Current-driven skyrmion motion along disordered magnetic tracks

The motion of skyrmions along ferromagnetic strips driven by current pulses is theoretically analyzed by means of micromagnetic simulations. Analytical expressions describing the skyrmion dynamics during and after the current pulse are obtained from an extended rigid skyrmion model, and its predictions are compared with full micromagnetic simulations for perfect samples with a remarkable agreement. The dynamics along realistic samples with random disorder is also studied by both models. Our analysis describes the relevant ingredients behind the current-driven skyrmion dynamics, and it is expected to be useful to understand recent and future experimental.


I. INTRODUCTION
Multilayers consisting in a thin ferromagnetic (FM) strip sandwiched between a heavy metal (HM) and an oxide, or between two different heavy metals, are promising systems to develop novel magnetic memory devices such as the race-track 1 .
Nowadays, Sks are the focus of active research because they offer great potential as information carriers in robust, high-density, and energy-efficient spintronic devices. Bloch-like skyrmions have been observed in some ferromagnets without inversion symmetry 12,13,14 under the presence of an out of plane field ( ). Their magnetization is antiparallel to the at their center and parallel to at the periphery, and it points along the azimuthal direction (⃗ ) in the transition region 15,16 . The dynamics of Bloch-like Sk under direct currents driven by the adiabatic and non-adiabatic spin transfer torques has been studied theoretically by several works 15,16,17,18 . On the other hand, recent experiments 10 have shown that Néel-like Sks can be stabilized at room temperature and zero magnetic field in ultrathin FM/HM films deposited by sputtering, which makes their use appealing for the study of skyrmion structure and dynamics. Differently from Bloch-like Sks [12][13][14][15][16][17][18] , the magnetization of these chiral Néel Sks points radially (⃗ ) in the transition region between the inner and the outer parts 10,18 and its size can be tuned by the nature and the thickness of the materials that comprises the multilayers. Moreover, the current-driven dynamics of Néel-like is driven due to the SHE by short current pulses at speeds exceeding 100 m/s 11 . These observations promise an industrial integration but several challenges must be addressed before these objects can be integrated into spintronic devices. For instance, due to the sputtering, the pinning that each Sk experiences is random and local. Consequently, adjacent Sks can move with different velocities 19 , and they can even collapse when one Sk is highly pinned at a strong local defect 11 . Therefore, a study of the motion of Néel Sk by current pulses under realistic conditions is timely and demanding.
Here, we theoretically study the motion of a single Néel Sk under current pulses along a perfect FM by means of micromagnetic simulations ( ). The current-driven Skyrmion dynamics (CDSD) is described analytically in the framework of the Thiele model 20 , which considers the Sk as a rigid object (Rigid Skyrmion Model, RSM). Realistic FM tracks, where the material parameters depict a random dispersion in the form of grains, are also evaluated. The RSM is extended to take into account the pinning generated by the random grains, and a good quantitative agreement with the results is demonstrated.
This study provides a simple framework to describe recent and further experiments, and it will be useful to control and develop skyrmion-based devices.

II. SKYRMION EQUILIBRIUM STATE AND MODELS.
A FM strip with a width of = 128 and thickness of = 1 is studied. Typical parameters for a HM/FM/Oxide multilayer with strong iDMI are considered 5,8,10,11 : saturation magnetization = 10 6 / , exchange constant = 20 / , uniaxial magnetocristalline anisotropy = 0.8 × 10 6 / 3 and iDMI parameter = 1.8 / 2 . Fig. 1(a) shows the equilibrium state of a Néel-like Sk, which was obtained by minimizing the total energy of the system using Mumax. 21 A 2D grid with 2 side cells was adopted. To extract the skyrmion size, the magnetization profile is described by a 360° where 0 denotes the gyromagnetic ratio and the Gilbert damping constant ( = 0.3) respectively. ⃗ ⃗ is the deterministic effective field which includes the exchange, the magnetostatic, the uniaxial anisotropy and the DMI. The last term in eq. (1)  The CDSD can be also described with the formalism introduced by Thiele 18,20,24 : be described as = − ⃗ with being an elastic constant, which is estimated from a single simulation 25 . is the driving force due to the SHE, which is given by ≈ − ℏ 2 ( ) 2 ⃗ . Eq. (2) describes the motion of a rigid skyrmion with a characteristic size of given by 25 .

