The current-induced domain wall motion along a ferromagnetic strip with high perpendicular magnetocristalline anisotropy sandwiched in an multilayer stack is theoretically studied, by means of micromagnetic simulations and the one-dimensional model, with emphasis on the roles of the Rashba spin-orbit coupling and the spin Hall effect. The results point out that in the presence of a strong Rashba field the domain wall motion can be either in the direction of the current or opposing to it depending on the amplitude of the spin Hall effect. The predictions are in agreement with the experiments only in a reduced range of positive spin Hall angles under a strong Rashba torque.

The current-induced domain wall motion (CIDM) along thin ferromagnetic strips with high perpendicular magnetocristalline anisotropy (PMA) is nowadays a subject of great fundamental and technological interest.1 A spin-polarized current traversing a domain wall (DW) transfers angular momentum to the internal DW magnetization, thereby pushing it in the direction of the electron flow in accordance with the adiabatic and the non-adiabatic spin transfer torques (STTs).2 Although both the origin and the amplitude of the non-adiabatic STT are still controversial,1,2 several experiments on thick PMA strips with negligible interfacial effects3–5 are consistent with the STT theory, which predicts a DW propagation along the electron flow (against the electric current) if both the spin polarization factor (P) and non-adiabatic parameter (ξ) are positive quantities. However, other experimental observations6–9 indicate that the CIDM is in the direction of the electric current (against the electron flow) in ultrathin PMA strips sandwiched between nonmagnetic metal layers. Particularly relevant are the experiments by Miron et al.,6 where narrow DWs (∼3 nm) were rigidly displaced with high velocity (∼500 m/s) under relatively low current densities (∼1 A/μm2) in an asymmetric Pt/Co/AlO stack. This observation is not only attractive for applications, but also from a fundamental point of view. Indeed, according to the STT theory the direction of the CIDM depends on the product ξP, and therefore, a possible theoretical explanation for this reverse CIDM could be negative values of either P10 or ξ.11,12 Although direct and independent measurements of P and ξ have not been carried out yet, positive values of P have been measured recently,13 and a negative non-adiabaticity seems to be very unlikely because it would mean that ξ varies strongly between nearly identical materials systems. Therefore, additional phenomena are required to understand the reverse CIDM (along the current direction), which is the aim of the present study.

A recent investigation14 has shown that the direction of the wall motion can be tuned by using two Pt layers with different thickness sandwiching the Co layer. In the presence of this structural inversion asymmetry and/or heavy elements like Pt, strong spin-orbit coupling (SOC) at both Co interfaces, and/or in the Pt layer can lead to additional spin-orbit torques (SOTs) qualitatively different from the usual STTs. The origin of these SOTs is still under debate, and it could be due to either the Rashba spin-orbit coupling (RSOC) at the Co interfaces15–22 or due to the spin Hall effect (SHE) in the Pt layers.23–28,30,31 The RSOC torque is mediated by a transverse Rashba field HR, which stabilizes the up or down Bloch DW configurations depending on the sign of the injected current

$\vec{j}_{a}$
ja⁠, such that they move rigidly rather than by precession.32 This Rashba field can explain the observed highly-efficient CIDM6 if a high non-adiabaticity is assumed (ξ ∼ 1), but it can not account for the reverse CIDM unless either the P or ξ are negative. In addition to the transverse Rashba field, RSOC in asymmetric stacks should generate a Slonczewski-like torque similar to that induced by perpendicular current injection in multilayers.16,22 Although a recent theoretical study22 has addressed that these two RSOC torques could account for the reverse CIDM in the high SOC limit, a Slonczewski-like torque on the Co layers could also arise through spin pumping from the adjacent Pt metal via Spin Hall effect (SHE).23–26 The amplitude of this SHE depends on the relative thickness of the ferro and the non-ferromagnetic layers,27,28,30 and the SHE can not explain the reverse CIDM for both current polarities.31 Therefore, in order to identify the dominant SOT mechanism responsible of the CIDM in these high PMA stacks, it is essential to perform a complete and realistic study taking into account both the RSOC and the SHE, which together, have not been evaluated yet.

In this letter, the influence of both the Rashba and the Spin Hall torques is theoretically studied by means of both full micromagnetic modeling and a one-dimensional model points of view. Our realistic analysis includes pinning and thermal effects, and it indicates that CIDM can be either in the direction of the current or opposing to it depending on the amplitude of the SHE in the presence of RSOC torque.

