Nico Kerber

Dmitriy Ksenzov

Frank Freimuth

Flavio Capotondi

Emanuele Pedersoli

Ignacio Lopez-Quintas

Boris Seng

Joel Cramer

Kai Litzius

Daniel Lacour

Hartmut Zabel

Yuriy Mokrousov

Mathias Kläui
email: klaeui@uni-mainz.de

Christian Gutt
email: christian.gutt@uni-siegen.de

Chiral versus collinear magnetic order dynamics: faster chiral recovery after optical excitation revealed by femtosecond XUV scattering

come

Introduction

In the field of magnetism and spintronics chiral magnetic structures, such as spin spirals, domain walls and skyrmions [START_REF] Rößler | Spontaneous skyrmion ground states in magnetic metals[END_REF][START_REF] Mühlbauer | Skyrmion lattice in a chiral magnet[END_REF][START_REF] Bode | Chiral magnetic order at surfaces driven by inversion asymmetry[END_REF][START_REF] Heinze | Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions[END_REF][START_REF] Jiang | Skyrmions in magnetic multilayers[END_REF][START_REF] Everschor-Sitte | Perspective: Magnetic skyrmions-Overview of recent progress in an active research field[END_REF], are intensively investigated due to their fascinating properties such as potentially enhanced stability and efficient spin-orbit torque driven dynamics [START_REF] Brataas | Chiral domain walls move faster[END_REF][START_REF] Emori | Current-driven dynamics of chiral ferromagnetic domain walls[END_REF][START_REF] Fert | Skyrmions on the track[END_REF][START_REF] Woo | Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets[END_REF]. It has been shown that these structures are stabilized by the Dzyaloshinskii-Moriya interaction (DMI) [START_REF] Dzyaloshinskii | A thermodynamic theory of weak ferromagnetism of antiferromagnets[END_REF][START_REF] Moriya | New mechanism of anisotropic superexchange interaction[END_REF] that favours a chiral winding of the magnetisation. This antisymmetric indirect exchange interaction requires materials with large spin orbit coupling as well as a broken inversion symmetry, present in special bulk systems such as B20 compounds where skyrmions have been first discovered experimentally [START_REF] Rößler | Spontaneous skyrmion ground states in magnetic metals[END_REF][START_REF] Mühlbauer | Skyrmion lattice in a chiral magnet[END_REF] or in interfacial systems such as heavy metal/ferromagnet multilayer stacks [3-6, 8, 10]. The domain wall type (Néel or Bloch) and chirality of the spin textures can be accessed in real space by imaging techniques [START_REF] Heinze | Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions[END_REF][START_REF] Tetienne | The nature of domain walls in ultrathin ferromagnets revealed by scanning nanomagnetometry[END_REF][START_REF] Pollard | Observation of stable Néel skyrmions in cobalt/palladium multilayers with Lorentz transmission electron microscopy[END_REF][START_REF] Chen | Out-of-plane chiral domain wall spin-structures in ultrathin in-plane magnets[END_REF] or in reciprocal space by resonant magnetic X-ray scattering [START_REF] Dürr | Chiral Magnetic Domain Structures in Ultrathin FePd Films[END_REF][START_REF] Zhang | Direct experimental determination of spiral spin structures via the dichroism extinction effect in resonant elastic soft x-ray scattering[END_REF][START_REF] Chauleau | Chirality in Magnetic Multilayers Probed by the Symmetry and the Amplitude of Dichroism in X-Ray Resonant Magnetic Scattering[END_REF]. The chiral wall spin structure is of key importance as it governs the dynamical properties of domain walls and skyrmions [START_REF] Brataas | Chiral domain walls move faster[END_REF][START_REF] Emori | Current-driven dynamics of chiral ferromagnetic domain walls[END_REF][START_REF] Fert | Skyrmions on the track[END_REF][START_REF] Woo | Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets[END_REF][START_REF] Kim | Asymmetric skyrmion Hall effect in systems with a hybrid Dzyaloshinskii-Moriya interaction[END_REF]. While the investigation of static structures and slow (ns) dynamics of chiral magnetic structures has been intensified recently, experimental studies addressing the ultimate fs-ps dynamics of the chirality have been elusive so far as ultrafast pump-probe experiments have concentrated on the collinear order in magnetic systems only [START_REF] Vaterlaus | Spin-lattice relaxation time of ferromagnetic gadolinium determined with time-resolved spin-polarized photoemission[END_REF][START_REF] Beaurepaire | Ultrafast spin dynamics in ferromagnetic nickel[END_REF][START_REF] Rhie | Femtosecond electron and spin dynamics in Ni/W(110) films[END_REF][START_REF] Stamm | Femtosecond modification of electron localization and transfer of angular momentum in nickel[END_REF][START_REF] Malinowski | Control of speed and efficiency of ultrafast demagnetization by direct transfer of spin angular momentum[END_REF][START_REF] Koopmans | Explaining the paradoxical diversity of ultrafast laser-induced demagnetization[END_REF][START_REF] Battiato | Superdiffusive Spin Transport as a Mechanism of Ultrafast Demagnetization[END_REF][START_REF] Pfau | Ultrafast optical demagnetization manipulates nanoscale spin structure in domain walls[END_REF][START_REF] Sant | Measurements of ultrafast spin-profiles and spin-diffusion properties in the domain wall area at a metal/ferromagnetic film interface[END_REF]. In particular, as the ferromagnetic alignment is governed by the Heisenberg exchange, while the chiral order is resulting from the DMI, the ultrafast dynamics of both needs to be probed individually. And since the characteristic time scale for the onset of the chiral magnetic order and its ultrafast dynamics are unexplored up to now, we need to ascertain both, as they hold fundamental insights into the underlying physical mechanisms and allow us to gauge the ultimate speed for the manipulation of chiral magnetism, e.g. for ultrafast writing of chiral spin textures. In this work we employ circularly polarized light pulses from an XUV free-electron laser and investigate time-resolved the evolution of the chirality of domain walls in magnetic thin film samples by an IR pump -X-ray magnetic scattering (XRMS) probe experiment. Using samples with interfacial DMI and perpendicular magnetic anisotropy exhibiting labyrinth-like domain patterns, we measure in the same experiment both the sum signal corresponding to the ferromagnetic order in the domains and the difference signal corresponding to the average chiral order in the domain walls. We find an ultrafast intensity decrease of both signals in the sub-ps regime with similar time constants. However, a significantly faster recovery of the chiral signal in the sub-ns timescale is observed. We subsequently investigate the origin of the faster recovery of the chiral signal by performing numerical simulations of the scattering signal, which reproduce the experimental findings.

Results

Magnetic small angle X-ray scattering

The experimental setup for the scattering experiment is shown schematically in Fig. 1 a. Circularly polarized extreme ultraviolet (XUV) radiation of 60 fs pulse duration was tuned to a wavelength of 23.0 nm corresponding to the Fe M2,3 dichroic transition, which exhibits magnetic scattering contrast due to the X-ray magnetic circular dichroism (XMCD) effect [START_REF] Pfau | Ultrafast optical demagnetization manipulates nanoscale spin structure in domain walls[END_REF][START_REF] Sant | Measurements of ultrafast spin-profiles and spin-diffusion properties in the domain wall area at a metal/ferromagnetic film interface[END_REF][START_REF] Schütz | Absorption of circularly polarized x rays in iron[END_REF][START_REF] Hannon | X-ray resonance exchange scattering[END_REF]. The limited transmission in the XUV regime required us to perform the experiment in reflection geometry with an incident angle of Θ = 44° [START_REF] Sant | Measurements of ultrafast spin-profiles and spin-diffusion properties in the domain wall area at a metal/ferromagnetic film interface[END_REF] yielding an effective XUV penetration depth of 8 nm. Due to an isotropic disordered domain structure we observe the typical ring-like diffraction feature well known from magnetic transmission SAXS experiments representing a broad distribution of Fourier components of the magnetic domain pattern [START_REF] Pfau | Ultrafast optical demagnetization manipulates nanoscale spin structure in domain walls[END_REF][START_REF] Sant | Measurements of ultrafast spin-profiles and spin-diffusion properties in the domain wall area at a metal/ferromagnetic film interface[END_REF]. The sample is pumped with a 60 fs IR laser pulse of 780 nm wavelength impinging on the sample with a small 2 degree offset with respect to the XUV beam (further setup information can be found in the Methods section). The SAXS pattern was detected on a 2D CCD-detector and investigated as a function of the pumpprobe delay over a time span of 100 ps. For each time delay and helicity, 7000 scattering patterns have been measured, normalized to the incoming flux and averaged. The area around the beamstop and charge scattering streaks have been masked. The background from the charge scattering due to the reflection geometry has been subtracted for each pattern leading to the corrected patterns used in the further analysis. The investigated [Ta(5.3 nm)/Co20Fe60B20(0.93 nm)/Ta(0.08 nm)/MgO(2.0 nm)]x20/Ta(1.6 nm) multilayer stack was produced by dc magnetron sputtering and exhibits perpendicular magnetic anisotropy (PMA). The material was imaged via magnetic force microscopy (MFM) as shown in Fig. 1b and exhibits, after out-of-plane demagnetisation, labyrinth-like magnetic domains at zero field. The domain pattern extended through all ferromagnetic layers due to stray field coupling exhibits a domain periodicity of (455±30) nm. In reciprocal space the MFM images yields a first order peak at q = (13.8±0.9) μm -1 . Due to partial alignment of the magnetic stripes, higher order peaks are visible as well [START_REF] Hellwig | X-ray studies of aligned magnetic stripe domains in perpendicular multilayers[END_REF]. Further material analysis with a superconducting quantum interference device (SQUID) yields a saturation magnetisation of Ms = (844±28) kA/m and an effective perpendicular anisotropy Keff = (133±17) kJ/m 3 of the sample at room temperature (for more sample details see Methods section). The Ta(5.3 nm)/Co20Fe60B20 interface leads to a positive DMI, with reported values for such stacks ranging from 0.06 to 0.30 mJ/m 2 [START_REF] Zázvorka | Thermal skyrmion diffusion used in a reshuffler device[END_REF][START_REF] Conte | Role of B diffusion in the interfacial Dzyaloshinskii-Moriya interaction in Ta/Co20Fe60B20/MgO nanowires[END_REF][START_REF] Casiraghi | Individual skyrmion manipulation by local magnetic field gradients[END_REF], which favours right-handed Néel-type domain walls (see Fig. 1c). The exact domain wall arrangement is finally determined by an interplay between interfacial DMI and dipolar interactions, which can lead to complex domain wall arrangements as shown recently [START_REF] Legrand | Hybrid chiral domain walls and skyrmions in magnetic multilayers[END_REF]. We also note that considering the small penetration depth of the XUV radiation, we are sensitive to the magnetisation of the topmost Co20Fe60B20 layer only. X-ray scattering from such magnetic structures leads to the ring shaped SAXS patterns shown in Fig. 1(d-g). In Fig. 1f the sum image of the circular left (CL) and right (CR) polarized light (Fig. 1d &1e) is shown, which is predominantly connected to the average domain magnetisation magnitude (I ∝ M 2 ) [START_REF] Pfau | Ultrafast optical demagnetization manipulates nanoscale spin structure in domain walls[END_REF][START_REF] Gutt | Resonant magnetic scattering with soft x-ray pulses from a free-electron laser operating at 1.59 nm[END_REF]. This signal shows a ring-like diffraction pattern at a q-value of (14.3±0.1) μm -1 that we assign to the 1 st order scattering peak of the magnetic stripe domains. The dichroic signal (CL-CR) defined as the difference of the diffracted left-and right-circularly polarized light [START_REF] Zhang | Direct experimental determination of spiral spin structures via the dichroism extinction effect in resonant elastic soft x-ray scattering[END_REF] shown in Fig. 1g contains the information about the average chiral magnetisation magnitude and exhibits a pronounced angular asymmetry with an amplitude of 14 % of the sum signal. This provides us a tool to individually probe the time resolved dynamics of the chiral magnetic order in the domain walls. It has recently been shown that one can directly determine the domain wall chirality (left/right-handed) and character (Bloch (helical), Néel (cycloidal)) from the dichroic signal [START_REF] Zhang | Direct experimental determination of spiral spin structures via the dichroism extinction effect in resonant elastic soft x-ray scattering[END_REF][START_REF] Chauleau | Chirality in Magnetic Multilayers Probed by the Symmetry and the Amplitude of Dichroism in X-Ray Resonant Magnetic Scattering[END_REF]. Fig. 2g shows the orthoradial profile of the dichroic signal for different pump-probe delay times. We used a fitting model of A*cos(Ψ-φ), where A is the amplitude, Ψ the azimuthal angle and the phase φ determines the domain wall angle and with that the domain wall chirality and character [START_REF] Zhang | Direct experimental determination of spiral spin structures via the dichroism extinction effect in resonant elastic soft x-ray scattering[END_REF]. Fitting the azimuthal profile leads to a value of φ = 90° which indeed confirms the presence of fully right-handed Néel-type domain walls [START_REF] Zhang | Direct experimental determination of spiral spin structures via the dichroism extinction effect in resonant elastic soft x-ray scattering[END_REF][START_REF] Chauleau | Chirality in Magnetic Multilayers Probed by the Symmetry and the Amplitude of Dichroism in X-Ray Resonant Magnetic Scattering[END_REF]. The experimental (dots) and fitted (lines) orthoradial dichroic profiles for different pump-probe delay times showing a maximum at 90° and a minimum at 270° corresponding to a right-handed cycloidal winding of the domain walls. While the amplitude of the dichroic signal drops within a ps upon pumping and recovers afterwards, no significant angle shift in the fitted phase φ can be detected. The beamstop is located at an azimuthal angle of 90°.

