Kaushalya Jhuria

Julius Hohlfeld

Akshay Pattabi

Elodie Martin

Aldo Ygnacio

Arriola Córdova

Xinping Shi

Roberto Lo Conte

Sebastien Petit-Watelot

Juan Carlos Rojas-Sanchez

Gregory Malinowski

Stéphane Mangin

Aristide Lemaître

Michel Hehn

Jeffrey Bokor

Richard B Wilson

Jon Gorchon
email: jon.gorchon@univ-lorraine.fr

Picosecond Spin Orbit Torque Switching

Device, setup & quasi-static switching

.c. The bottom Pt and top Ta were chosen for their opposites signs of spin Hall angles, in order to

Reducing energy dissipation while increasing speed in computation and memory is a long-standing challenge for spintronics research 1 . In the last 20 years, femtosecond lasers have emerged as a tool to control the magnetization in specific magnetic materials at the picosecond timescale [2][3][4] . However, the use of ultra-fast optics in integrated circuits and memories would require a major paradigm shift. An ultrafast electrical control of the magnetization is far preferable for integrated systems. Here we demonstrate reliable and deterministic control of the out-of-plane magnetization of a 1 nm-thick Co layer with single 6 ps-wide electrical pulses that induce spin orbit torques on the magnetization. We monitor the ultrafast magnetization dynamics due to the spin orbit torques with sub-picosecond resolution, thus far accessible only by numerical simulations. Due to the short duration of our pulses, we enter a counter-intuitive regime of switching where heat dissipation assists the reversal. Moreover, we estimate a low energy cost to switch the magnetization, below 50 pJ for our micrometer sized device. These experiments show that spintronic phenomena can be exploited on picosecond timescales for full magnetic control and should launch a new regime of ultrafast spin torque studies and applications.

There is a largely held belief 5,6 , that the time it takes for the magnetization of a system to be reversed coherently is limited to half of its natural precessional period (i.e. its ferromagnetic resonance, FMR). However, if a strong enough effective field is induced the Landau-Lifshitz-Gilbert (LLG) equation that governs the magnetization dynamics predicts switching on time-scales shorter than the FMR half-period. It is also commonly claimed 5,6 that ferromagnetic materials have FMR frequencies of a few GHz, limiting switching to a fraction of a nanosecond (ns). But this view neglects the fact that most industrially relevant ferromagnetic systems are thin films with strong perpendicular magnetic anisotropy fields (𝐻 ), typically 7 around 1 T (going up to 5 T), which have much higher FMR frequencies and half-periods below 𝜋/𝛾𝐻 ~20 ps (𝛾 being the gyromagnetic ratio) 8 .

Experimentally, current induced magnetization switching has been demonstrated 9 with current pulses as short as 50 ps using the spin transfer torque (STT), albeit only in in-plane magnetized samples. In STT devices a first magnetic layer polarizes the current, which is then injected into a second softer magnetic layer to impart a torque and switch its magnetization 10 . Generally, the two magnetic layers are spaced by a thin insulating barrier, in order to allow a readout of the magnetic state via the tunneling magneto resistance effect. In practice, switching in STT devices is typically limited to the nanosecond 11 (ns), since shorter current pulses would require increased current densities that would lead to damage of the tunnel barrier 12 . A recent alternative to STT are spin-orbit torques 13,14 (SOT), where the spin polarization is obtained by flowing a current in high spin-orbit materials or interfaces 15 . Because of the geometry of SOT devices, the current does not flow through the barrier, and switching in devices has been demonstrated with pulses as short as 200 ps 12,[START_REF] Garello | SOT-MRAM 300MM Integration for Low Power and Ultrafast Embedded Memories[END_REF] .

The limitations of scientific instruments are a major obstacle to switching on time-scales shorter than 200 ps. Commercial electrical current do not have sufficient bandwidth and/or amplitude to switch SOT devices on ps time-scales. However, picosecond-ready CMOS transistors exists since 2007 [START_REF] Mistry | A 45nm Logic Technology with High-k+ Metal Gate Transistors, Strained Silicon, 9 Cu Interconnect Layers, 193nm Dry Patterning, and 100% Pb-free Packaging[END_REF] in commercial technology. Therefore, if methods are discovered for using picosecond electrical pulses and SOT to control magnetic order, these methods have the potential to be integrated into mainstream integrated circuits.

To investigate picosecond SOT phenomena, we use an experimental platform that employs photoconductive switches [START_REF] Auston | Picosecond optoelectronic detection, sampling, and correlation measurements in amorphous semiconductors[END_REF] to generate picosecond electrical pulses. Similar devices have been previously used to demonstrate the reversal of the magnetization of a GdFeCo thin-film with a sub-10ps electrical pulse [START_REF] Yang | Ultrafast magnetization reversal by picosecond electrical pulses[END_REF] . However, the switching mechanism in GdFeCo relies on ferrimagnetic order and is therefore not generalizable to most magnetic materials. The magnetic moment of GdFeCo toggles back and forth on repeated pulses independent of the current polarity due to the effects of rapid Joule heating [START_REF] Yang | Ultrafast magnetization reversal by picosecond electrical pulses[END_REF][START_REF] Wilson | Electric current induced ultrafast demagnetization[END_REF] . For application in devices, one would rather have a mechanism in which the final state depends on the polarity of the current, not on the previous state of the magnetic bit. It is also desirable for the switching mechanism to be compatible with a wide range of materials. Here, we generate and inject 6 ps-wide electrical pulses to trigger SOT dynamics in a prototypical thin Co magnetic film, leading to ultrafast magnetization dynamics and a complete reversal of its magnetic moment. enhance the torques on the Co layer [START_REF] Woo | Enhanced spin-orbit torques in Pt / Co / Ta heterostructures Enhanced spin-orbit torques in Pt / Co / Ta heterostructures[END_REF] (see methods for sample details). We fabricate the Hall structures shown in Figure 1.b, and perform field driven (Figure 1.c) and current driven (Figure 1.d) magnetic hysteresis loops by monitoring the Hall resistance. For the current driven case we use 100 µs-long current pulses while a uniform and constant in-plane symmetry-breaking magnetic field 𝐻 is applied. We find that the critical current density 𝐽 for switching the magnetization is equal to ~2 • 10 A/m 2 for a 160 mT field, and is inversely proportional to the in-plane field as expected from SOT-driven switching 13,[START_REF] Cubukcu | Spin-orbit torque magnetization switching of a three-terminal perpendicular magnetic tunnel junction[END_REF][START_REF] Lo Conte | Spin-orbit torque-driven magnetization switching and thermal effects studied in Ta\CoFeB\MgO nanowires[END_REF] (see supp. mat.). The sample switches to -𝑀 (+𝑀 ) when the in-plane magnetic field and the charge current are parallel (antiparallel), in agreement with a SOT rising from the combination of the spin Hall effect from both heavy metals [START_REF] Liu | Current-Induced Switching of Perpendicularly Magnetized Magnetic Layers Using Spin Torque from the Spin Hall Effect[END_REF][START_REF] Baumgartner | Spatially and time-resolved magnetization dynamics driven by spin-orbit torques[END_REF] .

We use low-temperature GaAs (LT-GaAs) photoconductive switches to generate picosecond pulses of high intensity [START_REF] Auston | Picosecond optoelectronic detection, sampling, and correlation measurements in amorphous semiconductors[END_REF] (shown in Figure 2, see methods for sample fabrication and pulse generation details). The bias voltage Δ𝑉 allows us to select the amplitude and polarity of the current. In order to generate high current pulses we excite the switches with an amplified 5 kHz repetition rate laser system with 30 fs laser pulses centered at 800 nm. The use of this laser system with our photoswitches results in 6 ps duration (τ ) high intensity current pulses (shown in the graph of Figure 2). We measure the pulse duration with a Teraspike® free-standing electric field detector (see methods). After excitation, the electrical pulses propagate (see Figure 2) on Au coplanar waveguides and are focused into the magnetic structure by an impedance matched taper. Switching with a single 6ps -wide pulse Figure 3 shows polar magneto optical Kerr effect (MOKE) micrographs of the initial configuration and final state after a single 6ps electrical pulse, for various configurations. In each of the four quadrants of Figure 3 we test combinations of current 𝐼 and in-plane magnetic field 𝐻 directions. Again, we observe that parallel (antiparallel) current pulses and field result in -𝑀𝑧 (+𝑀𝑧 ), just as expected by the symmetries of the SOTs in the prepared stack. Moreover, we observe that the final state is independent of the initial magnetic state of our stack. We injected up to 10 successive pulses of the same polarity and saw no difference in the final state. We successfully repeated the experiment (initial saturation + single shot) at the switching voltage for 𝑛 =35 times. We can thus estimate a >91% switching probability with a 95% confidence interval (as per the "rule of three", 𝑃 > 1 -3/𝑛 ). As soon as the bias voltage Δ𝑉 is decreased below 40V, no more reversal is observed. When the inplane field is reduced below ( 𝐻 ~120 mT) no reversal is observed, likely requiring higher current densities, as also observed by Garello et al. 12 . In this work, we did not explore higher current densities in order to avoid the risk of degradation or permanent damage of the photoswitch. To test the device endurance, we subjected the device to electrical pulses at switching conditions (Δ𝑉~40V), a repetition rate of 5kHz, for 5 hours (> 10 pulses). After the endurance testing we noticed no degradation or change to the electrical and magnetic properties.

Time-resolved dynamics due to a 3.7ps-wide pulse

To measure the ultrafast magnetic response to the current pulses, we perform time-resolved MOKE measurements under various configurations of current polarity and magnetic fields. We were unable to obtain time-resolved switching dynamics due to technical limitations (see supp. mat.). We therefore perform low intensity time-resolved studies with an 80 MHz oscillator laser with ~250 fs-duration pulses centered at 780 nm (see methods). With this system we obtain 3.7 ps-duration electrical pulses (inset in .a). In these experiments, we monitor the change in the out-of-plane component of the magnetization (Δ𝑀 ) via polar-MOKE with a time-delayed (Δ𝑡) probe pulse. The typical magnetic response to the pulses is shown in .a.

We first focus our attention on the dynamics under zero inplane field (black symbols in .a). The response of the Co magnetization to the current is instantaneous (see methods for time-delay Δ𝑡 calibration). Both for +𝑀 and -𝑀 , the out-ofplane magnetization abruptly decreases, and slowly recovers. The decrease of |𝑀 | is due to two mechanisms. First, the SOT pulls the magnetization towards the plane. Second, the picosecond charge current induces Joule heating which leads to an ultrafast loss of magnetic order [START_REF] Wilson | Electric current induced ultrafast demagnetization[END_REF] , commonly known as ultrafast demagnetization [START_REF] Wilson | Electric current induced ultrafast demagnetization[END_REF][START_REF] Beaurepaire | Ultrafast spin dynamics in ferromagnetic nickel[END_REF] . The latter mechanism, and the subsequent slow cooling by heat diffusion, explain the slow recovery at long time delays (350 ps, see black curve in .a and .b).

We now focus on the dynamics under in-plane magnetic fields (blue and red symbols in .a). Unlike in the zero field case (black symbols in .a), the magnetic moment is not initially oriented along 𝑧 . In the presence of an in-plane field pointing along 𝑥 , the moment at negative time-delays is tilted in the 𝑥𝑧 plane, to be parallel with the effective field 𝐻 (Δ𝑡 < 0), see .c. The in-plane field breaks the symmetry of the system and, together with the injected spin polarization 𝝈 𝒚 , determines the sign for the observed coherent precession. A parallel (antiparallel) in-plane field and current causes the moment to precess towards (away from) -𝑧, causing Δ𝑀 to decrease (increase) on a 10 ps timescale. This occurs regardless of the initial up (+𝑀 ),or down (-𝑀 ) state, as expected from SOT and in agreement with the result of our quasi-static SOT switching experiments from Figure 1.d.