A ferromagnetic strip of rectangular cross-section Ly × Lz = 120 nm × 3 nm with the easy-axis along the z-axis (

$\vec{u}_{K}=\vec{u}_{z}$
uK=uz⁠), and archetypal high PMA material parameters have been considered: saturation magnetization Ms = 3 × 105 A/m, exchange constant A = 10−11 J/m, and anisotropy constant K = 2 × 105J/m3, damping α = 0.2, polarization factor P = 0.5 and non-adiabatic parameter ξ = 0.04.

The equilibrium DW states at rest, either Bloch Up (BU) or Bloch Down (BD) are shown Fig. 1(a) and 1(b) respectively. The sign criterion for ja is given in Fig. 1(c). The dynamics of the normalized local magnetization

$\vec{m}(\vec{r},t)=\vec{M}/M_{s}$
m(r,t)=M/Ms under the injection of current densities along the x-axis
$\vec{j}_{a}=j_{a}\vec{u}_{x}$
ja=jaux
spatially uniform and instantaneously applied at t = 0 is described by augmented Landau-Lifshitz-Gilbert equation,22 

\begin{eqnarray}\frac{d\vec{m}}{dt}&=&-\gamma _{0}\,\vec{m}\times \vec{H}_{eff}+ \alpha \left(\vec{m}\times \frac{d\vec{m}}{dt}\right)+ \nonumber \\&+& b_{J} \left(\vec{u}_{x}\cdot \nabla \right)\vec{m}-\xi b_{J}\vec{m}\times \left(\vec{u}_{x}\cdot \nabla \right)\vec{m}+ \nonumber \\&-& \gamma _{0}\vec{m}\times \vec{H}_{R}+\eta \gamma _{0}\xi \vec{m}\times \left(\vec{m}\times \vec{H}_{R}\right) + \nonumber \\&+& \gamma _{0}H_{SH}\vec{m}\times \left(\vec{m}\times \vec{u}_{y}\right)\end{eqnarray}
dmdt=γ0m×Heff+αm×dmdt++bJux·mξbJm×ux·m+γ0m×HR+ηγ0ξm×m×HR++γ0HSHm×m×uy
(1)

where γ0 is the gyromagnetic ratio. Apart from the standard precession around the effective field

$\vec{H}_{eff}$
Heff and the dissipation torques, Eq. (1) includes the adiabatic and non-adiabatic STTs. The STT coefficient is
$b_{J}=j_{a}\frac{\mu _{B}P}{eM_{s}}$
bJ=jaμBPeMs
where μB is the Bohr magneton, e < 0 is the electron's charge. The fifth and the sixth terms at the right hand side of Eq. (1) correspond to the RSOC torque. The Rashba field
$\vec{H}_{R}$
HR
is given by6,16

\begin{equation}\vec{H}_{R} = \frac{\alpha _{R}P}{\mu _{0}\mu _{B}M_{s}}\left(\vec{u}_{z}\times \vec{j}_{a} \right)\end{equation}
HR=αRPμ0μBMsuz×ja
(2)

where αR describes the strength of the RSOC,16 and the parameter η has been introduced to take (η = 1) or not (η = 0) into account the non-adiabatic Slonczewski-like contribution of the RSOC torque. Finally, the last term of (1) is the Spin-Hall torque. The coefficient HSH is given by31 

\begin{equation}H_{SH} = \frac{\hbar \theta _{SH}}{\mu _{0}2eM_{S}L_{z}}j_{a} = \frac{\mu _{B}\theta _{SH}}{\gamma _{0}eM_{s}L_{z}}j_{a}\end{equation}
HSH=θSHμ02eMSLzja=μBθSHγ0eMsLzja
(3)

where θSH is the Spin Hall angle, which is defined as the ratio between the spin and charge currents.27 

On the other hand, the CIDM can be also analyzed from the one-dimensional model (1DM) point of view, which is described by the following equations2,32

