Measurement of the tilt of a moving domain wall shows precession-free dynamics in compensated ferrimagnets

One fundamental obstacle to efficient ferromagnetic spintronics is magnetic precession, which intrinsically limits the dynamics of magnetic textures. We experimentally demonstrate that this precession vanishes when the net angular momentum is compensated in domain walls driven by spin–orbit torque in a ferrimagnetic GdFeCo/Pt track. We use transverse in-plane fields to provide a robust and parameter-free measurement of the domain wall internal magnetisation angle, demonstrating that, at the angular compensation, the DW tilt is zero, and thus the magnetic precession that caused it is suppressed. Our results highlight the mechanism of faster and more efficient dynamics in materials with multiple spin lattices and vanishing net angular momentum, promising for high-speed, low-power spintronic applications.

In magnetic materials, the exchange interaction aligns the magnetic moments producing ferromagnetic or antiferromagnetic orders. Even if ferromagnets have numerous applications in spintronics, two effects limit the development of higher-density and faster devices. Firstly, the stray fields couple adjacent magnetic textures and limit their density. Secondly, the magnetic precession changes the internal magnetisation of moving textures, resulting in e.g. a continuous precession in field-or spin-transfer-torque-driven DWs above Walker threshold 1-3 , in a steady-state internal angle in SOT-driven DWs 4,5 , or in the topological deflection of skyrmions [6][7][8][9] . All these effects limit the texture's velocity. Antiferromagnetic order leads to faster dynamics and robustness against spurious fields, and is emerging as a new paradigm for spintronics 10 . However, perfect antiferromagnets with exactly compensated magnetic sub-lattices are hard to probe and manipulate, and therefore have been rarely studied or used in applications. Rare Earth-Transition Metal (RETM) ferrimagnetic alloys allow to benefit from both antiferromagnetic-like dynamics and ferromagnetic-like spintronic properties. Indeed, they have two antiferromagnetically-coupled sublattices, corresponding roughly to the RE and TM moments, and their spin transport is carried mainly by the TM sub-lattice 11 . Furthermore, RETM thin films can exhibit perpendicular magnetic anisotropy, are conductors, and present large spin transport polarization and spin torques, even when integrated in complex stacks 12 . Additionally, their magnetic properties can be controlled by changing either their composition or temperature, as described by the mean-field theory 11 . For a given composition, they can exhibit two characteristic temperatures: the angular momentum compensation temperature (T AC ) and the magnetic compensation temperature (T MC ), for which the net angular momentum or the net magnetisation (M S ) are respectively zero (Fig. 1b). Interestingly, due to the different Landé factors of RE and TM, these two temperatures are different (with T MC < T AC for GdFeCo). At T MC , the magnetostatic response vanishes (as observed in the divergence of the coercivity, anisotropy field, …). In contrast, at T AC the dynamics is affected. Although these effects are challenging to evidence, the singular and promising behaviour of RETM at T AC was observed in current-induced switching 13 , magnetic resonance 14 , and time-resolved laser pump-probe measurements 15,16 . In very recent reports the signature of the dynamics at T AC was assigned to a DW mobility peak, under field 17,18 , under current by spin-orbit torques (SOT) [19][20][21][22] , or by spin transfer torque 23 . However, even if this mobility peak is a signature of angular compensation, it is affected by the strong sensitivity of DW propagation to Joule heating and pinning 24,25 . Furthermore, none of the latter experiments gives a direct access to the internal DW magnetisation angle that is an intrinsic signature of the magnetisation precession. In this paper, we use a robust www.nature.com/scientificreports/ measurement of the variation of the DW velocity with a transverse bias field to determine the DW internal magnetisation angle across the compensation temperatures, and we show that there is no DW magnetisation tilt, and therefore no magnetic precession, at the angular moment compensation.

Results
DWs driven by SOT have been observed in thin ferrimagnetic films with a heavy-metal adjacent layer, like Pt, which induces three main interfacial effects: perpendicular anisotropy, Dzyaloshinskii-Moriya exchange interaction (DMI), and vertical spin current generated by the spin Hall effect (SHE) (Fig. 1a). Such systems present chiral Néel DWs 26,27 , which is the configuration for which the SOT DW driving is most effective (Fig. 1a) 4 .
