10)/Ti(3), where Ni(Cu) denotes the Ni, Ni80Cu20, or Ni60Cu40 “spin sink.” The Ti/Cu seed bilayer promotes narrow FMR linewidths (minimizing two-magnon scattering21) in the NiFe “spin source,” crucial for straightforward spin pumping measurements. The Ag spacer suppresses direct magnetic coupling between the NiFe source and Ni(Cu) sink, such that spin transport from the source to the sink is mediated solely by electrons without complications from magnon interactions.22 Ag is selected as the spacer, instead of the oft-used Cu, to reduce atomic intermixing at the spacer/Ni(Cu) interface.

In the spin pumping scheme [Fig. 3(a)], a microwave field from a coplanar waveguide excites FMR in the NiFe source, such that the magnetization oscillates about the in-plane applied magnetic field. FMR generates a coherent ac spin current polarized transverse to the oscillation axis. This spin current is pumped through the nonmagnetic Ag spacer and into the Ni(Cu) sink. Since the thickness of Ag here is much smaller than the spin diffusion length of ∼100 nm,12,23 the coherent spin current propagates with negligible absorption in the spacer.14,24 The polarization of the spin current is transverse to the magnetization of the Ni(Cu) sink, which is set by the applied field. The FMR condition of the Ni(Cu) layer is sufficiently far from that of the NiFe source, so Ni(Cu) serves as a passive sink that receives the spin current from the NiFe source.

FIG. 3.

(a) Illustration of FMR spin pumping with the NiFe spin source and the Ni(Cu) spin sink. (b) Frequency dependence of the FMR linewidth for different Ni80Cu20 spin sink thicknesses d. (c)–(e) Nonlocal damping enhancement Δα as a function of d, where the spin sink is (c) Ni, (d) Ni80Cu20, and (e) Ni60Cu40. The solid black lines indicate the fits with Eq. (1). The vertical dashed lines indicate the coherence length λc extracted from the fits.

FIG. 3.

(a) Illustration of FMR spin pumping with the NiFe spin source and the Ni(Cu) spin sink. (b) Frequency dependence of the FMR linewidth for different Ni80Cu20 spin sink thicknesses d. (c)–(e) Nonlocal damping enhancement Δα as a function of d, where the spin sink is (c) Ni, (d) Ni80Cu20, and (e) Ni60Cu40. The solid black lines indicate the fits with Eq. (1). The vertical dashed lines indicate the coherence length λc extracted from the fits.

Close modal

Any spin-current absorption in the Ni(Cu) sink constitutes an additional loss of spin angular momentum, which manifests in an enhancement of Gilbert damping Δα in the NiFe source.14,25 As shown in Fig. 3(b), the total measured Gilbert damping parameter α is obtained from the linear slope of the FMR linewidth ΔH plotted against the microwave frequency f, μ0ΔH=μ0ΔH0+2πγαf, where μ0ΔH0<0.1 mT is the inhomogeneous linewidth broadening and γ2π=29.8 GHz/T is the gyromagnetic ratio for NiFe. By averaging samples from seven deposition runs, the baseline Gilbert damping parameter of NiFe/Ag without a Ni(Cu) sink is found to be α0=0.00693 ± 0.00014, similar to other reports on NiFe thin films.26,27Figure 3(b) shows an increased slope of ΔH vs f with finite Ni(Cu) sink thickness. This observation signifies a nonlocal damping contribution, Δα=αα0, due to spin absorption in the sink. Figures 3(c)–3(e) summarize the dependence of spin absorption, captured by Δα, on the spin-sink thickness d. For each d, an averaged α was obtained by measuring at least three separate sample pieces. The error bars for Δα are primarily from the scatter in α0.

For each Ni(Cu) sink composition, Δα rises at small d and then saturates [Figs. 3(c)–3(e)]. This behavior is consistent with spin-current absorption within a finite depth in the sink, such that there is essentially no additional absorption at dλc. We quantify λc by fitting our experimental data of Δα vs d. One possible approach is to employ a modified drift-diffusion model,28–30 but this involves multiple free parameters (e.g., complex transmitted spin-mixing conductance11,31) that could produce overdetermined fits. Instead, we employ a simpler empirical fitting function employed by Bailey et al.13,32,33 with only two parameters, i.e., λc and Δαsat,

Δα=Δαsatλc1Hdλcd+ΔαsatHdλc,
(1)

where Hdλc is the Heaviside step function centered at d=λc. From the resulting fits in Figs. 3(c)–3(e), we note that Δαsat is slightly higher for the Ni80Cu20 sink whereas it is lower for Ni60Cu40. We attribute this variation in Δαsat to the different spin-mixing conductances that depend on the effective spin susceptibilities in these magnetic spin sinks.34–37 We emphasize, however, that our focus here is on the length scale of transverse spin-current absorption, λc.

