Abstract
It is well known that skyrmions can be driven using spinorbit torques due to the spinHall effect. Here we show an additional contribution in multilayered stacks arises from vertical spin currents due to interlayer diffusion of a spin accumulation generated at a skyrmion. This additional interfacial spin torque is similar in form to the inplane spin transfer torque, but is significantly enhanced in ultrathin films and acts in the opposite direction to the electron flow. The combination of this diffusive spin torque and the spinorbit torque results in skyrmion motion which helps to explain the observation of small skyrmion Hall angles even with moderate magnetisation damping values. Further, the effect of material imperfections on threshold currents and skyrmion Hall angle is also investigated. Topographical surface roughness, as small as a single monolayer variation, is shown to be an important contributing factor in ultrathin films, resulting in good agreement with experimental observations.
Introduction
Skyrmions are topologically protected particlelike magnetic textures^{1}, which are of great interest for potential technological applications. Skyrmions have been observed in materials with broken inversion symmetry and stabilised at room temperature through the DzyaloshinskiiMoriya interaction (DMI)^{2,3,4,5}. As carriers of information it is important to effectively move, as well as detect, skyrmions using electrical signals and readout. To this end, recent experiments have revealed the fascinating physics behind the interaction of skyrmions with spin currents. Currentinduced skyrmion movement was demonstrated at room temperature in a number of recent experiments on ultrathin multilayered stacks^{6,7,8,9,10,11,12}, whilst electrical readout is made possible through the discrete Hall resistivity^{13}. The principal source of spin currents in these devices is the spinHall effect (SHE), which converts a charge current flowing in the plane into transverse pure spin currents. The resultant spinorbit torque (SOT) gives rise to skyrmion motion, with direction set by the charge current direction as well as skyrmion chirality^{14}. The skyrmion Hall effect, whereby the direction of skyrmion movement deviates from that of the charge current direction due to the Magnus force, was also demonstrated^{8,9,10}. In some spintronics devices it is desirable to obtain zero skyrmion Hall angles, allowing for movement only along the current direction. Strategies to reduce the skyrmion Hall angle to zero have been proposed, using antiferromagnetically exchangecoupled bilayer systems^{15}, antiferromagnetic skyrmions^{16}, ferrimagnetic skyrmions^{11,17}, as well as using skyrmionium magnetic textures^{18}.
Due to spin precession of spinpolarised electrons flowing through a magnetic texture, a spin accumulation is generated at magnetisation texture gradients resulting in adiabatic and nonadiabatic spin transfer torques (STT)^{19,20}. Furthermore spin diffusion was also shown to play a role, resulting in modified diffusive spin torques when considering twodimensional magnetic textures^{21}. On the other hand vertical spin currents have been shown to play a more important role in driving skyrmions in nanostructures^{22}, whilst the importance of interfaces and interfacegenerated spin currents has also been recognised^{23}. Here we show, using micromagnetics modelling coupled with a selfconsistent spin transport solver in multilayers, that the spin accumulation generated at the magnetisation gradients of a skyrmion results in additional vertical spin currents due to spin diffusion in adjacent nonmagnetic layers. These diffusive spin currents result in additional interfacial spin torques which can be comparable to the SOT, significantly reducing the calculated skyrmion Hall angle even for small magnetisation damping values. In experiments it was found the skyrmion Hall angle strongly depends on the skyrmion velocity, evidencing the important role material imperfections play^{8,9,10}. Using the selfconsistent spin transport solver we also study the effect of SOTs and interlayer diffusion in the presence of magnetic defects, as well as topographical surface roughness. In particular surface roughness is shown to result in strong confining potentials, resulting in a dependence of the skyrmion Hall angle with driving current, as well as threshold current densities comparable to those found in experiments. These results may indicate an alternative method of designing devices with zero skyrmion Hall angle, by purposely creating surface confining potentials.