To investigate SOT-driven DW dynamics in RETM, a 10 µm-wide track of amorphous Gd .4 (Fe .85 Co .15 ) 0.6 (5 nm) capped with Pt (7 nm) with perpendicular magnetic anisotropy was fabricated (Fig. 1a). Ms(T) was measured by SQUID magnetometry (Fig. 1b). Due to the migration of Gd during patterning 12 the M S values have changed. By measuring the T MC pre-and post-patterning, we corrected this M S temperature shift of − 31 K. A relatively low and temperature-dependent magnetisation has been measured as expected 12 . The magnetic compensation temperature where the net magnetisation vanishes is clearly visible. Transport measurements of the extraordinary Hall effect (EHE) versus field were made on 5 μm wide crosses at different temperatures for both in-plane and out-of-plane magnetic fields. The T MC of the track, 312 K, was determined by measuring the coercivity divergence (Fig. 1b). It diverges at T MC as the applied field produces opposite and balanced effects on the two compensated sub-networks. The magnitude of the SOT was determined with the second harmonic Hall voltage method 28,29 .
DW velocity measurements were performed using a Kerr microscope with a controlled temperature sample holder (at temperature set-point T SP ). 25 ns pulses of current density J were applied in the track containing a DW. After each pulse, a Kerr image is recorded. The DWs move against the electron flow, which is compatible with SOT-driving of chiral Néel DWs with the same relative sign of DMI and SHE as the one found in ferromagnetic Pt/Co 4,30 and which rules out any significant spin-transfer torque 25 . The linearity of the DW displacement with www.nature.com/scientificreports/ the pulse number and duration (see suppl.) allows the robust determination of the propagation velocity v. The magnitude of DMI was determined by analysing the DW velocity driven by electrical current under an in-plane field (H x ) collinear to the current 31 (see suppl.). Since in perpendicularly-magnetised tracks SOT induced propagation depends on the DW in-plane magnetisation, the reversal of the DW propagation induced by the in-plane field also validates the SOT-driven mechanism. High DW velocities (> 700 m/s; see velocity curves in suppl.) are observed for low J (~ 600 GA/m 2 ), as previously reported in similar alloys 19,20 . The DW mobility µ = v/J exhibits a peak that depends on the T SP and the current density J. Figure 1d shows measured mobilities in a (J,T sp ) colour-plot, and for each J the maximum µ is marked with a star. The coordinates of the maxima µ follow T − T SP ∝ J 2 (solid line in Fig. 1c), which suggests that they all occur at a single track temperature T = 345 K. In ferromagnets, models predict that the SOT-driven DW steady-state velocity follows where ϕ is the angle of the internal DW magnetisation 4 . The angle φ is determined by the balance between DMI, which favours the Néel configuration (φ = 0), and the precession induced by SOT, which increases |ϕ| . In ferrimagnets, it is expected that the precession depends on temperature and vanishes at T AC with a peak in velocity (See suppl.). If the effects of pinning and Joule heating are neglected, it is possible to attribute the observed mobility peak with minimal |ϕ| , and it can be deduced that the temperature of the maxima is T AC (345 K according to the fit in Fig. 1c), as previously done in Refs. 19,20 .
In order to overcome these limitations, we propose a new method based on the application of a transverse field H Y that reveals the internal magnetic dynamics of the DW. It provides both a qualitative and quantitative evaluation of ϕ , including its sign, across both compensation points, without requiring any additional sample magnetic parameters. Simultaneously, it determines the Joule heating amplitude.
We measured the DW velocity v versus T SP with an applied in-plane field H Y perpendicular to the current flow (see inset of Fig. 2a). Figure 2a shows the velocity v(T SP , H Y ) without field (µ 0 H Y = 0) and with two opposite fields (µ 0 H Y = ± 90 mT) for positive and negative current density (J = ± 360 GA/m 2 ). Whatever the H Y field, the DW moves along the current direction. Two crossing points, at T SP = 300 K and On the first crossing point, the velocity without field is the same as with field, v(300 K, 0) = v(300 K, ± H Y ), while on the second crossing point the velocity without field is larger than with field, v(328 K, 0) > v(328 K, ± H Y ). The crossing points are more readily distinguished by plotting Δv( Fig. 2b, and are the same for both current polarities.
