Nanoscale Advances PAPER Survival of skyrmions along granular racetracks at Cite this: Nanoscale Adv., 2023, 5, room temperature† 4728 Josep Castell-Queralt, Guillermo Abad-López, Leonardo González-Gómez, Nuria Del-Valle and Carles Navau * Skyrmions can be envisioned as bits of information that can be transported along nanoracetracks. However, temperature, defects, and/or granularity can produce diffusion, pinning, and, in general, modification in their dynamics. These effects may cause undesired errors in information transport. We present simulations of a realistic system where both the (room) temperature and sample granularity are taken into account. Key feasibility magnitudes, such as the success probability of a skyrmion traveling a given distance along the Received 27th June 2023 Accepted 27th July 2023 racetrack, are calculated. The results are evaluated in terms of the eventual loss of skyrmions by pinning, destruction at the edges, or excessive delay due to granularity. The model proposed is based on the DOI: 10.1039/d3na00464c Fokker–Planck equation resulting from Thiele's rigid model for skyrmions. The results could serve to rsc.li/nanoscale-advances establish error detection criteria and, in general, to discern the dynamics of skyrmions in realistic situations. current: either the skyrmion is annihilated at the edge or it is 1 Introduction channeled along the racetrack. Magnetic skyrmions are whirling magnetic structures that can However, real skyrmionic racetracks are granular, and one be found in certain magnetic materials.1 Their small size and would like to operate at room temperature. It is known that high mobility have promoted them as promising information granularity acts as an array of pinning potentials for skyrmions carriers, as well as building blocks in ultradense magnetic that results in a minimum applied current density for the memories, logic devices, and computational systems.2–6 In activation of skyrmion motion.22–30 At room temperature, ferromagnetic ultrathin lms, with a heavy-metal substrate, it stochastic effects on the skyrmions' position31–33 could has been found that skyrmions can be stabilized with the aid of compromise their stability when approaching the borders or the interfacial Dzyaloshinskii–Moriya interaction.7–10 The same defects.19,34 Moreover, even above the minimum threshold, mechanism allows the formation of skyrmions in multilayers there is a certain probability of trapping the skyrmion.35–37 with alternating ferromagnets (FMs) and heavy-metals.11,12 The Consequently, in a real racetrack, there is no longer a binary experimental nding of room-temperature skyrmions13 has scenario for the survival of skyrmions aer a given length of boosted the potentiality of skyrmions for applications and, track, for a given time. The problem becomes probabilistic. consequently, the study of their spin-current driven dynamics at Here we address this problem by realistically simulating non-zero temperatures to predict the feasibility of the afore- a nanoracetrack for skyrmion transport, taking into account mentioned applications.14,15 both the (room) temperature and granularity. We use a deter- One of the most promising skyrmion-based applications ministic, yet probabilistic, approach to study the dynamics of proposed to date is the skyrmionic racetrack memory. It is a skyrmion in a granular racetrack at room temperature. In designed to drive skyrmions along the racetrack, with the spin– the present work, we rst model the interaction of the orbit torque produced by a spin-polarized current fed into grains of the system with a skyrmion and derive the stochastic a heavy-metal substrate.2,16,17 In such systems, the borders of the Thiele's equation (STE); second, we obtain the corresponding track create a conning potential that sets a driving velocity Fokker–Planck equation (FPE), which includes the temperature, upper threshold above which the skyrmions would escape the and solve it numerically; and third, we study the feasibility of track.18–21 Hence, when transporting stable skyrmions at zero skyrmionic racetracks and how granularity affects the perfor- temperature along a clean (defect-free and grain-free) racetrack mance of such devices at room temperature. we have two possible scenarios depending on the driving 