IV. SKYRMION DYNAMICS UNDER CURRENT PULSES ALONG IDEAL STRIPS
The CDSD under current pulses with zero rise and falling time is evaluated both by and RSM models. The The CDSD under a current pulse can be also analytically described by the RSM, eq. (2). Its analytical solution during the current pulse ( ≤ ) is: where is the characteristic relaxation time given by = | 2 +( ) 2 |. The post-pulse ( > ) dynamics is determined by: where ( ) and ( ) represent the skyrmion position at the end of the pulse ( = ) as obtained from eqs. (3) and (4). These analytical predictions (solid lines) are compared to the results (open symbols) in Fig. 2

(a) and (b). A remarkable agreement
is observed between both models, which allows us to predict some interesting points. In particular, we have studied how the terminal longitudinal displacement ( ≡ ( ≫ )) depends on both the amplitude ( ) and the duration ( ) of a single current pulse. It has to be taken into account that scales with the transverse displacement at the end of the pulse ( ). If during the pulse the skyrmion approaches to the edge ( ( ) → 2 ), the repulsion from it cannot balance the transverse pushing force due to the current, and consequently, the skyrmion is expelled from the strip 18,19,24 . This imposes a limit in the maximum transverse displacement ( ℎ ) which can be reached without skyrmion annihilation. as function of and can be seen in Fig. 2(c), where it was assumed that the skyrmion annihilates if ℎ ≥ 50 (black area on top in Fig. 2(c)). See Ref. 26 for an analytical estimation of the product ( ) above which the Sk reaches the threshold transverse value ℎ . These results can be inferred from eqs. (3)-(6): the terminal longitudinal distance without expulsion is = , which is proportional to the product ( ). Therefore, the same can be achieved under different combinations of and , as it clearly seen in Fig. 2(c).
Other interesting observation is that the same can be achieved with a single pulse of length = or with pulses each one with = . This is shown in Fig. 2(d), where the displacement for a single pulse with ( , = ) = (10 10 / 2 , 5 ) is compared to the one achieved by a train of = 5 pulses of the same but length = = 1 , when they are applied every 30 . This is interesting for applications, because short pulses are required to minimize unwanted Joule heating effects due to the current injection 27,28 .

V. SKYRMION DYNAMICS UNDER CURRENT PULSES ALONG REALISTIC STRIPS
Former analysis was performed under ideal conditions assuming a perfect strip. However, real samples present unavoidably imperfections. In order to evaluate realistic conditions, the disorder is taken into account in the by considering that the FM strip consists on grains 29 . We assume the easy axis anisotropy direction (⃗ ) is distributed among a length scale defined by a characteristic grain size of 50 . ⃗ of each grain is mainly directed along the perpendicular direction ( -axis) but with a small in-plane component, which is randomly generated over the grains. The maximum percentage of the in-plane component of the uniaxial anisotropy unit vector is 5%. A typical grain pattern (GP) is shown in Fig. 3(c). Four different grain patterns have been evaluated to obtain statistic results. results of the temporal evolution of ( ) under a train of = 5 pulses and = 6 are shown in Fig. 3(a)-(c) for different . Here, the time between consecutive pulses is equal to . Each thin black line corresponds to one of the four GPs evaluated. The presence of disorder imposes a threshold pinning current density ~1.5 × 10 10 / 2 below which the skyrmion is hardly displaced from its initial location ( Fig. 3(a)). Note that the Sk stops at ~20 , well before the end of the train of pulses = 2 = 60 . On the contrary, the Sk is significantly driven by the current pulses for > ( Fig. 3(b)-(c)).
These results can be also described by the RSM, where the pinning can be accounted by adding a pinning force

VI. CONCLUSIONS
The CDSD has been studied by . The results can be analytically described by a RSM, which allows us to obtain the relevant parameters governing the dynamics. In particular, the acceleration and deceleration under current pulses is determined by the same characteristic time , which scales with the inverse of the damping of the system (~1/ ). Our analytical description indicates that the terminal longitudinal displacement under current pulses scales with the pulse length , and the same can be a achieved with a single pulse of a given duration = or with consecutive pulses of the same amplitude of with length = / . These observations are interesting for applications to minimize the unwanted Joule heating effect, which scales with 2 and . results under realistic conditions, including disorder, can be also described by