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Resolving Discrepancies in Spin-Torque Ferromagnetic Resonance Measurements: Lineshape vs. Linewidth Analyses

When spin-orbit torques are measured using spin-torque ferromagnetic resonance (ST-FMR), two alternative ways of analyzing the results to extract the torque efficiencies -- lineshape analysis and analysis of the change in linewidth versus DC current -- often give inconsistent results. We identify a source for these inconsistencies. We show that fits of ST-FMR data to the standard analysis framework leave significant residuals that we identify as due to (i) current-induced excitations of a small volume of magnetic material with magnetic damping much larger than the bulk of the magnetic layer, that we speculate is associated with the heavy-metal/magnet interface and (ii) oscillations of the sample magnetization at the modulation frequency due to heating. The dependence of the residual signals on DC current can interfere with an accurate extraction of spin-torque efficiencies by the linewidth method. We show that the discrepancies between the two types of analysis can be largely eliminated by extrapolating the window of magnetic fields used in the linewidth fits to small values so as to minimize the influence of the residual signals.


I. INTRODUCTION
Magnetic manipulation by spin-orbit torques (SOTs) [1,2] is a promising candidate mechanism for nextgeneration magnetic memory technologies [3,4]. Accurate, consistent measurements of the efficiencies of SOTs are essential for fully understanding the microscopic origin of SOTs and for the development of SOT memory technologies; yet despite the large body of work that SOTs have amassed, quantitative discrepancies between different measurement techniques persist. Here we will focus on inconsistencies that result from two approaches for analyzing spin-torque ferromagnetic resonance (ST-FMR) measurements, the most common experimental method used to quantify SOTs in heterostructures that have in-plane magnetic anisotropy [2,[5][6][7]. These two approaches are lineshape (LS) analysis in which the inplane and out-of-plane spin-orbit torques are determined from the amplitudes of the symmetric and antisymmetric resonance components [2,5,8], and linewidth (LW) analysis in which the damping-like in-plane component is determined from the dependence of the resonant linewidth on DC current [5,[9][10][11]. These approaches often give results for the damping-like spin-orbit torque that differ by large factors (e.g., in the data we present below a difference by more than a factor of 3).
Here, we investigate the cause of the discrepancy by performing ST-FMR measurements on Pt/Permalloy and Pt/CoFeB samples, comparing data acquired using amplitude modulation of the microwave current (the most-common approach) and using frequency modulation, and by analyzing carefully the residuals in fits to the standard ST-FMR analysis for both modulation schemes. We identify that the discrepancies between the lineshape and linewidth analyses arise primarily from * Correspondence email address: sk2992@cornell.edu current-induced excitation of a small volume of magnetic material with magnetic damping much greater than the majority of the magnetic film. This produces a lowamplitude background ST-FMR resonance with a broad tail, that can interfere with accurate determination of DC-current-induced changes in the linewidth of the primary resonance. We demonstrate that discrepancies between the lineshape and linewidth analyses can be reduced by employing frequency modulation rather than amplitude modulation of the microwave current, and by extrapolating the range of magnetic field used in the linewidth fits to small values so as to minimize the disruption from the low-amplitude, large-linewidth artifact signal.

II. BACKGROUND ON THE ST-FMR TECHNIQUE
During ST-FMR, a microwave current is applied inplane into a heavy metal/ferromagnet bilayer, so that current-induced SOTs and Ørsted fields induce ferromagnetic precession. Magnetic precession causes resistance variations in the device due to anisotropic magnetoresistance. Mixing between the microwave current and resistance oscillations then produces a DC voltage that is measured. The signal-to-noise ratio (SNR) is vastly improved if an endogenous parameter of the technique is modulated and the voltage is measured with a lock-in amplifier. There are a number of choices for the modulated parameter: amplitude modulation (AM) [2,5], magnetic field modulation [12], frequency modulation (FM), or phase modulation (the latter two of which are equivalent). Save a few works that have employed magnetic field modulation [12][13][14][15][16][17], almost all experiments featuring ST-FMR employ AM because it is the simplest parameter to modulate -it does not complicate the experimental apparatus nor the fitted model.
