Abstract
Spin orbit interactions are rapidly emerging as the key for enabling efficient currentcontrolled spintronic devices. Much work has focused on the role of spinorbit coupling at heavy metal/ferromagnet interfaces in generating currentinduced spinorbit torques. However, the strong influence of the spinorbitderived DzyaloshinskiiMoriya interaction (DMI) on spin textures in these materials is now becoming apparent. Recent reports suggest DMIstabilized 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 currentinduced magnetization switching has not been explored nor is yet wellunderstood, due in part to the difficulty of disentangling spin torques and spin textures in nanosized 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 nonuniform reversal, even for small samples at the nanoscale. We show that ultrafast currentinduced and fieldinduced 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.
Introduction
Understanding and controlling the currentinduced magnetization dynamics in high perpendicular magnetocristaline anisotropy heterostructures consisting of a heavymetal (HM), a ferromagnet (FM) and an oxide (HM/FM/O) or asymmetric HM1/FM/HM2 stacks, is nowadays the focus of active research^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}. Apart from their interest for promising spintronics applications, these systems are also attracting growing attention from a fundamental point of view due to the rich physics involved in the currentinduced magnetization switching (CIMS)^{1,2,3,4,5} and in the currentinduced domain wall motion (CIDWM)^{7,8,9,10,11}. Indeed, the combination of a HM and a thin FM film gives rise to new phenomena which normally vanish in bulk, but play an important role as the thickness of the FM is reduced to atomistic size.
Currentinduced torques arising from spinorbit phenomena can efficiently manipulate magnetization. In particular, the Slonczewskilike spinorbit torque (SLSOT)^{1,2,3,4,5,6,7,8,9,10,11} can switch the magnetization from up () to down () states and vice versa under the presence of small inplane fields. The SLSOT is expressed as
where is the gyromagnetic ratio, the unit vector along the magnetization, the unit vector along the polarized current which is perpendicular to both the easy axis () and current direction given by and parameterizes the torque. CIMS in ultrathin Pt/Co/AlO, where the Co layer is only thick (around three atomic layers), was experimentally observed first by Miron and coworkers^{1}, where the switching was attributed to SLSOT due to the Rashba field^{17,18}. The Rashba effect would generate both fieldlike (FLSOT)^{17,18} and Slonczewskilike (SLSOT)^{19,20} spinorbit torques. Similar to the conventional spin transfer torque (STT)^{21}, both Rashba FL and SL SOTs have magnitudes proportional to the spin polarization of the current () flowing through the FM and therefore, they are expected to be negligible for an ultrathin FM, as reported in experimental studies^{22,23,24}. Indeed, Liu et al.^{3} studied CIMS in Pt/Co/AlO, similar to the study by Miron et al.^{1} but they did not find any significant dominant Rashba FL torque and therefore the Rashba contribution to the SLSOT should be even vanishingly small. This was also the conclusion from switching experiments in asymmetric Pt/Co/Pt^{8} and for Pt/CoFe/MgO^{9}. Instead of the Rashba SLSOT, the switching is consistent with an alternative SLSOT based on the spin Hall effect (SHE)^{25,26}. The SLSOT due to the SHE is physically distinct from other torques STTs and RashbaSOTs: it is independent of because it arises from the spin current generated in the HM, rather than the spin polarization of the charge current in the FM.
The key to the existence of the SOTs is a high spinorbit coupling combined with structural inversion asymmetry (SIA) in these heterostructures: if the top and bottom interfaces/layers sandwiching the FM were completely symmetric, all the mentioned effects should cancel out. However, not only the SIA plays a role in these currentinduced magnetization dynamics but, it can also influence the static magnetization state through the interfacial DzyaloshinskiiMoriya interaction (DMI)^{27,28,29,30}. In systems with SIA, the interfacial DMI is an anisotropic exchange contribution which directly competes with the exchange interaction and when strong enough, it promotes nonuniform magnetization textures of a definite chirality such as spin helixes^{31}, chiral domain walls (DWs)^{8,9,10,11,30} and skyrmions^{32,33,34}. In particular, the experiments on currentinduced DW motion along Pt/Co/AlO^{7} or Pt/CoFe/MgO^{9,11} can be explained by the combined action of the DMI and the SHE. The strong DMI in these Pt systems is the responsible of the formation of the Neel walls with a given chirality, which are driven by the SHE^{9,10,11}. However, the influence of the DMI on the CIMS has not been explored nor is yet wellunderstood, due in part to the difficulty of disentangling spin torques and spin textures in nanosize confined dots.