Ultrafast time resolved pump-probe experiments

In a next step we investigated the time evolution of the scattering signals upon IR laser excitation. Fig. 2(a-f By azimuthal integration of the dichroic signal we can identify that within our experimental accuracy no significant shift in the phase φ of the orthoradial profile (Fig. 2g) has been observed upon pumping. Thereby we can exclude any significant ultrafast change of the domain wall angle. This means that the DMI is even at the ultrafast timescale contributing to the wall chirality and therewith the stability of chiral spin structures is retained.

The key step is the comparison of the ferromagnetic order dynamics and the chiral order dynamics. To ascertain both, we calculated the average intensity for the sum and the difference signal as a function of the wavevector Qr for different time delays (Fig. 3a &3b). The error bars from counting statistics are of the order of 0.01 % and the fluctuations visible are due to speckles from the coherent illumination. Numerical integration of the radial profiles for each time delay leads to the data shown in Fig. 4a & 4b which demonstrates the evolution of the total intensity of the sum and the difference signal as a function of delay time on linear (a) and logarithmic time scale (b) normalized to the unpumped total intensity. We used the model reported in equation ( 1) of the Methods section to fit the experimental data [START_REF] Pfau | Ultrafast optical demagnetization manipulates nanoscale spin structure in domain walls[END_REF]. The model includes the time constant 𝜏d of the demagnetisation process of the ferromagnetic collinear order respectively the dechiralisation process (vanishing of the chiral order in the domain walls), and the time constants τr1 and τr2 of the fast and slow recovery processes. The initial ultrafast demagnetisation time constant τd = 0.37±0.10 ps is in agreement with the typical demagnetisation times of ferromagnetic 3d-transition metals [START_REF] Beaurepaire | Ultrafast spin dynamics in ferromagnetic nickel[END_REF][START_REF] Rhie | Femtosecond electron and spin dynamics in Ni/W(110) films[END_REF][START_REF] Stamm | Femtosecond modification of electron localization and transfer of angular momentum in nickel[END_REF][START_REF] Pfau | Ultrafast optical demagnetization manipulates nanoscale spin structure in domain walls[END_REF][START_REF] Sant | Measurements of ultrafast spin-profiles and spin-diffusion properties in the domain wall area at a metal/ferromagnetic film interface[END_REF]. The main results of these experiments are firstly a dechiralisation process (τd = 0.33±0.10 ps) that occurs on similar timescales as the demagnetisation of the collinear domains, and secondly we observe a clear difference in the slower recovery process of the two signals. The time constant τr2 = 312±6 ps of the difference signal is significantly smaller than the one of the sum signal (τr2 = 905±28 ps) demonstrating a faster recovery of the dichroic signal that reflects the chiral magnetic order in the domain walls.

To understand the origin of this difference in the recovery, we next investigate the time evolution of the 2 nd moments of the radial distributions as displayed in Fig. 4 c &4d. The 2 nd moments (calculated using equation (2) in the Methods section) entail additional information about possible ultrafast changes of the radial distribution widths upon IR excitation. The 2 nd moment of the sum signal increases significantly upon pumping within 1 ps, indicating a wider radial distribution and therefore we can identify stronger fluctuations leading to a reduced correlation length of the domain-domain correlation function. After 100 ps it has just recovered partially to its unpumped value. In contrast, the 2 nd moment of the difference signal decreases slightly within 1 ps to about 90 % after optical excitation but recovers afterwards within a few ps to values around the unpumped value.

Discussion

The experimental results show an ultrafast decrease within the first ps of the total intensity of the sum signal that reflects the conventional collinear magnetic order inside the domains, as well as the dichroic signal, that corresponds to the chiral order of the domain walls. Both signals decay on similar timescales, indicating that for the ultrafast initial demagnetisation process the order of the magnetisation itself does not play a major role in our sample system.

In order to explain the key finding of the experimentally observed faster recovery of the chiral signal after laser excitation we can envisage two different mechanisms: (1) a change in the size ratio between domain walls and domains caused by an increase of the domain wall width during the whole investigated time frame or (2) a faster recovery dynamics of the chiral order within the domain walls compared to the ferromagnetic order in the domains leading to a faster build-up of the chiral magnetisation.

The increase of the domain-wall width as required for mechanism (1) can results from the temperature dependence of the micromagnetic parameters, more precisely the effective anisotropy and the exchange constant, as shown in the Supplementary S3. Since the average domain magnetisation is directly proportional to the square root of the total intensity of the sum signal one can estimate the temperature evolution of the spin system (assuming the 3-temperature model [START_REF] Beaurepaire | Ultrafast spin dynamics in ferromagnetic nickel[END_REF]) during the pumpprobe experiment using the fitted temperature dependence of the saturation magnetisation of the sample (see Supplementary S2). This leads to an increased spin system temperature of (493±22) K at a delay time of 1 ps, where the sample is significantly demagnetized, and an already decreased temperature of (469±13) K at 100 ps. Such a sudden change of the micromagnetic parameters is expected to excite a breathing mode of the domain walls. The breathing mode corresponds to a damped ps-scale oscillation of the width of the domain wall around the new equilibrium wall width as determined by the temperature-scaled micromagnetic parameters. We estimate the equilibrium wall width at the increased spin system temperature to be 37 nm (see Supplementary S3). Therefore, we estimate that for our system the wall width oscillates between values of 27 nm and 47 nm due to the sudden change of the micromagnetic parameters. Domain-wall breathing frequencies lie in the range of several GHz. Assuming a breathing frequency of 5 GHz, one-half period of the breathing mode amounts to 100 ps, in which the wall expands by 20 nm from 27 nm to 47 nm.

