SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION doi:.38/nphys436 Non-adiabatic spin-torques in narrow magnetic domain walls C. Burrowes,2, A. P. Mihai 3,4, D. Ravelosona,2, J.-V. Kim,2, C. Chappert,2, L. Vila 3,4, A. Marty 3,4, Y. Samson 3, F. Garcia-Sanchez 5, L. D. Buda-Prejbeanu 5, I. Tudosa 6, E. E. Fullerton 6 & J.-P. Attané 3,4 Institut d Electronique Fondamentale, CNRS, UMR 8622, 945 Orsay, France 2 Université Paris-Sud, 945 Orsay, France 3 CEA, Inac, SP2M, 3854 Grenoble, France 4 Université Joseph Fourier, 384 Grenoble, France 5 SPINTEC, URA252 CEA/CNRS/UJF/INPG, 3854 Grenoble, France 6 Center for Magnetic Recording Research, University of California at San Diego, La Jolla CA , USA In this document, we present supplementary information concerning the experimental methodology behind the study described in the paper.. Experimental constraints on applied fields and currents The data in Figures 3 and 4 of the article, which show the measured cumulative distribution function for the depinning times and the variation of the associated Arrhenius transition rate τ as a function of applied current, respectively, are representative of the trends observed. However, there are three important experimental constraints that limit the range of fields and currents we can use in the present study. First, problems associated with electrical discharge limit the lifetime of the samples and have important consequences on the range of electrical currents we can apply. We recall that the distribution of depinning times at each value of applied field and current consists of 2-4 measurements. As the narrow wires we used are very sensitive to any electrostatic discharge, we have deliberately limited each measurement of the depinning time to 2 s and to the low current densities used. Despite such precautionary measures, each sample we have studied is eventually destroyed; samples can typically support continuous measurements of up to one month before breaking down. We have made measurements on different samples for each material, which all exhibit the same behaviour as that reported in the paper. The data set presented in the manuscript corresponds to the sample for which the largest data set of depinning times was obtained. Second, the rise time of the electromagnet used limits the strength of the magnetic field we can apply; it takes around.5- s to reach fields in excess of koe. Therefore, it is only possible to obtain a nature physics

2 supplementary information doi:.38/nphys436 good estimate of the depinning time if the propagation time of the domain wall from the nucleation reservoir exceeds this rise time. Third, the applied current densities are limited in magnitude because of Joule heating. As shown in Figure S, the current density is higher at the constrictions in the CoNi system, which leads to more pronounced effects due to Joule heating. In Figure S2a, we present the cumulative probability distribution function for applied current densities up to A/m 2 (ΔT = 2K). In contrast to the behaviour at low current densities, Joule heating progressively masks the asymmetry due to spintransfer effects and the parabolic increase in temperature becomes the dominant contribution to the distribution of the depinning times even for such small temperature rise. As shown in Figure S2b, this is also the case for FePt samples for applied current densities over 7 A/m 2. As such, this makes the extraction of the spin-transfer parameters more difficult with additional uncertainties. a Jx b Jy Figure S. Spatial distribution of current densities (in A/m 2 ) at a constriction showing higher local Joule heating (a) along the direction of the wire and (b) in the direction perpendicular to the wire. The current density flowing in the wire out of the constriction is.6 A/m 2..8 (a) -7. A/m 2-3. A/m 2 3. A/m 2 7. A/m 2 F(t) A/m A/m 2.7 A/m 2. A/m 2 (b) Figure S2. Cumulative probability functions for high current densities for the (a) CoNi and (b) FePt system. The data are consistent with Joule heating: for low negative currents, the mean pinning time decreases due to spin torques, for higher negative currents it increases due to Joule heating. 2 nature physics

3 NPHYS B doi:.38/nphys436 supplementary information 2. Depinning fields As shown in Figure S3, the depinning field at short times for both the FePt and CoNi spin valves can be estimated from hysteresis loop measurements. The plateau corresponds to domain-wall pinning and allows the depinning field Hp to be estimated. This field can be controlled through the constriction shape (CoNi) or the intrinsic pinning defect (FePt). The measured depinning field Hp fluctuates between measurements because the depinning transition is driven by thermal activation. As such, Hp depends also on the rate at which the magnetic field is swept during the hysteresis loop measurement. For the samples considered in the manuscript, at dh/dt Oe/s and after averaging over 2 measurements, we find Hp = 4.5 koe for FePt and Hp =.58 koe for CoNi. A more accurate method to determine Hp is from the field variation of ln(τ) deduced from the cumulative probability function F(t). As usual for films with perpendicular magnetic anisotropy, we find τ=τ exp[e(h)/kt] with E(H)=-2µMSV(H-Hc) where V is the activation volume and Hc corresponds to the depinning field without thermal fluctuations in contrast to the hysteresis loops measurements. We find Hc =.9 koe for CoNi and Hc = 4.7 koe for FePt, which underlines once again the importance of thermal activation (Hc > Hp). V GMR (mv) (a) H p H (Oe) R () (b) H P H (koe) Figure S3. (a) Reversal of the free layer for the (a) CoNi system and (b) FePt system. These loops have been determined by measuring the GMR signal as a function of field with a typical sweep rate of Oe/s. The plateaus correspond to domain wall pinning at a constriction (CoNi) or on a large intrinsic defect (FePt). Arrows indicate the direction of the field sweep. nature physics 3