The precessions in .a are offset by the heating induced demagnetization (reduction of |𝑀 |). In addition to reducing the magnetization, ultrafast Joule heating reduces the magnetic anisotropy. We also depict this effect in .e. As temperature rises, the anisotropy field 𝐻 (𝑇) drops and, under a constant external field, the angle (and amplitude) of the effective field 𝐻 (Δ𝑡 > 4ps) changes. The moment experiences a torque 𝝉 𝑯 𝒆𝒇𝒇 due to the change in angle of 𝐻 . The torque 𝝉 𝑯 𝒆𝒇𝒇 is enhanced by the SOT 𝝉 𝑫𝑳 which pulls the moment even further from 𝐻 . The torque 𝝉 𝑯 𝒆𝒇𝒇 , which we call the thermal anisotropy torque, is commonly used in ultrafast pump-probe FMR experiments as a way to trigger oscillatory dynamics [START_REF] Koopmans | Unifying Ultrafast Magnetization Dynamics[END_REF][START_REF] Lattery | Quantitative analysis and optimization of magnetization precession initiated by ultrafast optical pulses[END_REF] . Surprisingly, the thermal anisotropy torque alone can lead to a complete switching of the magnetization of a ferromagnet, as it was recently demonstrated using femtosecond optical pulses [START_REF] Davies | Anomalously Damped Heat-Assisted Route for Precessional Magnetization Reversal in an Iron Garnet[END_REF] . In our experiments, the thermal anisotropy torque can assist the SOT in the switching of the magnetization.

Model

To understand the ultrafast dynamics, we use a simple LLG macrospin model. The model includes SOTs and ultrafast Joule heating (for details see suppl. mat.). The model assumes simplistic temperature dependence laws for the anisotropy and magnetization to calculate the thermal anisotropy torques. We set the spin Hall angle to 0.3 from Ref [START_REF] Woo | Enhanced spin-orbit torques in Pt/Co/Ta heterostructures[END_REF] , 𝑀 from vibrating sample magnetometry (VSM) measurements. We set the damping and anisotropy at room temperature using optically excited time-resolved MOKE measurements [START_REF] Koopmans | Unifying Ultrafast Magnetization Dynamics[END_REF] . We also had to include a number of electrical reflections of the current pulses from the end of the transmission lines which affect the dynamics. The resulting best fits are shown in .b. The quality of the fit is remarkable for such a simple model.

We now describe the model predictions of the dynamics in response to current pulses in the presence of an in-plane field. The model predictions are shown in .b-e: The magnetization 𝒎 is initially in its equilibrium position, along 𝐻 (Δ𝑡 < 0), as in .c. As soon as the current pulse arrives, a damping-like SOT [START_REF] Garello | Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures[END_REF] 𝝉 𝑫𝑳 ~𝒎 × 𝒎 × 𝝈 𝒚 brings the magnetization towards the 𝑦 axis regardless of the injected spins 𝝈 𝒚 (initial drop in Δ𝑀 on blue & red curves in .b), as depicted in .d. At the same time heating changes the effective field by decreasing 𝐻 . As 𝒎 is torqued away from its initial position, precession around the evolving effective field begins (the so called thermal anisotropy torque, shown in .e). The two current polarities will lead to a 180° phase difference in the precessional dynamics, resulting in opposite Δ𝑀 (red and blue trajectories in .b-c). It is interesting to note that a field-like SOT [START_REF] Garello | Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures[END_REF] 𝝉 𝑭𝑳 ~𝒎 × 𝝈 𝒚 cannot reproduce the initial drop in Δ𝑀 that leads to a kink close to Δ𝑡 = 0 on the blue curve. A damping-like dominant torque agrees well with reports on similar structures [START_REF] Woo | Enhanced spin-orbit torques in Pt/Co/Ta heterostructures[END_REF] .

Even though the LLG model describes the overall dynamics, its predictions don't agree with certain features of the data. In particular, the model does not match the dynamics of the black trace at 𝐻 =0 between 3 and 10 ps. A possible explanation is inhomogeneous broadening. In the time-resolved MOKE measurements of damping and anisotropy we observed that the samples have large inhomogeneous broadening in the effective damping, which is consistent with prior pump/probe studies of dynamics of perpendicularly magnetized films [START_REF] Lattery | Quantitative analysis and optimization of magnetization precession initiated by ultrafast optical pulses[END_REF] . In addition to inhomogeneities in anisotropy, it is possible that there are spatial inhomogeneities in the excitation, either due to the (spin) current distribution or hot spots. More experimental and theoretical work will be required to better understand these discrepancies.

Simulations performed with the same parameters at higher currents are presented in the suppl. materials, and suggest the thermal anisotropy torque plays an important role for switching. Including the effect of thermal anisotropy torques in the LLG simulations reduces the energy required for switching by a factor of two. A switching (crossing 𝑀 = 0) as fast as 16ps is predicted, with a recovery of 80% of the magnetization reached after 50ps. Finally, we highlight that just as in Ref [START_REF] Davies | Anomalously Damped Heat-Assisted Route for Precessional Magnetization Reversal in an Iron Garnet[END_REF] , our simulations can lead to switching even without spin-torques (setting spin Hall angles to zero), only due to the thermal anisotropy torque (see supplementary materials).

Energy estimation

We can set an upper limit to the energy dissipated at the load by estimating the total initial energy stored in the photoswitch capacitor (see methods), which is the energy that drives both the THz current pulse in the lines and the losses to THz radiation [START_REF] Hoffmann | Intense ultrashort terahertz pulses: Generation and applications[END_REF][START_REF] Bonvalet | Terahertz Femtosecond Pulses[END_REF][START_REF] Gregory | Optimization of photomixers and antennas for continuous-wave terahertz emission[END_REF][START_REF] Burford | Review of terahertz photoconductive antenna technology[END_REF] . The estimated maximum energy consumption at the magnet (neglecting all losses) is of 50 pJ. This in turn corresponds to a maximum critical current density of 𝐽 ~ 6 • 10

A/m 2 . Therefore, the energy requirements in this ultrafast SOT regime compare extremely favorably with state of the art, ns and sub-ns switching SOT results [START_REF] Garello | SOT-MRAM 300MM Integration for Low Power and Ultrafast Embedded Memories[END_REF][START_REF] Wong | Standford Memory Trends[END_REF] and other types of memory [START_REF] Wong | Standford Memory Trends[END_REF] , even though we are using a non-optimized stack with relatively large µm dimensions.

Discussion of possible mechanisms

Switching via a spin (orbit) torque on such a fast timescale brings up interesting questions about the mechanisms involved. According to our fits, the low current dynamics are almost fully coherent. However, as we approach the switching currents, more complex mechanisms such as magnetic domain nucleation and magnetic domain wall propagation might play a role. We will discuss some of these mechanisms in the following paragraphs. In particular, we will distinguish stochastic nucleation due to thermal noise and deterministic nucleation due to geometry, defects or spatial inhomogeneities in the sample properties.

For long pulses it is well known that thermal activation plays an important role in the nucleation of reversed magnetic domains, which followed by domain wall motion allows for full switching 12 . Thermal activation, and in particular the attempt frequency, is generally thought to be correlated to the FMR frequency [START_REF] Brown | Thermal Fluctuations of a Single-Domain Particle[END_REF] . In fact, the noise spectrum of a ferromagnet under a DC current has been experimentally found to peak at the FMR [START_REF] El Hadri | Two types of all-optical magnetization switching mechanisms using femtosecond laser pulses[END_REF][START_REF] Guan | Thermal-magnetic noise measurement of spin-torque effects on ferromagnetic resonance in MgObased magnetic tunnel junctions[END_REF] . Therefore, when using current pulses wider than the FMR period (~40 ps, for 𝐻 ~1 T), some degree of stochasticity should be expected. It is thus surprising that many works assume thermal activation should be negligible in the sub-ns regime 12,[START_REF] Garello | SOT-MRAM 300MM Integration for Low Power and Ultrafast Embedded Memories[END_REF][START_REF] Cubukcu | Ultra-Fast Perpendicular Spin-Orbit Torque MRAM[END_REF][START_REF] Decker | Time Resolved Measurements of the Switching Trajectory of Pt/Co Elements Induced by Spin-Orbit Torques[END_REF][START_REF] Grimaldi | Single-shot dynamics of spin-orbit torque and spin transfer torque switching in threeterminal magnetic tunnel junctions[END_REF] . Stochasticity in the context of sub-ns SOT switching has only been very recently reported [START_REF] Decker | Time Resolved Measurements of the Switching Trajectory of Pt/Co Elements Induced by Spin-Orbit Torques[END_REF][START_REF] Grimaldi | Single-shot dynamics of spin-orbit torque and spin transfer torque switching in threeterminal magnetic tunnel junctions[END_REF] . If we now consider our experiments with 6ps pulses, well below the FMR period of our samples, stochasticity becomes more questionable. An important point to keep in mind is that with ps pulses the transient temperature rise (and thus thermal noise amplitude) in the film can be much larger than with ns pulses. In fact, thermal noise terms are successfully included in models for optically induced ps switching dynamics 2,[START_REF] Graves | Nanoscale spin reversal by non-local angular momentum transfer following ultrafast laser excitation in ferrimagnetic GdFeCo[END_REF][START_REF] Iacocca | Spin-current-mediated rapid magnon localisation and coalescence after ultrafast optical pumping of ferrimagnetic alloys[END_REF][START_REF] Atxitia | Landau-Lifshitz-Bloch equation for ferrimagnetic materials[END_REF] , but ultimately have a macroscopic deterministic role. Experimentally, a few-nm sized nucleation points have been observed at the ps timescale [START_REF] Graves | Nanoscale spin reversal by non-local angular momentum transfer following ultrafast laser excitation in ferrimagnetic GdFeCo[END_REF][START_REF] Iacocca | Spin-current-mediated rapid magnon localisation and coalescence after ultrafast optical pumping of ferrimagnetic alloys[END_REF] , but are explained as a magnon localization [START_REF] Iacocca | Spin-current-mediated rapid magnon localisation and coalescence after ultrafast optical pumping of ferrimagnetic alloys[END_REF] rather than classical domain nucleation (we remind that domain wall widths are typically ~5-10nm in equilibrium, and that a domain should be larger than two walls). At those length-scales, the dynamics resemble those of a temperature dependent macrospin model [START_REF] Atxitia | Landau-Lifshitz-Bloch equation for ferrimagnetic materials[END_REF] . In summary, the question of the role of thermal fluctuations at ultra-fast timescales is far from resolved, but we believe it is unlikely that stochastic nucleation of classical domains is taking place in our experiments.

If we then consider the switching to be purely deterministic, a few different scenarios are possible. We can first consider an edge-to-edge single domain wall sweeping scenario, assuming a deterministic nucleation at one sample edge. Recently, it was experimentally shown [START_REF] Cai | Ultrafast and energy-efficient spin-orbit torque switching in compensated ferrimagnets[END_REF] that even in large (4µm long) devices, SOT reversal with 0.2ns pulses could be driven by a single domain wall motion. Given our ultra-short pulses, this scenario would imply unphysical domain wall velocities [START_REF] Miron | Fast current-induced domain-wall motion controlled by the Rashba effect[END_REF][START_REF] Thiaville | Dynamics of Dzyaloshinskii domain walls in ultrathin magnetic films[END_REF] of 10 5 m/s, too large for the estimated current densities.