Typically, one assumes that a macrospin approxima-tion is appropriate for describing the current-induced magnetic dynamics for experiments performed at suffficiently large microwave frequencies and magnetic fields, in which case the results of ST-FMR are modeled by the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equatioṅ where m is the magnetic moment, γ = 2µ B / is the gyromagnetic ratio, B is the external field, α is the Gilbert damping constant, and τ = τ DL +τ z describes the torque present in our system. The torques produced by a polycrystalline thin film must obey the Rashba symmetry [18]. For a film spanning the X − Y plane, with current flowing along the X-direction we have (3) ξ DL(FL) is the damping-like (field-like) SOT efficiency, µ B is the Bohr magneton, M s is the saturation magnetization of the ferromagnet, t F(N) is the thickness of the ferromagnet (normal metal) layer, µ 0 is the vacuum permeability, J e is the electric current density flowing through the heavy metal, and φ 0 is the angle between the direction of applied field and current flow (X-direction). Solutions to the LLGS equation for a sample with inplane magnetic anisotropy predict that resonant ferromagnetic precession will occur when the Kittel equation, ω 0 = γ B(B + µ 0 M eff ) [19], is satisfied. Here, µ 0 M eff = µ 0 M s −2K ⊥ /M s accounts for shape anisotropy minus any out-of-plane anisotropy. The total resonance lineshape will have contributions from symmetric (S) and antisymmetric (A) Lorentzians [5,7], which we define as where B 0 is the resonance field, and ∆ is the half-widthat-half-maximum linewidth related to the Gilbert damping by ∆ = αω/γ. The DC mixing signal is a weighted sum of these two lineshapes with coefficients V S and V A determined by the the torques and material parameters in our system [5,7,20] with I rf is the total microwave current that flows through the bilayer, ω + = γ(2B 0 +µ 0 M eff ), R AMR is the amplitude of the anisotropic magnetoresistance of the whole bilayer, and ω 2 = γ(B 0 + µ 0 M eff ). C is a constant voltage offset that is included to account for non-resonant signals. For samples with thick magnetic layers, there can also be a significant additional contribution to the symmetric resonance component from spin pumping/resonant heating and the inverse spin Hall effect [20] that we will mention below.
The experimental signal-to-noise ration (SNR) is significantly improved by modulating the microwave amplitude; this is captured by letting V mix (I rf ) → V mix (I rf (1 + µ cos ω m t)) ≈ V mix + 2µV mix cos ω m t where µ ∈ [0, 1] is the AM-depth. A lock-in amplifier demodulates the total signal by mixing with a cos ω m t reference and applying a low-pass filter. The AM signal is therefore simply: 2µV mix .
If, alternatively, frequency modulation is used instead of amplitude modulation, the expected FM signal can be derived in a similar manner. We let V mix (ω) → V mix (ω + δω cos ω m t) where δω ω; this admits the simple expansion near the microwave carrier frequency, V FM mix is again demodulated by a lock-in amplifier, which leaves us with only dVmix dω ω=ωc δω. Therefore, the ratio of the detected mixing signal to the amplitude of the frequency modulation is Here we have used that Equation (9) is nearly identical to a previously derived result where the magnetic field was modulated [12]. Compared to the AM result, Eq. (6), the FM result has two additional fit parameters, dV S /dω and dV A /dω to account for possible frequency dependence of microwave transmission through the measuring circuit to the device.

A. Lineshape (LS) Analysis
After measuring with either amplitude or frequency modulation and then fitting the ST-FMR resonance to determine V S and V A using either Eq. (6) or Eq. (9), the torque efficiencies may be determined directly from Eq. (7) if I rf is well-calibrated, since the other parameters in Eq. (7) are independently-measurable. However, since it is often challenging to determine accurately the value of I rf within the sample, we generally prefer to determine the torque efficiencies by taking appropriate ratios of V S and V A [8]. We first calculate an intermediate quantity, ξ FMR , defined as By using Eqs. (2), (3), & (10), as long as the torque efficiencies are independent of the ferromagnetic-layer thickness in the range of thickness we analyze, then ξ FMR can be related to the damping-like and field-like torque efficiencies as [8] 1 Therefore, by taking results form a series of samples with different ferromagnet thicknesses, t F , we can then determine ξ DL and ξ FL from a linear fit of 1/ξ FMR versus 1/t F .

