Universal chiral-triggered magnetization switching in confined nanodots

Spin orbit interactions are rapidly emerging as the key for enabling efficient current-controlled spintronic devices. Much work has focused on the role of spin-orbit coupling at heavy metal/ferromagnet interfaces in generating current-induced spin-orbit torques. However, the strong influence of the spin-orbit-derived Dzyaloshinskii-Moriya interaction (DMI) on spin textures in these materials is now becoming apparent. Recent reports suggest DMI-stabilized homochiral domain walls (DWs) can be driven with high efficiency by spin torque from the spin Hall effect. However, the influence of the DMI on the current-induced magnetization switching has not been explored nor is yet well-understood, due in part to the difficulty of disentangling spin torques and spin textures in nano-sized confined samples. Here we study the magnetization reversal of perpendicular magnetized ultrathin dots, and show that the switching mechanism is strongly influenced by the DMI, which promotes a universal chiral non-uniform reversal, even for small samples at the nanoscale. We show that ultrafast current-induced and field-induced magnetization switching consists on local magnetization reversal with domain wall nucleation followed by its propagation along the sample. These findings, not seen in conventional materials, provide essential insights for understanding and exploiting chiral magnetism for emerging spintronics applications.

spin polarization (P = 0.5), the conventional adiabatic and non-adiabatic spin transfer torques (STT) play a negligible role in the current induced magnetization switching. The current-induced magnetization switching of the ultrathin square (L = 90nm, L z = 0.6nm) was also analyzed considering the influence of the conventional adiabatic and non-adiabatic spin transfer torques (STT) 12,22 . The results are shown in Fig. S.3. In the absence of SHE (θ SH = 0), these STTs can not drive the switching by themselves. In the presence of SHE (θ SH = 0.11), the STTs only slightly delay the reversal but do not modify the reversal mechanism.

IV. CURRENT INDUCED MAGNETIZATION SWITCHING IN THIN DISK
Similar CIMS reversal mechanism was also observed for a thin disk with diameter 90nm, the same as the side of the square of the main text. The thickness is also the same (L z =

MICRO-SIZE SQUARE
The field-induced magnetization switching was studied for an extended L = 1000nm square dot with the same thickness 0.6nm as the small nanodot square of the main text.
The field-driven magnetization reversal is studied under a static longitudinal in-plane field

A. Assuming the same conductivity in the heavy-metal and the ferromagnet
The results shown in the main text were computed by assuming that the current flows uniformly distributed along Pt/Co layers, so the effective conducting thickness is 3.6nm. This is the common assumption in the experimental studies to estimate the spin Hall angle 3,7 , in particular, in the experimental work by Garello et al. 5 that we have reproduced, not only qualitatively but also quantitatively.
The electrical current distribution in the Pt/co bilayer was numerically computed by means of COMSOL 24 simulations assuming that the Co has the same resistivity as the Pt:

B. Considering the different conductivities in the heavy-metal and the ferromagnet
As it was mentioned above, for our quantitative description of the experiments by Garello 5 we have assumed that the current flows uniformly distributed along Pt/Co layers, which is also the conventional assumption in most of the experimental studies to estimate the spin Hall angle 3,5,7 . Here we point out that a more precise analysis would need to be adopted accounting for the different electrical resistivity of the Co and the Pt: ρ(Co) =  is negligible (see Fig. S.9). B Oe in the Co layer reaches its higher magnitude at top (y = L) and bottom (y = 0) edges where |B Oe,max | ≈ 3mT for j a = 10 12 A/m 2 , which would result in |B Oe,max | ≈ 12mT for j a = 4 × 10 12 A/m 2 . As in the previous case, this value is ≈ 20 times smaller than in-plane and SHE effective fields, so it does not modify significantly the results, as it was confirmed by full micromagnetic simulations including it for some tested cases.