On the other hand, experiments on CIMS in these asymmetric multilayers are usually interpreted in the framework of the singledomain model (SDM) which neglects both the exchange and DMI contributions and only a few recent studies in extended samples at the microscale () have considered the nonuniform magnetization by full 3D micromagnetic simulations^{35,36,37,38}. Here we focus on CIMS of a ultrathin Pt/Co/AlO with inplane dimensions two orders of magnitude below (). Although these dimensions should be amenable for the uniform magnetization description, our study indicates that the DMI is also essential to describe the CIMS at these dimensions, which occurs through chiral asymmetric DW nucleation and propagation. We analyze the key ingredients of the switching and confirm that a full micromagnetic analysis is necessary to describe and quantify the spin Hall angle under realistic conditions.
Results
The considered heterostructure here consists on a thin ferromagnetic Co nanosquare with a side of and a thickness of sandwiched between a AlO layer and on top of a Pt cross Hall (Fig. 1(a)). The thickness of the Pt layer is . Typical high PMA material parameters were adopted in agreement with experimental values^{5,6,38}. Details about the physical parameters can be found in Methods.
Cuasiuniform currentinduced magnetization switching in the absence of the DMI: single domain approach and micromagnetic results
The current induced magnetization dynamics under static inplane longitudinal field and current pulses is studied from both Single Domain Model (SDM) and full micromagnetic Model () points of view (see Methods). We first review the CIMS in the framework of the SDM, where the magnetization is assumed to be spatially uniform (). Within this approach the conventional symmetric exchange and interfacial DMI are not taken into account (). In the absence of inplane fields () or thermal fluctuations, with the magnetization initially pointing along the easy axis (, ), a moderate current along the longitudinal direction (axis) only generates an effective SHE field along the axis which does not promote the out  ofplane magnetization reversal (). However, in the presence of a longitudinal field below the saturating inplane field (), acquires a finite longitudinal component parallel to and the current pulse generates an outofplane component effective SHE field . If is parallel to (either and as in Fig. 1(c), or and as in Fig. 1(e)) and their magnitudes are sufficiently strong, the magnetization is stabilized pointing parallel to the outofplane component of , i.e. along the axis (Fig. 1(c) and (e)). On the contrary, if the field and the current pulse are anti parallel to each other (either and , or and ), is stabilized along the axis (Fig. 1(d)).
Fig. 1(f) shows the 3D magnetization trajectories for CIMS starting from the up state () with and for in the absence of DMI (). In this case, the reversal occurs via quasiuniform magnetization precession and therefore, the SDM reproduces accurately the magnetization dynamics (solid red line in Fig. 1(f)) computed from a full point of view (black dots in Fig. 1(f)), confirming the validity of the uniform magnetization approach in the absence of DMI ().
The SDM stability phase diagrams showing the terminal outofplane magnetization direction as function of and (with , and different amplitudes ) are depicted in Fig. 1(g) and (h) for a high and a more realistic value of spin Hall angle respectively. These results were computed at room temperature by averaging over stochastic realizations. The same results were also obtained at zero temperature (see open circles in Fig. 1(g) and (h)). Note that is around twice the value experimentally deduced for the Pt/Co from efficiency measurements^{6}, where was estimated . Therefore, these experiments^{6} cannot be reproduced by the SDM unless unrealistic values of are assumed^{6}. As it will be shown later, the key ingredient to achieve quantitative agreement is the presence of DMI, which can only be taken into account in a full analysis.