\begin{eqnarray}\frac{\dot{X}}{\Delta } & = & \alpha \gamma _{0}^{\prime }H_{T} + \frac{\gamma _{0}^{\prime }H_{K}}{2} \sin (2\Phi ) - \frac{(1+\alpha \xi )}{1+\alpha ^{2}}\frac{b_{J}}{\Delta } \nonumber \\& + & \frac{\pi }{2}\gamma _{0}^{\prime } \left[ (1+\alpha \xi \eta )H_{R} -\alpha H_{SH}\right]\sin (\Phi )\end{eqnarray}
ẊΔ=αγ0HT+γ0HK2sin(2Φ)(1+αξ)1+α2bJΔ+π2γ0(1+αξη)HRαHSHsin(Φ)
(4)
\begin{eqnarray}\dot{\Phi } & = & \gamma _{0}^{\prime }H_{T} - \alpha \frac{\gamma _{0}^{\prime }H_{K}}{2} \sin (2\Phi ) - \frac{(\xi - \alpha )}{1+\alpha ^{2}}\frac{b_{J}}{\Delta } \nonumber \\& + & \frac{\pi }{2}\gamma _{0}^{\prime } \left[(\eta \xi - \alpha )H_{R} - H_{SH} \right]\sin (\Phi )\end{eqnarray}
Φ̇=γ0HTαγ0HK2sin(2Φ)(ξα)1+α2bJΔ+π2γ0(ηξα)HRHSHsin(Φ)
(5)

where

$\gamma _{0}^{\prime }=\gamma _{0}/(1+\alpha ^{2})$
γ0=γ0/(1+α2)⁠, X = X(t) is the DW position and Φ = Φ(t) is the DW angle: Φ(0) = 0 or Φ(0) = π for BU and BD initial DW configurations. Δ is the DW width, and HK is the hard-axis anisotropy field of magnetostatic origin. The total field HT = He + Hp(X) + Hth(t) includes: (i) the applied magnetic field along the easy z-axis (He), (ii) the spatial dependent pinning field (Hp(X)), which accounts for local imperfections and can be derived from an effective spatial-dependent pinning potential Vpin(X) as
$H_{p}(X)=-\frac{1}{2\mu _{0}M_{s}L_{y}L_{z}}\frac{\partial V_{pin}(X)}{\partial X}$
Hp(X)=12μ0MsLyLzVpin(X)X
, and (iii) the thermal field (Hth(t)), which describes the effect of thermal fluctuations,32 and is assumed to be a random Gaussian-distributed stochastic process with zero mean value (⟨Hth(t)⟩ = 0) and uncorrelated in time (
$\langle H_{th}(t)H_{th}(t^{\prime })\rangle =\frac{2\alpha K_{B}T}{\mu _{0}\gamma _{0}M_{s}\Delta L_{y}L_{z}}\delta (t-t^{\prime })$
Hth(t)Hth(t)=2αKBTμ0γ0MsΔLyLzδ(tt)
, where KB is the Boltzmann constant and T the temperature).

In order to elucidate the role of the different SOC torques (Rashba and/or SHE) on the CIDM, we firstly consider a perfect strip (Hp = 0) at zero temperature (T = 0) and in the absence of applied field (He = 0). The 1DM Eqs. (4) and (5) were numerically solved with Δ = 8.32nm and HK = 12533A/m, which were obtained from a preliminary micromagnetic study.33 The temporal evolution of the velocity of BU and BD walls for both positive and negative currents of |ja| = 0.25 A/μm2 are displayed in Fig. 2. The panels A-F correspond to the Rashba (αR, η) and Spin Hall (θSH) parameters sumarized in Table I. As it is well known, above the Walker current (

$j_{a}>j_{W}(\xi )$
ja>jW(ξ)⁠), in the absence of SOTs (Fig. 2(A)), both BU and BD move along the electron flow direction (against the current direction) by DW precessing around the easy z-axis. If only the y-component of the Rashba field (η = 0 with θSH = 0) is taken into account, the DW reaches a stationary behavior with a finite velocity along the electron flow direction (Fig. 2(B)). In this case, a positive (negative) current drives the BU (BD) motion without precession. However, the initial BD (BU) DW configuration transforms to a BU (BD) under negative (positive) current before reaching the stationary regime. This half-rotation is observed in all cases including the RSOC torque (B, C, E and F). As it is shown in Fig. 2(C), for the evaluated parameters here, the DW finally stops independently of current direction if the non-adiabatic Slonczewski-like contribution to the RSOC torque is taken into account (η = 1). When the Rashba field is switched off (αR = 0) and only the SHE is taken into account with θSH = 0.1 (see Fig. 2(D)), the DWs reach a finite velocity close to the averaged value for αR = θSH = 0, but again in this case, both BU and BD propagate against the electric current (in the electron flow direction) for both current polarities. Finally, when both Rashba and SHE torques are taken into account (see Fig. 2(E) and 2(F) for η = 0 and η = 1 respectively) the DWs move against the electron flow along the current direction. In both cases E (η = 0) and F (η = 1), both BU and BD reach a positive (negative) velocity under negative (positive) currents, and the terminal DW velocity is higher in the presence of the non-adiabatic Rashba torque (η = 1). These 1DM results are similar to the full micromagnetic ones, obtained by considering the 3D spatial dependence of the magnetization by solving Eq. (1)(see Supplementary Material,35 I-II).