Spin Transport Model
Spin torques included in the magnetisation dynamics equation can be computed selfconsistently using a driftdiffusion model^{24,25}. Within this model the charge and spin current densities are given as:
Here J_{S} is a rank2 tensor such that J_{Sij} signifies the flow of the j component of spin polarisation in the direction i, J_{C} is the charge current density, E is the electric field, S is the spin accumulation, and m is the normalised magnetisation. Equation (2) contains contributions due to (i) drift included in ferromagnetic (F) layers, where P is the current spinpolarisation and σ the electrical conductivity, (ii) diffusion, where D_{e} is the electron diffusion constant, and (iii) spinHall effect, included in nonmagnetic (N) layers, where θ_{SHA} is the spinHall angle and ε is the rank3 unit antisymmetric tensor. The inverse spinHall effect is included for completeness as a contribution in Eq. (1). The spin accumulation satisfies the equation of motion:
Here λ_{sf} is the spinflip length which governs the decay of spin accumulation. In F layers the decay of transverse components of S are governed by the exchange rotation length λ_{J}, and the spin dephasing length λ_{φ}. Solving Eqs (1–3), we obtain a Poissontype equation for the steadystate spin accumulation:
Thus for each magnetisation configuration m the resulting spin accumulation is obtained by solving Eq. (4). This is justified since m and S vary on very different timescales (ps vs fs respectively). In Equation (1) we have the usual relation E = −∇V. For boundaries containing an electrode with a fixed potential, differential operators applied to V use a Dirichlet boundary condition. For other external boundaries we require both the charge and spin currents to be zero in the direction normal to the boundary^{26}, i.e. J_{C}.n = 0 and J_{S}.n = 0. This results in the following nonhomogeneous Neumann boundary conditions:
At the interface between two N layers we obtain composite media boundary conditions for V and S by requiring both a potential and associated flux to be continuous in the direction normal to the interface, i.e. V and J_{C}, and S and J_{S} respectively. At an N/F interface we do not assume such continuity, but instead model the absorption of transverse spin components using the spinmixing conductance^{27}:
Here ΔV is the potential drop across the N/F interface (ΔV = V_{F} − V_{N}) and ΔV_{S} is the spin chemical potential drop, where \({{\bf{V}}}_{S}=({D}_{e}/\sigma )(e/{\mu }_{B}){\bf{S}}\), and G^{↑}, G^{↓} are interface conductances for the majority and minority spin carriers respectively. The transverse spin current absorbed at the N/F interface results in a torque on the magnetisation as a consequence of conservation of total spin angular momentum. If the F layer has thickness d_{F}, this interfacial torque is obtained as:
In the equation of motion for m, the interfacial torque is included as:
Here \(\gamma ={\mu }_{0}{g}_{rel}{\gamma }_{e}\), where \({\gamma }_{e}=\,g{\mu }_{B}/\hslash \) is the electron gyromagnetic ratio, g_{rel} is a relative gfactor, and M_{s} is the saturation magnetisation, such that M = mM_{s} is the magnetisation vector. Using Eq. (6) we can also include spin pumping on the N side of the equation as^{28}:
For an N/F interface with current in the plane, if diffusion effects are negligible, the driftdiffusion equations may be solved analytically to obtain the resulting interfacial spin torques due to SHE^{25}. These are given as a combination of dampinglike and fieldlike spinorbit torques as:
Here p = z × e_{Jc}, where e_{Jc} is the charge current direction. The quantity θ_{SHA,eff} is proportional to the real spinHall angle θ_{SHA}, scaled by transport and interface parameters, and is given by:
where \({N}_{\lambda }=tanh({d}_{N}/{\lambda }_{sf}^{N})/{\lambda }_{sf}^{N}\), and \(\tilde{G}=2{G}^{\uparrow \downarrow }/{\sigma }_{N}\). The fieldlike torque coefficient is given by \({r}_{G}={N}_{\lambda }\text{Im}\{\tilde{G}\}/({N}_{\lambda }\mathrm{Re}\{\tilde{G}\}+\tilde{G}{}^{2})\). The selfconsistent spin transport solver reproduces the SOT in Eq. (10), thus including both dampinglike and fieldlike components. For simulations using the LLG equation complemented by the SOT in Eq. (10) directly, the calculated fieldlike and dampinglike SOT coefficients must take into account the role of interface transparency^{29} as given by the above equations.
