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 threshold1,2,3, in a steady-state internal angle in SOT-driven DWs4,5, or in the topological deflection of skyrmions6,7,8,9. 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 spintronics10. 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-lattice11. 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 stacks12. Additionally, their magnetic properties can be controlled by changing either their composition or temperature, as described by the mean-field theory11. For a given composition, they can exhibit two characteristic temperatures: the angular momentum compensation temperature (TAC) and the magnetic compensation temperature (TMC), for which the net angular momentum or the net magnetisation (MS) are respectively zero (Fig. 1b). Interestingly, due to the different Landé factors of RE and TM, these two temperatures are different (with TMC < TAC for GdFeCo). At TMC, the magnetostatic response vanishes (as observed in the divergence of the coercivity, anisotropy field, …). In contrast, at TAC the dynamics is affected. Although these effects are challenging to evidence, the singular and promising behaviour of RETM at TAC was observed in current-induced switching13, magnetic resonance14, and time-resolved laser pump-probe measurements15,16. In very recent reports the signature of the dynamics at TAC was assigned to a DW mobility peak, under field17,18, under current by spin–orbit torques (SOT)19,20,21,22, or by spin transfer torque23. 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 pinning24,25. 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 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.

Figure 1
figure 1

GdFeCo/Pt sample properties and SOT-driven DW propagation in tracks. (a) Sketch of the track containing a SOT-driven DW and of the magnetisation of the two sublattices (in red and blue) below TMC. The size of the arrows represents their relative magnitude. The grayscale corresponds to the domain Kerr contrast while the DW is depicted in white. The angle of the DW magnetisation is given by φ. (b) Measured net magnetisation MS (squares) of the virgin film and coercivity HC (dots) of the patterned track versus temperature T. The MS points were shifted by − 31 K to account for migration of Gd during patterning12. The continuous line is the result of the mean-field model (see suppl.). (c) Kerr images of a DW driven by 300 GA/m2, 25 ns current pulses in at temperature set-point of TSP = 300 K. (d) (J,Tsp) colour-plot of measured mobilities (black points). Stars mark the peak of mobility µ versus TSP (interpolated; see suppl.). The solid line is a quadratic fit of the maximum mobilities, with 345 K = TSP + k J2 (k = 1.3 10–4 K/(GA m−2)2).


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 DWs26,27, 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.85Co.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 patterning12 the MS values have changed. By measuring the TMC pre- and post-patterning, we corrected this MS temperature shift of − 31 K. A relatively low and temperature-dependent magnetisation has been measured as expected12. 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 TMC of the track, 312 K, was determined by measuring the coercivity divergence (Fig. 1b). It diverges at TMC 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 method28,29.

DW velocity measurements were performed using a Kerr microscope with a controlled temperature sample holder (at temperature set-point TSP). 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/Co4,30 and which rules out any significant spin-transfer torque25. The linearity of the DW displacement with 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 (Hx) collinear to the current31 (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/m2), as previously reported in similar alloys19,20. The DW mobility µ = v/J exhibits a peak that depends on the TSP and the current density J. Figure 1d shows measured mobilities in a (J,Tsp) colour-plot, and for each J the maximum µ is marked with a star. The coordinates of the maxima µ follow \(T - T_{SP} \propto J^{2}\) (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