2 General model Departament de Fı́sica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain. E-mail: carles.navau@uab.cat The rigid (Thiele's) model38 assumes that, during motion, the † Electronic supplementary information (ESI) available. See DOI: internal structure of the skyrmion is not modied, so its posi- https://doi.org/10.1039/d3na00464c tion and velocity can be described by generalized magnitudes, rs 4728 | Nanoscale Adv., 2023, 5, 4728–4734 © 2023 The Author(s). Published by the Royal Society of Chemistry Paper Nanoscale Advances extensively to explain the movement of skyrmions when driving currents are moderate or when the skyrmions are far from annihilation (including defects and temperature) both theo- retically and for understanding experimental measurements.15,23,30,31,36,41–44 The present model works in the regimes in which the skyrmions' deformation does not substantially affect their motion. 2.1 Temperature The temperature T is introduced by considering the term Fst as 2aDkB T 0 a white noise with hFst,ji = 0 and hFst;i ; Fst;j i ¼ dij dðt  t Þ, gm0 Ms where i, j = x, y, z. m0 is the vacuum permeability, kB the Boltzmann constant, dij the Kronecker delta and d(t − t′) the temporal Dirac's delta.31 2.2 Borders and granularity The term Fext in eqn (1) has two contributions, Fext = FR + FG. Fig. 1 Sketch of the modeled racetrack. (a) Clean racetrack: the sky- The force‡ coming from the conning potential of the racetrack rmion is driven along the track a given length L and confined by the is FR and the force produced by the granularity is FG. borders at y = ±W. (b) A Voronoi tessellation, showing a possible grain structure. (c) The grains are modeled as square regions randomly 2.2.1 Borders. The edges of a racetrack create a conning distributed in the plane with Gaussian-distributed anisotropy constants potential that repels the skyrmion. However, if a large enough (represented by the different blue tones in the sketch). driving current is applied, this potential can be overcome and the skyrmion is annihilated when it reaches the edge. An and vs respectively. We consider that the skyrmion is moving exponential function decaying with the edge-skyrmion distance along a thin ferromagnetic racetrack of length L and width 2W. has been proposed to account for these facts.18,45 Thus, the force The thickness of the ferromagnetic layer is d  W, L. Thus, the created by the borders of the racetrack over a skyrmion, whose magnetization is considered uniform across the thickness and generalized position is rs = xs^x + ysŷ, is modeled as we locate the FM layer at z = 0 (see the sketch in Fig. 1a). Considering symmetric Néel-type skyrmions on a background FR = f0[e−(ys+W)/R − e−(ys−W)/R]ŷ, (2) magnetization pointing to −ẑ, the STE for the movement of where f0 is a parameter that controls the strength of the skyrmions driven by damping-like torques (produced by spin- repulsion. polarized currents arising from the spin-Hall effect) aer Since the rigid model does not allow us to consider the feeding an in-plane current JH in a heavy-metal substrate is39,40 skyrmion deformation near the edges, we also assume that ðG  Ms aDÞvs þ Ms N vH þ gMs 2 ðFext þ Fst Þ ¼ 0; (1) a skyrmion is annihilated when it “touches” any racetrack       0 G D 0 0 N border, that is when the distance between its center and the where G ¼ , D¼ , and N ¼ , G 0 0 D N 0 edge equals the radius of the skyrmion. with G, D, and N being constants that depend on the internal 2.2.2 Granularity. To model the granularity force term, FG, structure of the skyrmion. a is the Gilbert damping constant, is a more intricate task. The physical origin of this term arises g (g = 2.21 × 105 m A−1 s−1) the gyromagnetic constant, from the different physical-property values of the different m qH grains that form the FM lms. Although all the relevant vH ¼  B ð^z  JH Þ, with mB being the Bohr magneton, qH the eMs magnetic parameters can vary slightly between grains (e.g., Hall angle, e (>0) the charge of the electron, and Ms the satu- exchange constant, Dzyaloshinskii–Moriya constant, or Ms) the ration magnetization. We use G = 4pdMs3, D = 4pdMs2 and most (usually) modied one is the uniaxial anisotropy constant, N = 4pRMs2, where R is the radius of the rigid skyrmion. These as the strength and the