Nonuniform magnetization patterns and current induced magnetization switching (CIMS) in the presence of finite DMI: micromagnetic results
Although the SDM could qualitatively describe the stability phase diagrams, it fails to provide a quantitative description of the experiments^{3,6} and the spatial magnetization dependence () needs to be taken into account for a realistic analysis. Indeed, it has been argued that the DzyaloshinskiiMoriya interaction (DMI) arises at the interface between the HM (Pt) and the FM (Co) layers^{9,38}. In particular, it was confirmed that apart from SLSOT due to the SHE, also the DMI is a key ingredient in governing the statics and dynamics of DWs along ultrathin FM strips sandwiched in asymmetric stacks^{9,10,30}. Similarly to the conventional symmetric exchange interaction () responsible of the ferromagnetic order, the interfacial DMI effective field is only different from zero if the magnetization is a nonuniform continuous vectorial function . Apart from promoting nonuniform magnetization textures of a definite chirality in the bulk of the FM, the interfacial DMI also imposes specific boundary conditions (DMIBCs) at the surfaces/edges of the sample^{34}. Indeed, for finite DMI (), the DMIBCs ensure that the local magnetization at the edges rotates in a plane containing the edge surface normal and therefore, in a finiteferromagnetic dot the uniform state is never a solution, so the SDM does no longer apply. Further details of the are given in Methods.
Nonuniform equilibrium states under and in the absence of current
In the equilibrium state at rest (), the average magnetization (, where represents the average in the FM volume) points mainly along the easy axis, either along (, Fig. 2(a)) or (, Fig. 2(b)). However, deviates from this easy axis direction at the edges (see Fig. 2). For the up state (), the local magnetization depicts a finite longitudinal component (), with and at the left and at the right laterals respectively (see Fig. 2(a)). Similarly, has a nonzero transversal component (), with and with at the bottom and top edges respectively. Instead of pointing inwards (Fig. 2(a)), the directions of the inplane components at the edges reverse to outwards for the state (, Fig. 2(b)). The deviations from the perfect outofplane state are maximum at the edges and decrease over a distance given by toward to the sample center.
A moderate positive longitudinal field well below the inplane saturating field slightly modifies the outofplane magnetization in the central part of the FM sample, but it introduces significant changes in the local magnetization at the edges, as it can be seen in Fig. 2(c)(d). A finite longitudinal component parallel to arises at both bottom and top transverse edges (see in Fig. 2(c)(d)). Importantly, the effect of the positive field with is opposite at the longitudinal left and right edges. Whereas supports the positive longitudinal magnetization component at the left edge , it acts against the negative longitudinal magnetization component at the right edge for the state, as it is clearly seen in Fig. 2(c). For the state, supports the positive and acts against the negative (Fig. 2(d)).
Nonuniform CIMS from to with and for finite DMI ()
Since for finite DMI () the equilibrium states of Fig. 2 depict nonuniform magnetization patterns and the SHE effective field depends on the local magnetization (), the magnetization dynamics must be also nonuniform, even for the small nanosized confined dots with with strong DMI. The nonuniform magnetization dynamics under static longitudinal field () was studied under injection of current pulses (Fig. 1(b)) with , and (corresponding to an uniform current through the Pt/Co section, ) by solving the dynamics equation (Methods). The value for the spin Hall angle is as deduced experimentally by Garello et al.^{6} for similar samples. The temporal evolution of the Cartesian magnetization components averaged over the volume of the FM ( with ) and the current pulse temporal profile () are shown in Fig. 3 for different combinations of and which promote the CIMS from to ( and ) and from to ( and ). Representative transient magnetization snapshots during the CIMS are also shown in Fig. 3, which clearly indicate that the switching is nonuniform as opposed to SDM predictions.
We focus our attention on the CIMS from to with and (left graphs in Fig. 3) in the presence of strong DMI (). The temporal evolution of the Cartesian magnetization components over the ferromagnet volume ( with ) is shown in Fig. 3(a), whereas representative transient magnetization snapshots are shown in Fig. 3(b)(f). The reversal takes place in two stages. The first one consists on the magnetization reversal at the top left corner of the square resulting in DW nucleation and the second one occurs via currentdriven domain wall (DW) propagation from the left to right due to the SHE. Apart from the snapshots of Fig. 3(b)(f), these two stages are also evident in the temporal evolution of the outofplane magnetization shown in Fig. 3(a). From to , decreases gradually, whereas it decreases almost linearly from to , consistent with the currentdriven DW propagation where its internal structure is seen in Fig. 3(d).