Mechanism (2) is based on a faster recovery of the chiral order parameter of the system in comparison to the collinear order parameter [START_REF] Villain | Two-level systems in a spin-glass model. I. General formalism and two-dimensional model[END_REF][START_REF] Cinti | Two-Step Magnetic Ordering in Quasi-One-Dimensional Helimagnets: Possible Experimental Validation of Villain's onjecture about a hiral Spin iquid State[END_REF][START_REF] Menzel | Information Transfer by Vector Spin Chirality in Finite Magnetic Chains[END_REF]. The chiral order parameter is defined here as the in-plane component of the vector 〈𝑺 𝑖 × 𝑺 𝑖+1 〉/| sin 𝜃 0 | with 𝜃 0 denoting the average angle between the neighbouring spins 𝑺 𝑖 and 𝑺 𝑖+1 in the wall [START_REF] Villain | Two-level systems in a spin-glass model. I. General formalism and two-dimensional model[END_REF][START_REF] Cinti | Two-Step Magnetic Ordering in Quasi-One-Dimensional Helimagnets: Possible Experimental Validation of Villain's onjecture about a hiral Spin iquid State[END_REF][START_REF] Menzel | Information Transfer by Vector Spin Chirality in Finite Magnetic Chains[END_REF]. The chiral order parameter defined in this way is a pseudo-scalar variable analogous to an Ising spin, which takes values of +1 for right-rotating walls and -1 for left-rotating walls, before the pump pulse arrives. Between 1 ps and 10 ps the electronic part of the system recovers its ground-state properties to a large extent, while the spin system is still excited with a relatively high spin temperature of about 490 K, which slowly decays on the scale of 10 -1000 ps towards room temperature. It is in this time window that the recovery rate of the chiral order parameter is significantly higher than that of the magnetization. This is based on different temperature dependences of the chiral and scalar order parameters as observed in various spin systems, with the chiral order parameter restoring to its ground state value faster as the temperature is lowered [START_REF] Menzel | Information Transfer by Vector Spin Chirality in Finite Magnetic Chains[END_REF]. The faster recovery behaviour can be traced back to the properties of the chiral order parameter as an essentially Ising type variable [START_REF] Villain | Two-level systems in a spin-glass model. I. General formalism and two-dimensional model[END_REF][START_REF] Cinti | Two-Step Magnetic Ordering in Quasi-One-Dimensional Helimagnets: Possible Experimental Validation of Villain's onjecture about a hiral Spin iquid State[END_REF][START_REF] Harada | One-dimensional Classical Planar Model with competing interactions[END_REF], which is also the basis of exotic spin states such as chiral spin liquids [START_REF] Villain | Two-level systems in a spin-glass model. I. General formalism and two-dimensional model[END_REF][START_REF] Cinti | Two-Step Magnetic Ordering in Quasi-One-Dimensional Helimagnets: Possible Experimental Validation of Villain's onjecture about a hiral Spin iquid State[END_REF][START_REF] Wen | Chiral spin states and superconductivity[END_REF].

In order to check the appropriateness of these two mechanisms for our case, we performed numerical simulations of the scattering signal and evaluated -similar to the experiment -the total intensity and the 2 nd moments of the sum and difference signal under the influence of these two mechanisms. We start from an initial labyrinth stripe state shown in Fig. 1b with a domain periodicity of 450 nm and a domain wall width of 27 nm as estimated from the magnetic parameters (S3). For mechanism (1) the domain wall width was altered, while keeping the domain periodicity fixed. Mechanism (2) was implemented by introducing smaller amplitudes of the (transversal) magnetisation fluctuations to the domain walls compared to the domains emulating the faster build-up of the chiral order parameter.

In Fig. 5a we show the simulated total scattering intensities of CL+CR and CL-CR as a function of the domain wall width. For an estimated maximum increase of the wall width towards 47 nm the signal strength of CL-CR increases to 125 %, while CL+CR decreases to 95 %. The ratio of both signals thus increases by a factor of 1.3 which fits very well the experimental values of 1.33 as measured at 100 ps (Fig. 4b).

In Fig. 5b we show the dependence of the simulated total scattering intensities of CL+CR and CL-CR on the fluctuation tilt angle, which characterizes the strength of the transverse spin fluctuations. The dashed and dotted lines in Fig. 5b correspond to a homogeneous distribution of fluctuations throughout the system, while the solid lines represent the case of reduced fluctuations in the walls, but not in the domains. First, we note that the experimentally observed drop of the intensities to 20 % during the first ps in Fig. 4b) correspond to a fluctuation tilt angle of 63° in Fig. 5b (dashed and dotted lines). We assume that during the following 100 ps the fluctuations in the walls reduce faster than in the domains caused by the above discussed temperature (and by that delay time) scaling of the order parameters such that finally the solid lines in Fig. 5b describe the intensities. At 100 ps the value of CL+CR observed in Fig. 4b) is 30 % and CL-CR is 41 % in good agreement with the numerical intensities at a tilt angle of 62° (solid blue and red line in Fig. 5b).

Fig. 5c displays the evolution of the 2nd moment of the scattering peak S(Q) from the simulation as a function of wall width. In contrast to the overall intensity (Fig. 5a) we do not observe significant changes in the second moments on increasing the relevant domain widths between 27 nm and 47 nm.

Fluctuations, however, can change the 2 nd moments significantly (Fig. 5d). While the 2 nd moments of the CL-CR signal are insensitive to the tilt angle (red lines), the CL+CR signal increases with increasing fluctuations -in good agreement with the experimental findings (Fig. 4d). Therefore, the experimentally observed decrease of CL+CR can be explained by a reduction of the fluctuation strength with increasing delay time and a faster reduction of fluctuations in the domain wall leading to a faster build-up of the chiral order.

Thus, the findings obtained from the second moments support the presence of mechanism (2): the fluctuation mechanism, while not necessarily ruling out the additional contribution of (1) the breathing mode mechanism to the faster recovery of the total intensity of the difference signal, since we showed that both mechanisms can explain the observed faster recovery of the total intensity of CL-CR. However, this implicates the necessity of breathing frequencies lower than 5 GHz, since otherwise also a decrease of the DW width and accordingly CL-CR should be observed within the 100 ps timeframe, while frequencies below 5 GHz would lead to a reduced domain wall increase within the 100 ps timeframe. Analytical calculations of the domain wall oscillation frequency [START_REF] Liu | Theory of magnetoelastic dissipation due to domain wall width oscillation[END_REF] using the measured parameters suggest that even the strongly reduced anisotropy values at the high spin system temperatures after the IR excitation lead to oscillation frequencies in the range of 13-17 GHz. This does not fit the experimental data that could only be explained by an oscillation of < 5 GHz.

Therefore, we conclude that the main driver behind our experimental findings is likely mechanism (2) leading to a faster recovery of the chiral order in the domain walls in comparison to the ferromagnetic order in the domains. The responsible mechanism(s) might in addition be also material system dependent. So for the future, one can explore further multilayer stacks and the time dependence of the total intensity on even longer timescales to probe if one can identify additionally the effects of a damped breathing mode of the domain wall excited by the IR laser.

Conclusion

In conclusion, using resonant magnetic scattering at an XUV FEL we identify a distinctly faster recovery of the scattering intensity signal stemming from the chiral magnetic order in the domain walls compared to the signal from the collinear magnetic order in the domains. We also observe an increase of the width of the structure factor peak for the collinear signal while the chiral scattering peak does not increase in width upon IR pumping indicating strong fluctuations. We explain both observations with the onset of strong transversal magnetic fluctuations after pumping in the domain systems. Importantly, however, our experimental findings imply reduced fluctuations within the chiral domain walls leading to a chiral order that is restored faster. We can connect this to the realisation that the characteristic time scale for the DMI-driven onset of the chiral magnetic order is significantly lower than the onset of the ferromagnetic order driven by the Heisenberg exchange interaction. In the future further fundamental aspects can be studied in detail e.g. the dependence of the timescales of the chiral order build-up on the absolute strength of the DMI by varying the heavy metal layers. The better control of the DMI and the chirality of spin structures on the ultrafast timescale can finally allow the controlled ultrafast manipulation of chiral magnetism, e.g. ultrafast writing of chiral topological objects such as skyrmions and pave the path to applications in the field of ultrafast chiral spintronics.

Methods

Sample fabrication and characterisation

The magnetic multilayer sample has been grown at room temperature by dc magnetron sputtering and consist of (thicknesses in nm) Si/SiO2/[Ta(5.3)/Co20Fe60B20(0.93)/Ta(0.08)/MgO(2.0)]x20 /Ta(1.6).

The Ta(0.08 nm) interlayer was inserted to tune the anisotropy stemming from the Co20Fe60B20/MgO interface, while the 1.6 nm Ta capping was reduced to a thickness that prevented the sample from oxidation, but simultaneously led to sufficient penetration of the XUV light into the topmost Co20Fe60B20 layer to ensure sufficient statistics. The magnetic properties of the sample were characterized using superconducting quantum interference magnetometry (SQUID) in the temperature range 10 K to 390 K and a vibrating sample magnetometer (VSM) in the temperature range 300 K to 600 K. Both out-of-plane and in-plane hysteresis loops were measured (see Supplementary S1) for each temperature. From those the saturation magnetisation Ms and the effective perpendicular anisotropy constant Keff were determined. The latter was ascertained from the difference in the areas of the in-plane and out-of plane loops [START_REF] Lemesh | Accurate model of the stripe domain phase of perpendicularly magnetized multilayers[END_REF]. This leads to the temperature dependencies shown in Supplementary S1 yielding a Curie temperature of (553±1) K and parameters Ms(300 K) = (844±28) kA/m and Keff(300 K) = (133±17) kJ/m 3 at room temperature. Additionally, a 20x20 μm 2 MFM image of the multilayers was performed at room temperature as shown in Fig. 1b. The sample exhibits labyrinth-like stripe domains with a domain periodicity of (455+/-30) nm, as extracted from the Fourier spectrum which exhibits a first order peak at q = (13.8+/-0.9) μm -1 (see Supplementary S4). The interfacial DMI arises from the Ta(5.3 nm)/ Co20Fe60B20 interface, which favours right-handed chiral Néel domain walls, as observed in the dichroic SAXS signal in Fig. 1f. One must take into consideration that the Ta(0.08 nm) insertion layer is not even a complete Ta monolayer and therefore a net interfacial DMI is still prominent in the sample. Finally, the interplay between interfacial DMI and dipolar interactions determines the equilibrium domain and domain wall arrangement in the sample [START_REF] Legrand | Hybrid chiral domain walls and skyrmions in magnetic multilayers[END_REF].