FIG. 2.

Temporal evolution of the velocity as computed by the 1DM Eq. (4) and (5) under positive (solid lines, along the x < 0-axis) and negative (dashed lines, along the x > 0-axis) density currents with |ja| = 0.25 A/μm2. Two initial states BU and BD are studied. Strip dimensions, material parameters, 1DM inputs and the cases A-F are described in the text and in Table I.

FIG. 2.

Temporal evolution of the velocity as computed by the 1DM Eq. (4) and (5) under positive (solid lines, along the x < 0-axis) and negative (dashed lines, along the x > 0-axis) density currents with |ja| = 0.25 A/μm2. Two initial states BU and BD are studied. Strip dimensions, material parameters, 1DM inputs and the cases A-F are described in the text and in Table I.

Close modal
Table I.

Rashba (αR, η) and Spin Hall (θSH) parameters corresponding to cases A-F.

CaseABCDEF
αR( × 10−30 Jm) 1.6 1.6 1.6 1.6 
η 
θSH 0.1 0.1 0.1 
CaseABCDEF
αR( × 10−30 Jm) 1.6 1.6 1.6 1.6 
η 
θSH 0.1 0.1 0.1 

The terminal DW velocity as a function of ja is shown in Fig. 3(a) for the cases A-F. These results indicate that the CIDM can take place in the direction of the applied current ja if both the RSOC and the SHE SOTs are taken into account (cases E and F).

While the Rashba spin-orbit interaction is expected to be localized at the interface between the ferromagnetic and the heavy metal layers, the SHE torque is due to the current flowing in the bulk of the heavy metal Pt layer. Therefore, for a given current, varying the thickness of the heavy metal Pt layer will results in a variation of the SHE torque strength, while keeping the Rashba torque unchanged. The strength of the Rashba field, which is given by αR, has been extensively studied for semiconductors and for a normal metals in contact with a heavy metal resulting in values around 10−31–10−29 Jm for the conduction electrons near the interface.20 Experimental knowledge of αR for a ferromagnet in contact with other materials is still quite limited. A typical value for a 2D electron gas with structural inversion asymmetry has been used here (αR = 1.6 × 10−30 Jm), which is believed to be a realistic value for the system under consideration.6,16 On the other hand, the Spin Hall angle θSH has been measured recently and it depends on the thicknesses of both the ferromagnetic and non-ferromagnetic layers of the stack.27,28,30 In order to characterize the CIDM in the presence of the Rashba field (αR = 1.6 × 10−30 Jm) for different stack configurations with different thickness of the ferromagnetic and the non-ferromagnetic layers, several values of θSH have been studied. The DW velocity as a function of θSH under a fixed current ja = 1 A/μm2 is shown in Fig. 3(b), for both cases E (η = 0) and F (η = 1) respectively. There is a range of θSH where the DW moves with negative velocity, that is, along the current direction, or against the electron flow. This range varies between 0.01 ⩽ θSH ⩽ 0.12 if the non-adiabatic contribution to the Rashba torque is taken into account (η = 1, case F), and between 0.04 ⩽ θSH ⩽ 0.14 if not (η = 0, case E). Note that although the maximum DW velocity in the direction of the current is reached for different values of θSH, its magnitude does not depends significantly on considering (η = 1) or not (η = 0) the non-adiabatic contribution to the Rashba torque.

The results presented so far were obtained by assuming a perfect strip (Hp(X) = 0) at zero temperature (T = 0). However, real samples always present some degree of imperfections such as impurities, local defects, surface roughness or edge roughness. In order to take into account these local pinning effects in the 1DM,32,33 a space-dependent periodic pinning potential given by Vpin(X) = V0sin 2X/p) is considered, where V0 represents the energy barrier between adjacent minima and p the spatial periodicity. The values V0 = 1.65 × 10−20 J and p = 30 nm were selected to reproduce the full micromagnetically computed depinning field at T = 0 along the strip with a characteristic edge roughness size of 3nm.33 On the other hand, real experiments are performed at room temperature (T = 300 K). In order to account for these effects, both the pinning field Hp(X) and the thermal field Hth are included in HT in both (4) and (5) as discussed above.