A second possible scenario would be one considering multiple domain nucleations at specific sample sites, due to spatial inhomogeneities in material properties. Based on our estimated peak current density (𝐽 ~ 6 • 10 A/m 2 ), if we extrapolate the expected domain wall velocity based on results of the fastest measured velocities in ferromagnets [START_REF] Miron | Fast current-induced domain-wall motion controlled by the Rashba effect[END_REF] we obtain ~1000 m/s. A Gaussian-like current pulse of 6 ps would allow for less than ~6nm of displacement, which is comparable to the domain wall width [START_REF] Thiaville | Dynamics of Dzyaloshinskii domain walls in ultrathin magnetic films[END_REF] . Therefore, under this scenario switching would be driven by nucleation. Nucleation would then need to cover at least 50% of the area, in order to determine the final state. At longer timescales, these domains could merge leading to a full switching [START_REF] Decker | Time Resolved Measurements of the Switching Trajectory of Pt/Co Elements Induced by Spin-Orbit Torques[END_REF] . We note that we did observe some domain formation observed in a different device, at currents close to the switching threshold, possibly correlated to the current distribution (see suppl. mat.). In order to check for a signature of non-coherent switching at ultrafast timescales, we performed low intensity time-resolved dynamics at various positions on the magnet with our ~1.5 µm diameter probe and found negligible differences (see suppl. mat.). This measured homogeneity could mean nucleated domains are too small and numerous, or just that spatial inhomogeneities in the dynamics are below the sensitivity at such low intensities.

A final alternative scenario would be to consider some form of heat-assisted magnetic recording due to Joule heating. In such scenario the anisotropy barrier and/or magnetic order would be completely reduced by heating up to the Curie temperature 𝑇 and any tiny torque could then determine the switching direction. To test for this scenario, we performed experiments where the in-plane field was replaced by an out-of-plane field, just below the coercive field and favoring a reversal (see suppl. mat.). No reversal was observed after injection of a single current pulse at the critical current, which allows us to discard the HAMR scenario.

Therefore, we believe that, depending on the homogeneity of the sample properties, the switching in our samples could be either nearly-coherent or governed by multiple nucleation events that happen within the current pulse and cover at least 50% of the area. Tracking the dynamics during the reversal could help in distinguishing between these mechanisms.

Conclusions

In conclusion, we have demonstrated spin orbit torque switching of a thin Co film with a single 6 ps electrical pulse. We show that picosecond-duration electrical pulses can inject spin into a magnet at ultrafast timescales. We can then probe the generated magnetization torques with picosecond resolution. All our experiments are in agreement with the symmetries expected from SOT. Macrospin simulations can accurately predict the observed dynamics, showing a damping-like torque dominated effect. Finally, we have shown that the reversal process is extremely energy efficient. Future work will include tracking different components of the magnetization via different magneto-optical effects, in order to spatially reconstruct the time-dependent spin torque dynamics. We believe our approach will trigger new interest in ultrafast electrical studies of spin torque dynamics, opening the door for the possible observation of elusive phenomena such as inertial dynamics in ferromagnetic materials [START_REF] Kikuchi | Spin dynamics with inertia in metallic ferromagnets[END_REF][START_REF] Wegrowe | Magnetization dynamics, gyromagnetic relation, and inertial effects[END_REF][START_REF] Neeraj | Experimental evidence of inertial dynamics in ferromagnets[END_REF] and offer a new way of triggering resonant dynamics in antiferromagnetic materials [START_REF] Němec | Antiferromagnetic opto-spintronics[END_REF][START_REF] Baltz | Antiferromagnetic spintronics[END_REF] .

Methods

Samples

The LT GaAs substrate was obtained by first depositing, in a molecular beam epitaxy chamber, at high temperature (550°C) a 300 nm thick GaAl0.8As buffer followed by a 5nm thick GaAs layer on a semi-insulating GaAs (100) substrate. Then the LT GaAs layer (1 µm thick) was deposited at 260°C with a As/Ga beam equivalent pressure ratio of 50. The magnetic stacks were grown by DC magnetron sputtering in an AJA system. The Ta(5nm)/Pt(4) buffer layer ensures a welldefined (111) texture for the growth of Co(1) and guarantees an interface anisotropy that promotes PMA for Co (1). The Cu(1)/Ta(4) bilayer, capped by Pt(1) to prevent Ta oxidation, has been added to preserve PMA and enhance the torques on the Co layer since Pt and Ta have spin Hall angles with opposite sign. First trials with Pt/Co/Ta stacks resulted in non-square magnetic hysteresis curves with small remanence indicating a possible large effective Dylazhosinkii-Moriya interaction (DMI) as in Ref. [START_REF] Woo | Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets[END_REF] . In order to obtain two well-defined remanent states at zero field we inserted the Cu layer to reduce the DMI at the top Co interface. Due to the long spin-diffusion length of Cu, spin currents generated in the Ta are expected to contribute to the SOT [START_REF] Nan | Comparison of spin-orbit torques and spin pumping across NiFe/Pt and NiFe/Cu/Pt interfaces[END_REF][START_REF] Ostwal | Spin-orbit torque based magnetization switching in Pt/Cu/[Co/Ni]5 multilayer structures[END_REF] . The choice of the stack was also determined by the necessity of having a top metallic layer (Cu/Ta + Pt capping) in order to get a good electrical contact with the transmission lines shown in Figure 2. The 𝑇 of the sample is estimated as ~800 K due to previous experience with extremely similar samples grown and characterized in the group over the years. The sample was fabricated using a 3 step-based UV-lithography technique where the SiO2 layer, magnetic load, and transmission lines were patterned at each step. The SiO2 layer allows for a good insulation of the transmission lines from the substrate in order to suppress leakage currents. A single recipe was used to perform lithography and lift-off for all three steps. An AC 450 (Alliance Concept) sputtering system was used to deposit 100 nm of SiO2 in the presence of 20sccm Ar and O2 flow at a base pressure of 6.1×10-3 mbar. E-beam evaporation was used to deposit Ti(20 nm)/Au(300 nm) for the transmission lines. The coplanar waveguide has a center-line to side-line distance of 60 µm. The waveguides have a 60 µm-wide center-line, that tapers down to a 5.5 µm spaced 4 µm-wide center-line, as depicted in Figure 2. The magnetic load is a 20 µm x 4 µm strip partially covered by the Au transmission lines, so current only flows through the magnetic stack only in the uncovered 5 x 4 µm opening. The Hall bars where patterned via similar lithography process, as published elsewhere [START_REF] Pham | Thermal Contribution to the Spin-Orbit Torque in Metallic-Ferrimagnetic Systems[END_REF] .

Generation of picosecond-duration electrical pulses

In order to generate the picosecond pulses (schematic shown in Figure 2) we contact the left side of the transmission lines with a CPW 40GHz GBB® probe tip. We also contact the right side with another CPW 40GHz GBB® tip, and add a 50Ω resistor to close the circuit. We apply a constant voltage bias (Δ𝑉) through the left tip between -50V and +50V via a Keithley 2400 voltage source, while reading the average current. If no laser irradiation is incident, we can measure a dark (i.e. leakage) current due to the finite switch resistance (>10 MΩ). We then irradiate the photoswitch with either 1.5mW (0.3 µJ per pulse) from our 5kHz amplified laser or 30mW (0.37 nJ per pulse) from the 80MHz oscillator system. The pump is focused by a 15 cm lens to a (FWHM) radius of about 150 µm. When the switch is irradiated a photocurrent is generated, which we optimize by finely tuning the pump mirror. We note that photoswitch excitation with the high voltages and high pulse energies used for the switching experiments can result in longer electrical pulses durations (6 ps for switching experiments vs 3.7 ps for time-resolved experiments) [START_REF] Burford | Review of terahertz photoconductive antenna technology[END_REF] .

MOKE micrographs

The images where obtained with a home-made magneto-optical Kerr effect microscope. We use a 633nm light source in the Köhler configuration, a long-working distance 50x objective and a 6fps monochrome CCD. Due to the highly reflective Au transmission lines that are next to the magnetic section, we are very susceptible to pixel overflowing, which means the exposure time has to be kept short, which results in an important degradation of magneto-optical signal. As a solution we use 8x binning, without loss of spatial resolution, to speed up the camera up to 15fps and average 60 images to reduce noise and boost back the signal to noise ratio. Finally, mechanical drift and vibrations importantly degrade the quality of the images, in particular when live background subtraction is used in order to the enhance the magneto-optical contrast.

Measurement of time-resolved dynamics

All the presented small-current induced dynamics where measured with the 80MHz oscillator laser system focused through a 50x objective into a ~1µm sized spot. The experiments where performed with no out-of-plane field, since at small excitations the sample naturally relaxed back between pulses, as is typical with low-excitation optical pump-probe experiments. We determined the zero delay time, i.e. the arrival of the electrical pulse, by monitoring the time-domain thermoreflectance (TDTR) response (see supp. mat.).

Measurement of picosecond pulse electric field

A pump beam is focused on the photoconductive switch to generate the pulse. The pump beam is chopped by an optical chopper at about 300 Hz. We place the Teraspike® probe on top of the transmission lines. A probe beam is focused at the tip of Teraspike® probe, exciting the free-standing photoconductive switch. A Stanford Research Systems 865A Lock-In amplifier is used to directly measure the change in current induced by the transient electric field at the tip of the probe. This change in current is only present when both the electric field and probe beams coincide in time. Therefore, changing the delay between optical pump and probes allows us to sample the transient electric field associated with the picosecond current pulses. The resulting trace is a convolution of the real electric field and the Teraspike®'s response. We note that we do not measure the electric pulse at the sample position.

Estimation of switch capacitance and energy dissipation

The capacitance of an interdigitated electrode (IDE) capacitor 𝐶 is roughly 𝐶 = (𝑁 -1)𝜖 𝜖 𝐴/𝑑 where 𝑁 is the number of electrodes, 𝜖 is the effective relative permittivity due to LT-GaAs substrate and air (measured as 15), 𝜖 is the vacuum permittivity, 𝐴 is the surface area of an electrode and 𝑑 is the center-to-center distance between electrodes. We find a capacitance of about ~10 F for our photoconductive switch. As a second verification, it is also well known that the RC time constant due to the capacitance of the photoswitch limits the pulse duration of the generated pulses [START_REF] Auston | Picosecond optoelectronic detection, sampling, and correlation measurements in amorphous semiconductors[END_REF] , which means we can also set an upper bound for the capacitance given by RC<3.7ps (which is our smallest measured pulse duration). Here, the characteristic impedance (𝑍 ) of the line plays the role of the resistor. The CPW line impedance was designed [START_REF] Gupta | Microstrip Lines and Slotlines[END_REF] to be 𝑍 = 70Ω , which means the capacitance is at most around 5.3 • 10 F, consistent with our initial estimation. We will consider this upper-bound value as the capacitance of our switch to calculate the upper-bound energy dissipation in our experiments. For the measured bias voltage at the switching threshold voltage of Δ𝑉 = 40 V, the energy stored is 𝐶 Δ𝑉 ~50 pJ. In a worst case scenario, if we assume a full discharge of the switch, no radiative losses, no transmission loses and assume a perfect absorption at the magnet, then all of the energy stored in the switch capacitor would be dissipated at the load. For our magnetic load dimensions, this corresponds to an energy density of ~150MJ/m 3 . Because the energy dissipation for a Gaussian current pulse is 𝐸 = ∫ 𝐽(𝑡) ρ • d𝑡 = 0.75 • 𝐽 ρτ , where ρ (= 81 µΩ • cm ) is the measured resistivity of the magnet, we estimate the maximum peak current density for switching with τ = 6 ps pulses to be about 𝐽 ~ 6 • 10 A/m 2 .

