Current-Driven Skyrmion Dynamics and Drive-Dependent Skyrmion Hall Effect in an Ultrathin Film

TL;DR

This study demonstrates current-driven Ne9el skyrmion motion in an ultrathin film, revealing a drive-dependent skyrmion Hall effect influenced by pinning, supported by experiments, modeling, and simulations.

Contribution

It provides new insights into skyrmion dynamics in ultrathin films with simple spin textures, highlighting the impact of pinning on the skyrmion Hall effect.

Findings

Skyrmions can be driven at 100 m/s in ultrathin films.

The skyrmion Hall effect depends on the drive due to pinning effects.

Experimental results agree with analytical and micromagnetic models.

Abstract

Magnetic skyrmions are chiral spin textures that hold great promise as nanoscale information carriers. Since their first observation at room temperature, progress has been made in their current-induced manipulation, with fast motion reported in stray-field-coupled multilayers. However, the complex spin textures with hybrid chiralities and large power dissipation in these multilayers limit their practical implementation and the fundamental understanding of their dynamics. Here, we report on the current-driven motion of N\'eel skyrmions with diameters in the 100-nm range in an ultrathin Pt/Co/MgO trilayer. We find that these skyrmions can be driven at a speed of 100 m/s and exhibit a drive-dependent skyrmion Hall effect, which is accounted for by the effect of pinning. Our experiments are well substantiated by an analytical model of the skyrmion dynamics as well as by micromagnetic simulations including material inhomogeneities. This good agreement is enabled by the simple skyrmion spin structure in our system and a thorough characterization of its static and dynamical properties.

Tables

Table 1. Table 1: A summary of the parameters used in the analytical model and the simulations. The saturation magnetisation M s subscript 𝑀 𝑠 M_{s} and uniaxial anisotropy K u subscript 𝐾 𝑢 K_{u} obtained from SQUID magnetometry measurements. The DMI parameter D 𝐷 D measured using Brillouin light scattering (BLS) spectroscopy. The damping parameter α 𝛼 \alpha extracted from the DW mobility in the field-driven steady-flow regime. The exchange constant A 𝐴 A extracted from micromagnetic simulations and BLS measurements. The DW width parameter Δ Δ \Delta deduced from micromagnetic simulations. The average skyrmion radius R 𝑅 R measured at μ 0 ​ H z ≈ − 5 subscript 𝜇 0 subscript 𝐻 𝑧 5 \mu_{0}H_{z}\approx{}-5 mT. The DL-SOT and FL-SOT effective fields per unit current density, C D ​ L subscript 𝐶 𝐷 𝐿 C_{DL} and C F ​ L subscript 𝐶 𝐹 𝐿 C_{FL} respectively (given in mT per 10 11 superscript 10 11 10^{11} A m -2 ), extracted from harmonic Hall voltage measurements. The current density is calculated from the measured current transmitted through the device, assuming that it flows uniformly in Pt(3)/Co(0.9), owing to the larger resistivity of the Ta(3) under-layer. Additional details can be found in the Supplementary Material.

$M_s$ (MA m-1)	$K_u$ (MJ m-3)	$\left|D\right|$ (mJ m-2)	$\alpha$	$A$ (pJ m-1)	$\Delta$ (nm)	$R$ (nm)	$C_{DL}$ (T A-1 m2)	$C_{FL}$ (T A-1 m2)
$1.42\pm 0.05$	$1.34\pm 0.12$	$1.27\pm 0.04$	$0.43\pm 0.08$	$16\pm 6$	11.5	$78\pm 23$	$2.1$	$0.9$