The current dependence of the DW velocity computed at T = 300 K along a rough (Hp(X) ≠ 0) strip is depicted in Fig. 4(a) for the same cases A-F of former Fig. 3. These results were averaged over N = 10 stochastic realizations over a temporal window of tw = 50 ns. The more significant differences with the deterministic (T = 0) perfect-strip (Hp(X) = Vpin(X) = 0) results of former Fig. 3 occur in the cases B (η = 0, αR = 1.6 × 10−30 Jm, θSH = 0), E (η = 0, αR = 1.6 × 10−30 Jm, θSH = 0.1) and F (η = 1, αR = 1.6 × 10−30 Jm, θSH = 0.1). As it was already studied,34 the transversal component of the Rashba field (case B) supports the pinning and increases the critical depinning threshold, which in this particular case is out from the evaluated currents range. In the presence of both RSOC and SHE (cases E and F), there is a depinning threshold current jd below which the DW does not propagate at zero temperature. These deterministic threshold currents are not symmetric with respect to the current polarity: jd + = +1.4 A/μm2 and jd = −0.9 A/μm2 for case E (η = 0), and jd + = +0.9 A/μm2 and jd = −0.75 A/μm2 for case F (η = 1). Above this values (ja > jd + or ja < jd) the DW propagates as in the free-defect case. At room temperature the DW dynamics is not deterministic, and there is a no null probability of DW propagation for currents below the deterministic threshold. As it is depicted in Fig. 4 for both cases E and F, a low-current creep regime and a high-current flow regimes are clearly evidenced. For very low currents, the DW does not propagate because the applied current is still very low to overcome the local energy barrier, even at T = 300 K. For larger currents but smaller than the deterministic depinning threshold (jd), the DW also gets pinned due to the local pinning, but due to thermal fluctuations, it eventually depins and propagates along the strip. Finally, for currents above the deterministic threshold, the DW moves similarly to the free-defect case because the applied current is high enough to overcome the local pinning barrier independently on thermal activation. These creep, depinning and flow regimes are similar to the field-driven or the current-driven cases in the absence of RSOC and SHE,32–34 and an extended description can be seen in the Supplementary Material35(III). It was also verified that in both cases E and F, the DW propagates rigidly with a Bloch DW configuration.

FIG. 1.

(a) Bloch Up (BU), (b) Bloch Down (BD) equilibrium states at rest. (c) Sign criterion for ja (positive along the x < 0-axis) corresponding to the electron flow toward the right (x > 0-axis). A positive velocity indicates a DW motion along the x > 0-axis.

FIG. 1.

(a) Bloch Up (BU), (b) Bloch Down (BD) equilibrium states at rest. (c) Sign criterion for ja (positive along the x < 0-axis) corresponding to the electron flow toward the right (x > 0-axis). A positive velocity indicates a DW motion along the x > 0-axis.

Close modal
FIG. 3.

(a) DW velocity as a function of ja for the cases A-F. (b) DW velocity as a function of θSH under a fixed ja = 1 A/μm2 for the cases E (η = 0) and F (η = 0). The results were computed by solving the 1DM eqs. (4) and (5) in the absence of any driving field (He = 0) for a perfect strip (Hp = 0) at zero temperature (T = 0).

FIG. 3.

(a) DW velocity as a function of ja for the cases A-F. (b) DW velocity as a function of θSH under a fixed ja = 1 A/μm2 for the cases E (η = 0) and F (η = 0). The results were computed by solving the 1DM eqs. (4) and (5) in the absence of any driving field (He = 0) for a perfect strip (Hp = 0) at zero temperature (T = 0).

Close modal
FIG. 4.

(a) DW velocity as a function of ja for the cases A-F of formers Figs. 1 and 2. The results were computed by solving the 1DM eqs. (4) and (5) for a 120 × 3 nm2 rough strip (V0 = 1.65 × 10−20 J, p = 30 nm) at room temperature T = 300 K. (b) 1DM reproduction of the experiment by Miron et al.6 In both cases (a) and (b), the results were averaged over N = 10 stochastic realizations and a temporal window of tw = 50 ns.

FIG. 4.

(a) DW velocity as a function of ja for the cases A-F of formers Figs. 1 and 2. The results were computed by solving the 1DM eqs. (4) and (5) for a 120 × 3 nm2 rough strip (V0 = 1.65 × 10−20 J, p = 30 nm) at room temperature T = 300 K. (b) 1DM reproduction of the experiment by Miron et al.6 In both cases (a) and (b), the results were averaged over N = 10 stochastic realizations and a temporal window of tw = 50 ns.