With N/F multilayers another important source of vertical spin currents, resulting in an interfacial spin torque contribution, is due to N/F interlayer diffusion of a spin accumulation generated in the F layer at spatial gradients in the magnetisation texture, e.g. a skyrmion. This is in some ways similar to the inplane STT arising in the F layer alone^{19,20}, but in ultrathin films the interlayer diffusion results in much stronger spin torques partly due to the inverse dependence on d_{F}. It can be shown this additional interfacial spin torque is given by (see Supplementary Material for Derivation):
This interfacial spin torque has a very similar form to the wellknown ZhangLi STT, with the exception it acts in the opposite direction, i.e. results in motion along the current direction, and the spindrift velocity and nonadiabaticity parameters are replaced by effective perpendicular spindrift velocity and perpendicular nonadiabaticity parameters. In particular the perpendicular spindrift velocity is given by:
where P_{⊥} is an effective perpendicular spin polarisation parameter. These parameters are not dependent on a single material alone, but are effective parameters for the entire multilayered stack.
Spin Torques in Multilayers
Currentinduced Néel skyrmion movement has been observed in a number of ultrathin multilayered stacks, including Ta/CoFeB/TaO_{x}^{6,9}, [Pt/Co/Ta]_{x}^{7}, Ta/[Pt/Ir/Co]_{x}/Pt^{8}, [Pt/CoFeB/MgO]_{x}^{7,10}, [Pt/GdFeCo/MgO]_{x}^{11}, and symmetric bilayer stacks^{12}. To study the effect of the spin torques described in the previous section on skyrmion motion, a multilayered disk geometry was chosen, with the structure [Pt (3 nm)\Co (1 nm)\Ta (4 nm)]_{x}, which has been wellcharacterised experimentally^{7,30,31,32}. The disk geometry was chosen so the influence of sample boundaries is the same irrespective of the skyrmion Hall angle. The studied geometry is shown in Fig. 1 for a repetition of 6 Pt/Co/Ta stacks (x = 6). The bottom Pt layer was extended and a current applied to the structure through electrodes on its xaxis ends. This configuration ensures that, apart from the edges of the disks, the current density is approximately uniform (less than 2% variation in the region where skyrmion motion is simulated). A table with the full list of material parameters used is given in the Methods section.
First we investigate the effect of spin torques on a skyrmion in a single Pt/Co/Ta stack with fixed current density, for different chiralities and topological charges. Using a fixed outofplane field H_{z} = 15 kA/m the skyrmion diameter is fixed to 60 nm, similar to that observed experimentally^{7}. The results are shown in Fig. 2. For the spin torque obtained with the selfconsistent spin transport solver we see two distinct contributions. In addition to the spin torque due to SHE alone, namely SOT, an equally important contribution is obtained due to interlayer spin diffusion. To demonstrate this, skyrmions have been driven with and without SHE contribution. Without SHE (θ_{SHA} = 0 in both Pt and Ta) the only torque acting on the skyrmions is due to interlayer spin diffusion, as seen from the good agreement between spin transport solver computations with θ_{SHA} = 0, and simulations using the LLG equation complemented by the diffusive spin torque in Eq. (12). The vertical spin current due to diffusion is shown in the Pt layer in Fig. 1, as well as the resulting interfacial spin torque acting on the skyrmion. Note, in this work we didn’t consider ZhangLi inplane STTs since their effect is much smaller in ultrathin films compared to interfacial spin torques^{22}.
For Equation (12) we find P_{⊥} = 0.87 and β_{⊥} = −0.13 for the interfacial spin torque. The large effective perpendicular spin polarisation and perpendicular nonadiabaticity parameters result in a total spin torque comparable to the SOT. In contrast to SOTs however, the direction of motion is opposite to the electron flow in all cases, for both D < 0 and D > 0, where D is the DMI exchange constant. Further, by subtracting results obtained using the spin transport solver for θ_{SHA} ≠ 0 and θ_{SHA} = 0 we obtain a good agreement with simulations using the LLG equation complemented by the SOT in Eq. (10). As expected, with the SOT alone the direction of motion depends on the sign of the DMI. However, when interlayer spin diffusion is taken into account, the overall effect is for skyrmion motion opposing the flow of electrons in all cases.