$$ v/J \propto \cos \left( \varphi \right) $$

where \(\varphi\) is the angle of the internal DW magnetisation4. 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 \(\left| \varphi \right|\). In ferrimagnets, it is expected that the precession depends on temperature and vanishes at TAC 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 \(\left| \varphi \right|\), and it can be deduced that the temperature of the maxima is TAC (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 HY that reveals the internal magnetic dynamics of the DW. It provides both a qualitative and quantitative evaluation of \(\varphi\), 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 TSP with an applied in-plane field HY perpendicular to the current flow (see inset of Fig. 2a). Figure 2a shows the velocity v(TSP, HY) without field (µ0HY = 0) and with two opposite fields (µ0HY =  ± 90 mT) for positive and negative current density (J =  ± 360 GA/m2). Whatever the HY field, the DW moves along the current direction. Two crossing points, at TSP = 300 K and TSP = 328 K, are observed where v(TSP, + HY) = v(TSP, − HY). On the first crossing point, the velocity without field is the same as with field, v(300 K, 0) = v(300 K, ± HY), while on the second crossing point the velocity without field is larger than with field, v(328 K, 0) > v(328 K, ± HY). The crossing points are more readily distinguished by plotting Δv(TSP) ≡ v(TSP, + HY) − v(TSP, − HY), shown in Fig. 2b, and are the same for both current polarities.

Figure 2
figure 2

SOT-driven DW under HY: determination of the internal DW dynamics, TMC and TAC. (a) Measured DW velocity v versus sample holder set-point temperature TSP with µ0HY =  ± 90 mT or 0 mT, for J =  ± 360 GA/m2. (b) Velocity difference Δv(TSP) ≡ v(J, + HY) − v(J,-HY) from (a). (c) Diagram of the sublattice orientations in a SOT-driven DW under HY across compensation points. The red and blue arrows correspond to RE and TM, respectively. (d) Colour-plot of Δv/‹v›(J, TSP). Black points correspond to measurements. Grey circles correspond to crossing points where Δv = 0 (see (a)). The black lines are parabolic fits of these crossing points (T = TSP + k J2, k = 0.8 10–4 K/(GA/m2)2, TMC = 312 K, TAC = 334 K). Above, the action of the SOT on the internal DW angle \(\varphi_{J}\) is sketched, to illustrate the sense of rotation of \(\varphi_{J}\) without field. Throughout, pink and green backgrounds mark the sign of Δv.

To understand the effect of HY on SOT-driven DWs in ferrimagnets, we first consider the well-understood ferromagnetic case. The HY couples to the internal magnetisation of the DW and changes the equilibrium φ. As v/J cos(φ) (Eq. 1), if + HY rotates φ closer to Néel configuration, it will increase the velocity. The sign of Δv shows whether + HY rotates φ closer to or farther from the Néel configuration compared to − HY. In particular, a positive Δv means that + HY and J have opposite contributions to φ (and Δv < 0 means + HY and J push φ in the same direction). Since the sign of the effect of HY is known, we can deduce the sign of the φ angle without field, that we note φJ.

In the RETM ferrimagnetic case, the DW velocity can still be described with the same model32. Since the spin current interacts mainly with the TM sub-lattice (12and references therein), φ in Eq. (1) corresponds to the DW angle of the TM sub-lattice (see Fig. 1a). The Zeeman contribution of HY depends now on the net magnetisation MS = MTMMRE, which changes sign at TMC.

Figure 2c shows a sketch of the in-plane magnetisation of the RE and TM sublattices at the centre of the SOT-driven DW at different temperatures. At T < TMC, the RE sublattice is dominant (MTM < MRE) and + HY rotates φ clockwise. At T = TMC, MRE = MTM and HY does not affect φ nor v, and v(HY = 0) = v(± HY). Above TMC, MTM > MRE and the effect of external fields is reversed: + HY rotates φ counterclockwise.

In Fig. 2b, Δv < 0 below TSP = 300 K, so we conclude that the current acts on φ in the same direction as + HY, i.e. φJ < 0. We observe that TMC occurs at TSP = 300 K, as v(HY = 0) = v(± HY). At this point, it is not possible to determine the φJ. Above TMC, interestingly, the measured Δv crosses zero once more (TSP = 328 K). Below this point, Δv > 0 so φJ < 0, and above it Δv < 0 so φJ > 0. At the crossing point, the current does not affect φ: φJ = 0. The fact that the velocity with ± HY are smaller than without field confirms the symmetrical configuration shown in Fig. 2c with φJ = 0 (see suppl. mat. for other values of HY). The observed reversal of the direction of the precession, and the precession-free point, is characteristic of the angular compensation, TAC.