Pump-probe experiments

The measurements have been performed at the DiProI beamline [START_REF] Capotondi | Multipurpose end-station for coherent diffraction imaging and scattering at FERMI@ Elettra free-electron laser facility[END_REF][START_REF] Capotondi | Coherent imaging using seeded free-electron laser pulses with variable polarization: First results and research opportunities[END_REF] at the FERMI FEL facility in Trieste, Italy [START_REF] Allaria | Highly coherent and stable pulses from the FERMI seeded free-electron laser in the extreme ultraviolet[END_REF][START_REF] Allaria | Two-stage seeded soft-X-ray free-electron laser[END_REF]. For the experiment, the FEL was tuned to the Fe M2,3-edge at a wavelength of .0 nm, a pulse duration of 0 fs, a repetition rate of 50 Hz, and an attenuated pulse energy of 0.94 μJ. Using a Kirkpatrick-Baez (KB) optics, the beam was focused down to a size of (200) × (200) μm 2 leading to an energy density of 3,0 mJ/cm 2 . The optical laser for pump-probe experiments is the same as the Ti:sapphire seed laser used for generating the FEL pulses in the HGHG scheme and therefore is intrinsically synchronized to the XUV-FEL pulses with a jitter of less than 10 fs. We used as a pump a 780 nm IR pulse of 00 fs duration, with an energ of . μJ and a size of (300) × (300) μm 2 , leading to a pump energy density of 3,7 mJ/cm 2 , well below the material damage threshold and the intra-pulse self-demagnetization process observed with FEL radiation on similar magnetic structures [START_REF] Müller | Breakdown of the X-Ray Resonant Magnetic Scattering Signal during Intense Pulses of Extreme Ultraviolet Free-Electron-Laser Radiation[END_REF]. The IR and FEL beams are nearly parallel with a small angular offset of 2° which allows for a constant temporal resolution during the rotation of the sample through the beam. The resonant magnetic XUV reflectivity experiments were performed for left-and right-hand circular polarization with a charge-coupled device (CCD) area detector (2,048×2,048 pixels, 13.5×13.5 µm 2 pixel size) placed 145 mm behind the sample. The circularly polarized incident X-rays hit the sample at an angle of 44 degrees yielding an effective penetration depth of 8 nm. The intense reflected primary beam is blocked by a beam stop. The CCD is IR protected by an Al filter.

SAXS data treatment

For each measured pump probe time delay, 7000 SAXS patterns have been measured for each helicity. The patterns have been normalized to the incoming flux and averaged. The area around the beamstop and charge scattering streaks have been masked. To remove the contribution of the charge component, the background signal must be calculated for each pattern. Using the areas inside and outside the diffraction ring, we can reconstruct the whole pattern without the magnetic part. As a result of the background removal, we obtain the corrected patterns for post-proceeding analysis. Afterwards the average intensity of the sum (CL+CR) and difference (CL-CR) diffraction images was determined as a function of the wavevector Qr for different time delays as depicted in Fig. 3. The radial profiles for each time delay were then integrated in order to obtain the evolution of the total intensity of the sum and the difference signal as a function of delay time. The time evolution of the total intensity normalized to the unpumped total intensity is displayed in Fig. 4 a &4b. The time dependence of the total intensities was fitted with the sum of three exponential functions convolved with a Gaussian distribution containing the time resolution σt of the experiment given by the pulse duration of the FEL: 2 )],

𝐼(𝑡) 𝐼 𝑢𝑛𝑝𝑢𝑚𝑝𝑒𝑑 ⁄ = {1 -𝐻(𝑡 -𝑡 0 ) • [𝐴 𝑑 • 𝑒𝑥𝑝 (- 𝑡-𝑡 0 𝜏 𝑑 ) + 𝐴 𝑟1
(

) 1
where τd denotes the time constant of the demagnetisation respectively the dechiralisation process and τr1 and τr2 the time constants of the fast and slow recovery processes, whereas Ad, Ar1 and Ar2 denote the strength of these processes (fit parameters displayed in Supplementary S5). H(t) is the Heaviside step function and time zero t0 was obtained from the fit. The 2 nd moments of scattering peak S(Q) (radial scattering distribution) displayed in Fig. 4c & 4d are calculated by:

2 𝑛𝑑 𝑚𝑜𝑚𝑒𝑛𝑡 = ∫ (𝑞 -𝑞 0 ) 2 𝑆(𝑞) 𝑑𝑞 ∫ 𝑆(𝑞) 𝑑𝑞 (2)
where the peak center q0 is determined for each delay time individually.

Numerical calculation of the diffracted intensity

In the simulation setup we assume a system of homochiral walls, where the -+ and +-walls are modelled by:

We model the diffracted intensity by:

𝐼(𝑄) = |(𝜀 × 𝜀 ′ ) • ∫ 𝒎(𝑥)𝑒 𝑖𝑄𝑥 𝑑𝑥 𝐿/2 -𝐿/2 | 2 , ( 4
)
where L is the size of the simulation box. We used L=225µm, corresponding to 500 domain periodicities of 450 nm. The magnetization m(x) in the system without fluctuations is obtained by alternating -+ and +-walls spaced 225 nm apart. In order to model the effect of transverse fluctuations we first tilt the magnetization at every point by a polar tilt angle. After that we rotate the magnetization at every point by an azimuthal angle obtained from a random number generator. We characterize the fluctuation strength by the polar tilt angle.

In order to model the case where fluctuations are reduced in the walls (due to the proposed faster recovery of the chiral order parameter) we switch off the fluctuations in the region -1.1𝑤 < 𝑥 -𝑥 𝑖 < 1.1𝑤 for every wall i.

The temperature dependence of the effective anisotropy constant was fitted using:

𝐾 𝑒𝑓𝑓 (𝑇) = 𝐾 𝑢 (𝑇) -µ 0 2 (𝑀 𝑠 (𝑇)) 2 , [START_REF] Everschor-Sitte | Perspective: Magnetic skyrmions-Overview of recent progress in an active research field[END_REF] where the magnetisation dependence of the anisotropy constant was fitted by a power law:

𝐾 𝑢 (𝑇) 𝐾 𝑢 (0 𝐾) = ( 𝑀 𝑠 (𝑇) 𝑀 𝑠 (0 𝐾) ) 𝑐 , [START_REF] Brataas | Chiral domain walls move faster[END_REF] leading to an anisotropy constant Ku(0 K) = (1031±11) kJ/m 3 and c = (2.16±0.02).

Using the fit parameters, we obtain finally a room temperature saturation magnetisation of Ms(300 K) = (844±28) kA/m and an effective anisotropy of Keff(300 K) = (133±17) kJ/m 3 . 5) respectively equation ( 6).

Figure 1 :

 1 Figure 1: Experimental setup and diffraction images. (a) Measurement geometry: A magnetic thin film sample is pumped by an optical infrared laser pulse and probed by a circularly polarized X-ray FEL pulse with an incident angle of Θ = 44° (wavevector k, s-polarisation(σ) and p-polarisation(π)) that scatters on the sample. Afterwards an IR-protected charge-coupled device (CCD) detector records the magnetic SAXS pattern. (b) MFM image of a typical labyrinth domain pattern of the [Ta(5.3 nm)/Co20Fe60B20(0.93 nm)/Ta(0.08 nm)/MgO(2.0 nm)]x20/Ta(1.6 nm) sample. The inset shows the FFT with the first order peak corresponding to isotropic distributed labyrinthine stripes with a domain periodicity of (455+/-30) nm. These magnetic structures lead to the SAXS signals for (d) left-hand (CL) and (e) right-hand (CR) circular polarized incident x-rays. (f) The resulting sum = CL+CR of the diffraction pattern, confirms that the diffraction corresponds to the magnetic domains observed by MFM. (g) The azimuthal asymmetry in the dichroic signal (difference = CL-CR) confirms the presence of (c) right-handed chiral Néel (cycloidal) domain walls.

Figure 2 :

 2 Figure 2: Time dependence of diffraction images. Diffraction patterns for the sum CL+CR (top) and the difference CL-CR (bottom) images for the initial (a, d), +1.0 ps (b, e) and +100 ps (c, f) states. (g)The experimental (dots) and fitted (lines) orthoradial dichroic profiles for different pump-probe delay times showing a maximum at 90° and a minimum at 270° corresponding to a right-handed cycloidal winding of the domain walls. While the amplitude of the dichroic signal drops within a ps upon pumping and recovers afterwards, no significant angle shift in the fitted phase φ can be detected. The beamstop is located at an azimuthal angle of 90°.

  ) shows the diffraction patterns of the sum and the dichroic signals at different pump-probe delays. At +1.0 ps (Fig.2b & 2e)) the intensity in both signals has dropped down significantly in comparison to the initial states (Fig.2a & 2d) with a visible partial recovery of both signals after +100 ps (Fig.2c & 2f).

Figure 3 :

 3 Figure 3: Time dependence of radial scattering intensities. The radial profiles of the sum CL+CR (a) and difference CL-CR (b) signals for selected delay times. Error bars from counting statistic are smaller than the symbol sizes. The visible fluctuations in the profiles are due to speckles originating from the coherent illumination.

Figure 4 :

 4 Figure 4: Time dependence of total scattering intensities and second moments. Time evolution of the (a,b) radially integrated intensity and(c,d) the 2 nd moments of the sum signal (ferromagnetic order) and the difference signal (chiral magnetic order) in linear scale (a,c) and logarithmic scale (b,d). The solid lines in (a,b) depict the fitted dependences using equation (1) and highlight the faster recovery of the chiral signal. The dashed lines in (c,d) provide a guide to the eye. Error bars based on photon counting statistics and numerical integration of the peaks in Fig.3are smaller than the symbol sizes.

Figure 5 :

 5 Figure 5: Numerical calculations of total scattering intensities and second moments. The numerically calculated (a,b) integrated intensity and (c,d) 2 nd moments of the sum signal (ferromagnetic order) and the difference signal (chiral magnetic order) as a function of the (a,c) domain wall width and (b,d) the transversal magnetisation fluctuation strength (cone angle in degrees). The 100 % value in (a,c) corresponds to the room temperature domain wall width of 27 nm. The dashed lines in (b,d) correspond to the case of fluctuation in the whole system, while the solid lines correspond to the case of reduced fluctuations within the domain walls.

  𝑚 𝑥 𝑊-+ (𝑥) = -𝑚 𝑥 𝑊+-(𝑥) = -√1 -[tanh 𝑥 -𝑥 𝑖 𝑤 ] 2 𝑚 𝑦 𝑊-+ (𝑥) = 𝑚 𝑦 𝑊+-(𝑥) = 0 𝑚 𝑧 𝑊-+ (𝑥) = -𝑚 𝑧 𝑊+-(𝑥) = tanh 𝑥 -𝑥 𝑖 𝑤 ,

Figure S2 :

 S2 Figure S2: Temperature dependence of Ms and Keff. Temperature dependence of the saturation magnetisation Ms (a) and effective anisotropy Keff (b). The datapoints were fitted by the models displayed in equation (5) respectively equation[START_REF] Everschor-Sitte | Perspective: Magnetic skyrmions-Overview of recent progress in an active research field[END_REF].