direction of the uniaxial anisotropy values correspond to ideal Néel skyrmions with z-magnetization strongly depend on the crystallographic orientation of the Mz = Ms(R2 − r2)/(r2 + R2). Fst and Fext come from the stochastic material.27,46 For simplicity, we only consider variations of and external forces, respectively. They will contain information anisotropy in our model. about the temperature and granularity, as explained in the A common approach to model granularity within the following subsections. micromagnetic model is to generate a random set of grains, Note that we are assuming that the rigid model holds for the using Voronoi tessellation,36,47,48 with an average size and some parameters used in this paper. As in all Thiele's approaches, randomly distributed variation of anisotropy constant in each this is valid as long as the skyrmions do not change consider- ably their internal structure. Thiele's approach has been used ‡ Strictly, Fext and Fst do not have units of force. Nevertheless, we call them “force terms”, following the usual nomenclature. © 2023 The Author(s). Published by the Royal Society of Chemistry Nanoscale Adv., 2023, 5, 4728–4734 | 4729 Nanoscale Advances Paper grain (see Fig. 1b). However, there is no clear way of how to where rs is the position of the skyrmion and ri represents the translate this into the rigid model. positions of the centers of the different grains. Here we use a slightly different approach. In ref. 49, an To set the value of DK (needed for evaluating fG) we follow the analytical expression for the Thiele's force term generated by same procedure as in the Voronoi tessellation models:36 we a single atomic defect that consists of a local modication of assume that the distribution of the values of the uniaxial anisotropy, Fld, was found as anisotropy constant of the grains follows a normal distribution   with mean value K and standard deviation sK. The value of DK is a2 g 4R2 r R2  r2 Fld ¼ DK  3 ^r; (3) given to each grain randomly following the aforementioned m0 Ms R 2 þ r2 normal distribution. Since FG is randomly generated, for each where DK is the difference between the value of the anisotropy set of parameters several simulations with different FG are constant at the defect and the one of the plain sample, K. a is the performed and averaged to get statistically signicant results. (squared) lattice constant and r is the distance from the position The granularity model presented yields similar results to of (the center of) the skyrmion to the defect. those presented, for example, in ref. 27 using a Voronoi We consider each grain as a surface with an atomic defect tessellation and micromagnetic calculations, justifying the use density given by sld = 1/a2. Each surface differential dS gener- of Thiele's rigid model approach. However, the key point is that ates a differential of force given by sldFlddS. Then, the force that we are able to analytically model the forces due to the grains. a grain exerts over a skyrmion is This simplies the numerical treatment of granular systems ð within the rigid model. Fgr ¼ sld Fld dS; (4) Sg 3 Fokker–Planck equation where the integration is done over the grain surface, Sg. The problem now is that the position and shape of the Eqn (1) is a stochastic equation whose dependent variable is the grains have to be known in order to evaluate this force term. position rs of the skyrmion. Its corresponding Fokker–Planck The grain distribution is modeled as follows. Consider that equation can be derived (in a similar way as described in ref. the whole surface of the track (of total surface area 2WL) is 35): formed by Ng grains of square shape and surface area Lg2. v Thus, Ng = 2WL/Lg2. Those grains are randomly distributed pðr; tÞ ¼ V$½pðr; tÞðvdrv þ vext Þ þ Dd V2 pðr; tÞ; (7) vt over the track with the constraint that there cannot be more with the denitions than four grain centers in any given square surface of side Lg (see Fig. 1c). gM 3 aDkB T Dd ¼  s ; (8) The typical size of grains is Lg x 10–20 nm while the m0 G2 þ D2 a2 Ms 2 radius of skyrmions with the parameters used here (see below) is R x 80 nm. Thus, when evaluating the force over a skyrmion vdrv ¼ Ms ðG  aMs DÞ1 ℕvH ; (9) due to a given grain its relative orientation with respect to the skyrmion and its shape