Experimental investigations of currentinduced skyrmion movement have revealed skyrmion displacement in the direction opposing the flow of electrons^{6,7,8,9,10,11,12}, and these results have been analysed principally based on the SOT due to SHE. We show here however, interlayer spin diffusion could also have a significant effect and should be considered when analysing skyrmion motion. The implications are both qualitative and quantitative. Since the skyrmion motion direction due to the diffusive spin torque is always in the direction opposing the flow of electrons, if interlayer spin diffusion is significant, the exact topology of skyrmions cannot be determined purely based on observing their motion direction (with or against the electron flow). Quantitatively, whilst the skyrmion velocities are not greatly affected by inclusion of the diffusive spin torque, due to the nearly orthogonal skyrmion movement directions under these two torques respectively, the skyrmion Hall angle obtained varies markedly and could have significant implications in explaining experimental results. To experimentally verify the interfacial diffusive spin torque directly, material stacks having both low total SOT and high DMI could be used, whilst still preserving the lack of inversion symmetry required for stable skyrmions; moreover the metallic underlayers used should be good spin sinks (small spin diffusion length) in order to maximise the diffusive spin torque. This presents a materials engineering challenge since SHE and DMI strengths tend to be correlated. One suggested possibility is to use interface doping to change the efficiency of the SOT^{33,34}, and it is hoped the results shown here will stimulate further experimental work.
The Onsager reciprocal process to absorption of transverse spin currents is the generation of spin currents via dynamical magnetisation processes, known as spin pumping^{28} – see Eq. (9). The effect of spin pumping on magnetisation precession is an increase in the effective magnetisation damping. As expected from the Thiele equation^{35}, larger damping values should result in reduced skyrmion velocities, thus it is interesting to observe its effect on skyrmion motion. First we keep the current density fixed, and later analyse the effect of varying the current density. With spin pumping enabled in the spin transport solver, a spin drag effect is observed, resulting in a slight reduction in velocity as shown in Fig. 2, as well as a slight deviation of the skyrmion path. This effect is quite small however and could be ignored in simpler simulations using only the analytical form of the SOT and diffusive spin torque.
The results discussed thus far used an intrinsic damping value in Co of 0.03, as obtained using ferromagnetic resonance measurements in ultrathin films^{36}. On the other hand, much higher damping values of up to 0.3 have been obtained in Pt/Co/Pt films from magnetic domainwall motion experiments^{37}. Increasing damping results in a reduced skyrmion velocity as expected, however more significantly the direction of motion is strongly affected, resulting in a clockwise rotation of the skyrmion paths with increasing damping as seen in Fig. 3. This holds for both the SOT and diffusive spin torque, and again a good agreement is obtained between the spin transport solver results and simulations using the analytical forms of the SOT and diffusive spin torque. Small skyrmion Hall angles have been observed experimentally^{9,10}. Based on the SOT alone the skyrmion Hall angle is inversely dependent on the magnetisation damping^{38}, namely \(\tan \,{\theta }_{SkH}\propto 1/\alpha \), and as noted in ref.^{10} the experimentally observed skyrmion Hall angles are smaller than those obtained using micromagnetics modelling with the SOT alone, under realistic parameters. As shown in ref.^{9} disorder plays a very significant effect on the skyrmion Hall angle, particularly in explaining its dependence on the skyrmion velocity, and this aspect is also analysed in this work in the following section. The results in Fig. 3 show that inclusion of diffusive spin torque can result in small skyrmion Hall angles even at moderate magnetisation damping values. We propose here the diffusive spin torque may help to explain the experimentally observed small skyrmion Hall angles and it is hoped these results will encourage further work in this direction.