The values of λc from the fits with Eq. (1) are well over λc=1.2 ± 0.1 nm of the Ni80Fe20 alloy from Ref. 13. Specifically, we obtain λc=2.0 ± 0.2 nm for Ni, 3.0 ± 0.2 nm for Ni80Cu20, and 4.3 ± 0.5 nm for Ni60Cu40. These values exceed several atomic monolayers, strongly pointing to spin absorption in the bulk of the sink layer rather than at its interface.

We now consider which absorption mechanism in the bulk of Ni(Cu) is most consistent with the observation of longer λc with increasing Cu content. (i) Dephasing due to the ferromagnetic exchange field—A higher content of nonmagnetic Cu dilutes the ferromagnetic exchange field, hence slowing down the dephasing of the spin current. If dephasing dominates transverse spin absorption, λc should become longer with more Cu impurities. This scenario is indeed consistent with our observation. (ii) Spin-flip scattering due to impurities—A higher Cu impurity content enhances the momentum scattering of electrons [e.g., as evidenced by the increasing resistivity in Fig. 2(b) and a shorter mean free path38], which in turn increases the rate of spin-flips. The dominance of such spin-flip scattering (i.e., Elliott–Yafet spin relaxation expected in centrosymmetric metals at room temperature1,39,40) would yield shorter λc with more Cu impurities. This spin-flip-dominant scenario is contrary to our observation. We then deduce that dephasing, rather than spin-flip scattering, dominates the absorption of the transverse spin current in Ni(Cu) examined here.

It is worth noting that the Dyakonov–Perel spin-relaxation mechanism can also result in longer λc with increasing scattering.1,41 Yet, Dyakonov–Perel spin relaxation is another manifestation of dephasing, particularly from spins precessing about a spin–orbit field. Moreover, the dominance of Dyakonov–Perel spin relaxation would be surprising in centrosymmetric, polycrystalline Ni(Cu) at room temperature.39,40 We thus posit that the dephasing is primarily driven by the ferromagnetic exchange field.

To gain further insight into how λc scales with the diluted ferromagnetic exchange (i.e., decreasing TC), we plot λc against the inverse of TC for the Ni(Cu) compositions investigated in our work, along with Ni80Fe20 from Ref. 13. Figure 4 illustrates the central finding of this study: λc scales inversely with the ferromagnetic exchange strength (represented by TC). Again, the consistent explanation is that decreasing exchange—hence weaker dephasing—from the nonmagnetic Cu impurities enables the transverse spin current to remain coherent over a distance well above ≈1 nm. Our finding indicates that in these Ni-based systems, spin dephasing in the bulk remains dominant over interfacial or impurity-induced spin-flip scattering.

FIG. 4.

Transverse spin-current coherence length λc plotted against the inverse of the Curie temperature TC. The data point for Ni80Fe20 is from Ref. 13.

FIG. 4.

Transverse spin-current coherence length λc plotted against the inverse of the Curie temperature TC. The data point for Ni80Fe20 is from Ref. 13.

Close modal

The bulk nature of dephasing in these ferromagnets is distinct from prior reports on proximity-magnetized Pd and Pt films, in which the induced magnetic order is confined to a few monolayers at the interface.33,42,43 It is also noteworthy that Ni60Cu40 in our study is essentially on the trend line in Fig. 4, even though its TC is somewhat below room temperature (see Fig. 2) where the FMR spin pumping measurements were performed. This result suggests that spin-current dephasing may occur even in the bulk of a metal that is “almost” ferromagnetic with fluctuating magnetic order.44 Alternatively, the fact that λc for Ni60Cu40 is slightly below the trend line in Fig. 4 may signify that the spin-flip length scale in Ni60Cu40 is ≈4 nm, comparable to the dephasing length scale. Though beyond the scope of our present work, the evolution of λc for Cu content beyond 40 at. % would be an interesting subject for future experiments.