B. Linewidth (LW) Analysis: Change of Linewidth Versus DC Current
In DC-biased ST-FMR, a DC current is applied parallel to the microwave current, such that the damping-like torque from the DC current rescales the effective Gilbert damping of the magnetic layer and causes the resonance linewidth to change linearly as a function of I DC . The damping-like SOT efficiency can be calculated from DCcurrent linewidth modulation as [5,11] where W is the width of the current-carrying channel and x is the fraction of the total DC current that flows through the HM.

III. MEASUREMENTS
All our samples are grown using DC-magnetron sputtering (in a system with base pressure < 4 × 10 −8 torr) onto a surface-passivated high-resistivity Si wafer (ρ > 20,000 Ωcm). Each sample is grown in an independent deposition. Samples shown in the main text have the stacking order: Substrate/Ta(1 nm)/Pt(6 nm)/Py(t F )/Al(1 nm), with the magnetic layer being Permalloy (Py = Ni 81 Fe 19 ). The Ta is used as a seed layer to promote smooth growth and the Al is oxidized upon exposure to air an is used as a capping layer to prevent oxidation of the Py. The Pt (ρ = 20.4 µΩcm) and Py (ρ = 25 µΩcm) are far more conductive than the Ta or oxidized Al so we assume all of the current flows through just the Pt and Py layers. Analogous results for which the Py is substituted with Co 40 Fe 40 B 20 can be found in the supplemental information.
After growth, we pattern the samples into rectangular bars of varying dimension using photolithography and Ar ion milling. The devices have dimensions: 40 µm × 80 µm, 20 µm × 60 µm, or 20 µm × 80 µm. All measurements shown in this work are taken on 20 µm × 80 µm devices, and the quantitative conclusions do not depend on the device geometry. We attach Ti(3 nm)/Pt(75 nm) contacts to the devices by another step of photolithography, DC-magnetron sputtering, and liftoff.
All data shown are measured on microwavecompatible Hall-bar structures that allow measurements of both longitudinal and transverse mixing voltages, as described in ref. [20]. Here, we will analyze only the longitudinal mixing voltages, as that is the usual ST-FMR measurement geometry. The devices are connected to the circuit shown in Fig. 1(b). A RF source inputs a microwave current into the device through the AC port of a bias tee with either amplitude or frequency modulation ( Fig. 1(a)), while the magnitude of an external magnetic field is swept at a fixed angle φ 0 through the Kittel resonance condition. The DC voltage along the longitudinal direction generated by mixing is detected with a lockin amplifier that references the modulating signal. For the DC-biased measurements, an additional DC current is applied through the DC port of the bias tee to flow through the device in addition to the microwave current. All measurements are performed at room temperature.
Measurements in the main text are performed at a 10 GHz carrier frequency (f c = ω c /2π). The modulating signal for both AM and FM measurements is applied at 1.7 kHz modulation frequency (f m = ω m /2π). The external magnetic field is applied at φ 0 = 45 • with respect to the direction of applied microwave current. The AM measurements are done with 100% AM-depth as depicted in Fig. 1(a) to maximize the measured signal. We find that reducing the AM-depth has no effect on the results shown (see supplemental information). The FM measurements are done with a frequency deviation (δf = δω/2π) of 16 MHz. Both the AM and FM are applied by the internal circuitry of the RF source, an Agilent 8257D. Both the 16 MHz frequency deviation and the 1.7 kHz modulation frequency are far smaller than the carrier frequency of 10 GHz, so that within either measurement mode the modulation has negligible effect on the microwave current over one precession cycle, a key assumption of the modeling.

A. Results of Lineshape Analyses
Examples of the longitudinal resonant mixing signals from a Pt(6 nm)/Py(5 nm) sample for both the AM and FM measurements are shown in Fig. 2. The AM measurement ( Fig. 2(a)) is fit to Eq. (6) with the five fit parameters V S , V A , C, B 0 , ∆ while the FM measurement ( Fig. 2(b)) is fit to Eq. (9) including the two additional fit parameters dV S /dω, dV A /dω. The fit to the AM measurement looks good by eye, but the best fit nevertheless produces significant systematic residuals, which hints that the framework of conventional ST-FMR analysis (Eq. (6)) gives an incomplete description. To rule out spurious measurement artifacts, we have repeated the AM measurements on three independent ST-FMR apparatuses at Cornell and have also performed measurements on different sample stacks; all of these measurements show the same systematic residuals for the AM fits. In contrast, for the FM measurements the scale of the residuals after fitting to Eq. (9) is significantly smaller relative to the full signal magnitude. A more complete discussion of the fit quality, residuals, and statistical details can be found in the supplemental information.