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 TSP and J, normalized by the average velocity with + HY and -HY. This normalization removes the first-order dependence on |J| of Eq. (1) (\(v \propto \cdot J \cdot \cos \varphi \left( J \right)\)), as \({\Delta }v/v = 2\frac{{\cos \left( {\varphi_{{ + H_{Y} }} } \right) - \cos \left( {\varphi_{{ - H_{Y} }} } \right)}}{{\cos \left( {\varphi_{{ + H_{Y} }} } \right) + \cos \left( {\varphi_{{ - H_{Y} }} } \right)}}\) 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 TMC = 312 K and TAC = 334 K. These observations hold for different magnitudes of HY (see suppl.). Also, spurious external fields have low impact on the crossing points (see calculations in suppl.). Note that TMC and TAC are consistent with the previous measurements of HC(T) in Fig. 1b (TMC = 312 K) and µ(J) in Fig. 1d (TAC = 345 K). Furthermore, the measured values of Δv are large (few hundreds of m/s), give directly the sense of precession of the magnetisation and show that the angle \(\varphi\) of the moving DW changes sign and vanishes at TAC.


In ferromagnets, the angle φ can be described with the 1D model4, extended to include external magnetic fields:

$$ \varphi = \arctan \left( {\frac{\Delta }{D}\left( {\frac{{ \hbar \theta_{SHE} }}{2\,e t}\frac{ J}{\alpha } + \mu_{0} {\text{M}}_{s} {\text{H}}_{y} } \right)} \right) $$

where \(\Delta\) is the DW width parameter, D is the DMI parameter, \(\alpha\) is the Gilbert damping parameter, ħ is the reduced Planck constant, e is the fundamental charge, θSHE is the spin Hall angle of the Pt layer, and t is the magnetic film thickness. For a ferrimagnet, \(\varphi\) corresponds to the DW angle of the TM sub-lattice (see Fig. 1a), MS to the net magnetisation, and α is the effective Gilbert damping parameter. The observed reversal and vanishing of the precession dynamics (φJ = 0) at TAC 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 TAC. 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 TAC. This effective parameter approach was successfully used to describe ferrimagnetic dynamics observations15,16,20,34.

Figure 3a,b show analytical calculations of the DW angle φ and related velocity v as a function of T and HY. All material parameters were taken from measurements (see Fig. 1b,c and suppl.), except for effective damping parameter α(T) which was approximated by \(\propto\) 1/(T − TAC) to account the expected divergence at TAC. All other quantities (Δ, D, θSHE) are taken as constant in the narrow investigated range. We observe an excellent agreement between the theoretical curves and the experimental data in Fig. 2a. We can also verify the explanation given above (Fig. 2c): at TMC, the φ and v are the same for all HY and, at TAC, φ are opposite for + HY and − HY.

Figure 3
figure 3

taken from measurements (see Fig. 1b,c and suppl.). µ0HY =  ± 90 mT, and J =  ± 360 GA/m2. (c) Experimental Δv/‹v› for different current densities. The thin lines correspond to fits using a simplified version of Eq. (2) (see text). The data were horizontally shifted so that their first crossing points are juxtaposed at TMC = 312 K. (d) Obtained φJ from the fits in (c). The thick lines and envelopes in (c) and (d) correspond to the theoretical curves, obtained with the same parameters of (a) and (b).