The magnetization reversal during the first stage is nonuniform due to the DMI imposed boundary conditions (DMIBCs, see Methods), but to understand in depth the underlaying reasons, it is needed to take into account the chiralinduced nonuniform magnetization () in the presence of the applied field () and current (). As it can be seen in Fig. 2(c) or in Fig. 3(b), and DMIBCs support the positive longitudinal magnetization component () at the leftedge , whereas the negative is very small at the right edge . An schematic view of the local equilibrium magnetization at relevant locations is shown in Fig. 3(g) for the state under and zero current. The effective SHE field is also nonuniform: with and . As the outofplane component is negative (note that for ) and proportional to the local , which is maximum and positive at the left edge (), the reversal starts from the left edge (see Fig. 3(h)). However, in addition to this asymmetry along the longitudinal axis imposed by the DMIBCs and supported by (left right edges), other chiral asymmetry arises along the transverse axis in the left edge: the reversal is first triggered from the top left corner (), whereas the local CIMS is delayed at the bottomleft corner (), as it clearly seen in Fig. 3(c). The reason for this transverse asymmetry relies in the different direction of local torque at the initial state (Fig. 3(i)). The relevant torque is the one experienced by the local magnetization at the left edge due to , which is also supported by : . As the local transverse magnetization has different sign at the top () and bottom () corners of the left edge, both the longitudinal component () and the outofplane component of this torque () point in opposite directions at the top and the bottom corners of the left edge (see Fig. 3(i)). The relevant component of to understand the local reversal is the outofplane one: as at the top left corner but at the bottom left corner, the reversal is firstly triggered from the top corner, where opposes to the initial outofplane component of the magnetization (). Once the local reversal is achieved at the top left corner, the switching expands from left to right and from top to bottom: the local inplane magnetization at the bottom left edge rotates clockwise due to and once becomes negative, also promotes the local reversal.
When all points at the left edge have reversed their initial outofplane magnetization () a lefthanded () downup DW emerges, separating the reversed (with ) from the nonreversed (with ) zones. Note that once the local magnetization has reversed its initial outofplane direction, it experiences little torque due to (see Supplementary Information), so it is stable for the rest of the switching process, which takes place by currentdriven DW propagation during the second stage.
The internal structure of the propagating DW is shown in Fig. 3(d). Even in the presence of the longitudinal field (), its internal moment () and its normal () do not point along the positive axis and the DW depicts tilting or a rotation of its normal due to the SHE currentdriven propagation. The DW tilting has been experimentally observed in the absence of inplane field under high currents^{39} and theoretically studied, both in the absence and in the presence of inplane fields, in elongated strips along the axis^{11,40,41,42}. If the only driving force on the downup DW () were a strong positive (negative) current () with , both and would rotate clockwise (counterclock wise)^{41}. Here, we observe that the DW tilting is also assisted during the DW nucleation due to the DMIBCs, and . would support the internal longitudinal magnetization of the lefthanded downup DW if its normal points along the axis (), as it would be the case of currentdriven DW motion along an elongated strip along the axis^{11}. However, due to the nonuniform local CIMS at the left edge in our confined dots, the DW normal has a nonzero negative transverse component () for and . As it is shown in the snapshot of Fig. 3(d), in addition to a positive longitudinal component (), the internal DW moment also has a nonull negative transverse component (). Note that the direction of both and during the DW propagation is also the direction of the local magnetization at the topleft corner, where the reversal was initially launched (see Fig. 3(c),(i)).