For the AM fits in Fig. 2(a), we see that the residuals have a lineshape near the resonance field that closely resembles a Lorentzian derivative, suggesting that an additional parameter in Eq. (6) is varying at the modulation frequency and contributing to the homodyne mixing signal. Quantitative estimates suggest that a varying M eff will contribute far more than other candidate sample parameters and that an M eff oscillating at the AM frequency, presumably due to heating, can result in the residual lineshape we observe near the resonance field. That is, suppose (in addition to the amplitude modulation of I RF ) that M eff also varies periodically as M eff → M eff + δM eff cos ω m t; this, analogously to the frequency modulation, would allow the expansion The total mixing signal will thus consist of the sum of two terms that vary periodically with the AM and both will be demodulated by the lock-in amplifier. A homodyne signal from an oscillating value of M eff cannot by itself explain the full residual in the AM fits; in addition the AM residuals appear to contain an ordinary AM resonance lineshape (Eq. (6)) with a very large linewidth. In Fig. 3 we show the fit residuals of an AM measurement taken on a Py(3 nm) sample (with no DC current bias). We fit the residuals to the sum of a homodyne signal corresponding to an oscillating value of M eff (green curve) and a large-∆ resonant background (Eq. (6)) (blue) with ∆ large = 41.6 mT, much larger than the value ∆ = 7.3 mT for the primary resonance. The sum of the two contributions (red curve) fits the residuals very well. Based on direct measurements of M eff versus temperature on the same device [d(µ 0 M eff )/dT = 8×10 −4 T/ • C], the scale of the temperature oscillations needed to produce the oscillating-M eff homodyne signal is approximately 1 • C (see supplemental information). V S and V A for the large-∆ resonance for the data in Fig. 2 are 12 µV and 7 µV, while for the primary resonance V S = 311 µV and V A = 684 µV.
We have considered two options for the origin of the large-∆ resonance: a region of increased damping (a) near the sample edges or (b) near a magnetic interface. If the origin were due to increased damping near the sample edges, we would expect the ratio of the amplitudes for the large-∆ and primary resonances to scale inversely with the sample width and to be approximately independent of the ferromagnetic-layer thickness. Instead, we find that this ratio is insensitive to the sample width (a change of < 10% in the symmetric and < 4% in the antisymmetric component upon changing the sample width by a factor of 2), while it is sensitive to the ferromagnetic layer thickness (see supplemental information). This suggests that the portion of the sample  with increased damping is an interfacial region. Additional evidence for an origin associated with the heavymetal/ferromagnet interface comes from the fact that the large-∆ linewith is very sensitive to applied DC current (see supplemental information), consistent with a very thin and/or low-moment region under the influence of the spin current generated by the heavy metal. Our observations might be related to recent findings from the IBM group of interfacial regions in CoFeB/MgO/CoFeB magnetic tunnel junctions whose dynamics can become partially decoupled from the bulk of the magnetic films [21,22]. The two experiments differ, however, in that the IBM work deduced a difference in effective magnetic anisotropy (compared to the bulk of the magnetic film) for the interfacial layers at CoFeB/MgO interfaces, while in our devices the large-∆ resonance corresponds to an increased damping near a Pt/ferromagnet interface without a large difference in anisotropy.
We suggest that there are two reasons why the fit residuals for the FM measurements are reduced compared to the AM measurements. First, temperature oscillations at the modulation frequency will be smaller for the FM measurements because the magnitude of I RF will be approximately constant in time, so Ohmic heating caused by I RF will also be approximately constant rather than oscillating at the modulation frequency. Temperature oscillations will not be eliminated completely however, since FM near the resonance will cause the energy absorbed by resonant heating of the magnetic layer (energy transfer associated with magnetic excitation by the current-induced torques) to oscillate at the modulation frequency. We suggest that this resonant heating is likely the main cause of the small remaining systematic residuals near the resonance field in the fits to the FM data ( Fig. 2(b)). Second, contributions from the large-∆ resonance to the FM measurements are reduced precisely because the linewidth is so broad, so this part of the signal is relatively insensitive to variations in applied frequency.