Dependence of quasi-static critical current density on in-plane field

Figure S1: The critical current density for SOT switching with 100 µs pulses is inversely proportional to the in-plane 𝐻 field, as reported previously [START_REF] Lo Conte | Spin-orbit torque-driven magnetization switching and thermal effects studied in Ta\CoFeB\MgO nanowires[END_REF] .

Estimation of resistivity and quasi-static critical current density

A four point resistivity measurement yielded an effective resistivity of ρ = 81 µΩ • cm. Using previously measured values of resisitivity 4-5nm Pt and Ta films deposited with the same sputtering system (24 and 200 µΩ • cm respectively), and a parallel resistor model, we determine the correct effective resistivity by assuming that most of the current flows in the 13nm comprised by the 5nm Ta buffer, bottom 4nm Pt and top 4nm Ta layers, and neglecting the currents through the 1nm Co, Cu and Pt capping layers. We then estimate the critical current density by using again a parallel resistor model to calculate the amount of current going through Pt and Ta layers surrounding the Co/Cu bilayer.

Switched area vs current amplitude

Figure S2: The switched area as a function the current density normalized by the critical current density (𝐽 ). The reversal process is first happening where the current density is the highest, at mid-height, right in between the tips of the rounded gold electrodes. In order to fully switch the device a little more current density than 𝐽 is needed. We note that these experiments were performed on a different sample with a coplanar stripline design. Moreover, the magnetic section had a ~5𝜇m width, slightly wider than the main sample of the article (~4𝜇m). In fact, in the main sample, these partial reversals were not clearly evidenced. We believe this could be due to the narrower section, or also due to the gold contacts being flatter, possibly resulting in a more homogeneous current distribution. We also note that in the experiments of Figure S2 the current pulse duration is unknown (not measured). We estimate it in between 3-10ps, from experience with similar devices. The electrons immediately respond to the heat pulse (negative peak at time-zero). The magnetic dynamics (red) equally start at the arrival of the pulse with no noticeable delay. Further work is needed to fully interpret the TDTR response.

Determination of zero-time delay from Time Domain Thermoreflectance

Time-resolved switching dynamics

Time-resolved pump/probe measurements of the switching dynamics are not yet possible with our 80 MHz laser system, since the energy per pulse is not high enough to generate sufficiently intense current pulses. The time-resolved dynamics shown in was taken with almost the full pump power irradiating the switch, and the strongest in-plane field available in our setup. Probing the switching dynamics was also not possible with our 5 kHz amplified laser system. On our MOKE setup, measurements require a minimum probe power of ~60 µW to resolve Kerr rotations of ~100 µrad. The full amplitude of the hysteresis (2𝑀 ) measured with the 80MHz oscillator system, shown in Figure S4, is ~485 µrad. Therefore, 60 µW would allow us to resolve the dynamics with a SNR of less than ~5:1 (without accounting for drift issues). However, at a 5 kHz rep rate, the per pulse energy for 60 µW causes damage to the sample when focused to sub-3 µm dimensions. Possible solutions could be to make the sample area bigger, and defocus the beam, or to use a higher repetition rate laser.

HAMR-scenario and dependence of coercivity on number of pulses

In order to check for a heat assisted magnetic recording-like scenario, we injected single pulses at the switching threshold current, under no in-plane field, and monitored the variation of the coercive field (the applied out-of-plane field leading to ~50% reversal or more) with the number of applied current pulses. If the sample was to be heated near 𝑇 , then any out-of-plane magnetic field would lead to reversal even after excitation with a single current pulse. However, we did not observe any switching when injecting a single pulse and applying an out-of-plane field as large as 93% of the switching field. In fact, we only observe a small decrease of ~30% in the coercivity when increasing the number of pulses by a factor of 10 (see Figure S5). We conclude that the dissipation by the electrical pulse does not heat the Co film near 𝑇 .

Spatial dependence of magnetic dynamics

Figure S6: Spatial dependence of dynamics. Inset shows the peak (at 11ps) as a function of the y position (across the sample width). The signal drops as we get close to the edges because the probe no longer fully overlaps the magnet (the probe width is about 1.5 µm (FWHM), and the sample width is 4 µm. The dynamics are extremely similar across the surface of the sample. Experiments along the length of the magnet (x direction) also showed no major differences.

8. Macrospin model (table of parameters in section 7.4)

LLG dynamics

To model the ability of picosecond electrical pulses to reverse the magnetization via spin orbit torques, we solve the Landau-Lifshitz-Gilbert equation of motion with anti-damping like and field-like damping terms from the spin-current in the following form:

⃗ = -𝛾𝜇 𝑀 ⃗ × 𝐻 ⃗ + 𝑀 ⃗ × ⃗ -𝜃 𝑀 ⃗ × 𝑀 ⃗ × 𝜎 ⃗ + 𝜃 𝐶 𝑀 ⃗ × 𝜎 ⃗ (eq.S1)
where 𝐶 = (eq.S2)

Here, 𝛾 is the gyromagnetic ratio, 𝜇 is the vacuum permeability, 𝛼 is the damping parameter, 𝑀 is the saturation magnetization, 𝜎 ⃗ is the direction of spin-polarization, 𝜃 is the damping-like spin Hall angle, 𝜃 is the field-like spin Hall angle, 𝐽 is the current density, 𝑑 is the thickness of the magnetic layer, 𝑞 is the charge of the electron and 𝜇 is the Bohr magneton. The effective field in the first term consists of a magneto-crystalline anisotropy field, a demagnetization field, and any applied external field,

𝐻 ⃗ = - ⃗ = 𝐻 𝐻 𝐻 + -𝑀 𝑚 (eq.S3)
Here, 𝐻 ⃗ , 𝐻 ⃗ and 𝐻 ⃗ are the x, y, and z-components of the external field, 𝐾 is the perpendicular anisotropy constant, and 𝑀 ⃗ = 𝑀 𝑚 ⃗ 𝑚 ⃗ 𝑚 ⃗ . The -𝑀 𝑚 ⃗ term in the z-direction is the demagnetization field due to thin-film shape anisotropy. In this work, we use the macrospin approximation, i.e. we assume that the properties in Eq. (eq.S1) are independent of position. To solve for the dynamics, we first set 𝑑𝑀 ⃗ /𝑑𝑡 = 0 and solve for the equilibrium orientation of the moment, 𝑀 ⃗ (𝑡 < 0) = 𝑀 ⃗ . We identify the stable solution to 𝑑𝑀 ⃗ /𝑑𝑡 = 0 by choosing the solution with the lowest free energy. We then use a finite-difference scheme to evolve 𝑀 ⃗ (𝑡) forwards in time in response to a charge current 𝐽 (𝑡). We evolve the magnetization forward in time with time-increments of Δ𝑡 = 1 fs (we verified that the results were unchanged with smaller time-increments).

Temperature Dynamics

Ultrafast heating of a magnetic material leads to temperature-induced changes to the magnetic moment and the interfacial anisotropy. We estimate the temperature response of the metal film in our experiments by solving the heat-diffusion equation,

𝐶 = Λ + 𝑞(𝑡) (eq.S4)
Here, 𝑇 is the temperature, 𝐶 is the heat-capacity per unit volume, Λ is the thermal conductivity, and 𝑞(𝑡) is the volumetric heating from either a laser or electrical pulse. Based on literature values of the heat-capacity of metals [1] and the thickness of each layer, we estimate an average value for 𝐶 of the multilayer of 2.6 J m -3 K -1 for our stack at room temperature. We fix Λ according to the Wiedemann-Franz Law Λ = L 𝑇/𝜌 ~9 Wm -1 K -1 , where 𝜌 = 81 µΩ • cm is the measured electrical resistivity of the film. For the optical experiments described in section 7.3., we assume 𝑞(𝑡) = 𝑃 (𝑡)/(𝜋𝑤 𝑑), where 𝑃 (𝑡) the absorbed laser power vs. time, 𝑤 is the 1/e 2 radius, and 𝑑 is the total film thickness of 16 nm. For electrical experiments, we set 𝑞(𝑡) = 𝜌 𝐽 (𝑡), where 𝐽(𝑡) is the charge current density. Solving Eq. (eq.S4) for 𝑇(𝑡) requires boundary conditions. We assume an adiabatic boundary condition at the metal film surface. We assume the heat-current 𝐽 at the bottom of the metal film is limited by the interfacial thermal conductance 𝐺 between the Ta and sapphire, 𝐽 = 𝐺 𝑇(𝑧 = 𝑑) (eq.S5)

Typical values for the conductance between metal films and oxide substrates are 100-300 MWm -2 -1 [2-4]. We treat 𝐺 as a fit parameter and deduce 𝐺 ~170 MWm -2 K -1 .

By using Eq. (eq.S4) to model the temperature response of the stack to heating, we are assuming that electrons, phonons, and spins are in thermal equilibrium with one another. Such an assumption is not always valid on picosecond time-scales, and nonequilibrium between thermal resevoirs can drive ultrafast magnetic phenomena [5]. In our experiments, nonequilibrium effects should be small due to the 4-6 picosecond pulse duration of the electrical experiments, together with the strong thermal coupling between electrons and phonons in the Co layer [6]. The picosecond time-scale for heating is much greater than the electron-phonon relaxation time in transition metals [7]. Therefore, we can estimate the nonequilibrium between electrons and phonons by assuming a quasi-steady-state condition where rate of heat absorption of electrons equals the rate of heat-loss to the phonons. In other words, we assume 𝑞(𝑡)~𝑔 𝛥𝑇 (𝑡), where 𝑔 is the electron-phonon volumetric energy transfer coefficient [8], which we take from Ref. [6]. For our estimated maximum electrical current density of 6 • 10 A m -2 , we can estimate the upper bound for the nonequilibrium during our experiments to be 𝛥𝑇 (𝑡)~15K, or about 25% of the average peak temperature rise at those currents.

Anisotropy Torques and Precessional Dynamics Caused by Ultrafast Heating

The anisotropy field and magnetization are both temperature dependent [9]. As a result, picosecond changes in the temperature of the Co film induce precessional dynamics [10], even in the absence of spin-orbit-torques or Oersted fields. To experimentally investigate the effect of temperature, we perform time-resolved magneto-optic Kerr effect measurements on the sample in the presence of an in-plane magnetic field. Prior to these time-resolved measurements, we orient the magnetic moment of the Cobalt with an out-of-plane magnetic field of 0.3 Tesla. After removing the out-of-plane magnetic field, we optical heat the sample surface with 250 fs duration pump pulses at a fluence of 0.7 J m -2 under the application of an in-plane magnetic field. The transient temperature response causes precessional dynamics. We track the resulting out-of-plane component of the magnetic moment by monitoring the polar Kerr angle with a timedelayed probe pulse. We repeat this experiment with varied in-plane magnetic fields.

Fig. S7 shows the results of time-resolved magneto-optic Kerr measurements with an in-plane field of 0, 0.15, and 0.3 T on an identical magnetic stack which was co-grown on a glass substrate (due to the glass substrate, the interfacial conductance is lowered to 𝐺 ~100 MWm -2 K -1 The lines in Fig. S7 are best-fit model predictions for the data based on the LLG equations, including the thermal model of section 7.2. We describe those predictions in more detail below.