Increasing the number of stack repetitions results in modified demagnetising fields, and these are known to have an effect on skyrmion motion under a SOT^{22,39,40}. In particular multilayered stacks can accommodate hybrid skyrmion structures^{39,40} where the chirality changes with layer number. For the Pt/Co/Ta stacks studied here, with relatively large spacing between the Co layers, we have verified the chirality does not change along the thickness (see Supplementary Material for details). Additionally, the total effective diffusive spin torque is affected by the number of repetitions in the multilayered stack. The spin accumulation generated at a skyrmion diffuses across the Pt and Ta layers, with the transverse components of the spin current absorbed in neighbouring Co layers. Due to the symmetry of the structure, a decrease in the overall diffusive spin torque is expected, reflected in a decrease of the effective perpendicular spin polarisation parameter P_{⊥}. The results for stacks with up to 6 number of repetitions are shown in Fig. 4. With a single stack repetition the only contribution to the total effective diffusive spin torque is the main contribution due to diffusion from the Co layer into the adjacent Pt and Ta layers. From 2 stack repetitions up we have the additional contributions due to diffused spin currents between adjacent Co layers – with 2 stacks each Co layer has contributions due to the other Co layer only; with 3 stack repetitions up, the inner Co layers experience contributions from the 2 adjacent layers. The total effect is a decrease in the total effective diffusive spin torque as the number of layers is increased – this is reflected in the decrease of P_{⊥} as shown in the inset to Fig. 4. Further details are given in the Supplementary Material, where the current densities through the multilayered stack are also shown. It is interesting to note that as the number of repetitions is increased, the total spin torque, consisting of the combination of SOT and diffusive spin torque, results in the same skyrmion motion above 3 repetitions – thus for x = 4, 5, 6 the skyrmion paths and velocities are nearly identical, as seen in Fig. 4. It must be stressed however this is not generally true since the skyrmion motion is a result of the interplay between the spin torques and demagnetising fields.
Finally, before analysing the effect of defects on skyrmion motion and threshold currents, the velocities are computed in perfect structures using the selfconsistent spin transport solver. The results for up to 3 stack repetitions for both D < 0 and D > 0 are shown in Fig. 5. Due to the symmetry of the various spin torques (see Fig. 2) the velocities for D > 0 are greater than for D < 0, and moreover the motion in both cases opposes the drift direction of electrons. The skyrmion reaches its steady velocity almost instantaneously in these disk structures – any acceleration period is below the numerical error. The velocities obtained are very similar to those obtained in experiments on similar stack compositions with similar skyrmion diameters^{7}. It must be stressed however that a precise comparison is difficult largely due to the unknown skyrmion Hall angle. Moreover the movement of skyrmions is also affected by the shape anisotropy of track structures, and disorder plays a very significant effect on the skyrmion movement path. Comparable skyrmion velocities have also been observed in other experimental studies^{10,11,12}, although the stack compositions are different. When spin pumping is also taken into account, a small decrease in velocity is observed in Fig. 5 which is proportional to the driving current. This is explained as an increased spin drag effect as the skyrmion velocity increases, resulting in larger pumped spin current in Eq. (9).
Threshold Currents
Experimental results on currentinduced skyrmion motion show the existence of threshold currents required to initiate and sustain motion^{7,8,9,10,11,12}. Further, the skyrmion Hall angle has been found to vary with skyrmion velocity^{8,9,10}. These effects are difficult to explain using simulations with perfect structures and constant material parameters. Material imperfections seem to play a very significant role in explaining these experimental observations. Previous studies have shown how a threshold current arises due to confining pinning potentials^{41,42}, defect scattering^{43,44}, polycrystalline structures with crystallites of varying anisotropy axes orientation^{45}, disorder originating from M_{S} fluctuations using a granular structure^{46}, as well as disorder in the DMI^{7,8} and anisotropy constant^{7,11}. Imperfections have also been shown to result in a change of the skyrmion Hall angle with skyrmion velocity due to sliding motion along grain boundaries^{45}. Another mechanism which results in variation of the skyrmion Hall angle with velocity is due to the fieldlike SOT component in combination with breathing skyrmion modes^{47,48}, or deformations and internal mode excitations^{10}. The fieldlike SOT in Eq. (10) has also been included in this work. Moreover Brownian motion of skyrmions due to thermal effects can result in distortions and diffusion of skyrmions^{49,50}.