If one proceeds with the standard ST-FMR macrospin analysis -Eqs. (10), (11) -(ignoring the residuals for now) the resulting values of 1/ξ FMR for both the AM and FM measurements are shown in Fig. 4 for samples with ferromagnet layer thicknesses t F varying from 2 nm to 10 nm. The samples with the thickest ferromagnet layers (t F ≥ 8 nm) show deviations from a linear dependence of 1/ξ FMR vs. 1/t F that can be understood as due to the effect of an inverse spin Hall voltage resulting from spin pumping or resonant heating [8,20]. We therefore perform the linear fits only to the four samples with the thinnest F layers, extracting the values shown in Table  I. ξ DL and ξ FL are calculated from the y-intercept and slope of the fits, respectively, following the prescription of Eq. (11). The FM and AM methods yield values for both ξ DL and ξ FL that differ by considerably more than the estimated statistical uncertainty in the results. The difference in the values of ξ DL is about 30%, while for ξ FL the FM result nearly double that of the AM.   Fig. 4.
We suggest that the differences between these AM and FM LS results can be explained by the neglect of the residual terms. If we take the values of ξ DL and ξ FL determined by the FM measurements and use them in fitting to the AM data, the result is a residual similar to that shown in Fig. 3 that can be fit just as well to a sum of a signal due to an oscillating value of M eff plus a large-∆ resonance (see supplemental information). Fits to the AM data that include both the primary resonance and the two artifact contributions therefore possess neardegenerate fitting parameters, that can make determination of the spin-torque efficiencies imprecise.

B. Results of Linewidth Analyses
The LW measurement proceeds identically to the LS measurement, but for the application of a DC current parallel to the microwave current. A full resonance lineshape (e.g., Fig. 2) is collected for DC currents ranging from −4 mA to 4 mA and the resonances are fit to Eq. (6) for the AM measurements or Eq. (9) for the FM measurements to extract the linewidth, ∆ (ignoring residuals for now). We find that the value of ∆ that we get from the fits for the AM measurement depends strongly on whether and to what extent we include the tails of the resonance. Figure 5 shows the current dependence of linewidths for a Pt(6 nm)/Py(5 nm) sample extracted from fits over the field range [B 0 − 15∆, B 0 + 15∆] (with ∆ adjusted for each sample corresponding to the linewidth of the primary resonance at zero DC current). This is the largest fit window that is possible while consistently excluding artifacts associated with deviations from magnetic saturation at low field for all samples. The zero-current value of ∆ is subtracted from each of the plots in Fig. 5 to highlight the difference in the slopes of the best-fit lines. We apply Eq. (12) to the slopes of the best-fit lines and get the results for ξ DL shown in Table II.
For this sample, we see that the FM LW measurements agree with the FM LS result within the experimental uncertainties (Table I), while the AM LW measurements differ by more than a factor of 3 from both the FM results and the AM LS measurements. Figure  6 compares the results of similar LW analyses for all of the Pt(6 nm)/Py(t F ) samples with different magneticlayer thicknesses using the same fit window [B 0 − 15∆, B 0 + 15∆]. The AM LW measurements (black points) give far larger values for ξ DL compared to any of the other techniques. The FM LW measurements are reasonably consistent with the FM LS value in the range t F = 4 -10 nm (with small deviations for t F = 10 nm possibly due to the neglect of an inverse spin Hall voltage generated by spin pumping or a spin Seebeck effect), but the FM LW measurements also differ increasingly from the the LS results for Py thicknesses below 4 nm.   Fig. 5 and Eq. (12).