are not relevant. Fgr can be thus evalu- vext ¼ gMs 2 ðG  aMs DÞ1 ðFG þ FR Þ: (10) ated as 2 3rþ     4 Lg 2 2 R2 R2 L g Eqn (7) is a deterministic equation whose solution is p(r,t), Fgr ¼ fG arctan þ þ h   i5 ^r: the probability density of nding the center-of-mass of the r0 r0 r0 3 2 2r0 2 Lg 2 þ r0 2 r skyrmion at position r = (x, y) at time t. It is a convection– (5) diffusion equation. The rst term on the right-hand side indi- cates that the probability density is transported at a velocity vdrv + vext, whereas the second term is a linear, homogeneous, Eqn (5) is found aer eqn (4), by arbitrarily choosing that the and isotropic diffusion term with constant Dd. Note that each of line segment that goes from the center of the grain to the sky- the solutions of the FPE [eqn (7)] corresponds to innitely many rmion position crosses perpendicularly one of the sides of the solutions of the STE [eqn (1)] with the same grain distribution. grain. We have checked that considering other orientations and To achieve the same accuracy as obtained by solving the FPE, shapes the difference in the force is less than 6%. The constant the calculation time needed using the STE would be prohibitive. fG is fG = (2DKgR2)/(m0Ms). The variable r0 should be evaluated at qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Some recent studies show direct measurements of the proba- r ¼ ðr  Lg =2Þ2 þ R2 . r is now the distance between the grain bility density cloud.37 center and the position of the skyrmion. Finally, the total force over a skyrmion is the sum of all the 4 Results grains, Consider a skyrmion, initially at the position (xs, ys) = (0, 0), XNg   FG ¼ Fgr ðjrs  ri jÞ rs  ri ; (6) traveling along the racetrack while driven by a vH = (0, −vH). Our i¼1 jrs  ri j purpose is to study how the granularity of the sample, charac- terized by Lg and sK, affects the skyrmion dynamics at room 4730 | Nanoscale Adv., 2023, 5, 4728–4734 © 2023 The Author(s). Published by the Royal Society of Chemistry Paper Nanoscale Advances temperature. To x numbers, we use vH = (0, −426.8) m s−1 and ðW ðt 0 0  0 0 f0 = 8.71 × 10−14 m2 A−1. These parameters are chosen so that, Ps ðx; tÞ ¼ dy dt x^$Jp x; y ; t ; (11) W 0 in the clean sample and at T = 0 K, the skyrmion travels at about 35 m s−1 when far from any border, speeding up to about where Jp(x,y,t) is the probability current density. It can be ob- 100 m s−1 when reaching the borders. The probability of tained by considering that vp(r,t)/vt = −VJp. Aer eqn (7), surviving along the track at a distance of 1.2 mm is found to be Jp = (vdrv + vext)p(r,t) − DdVp(r,t). (12) higher than 0.99998. The rest of the parameters used in the simulations are a = 0.3, Ms = 580 kA m−1, K = 0.425 MJ m−3, To have the potential to become a real device, the skyrmionic R = 80 nm, T = 300 K, W = 150 nm, d = 0.6 nm, and L = 1.2 mm. racetrack should ideally achieve a Ps(L,s) = 1 for reasonable The radius of the skyrmion is a xed parameter. One could times s to be determined by the particular device in mind. This consider the radius dependence by changing some parameters is because in general, one does not desire the skyrmions as a function of radius.21,50 carrying information to “disappear”, nor to reach their desti- In Fig. 2 we show four simulations considering the same nation beyond the expected time. grain distribution, but with different dispersions in the DK In Fig. 3 we show the skyrmion success probability as it value, from sK = 0 (no granularity) to sK = 0.2K. Each plot shows travels along the track for different sK. These results have been a superposition of four snapshots (at different times) corre- obtained aer averaging the results of 30 simulations [solutions sponding to the probability density of nding the skyrmion. In of eqn (7)] considering different distributions of grains with the this way, the translation and the diffusion effects are clearly same size Lg = 20 nm and sK. visualized. In particular, as sK becomes larger, skyrmions can In general, nding a skyrmion at a given distance and time is be trapped or delayed more easily at some grains, stretching the less likely for more heterogeneous samples (larger sK) indi- probability density cloud. As a result, the probability of a sky- cating that the granularity somewhat delays the skyrmion. As rmion reaching a given distance along the track for a given time time goes, the probability of reaching a given distance increases is expected to be reduced due to this grain-induced delay effect. (naturally). Aer 100 ns (well aer the longest time shown in In addition, even for innite times, the probability of a sky- Fig. 3 and 4) we have observed that for the sK = 0.2K case (red rmion reaching a given position is reduced due to the pinning at line in Fig. 3), the probability of reaching the 1.2 mm is about some grains. 