Using an optical multilayer calculation like described in Ref. [11], we estimate the absorbed fluence per pump pulse (𝐹) is 0.3 J m-2. We estimate that this absorbed fluence should cause a per-pulse temperature rise of approximatively 𝐹/(𝑑 • 𝐶)~7 K. Comparing this temperature rise to our experimental measurements of the resulting Δ𝑀 , allows us to quantify 𝑑𝑀 /𝑑𝑇 close to room temperature (i.e. for small heatings, just like in the SOT time-resolved dynamics). The amplitude of precession of 𝑀 (𝑡) following heating provides information about 𝑑𝐾 /𝑑𝑇. The frequency and decay rate of precession allow us to determine the total anisotropy field and effective damping.

To theoretically quantify the effects of the temperature-evolution after optical heating we added temperature effects to Eq. (eq.S1-S3).

We allow 𝑀 and 𝐾 in Eq. (eq.S1-S3) to evolve in time based on the predictions of our thermal model described in Eq. (eq.S4-S5). We follow Ref. [9], and assume the temperature dependencies of the magnetization and magneto-crystalline anisotropy to be described by

𝑀 (𝑇) = 𝑀 (0)[1 -(𝑇 𝑇 ⁄ ) . ] (eq.S6) 𝐾 (𝑇) = 𝐾 (0)[𝑀 (𝑇)/𝑀 (0) ] (eq.S7)
Here, 𝑇 is the Curie temperature, 𝑀 (0) is the magnetization at absolute zero, and 𝐾 (0) is the anisotropy constant at absolute zero.. We fix 𝑀 (𝑇 = 300 𝐾) to 10 A/m based on VSM measurements. We treat 𝐾 (𝑇 = 300 𝐾) a fit-parameter. The precessional frequency in Fig. eq.S2 depends only on the magnetic moment and total anisotropy field. With 𝑀 (300 K) fixed from VSM measurements, the only unknown parameter that affects the precessional frequency is 𝐾 (300 K). The best-fit value for 𝐾 (300 K) is 10 Jm -3 , corresponding to an out-of-plane anisotropy field 𝐵 (300 K) = 2𝐾 /𝑀 -𝜇 𝑀 of ~0.8 T. The fitting is only carried for data after several picoseconds, since the simple thermal model we are using (to avoid including too many parameters) does not account for the absorption profile of the laser pulse and assumes an instantaneous homogeneous heating of the magnetic stack, not capturing the initial thermal equilibration between layers. We fit for the effective value of 𝛼 in eq.S1 by matching the model's prediction for the decay of oscillations to the data, and obtain a surprisingly large 𝛼~0.23. We note that the effective 𝛼 includes effects such as inhomogeneous broadening [12]. The large damping could also explain why we could not detect the ferromagnetic resonance with our vector network analyzer. The only remaining model parameter is 𝑇 . The value of 𝑇 determines 𝑑𝑀 /𝑑𝑇 and 𝑑𝐾 /𝑑𝑇 at room temperature. A best-fit to our data in Fig. S6 yields 𝑇 ~800 K, in agreement with our extensive experience with similar samples grown in the same sputtering system. Then, eqs.S6-S7 predict 𝑑𝑀 /𝑑𝑇~-10 Am -1 K -1 and 𝑑𝐾 /𝑑𝑇~-4 • 10 Jm -3 K -1 , around room temperature.

critical current density (Fig S8a) and required energy (Fig S8b) is smaller when including thermal effects. The ratio between the required critical current densities (energies) in both models is plotted as a dotted orange line in FigS8a (b). With the used parameters, this ratio is at least ~1.5 thereby yielding at least a factor of 2 in energy gains.

When using the macrospin LLG model (red line in Figure S8), we find a critical current proportional to Δ𝑡 (+ an offset), which could mean that the spin angular momentum for switching is constant below a certain pulse duration, as suggested in Ref.14. Surprisingly, also in Ref.14, a Δ𝑡 relationship was found using similar macrospin simulations. When we include thermal effects (black empty circles) the relationship deviates from Δ𝑡 . However, we note that if we were to make a fit of the thermal estimates (black empty cirlces) using a Δ𝑡 relationship within a reduced range (between ~100ps and ~1ns as in Ref.14) the fit would still work quite well (see blue dashed line) for that narrow range. This exercise highlights the difficulty in extracting the exact exponent when the range of the data is narrow.

Finally, it is interesting to look at the energy requirements on the right Figure S8b. Because of damping and anisotropy, as the current pulse gets wider than 10 ns, a large part of the angular momentum injected by the current pulses gets dissipated and thus even more current is required to keep increasing the precession angle, until switching is achieved. This results in an important increase in the required energy. On the other hand, when going into the very short pulse regime, even if less angular momentum is "wasted", more and more current density is required, resulting in stronger Joule heating. In between, around 10-20 ps for the used parameters (not far from the FMR half-period), a minimum of required energy will be found. Curiously it is around this minimum point that we also find the least difference between the two model's estimated critical energies. ) corresponds to the ratio of critical current densities (energies) between the LLG and the LLG + thermal model. In plane field is 0.16T and the rest of parameters are presented in table T1.

Effect of Heating on Ultrafast SOT Dynamics

To evaluate the role of temperature rises on the switching dynamics, we use the model parameters described above, to estimate the change in dynamics that results from the addition of temperature-induced precessional dynamics. We set an in-plane magnetic field of 0.16T and a 6ps wide pulse. The results of these simulations for two opposite current directions (plotted as blue and red) are shown in Figure S9. Using the full thermal model (Figure S9b) the critical current density is estimated to around 6 • 10 A/m 2 whereas when using a pure LLG with no temperature dependence (Figure S9a) we need to increase the current up to 9 • 10 A/m 2 in order to observe switching. This corresponds to about a twofold increase in energy dissipation/consumption. The speed of reversal can vary quite a bit depending on the current density, but we observe minimum switching times around 16ps (for the crossing of zero magnetization), for the shown parameters. T1.

Interestingly, even if we set the spin Hall angles to zero (and thus have zero SOT), if we increase the current density to have enough of a temperature rise, then a strong enough thermal anisotropy torque will be present and lead to switching. We show this in the simulation of Figure S10. This was also experienced in YIG by using ultrafast optical pulses [15], and the important requirementss are: a) having dissimilar thermal derivatives of M and K, and b) to perform the experiment under a constant field perpendicular to the magnetic anisotropy axis.

Figure S10. Switching with a 6 ps pulse purely via the thermal anisotropy torque. We set the spin Hall angle to zero, and simulate the dynamics of out-of-plane magnetization under the influence of Joule heating due to a 6 ps pulse. In case an in-plane 160mT field is applied (blue line) we observe switching, but if no in-plane is applied (red line), switching is no longer possible.

Figure 2 :

 2 Figure 2: Setup for generation of picosecond electrical pulses and magnetooptical detection. The optical pump excites the photoconductive switch in order to generate ~6 ps duration electrical pulses, that are guided and focused by a coplanar waveguide into the magnetic stack, resulting in ultrafast spin orbit torques. The sampled picosecond current pulse is shown on the back of the figure. The solid green line is a guide for the eyes.

Figure 3 :

 3 Figure 3: MOKE micrographs of single 6 ps electrical pulses switching the magnetization via SOT. The four quadrants show 2 before and 2 after-pulse images under different in-plane field and current directions. The inversion of the final state with current or in-plane field is a clear signature of SOT switching. Bias voltages used for switching were slightly above the critical threshold (Δ𝑉~40 V). Light (dark) grey indicates magnetization down (up).

Figure 1 :

 1 Figure 1: Sample and switching behavior via field and current. a) Magnetic sample stack b) Patterned Hall bar using the magnetic stack and gold contact pads. The schematic shows the electrical connections used for Anomalous Hall resistance (𝑅 = 𝑉 /𝐼) detection as a function of c) outof-plane magnetic field (𝐻 ) and d) 100 µs current pulses under a 160mT in-plane field (𝐻 ).

Figure 4 :

 4 Figure 4: Time-resolved MOKE response due to a 3.7ps electrical pulse and macrospin simulation. a) The dynamics include spin orbit torques and thermal effects (demagnetization). The phase and sign of the torques is dependent on the in-plane field (𝐻 ) and current direction, as expected from SOT. Without the symmetry-breaking in-plane field the oscillations disappear. The solid green line is a fit of current trace with a sech 2 function of 3.7ps (FWHM). Experiments were done after ±𝑀 𝑧 saturation of the sample, under 𝐻 = 0 mT , 𝐻 = ±160 mT and a bias voltage of Δ𝑉~30 V. b) Macrospin simulation (lines) including ultrafast demagnetization and SOTs on top of the +𝑀 data (circles). All data is normalized by the saturation magnetization 𝑀 at room temperature. c-e) Schematic of the magnetization dynamics induced by the ps current pulse as predicted by the model. Description in the text.

Figure S3 :

 S3 Figure S3: Time-domain thermoreflectance (TDTR) in black and polar MOKE (PKR) response in red. The TDTR allows us to set time-zero in our experiment.

Figure S4 :

 S4 Figure S4: Polar MOKE hysteresis on the CPW-embedded magnetic sample taken with the 80MHz laser system.

Figure S5 :

 S5 Figure S5: Coercivity as a function of the number of single 6ps pulses spaced every 200 µs.

Figure S8 .

 S8 Figure S8. Simulated critical current densities as a function of current pulse duration. a. Critical current density and b. critical energy as a function of the pulse duration. Red filled circles are obtained using the LLG model and the black open circles are obtained when adding the thermal model. Lines are ~1/Δ𝑡 + 𝐽 fits, as discussed in the text, for comparison with previous work from Ref.14. The orange dot line in a. (b.) corresponds to the ratio of critical current densities (energies) between the LLG and the LLG + thermal model. In plane field is 0.16T and the rest of parameters are presented in tableT1.

Figure S9 .

 S9 Figure S9. Strong current dynamics predicted with a 6 ps pulse. Dynamics of the normalized out-of-plane magnetization Mz as obtained from numerical solutions of the LLG model that a) exclude and b) include the variations of material parameters with temperature. All blue curves correspond to a positive current flow, and red curves to a negative current flow. The fastest reversal (crossing of zero) takes about 16ps in our simulations, but only reaches 80% of saturation after 50ps. We assume an in-plane field of about 0.16T. The rest of parameters are in tableT1.

Acknowledgments

We would like to thank Emmanuel Vatoux, Tom Ferté and both Laurent Badie and Gwladys Lengaigne for the electromagnet construction, VSM measurements and sample preparation, respectively. We would like to specially thank Yang Yang and Charles-Henri Lambert for their help in the first SOT switching trials years ago. This work was supported by the impact project LUE-N4S part of the French PIA project "Lorraine Université d'Excellence", reference ANR-15IDEX-04-LUE and the "FEDER-FSE Lorraine et Massif Vosges 2014-2020", a European Union Program. This work was also partly supported by the French RENATECH network. R.L.C. and J.B. gratefully acknowledge support from the National Science Foundation (NSF) through the Cooperative Agreement Award EEC-1160504 for Solicitation NSF 11-537 (TANMS). A.P. and J.B. also gratefully acknowledge support from the NSF Center for Energy Efficient Electronics (E3S). Work by X. S. and R. W. was supported by the U.S. Army Research Laboratory and the U.S. Army Research Office under contract/grant number W911NF-18-1-0364. J. B. also gratefully acknowledges support by ASCENT (one of the SRC/DARPA supported centers within the JUMP initiative). Preliminary experiments in this work were supported by U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05-CH11231 within the Nonequilibrium Magnetic Materials Program (MSMAG).