Close modal

Considering the spin Hall range where the DWM was propagated in the direction of the applied current, former results are in good qualitative agreement with very recent experimental observations by Miron and coworkers,6 where the DW is driven along a Co strip with a cross section of Ly × Lz = 500 nm × 0.6 nm and sandwiched between Pt and AlO layers. Our final aim consists on trying to find under which conditions it is posible to fit them also quantitatively. In order to do it, the same material parameters as deduced in6 for the sandwiched Co layer are considered: Ms = 1.09 × 106 A/m, A = 10−11 J/m, K = 1.19 × 106 J/m3, α = 0.2, P = 0.5, ξ = 0.1, and αR = 10−29 Jm with η = 1. The rest of inputs for the 1DM are DW width

$\Delta =\sqrt{A/K} = 3\; \mathrm{nm}$
Δ=A/K=3 nm ⁠, and the hard-axis field HK = 27852 A/m, which were deduced from a preliminary micromagnetic characterization as described elsewhere.33 The effect of disorder is modeled by assuming a periodic pinning potential Vpin(X) = V0sin 2X/p) with V0 = 6 × 10−19 J, and p = 30 nm. The 1DM results, which were obtained by considering tw = 50 ns and N = 10 at T = 300 K, are compared to the experimental data6 in Fig. 4(b). As it can be clearly seen, a good qualitative and quantitative agreement is achieved for a Spin Hall angle of θSH = 0.13. The only fitting parameters are non-adiabatic parameter and the Spin Hall angle, and both values are reasonable with experimental data.3,27–29

In summary, we have explored the influence of the Rashba and spin Hall spin-orbit torques on the current-induced DW motion by means of realistic simulations including pinning and thermal effects. The results point out that, in principle, the inclusion of both the Rashba and the Spin Hall effects along with the standard spin transfer torques, could explain recent experimental observations by considering positive values of the polarization factor and the non-adiabatic parameter. In particular, they show that a DW can propagate either in the direction of the current or opposing to it depending on the strength of the Spin Hall angle in the presence of a strong Rashba field, but the reversed DW motion (along the current) only occurs for a reduced range of positive spin Hall angles. Under these conditions, the results are also in good quantitative agreement with the experiments by Miron et al.6 by assuming the same value for the Rashba parameter deduced in their experiments (αR = 1 eVA = 10−29 Jm). This value is in the same order of magnitude than the one recently deduced from a first-principles calculation by Park et al.,36 for an ultrathin magnetic layer in contact with a nonmagnetic heavy metal layer. According to,36 the Rashba interaction could significantly enhance the spin-transfer torque, in agreement with the results addressed here. However, other very recent experimental measurements for Pt/Co/AlO,29 for Pt/Co/Pt stacks37 and for Pt/CoFe/Mg0 stack38 indicate that the Rashba parameter is indeed two orders of magnitude smaller than the one deduced in.6 The experiments by Haazen et al.37 also demonstrate that when the ultrathin Co layer is sandwiched simmetrically by the Pt layers, the spin transfer torque plays a negligible role, and that the spin Hall is the driving force only under the application of longitudinal in-plane fields which promote Neel domain wall configurations. The work by Emori et al.38 points out experimentally and computationally, that a strong Dzyaloshinskii-Moriya interaction39 stabilizes the Neel DW with a left-handed chirality, such that even in the absence of spin-transfer torque, the SHE alone drives it along the current, uniformly and with high efficiency. On the other hand, other authors40 have suggested that the Rashba interaction have similar effects as the Dzyaloshinskii-Moriya interaction. Although at the current state of the art this problem is not to be completely understood, and much more experimental and theoretical efforts are needed to get a full understanding of the puzzle physics behind the current-induced domain wall motion along a ferromagnetic strips with high perpendicular magnetocristalline anisotropy sandwiched in multilayer stacks, the scenario addressed in38 seems to be a pausible explanation, as it does not require Rashba nor spin transfer torques in agreement with other experiments.

The authors would like to thank D. C. Ralph for a critical reading of the manuscript, and M. Miron and G. Gaudin for providing their experimental data of Fig. 4(b). This work was supported by project MAT2011-28532-C03-01 from Spanish government and project SA163A12 from Junta de Castilla y Leon.

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Supplementary Material