The full magnetization switching is completed before the current pulse has been switched off (see Fig. 3(a)), when the propagating downup DW () reaches the right edge. Due to the DW tilting, the reversal occurs first at the top right corner () with respect to the bottom right corner () (see snapshot of Fig. 3(e)). Although this second stage, consisting on currentdriven DW propagation, is similar to the one already explained for elongated thin strips as driven by the SHE^{11,41,42}, the DW nucleation during the first stage has not been addressed so far for such small nanosized confined dots and as it was explained above it is mainly due to the longitudinal field which supports the longitudinal magnetization component at the left edge imposed by the DMIBCs.
Discussion
Universal chiral promoted currentinduced magnetization switching (CIMS) in strong DMI systems
The CIMS from to can also be achieved if both and reverse their directions ( and ). As it is straightforwardly understood from the former description, in this case the reversal is triggered from the bottom right corner (, where opposes to the initial outofplane magnetization) and an updown DW is driven toward the left (not shown). The CIMS from to under anti parallel field and current is shown at the right panel of Fig. 3(j)(r).
In general, the CIMS can be described as follows: () the initial outofplane magnetization direction ( or ) determines the direction (inwards or outwards) of the local inplane at the edges imposed by the DMIBCs. () The longitudinal field supports the longitudinal inplane magnetization component () at one of the two lateral edges and acts against it at the opposite one. () For the favored lateral edge, the local magnetization reversal is triggered at the corner where the outofplane torque due to and opposes to the initial outofplane magnetization component (). After that, the reversal also takes place in the middle part of the selected edge and finally, the other corner is also dragged into the reversed region with the formation of a tilted DW. () The CIMS is completed by the currentdriven DW propagation.
Also remarkable is the fact that for the same current pulses as in Fig. 3 the CIMS is not achieved in the framework of the SDM if a realistic value for the spin Hall angle is adopted ()^{6} and the same limitation was also observed by full simulations in the absence of the DMI (). All these simulations point out that, even for the small confined dots considered here (), the strong DMI and the BCs imposed by it are essential to describe the CIMS driven by the SHE from both quantitative and qualitative points of view. The DMItriggered switching () was also studied for other ultrathin () squares () with different inplane dimensions () and reversal mechanism remains similar to the one already described and depicted in Fig.3. Note that the smallest evaluated side () is small than the minimum side required to achieve thermal stability () according to the conventional criterion: in order to maintain sufficient stability of the data storage over at least five years, the effective energy barrier given by (with the effective uniaxial anisotropy constant from Ref.^{6} and the volume of the sample) should be larger than , where Boltzmann constant. The reversal was also similar under realistic conditions including disorder due to the edge roughness and thermal effects (see Supplementary Information). Moreover, this chiral CIMS, either from to or from to , does not change when the FM Co layer is patterned with a disk shape (see Supplementary Information). It was also verified that this nonuniform reversal mechanism, consisting on DW nucleation and propagation, does not depend on the specific temporal profile of the applied pulse, provided its magnitude () and duration () are sufficient to promote the complete reversal for each .