Acknowledgments

N.K., B.S., K.L. and M.K. gratefully acknowledge financial support by the Graduate School of Excellence Materials Science in Mainz (MAINZ, GSC266). M.K. and the groups in Mainz acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)-projects 290319996/TRR173, 403502522/SPP 2137 Skyrmionics and 290396061/TRR173. C.G. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG) -projects GU 535/4-1 and KS 62/1-1. Y.M. acknowledges funding from the DARPA TEE program through Grant MIPR (No. HR0011831554) from DOI. We also gratefully acknowledge the Jülich Supercomputing Centre and RWTH Aachen University for providing computational resources under project jiff40. Measurements were carried out at the DiProI beamline at the FERMI FEL facility in Trieste, Italy. The authors thank FERMI FEL facility for the allocation of the beamtime and the technical support offered during the measurements.

Author Contributions

C.G. and M.K. proposed the study. C.G., M.K., and Y.M. supervised the respective members of the study. N.K. and J.C. fabricated samples. N.K., C.G., H.Z., F.C., E.P. and I.L.-Q. performed the FEL experiment. D.K. and C.G. analysed the beamtime data. N.K., B.S., K.L. and D.L. performed the sample characterization. F.F., Y.M., N.K., C. G., M. K. and D.K. developed the two discussed mechanism. F.F. performed numerical calculations of the scattering signal. All authors participated in the discussion and interpreted results. N.K. drafted the manuscript with the help of D.K., F.F. and assistance from C.G., M.K. and Y.M. All authors commented on the manuscript.

Competing interests

The authors declare no competing interests.

Supplementary S1: Determination of material parameters and their temperature dependence by SQUID/VSM

Hysteresis loops were recorded by SQUID and VSM in order to determine the saturation magnetisation Ms as well as the effective anisotropy Keff of the sample. The curves displayed in Fig. S1 show the outof-plane (OOP) and in-plane (IP) hysteresis loops at 300 K. The effective perpendicular anisotropy Keff was determined by the difference in the areas of the in-plane and out-of-plane hysteresis loops and corresponds to Ku -μ0/2 Ms 2 with the uniaxial anisotropy Ku [START_REF] Lemesh | Accurate model of the stripe domain phase of perpendicularly magnetized multilayers[END_REF]. The hysteresis loops displayed in Fig. S1 measured at room temperature (300 K) via SQUID were also performed at temperatures of 10 K, 50 K, 100 K, 200 K, 300 K, 350 K and 390 K by SQUID and 300 K, 350 K, 390 K, 450 K, 500 K, 550 K, and 600 K by VSM. The saturation magnetisation was determined and afterwards the following model was used to fit the temperature dependence of the saturation magnetisation: [START_REF] Jiang | Skyrmions in magnetic multilayers[END_REF] with a saturation magnetisation Ms(0 K) = (976±10) kA/m, a Curie temperature Tc = (553±1) K, 𝛼 = (1.73±0.13) and 𝛽 = (0.63±0.04).

S2: Pump-probe-delay time dependent spin system temperature

The total intensity of the sum signal displayed in Fig. 4a & 4b is connected to average domain magnetisation (I ∝ M 2 ). Using the temperature dependence of the saturation magnetisation according to Fig. S2 one can estimate the temperature evolution of the spin system during the pump-probe experiment displayed in Fig. S3. Upon optical excitation the spin system temperature increases within a ps to (493±22) K and relaxes afterwards slowly back to room temperature with an already decreased temperature of (469±13) K at 100 ps.

Figure S3

: Time dependence of spin system temperature. The evolution of the spin system temperature as a function of pump-probe delay time in logarithmic scale.

S3: Temperature dependence of the domain wall width

Using the temperature dependence of the effective anisotropy constant obtained in Fig. S2 one can estimate the temperature dependence of the (equilibrated) domain wall width given by: ∆= 𝜋 √ 𝐴/𝐾 𝑢,𝑒𝑓𝑓 .

The temperature dependence of the exchange stiffness constant A is in mean-field approximation described by a power law A ∝ Ms 2 leading to the temperature scaling of the domain wall width displayed in Fig. S4. One can observe that the domain wall width increases with increasing temperature due to a stronger scaling of the effective anisotropy constant in comparison to the exchange constant. It was also shown that fluctuation corrections from nonlinear spin-wave effects lead to a scaling A ∝ Ms κ with κ< depending on the lattice structure [49], which would lead to an even stronger scaling of the domain wall width with the temperature. Using the assumed exchange stiffness constant of A=10 pJ/m at room temperature (300 K) leads to the absolute domain wall widths displayed in Fig. S5. Combining the time delay dependence of the spin system temperature displayed in Fig. S3 with the here obtained temperature dependence of the domain wall width one can estimate the increase of the equilibrium domain wall width to approximately 10 nm for delay times between 1 ps to 100 ps.

S4: Fourier spectrum of the MFM image

The magnetic force microscopy image displayed in Fig. 1b was Fourier transformed leading to the spectrum displayed in Fig. S6. One can see a clear first order peak at a value of q=(13.8+/-0.9) μm -1 as well as a second order peak at q=(27.6+/-0.9) μm -1 and a third order peak at q=(41.4+/-1.0) μm -1 due to partial short range stripe alignment of the OOP demagnetized magnetic labyrinth pattern. The first order peak indicates a stripe periodicity of (455+/-30) nm, which agrees well with the real space stripe data measured. Table S1: Pump-probe fit parameters. Fit parameters obtained by fitting the pump-probe data shown in Fig. 4a & 4b with the model described in equation [START_REF] Rößler | Spontaneous skyrmion ground states in magnetic metals[END_REF].

Continued References:


K Katcko

E Urbain

L Kandpal

B Chowrira

F Schleicher

U Halisdemir

F Ngassam

D Mertz

B Leconte

N Beyer

D Spor

P Panissod

A Boulard

J Arabski

C Kieber

E Sternitsky

V Da Costa

M Alouani

M Hehn

F Montaigne

A Bahouka

W Weber

E Beaurepaire

D Lacour

S Boukari

M Bowen
email: bowen@unistra.fr

Magnetoresistance and spintronic anisotropy induced by spin excitations along molecular spin chains

Magnetoresistance and spintronic anisotropy induced by spin excitations along molecular spin chains

Overall, this differing level of maturity between these model (STM, lateral) and solid-state junctions reflects not only a technological bottleneck toward the widespread study of high-quality vertical nanojunctions, but also a challenge to precisely determine the effective nanotransport path [START_REF] Studniarek | Probing a Device's Active Atoms[END_REF] within a solid-state device. In this Letter, we bridge these technological and knowledge gaps through spectroscopic temperature-dependent magnetotransport experiments. Herein, we craft solid-state vertical nanojunctions from full in-situ grown FM/CoPc/FM heterostructures using a lowtech nanojunction process that does not structurally degrade the active organic layer, or expose it to resist, solvents or air. We find that the magnetoresistance generated upon flipping the magnetic orientation of the two FM electrodes tracks the increase in junction conductance resulting from excitations along spin chains in the intervening CoPc molecular magnetic layer. According to a three-macrospin model that fits our magnetotransport data, the junction's FM electrodes are magnetically coupled to a central magnetic unit comprising MSCs coupled to the junction's lower interface. This strength of this unit's unidirectional anisotropy at 17K nearly doubles with spin excitations to effectively reach 18T at |V|>100mV, while its temperature dependence mimicks that of exchange bias seen in magnetometry [START_REF] Gruber | Exchange bias and room-temperature magnetic order in molecular layers[END_REF][START_REF] Boukari | Disentangling Magnetic Hardening and Molecular Spin Chain Contributions to Exchange Bias in Ferromagnet/Molecule Bilayers[END_REF] . The magnetic coupling between this unit and the bottom FM layer abruptly drops with increasing T at ~70K, which pegs a 5.9+-0.3meV thermal energy threshold of the so-called 'magnetic hardening' effect [START_REF] Raman | Interface-engineered templates for molecular spin memory devices[END_REF] induced at the Fe/CoPc interface by molecule-induced charge transfer. We find that spin excitations dynamically induce a mutual magnetic coupling of all magnetic units, and alter their magnetic anisotropy as direct experimental evidence of spintronic anisotropy [START_REF] Misiorny | Spintronic magnetic anisotropy[END_REF] due to this transport mechanism.

VSubmit

Save trees and the air, please don't print! bowen@unistra.fr 3 Our junctions are crafted from entire in-situ grown FM/molecular layer/FM stacks, thereby preserving nominal structural/magnetic properties, especially at interfaces, using a novel, solvent-and resist-free processing technique inspired by nanosphere lithography [START_REF] Wang | Parallel fabrication of magnetic tunnel junction nanopillars by nanosphere lithography[END_REF] (see Methods for growth and processing details). After depositing the entire heterostructure through a shadow mask to define the lower electrode, 500nm-diameter SiO 2 nanobeads synthesized using a surfactant-mediated sol-gel reaction [START_REF] He | Size-controlled synthesis of monodispersed mesoporous silica nanospheres under a neutral condition[END_REF] are randomly distributed on the surface (Fig. 1a). Once Ar atom milling to the organic layer is complete (Fig. 1b), the sample is encapsulated in sputtered SiO 2 and the beads are blown off, leaving a sub-diameter access (not shown) to the nanopillar (Fig. 1c). In a final step, top metallic contacts are deposited through a shadow mask (Fig. 1d).

The nanotransport path [START_REF] Studniarek | Probing a Device's Active Atoms[END_REF] across the CoPc thin film nanojunctions that we infer from our combined experimental/analytical results is schematized in Fig. 1e. The effective nanotransport path proceeds across three coupled magnetic units. The energy density E of the nanotransport path is that of the top (i=t), central (i=c) and bottom (i=b) units and can be written as:

𝐸 = -µ 0 ∑ 𝑯. 𝑴 𝒊 𝒊=𝒕,𝒄,𝒃 + 1 2 ∑ 𝐾 𝑖 sin 2 (𝜃 𝑀 𝑖 -𝜃 𝐾 𝑖 ) 𝒊=𝒕,𝒄,𝒃 -∑ 𝐶 𝑐𝑖 𝒎 𝒄 . 𝒎 𝒊 𝒊=𝒕,𝒃 .
where 𝐻 is the applied magnetic field and, for each unit, 𝑀 𝑖 is the magnetization, 𝑚 i the reduced magnetization, 𝜃 𝑀 𝑖 the magnetization angle, and 𝐾 𝑖 its uniaxial anisotropy with an easy axis angle 𝜃 𝐾 𝑖 . Finally, 𝐶 𝑐𝑖 is the coupling strength between the central (c) and outer (i=b,t) magnetic units. We define 𝐶 𝑐𝑡 = ECTop and 𝐶 𝑐𝑏 =MHBott. We justify this nomenclature hereafter. For each H step, E is minimized to yield, for each magnetic unit, the magnetization's in-plane orientation.