The above-described measurements of Δα (Fig. 3) detect spin absorption in the sink, but they provide no direct insight into what the spin current does inside the sink. Therefore, we examine the by-product of the transverse spin current interacting with the magnetization: spin-transfer torque. To this end, we employed the synchrotron-based x-ray ferromagnetic resonance (XFMR) technique24,45–47 at the Advanced Light Source Beamline 4.0.2,48 which leverages the element-specificity of x-ray magnetic circular dichroism (XMCD). This XFMR technique can directly detect the magnetization dynamics of a specific layer. Moreover, the out-of-plane spin transport here does not involve in-plane net charge transport, hence eliminating ambiguities from coexisting charge-to-spin conversion processes that plague standard electrical spin-torque measurements.49–51 

We conducted XFMR measurements on samples with a stack structure MgO(substrate)/Ti(3)/Cu(3)/Fe80V20(10)/Ag(5)/Ni(Cu)(5.3)/Ti(3). The (001)-oriented MgO crystal substrate permits high XMCD signals from luminescence yield.48 As illustrated in Figs. 5(a) and 5(b), Fe80V20 (instead of Ni80Fe20) is the soft low-damping spin source52,53 for detecting magnetization dynamics via XMCD at the Fe L3 edge—separately from the Ni L3 edge for the Ni(Cu) sink (i.e., Ni or Ni80Cu20). The thickness of the Ni(Cu) sink is greater than λc to ensure complete spin absorption. Our measurements were performed at a microwave excitation frequency of 4 GHz using a protocol similar to Ref. 54. We detected the magnetic oscillations transverse to the in-plane applied field by acquiring the XMCD response vs time. Examples of such time-resolved traces, obtained separately for the FeV source and the Ni(Cu) sink, are shown in Figs. 5(c) and 5(d).

FIG. 5.

(a) and (b) Stack structure for XFMR spin pumping, where the FeV spin source pumps a spin current into the (a) Ni or (b) Ni80Cu20 spin sink. (c) and (d) XMCD response as a function of the microwave delay time at the Fe and Ni L3 edges for the sample with the (c) Ni or (d) Ni80Cu20 spin sink. The applied field here is μ0Hx14 mT. (e) and (f) Field (Hx) dependence of the oscillation phase for the FeV spin source and the (e) Ni or (f) Ni80Cu20 spin sink. The solid red curve represents the fit modeling the total torque in the spin sink; the dashed gray curve represents the contribution from the dipolar field torque [with βST=0 in Eq. (2)], and the solid green curve represents the contribution from the spin-transfer torque [with βdip=0 in Eq. (2)].

FIG. 5.

(a) and (b) Stack structure for XFMR spin pumping, where the FeV spin source pumps a spin current into the (a) Ni or (b) Ni80Cu20 spin sink. (c) and (d) XMCD response as a function of the microwave delay time at the Fe and Ni L3 edges for the sample with the (c) Ni or (d) Ni80Cu20 spin sink. The applied field here is μ0Hx14 mT. (e) and (f) Field (Hx) dependence of the oscillation phase for the FeV spin source and the (e) Ni or (f) Ni80Cu20 spin sink. The solid red curve represents the fit modeling the total torque in the spin sink; the dashed gray curve represents the contribution from the dipolar field torque [with βST=0 in Eq. (2)], and the solid green curve represents the contribution from the spin-transfer torque [with βdip=0 in Eq. (2)].

Close modal

Figures 5(e) and 5(f) summarize the oscillation phase at several values of in-plane applied field Hx. The FMR of the FeV source is seen as a 180° shift in the phase, ϕsrc=atan(ΔH/(HxHFMRsrc)), centered at the resonance field μ0HFMRsrc14 mT with linewidth μ0ΔH0.95 mT. For the Ni(Cu) sink, we observe a qualitatively distinct shift in the phase ϕsnk around HxHFMRsrc. We fit ϕsnk vs Hx with the following function:45,55

ϕsnkϕ0snk=atanβdipsin2ϕsrcβSTsinϕsrccosϕsrc1+βdipsinϕsrccosϕsrc+βSTsin2ϕsrc,
(2)

where ϕ0snk is the baseline phase that depends on the saturation magnetization of the spin sink. The unitless coefficient βdip represents the dipolar field torque (e.g., from the interlayer orange-peel coupling56 with the precessing source magnetization) normalized by the off-resonant microwave field torque. Similarly, βST represents the spin-transfer torque (driven by the pumped spin current24) normalized by the off-resonant torque. Since the off-resonant torque scales with the magnetization, βST is also proportional to the efficiency of spin-transfer torque per unit magnetization in the Ni(Cu) sink.