Chiral nature of the fieldinduced magnetization switching (FIMS)
An analogous CIMS mechanism to the one described here for nanosize samples () was recently observed by Yu et al.^{43} using Kerr microscopy for an extended Ta()/CoFeB()/TaO() stack with microsize inplane dimensions (). In that work, righthanded DWs () were nucleated assisted by the inplane field and displaced along the current direction due to the negative spin Hall angle of the Ta. More recently, Pizzini et al.^{38} also used Kerr microscopy to visualize the asymmetric chiral DW nucleation under inplane field and its subsequent propagation along extended (≈70 μm) Pt()/Co()/AlO() thinfilms driven by outofplane field (). Similar to our study, starting from the up state (), a positive (negative) inplane field () promotes the local magnetization reversal at the left (right) edge, which was propagated to the right (left) driven by a negative outofplane field . Their images indicate the nucleated DW has a lefthanded chirality and it propagates without significant tilting due to the extended unconfined inplane dimensions (≈70 μm). In order to understand these observations, the fieldinduced magnetization switching (FIMS) has been also studied for confined small squares with (the same geometry as in the former CIMS analysis) and others with lateral dimensions one order of magnitude larger (). Static longitudinal fields with and () are applied along with short outofplane field pulses with with , and (the temporal profile of this pulse is the same as for the currentinduced magnetization switching). The results for the confined square dot are shown in Fig 4 for different combinations of the initial state ( and ), inplane static field ( and ) and outofplane field pulse ( and ). Similarly to the CIMS, the FIMS starts from an edge selected by the direction of , with an even more evident chiral asymmetry between the two corners. Note again that the corner where the reversal starts has a transverse magnetization component () pointing in the same direction as the transverse internal magnetization of the nucleated DW. Once the local switching has been triggered, the reverse domain (pointing along the opposite direction with respect to the initial state) expands asymmetrically along the longitudinal () and transverse () directions (see for instance snapshots at and in Fig. 4). Although here just a quarterofbubble is developed due to the confined shape at the corner, this asymmetric fielddriven chiral expansion is similar to the one recently observed^{12,14} in extended thin films. Moreover, our study also points out a qualitative difference between the currentdriven and the fielddriven nucleation: while the first one is driven by a nonuniform SHE outofplane effective field ( which depends on local ), the second one is promoted by a uniform outofplane field . Therefore, the currentinduced nucleated DW propagates along the current direction (axis, see yellow arrows in Fig. 3(d) and (m)), whereas the fielddriven DW expands radially from the corner (see yellow arrows in Fig. 4). Nevertheless, the fact that similar chiral local magnetization reversal occurs also at the corners of nanosize confined dots ( and below) clearly confirms the universality of the chiral reversal mechanism in these nanosize confined dots with strong DMI.
On the other hand, the Kerr images by Pizzini et al.^{38} do not show the corners of their extended thinfilm (which is unconfined along the transverse axis) which are precisely where our modeling points out additional chiral asymmetry in the DW nucleation for confined dots (Fig. 4). Moreover, in their thinfilms the fielddriven DW does not depict tilting. In order to contrast these observations with our predictions, the fielddriven nucleation and propagation in an confined square dot has been also analyzed here, but with lateral inplane dimensions one order of magnitude larger (). We note that as is increased to the microscale, the nucleated DW is almost straight, with its normal oriented along the axis (no DW tilting), in the middle part of the nucleating edge (far form the corners). However, an asymmetry between the top and bottom corners is still present even for (see Supplementary Information): the reversal from to (from to ) is anticipated at the topleft (topright) corner with respect to the bottom one under and ( and ). This chiral asymmetry at the corners of the extended microsize sample is similar to the observed for a confined dot (see. Fig. 4) and although it has not been addressed before, it could be observed by high resolution techniques^{44}.
CIMS in confined nanodots with rectangular shape
The CIMS was also studied in rectangles with different inplane aspectratios (Fig. 5(a)(d)). The thickness is fixed () as before. Again the switching takes place by DW nucleation followed by its currentdriven propagation along the axis, which further supports the universality of the reversal mechanism in systems with strong DMI. In this case, the nucleation takes place during the first independently of the rectangle aspectratio , but the critical pulse duration () for fixed and , increases linearly with (see the inset in Fig. 5(c)), a prediction which could be experimentally validated to estimate both the spin Hall angle () and the DMI parameter () if the rest of material parameters (, , , ) are known by other means.
Comparison to experiments of currentinduced magnetization switching
Although our study goes further than a mere comparison to available experimental results, it is interesting to show how the nonuniform CIMS can explain quantitatively the experimental measurements by considering realistic material parameters (see Methods and Supplementary Information). With the aim of providing an explanation of experimental observations^{6} for the ultrahin Co square with in a Pt()/Co()/AlO() stack, we have repeated the former study for several values of the applied field () and different different magnitudes of the current pulse (). The rise and fall times () and the duration () of the pulse were maintained fixed as in the experimental study^{6}. Here we consider the up state (, ) as the initial one. For each , the switching probability at room temperature was computed as the averaged over stochastic realizations. Realistic conditions were taken into account by considering random edge roughness with characteristic sizes ranging from  (see Methods). The results are collected in Fig. 6 which indicates a good quantitative agreement with recent experimental measurements^{6}.