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Save trees and the air, please don't print! bowen@unistra.fr 4

The adsorption of Pc molecules onto a FM surface induces a change in sign of spin polarization [START_REF] Djeghloul | High Spin Polarization at Ferromagnetic Metal-Organic Interfaces: a Generic Property[END_REF] , as well as a magnetic hardening [START_REF] Raman | Interface-assisted molecular spintronics[END_REF] for temperatures lower than ~70K [START_REF] Boukari | Disentangling Magnetic Hardening and Molecular Spin Chain Contributions to Exchange Bias in Ferromagnet/Molecule Bilayers[END_REF] of the FM's top monolayer, which can rotate independently of the buried FM monolayers thanks to a weakened FM coupling that we call MHBott. The term 'spinterface' encompasses these interfacial properties, and sets the magnetic referential of the central magnetic unit, including that of MSCs that extend away from the interface and are formed by CoPc molecules with AF interactions [START_REF] Serri | High-temperature antiferromagnetism in molecular semiconductor thin films and nanostructures[END_REF] . Magnetometry confirms that these spin chains also contribute to the anisotropy below 100K [START_REF] Boukari | Disentangling Magnetic Hardening and Molecular Spin Chain Contributions to Exchange Bias in Ferromagnet/Molecule Bilayers[END_REF] . The remainder of the bottom FM forms a lower magnetic unit that is modelled as a free layer, i.e. with a reduced anisotropy K/M=5mT. [START_REF] Chen | Probing Superexchange Interaction in Molecular Magnets by Spin-Flip Spectroscopy and Microscopy[END_REF] As we will see, the top magnetic unit is comprised of the Co layer directly coupled to a MSC. The absence of a spinterface at the top interface presumably reflects metal interdiffusion during top electrode deposition. The presence in the nanotransport path of disjointed chains across the film thickness is expected from both structural and magnetic studies [START_REF] Serri | High-temperature antiferromagnetism in molecular semiconductor thin films and nanostructures[END_REF][START_REF] Boukari | Disentangling Magnetic Hardening and Molecular Spin Chain Contributions to Exchange Bias in Ferromagnet/Molecule Bilayers[END_REF] , such that a weak AF exchange term ECTop is present between the top and central magnetic units. The following dataset, which is typical of results found on several junctions (see Methods), was entirely acquired on a Fe/CoPc(20nm)/Co junction with resistances R(300K)=32k and R(17K)=11G at 20mV. This underscores thermally activated hopping transport [START_REF] Tong | Defect states dependence of spin transport in iron phthalocyanine spin valves[END_REF][START_REF] Boukari | Electrical transport across a structurally ordered phthalocyanine film: Role of defect states[END_REF] across the thin CoPc layer.

We first examine magnetotransport at 60K, i.e at the temperature onset of the magnetic hardening effect, and with MSCs already promoting exchange bias. Fig. 1 shows R(H) loops at 58mV following a cooldown at H=-1T. As the positive maximum H is increased, one notices a 1 st resistance jump near H=0, and a 2 nd resistance change centered around H=1.17T that is reversible as long as the resistance baseline at H=-2T isn't exceeded. For higher positive field sweeps, a third resistance change is observed and the R(H) loop becomes field-symmetric.

The three resistance levels observed justify our model's three magnetic units and suggest, consistently with further data/analysis, that two MR terms describe magnetotransport. We therefore write the resistance R due to noncollinear magnetizations M t , M b and M c as:

𝑅 = 𝑅 0 . [1 - 𝑀𝑅𝑇𝑜𝑝 2
• (𝐦 𝒕 . 𝐦 𝒄 -1) -𝑆𝑝𝑖𝑛𝐹𝑙𝑖𝑝𝑀𝑅 2

• (𝐦 𝒕 . 𝐦 𝒃 -𝟏)].

MRTop refers to the impact on nanojunction resistance of flipping the magnetization of the top magnetic unit relative to that of harder magnetic units, i.e. to the central magnetic unit. SpinFlipMR considers the MR due to flipping both the top and bottom magnetic units (see Suppl. Note 1 for details on pairing magnetic units to MR terms). We justify this nomenclature hereafter. By successfully fitting this unusual, symmetric R(H), we identify the sequential magnetization reversal of the top, bottom and central magnetic units (see green arrows in Fig. 1f). Despite its low anisotropy, the bottom magnetic unit reverses at |H|>1T due to the FM coupling term MHBott arising from magnetic hardening (see Fig. 1e). 3), and schematize these MR contributions in the R(H) at 80mV and T=17K (see Fig. 2a), for which the magnetic states within |H|<2T are better defined. Due to our experimental limitation |H|<2T, when comparing the model with experiment, we make the approximation that 𝑅 0 = 𝑅(-2𝑇), while small deviations between experimental and modelled values of MRTop and SpinFlipMR can occur because full magnetization reversal of the bottom magnetic unit can be incomplete within |H|<2T. Nevertheless, dataset consistency criteria and the shape of the R(H) data strongly limit possible errors. See Suppl. Note 1 for details. The ECTop and MHBott coupling terms are also shown in Fig. 2a next to the reversal process for the top and bottom magnetic units, respectively.

We experimentally define

MRTop=R(2T)/R(-2T)-1 and SpinFlipMR= R(H f )/R(2T)-1 (H f = 1T for 17 < T(K) < 50, H f < 1T for T > 50K, see Fig.

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Save trees and the air, please don't print! bowen@unistra.fr 5 We now examine magnetotransport at 17K. For 43 ≤ V(mV) ≤ 150, only two magnetization reversals are observed, as illustrated for R(H) loops in Fig. 2a-e. According to the model's fits (black lines of Fig. 2a-e), also shown in Fig. 2f for a larger field sweep, the central unit flips at H>2T, i.e. beyond our experimental limit. At 20mV, the R(H) loop (Fig. 2f) exhibits only one resistance change, even though the model's fit indicates that the bottom FM electrode has flipped during the sweep (data not shown). This implies that, at 20mV, SpinFlipMR=0. We will discuss this effect in what follows. Similar effects are observed for V<0 (data not shown).

Varying the in-plane orientation angle   of the applied magnetic field causes MRTop to smoothly switch sign between 0º and 180º for SpinFlipMR=0 (see Suppl. Note 2). We refrain from studying SpinFlipMR(  ) since the magnetization state at H=1T evolves with   . Nevertheless, we observe that, after cooling from 120K at H=-1T, the R(H) at 17K and   =0º is flipped along H when   = 180º (see Suppl. Note 2). This flipping also occurs when comparing field cooling at H=-1T with field cooling at H=0 + after applying H=2T at T=120K (see Suppl. Note 2). Varying the amplitude of the bias voltage that is applied during cooldown does not significantly alter the R(H) loop (data not shown). We thus infer that the device's magnetic state during cooling defines a unidirectional axis for magnetotransport. As we will see, this unidirectional character arises from the coupling of the outer magnetic units to the central magnetic unit with very high anisotropy strength.

While magnetometry on FM/MPc bilayers revealed a fatigue effect upon repeatedly sweeping the external magnetic field [START_REF] Gruber | Exchange bias and room-temperature magnetic order in molecular layers[END_REF][START_REF] Boukari | Disentangling Magnetic Hardening and Molecular Spin Chain Contributions to Exchange Bias in Ferromagnet/Molecule Bilayers[END_REF] , no such fatigue effect was seen in magnetotransport (see Fig. 1f). This, and the ability to reproduce magnetotransport data using a macrospin model (see Fig. 1f, Fig. 2a-e and Fig. 3a-e), suggest that the nanotransport path within the 500nm-nominal diameter is proceeding through a reduced number of grains.

Referring to Fig. 2g, the bias dependence of dI/dV reveals a mostly constant amplitude for |V|<35mV, and large increases for |V|>35mV, punctuated by peaks at |V|≈70mV. The dI/dV amplitude further increases at higher bias. In line with previous literature [START_REF] Chen | Probing Superexchange Interaction in Molecular Magnets by Spin-Flip Spectroscopy and Microscopy[END_REF][START_REF] Muenks | Correlation-driven transport asymmetries through coupled spins in a tunnel junction[END_REF][START_REF] Barraud | Phthalocyanine based molecular spintronic devices[END_REF] on STM-assembled and solid-state-based transport across spin chains, we interpret these dI/dV features as the signature of spin excitations. We note that the complicated bias drop across the nominal 20 nm-thick CoPc junction due to the hopping transport regime could explain the higher bias onset compared to STM

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studies [START_REF] Chen | Probing Superexchange Interaction in Molecular Magnets by Spin-Flip Spectroscopy and Microscopy[END_REF] and impedes a discussion on the effective MSC length and exact spin excitations processes. This complicated bias drop, and the likely presence of magnetic disorder across the molecular layer (see hereafter), could also account for the different amplitudes in the dI/dV peaks for V>0 and V<0, instead of an interpretation as a signature of spinpolarized transport [START_REF] Muenks | Correlation-driven transport asymmetries through coupled spins in a tunnel junction[END_REF] . For this reason, we refrain from

We observe that SpinFlipMR spectroscopically tracks dI/dV, both from R(H) and I(V) data (see Fig. 2g). The small voltage lag originates from spectroscopic averaging effects for current compared to conductance [START_REF] Bowen | Half-metallicity proven using fully spin-polarized tunnelling[END_REF] . To the best of our knowledge, this is the first observation of a MR signal that is driven to appear due to bias voltage, and whose amplitude tracks junction conductance. Given the above interpretation of dI/dV, this means that we have successfully measured MR between the two FM electrodes due to opening of spin-flip channels of transport across MSCs. The opposite signs of MRTop and SpinFlipMR are consistent with the change in sign of the spin polarization of the current that is expected [START_REF] Chen | Probing Superexchange Interaction in Molecular Magnets by Spin-Flip Spectroscopy and Microscopy[END_REF][START_REF] Muenks | Correlation-driven transport asymmetries through coupled spins in a tunnel junction[END_REF][START_REF] Ormaza | Controlled spin switching in a metallocene molecular junction[END_REF][START_REF] Loth | Controlling the state of quantum spins with electric currents[END_REF][START_REF] Barraud | Phthalocyanine based molecular spintronic devices[END_REF] due to the spin-flip process across the MSC with S=1/2 spin moments. Indeed, the spin angular momentum thus conferred to the MSC causes the transport electron to flip its spin. This is then analyzed using the fixed spin referential of the collecting FM electrode in the two magnetization orientations. As a complementary effect, flipping the orientation of the injecting FM electrode's magnetization reverses the spin referential of the transport electron prior to a possible spin flip event. Thus, spin-flip transport channels effectively increase the junction conductance, but with an opposite sign of spin polarization, which is detected through the sign change between MRTop and SpinFlipMR.