0.85, indicating that the skyrmion has been lost, pinned, or To quantify these results more accurately, the probability greatly delayed. It is also worth noting that some of the results that a skyrmion reaches the position x of the track before a time found here have also been predicted for antiferromagnetic t, or the skyrmion success probability, is calculated as skyrmions.43 The presence of disorder and temperature Fig. 2 Overlapped snapshots of the probability density p(r,t) for the presence of a skyrmion at a given position r along the track for different times (t = 1.7, 4.2, 6.7, and 11.7 ns). The distribution of the grains is the same in all figures but sK changes: sK = 0 (a), 0.05K (b), 0.1K (c), 0.2K (d). The side of the grains is Lg = 20 nm. The dots indicate the grain centers. The gray region indicates where the skyrmion has reached the border, and thus disappeared. © 2023 The Author(s). Published by the Royal Society of Chemistry Nanoscale Adv., 2023, 5, 4728–4734 | 4731 Nanoscale Advances Paper large probability of nding skyrmions at larger distances along x, with respect to considering a plain sample. At larger times, this effect disappears since the p(r,t) distribution is already mostly concentrated near the borders in all cases. We also studied how the defect size affects the performance of this device. In Fig. 4 we compare the results already shown in Fig. 3 for sK = 0.1K and 0.2K with those using a different grain size, Lg = 10 nm. It is observed that the smaller the grains are, the weaker their effect is. This is because, when the grains are small, the skyrmions are much larger than them, and thus, they seldom perceive the granularity. Actually, from the skyrmion perspective, it moves over an effectively uniform lm. This is clearly seen in Fig. 4 where the small-grain results (dashed lines) practically match with the plain sample calculation (green solid line). In this case, if sK were very high, there could be some grain acting basically as a local point defect.49 If the grains are Fig. 3 Probability of finding a skyrmion as a function of the traveled much larger than the skyrmions then one would expect that the length x along the racetrack. Each plot corresponds to a different grain boundaries act as extended defects.40 time, from (a) to (d): t = 6.67, 10, 13.33, and 20 ns. The different lines correspond to averaging 30 different grain distributions of side In order to validate this approach we have performed several Lg = 20 nm with the same sK. sK/K = 0 (green), 0.05 (purple), 0.1 (blue), micromagnetic calculations using the same geometry and and 0.2 (red). parameters (exchange, saturation magnetization, DM constant, temperature, and a) as above and by generating the grains in the track following also the same procedure as described using produces a “hindering” phase, similar to our delay, or to the a position dependent anisotropy constant (see ESI for videos of creep regime described in other studies. the simulation results†). The summary of some of these There is a curious effect for small times. As seen in Fig. 3a the micromagnetic runs can be seen in Fig. 5 where the x-position distances at which a skyrmion can be found are larger for more of the center of the skyrmion as a function of time is plotted. heterogeneous samples (the red line overcomes the other ones The results show that: (i) the skyrmion moves approximately at x x 0.6 mm). This is because, when far from the edges, there rigidly. Small deformation due to the temperature and the is a substantial diffusion along the y axis and the cloud of crossing between grains do not destroy the global shape of the probability density stretches signicantly along y. When the skyrmion (although the radius of the skyrmion decreases when skyrmion reaches the