4 supplementary information doi:.38/nphys Fits to cumulative distribution functions For both systems and all the wires that we have studied, we have used the same fitting procedure. Depending on the shape of the constriction (CoNi) and the role of intrinsic pinning centres (FePt), the results of the fit were always consistent with one of two scenarios: (a) Thermal activation over a single energy barrier; (b) Thermal activation involving two energy barriers, with the barriers either in series, in parallel or involving a Markov process (see below). An example of an exponential fit to the cumulative distribution function is shown in Figure S4, for which we obtain typical error of % for τ. We obtain similar error bars for other fields and currents. As such, this result is consistent with the picture that domain wall depinning from the constrictions considered in the paper is dominated by thermal activation involving a single energy barrier. F(t) F (t)= e t/τ τ = (.2 ±.9) s R 2 = Figure S4. Example of an exponential fit of the cumulative function for H = 55 Oe and J =.7 x A/m 2 for the CoNi system. The circles represent experimental data and the solid line represents a fit based on the equation given in the figure, along with the best-fit parameter and correlation coefficient. In contrast for sharper constrictions (< nm in lateral extension), we find that it is often necessary to use more than one exponential function to fit the cumulative distribution function. Micromagnetics simulations for these constriction sizes show that the depinning transition in such systems takes place through two different domain wall configurations, each with a different characteristic time τ (Markov process). If the values of the two characteristic times are close, it is more difficult to extract a measure of the spin-transfer parameter β from the experimental data, which is why these data sets have been excluded in the present manuscript. For FePt systems, the fits are consistent with a Markov process [] that involved two possible domain wall configurations as shown in Figure S6. The exact micromagnetic configuration corresponding to these states is unknown, however we do know that both states and 2 correspond to a single defect, as they correspond to identical GMR values and the dominant pining defects are well separated from each other. For example, the configuration difference could be linked to small deformations of the wall (depinning in two times, as shown in Figure S6), or to domain wall chirality (presence of a small in-plane component, presence or absence of a vertical Bloch line ). Basically, 4 nature physics

5 doi:.38/nphys436 supplementary information the idea is that it is more difficult to depin from the defect in state than in stage 2. Such a generic hypothesis gives account of our results, and is supported by the fact that in other experiments, the reduction of the width of the wire, reducing the available configurations of the domain wall, transformed experimental results similar to ours in more classical exponential laws. Figure S5. Example of DW depinning for sharper constrictions. The domain wall can be depinned through two different configurations involving different characteristic time. Figure S6. DW depinning process involving two different configurations. To describe this model more precisely, let s consider the initial (state ) and final (state 3) states. The domain wall can be depinned following two different paths: ( 3) or ( 2 and then 2 3). The 2 3 transition revealed to involve a very long characteristic time τ23 > τ3, which explains why the probability distribution curve does not saturate to for long times. The evolution of the system, represented by the vector P(t), whose 3 components are the probabilities to be in the i th state at time t, may be described by the differential equation dp (t) dt = M P (t), where the matrix M is given by (/τ 3 +/τ 2 ) /τ 2 /τ 3 M = /τ 23 /τ 23. This leads to F (t) = P 3 (t) = exp[tm] 3, = ( r)e t(/τ 2+/τ 3 ) re t/τ 23, nature physics 5

6 supplementary information doi:.38/nphys436 where r = /τ 2 /τ 2 +/τ 3 /τ 23. This expression was successfully used to fit the experimental data at room temperatures, as shown in Figure S7. For all fields and currents, τ23 is significantly larger than τ3 and τ2, meaning that the system is indeed frozen when going through the state 2. At room temperature, τ3 is by far smaller than τ2, which means that the system follows mainly the direct path 3. In all cases, τ3 plays the dominant role, and its value is quite accurately derived (±%) and represented in Figure 4b of the manuscript. Conversely, the alternative path through state (2) can be considered as a small correction. Consequently, τ2 and τ23 play a lesser role in the fits, and the given values should be seen as indicative..5 (a) (b).4.8 F(t) Figure S7. Example of a Markov fit (solid line) to the cumulative functions for two different applied fields (FePt), (a) H = 4.27 koe and (b) H = 4.49 koe. Experimental data are shown as circles. 6 nature physics

7 doi:.38/nphys436 supplementary information 4. Critical currents Current-driven wall motion has been observed in our CoNi patterned wires without constrictions. In these systems, the typical propagation field is around Hp = 5 Oe. In this case, we have been able to measure the dc threshold current to move the domain wall at zero field. By ramping the current up rapidly in order to minimize Joule heating, we find a value of about Jc = 8 A/m 2 that compares well with the result of Tanigawa et al. [2]. For patterned wires with constrictions, as those presented in the manuscript, the pinning field (as discussed above) is much higher (Hp = 6 e). While we have not measured zero field current-driven wall motion in these systems, we have extrapolated the magnetic field variation of the threshold current, Jc(H), to zero magnetic field and we find a typical value of Jc 4 2 A/m 2 for CoNi. Using a similar method, we find an efficiency of 3-3 Tm 2 A - for the FePt system, which extrapolates to a critical current of ~2 2 A/m 2. References. Attané, J. P., Ravelosona, D., Marty, A., Samson, Y. & Chappert, C. Thermally Activated Depinning of a Narrow Domain Wall from a Single Defect. Phys. Rev. Lett. 96, 4724 (26). 2. Tanigawa, Hironobu et al. Domain Wall Motion Induced by Electric Current in a Perpendicularly Magnetized Co/Ni Nano-Wire. Appl. Phys. Express 2, 532 (29). nature physics 7