In Fig. 7 we show the results of the same LW analysis using different sizes for the window of magnetic field included in the fits. The panels on the left show the values of ξ DL extracted for window sizes from [B 0 − 15∆, B 0 + 15∆] to [B 0 − 2∆, B 0 + 2∆]. For both the AM and FM data sets, the extracted values of ξ DL decrease with decreasing window size. We interpret this dependence as a clear indication that the LW analysis can be  disrupted by the long tails of the residual terms that are not included as part of the standard linewidth analysis. For a fixed value of fit-window size, the disruption is most severe for magnetic layers thinner than 4 nm because the linewidth of the primary resoance increases for thin layers, making the primary resonance more difficult to disentangle from the large-linewidth residual signal. The right panels of Fig. 7 show zoom-ins of the same LW results to better visualize the extrapolation of the measurements to zero linewidth. We find that this extrapolation brings the results of both the AM LW and FM LW analyses into reasonable quantitative agreement with the lineshape results. We emphasize that the sensitive dependence on fitwindow size shown by Fig. 7 occurs despite the fact that the individual fits look quite good by eye for any choice of window size. The LW analyses are based on quite subtle changes in the resonance lineshape, e.g. about a 2% change in linewidth over the full range of I DC for the FM measurements shown in Fig. 5. Therefore, even small changes in V mix associated with current-dependent residuals can affect the LW analysis -the small tails of the ST-FMR resonances can be substantially affected even if the overall magnitude of the residual signals near the resonance field is small. The large-∆ resonance in particular has a large affect on the LW analyses because its linewidth is strongly current dependent (see supplemental Fig. 13). We have tried fitting the AM resonances to a generalized Eq. 6 that includes the models for the residuals directly in the fit, but this is not able to provide improved quantitative results because of near-degenerate fit parameters. We therefore recommend the procedure depicted in Fig. 7 as the simplest approach to improving ST-FMR linewidth analyses -performing the standard ST-FMR fits using a series of different fit-window sizes and then extrapolating to small windows to minimize the influence of the large-linewidth residuals.

IV. CONCLUSIONS
We have identified a cause of inconsistencies between measurements of spin-orbit torque determined via lineshape and linewidth analyses of ST-FMR data -that the standard model for analyzing ST-FMR data does not fully account for all of the magnetic dynamics that can affect the measurements. The standard analysis leaves residuals that we identify as due to (i) current-induced excitation of a magnetic mode with larger damping than the bulk of the magnetic layer and also (ii) temperature oscillations (≈ 1 • C) associated with the modulation schemes employed for lock-in amplifier measurements. The residuals are not large, with amplitudes of order 1% of the primary resonance, but nevertheless they can affect the current dependence of the resonance tails sufficiently to disrupt an extraction of the anti-damping spinorbit torque efficiency based on the current dependence of the ST-FMR linewidth. The influence of the largelinewidth residuals can be minimized by performing the standard lineshape analysis using different choices for the range of magnetic field values used to fit the ST-FMR resonances, and then extrapolating to zero fit window. We recommend this procedure for all future uses of the LW analysis. The effect of the residuals can also be reduced by performing ST-FMR using frequency modulation rather than amplitude modulation, but frequency modulation alone does not cure inconsistencies between the lineshape and linewidth results for our thinnest mag-netic layers without extrapolation of the fit window to small values.
It remains an interesting open question what is the microscopic origin of the large-linewidth mode that contributes to the residual signal. Based on the scaling of signal amplitudes with the widths and thicknesses of our samples, we identify this mode with the heavymetal/magnet interface rather than as due to increased damping at the lateral edges of our magnetic layers. It is possible that this mode is due to an interface magnon, magnetic impurities caused by intermixing near the interface, a magnetic proximity layer within the platinum, or coupled dynamics involving two or all three of these effects. We plan future experiments to begin to resolve this question by making samples with different heavy metals and performing measurements as a function of temperature, to vary the importance of magnetic proximity effects.      In this section, we show an AM resonance where the fit has been performed with a fixed torque ratio (S/A) determined by the fit to the FM data. The residuals are still fit well by the sum of a function describing an oscillating M eff and large-∆ resonance. Figure S5: An AM resonance with the corresponding residuals that can be fit to a fluctuating M eff and large-∆ resonance. The torque ratio (S/A) in the fit to the AM resonance is held fixed to the best-fit results from the corresponding FM resonance.  