border, it is accelerated along x. Thus, the approaching the border). We conclude that, while the skyrmion part of the p(r,t) cloud near the edge is then stretched along x. As a result, in more heterogeneous samples, there is a relatively Fig. 5 x-Position of the center of the skyrmion along the racetrack as Fig. 4 Probability of finding a skyrmion along a racetrack as a function a function of time calculated from four (red, green, blue, gray) identical of x. Each plot corresponds to a different time, from (a) to (d): t = 6.67, stochastic micromagnetic calculations with the same parameters as 10, 13.33, and 20 ns. Solid blue and red curves are the same as in Fig. 3. discussed in the text. In the insets there are snapshots of the position of The dotted curves of each color correspond to the respective sK but different skyrmions at the same time (for the green and blue simula- with smaller grains, Lg = 10 nm. The green solid lines correspond to the tion, together with the initial skyrmion for comparison). The full video clean racetrack and are included for comparison. for these and other cases can be found in the ESI.† 4732 | Nanoscale Adv., 2023, 5, 4728–4734 © 2023 The Author(s). Published by the Royal Society of Chemistry Paper Nanoscale Advances if moving along the track the rigid model is a good approxi- this sense, racetracks require some kind of “synchronization mation with the parameters used. (ii) The results reported in the point” for skyrmions. A possible solution could be considering micromagnetic calculations are consistent with the probabi- wedges at the borders.53 The overall velocity would be reduced, listic results. In particular, we have found cases in which there but feasibility would be ensured. is a delay in the movement of the skyrmions due to the granu- The methodology presented here can be useful for control- larity, cases where the skyrmions are destroyed at the border, ling and somewhat harnessing the randomness in magnetic etc. The quantitative evaluation of the probability of having systems.54 The “as realistic as possible” simulations are key these events would be impossible in practice with full micro- ingredients in the design and development of a particular magnetic approaches, but is just the information that the application. Neuromorphic computing systems,55 logic and Fokker–Planck solutions give us. probabilistic computing devices,56 or true random number generators57 can benet from the presented results and model. 5 Discussion and conclusions Conflicts of interest A direct comparison of the presented results with published experimental results is difficult. However, it is worth There are no conicts of interest to declare. mentioning that our model predicts a dependence, experi- mentally found, of the skyrmionic Hall angle with granu- Acknowledgements larity,15,29 as seen in Fig. 4 where the skyrmion approaches the border sooner for large granularities. It also predicts the pres- We acknowledge Catalan project 2017-SGR-105 and Spanish ence of different regimes in the movement of skyrmions: project PID2019-104670GB-I00 of Agencia Estatal de Inves- pinning, creep, and ow regimes at low, medium, and large tigación/Fondo Europeo de Desarrollo Regional (UE) for nan- driving currents, respectively.44 This is seen in the averaged cial support. J. C.-Q. acknowledges a grant (FPU17/01970) from delay of skyrmions (creep regime) used in the calculations; we Ministerio de Ciencia, Innovación y Universidades (Spanish also performed simulations at lower driving currents and found Government). that the probability of survival at a given distance went to zero while the probability density plot was xed around some defects Notes and references (indicating the pinning regime, not shown). Simulations at larger driving currents (ow regime) also showed a probability 1 K. Everschor-Sitte, J. Masell, R. M. Reeve and M. Kläui, J. of going to zero, but because of the overcoming of the conning Appl. Phys., 2018, 124, 240901. edge barriers (not shown). 2 A. Fert, V. Cros and J. Sampaio, Nat. Nanotechnol., 2013, 8, Here, we have only taken into account anisotropy variations 152–156. for the grains. In general, other parameters may change from 3 G. Bourianoff, D. Pinna, M. Sitte and K. 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