Author contributions

J.G. designed the experiments with inputs from R.B.W, J.B, S.P-W. & J.H. A.L. grew the LT-GaAs substrates. M.H. optimized and grew the samples by sputtering. K. fabricated the devices. K., J.H. and J.G., performed the ultrafast SOT experiments and characterized the picosecond pulses. E.M., A.Y.A.C. performed the anomalous Hall measurements and 100µs SOT switching experiments under the supervision of J.C.R.S. and S.P-W. who designed the mask and set up this experiment. R.B.W. built the numerical model and performed the simulations with inputs from J.G. R.B.W. performed optical time-resolved experiments to determine the damping and anisotropy of the samples. J.G. analyzed the experimental data with help from K., R.B.W., R.L.C., J.H. and S-P.W. J.G. wrote the manuscript with input from all authors.

Competing financial interests

The authors declare no competing financial interests.

Table of selected and fitted parameters

Below is the list of all used parameters during all the SOT simulations. We note that we fix most parameters using different sources, and only a few variable parameters are fitted (in red), with the benefit that each of them influences the dynamics differently. For example, the interface thermal conductance plays a big role in the recovery (long time-delay), whereas the field-like spin Hall angle is very important during the initial instants, while the current is on. Limitations of the model are discussed in the paper.

Table T1. Parameters of macrospin model

Critical current density vs pulse duration

In this section we compute the final outcome of the magnetic moment of the Co layer after a single current pulse, as a function of current density and pulse duration (Δ𝑡). We increase the current density for a given pulse duration until the magnetization's final state (end of the pulse + 200ps) is reversed and record that current density as the critical current density. As shown in Figure S8, we compute two different scenarios: Only LLG (red circles) and LLG + thermal effects (black open circles). We remind that the results for strong currents (such as here, for switching) when including thermal effects, are not representative of our samples necessarily, since we do not know the exact temperature dependence of 𝑀 and 𝐾 away from room temperature. However, it is clear that ultrafast heating should lead to extra torques (just as in the case of optical pump-probe experiments) that should help the switching. Indeed, as Figure S8


Hella Boumaiza

Patrick Dutournié

Jean-Marc Le Meins

Lionel Limousy

Jocelyne Brendlé

Christelle Martin

Nicolas Michau

Liva Dzene
email: liva.dzene@uha.fr

IRON-RICH CLAY MINERAL SYNTHESIS USING DESIGN OF EXPERIMENTS APPROACH

Keywords: iron-rich clay minerals, design of experiments, synthesis, nontronite, berthierine

The precipitation of iron-rich clay minerals has been reported in various geologic environments.

However, large quantities of these minerals are not widely accessible to study their thermodynamic properties and thus improve our understanding of these particular geological systems. To obtain sufficient amount of iron-rich minerals, their synthesis can be foreseen. A design of experiments and surface response methodology was used in this study to find optimum synthesis conditions for these clay minerals. Three parameters were considered: quantity of OH -, synthesis time and temperature. The quantity of OH -had the largest influence on the synthesis result. The amount of 2:1 type clay mineral with respect to 1:1 type clay mineral increased with OH/Fe molar ratio. Time and temperature had less significant impact on the synthesis outcome, yet the formation of 2:1 type phase was favored at short synthesis time (1 day) and high temperature (160°C). The findings suggested that the kinetic of 2:1 type clay mineral phase formation with respect to 1:1 type phase was faster, yet the precipitation was strongly dependent on the availability of the reactants.

Introduction

Clay minerals are layered silicate materials, where silica tetrahedra are arranged in continuous sheet combined with hexagonally coordinated metal-containing octahedral sheet. In general, two types of structures can be distinguished, a so-called 2:1 type clay mineral where the metalcontaining octahedral sheet is sandwiched between two silicate sheets forming a 2:1 layer, and a 1:1 type clay mineral, where one silicate sheet is combined with one octahedral sheet forming a 1:1 layer (Figure 1). These minerals are widely encountered in natural sedimentary rocks and soils. The most common elements found in natural clay minerals are Si, Al, Mg, Fe, O and H (i.e., montmorillonite, beidellite, vermiculite, kaolinite, saponite). Minerals containing solely Si, Fe, O and H are known, but less common (i.e., greenalite, cronstedtite, hisingerite, nontronite). The environments where the formation of iron-rich clay minerals have been reported include deep-sea sediments [START_REF] Baldermann | Substantial iron sequestration during green-clay authigenesis in modern deep-sea sediments[END_REF][START_REF] Tosca | An authigenic origin for Precambrian greenalite: Implications for iron formation and the chemistry of ancient seawater[END_REF], subduction zones [START_REF] Beaufort | Chlorite and chloritization processes through mixed-layer mineral series in lowtemperature geological systemsa review[END_REF][START_REF] Sforna | Abiotic formation of condensed carbonaceous matter in the hydrating oceanic crust[END_REF], transform faulting [START_REF] Kodolányi | Geochemistry of ocean floor and fore-arc serpentinites: Constraints on the ultramafic input to subduction zones[END_REF], the surface of Mars [START_REF] Chemtob | Synthesis and structural characterization of ferrous trioctahedral smectites: Implications for clay mineral genesis and detectability on Mars[END_REF] and meteorites [START_REF] Zolotov | Formation of brucite and cronstedtite-bearing mineral assemblages on Ceres[END_REF]. Their presence has also been reported at the interface between different materials such as steel and glass [START_REF] Schlegel | Alteration of nuclear glass in contact with iron and claystone at 90°C under anoxic conditions: Characterization of the alteration products after two years of interaction[END_REF] and steel and clay-rich rocks [START_REF] Lanson | Experimental investigation of smectite interaction with metal iron at 80°C: Structural characterization of newly formed Fe-rich phyllosilicates[END_REF] studied for the conception of radioactive waste repository sites.

For the different geological systems, it is of great interest to predict the evolution of these systems whether millions of years in the past, as in the case of Mars and Earth surfaces, or other thousands of years in the future as in the case of long-term radioactive waste repository sites.

In order to do this, geochemical modelling is required to predict the evolution with time of the mineralogical composition of these different systems. The prediction of this evolution is crucial as it enables assessing the changes of key properties such as porosity and permeability.

To assess the clay mineral thermodynamic properties for geochemical modelling, different approaches were developed [START_REF] Tardy | An ideal solid solution model for calculating solubility of clay minerals[END_REF][START_REF] Vieillard | A new method for the prediction of Gibbs free energies of formation of hydrated clay minerals based on the electronegativity scale[END_REF]. The thermodynamic properties of clay minerals are particularly challenging to constrain owing to the great chemical and structural variability of these minerals [START_REF] Gailhanou | Thermodynamic properties of saponite, nontronite, and vermiculite derived from calorimetric measurements[END_REF]. A common approach to assess them is the summation of oxide/hydroxide parameters, which on the other hand are wellknown. However, the knowledge of experimental thermodynamic and kinetic properties of iron-rich clay minerals would enhance the robustness of the models, as well as allow the comparison of experimental data with modelling [START_REF] Gailhanou | Thermodynamic properties of saponite, nontronite, and vermiculite derived from calorimetric measurements[END_REF].

As Fe-rich clay mineral phases and especially tri-octahedral ones are seldom encountered in environments and as only very small quantities can be extracted, their synthesis can be foreseen.

Two reviews have recently gathered the different methods carried out to prepare iron-rich clay minerals [START_REF] Dzene | Synthesis of iron-rich tri-octahedral clay minerals: A review[END_REF][START_REF] Petit | Synthesis of nontronite and other Fe-rich smectites : a critical review[END_REF]. The hydrothermal synthesis route remains the preferred one because it is a versatile and easy to use process which allows the attainment of well crystallized compounds [START_REF] Jaber | Synthesis of Clay Minerals[END_REF][START_REF] Kloprogge | Synthesis of smectites and porous pillared clay catalysts: a review[END_REF][START_REF] Forestier | Textural and hydration properties of a synthetic montmorillonite compared with a natural Na-exchanged clay analogue[END_REF][START_REF] Reinholdt | Synthesis and characterization of montmorillonite-type phyllosilicates in a fluoride medium[END_REF]. The key parameters governing the formation of clay minerals are the sources of reactants, synthesis time, temperature and the pH of the hydrogel. Regarding the preparative methods, the formation of a sol followed by a gel is expected to yield a homogeneous repartition of elements in the final product [START_REF] Dzene | Synthesis of iron-rich tri-octahedral clay minerals: A review[END_REF]. The use of Fe 2+ source for the preparation of precursor is expected to increase the kinetics of synthesis and to allow obtaining faster a well-crystallized product compared to the use of Fe 3+ source [START_REF] Petit | Synthesis of nontronite and other Fe-rich smectites : a critical review[END_REF]. The three other parameters, pH, time and temperature, have been widely varied in literature and no particular tendency can be extracted. Thus, the variation from neutral to alkaline pH has been previously reported, and syntheses have been performed in various time and temperature ranges. Nevertheless, the three later parameters can significantly influence the type of synthesized clay mineral and its crystallinity [START_REF] Jaber | Synthesis of Clay Minerals[END_REF][START_REF] Kloprogge | Synthesis of smectite clay minerals: A critical review[END_REF].

For this reason, the three parameters (time, temperature and the initial quantity of OH -) were chosen as the variables for this study. In general, time will intervene in reaction kinetics and crystal growth processes. The increase of synthesis time will allow attaining the equilibrium or pseudo-equilibrium conditions and will allow growing larger crystals. The temperature will intervene in both, in reaction kinetics and thermodynamics. On the one hand the increase of temperature is expected to accelerate the kinetics of reaction. Thus, higher temperature would lead to shorter synthesis time. On the other hand, the increase of temperature will influence the thermodynamics of the reaction. As the precipitation of clay minerals is in general an endothermic process [START_REF] Giffaut | Andra thermodynamic database for performance assessment: ThermoChimie[END_REF], the increase of the temperature is expected to favor the formation of products. Finally, the quantity of OH -will influence the condensation reactions [START_REF] Cundy | The hydrothermal synthesis of zeolites: Precursors, intermediates and reaction mechanism[END_REF].

The complexity of the studied reactions, make the systematic exploration rather challenging.

Therefore, the design of experiments (DOE) can be used. This approach largely applied in natural sciences and engineering has been only recently explored in the synthesis of inorganic materials such as zeolite, aluminophosphate and glass [START_REF] Carvalho | Synthesis of aluminophosphate by the ionothermal method using factorial design[END_REF][START_REF] Cichocki | Experimental designs applied to hydrothermal synthesis of zeolite ERI+OFF(T) in the Na2O-K2O-Al2O3-SiO 2-H2O system: Part 2. Regular study[END_REF][START_REF] Ji | Application of orthogonal experimental design in synthesis of mesoporous bioactive glass[END_REF][START_REF] Katovic | The factorial experimental design applied to the zeolite synthesis[END_REF][START_REF] Zhang | Application of fractional factorial design to ZSM-5 synthesis using ethanol as template[END_REF]. This approach has allowed to eliminate insignificant parameters and reveal the most significant ones, which influence material properties such as crystallinity, ion exchange capacity and solubility of the product. However, the method has not yet been explored for lamellar silicate synthesis. This might be due to the fact, that the characterization of synthesis products requires a set of several analytical techniques, thus choosing the response variable(s) for the DOE is not trivial. By using oriented preparation for X-ray diffraction, we took advantage of a particular feature of lamellar materials to characterize them. The objective of this study then is to test this characterization technique for the design of experiments and explore in an efficient way, how, the three parameters: quantity of OH -, synthesis time and temperature, influence the type of synthesized clay mineral and its crystallinity.