Here we consider, in addition to variations of M_{S} and K_{u} parameters, also the effect of topographical surface roughness, included in simulations as a roughness field^{51,52}. Topographical surface roughness results in an effective uniaxial anisotropy when averaged over the entire sample, however locally the roughness field has strong variations, which can result in confining potentials due to local fluctuations of the total effective anisotropy. Since the Co layers are very thin, variations in thickness of even a single monolayer can result in strong confining potentials. In this work we consider the effect of surface roughness up to 2 Å roughness per surface which is comparable to a single monolayer thickness variation. Roughness textures are generated as jagged granular profiles (see Methods section) with a 50 nm grain size, as shown in Fig. 6 for topographical surface roughness. It is known that threshold currents depend on the skyrmion diameter to grain size ratio, with the strongest pinning obtained when this ratio is close to unity^{8}. Here we keep the grain size fixed in order to investigate threshold currents and variation of the skyrmion Hall angle. Further analysis using combinations of various sources of imperfections as well as grain size variation is outside the scope of the current work.
Results for skyrmion motion using surface roughness are shown in Fig. 6, both for zero temperature and room temperature. For the latter a thermal field was also introduced as outlined in the Methods section. For the simulations in this section the skyrmion was relaxed into a confining site, then current densities of various strengths were applied and the skyrmion motion was computed using the full spin transport solver. For small current densities the skyrmion tends to undergo an orbiting motion inside the confining potential, as shown in Fig. 6. As soon as the current is turned off, the skyrmion relaxes back to the initial position, representing the lowest energy configuration inside the twodimensional confining potential. As the current is increased, eventually the skyrmion is able to escape. The calculated threshold current of 10^{11} A/m^{2} for 2 Å surface roughness is very similar to that found in experiments^{7}. The skyrmion motion is strongly influenced by the local roughness profile and can differ considerably from that obtained in perfect structures. With surface roughness the skyrmion tends to follow a path with greatest layer thickness, since this represents the lowest energy path. As the current density is increased the skyrmion path tends towards that obtained in perfect structures, as shown in Fig. 6. Whilst the movement direction is strongly affected by the local roughness profile, the average skyrmion velocity above the threshold current is similar to that obtained in perfect structures, as seen in Fig. 7(a), especially for the full spin transport solver results which include both the SOT and interfacial diffusive spin torque. The skyrmion velocity shown in Fig. 7(a) was also obtained separately using these two contributions; with the interfacial diffusive spin torque the velocities are slightly larger compared to those in ideal structures since the movement direction due to this torque alone is close to the lowest energy path direction on average – compare the low currentdensity path in Fig. 6 with that obtained under the diffusive interfacial spin torque alone in Fig. 2(a). When a stochastic thermal field is introduced for roomtemperature simulations the skyrmion paths are largely unaffected, showing only a small random variation around the path taken without a thermal field. The threshold current is also unaffected. This suggests the additional Brownian motion of skyrmions, including diameter variations due to thermallyexcited breathing modes, is insufficient to overcome the pinning potentials in this case.
We further study the effect of magnetic defects, in particular considering variation of M_{S} and K_{u} parameters by changing the variation amplitude from 5% up to 15%. The results are shown in Fig. 7. As expected, increasing the variation amplitude results in increasing threshold currents, with the largest threshold currents obtained for 15% variation as 1.25 × 10^{11} A/m^{2}, comparable to that obtained for 2 Å surface roughness. It is unclear if such a strong parameter variation amplitude is likely in good quality samples, however a single monolayer variation at surfaces is possible, particularly in multilayered stacks considering the size of typical samples used to study skyrmion motion. The average skyrmion Hall angle is plotted in Fig. 7(c,d) as a function of both current density and parameter variation amplitude. As the current density is increased the skyrmion Hall angle tends to that obtained in the ideal structure, levelling off as the current density is increased. This behaviour is also observed under the SOT and diffusive interfacial spin torque separately as shown in Fig. 7(b). Such a strong influence of the skyrmion velocity on its motion direction has also been observed in experimental studies^{8,9,10}. Moreover, imaging of multiple skyrmions movement has shown simultaneously both negative and positive skyrmion Hall angles within the same driving current pulse^{8}. The results in Fig. 7 show how the sign of the skyrmion Hall angle can change depending on the level of disorder, as well as the driving current density, highlighting the effect local disorder can have on skyrmion movement.