Equations

$${\mathbf{F}}_{DL}+\mathbf{G}\times \mathbf{v}-\alpha \mathcal{D}\cdot \mathbf{v}=0$$

$$v$$

$$\tan {\Theta }_{SkH}$$

Full text

I Introduction

Magnetic skyrmions are fascinating spin textures that have recently attracted considerable attention. Their peculiar topology and nanometer size confer on them quasiparticlelike properties that, combined with their ability to be moved by an electrical current, make them promising candidates for the storage and manipulation of information. They are envisaged to be information carriers in racetrack memories Fert et al. (2013) and logic devices Zhang et al. (2015) combining high density, thermal stability, and high data flow. Among the different classes of materials hosting skyrmions, sputtered multilayered stacks comprising ultrathin heavy-metal (HM)/ferromagnet (FM) layers combine several interesting features. Their structural-inversion asymmetry along with the large spin-orbit coupling of the HM lead to a large interfacial Dzyaloshinskii-Moriya interaction (DMI), a key ingredient in the skyrmion stabilization that ensures their homochirality and their Néel nature Bogdanov and Hubert (1999). In addition, the current-induced spin-orbit torques (SOTs) in these systems are expected to provide an efficient way to drive the skyrmions Sampaio et al. (2013). In recent years, a concerted effort has led to the observation of stable magnetic skyrmions under ambient conditions in these structures Jiang et al. (2015); Moreau-Luchaire et al. (2016); Boulle et al. (2016); Woo et al. (2016); Yu et al. (2016); Soumyanarayanan et al. (2017); Woo et al. (2017); Hrabec et al. (2017); Caretta et al. (2018) as well as their current-driven dynamics Jiang et al. (2015); Woo et al. (2016); Hrabec et al. (2017); Woo et al. (2018). Their topological properties affect their dynamics in a nontrivial manner. Notably, due to their quantized topological charge, magnetic skyrmions are subject to the so-called skyrmion Hall effect (SkHE) Jiang et al. (2017); Litzius et al. (2017), which leads to their deviation from the trajectory imposed by the current. This dynamical signature of their topology needs to be characterized and taken into account for future applications. Numerous studies of skyrmion dynamics have been focused on multilayers with a large number of repetitions Woo et al. (2016); Litzius et al. (2017); Woo et al. (2018). In particular, fast motion (100 m s*-1*) of small skyrmions (100 nm) has been reported in [Pt/CoFeB/MgO]15 Woo et al. (2016); Litzius et al. (2017). However, in these stray-field-coupled multilayers, the large dipolar fields can outweigh the DMI and stabilize twisted spin structures with a nonuniform chirality across the film thickness Legrand et al. (2018a); Dovzhenko et al. (2018); Li et al. (2019). This leads to a complex current-driven dynamics due to layer-dependent SOTs and topologies Lemesh and Beach (2018); Legrand et al. (2018b). Notably, these hybrid textures may not conserve their topological charge during motion at high current density and therefore may not be skyrmions which cannot be described by existing models Lemesh and Beach (2018). Furthermore, for a given current density, a larger Joule dissipation is also expected, since it scales with the total film thickness. Therefore, the demonstration of fast current-driven skyrmion motion in nanometer-thick film systems is a prerequisite for the development of low-power skyrmion-based devices and for a simple understanding of their dynamics.

In this work, we report on the observation of the current-driven motion of isolated magnetic skyrmions in an ultrathin Pt/Co/MgO film. Previously, we have shown that this system exhibits a large interfacial DMI and hosts homochiral Néel skyrmions at room temperature Boulle et al. (2016). Here, we find that these skyrmions, with diameters in the 100-nm range, move at a velocity up to 100 m s*-1* and that the SkHE is markedly drive dependent. These observations are complemented by a detailed characterization of the static and dynamical properties of the stack. These measurements and the simple skyrmion spin structure in our system allow us to approach a quantitative comparison with the prediction of the Thiele equation at high current density. Real-scale micromagnetic simulations including material inhomogeneities reproduce the observed drive-dependence of the skyrmion mobility and the SkHE, allowing the identification of different dynamic regimes. The simulations reveal that the drive-dependence of the SkHE is not accounted for by the fieldlike SOT (FL-SOT), as previously claimed Litzius et al. (2017) but by pinning, which greatly impacts the skyrmion dynamics at low current density.