It was verified that the CIMS mechanism (local magnetization reversal with DW nucleation and subsequent currentdriven propagation) remains qualitatively unchanged even under these realistic conditions (see Supplementary Information). Moreover, although marginal discrepancies between these data (Fig. 6) and the experimental results shown in Fig. 2(d) of ref.^{6} can be seen, the quantitative agreement is remarkable considering similar material parameters as inferred experimentally^{6}: , , , , and (see Supplementary Information for detailed justification of these inputs). Note that with the SDM a quantitative agreement with the experimental data was only achieved with unrealistic values of the ()^{6}. Note that the DMI parameter was not determined experimentally^{6}, but the fact that this value provides reasonable quantitative agreement with their experiments and that this value is also in good quantitative agreement with very recent estimations by other means for similar Ptbased systems (Ref. 11) constitute additional evidences that our modeling is compatible with the dominant physics behind these CIMS processes.
Conclusions
In summary, the currentdriven magnetization switching in ultrathin HM/FM/Oxide heterostructures with high PMA and strong DMI has been studied by means of full micromagnetic simulations. Even for the small inplane dimensions (), the analysis points out that the magnetization reversal mechanism is nonuniform. It starts by local magnetization reversal induced by the SHE and assisted by the inplane field in collaboration with the DMI boundary conditions. The longitudinal field and the DMI imposed boundary conditions select the lateral/edge and the specific corner at which the nucleation is triggered, where the relevant torques due to the SHE and the longitudinal field accelerate the local reversal. After that, the switching is completed by currentdriven domain wall propagation driven by the SHE, where the current direction determines the direction of the wall motion and the internal magnetization of the propagating wall points closely to the local magnetization at the selected corner where the reversal was initially launched. Similar nucleation and propagation mechanisms were also observed under outofplane fields, confirming again the chiraltriggered magnetization reversal in these nanosize confined dots. These results clearly exclude the single domain approach as a proper model to describe these switching experiments and therefore, the estimations of the spin Hall angle based in this oversimplified model should be revised by adopting a much more realistic full 3D micromagnetic approach. Moreover, by analyzing the switching under realistic conditions including disorder and thermal effects, it was found that the mechanism is universal and for instance, it could be used to the quantify both the DMI and the spin Hall angle by studying the reversal of ferromagnetic layers with different length for fixed width and thickness. As the reversal mechanism occurs in a reliable and efficient way and more importantly, as it is also highly insensitive to defects and thermal fluctuations, our results are also very relevant for technological recording applications combining nonvolatility, high stability, ultradense storage and ultrafast writing.
Methods
Magnetization dynamics under SOT due to the SHE
Under injection of a spatially uniform current density pulse along the axis (see its temporal profile in Fig. 1(b)), the magnetization dynamics is governed by the augmented LandauLifshitz Gilbert eq.
where is the normalized local magnetization with saturation magnetization, is the gyromagnetic ratio and is the effective field derived from the energy density of the system (). The first term in equation (2) represents the precessional torque of around , where is the thermal field representing the effect of thermal fluctuations at finite temperature. is a whitenoise Gaussiandistributed stochastic random process with zero mean value (its statistical properties are given below). The second term in equation (2) is the damping torque with the dimensionless Gilbert damping parameter. The last term in equation (2) is the SLSOT from the spin Hall effect (SHE), where is the unit vector pointing along the direction of spin current polarization due to the SHE in the Pt layer and represents the magnitude of the effective spin Hall field given by
where is thickness of the FM layer, is Planck’s constant, is the electron charge and is the instantaneous value of the electrical density current. As in the experiment by Garello et al.^{6}, the current is assumed to flow uniformly through the HM/FM bilayer (see Supplementary Information for additional discussion). is the Spin Hall angle, which is defined as the ratio between the spin and charge current densities.