Materials and methods

Design of experiments

Preliminary experiments identified the three parameters that influence the type and amount of obtained clay mineral: synthesis time t, temperature T and (OH/Fe)ini molar ratio. They allowed also to set the relevant range of each of these three parameters (T=80 °C -180 °C, t=1 -7 days and (OH/Fe)ini=2 -4) for the design of experiments. To analyze the results in terms of synthesis parameters, a methodology based on Design Of Experiments (DOE) and Response Surface Methodology (RSM) was used [START_REF] Myers | Response Surface Methodology: Process and Product Optimization Using Designed Experiments[END_REF]. The main objective of DOE is to obtain the maximum of information from a minimum of experiments. RSM enables to explore relationships between the studied functions and the operating factors. To obtain an optimal response a linear quadratic model with 10 parameters is used to numerically approximate the response function. For this, a system of 15 equations with 10 variables is statistically solved by minimizing a quadratic criterion, i.e.

∑ (𝑦 𝑒𝑥𝑝 (𝑖) -𝑦 𝑐𝑎𝑙𝑐 (𝑖)) 2 𝑛 𝑖=1
, where yexp and ycalc are the experimental and calculated variable, respectively. The studied parameters (T, t and OH/Fe molar ratio) are standardized in the range [-1,1] to analyze the sensitivity of the calculated response functions. Additional experiments were performed to investigate experimental error. The relevance of the response functions (presented in paragraph 2.3.) was studied by statistical analysis via the calculation of five criteria, as follows:

-The determination coefficient r² -The standard error of estimate S = √ ∑ (𝑦 𝑒𝑥𝑝 (𝑖)-𝑦 𝑐𝑎𝑙𝑐 (𝑖))

2 𝑛 𝑖=1 𝑛-6 -The Marquard's percent standard deviation MPSD = √ 1 𝑛-6 ∑ (𝑦 𝑒𝑥𝑝 (𝑖)-𝑦 𝑐𝑎𝑙𝑐 (𝑖)) 2 𝑦 𝑒𝑥𝑝 (𝑖) 2 𝑛 𝑖=1 -The mean absolute error EABS = 1 𝑛 ∑ ⌊𝑦 𝑒𝑥𝑝 (𝑖) -𝑦 𝑐𝑎𝑙𝑐 (𝑖)⌋ 𝑛 𝑖=1 -The mean relative error RE = 1 𝑛 ∑ ⌊ 𝑦 𝑒𝑥𝑝 (𝑖)-𝑦 𝑐𝑎𝑙𝑐 (𝑖) 𝑦 𝑒𝑥𝑝 (𝑖) ⌋ 𝑛 𝑖=1
The five criteria are used to assess the pertinence of the numerical approximation. The studied functions are studied directly and in power, logarithmic, exponential, hyperbolic, etc. forms.

According to the five studied criteria, the best function is chosen and used for the numerical approximation.

Synthesis protocol and chemicals used

The following chemicals were used: iron (II) sulfate heptahydrate (FeSO4•7H2O, Merck, 99.9 %) as Fe(II) source, fumed silica (SiO2, Aerosil ® 380, Evonik) as silicon source, sodium hydroxide (NaOH, Sigma Aldrich, 97 wt.%) to provide an alkaline medium and sodium dithionite (Na2S2O4, Sigma Aldrich, ca. 85 wt.%,) to maintain reducing conditions. Double distilled water (DDW, 18.2 MΩ•cm) was used for all the experiments.

The procedure is based on a hydrothermal synthesis starting from a suspension having an Fe to Si molar ratio equal to 1.5. The procedure consisted of dissolving successively in 70 mL of distilled water 0.041 g of Na2S2O4, 4.510 g of FeSO4•7H2O, 0.647 g of SiO2 and a given quantity of NaOH at the desired OH/Fe molar ratio corresponding to each run of experiment (Table 1).

Thus, the initial theoretical concentrations were c(Fe 2+ ) = 0.23 mol•L -1 , c(Si)=0.15 mol•L -1 and c(OH -)=0.45 -1.00 mol•L -1 . The mixture was stirred during 2 hours at room temperature before being sealed in a Teflon-lined stainless-steel autoclaves (Top Industrie ®, 150 mL) and put at the desired temperature (T) ranging from 80 °C to 180 °C and time (t) ranging from 1 to 7 days.

After the synthesis, the autoclaves were cooled down at room temperature. The run products were then recovered and washed by centrifugation (8000 rpm for 10 min), and dried at 60 °C.

Choice of response for the design of experiments

Clay minerals are highly anisotropic, therefore, it is challenging to obtain truly un-oriented sample for powder XRD. To overcome this drawback, oriented preparations can be made, recording solely (00ℓ) reflections, which are characteristic for different clay mineral types. An The absence of other (hkℓ) plane reflections was systematically checked in the analyzed samples. The measured peak areas of different phases then represent the relative abundance of each phase in the sample. Moreover, it is also possible to assess the information regarding the coherent scattering domain size (CSDS) in the direction perpendicular to the corresponding (00ℓ) plane. In conventional powder X-ray diffraction, this information can be obtained from the Full Width at Half Maximum (FWHM) of a reflection. However, in this particular case of oriented preparations, the simplifications can be done so that the peak intensity can provide this information on the CSDS in the direction perpendicular to the corresponding (00ℓ) plane. The chosen synthesis protocol revealed the formation of both types of clay minerals (1:1 and 2:1).

The assumption was made that the chemical composition of 2:1 type phase is very similar in all samples and the chemical composition of 1:1 type phase is very similar in all samples. The objective of the study was to find optimum synthesis conditions for each type of phase. For this reason, the area and intensity ratios of (001) reflection, (respectively R(A) and R(I)), between 1:1 and 2:1 type structures were chosen as the studied responses in the design of experiments (Figure 4).

X-ray diffraction

The mineralogical composition of the samples and, more specifically, the type of clay mineral was determined using X-ray diffraction of powder and oriented preparations. The powders were previously ground and then introduced into PMMA sample holder with a cavity 25 mm in diameter and 1.2 mm deep, whereas the thin films of oriented samples were prepared from diluted suspensions by deposition onto a glass slide. All the samples, powder and oriented, were characterized on a Bruker D8 Advance X-ray diffractometer with a LYNXEYE XE-T high resolution energy dispersive 1-D detector (CuKα1,2) fully opened (2.938°). This kind of detector allows to avoid Fe fluorescence effects with Cu X-ray tube.

For the powder samples, the X-ray diffractograms were recorded from 3° to 70° 2θ at a step scan of 0.017° 2θ, a step time of 1.80 s with variable divergence slits that ensure that the irradiated sample length (i.e., 15 mm) is kept equal over a measured range of 2θ. For the oriented samples, the XRD patterns were recorded in the range from 3 to 40° 2θ at a step scan of 0.017° 2θ, a step time of 3.3 s with a fixed divergence slits width of 0.08°. For both kind of samples (powder and oriented), automatic motorized anti-scatter screen is used for effective suppression of air-scatter at low angles 2θ.

Results

Experiments used for modelling

Examples of powder XRD patterns for selected experiments are given in Figure 5. The experimental synthesis conditions corresponding to each run N° are reported in Table 1. The formed reaction products were not pure phases, but containing also iron oxides: hematite and Figure 6 shows the area ratio versus temperature for different times for a low OH/Fe molar ratio (Figure 6A), an average OH/Fe molar ratio (Figure 6B) and a high OH/Fe molar ratio (Figure 6C). The function increases significantly with OH/Fe molar ratio while it increases with time at low temperature and decreases at high temperature. The same applies for temperature, since an increase of temperature decreases the area ratio except for high OH/Fe molar ratio values.

The model represents well experimentally observed tendencies. Considering the significant experimental error, the tendencies given by the model are the same as for the experimental points thus validating the obtained model. to each run N° see Table 1.

Study of phase abundance and crystallinity: response function R(I)

This response function gives additional quantitative information and results about crystallinity.

For R(I) (eq.2), it is noted that the conclusions about the sensitivity analysis are almost the same, except for time contribution, which is, in this present case, more significant, especially, the coupled terms with temperature and initial OH/Fe molar ratio. Indeed, the diffracted intensity in the particular case of oriented preparations among other factors also includes the contribution of the size of coherent scattering domain. It is expected that the synthesis time influences the size of formed particles, and thus the coherent scattering domain [START_REF] Carrado | Chapter 4 Synthetic Clay Minerals and Purification of Natural Clays[END_REF]. Thus, the contribution of time in the function of R(I) being more important than in R(A) function is in agreement with expected results.

R(I) = 2.146 -0.283𝑇 ̅ + 0.531𝑇 ̅ 2 + 3.127 𝑂𝐻 𝐹𝑒 ̅̅̅̅ + 1.100 𝑂𝐻 𝐹𝑒 ̅̅̅ 2 -0.734𝑡 ̅ -0.141𝑡 ̅ 2 + 0.928 𝑇 ̅ × 𝑂𝐻 𝐹𝑒 ̅̅̅ -0.819𝑇 ̅ × 𝑡 ̅ -1.834 𝑂𝐻 𝐹𝑒 ̅̅̅̅ × 𝑡 ̅ (eq.2)
The pertinence of the model is investigated from all trials (20): r² = 0.962, S = 2.014, MPSD = 3.232, EABS = 1.403, RE = 1.422

For high OH/Fe molar ratio, i.e. (OH/Fe)ini = 3.6, it has to be underlined that whatever the temperature T, R(I) decreases with time (Figure 8). Likewise, the temperature increases the contribution of time on R(I) (Figure 8B). That is to say, the ratio of diffracted peak intensity increases with respect to the background (with no curvature, see Figure 4) without the increase of peak full-width at half-maximum when time decreases and temperature increases. It leads then to a conclusion that 2 :1 type clay mineral phase formation is favored in short times, and this becomes even more important with the increase of temperature. 1).

With this function (R(I)), the interpretation is limited for results obtained with low initial OH/Fe molar ratio because R(I) is close to zero (Table 1). To overcome this drawback, the last response function (1 /R(I)) was defined and studied to investigate the favoring of 1:1 type clay formation.

All the studied parameters influence this function and no contribution can be neglected (eq.3). representative values of investigated conditions. It was not possible to fix a higher value for 1/R(I) > 2 due to the experimental error. This objective is reached, when OH/Fe molar ratio is as low as possible, i.e., OH/Fe=2.4 (Figure 10). It is important to mention, that at OH/Fe < 2.2, the formation of crystalline 1:1 type clay mineral phase was not observed (Table 1). The optimal conditions for the formation of 1:1 type clay mineral phase (with respect to 2:1 phase) are reached for short time and high temperature. Interestingly, the formation of 1:1 type clay mineral phase can also be achieved at low temperature (below 140 °C) and short times (less than 3 days) with low OH/Fe molar ratio. In comparison, the synthesis of 2:1 phase is favored only at high temperatures and high OH/Fe molar ratio. This aspect can be relevant to environmentally friendly synthesis procedure, as less resources (chemicals and energy) are consumed.