II Results and discussion

II.1 Experimental results

We study the current-driven dynamics of magnetic skyrmions in a sputter-deposited Ta(3)/Pt(3)/Co(0.9)/MgO(0.9)/Ta(2) film (thicknesses in nanometers) at room temperature, using the high-spatial-resolution photoemission electron microscopy combined with x-ray magnetic circular dichroism (XMCD-PEEM). The sample is patterned into 3-µm-wide tracks with injection pads consisting of Ti(10 nm)/Au(100 nm), as represented in Fig. 1.a. An external out-of-plane magnetic field $\mu_{0}H_{z}\approx{}-5$ mT is applied on the initially demagnetized configuration in order to stabilize skyrmions with a core magnetized along $+\boldsymbol{\hat{z}}$. In Fig. 1.b-e and Fig. 1.f-j, two sequences of images display the characteristic current-driven motion of the skyrmions. Between each acquisition, a single 11-ns current pulse of amplitude $J=5.6\times{}10^{11}$ A m*-2* is injected along $+\boldsymbol{\hat{x}}$. Overall, some skyrmions exhibit a net motion of several hundreds of nanometers along the current direction, that is, against the electron flow (yellow and orange circles). The same directionality is also observed for skyrmions with the opposite core polarity, when $H_{z}>0$. In addition, these skyrmions experience a net motion perpendicular to the current direction, which is the signature of the SkHE. This kind of motion is consistent with the current-driven dynamics of left-handed Néel skyrmions governed by the spin Hall effect (SHE), as discussed hereafter. However, it also exhibits a stochastic character, with events of nucleation (black circle) and annihilation (red and blue circles). In addition, the sequences b-e and f-j reveal that the skyrmions displacements are not identical for each current pulse. Finally, some skyrmions appear to be distorted after a current pulse and sometimes acquire an elongated shape. This stochastic behavior can be attributed to the presence of pinning sites obstructing the skyrmion motion Kim and Yoo (2017); Legrand et al. (2017). This local disorder can also explain the large dispersion observed in the skyrmion size and shape Zeissler et al. (2017); Juge et al. (2018); Gross et al. (2018) with a measured average effective diameter $d_{Sk}=156\pm{}45$ nm (see Sec. 1.4 within the Supplementary Material).

Despite this irregular motion, systematic measurements of the current-induced displacements averaged over a large number of skyrmions allow us to extract an average skyrmion velocity and skyrmion Hall angle (SkHA) — namely, the angle $\Theta_{SkH}$ between the current and the skyrmion motion directions (see Fig. 2.a-c). Fig. 2.d displays the dependence of the skyrmion velocity with the current density. The velocity is calculated as the total displacement divided by the measured pulse width (8 ns and 11 ns in our experiments) and then averaged over multiple events at the same current density. The thermal drift between each acquisition is corrected from larger images using the right-angled corner of the track (see Fig. 1.a). Note that below a certain current density, highlighted by the shaded area in Fig. 2.d, no significant displacement is observed after a single current pulse. Above $J=3.5\times{}10^{11}$ A m*-2*, the skyrmion velocity increases monotonically with the current density and reaches 100 m$\cdot$s*-1* for $J=6.8\times{}10^{11}$ A m*-2*. This value is comparable to the largest velocities recorded for skyrmions of similar size in stray-field-coupled multilayers Woo et al. (2016). However, the use of a single repetition allows to lower the power dissipation by one order of magnitude. Furthermore, as previously mentioned, the current injection also leads to a motion perpendicular to the current direction. The direction of deflection with respect to the current direction is unchanged when reversing the current direction (see Fig. 2.a-c), which is characteristic of the SkHE. The measurements (Fig. 2.e, black squares) reveal that the SkHA depends on the skyrmion velocity and exhibits a monotonic increase up to approximately 50°.

In the following, we discuss and interpret these results in the light of the Thiele model as well as micromagnetic simulations. For this purpose, we realize a detailed characterization of the static and transport properties of the Pt/Co/MgO stack to approach a quantitative comparison with the experimental results. Details of the different measurements are given in the Supplementary Material and the parameters extracted are summarized in Table 1.