Determination of the internal DW angle. (a, b) Calculated φ and v according to Eqs. (1), (2), using the material parameters

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 (TMC = 312 K), we shifted all the curves in Fig. 3c so the crossing points overlap at TMC. Note that, for a given J, Δv/‹v› is only a function of φ(+ Hy) and φ(− Hy), with no other parameters28. We use the approximation of \(\alpha \propto 1/\left( {T - T_{AC} } \right)\) and \(M_{S} \propto T - T_{MC}\) to get a simplified version of Eq. (2): \(\varphi \left( {T, \pm H_{Y} } \right) = \arctan \left( {Ja\left( {T - T_{a} } \right) \pm H_{Y} b\left( {T - T_{b} } \right)} \right)\), 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 TAC, as the effective damping does, and the second term should change sign at TMC, like MS. The fitting indeed gives Ta = 336 ± 3 K ≈ TAC, Tb = 312 ± 2 K ≈ TMC (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 φJ in Fig. 3d, obtained directly from a and Ta. Without field, the angle φJ 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 φJ (Fig. 3d) are in the envelope that corresponds to the theoretical curve for J between 150 and 450 GA/m2.

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 Hy. This bias field changes the internal angle of the magnetisation of the DW (φJ) that affects its velocity (v ~ cos φJ). By analysing the sign of the velocity difference with + Hy and—Hyv), it is possible to determine the sign of φJ. We found that there are two temperatures for which Δv = 0 and we showed that they correspond to the TMC and TAC. These measurements also reveal the vanishing of the tilt of the magnetisation at TAC (φJ = 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 TMC and TAC 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 reports19,20,21. 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 angular-momentum-compensated materials are also interesting for skyrmion dynamics, and it has been shown that it leads to efficient manipulations35, and vanishing topological deflection22.

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 in12. 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 HDL was determined with the harmonic voltage method28,29. MS(T) was measured by SQUID magnetometry. The magnitude of DMI equivalent field HDMI was determined by analysing the SOT-driven DW velocity with a field collinear to the current as in31.

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.

Analytical model of DW velocity under SOT and field

Equations (1) and (2) and the theoretical plots in Figs. 2 and 3, were done using the 1D model described in4 in the steady-state regime (\(\dot{\varphi } = 0\)), extended to include external magnetic fields and neglecting the in-plane demagnetisation field:

$$ \left\{ {\begin{array}{ll} {\frac{\alpha v}{\Delta } = \gamma_{0} \left( {H_{Z} + \frac{\pi }{2}H_{SHE} \cos \varphi } \right)} \\ {\frac{v}{\Delta } = \gamma_{0} \frac{\pi }{2}\left( {\left( {H_{DMI} + H_{X} } \right)\sin \varphi + H_{Y} \cos \varphi } \right)} \\ \end{array} } \right. $$

with \(H_{SHE} = \frac{\hbar }{2e}\frac{{\theta_{SHE} }}{{\mu_{0} M_{S} t}}J,\;H_{DMI} = \frac{D}{{\Delta \mu_{0} M_{S} }}\). In the absence of HZ, this yields \(v = \frac{{\gamma_{0} \Delta }}{\alpha } \frac{\pi }{2}H_{SHE} \cos \varphi\), \(\varphi = \tan \left( {\frac{{H_{SHE} /\alpha + H_{Y} }}{{H_{DMI} + H_{X} }}} \right)\). These equations can be used for ferrimagnets using the effective parameters15,16,33 as described above. The calculated plots in Fig. 3 are obtained using a constant ratio D/Δ obtained from the determination of HDMI \(\left( {\frac{D}{\Delta } = \mu_{0} M_{S} \left( T \right)H_{DMI} \left( T \right) = 2 \;{\text{kJ}}/{\text{m}}^{3} } \right)\), and the SOT factor \(\frac{\hbar }{2e}\frac{{\theta_{SHE} }}{t}\) from the determination of HDL ( θSHE/(2 e t) = µ0 MS(T) HDL(T)/J = 4.0 J·m−3/(GA·m−2)). The only parameter that is not experimentally determined, α(T), is approximated by an inverse linear law α(T) = 13.0 K/(T − TAC), chosen to best reproduce the shape of the experimental curves (see Figs. 2a and 3b). See supplementary materials for more results.