Discussion

The observed tendencies in this study are in agreement with previously reported studies. Indeed, Fe-serpentines (1:1 type clay mineral) were observed for the experiments at low temperatures below 150°C [START_REF] Bertoldi | The Heat Capacity of the Serpentine Subgroup Mineral Berthierine (Fe2.5Al0.5)[Si1.5Al0.5O5](OH)4[END_REF][START_REF] Lantenois | Experimental study of smectite interaction with metal Fe at low temperature: 1. Smectite destabilization[END_REF][START_REF] Pignatelli | Iron-clay interactions: Detailed study of the mineralogical transformation of claystone with emphasis on the formation of iron-rich T-O phyllosilicates in a step-bystep cooling experiment from 90°C to 40[END_REF][START_REF] Rivard | Combination of multi-scale and multi-edge X-ray spectroscopy for investigating the products obtained from the interaction between kaolinite and metallic iron in anoxic conditions at 90 °C[END_REF][START_REF] Tosca | An authigenic origin for Precambrian greenalite: Implications for iron formation and the chemistry of ancient seawater[END_REF], whereas Fe-rich chlorites (2:1 type clay mineral) were identified at higher temperature of 300 °C [START_REF] Mosser-Ruck | Contribution of longterm hydrothermal experiments for understanding the smectite-to-chlorite conversion in geological environments[END_REF]. However, it has to be noted that in It can be expected that the formation of phyllosilicate phases is strongly governed by sol-gel process as in the case of other silicates such as zeolites. An important step is the precipitation of dissolved silicon. This condensation reaction is strongly governed by the amount of hydroxyl ions [START_REF] Cundy | The hydrothermal synthesis of zeolites: Precursors, intermediates and reaction mechanism[END_REF]. Then, it is not surprising that the quantity of OH -has the most significant impact on both studied functions. It has been noted previously that Si speciation in solution, influence the structure of synthesized nontronite (2:1 type clay mineral) [START_REF] Baron | Influence of Aqueous Si and Fe Speciation on Tetrahedral Fe(III) Substitutions in Nontronites: a Clay Synthesis Approach[END_REF]. In a similar way here, the quantity of OH -acts on the species and their amount in solution during the synthesis. Also, the study of [START_REF] Tosca | An authigenic origin for Precambrian greenalite: Implications for iron formation and the chemistry of ancient seawater[END_REF] notes the influence of pH. They performed the study at three different pH (7.0,7.5 and 8.0), and they noted that the concentration of Fe 2+ and SiO2 in solution decreased faster when pH increased. Thus, the precipitation of greenalite-like phase occurred faster when pH increased. Although the pH in our study was very different (13 to 14) compared to the study of [START_REF] Tosca | An authigenic origin for Precambrian greenalite: Implications for iron formation and the chemistry of ancient seawater[END_REF], both studies note the important influence of this parameter towards the precipitation of clay minerals.

For synthesis at high OH/Fe ratio, a high temperature favors the formation of 2:1 type clay mineral. Indeed, the thermodynamic constant of the precipitation reaction increases with temperature and favors the formation of clays (enthalpy of precipitation reactions being positive) [START_REF] Giffaut | Andra thermodynamic database for performance assessment: ThermoChimie[END_REF]. In the same way, the reaction kinetics increase with the increase of temperature and the preferential precipitation of 2:1 type clay mineral can be attributed to a faster kinetic. It is in agreement with observations reported in the study of [START_REF] Grubb | Silicates and their paragenesis in the brockman iron formation of Wittenoom Gorge, Western Australia[END_REF] This indicates a decrease of the ratio R(A) but the amounts can probably increase.

For synthesis performed at low OH/Fe ratios, the quantity of OH -is limited. There is no thermodynamic limitation but the kinetics are decreased by the limiting reactant. The reaction leading to the precipitation of 2:1 type clay mineral is favored at high pH and disfavored in these operating conditions. So, the reaction of precipitation of 1:1 phase clay mineral becomes dominating.

To summarize, the ratio of reaction kinetics (k2:1 / k1:1) increases with temperature and decreases when the available amount of hydroxyl ion in solution decreases. This amount of hydroxyl ion decreases as the precipitation reactions advances (time dependent).

Conclusion

The feasibility of synthesis of 2:1 and 1:1 type clay minerals was demonstrated at given physicochemical conditions. Depending on these conditions, the synthesis can be tailored towards favoured formation of one or another type of phases. To optimize the number of experiments and the amount of information, which can be extracted from them, a design of experiments was used. From few synthesis experiments, the predominant operating parameters and the conditions leading to the formation of 2:1 and 1:1 type clay minerals were identified.

From RSM (Response Surface Methodology) investigations the optimal conditions of synthesis were estimated for each type of phase.

Among the studied parameters (synthesis temperature, time and ratio OH/Fe), the ratio OH/Fe (proportional to the available amount of hydroxyl ions in solution) is preponderant. At high OH/Fe ratio, the precipitation of 2:1 type clay mineral is favoured with respect to 1:1 type clay mineral. The decrease of available hydroxyl ions with time slowed down the kinetic. For the same reason, the formation of 1:1 type clay mineral was favoured when the availability of hydroxyl ions decreased (low OH/Fe ratio).

From these results, synthesis of several grams of desired products will be carried out to perform detailed characterizations (full set of thermodynamic properties, different spectroscopic measurements, structural model …). These investigations will allow to improve the understanding of many aspects related to the presence of either or both 2:1 and 1:1 type iron bearing clays in natural or anthropized environments: identification of each type of phases in experimental products or natural samples, understanding of their neoformation mechanisms, sensitivity to relevant or discriminant physico-chemical parameters, thermos-kinetic modelling of geochemical systems.

Figure 1 .

 1 Figure 1. Schematic representation of clay mineral structures.

Figure 2 .

 2 Figure 2. Scheme representing central composite design for k=3 factors.

  example of X-ray diffractogram of oriented preparation is shown in Figure 3. Thus, 1:1 type structure would have d(001) value of ca. 7 Å and 2:1 type structure would have d(001) value in the range of 10 Å to 15 Å.

Figure 3 .

 3 Figure 3. Example of XRD pattern of oriented preparation (Run N°01). Red dashed lines

Figure 4 .

 4 Figure 4. Illustration of R(A) and R(I) between 1:1 and 2:1 type clay minerals.

Figure 5 .

 5 Figure 5. Powder XRD patterns of selected experiments. For the experimental conditions

  4 runs (N°01-R, N°11-R1, R2 and N°26-R) were repeated to investigate the experimental error. 227 These tests were chosen to have an experiment corresponding to predominant 2:1 type clay 228 mineral (test N°01), an experiment resulting in a majority of 1:1 type clay mineral (test N°26) 229 and an experiment with intermediate proportions of 2:1 and 1:1 type clay minerals (test N°11). 230 The obtained experimental error was important but all the repeated tests show a parallel trend. 231 The relevance of RSM is studied on the basis of all the experimental tests. In the run N°15, 232 which was repeated several times, the only crystalline phases identified were iron oxides. No 233 crystalline clay minerals were identified in powder XRD pattern (Figure 5) of this run, thus no 234 values of R(A) and R(I) could be obtained for oriented preparations. 235 3.2. Study of response function R(A): ratio between relative quantities of 2:1 and 1:1 clay minerals This function is proportional to the relative quantity of each phase in the sample. The obtained quadratic model for the studied function R(A) is given below (eq. 1). The analysis of sensitivity of the quadratic model for R(A) shows that the response function is highly dependent on OH/Fe molar ratio (coefficients before linear linear, quadratic and coupled coefficients relative to time are relatively small, unlike the coupled coefficient 𝑇 ̅ × 𝑡 ̅ . ln (R(A)) = 2.292 -0.268𝑇 ̅ -0.059𝑇 ̅ 2 + 1the model is investigated from all trials (20) except run N°15. Averages of tests N°01, N°11 and N°26 are used for the calculation of all the tests for the experimental error. r² = 0.961, S = 4.706, MPSD = 0.512, EABS = 2.754, RE = 0.308

Figure 6 .

 6 Figure 6. Time and temperature influence on the obtained R(A) for (OH/Fe)ini molar ratios

Figure 7 .

 7 Figure 7. XRD patterns of oriented preparations for selected experiments (red dashed line 2:1

Figure 8 .

 8 Figure 8. XRD patterns of oriented preparations obtained for (OH/Fe)ini molar ratio = 3.6 at

  the model is investigated from 18 trials (set 1 + tests N°26 and N°27): r² = 0.981, S = 0.472, MPSD = 1.192, EABS = 0.310, RE = 0.621 Starting from these 3 mathematical models, optimal solutions are searched by minimizing an objective function. 3.4 Optimum parameters favoring the 2:1 type clay mineral formation The study of the favored formation of 2:1 type clay mineral requires to investigate the parameters (T, time and initial OH/Fe molar ratio) resulting in high values of R(A) and R(I). It was considered that R(A) > 30 and R(I) > 12 (objective functions) are representative values of investigated conditions. Figure 9 shows in a temperature / time diagram the optimal solutions for having a vast majority of 2:1 type clay for R(A) > 30 (Figure 9A) and for R(I) > 12 (Figure 9B) for (OH/Fe)ini = [3.6 -4.0]. The objective function is reached in the hatched area. This objective (R(A) > 30) is reached for high OH/Fe molar ratio and especially for short synthesis time and high temperature. The same trend is observed for R(I) with temperature but the influence of time is more moderate at low temperatures below 120 °C.

Figure 9 .

 9 Figure 9. Conditions of temperature and time favoring the formation of the 2:1 type clay

Figure 10 .

 10 Figure 10. Conditions of temperature and time favoring the formation of the 1:1 type clay

  other studies the duration of experiments was different. The experiments reported in Pignatelli et al. (2014) lasted 6 months and the experiment of Mosser-Ruck et al. (2016) went for 25 months. Also, in these experiments additional phenomena were in operation such as the dissolution of iron from metallic plates, which could have been the limiting step regarding the precipitation of clay mineral phase.

  , which used 1N NaOH solution in his experiments and observed that minnesotaite (2:1 type clay mineral) become the dominant constituent in the three year experiments. Nevertheless, in our study, when synthesis time increases this tendency is reversed (R(A) decreases) possibly due to a limited species (intermediate reaction leading to the precipitation of 2:1 phase clay mineral).

Table 1 .

 1 Experimental conditions and obtained R(I) and R(A) corresponding to each run. 225

	Set	Run	T (°C) (OH/Fe)ini	t	R(I) R(A)
		N°			(days)
		01 01-R	160	3.6	2.3	6.9 13.1	29.2 44.3
		02	160	2.4	2.3	0.1	1.2
		03	100	3.6	2.3	5.3	15.9
	1	04	100	2.4	2.3	0.2	2.0
		05	130	3.0	1.0	2.1	8.3
		06	130	3.0	7.0	1.4	5.9
		07	160	3.6	5.7	3.0	11.2
		08	160	2.4	5.7	0.2	0.4
		09	100	3.6	5.7	2.5	10.1
		10	100	2.4	5.7	0.9	5.1
		11				2.5	10.7
		11-R1	130	3.0	4.0	3.9	18.1
		11-R2				2.8	13.0
		12	180	3.0	4.0	1.1	6.0
		13	80	3.0	4.0	7.4	17.0
		14	130	4.0	4.0	13.1 31.8
		15	130	2.0	4.0	-	-
		16	150	3.0	7.0	0.6	11.4
	2	26 26-R	130	2.6	5.0	0.8 0.9	3.5 4.1
		27	130	2.8	5.0	0.6	3.5
		39	180	4.0	1.0	24.6 64.8
		40	180	4.5	1.0	32.1 68.2
	226

Acknowledgments

The study was conducted within the framework of R&D project between the French National Agency for Radioactive Waste Management (ANDRA) and Institut Carnot MICA.