To understand the effect of H Y on SOT-driven DWs in ferrimagnets, we first consider the well-understood ferromagnetic case. The H Y couples to the internal magnetisation of the DW and changes the equilibrium φ.
As v/J ∝ cos(φ) (Eq. 1), if + H Y rotates φ closer to Néel configuration, it will increase the velocity. The sign of Δv shows whether + H Y rotates φ closer to or farther from the Néel configuration compared to − H Y . In particular, a positive Δv means that + H Y and J have opposite contributions to φ (and Δv < 0 means + H Y and J push φ in the same direction). Since the sign of the effect of H Y is known, we can deduce the sign of the φ angle without field, that we note φ J .
In the RETM ferrimagnetic case, the DW velocity can still be described with the same model 32 . Since the spin current interacts mainly with the TM sub-lattice ( 12 and references therein), φ in Eq. (1) corresponds to the DW angle of the TM sub-lattice (see Fig. 1a In Fig. 2b, Δv < 0 below T SP = 300 K, so we conclude that the current acts on φ in the same direction as + H Y , i.e. φ J < 0. We observe that T MC occurs at T SP = 300 K, as v(H Y = 0) = v(± H Y ). At this point, it is not possible to determine the φ J . Above T MC , interestingly, the measured Δv crosses zero once more (T SP = 328 K). Below this point, Δv > 0 so φ J < 0, and above it Δv < 0 so φ J > 0. At the crossing point, the current does not affect φ: φ J = 0. The fact that the velocity with ± H Y are smaller than without field confirms the symmetrical configuration shown in Fig. 2c with φ J = 0 (see suppl. mat. for other values of H Y ). The observed reversal of the direction of the precession, and the precession-free point, is characteristic of the angular compensation, T AC .
We measured this quantity for different current densities and the obtained behaviour is very similar. Figure 2d shows all measured Δv/‹v› in a colour-plot as a function of T SP and J, normalized by the average velocity with + H Y and -H Y . This normalization removes the first-order dependence on |J| of Eq. (1) ( v ∝ ·J · cos ϕ(J) ), as is independent of J, enabling the direct comparison of data for different current densities. Three regions can be observed with, successively, Δv < 0, Δv > 0 and Δv < 0, separated by the two sets of crossing points. These crossing points depend on J but their difference is independent of J (see data in suppl.). Indeed, both follow a Joule heating parabolic relation with same heating parameter (within 1%), which can be associated to the isothermal lines of T MC = 312 K and T AC = 334 K. These observations hold for different magni-  www.nature.com/scientificreports/

Discussion
In ferromagnets, the angle φ can be described with the 1D model 4 , extended to include external magnetic fields: where is the DW width parameter, D is the DMI parameter, α is the Gilbert damping parameter, ħ is the reduced Planck constant, e is the fundamental charge, θ SHE is the spin Hall angle of the Pt layer, and t is the magnetic film thickness. For a ferrimagnet, ϕ corresponds to the DW angle of the TM sub-lattice (see Fig. 1a), M S to the net magnetisation, and α is the effective Gilbert damping parameter. The observed reversal and vanishing of the precession dynamics (φ J = 0) at T AC is directly associated with a divergence and change of sign of this effective Gilbert damping parameter α(T) in Eq. (2), as described in Refs. 15,16,33 (Note that in Ref. 17 it is stated that the α parameter does not diverge at T AC . However, the authors refer to their new and distinct definition of α that is not the conventional Gilbert's parameter. Gilbert's α does diverge, as it is discussed briefly in Ref. 17 ). Note that, even if α diverges and changes sign, the dissipation power, which is proportional to the product of α and the net angular momentum, remains finite and positive even across T AC . This effective parameter approach was successfully used to describe ferrimagnetic dynamics observations 15,16,20,34 . Figure 3a,b show analytical calculations of the DW angle φ and related velocity v as a function of T and H Y . All material parameters were taken from measurements (see Fig. 1b,c and suppl.), except for effective damping parameter α(T) which was approximated by ∝ 1/(T − T AC ) to account the expected divergence at T AC . All other quantities (Δ, D, θ SHE ) are taken as constant in the narrow investigated range. We observe an excellent agreement www.nature.com/scientificreports/ between the theoretical curves and the experimental data in Fig. 2a. We can also verify the