VII. DISCUSSION OF THE MICROMAGNETIC PARAMETERS
In this study we focus our attention in ultrathin FM Co layer (0.6nm) sandwiched between a HM Pt with thickness in the range 3 − 5nm and capped by AlO. The material parameters considered as inputs for the modeling are the saturation magnetization M s , the exchange stiffness constant A, the uniaxial PMA constant K u , the Gilbert damping α, the spin Hall angle θ SH and the DMI parameter D. Due to the atomic scale thickness of the FM Co layer, the polarization factor P of the spin current flowing through it is assumed to be negligible in agreement with several experiments 9,13-15 , and therefore, both the conventional adiabatic and non-adiabatic STTs along with the FL-SOT due to the Rashba are assumed to be negligible, and these effects are not taken into account in the results of the main text. Nevertheless, several micromagnetic tests were performed including the adiabatic and non-adiabatic STTs with P = 0.5 and β = 0.3. Although these STTs introduce a tiny delay in the CIMS (see Fig. S3), they do not modify substantially the CIMS driven by the SHE.
The SL-SOT due to the Rashba has the same symmetry than the corresponding spin Hall SL-SOT. However, here we are neglecting its effect based on direct experimental measurements 3 and also on the fact that it would need finite contribution from the conventional STTs with either P < 0 or a negative non-adiabatic parameter (β < 0) to explain the current-driven DW dynamics against the electron flow. Although some theoretical studies have suggested P < 0 or β < 0, no experimental evidence has been presented so far.
Moreover, our assumption is also justified by the fact that this Rashba SL-SOT would be also proportional to the spin polarization of the current P which is assumed to be vanishingly small for the atomistic scale of the Co thickness 7,9,[13][14][15] . Therefore, although we  .6 × 10 −11 J/m for Pt/Co/AlO systems, which is the same value we are considering here to reproduce the experiments by Garello 5 . Some trials were also done with A = 10 −11 J/m with analogous qualitative results, but the better quantitative agreement with the experiments for Pt/Co/AlO was achieved with A = 1.6 × 10 −11 J/m. The magnitude of the uniaxial PMA constant K u is commonly deduced from the measurement of in-plane field (B sat = µ 0 H sat ) needed to saturate the magnetization in the plane of the film. From this H sat , the effective uniaxial anisotropy K ef f is deduced. K ef f includes contributions from the uniaxial PMA K u and from the out-of-plane contribution shape anisotropy, K ef f = K u − 1 2 N z µ 0 M 2 s with N z the out-of-plane magnetostatic factor by Aharoni 20 . The typical values for K u obtained from K ef f = K u − 1 2 N z µ 0 M 2 s range from 1.1×10 6 J/m 3 (see 17 and 11 for unpatterned films) to 8.9×10 5 J/m 3 as would be deduced from ref. 5 for a patterned square dot with L ∼ 90nm. Although this way to estimate K u could be justified as a first approach, a better estimation can be performed for more accurately obtaining K u . Note that using K ef f = K u − 1 2 N z µ 0 M 2 s implies assuming N z ≈ 1 which could be justified if the sample depicted uniform-magnetization. However, this is not the case due to the finite DMI, and the full 3D space dependence of the demagnetizing field should be taken into account. Moreover, the value of B sat = µ 0 H sat could be influenced by the DMI, and it would play a significant role, specially for patterned dots at the nanoscale. We The value of the spin Hall angle for ultrathin Pt/Co/AlO stacks have been ranged from 0.056 < θ SH < 0.16 2,4 . An intermediate value of θ SH = 0.11 in this range has been selected for our study in main text, which is also the same as the one experimentally deduced by Garello et al. 5 . As it was already mentioned, this value was estimated from the SHE efficiency H SH j =h θ SH 2eµ 0 MsLz by considering that the electric current flows uniformly through the Pt/Co bilayer. As mentioned above (Fig. S.4), if the different electrical resistivity of the Pt and Co layers were taken into account, only 64% of the current would be flowing through the Pt, and therefore, this would result in a different spin Hall angle. In order to reproduce the experimental data by Garello et al. 5 by considering uniform current through both the Pt and Co layer, we have use their experimentally deduced value θ SH = 0.11. Note also that a very close value was also recently deduced for Pt θ SH = 0.098 by Ryu et al. 10 .
Finally, it remains to justify the assumed value for the DMI parameter. At the moment, the measurements of this parameter are still very few. For instance, using spin Hall magnetometry, Emori et al. 9  In short, all our inputs for the material parameters are well justified within the uncertainty and the scattered of the experimentally deduced values.

DMI PARAMETER AND THE SPIN HALL ANGLE
As discussed in former Sec. VII, the spin Hall angle θ SH and the DMI parameter D for

MENTAL DATA
Although our µM simulations describe the experimental data by Garello et al. 5 with a quantitative accuracy that is well within any reasonable expectation in experimental physics, the exact reproduction of their experiments is beyond the scope of the present work. Here, possible sources of these minor discrepancies between their experimental and our µM results can be enumerated as follows: i) As it is usually considered in experiments 5,7 , in our µM study the current is assumed to be uniformly distributed throw the Pt/Co bilayer. However, the tabulated resistivity of Co is ρ(Co) = 62.4 × 10 −9 Ωm whereas ρ(P t) = 105 × 10 −9 Ωm for Pt at room temperature.
In the studied stack the current density is not uniformly flowing in the Pt/Co bilayer, and a simple estimation confirmed by COMSOL 24 simulations indicates that even for the small Co thickness (0.6nm) as compared to the Pt layer (3nm), indeed the 36% of the current would flow through the Co layer. This non-uniform distribution of the current would reduce the electrical current in the Pt and the SHE efficiency, and it would result in a smaller spin Hall angle. If the different electrical resistivity of the Pt and Co layers were taken into account, only 64% of the current would be flowing through the Pt, and therefore, this would result in a smaller spin Hall angle θ SH = 0.07, which is precisely the value deduced by Liu et al. 3 .
ii) On the other hand, it is also known that the resistivity also depends on the magnetic texture 27 , and therefore, it is expected that the resistivity of the Co layer increases as soon as the DW is nucleated with respect to the quasi-uniform state. Contrary to the pure electrical resistivity, this magneto-resistive effect would increase the current that flows through the Pt in detriment to the Co layer, resulting in a net increase of the SHE.
iii) The non-uniform current along the x-axis would also generate a non-uniform Oer-  iv) It has to be taken into account that in the Co layer the thickness is only three atomic layers, and due to the fabrication process there must be a significant random disorder at the Pt/Co interface. This disorder must result in a random dispersion of some of the material parameters, such as K u , D and θ SH . v) Finally, Joule heating effects due to the electric current flowing through the Pt and/or the Co layers may also play a role on the current induced magnetization switching, for instance by modifying the nominal values of M s , A and K u .
Although all these effects (non-uniform distribution of the current due to both the electric and magnetic effects, the Oersted field, the random dispersion of the material parameters due to the disorder at the Pt/Co and Co/AlO interfaces and Joule heating) should be taken into account for extremely precise estimations of the spin Hall angle and the DMI parameter, they are second-order contributions which would not modify the qualitative reversal mechanism addressed here. Refining the model to include them is however, beyond the scope of this work. Indeed, analogous reversal mechanisms consisting on DW nucleation and subsequent DW propagation were also observed under the presence of the edge roughness and thermal effects, which clearly supports the universality of the switching mechanism for strong DMI systems.