The modelling of the bias-dependent R(H) data reveals that the reduced anisotropy strength K t /M t of the top magnetic unit is ≈0.2T, i.e. over an order of magnitude stronger than that of a free layer (see Fig. 2h). We infer that it contains not only the top Co FM electrode, but also MSCs (see Fig. 1g). The weak, mostly bias-independent coupling term ECTop is then attributed to AF coupling between two MSCs belonging to the top two magnetic units along the nanotransport path.

With K c /M c ≈10-20T, the central magnetic unit is magnetically very hard at 17K, and is coupled by MHBott~1T to the bottom magnetic unit. With a very low, almost always constant K b /M b =5mT, we infer that the bottom magnetic unit corresponds to the sub-interface atoms of the lower Fe electrode. MHBott describes the FM coupling expected between these layers and the spinterface owing to magnetic hardening. Comparing Figs. 2g and2j, we see that the central unit's K c /M c tracks the spin excitations. We thus infer that the central magnetic unit's spin referential is set by the spinterface and the MSC that is coupled to it. Our model indicates that, for V<35mV, the molecular layer's anisotropy must be at least ≈10T in order not to witness a symmetric R(H).

Now that the nanotransport path's three magnetic units and couplings are identified and summarized in Fig. 1g, we examine how they are affected by spin excitations. While the R(H) loop for V=65mV (Fig. 2c) resembles that of V=80mV (Fig. 2a), reaching the dI/dV peak at 73mV results in a strongly distorted R(H) (Fig. 2b). To the best of our knowledge, this is the first evidence of how small bias changes can so drastically alter magnetotransport. This can be modelled only though a strong deviation in the angle of the central unit's anisotropy. We also witness strong deviations in the anisotropy strength/angle of the top and bottom magnetic units, while the AF coupling ECTop jumps by one order of magnitude. Meanwhile, the FM coupling MHBott jumps from 1.2T to 1.4T. We propose that spin excitations drive a dynamical magnetic coupling of the two otherwise disjointed MSCs. This would explain, for V>73mV, not only the increase in MHBott due to an effectively longer single MSC, but also how the top magnetic unit's anisotropy increases while its angle jumps and further increases in order to dynamically accommodate the geometrical requirements of the 90° super-exchange interaction between the two MSCs.

Overall, this strong impact on all three magnetic units at the bias voltage corresponding to maximum spin-flip conductance (see dI/dV peak at 73mV of Fig. 2g) can be interpreted as a mutual magnetic coupling between all three units that is induced by spin excitations. This constitutes direct evidence in magnetotransport of spintronic anisotropy, i.e. a change in magnetic anisotropy caused by a spin-polarized current [START_REF] Misiorny | Spintronic magnetic anisotropy[END_REF][START_REF] Katcko | Spin-driven electrical power generation at room temperature[END_REF] , due to spin excitations.

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In light of these observations, we now discuss the absence of any MR due to flipping the bottom FM layer for |V|≤35mV. Since SpinFlipMR does not abruptly reach a saturated value at the threshold bias voltage for the dI/dV increase, we conclude that the spin-flip transport channel is not altering the magnetic anisotropy of the bottom FM layer such that it can abruptly switch orientations. This is in line with the model's treatment of the bottom FM layer as a free layer subject to the strong FM coupling MHBott to the central magnetic unit's spinterface. We propose that, in the |V|≤35mV bias range, the MSC that is coupled to the spinterface impedes spin-polarized transport because it is in its AFM ground state and is structurally of high quality (high K c /M c ). In that case, only the spin flip channel can reveal MR due to flipping the bottom FM layer's magnetization. We presume that this doesn't occur at the top interface because of structural imperfections that are embodied by the top magnetic unit's much lower K t /M t . Increasing temperature causes the nanojunction magnetics, and thus magnetotransport, to strongly change, as illustrated by the R(H) loops at 100mV for 17 < T(K) < 80 of Fig. 3a-e. The increasing noise despite the R decrease suggests that thermal excitations are destabilizing magnetism along the nanotransport path. For 17 < T(K) < 55, the same flipping of the top and bottom magnetic units is observed for |H|≤2T (data at 55K not shown). For 60 ≤ T(K) ≤ 72, a symmetric R(H) is observed (see also Fig. 1f). For T≥80K, the R(H) loop has collapsed, and only low-field MR is observed until 100K. Thus, only a 5-8K temperature increase around 55K and 75K can promote remarkable changes in the R(H) loop shape.

Modelling these R(H) loops (black lines of Fig. 3a-e and Fig. 3f) reveals that, while ECTop remains mostly constant, weak and negative (i.e. AFM coupling, see Fig. 3g), MHBott decreases only moderately up to 65K, and then abruptly decreases for T>65K (Fig. 3h). We thus peg a E MH =5.9±0.3meV threshold energy for the magnetic hardening effect at VSubmit Save trees and the air, please don't print! bowen@unistra.fr 8

the bcc Fe(110)/CoPc interface, which constitutes a first experimental energetic benchmark toward encoding information using the spinterface [START_REF] Cinchetti | Activating the molecular spinterface[END_REF] . For thermal fluctuations above this energy threshold, the spinterface and bottom FM electrode can no longer rotate independently, which explains the collapse for T>72K of the R(H) loop (Fig. 4e).

According to the model, the central unit's anisotropy K c /M c decreases steadily from ≈18T at 17K to ≈4.4T at 65K, and then to under 3T for T>70K (Fig. 4i). This anisotropy term is thus dominated by a MSC contribution, but also comprises a contribution due to magnetic hardening, in line with the model's description of the molecular layer (see Fig. 1a). Both this term and MHBott, both cause SpinFlipMR and MRTop to concurrently decrease with increasing T, and to disappear at T=100K, thereby mimicking the effective exchange bias effect seen in magnetometry data [START_REF] Gruber | Exchange bias and room-temperature magnetic order in molecular layers[END_REF][START_REF] Boukari | Disentangling Magnetic Hardening and Molecular Spin Chain Contributions to Exchange Bias in Ferromagnet/Molecule Bilayers[END_REF] .

To conclude, our work articulates the concepts of spin-flip spectroscopy, magnetic hardening and exchange bias at ferromagnetic metal/molecule interfaces (so-called 'spinterfaces'), and spintronic anisotropy within magnetotransport across solid-state nanojunction devices. By using FM electrodes with a fixed spin referential for transport, we isolated the magnetoresistance contribution arising from spin flip excitations along molecular spin chains. In the process, we identified three coupled magnetic units along the spintronic nanotransport path, and examined how bias voltage and temperature drive the nanopath's magnetism, thanks to a phenomenological macrospin model of transport. This also showed how spin excitations alter the nanopath's spintronic anisotropy.

Temperature-dependent studies reveal the energy threshold for magnetic decoupling of the spinterface, and the complementary impact on magnetotransport of both the MSC and of the magnetic hardening effect, beyond magnetometry data on FM/molecule bilayers [START_REF] Gruber | Exchange bias and room-temperature magnetic order in molecular layers[END_REF][START_REF] Boukari | Disentangling Magnetic Hardening and Molecular Spin Chain Contributions to Exchange Bias in Ferromagnet/Molecule Bilayers[END_REF] . This establishes an experimental benchmark into the energetics of encoding information using the spinterface [START_REF] Cinchetti | Activating the molecular spinterface[END_REF] . This nanotechnological progress is enabled by an innovative, low-tech, solvent-and resist-free processing technique that works with entire heterostructure stacks and can be further rationalized (e.g. using nanobead positioning techniques) toward industrial applications of quantum physics using nano-objects within solid-state devices.

Methods

Heterostructure stacks were grown in-situ and at room temperature in an ultra-high vacuum multichamber cluster by dc sputtering (metals) and thermal evaporation (CoPc). The SiO x substrate was annealed at 110°C and allowed to cool down prior to deposition. After nanojunction processing (see main text), the junctions were wirebonded to a sample chip and inserted onto a cryo-free magnetotransport bench. Measurements were performed in 4-point mode with (-) contacts on the lower electrode. In the main text, the junction stack was SiO x //Cr(5)/Fe(50)/CoPc(20)/Co(10)/Cr(5) (all numbers in nm). In addition to that junction, eight CoPc junctions with both top and bottom Fe electrodes showed MR for T<100K, fived exhibited spin-flip behavior, and MR tracking of junction conductance was observed on two junctions.

R(H=-2T

) between panels reflects a minor evolution of junction resistance during the ~100-hour measurement and across several field-cooling sequences. The fitting parameters in panel b were:

K t /M t =0.28T,  Mt =70º, K c /M c =21T,  Mc =-1º,
ECTop=-0.05T, MHBott=1.49T, R 0 =7.8G, MRTop=-1.4% and SpinFlipMR=4.8%. Small differences between these parameters and those found in the main text reflect minor evolutions in junction magnetotransport during the measurement run.