explanation given above (Fig. 2c): at T MC , the φ and v are the same for all H Y and, at T AC , φ are opposite for + H Y and − H Y . All Δv/‹v› are shown in the same graph versus T in Fig. 3c. Since we know that all the first crossing points occur at the same track temperature (T MC = 312 K), we shifted all the curves in Fig. 3c so the crossing points overlap at T MC . Note that, for a given J, Δv/‹v› is only a function of φ(+ H y ) and φ(− H y ), with no other parameters 28 . We use the approximation of α ∝ 1/(T − T AC ) and M S ∝ T − T MC to get a simplified version of Eq. (2): ϕ(T, ±H Y ) = arctan (Ja(T − T a ) ± H Y b(T − T b )) , that is used to fit the Δv/‹v› points for each current density (thin lines in Figs. 3c,d). The first term corresponds to the current contribution, and should change sign at T AC , as the effective damping does, and the second term should change sign at T MC , like M S . The fitting indeed gives T a = 336 ± 3 K ≈ T AC , T b = 312 ± 2 K ≈ T MC (and a = − (0.3 ± 0.1) 10 -3 (K·GAm −2 ) −1 and b = − 0.04 ± 0.01 (K·T) −1 ). We plot the temperature evolution of φ J in Fig. 3d, obtained directly from a and T a . Without field, the angle φ J follows the sketch of Fig. 2c, and is in agreement with the theoretical curve of Fig. 3a. All data Δv/‹v› (Fig. 3c) and the fitted φ J (Fig. 3d) are in the envelope that corresponds to the theoretical curve for J between 150 and 450 GA/m 2 .
In GdFeCo/Pt with Néel DWs and DMI, we measured how the velocity of an SOT-driven DW changes with an in-plane transverse bias field H y . This bias field changes the internal angle of the magnetisation of the DW (φ J ) that affects its velocity (v ~ cos φ J ). By analysing the sign of the velocity difference with + H y and-H y (Δv), it is possible to determine the sign of φ J . We found that there are two temperatures for which Δv = 0 and we showed that they correspond to the T MC and T AC . These measurements also reveal the vanishing of the tilt of the magnetisation at T AC (φ J = 0), which had been theoretically predicted but never directly observed. This novel approach determines precisely the magnitude and the sense of the DW tilt for a moving DW, which is a consequence of the magnetic precession of the spins through which the DW travels. This method gives T MC and T AC and is based on the intrinsic DW dynamics and so is unaffected by DW pinning. Finally, the velocity difference is easily observed (Δv ≈ 100 m/s), independent of the Joule heating and does not require knowledge of material parameters.
The suppression of magnetic precession opens new perspectives for fast and energy-efficient spintronics using any angular-momentum-compensated multi-lattice material. It induces a maximum of SOT-driven DW mobility in a compensated RETM ferrimagnet, as we observed in agreement with previous reports [19][20][21] . Here, for the first time, direct experimental evidence is provided that the SOT-driven DW propagation is tilt-free and the DW remains Néel (φ = 0) in angular momentum compensated ferrimagnets. These dynamics in angularmomentum-compensated materials are also interesting for skyrmion dynamics, and it has been shown that it leads to efficient manipulations 35 , and vanishing topological deflection 22 . www.nature.com/scientificreports/

Materials and methods
Sample deposition, fabrication. The film of amorphous GdFeCo(5 nm) capped with Pt(7 nm) was deposited by electron beam co-evaporation in ultrahigh vacuum on thermally-oxidised Si substrates. Details of the film growth and characterisation can be found in 12 . The tracks were patterned by e-beam lithography and hard-mask ion-beam etching.
characterisation of the magnetic properties. Transport measurements of the extraordinary Hall effect versus field were made on 5 μm crosses at different temperatures in a commercial QD PPMS. The magnitude of the SOT equivalent field H DL was determined with the harmonic voltage method 28,29 . M S (T) was measured by SQUID magnetometry. The magnitude of DMI equivalent field H DMI was determined by analysing the SOT-driven DW velocity with a field collinear to the current as in 31 .
Kerr microscopy. Kerr microscopy experiments were performed using an adapted commercial Schafer Kerr microscope, with a temperature regulated sample holder. The DW velocity was measured by taking Kerr images before and after each of about ten current pulses (see Fig. 1c). The linearity of the DW displacement with the number of pulses and with the pulse duration allowed a reliable determination of the propagation velocity v. (1) and (2)