The parameters derived from the fitting with Eq. (2) are summarized in Table I. The comparable values of βdip for the Ni and Ni80Cu20 sinks are reasonable, because the dipolar- and microwave-field torques scale similarly with the saturation magnetization of the sink. More importantly, βST also remains the same within experimental uncertainty between Ni and Ni80Cu20. We emphasize that βST is an efficiency metric for the spin-transfer torque per unit magnetization. Evidently, the Cu impurities do not diminish this spin-transfer torque efficiency. Our finding confirms that a sizable spin-transfer torque emerges from spin dephasing even in an alloy with a high nonmagnetic impurity content. It also implies that spin-transfer torque can be remarkably robust against electronic momentum scattering by impurities.

TABLE I.

Parameters for the fit curves of the total torque for the Ni and Ni80Cu20 sinks. ϕ0snk is the baseline phase; βdip and βST are coefficients proportional to the dipolar field torque and spin-transfer torque, respectively, normalized by the off-resonant microwave field torque.

ϕ0snk (°)βdipβST
Ni sink 90 ± 6 1.5 ± 0.5 1.3 ± 0.5 
Ni80Cu20 sink 142 ± 3 1.0 ± 0.2 1.7 ± 0.3 
ϕ0snk (°)βdipβST
Ni sink 90 ± 6 1.5 ± 0.5 1.3 ± 0.5 
Ni80Cu20 sink 142 ± 3 1.0 ± 0.2 1.7 ± 0.3 

In summary, we have experimentally investigated the mechanism behind the ultrashort coherence length λc of the transverse spin current in ferromagnetic Ni-based thin films. We find that λc scales inversely with the exchange strength in the ferromagnets examined here, even those with rather high Cu impurity contents. This central result strongly indicates that dephasing—not scattering—dominates transverse spin-current absorption in these nanometer-thick ferromagnetic metals. This result also highlights the ability to tune λc by engineering the magnetic exchange. While such tuning was previously explored for ferrimagnets and antiferromagnets,30,57,58 our study demonstrates that λc can be extended in ferromagnets as well by diluting the magnetic order. We further find that the efficiency of spin-transfer torque in a ferromagnet can remain invariant with its impurity content. Our findings provide crucial insights into transverse spin transport in the “bulk” of nanometer-thick ferromagnets, which may help enhance the performance of spin-torque devices by optimizing the length scale of spin dephasing.29 

See the supplementary material for additional information on film growth, the estimation of the Curie temperature, and the electrical resistivity of Ni(Cu).

Y.L. and S.E. were supported by the Air Force Office of Scientific Research (AFOSR) under Grant No. FA9550-21-1-0365. D.A.S. was supported by the National Science Foundation (NSF) under Grant No. DMR-2003914. This work was made possible by the use of Virginia Tech's Materials Characterization Facility, which is supported by the Institute for Critical Technology and Applied Science, the Macromolecules Innovation Institute, and the Office of the Vice President for Research and Innovation. This research used resources of the Advanced Light Source, a U.S. DOE Office of Science User Facility under Contract No. DE-AC02-05CH11231. S.E. thanks Xin Fan for helpful feedback.

The authors have no conflicts to disclose.

Youngmin Lim: Formal analysis (equal); Investigation (lead); Methodology (equal); Visualization (supporting); Writing – review & editing (supporting). Shuang Wu: Formal analysis (supporting); Investigation (equal); Methodology (equal); Software (equal). David A. Smith: Formal analysis (supporting); Investigation (equal); Software (supporting); Writing – review & editing (supporting). Christoph Klewe: Investigation (supporting); Methodology (equal); Resources (equal); Writing – review & editing (supporting). Padraic Shafer: Methodology (equal); Resources (equal); Writing – review & editing (supporting). Satoru Emori: Conceptualization (lead); Formal analysis (equal); Funding acquisition (lead); Methodology (equal); Project administration (lead); Supervision (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Supplementary Material