To test the conditions for, and the symmetry of, magnetic anisotropies along the nanotransport path, we first examined cooldown conditions. We present in Suppl. Fig. S4a R(H) data at 25K and 100mV mV after cooling the junction from 120K while maintaining  H =0. The red R(H) data is obtained after cooling in H=-1T. If, after applying H=2T at 120K, cooldown is instead performed at H=0+, this causes the R(H) trace to flip about the H=0 axis. This also occurs if, while at 17K, the inplane angle of the external magnetic field H,  H , is switched from 0° to 180° (Suppl. Fig. S4b). Cooling down with H=1T applied at an in-plane angle of 90° instead of 0° did not change the R(H) loop (not shown). The angular dependence of MRTop, shown in Fig. S4c, reveals a unidirectional behavior both without (V=-50mV; see Fig. 2g)) and with (V=100mV) the spin-flip conductance contribution. Due to the additional SpinFlipMR term in the latter case, and given our |H|<2T measurement window, it isn't possible to follow the angular dependence of MRTop for 30 <  H < 135 because there is no longer a R plateau for 1.7 < H(T) < 2. Note that this, and the fact that the 1T resistance level becomes undetermined at intermediate angles, are why it is not possible to study the angular dependence of SpinFlipMR. Generally, we conclude from these data that it is the orientation of magnetization of the FM electrodes during cooldown that determines a unidirectional anisotropy along the nanotransport path. To be complete, we show in Fig. S5 experimental and fitted R(H) data at (17K,100mV) for salient values of the in-plane angle  H of the applied magnetic field. For deviations from the field-cooling angle  H =0, the fitted R(H) obtained by simply adjusting the in-plane angle of H while conserving the same parameters only qualitatively reproduce features of the experimental R(H), whereas an exact fit is obtained when the fitting parameters found for  H =0 º are used for  H =180º (see also Suppl. Fig. S4b). This might reflect limitations of our model's macrospin assumption, or the presence of higher-order terms within a more complicated model. Nevertheless, our model captures all essential magnetotransport features (see Figures of main text).

Figure 1 :

 1 Figure 1: Organic nanojunctions: processing and magnetics. (a-d) Processing steps to transform entire heterostructure stacks into vertical nanopillars with dielectrically separated bottom and top contacts, without using resists/solvents. See text for details. (e) Schematic of the Fe/CoPc/Co junction's effective nanotransport path across three magnetic units (green boxes): a bottom Fe layer, the junction's lower interface forming a spinterface that is coupled to a MSC, and a top Co layer that is coupled to a MSC without a spinterface due to interdiffusion at the top interface. Not all molecular layers are shown (semi-transparent zone). The macrospin model's coupling terms ECTop and MHBott between the central and outer units are shown. See text for details. (f) Major and minor R(H) loops at T=60K and V=58mV reveal three R levels due to the magnetization reversal of the three magnetic units, which are schematized by green arrows. The fitting parameters were: K t /M t =0.26T,  Mt =60º, K c /M c =4.95T,  Mc =-4º, ECTop=-0.05T, MHBott=1.28T, R 0 =6.68G, MRTop=-2.4% and SpinFlipMR=5.1%.

Figure 2 :

 2 Figure 2: Current-induced alterations to magnetotransport. (a-e) R(H) loops at 17K for 20 < V(mV) < 80 and associated fits, also represented in panel f for ±20T. (g) Bias dependence of dI/dV, and of MRTop / SpinFlipMR inferred from I(V) and R(H) data. Bias dependencies of the strength and angle of the reduced anisotropy K/M for the (h-i) top and (j-k) central magnetic units. Bias dependencies of the (l) MH Bott and (m) EC Top coupling terms. The SpinFlipMR term tracks the dI/dV increase due to spin excitations. At the dI/dV peak, coupling between the MSCs of the top and central unit causes strong changes to their anisotropy parameters (panels h-k) and to the coupling parameters (panels l-m), which distorts the R(H) at 73mV (panel b). In the absence of the spin-flip conduction channel, flipping the bottom FM magnetization does not induce any MR (panel e) because the spinterface-stabilized MSC's AF ground state degrades spin-conserved transport. The error bars for data in panels h-m are discussed inSuppl. Note 1.

Figure 3 :

 3 Figure 3: Temperature weakens spin chain coupling and magnetic hardening. (a-e) R(H) loops at 100mV for 17 < T(K) < 80. Data are in blue, while the modelled fits are in black and are also shown in panel f for ±20T. Temperature dependencies of (g) ECTop, (h) MHBott, (i) K c /M c of the central magnetic unit and (j) of the SpinFlipMR and MRTop. The gray data in panel i for T≥80K are the minimum values required to obtain R(-2T)≠R(2T). The error bars for data in panels g-i are discussed in Suppl. Note 1.

Suppl. Fig. S5 :

 S5 Angle-dependent R(H): experimental & fitted data.Parameters are the same as those in Suppl. Fig.S4b.

Only at the (V=80mV,17K) and (V=100mV, 72K) critical points does K b /M b strongly increase, as expected in an exchange bias system near criticality

Acknowledgements

We thank Y. Henry for stimulating discussions, M. Gruber for reading our manuscript, S. Dornier for engineering assistance, the STNano technological platform staff for technical assistance with certain processing steps, and the remaining members of the IPCMS machine shop for support. We acknowledge financial support from the Institut Carnot MICA (project 'Spinterface'), from the Region Grand Est and Synchrotron SOLEIL, from CEFIPRA grant 5604-3, from the ANR (ANR-06-NANO-033-01, ANR-09-JCJC-0137, ANR-14-CE26-0009-01), the Labex NIE "Symmix" (ANR-11-LABX-0058 NIE), the EC Sixth Framework Program (NMP3-CT-2006-033370), the Contrat de Plan Etat-Region grants in 2006 and 2008, by the Region Grand Est 'NanoteraH' project supported by the EU through the European Regional Development Fund program, by the impact project LUE-N4S part of the French PIA project "Lorraine Université d'Excellence", reference ANR-15IDEX-04-LUE and by the « FEDER-FSE Lorraine et Massif Vosges 2014-2020 ", a European Union Program.

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Magnetoresistance and spintronic anisotropy induced by spin excitations along molecular spin chains

This Note details the procedure that was used in order to fit the R(H) data at 17K versus applied bias voltage, and at 100mV versus temperature. The macrospin 3-unit model contains several parameters : two for each magnetic unit (Top unit=t, Central unit=m, Bottom unit=b; anisotropy strength K/M and angle ), the two coupling terms ECTop and MHBott between the central and outer magnetic units, and two MR coefficients. MRTop is associated with flipping the magnetization of the top and central layers (called 'MR Top/Central in what follows), and SpinFlipMR with that of the top and bottom layers (called 'MR Top/Bottom' in what follows). This association proceeds from the observation of the R(H) curves at 17K (main Fig. 2) and 60K (main Fig. 1) under the assumption of the sequential magnetization reversal of the three magnetic units. Indeed, referring to Fig. S1, three MR terms may potentially define the effective R(H) data. Each MR term has a specific H-dependent signature (see Fig. S1a). We are able to fit our data using only MR Top/Central and MR Top/Bottom terms. As schematized in Fig. S1b, the (17K, 20mV) data can be fitted using only a MR Top/Central trace (compare with Fig. 2e). Furthermore, we experimentally observe that the MR upon reversing the top unit is made to eventually switch sign as the spin excitation conductance channel is opened with increasing bias. This is also associated 2 with a MR term upon flipping the bottom magnetic unit's magnetization. These two aspects are schematized in Fig. S1b for the case of (17K, 100mV; see Fig. 2a for a similar experimental trace at 80mV): the R(H) data can be reproduced by adding to the MR Top/Central trace a MR Top/Bottom trace.

Within the self-consistent description of the three magnetic units provided in the main text (see Fig. 1e), we constrained the fitting procedure by imposing 'free-layer' properties to the bottom magnetic unit: K b /M b =5mT. The external magnetic field at which the magnetization of the bottom magnetic unit reverses is then driven mostly by its coupling MHBott to the Central magnetic unit (see Fig. 2l). The shape of the reversal (sharpness of the onsets, slope of the reversal) is determined by the central layer's anisotropy strength K c /M c , while  m was almost always constant (see Fig. 2j-k). The top magnetic unit's K t /M t and  t were fitted in order to reproduce the minor loop due to the two magnetization reversals at low H. Here, care during fitting was taken to adhere to the 'squareness' of the loop, and to the evolution from saturation to remanence. The loop's bias shift was set through ECTop. This ability to compartmentalize the fitting parameters to segments of the R(H) trace constitutes an important measure of the confidence of the fitting procedure. In panels h-m of Fig. 2, the error bars show, with all other parameters constant, the parameter range within which the fit is still considered correct after visual inspection. We present in Suppl. Fig. S2 examples of the R(H,17K,115mV) data to showcase this aspect. This compartmentalization of our 3-macrospin model's parameters to R(H) features fails at the critical points (17K, 73mV; point 'A'; see Fig. 2b) and (17K, 80mV, point 'B'; see Fig. 2a) (i.e. on the dI/dV peak, see Fig. 2g), and at (72K ,100mV, point 'C'; see Fig. 3d) (i.e. when the magnetic hardening is overcome by thermal fluctuations). At points 'B' and 'C', it is impossible to maintain 'free-layer' properties for the bottom magnetic unit: K b /M b ~3000 and  b =13°. At points 'A' and 'B',  m ≈11°. Note also the deviations in ECTop and  t at point 'A', which are associated with a jump in MHBott to a higher, nearly constant value. These mutual couplings of otherwise independent model parameters indicate that the three magnetic units become dynamically coupled at these critical points due to changes induced by the spin-polarized current in the spin-flip conductance channel's fully spin-polarized current, i.e. by spintronic anisotropy 2 (points 'A' and 'B') and by thermal fluctuations (point 'C'). To fit the R(H) data, the two MRTop and SpinFlipMR terms are used. However, the R(H) traces do not necessarily guarantee full magnetization reversal within the |H|<2T experimental window, if at all. Fitting therefore proceeded starting with high bias voltage data (see e.g. (17K, 115mV) data in Suppl. Fig. S2), for which the resistance is mostly flat not only at large H<0, but also for 1.7 < H(T) < 2, under the assumption that these plateaus imply full magnetization reversal. As V or T was changed, the evolution of the plateau at 1.7 < H < 2 was monitored, and the MRTop and SpinFlipMR values adjusted accordingly to mimick the R(H) slope in this H range. Suppl.