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 effects^{3–5} 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 observations^{6–9} 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/μm^{2}) 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 ξ*P*, and therefore, a possible theoretical explanation for this reverse CIDM could be negative values of either *P*^{10} 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 investigation^{14} 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 interfaces^{15–22} 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 *H*_{R}, which stabilizes the up or down Bloch DW configurations depending on the sign of the injected current

^{32}This Rashba field can explain the observed highly-efficient CIDM

^{6}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 study

^{22}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 *L*_{y} × *L*_{z} = 120 nm × 3 nm with the easy-axis along the *z*-axis (

*M*

_{s}= 3 × 10

^{5}A/m, exchange constant

*A*= 10

^{−11}J/m, and anisotropy constant

*K*= 2 × 10

^{5}J/m

^{3}, 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 *j*_{a} is given in Fig. 1(c). The dynamics of the normalized local magnetization

*x*-axis

*t*= 0 is described by augmented Landau-Lifshitz-Gilbert equation,

^{22}

where γ_{0} is the gyromagnetic ratio. Apart from the standard precession around the effective field

_{B}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

^{6,16}

where α_{R} 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 *H*_{SH} is given by^{31}

where θ_{SH} 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 equations^{2,32}

where

*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

*H*

_{K}is the hard-axis anisotropy field of magnetostatic origin. The total field

*H*

_{T}=

*H*

_{e}+

*H*

_{p}(

*X*) +

*H*

_{th}(

*t*) includes: (i) the applied magnetic field along the easy

*z*-axis (

*H*

_{e}), (ii) the spatial dependent pinning field (

*H*

_{p}(

*X*)), which accounts for local imperfections and can be derived from an effective spatial-dependent pinning potential

*V*

_{pin}(

*X*) as

*H*

_{th}(

*t*)), which describes the effect of thermal fluctuations,

^{32}and is assumed to be a random Gaussian-distributed stochastic process with zero mean value (⟨

*H*

_{th}(

*t*)⟩ = 0) and uncorrelated in time (

*K*

_{B}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 (*H*_{p} = 0) at zero temperature (*T* = 0) and in the absence of applied field (*H*_{e} = 0). The 1DM Eqs. (4) and (5) were numerically solved with Δ = 8.32nm and *H*_{K} = 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 |*j*_{a}| = 0.25 A/μm^{2} are displayed in Fig. 2. The panels A-F correspond to the Rashba (α_{R}, η) and Spin Hall (θ_{SH}) parameters sumarized in Table I. As it is well known, above the Walker current (

*z*-axis. If only the

*y*-component of the Rashba field (η = 0 with θ

_{SH}= 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 (α

_{R}= 0) and only the SHE is taken into account with θ

_{SH}= 0.1 (see Fig. 2(D)), the DWs reach a finite velocity close to the averaged value for α

_{R}= θ

_{SH}= 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 3

*D*spatial dependence of the magnetization by solving Eq. (1)(see Supplementary Material,

^{35}I-II).

Case . | A . | B . | C . | D . | E . | F . |
---|---|---|---|---|---|---|

α_{R}( × 10^{−30} Jm) | 0 | 1.6 | 1.6 | 0 | 1.6 | 1.6 |

η | 0 | 0 | 1 | 0 | 0 | 1 |

θ_{SH} | 0 | 0 | 0 | 0.1 | 0.1 | 0.1 |

Case . | A . | B . | C . | D . | E . | F . |
---|---|---|---|---|---|---|

α_{R}( × 10^{−30} Jm) | 0 | 1.6 | 1.6 | 0 | 1.6 | 1.6 |

η | 0 | 0 | 1 | 0 | 0 | 1 |

θ_{SH} | 0 | 0 | 0 | 0.1 | 0.1 | 0.1 |

The terminal DW velocity as a function of *j*_{a} 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 *j*_{a} 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 α_{R}, 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 α_{R} 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 (α_{R} = 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 θ_{SH} 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 (α_{R} = 1.6 × 10^{−30} Jm) for different stack configurations with different thickness of the ferromagnetic and the non-ferromagnetic layers, several values of θ_{SH} have been studied. The DW velocity as a function of θ_{SH} under a fixed current *j*_{a} = 1 A/μm^{2} is shown in Fig. 3(b), for both cases E (η = 0) and F (η = 1) respectively. There is a range of θ_{SH} where the DW moves with negative velocity, that is, along the current direction, or against the electron flow. This range varies between 0.01 ⩽ θ_{SH} ⩽ 0.12 if the non-adiabatic contribution to the Rashba torque is taken into account (η = 1, case F), and between 0.04 ⩽ θ_{SH} ⩽ 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 θ_{SH}, 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 (*H*_{p}(*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 *V*_{pin}(*X*) = *V*_{0}sin ^{2}(π*X*/*p*) is considered, where *V*_{0} represents the energy barrier between adjacent minima and *p* the spatial periodicity. The values *V*_{0} = 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 *H*_{p}(*X*) and the thermal field *H*_{th} are included in *H*_{T} in both (4) and (5) as discussed above.

The current dependence of the DW velocity computed at *T* = 300 K along a rough (*H*_{p}(*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 *t*_{w} = 50 ns. The more significant differences with the deterministic (*T* = 0) perfect-strip (*H*_{p}(*X*) = *V*_{pin}(*X*) = 0) results of former Fig. 3 occur in the cases B (η = 0, α_{R} = 1.6 × 10^{−30} Jm, θ_{SH} = 0), E (η = 0, α_{R} = 1.6 × 10^{−30} Jm, θ_{SH} = 0.1) and F (η = 1, α_{R} = 1.6 × 10^{−30} Jm, θ_{SH} = 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 *j*_{d} below which the DW does not propagate at zero temperature. These deterministic threshold currents are not symmetric with respect to the current polarity: *j*_{d +} = +1.4 A/μm^{2} and *j*_{d −} = −0.9 A/μm^{2} for case E (η = 0), and *j*_{d +} = +0.9 A/μm^{2} and *j*_{d −} = −0.75 A/μm^{2} for case F (η = 1). Above this values (*j*_{a} > *j*_{d +} or *j*_{a} < *j*_{d −}) 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 (*j*_{d}), 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 Material^{35}(III). It was also verified that in both cases E and F, the DW propagates rigidly with a Bloch DW configuration.

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 *L*_{y} × *L*_{z} = 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 in^{6} for the sandwiched Co layer are considered: *M*_{s} = 1.09 × 10^{6} A/m, *A* = 10^{−11} J/m, *K* = 1.19 × 10^{6} J/m^{3}, α = 0.2, *P* = 0.5, ξ = 0.1, and α_{R} = 10^{−29} Jm with η = 1. The rest of inputs for the 1DM are DW width

*H*

_{K}= 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

*V*

_{pin}(

*X*) =

*V*

_{0}sin

^{2}(π

*X*/

*p*) with

*V*

_{0}= 6 × 10

^{−19}J, and

*p*= 30 nm. The 1DM results, which were obtained by considering

*t*

_{w}= 50 ns and

*N*= 10 at

*T*= 300 K, are compared to the experimental data

^{6}in Fig. 4(b). As it can be clearly seen, a good qualitative and quantitative agreement is achieved for a Spin Hall angle of θ

_{SH}= 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 (α_{R} = 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 stacks^{37} and for Pt/CoFe/Mg0 stack^{38} 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 interaction^{39} 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 authors^{40} 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 in^{38} 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.