Here we showed that in addition to magnetic defects, topographical surface roughness also plays a very important part. The results on surface roughness show it may be possible to design devices with skyrmion motion only along the current direction, by purposely enlarging the thickness of the structure in the center. This creates a strong confining potential in the center, whilst avoiding the sample boundaries, without significantly affecting the skyrmion speed. It is hoped these results will further stimulate experimental work in this direction.
Conclusions
In conclusion, we have studied single skyrmion motion in ultrathin multilayered Pt/Co/Ta disks by means of micromagnetics simulations coupled with a selfconsistent spin transport solver. Vertical spin currents can drive skyrmions very efficiently in such structures. One source of vertical spin currents is the SHE, resulting in SOTs acting on the Co layers. Another source of vertical spin currents was shown here, resulting from interlayer diffusion of a spin accumulation generated at a skyrmion. This diffusive spin torque was shown to act in the direction of electrical current irrespective of the skyrmion chirality or topological charge, and in ultrathin films can be comparable to the SOT. The combination of SOT and diffusive spin torque was found to result in small skyrmion Hall angles even for small magnetisation damping values. Further, the effect of magnetic defects and topographical surface roughness on the skyrmion Hall angle and threshold current was studied. In particular topographical surface roughness, as small as a single monolayer variation, was shown to have a marked effect, resulting in a dependence of the skyrmion Hall angle on the skyrmion velocity, with threshold currents comparable to those found in experiments.
Methods
All simulations were done using Boris Computational Spintronics software^{53}, version 2.2. Material parameters used in the simulations are summarised in Table 1.
In Table 1 the spin dephasing length is given is given by \({\lambda }_{\varphi }={\lambda }_{J}\sqrt{{l}_{\perp }/{l}_{L}}\), where l_{⊥} and l_{L} are the transverse spin coherence and spin precession lengths respectively^{54,55}, estimated as 4 nm for Co.
Computations were done using cellcentered finite difference discretisation. Differential operators are evaluated to second order accuracy in space, for both magnetisation and spin transport calculations. For magnetisation dynamics the computational cellsize used was (4 nm, 4 nm, 1 nm). For spin transport calculations the computational cellsize used was (4 nm, 4 nm, 0.5 nm) for the Pt and Ta layers, and (4 nm, 4 nm, 0.25 nm) for the Co layers. The LLG equation was evaluated using the RK4 evaluation with a 0.5 ps fixed time step. The Poisson equations for spin and charge transport, e.g. Eq. (4) for S, were evaluated using the successive overrelaxation algorithm. All computations were done on the GPU using the CUDA C framework.
In the LLG equation the contributing interactions are the demagnetising field, direct exchange interaction, the interfacial DzyaloshinskiiMoriya exchange interaction, uniaxial magnetocrystalline anisotropy and applied field. The roughness field resulting from topological surface roughness is described in^{51}. Roughness profiles were generated using a jagged granular generator algorithm. Equally spaced coefficients at 50 nm spacing in the xy plane are randomly generated. The remaining coefficients are obtained using bilinear interpolation from the randomly generated points. The resulting array of coefficients in the xy plane are used to locally multiply the base parameter values, M_{S} and K_{u}, or to obtain a topographical surface roughness profile – see Fig. 6. Further details are given in the user manual for Boris^{53}. Simulations with a thermal field at room temperature were done using the stochastic LLG equation, evaluated using Heun’s method, with a thermal field obtained as:
where V is the computational cell volume, Δt is the time step, and T is the temperature. The interfacial DMI effective field is introduced as shown below, where M_{x}, M_{y}, M_{z} are the components of magnetisation:
Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Lepadatu, S. Effect of interlayer spin diffusion on skyrmion motion in magnetic multilayers. Sci Rep 9, 9592 (2019). https://doi.org/10.1038/s41598019460911
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DOI: https://doi.org/10.1038/s41598019460911
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