Single Domain Model (SDM)
If the magnetization is assumed to be spatially uniform (), the deterministic effective field in equation (2) only includes the PMA anisotropy, magnetostatic and Zeeman contributions . The Zeeman contribution due to the longitudinal field is . The uniaxial PMA anisotropy effective field is
and the demagnetizing field in the SDM approach is expressed as
where is the diagonal magnetostatic tensor with and being the selfmagnetostatic factors^{45} for and .
The thermal field is a stochastic vector process whose magnitude is related to the temperature via the fluctuationdissipation theorem^{46}.
where is the Boltzmann constant, is the volume of the sample, is the time step and is a Gaussian distributed whitenoise stochastic vector with zero mean value ( for ) and uncorrelated in time (, where is the Kronecker delta and the Dirac delta). Here means the statistical average over different stochastic realizations of the stochastic process. Equation (2) was numerically solved with a order RungeKutta scheme with a time step of .
Micromagnetic Model ()
When the spatial dependence of the magnetization is taken into account (), the deterministic effective field in equation (2) includes the spacedependent exchange with the exchange constant and the interfacial DMI ^{30,34} where is a parameter describing the DMI magnitude. Both the local Zeeman and PMA uniaxial contributions to are computed similarly as in the SDM ( and ). Note also that in the the magnetostatic field is also spacedependent on everywhere. The Oersted field due to the current was also taken into account but it was found irrelevant and very small as compared to the other dominant contributions in . (see^{47,48} for the numerical details).
In the absence of DMI (), the symmetric exchange interaction imposes boundary conditions (BCs) at the surfaces of the sample^{49} so that does not change along the surface (, where indicates the derivative in the outside direction normal to the surface of the sample). However, in the presence of the interfacial DMI (), these BCs have to be replaced by^{11,34}
where represents the local unit vector normal to each sample surface.
In the the thermal field is also a stochastic vector process given by
where now is the volume of each computational cell and is a whitenoise Gaussian distributed stochastic vector with zero mean value ( for ) and uncorrelated both in time and in space (). Most of the simulations for perfect samples were performed with a 2D discretization using cells of in side and thickness equal to the ferromagnetic layer (). Several tests were performed with cell sizes of to confirm the numerical validity of the presented results. Realistic samples were also studied by considering edge roughness using cell sizes of . These realistic conditions are introduced by randomly generating edge roughness patterns with different characteristic sizes at all edges. Equation (2) was numerically solved with a order RungeKutta scheme with a time step of by using GPMagnet^{47}, a commercial parallelized finitedifference micromagnetic solver^{48}.
Material parameters
Typical high PMA material parameters were adopted for the results collected in the main text in agreement with experimental values for Pt/Co/AlO^{5,6,38}: saturation magnetization , exchange constant , uniaxial anisotropy constant . The spin Hall angle is assumed to be , also according to experiments by Garello et al.^{5,6}. Note that this value is also in the middle of the experimental bounds estimated by Liu et al.^{2} and Garello et al.^{5} and very close to the one deduced in^{13}. A DMI parameter of is assumed, which is similar to the one experimentally deduced by Emori et al.^{11}. The Gilbert damping is as measured in^{50}. Several tests were also performed by varying these inputs within the range available in the experimental literature (see Supplementary Information).
Additional Information
How to cite this article: Martinez, E. et al. Universal chiraltriggered magnetization switching in confined nanodots. Sci. Rep. 5, 10156; doi: 10.1038/srep10156 (2015).
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Acknowledgements
This work was supported by project WALL, FP7PEOPLE2013ITN 608031 from European Commission, projects MAT201128532C0301 MAT201452477C54P from Spanish government and projects SA163A12 and SA282U14 from Junta de Castilla y Leon.
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E.M. and L.T. conceived and coordinated the project. N.P., L.T., M.H., V.R., S.M. and E.M. performed the micromagnetic simulations. E.M., L.T. and N.P. analyzed and interpreted the results. E.M. wrote the manuscript. All authors commented on the manuscript.
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Martinez, E., Torres, L., Perez, N. et al. Universal chiraltriggered magnetization switching in confined nanodots. Sci Rep 5, 10156 (2015). https://doi.org/10.1038/srep10156
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DOI: https://doi.org/10.1038/srep10156
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