Abstract
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 inplane fields to provide a robust and parameterfree 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 highspeed, lowpower spintronic applications.
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Introduction
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 higherdensity 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 spintransfertorquedriven DWs above Walker threshold^{1,2,3}, in a steadystate internal angle in SOTdriven 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 sublattices are hard to probe and manipulate, and therefore have been rarely studied or used in applications. Rare EarthTransition Metal (RETM) ferrimagnetic alloys allow to benefit from both antiferromagneticlike dynamics and ferromagneticlike spintronic properties. Indeed, they have two antiferromagneticallycoupled sublattices, corresponding roughly to the RE and TM moments, and their spin transport is carried mainly by the TM sublattice^{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 meanfield 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 currentinduced switching^{13}, magnetic resonance^{14}, and timeresolved laser pumpprobe 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 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 heavymetal adjacent layer, like Pt, which induces three main interfacial effects: perpendicular anisotropy, DzyaloshinskiiMoriya 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 SOTdriven DW dynamics in RETM, a 10 µmwide 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 postpatterning, we corrected this M_{S} temperature shift of − 31 K. A relatively low and temperaturedependent 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 inplane and outofplane 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 subnetworks. 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 setpoint 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 SOTdriving 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 spintransfer torque^{25}. 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 inplane field (H_{x}) collinear to the current^{31} (see suppl.). Since in perpendicularlymagnetised tracks SOT induced propagation depends on the DW inplane magnetisation, the reversal of the DW propagation induced by the inplane field also validates the SOTdriven 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}) colourplot, 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 SOTdriven DW steadystate velocity follows
where \(\varphi\) 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 \(\left \varphi \right\). 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 \(\left \varphi \right\), 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 \(\varphi\), 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 T_{SP} with an applied inplane 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 T_{SP} = 328 K, are observed where v(T_{SP}, + H_{Y}) = v(T_{SP}, − H_{Y}). 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(T_{SP}) ≡ v(T_{SP}, + H_{Y}) − v(T_{SP}, − H_{Y}), shown in Fig. 2b, and are the same for both current polarities.
To understand the effect of H_{Y} on SOTdriven DWs in ferrimagnets, we first consider the wellunderstood 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 sublattice (^{12}and references therein), φ in Eq. (1) corresponds to the DW angle of the TM sublattice (see Fig. 1a). The Zeeman contribution of H_{Y} depends now on the net magnetisation M_{S} = M_{TM} − M_{RE}, which changes sign at T_{MC}.
Figure 2c shows a sketch of the inplane magnetisation of the RE and TM sublattices at the centre of the SOTdriven DW at different temperatures. At T < T_{MC}, the RE sublattice is dominant (M_{TM} < M_{RE}) and + H_{Y} rotates φ clockwise. At T = T_{MC}, M_{RE} = M_{TM} and H_{Y} does not affect φ nor v, and v(H_{Y} = 0) = v(± H_{Y}). Above T_{MC}, M_{TM} > M_{RE} and the effect of external fields is reversed: + H_{Y} rotates φ counterclockwise.
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 precessionfree 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 colourplot as a function of T_{SP} and J, normalized by the average velocity with + H_{Y} and H_{Y}. This normalization removes the firstorder 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 T_{MC} = 312 K and T_{AC} = 334 K. These observations hold for different magnitudes of H_{Y} (see suppl.). Also, spurious external fields have low impact on the crossing points (see calculations in suppl.). Note that T_{MC} and T_{AC} are consistent with the previous measurements of H_{C}(T) in Fig. 1b (T_{MC} = 312 K) and µ(J) in Fig. 1d (T_{AC} = 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 T_{AC}.
Discussion
In ferromagnets, the angle φ can be described with the 1D model^{4}, extended to include external magnetic fields:
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, θ_{SHE} 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 sublattice (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 \(\propto\) 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 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 \(\alpha \propto 1/\left( {T  T_{AC} } \right)\) 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 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 SOTdriven DW changes with an inplane 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 energyefficient spintronics using any angularmomentumcompensated multilattice material. It induces a maximum of SOTdriven 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 SOTdriven DW propagation is tiltfree and the DW remains Néel (φ = 0) in angular momentum compensated ferrimagnets. These dynamics in angularmomentumcompensated materials are also interesting for skyrmion dynamics, and it has been shown that it leads to efficient manipulations^{35}, and vanishing topological deflection^{22}.
Materials and methods
Sample deposition, fabrication
The film of amorphous GdFeCo(5 nm) capped with Pt(7 nm) was deposited by electron beam coevaporation in ultrahigh vacuum on thermallyoxidised Si substrates. Details of the film growth and characterisation can be found in^{12}. The tracks were patterned by ebeam lithography and hardmask ionbeam 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 SOTdriven 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.
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 in^{4} in the steadystate regime (\(\dot{\varphi } = 0\)), extended to include external magnetic fields and neglecting the inplane demagnetisation field:
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 H_{Z}, 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 parameters^{15,16,33} as described above. The calculated plots in Fig. 3 are obtained using a constant ratio D/Δ obtained from the determination of H_{DMI} \(\left( {\frac{D}{\Delta } = \mu_{0} M_{S} \left( T \right)H_{DMI} \left( T \right) = 2 \;{\text{kJ}}/{\text{m}}^{3} } \right)\), and the SOT factor \(\frac{\hbar }{2e}\frac{{\theta_{SHE} }}{t}\) from the determination of H_{DL} (ℏ θ_{SHE}/(2 e t) = µ_{0} M_{S}(T) H_{DL}(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 − T_{AC}), chosen to best reproduce the shape of the experimental curves (see Figs. 2a and 3b). See supplementary materials for more results.
Data availability
Raw data related to this paper may be requested from the authors.
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Acknowledgements
We are very thankful to S. Rohart and A. Thiaville for fruitful discussions, and to R. Mattana for the SQUID measurements of M_{S}. S. K. and E. H. acknowledge public grant overseen by the ANR as part of the “Investissements d’Avenir” programme (Labex NanoSaclay, reference: ANR10LABX0035) for the FEMINIST project and travelling grants. S. K. acknowledges funding by public grant overseen by the ANR (PIAF ANR17CE09003003). The transport measurements were supported by Université ParisSud Grant MRM PMP.
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E.H., J.S. and A.M. designed the experiment. E.H. and R.W. prepared the samples. E.H., S.K., L.B. performed the measurements. All authors analysed the data and prepared the manuscript.
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Haltz, E., Sampaio, J., Krishnia, S. et al. Measurement of the tilt of a moving domain wall shows precessionfree dynamics in compensated ferrimagnets. Sci Rep 10, 16292 (2020). https://doi.org/10.1038/s41598020730495
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DOI: https://doi.org/10.1038/s41598020730495
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