Figure S8 shows the current dependence of the ST-FMR linewidth for the same sample, comparing the results of amplitude modulation and frequency modulations. Like the results for Pt/Py shown in the main text, the slope of the amplitude-modulated results is much larger than for the frequency-modulated measurements. We also see strong non-linearity and noise in the AM data in this sample-making the FM LW measurement more robust.  Table II shows the results of ξ DL for the Pt(6 nm)/Co 40 Fe 40 B 20 (6 nm) sample calculated from the slopes of the lines in Fig. S8. Again like the results for Pt/Py shown in the main text, we see that the FM measurements are in much better agreement with the LS result of a Pt(6 nm)/Co 40 Fe 40 B 20 thickness series, as performed in ref. [20] using the same basic procedure as the LS method in this work. Figure S9 shows how the determination of ξ DL from the linewidth method for the Pt(6 nm)/Co 40 Fe 40 B 20 (6 nm) sample depends on the choice of fit window size used for determining the linewidths. With a reduced fit window size, the results for both the amplitude-modulation and frequency-modulation measurements extrapolate close to the value from a lineshape analysis, in a way similar to the results for the Pt/Py samples described in the main text.    We suggest that even a small temperature oscillation due to the AM can result in an oscillation in M eff that can be detected by the lock-in amplifier (see Main Text Fig. 6). If the value of M eff oscillates at the modulation frequency we have with M 0 eff being the equilibrium effective magnetization. We can simply differentiate the mixing signal to find In order to quantitatively estimate the magnitude of this contribution, we substitute in the fitted values of the parameters from the fits to the standard V mix signal (main text Eqs. (6) & (7)). We find that ∂Vmix ∂M eff ∼ 10 mV/T for the Pt(6 nm)/Py(3 nm) sample. Based on direct measurements of M eff versus temperature performed by measuring the ST-FMR resonance field while heating the sample externally (supplementary Fig. 12), a temperature oscillation of about 1 • C is required to fit this contribution to the AM residual signal Although the oscillation M eff gives a substantial contribution to the structure of the AM residuals, we find that it contributes little to the quantitative errors in the LW AM ST-FMR analysis. The large-∆ resonance contribution to the AM residuals has the dominant effect in altering the fits to the current-dependence of the ST-FMR linewidth.  Determination of the parameters of the large-∆ resonance requires some care because of the potential for neardegenerate fitting parameters if one attempts to perform a combined fit to the primary resonance and the large-∆ resonance all at once. For the following graphs, we have first determined the linewidth of the primary resonance by fitting the ST-FMR signal to a single resonance using different choices of fitting window for the magnetic field, and extrapolating the fit window to zero. We then determine the symmetric and antisymmetric Lorentzian amplitudes for the primary resonance from a fit to the full window of data, and subtract to yield the residual signal. The parameters of the large-∆ resonance are then determined from a fit to the residual signal that includes both an oscillating-magnetization contribution (with linewidth fixed to be the same as the primary resonance) and a large-∆ resonance signal. Figure S11: Parameters of the large-∆ resonance that contributes to the residual in the AM fits, for the Pt/Permalloy samples. (a) Amplitude of the symmetric component of the large-∆ resonance relative to the primary resonance, normalized by the respective damping parameters, as a function of Permalloy thickness. (b) Amplitude of the antisymmetric component of the large-∆ resonance relative to the primary resonance, normalized by the respective damping parameters, as a function of Permalloy thickness. (c) Dependence of the large-∆ linewidth on DC current for different thicknesses of the Permalloy layer.

VI. LW (IN)DEPENDENCE ON AM MEASUREMENT PARAMETERS
In this section, we show LW plots akin to the main text Fig. 3, but with some variations in the parameters of the AM modulation. First, we show the dependence on the AM depth in Fig. S12(a). All the data in the main text is taken at 100% AM depth to maximize the SNR. An AM current may be written I(t) = (1 + µ cos ω m t) sin ω c t. (S3) Here, the angular frequencies are as defined in the main text, and µ ∈ [0, 1] is the AM depth, which is usually expressed as a percentage like in Fig. S12(a). We can see that there is very little dependence of the LW result (slope of best-fit lines) on the AM depth and that this dependence is not even monotonic, indicating that the the AM depth does not systematically affect the discrepancy observed in the AM LW measurement. We also show dependence on the RF power applied in Fig. S12(b). All of the main text data is taken at 20 dBm to maximize the SNR. We can see that there is no significant dependence of the LW result on the applied RF power, which additionally confirms that our applied power is still within the linear regime.   Fig. 7 to zero window size.