II.2 Comparison with the analytical model

The experimental results are first compared with the prediction of the Thiele equation Thiele (1973), which captures the dynamics of magnetic skyrmions well. Under the assumption that the skyrmions behave as rigid spin textures, the steady-state velocity $\boldsymbol{v}$ results from an equilibrium between different forces acting on the skyrmion:

[TABLE]

Here, $\boldsymbol{F}_{DL}$ is the force due to the current injection. In the case of HM/FM ultrathin films, the force results from the action of the so-called dampinglike spin-orbit torque (DL-SOT): the charge current flowing in the Pt layer is converted into a spin current due to the SHE. The spin accumulation at the Pt/Co interface leads to a torque acting on the Co magnetization. In the case of Pt/Co/MgO, since the spin Hall angle (SHA) is positive for Pt Nguyen et al. (2016) and skyrmions are of Néel type and exhibit a left-handed chirality Boulle et al. (2016), the symmetries of the DL-SOT impose that the resulting force drives the skyrmion in the current direction (i.e. opposite to the electron flow) Hrabec et al. (2017). In addition, the topology of the skyrmion leads to a gyrotropic force, $\boldsymbol{G}\times\boldsymbol{v}$, where $\boldsymbol{G}=G\boldsymbol{\hat{z}}$ is the gyrovector, whose sign depends only on the topological charge $N_{Sk}$, with $G\propto{}N_{Sk}$ Thiele (1973, 1974). For instance, in Fig. 2.a-c, $\boldsymbol{G}$ points in the $-\boldsymbol{\hat{z}}$ direction. This second term thus describes the SkHE. The third term describes the dissipative force with $\boldsymbol{\mathcal{D}}$ the dissipative tensor and $\alpha$ the Gilbert damping parameter. Note that the effect of the FL-SOT is neglected. Indeed, the action of the FL-SOT is equivalent to that of an external magnetic field applied in a direction perpendicular to the current direction and a spatially homogeneous magnetic field does not result in a force on a rigid skyrmion in Thiele’s approach. This approximation is justified in the next section by micromagnetic simulations. Although simple, this equation captures most of the physical ingredients that drive the skyrmion dynamics and is particularly relevant in our ultrathin films with a homogeneous homochiral Néel spin configuration across the film thickness. Assuming that the skyrmion has a radial 360° Bloch wall profile 111In a previous work, we observed using XMCD-PEEM, that the skyrmion profile in Pt/Co/MgO was fitted well by a 360° Bloch wall profile Boulle et al. (2016), a rotational symmetry, and that its radius $R$ is much larger than the domain wall (DW) width parameter $\Delta$ ($R\gg\Delta$), the skyrmion velocity and the SkHA can be written as follows Tomasello et al. (2014); Hrabec et al. (2017):

[TABLE]

Here, $\gamma$ is the gyromagnetic ratio, $\alpha$ the Gilbert damping parameter, $J$ the current density flowing in the Pt layer and $C_{DL}$ the effective field (per unit current density) associated with the DL-SOT. We use the parameters listed in Table 1. The analytical solution for the velocity is plotted in Fig. 2.d (red solid line): it reveals a relatively good agreement with the experimental data at high current density, considering that the velocity depends critically on all the aforementioned parameters. Nevertheless, we observe that the experimental values are smaller than the one predicted by the Thiele equation at low current density. A similar observation has been pointed out in the study of the current-driven dynamics of DWs in Pt/Co/AlOx ultrathin films Miron et al. (2011). It can be accounted for by the effect of pinning on the skyrmion dynamics, not taken into account in this simple model, which is expected to lead to a thermally activated regime at low drive current, characterized by smaller velocities. This indicates that the skyrmion dynamics in our experiments is in a depinning regime and this points at the existence of a flow regime for the largest current densities injected.

With regard to the SkHA, we find a value $\Theta_{SkH}=$ 34° $\pm$ 8° using Eq. (3). The uncertainty includes the uncertainty on the skyrmion radius. Note that this value is independent of the skyrmion velocity or the applied current, in sharp contrast with our observations as well as previous experimental studies Jiang et al. (2017); Litzius et al. (2017). This disagreement with the experiments can also be explained by the presence of pinning sites Reichhardt et al. (2015a, b); Reichhardt and Reichhardt (2016); Kim and Yoo (2017); Legrand et al. (2017). The calculated SkHA (i.e. the steady-flow SkHA) is found to be smaller than the maximum measured values (see Fig. 2.e, red solid line). However, as detailed in the next section, one expects the SkHA to tend toward the calculated value for large skyrmion velocities. Note that the damping is a key parameter in the calculation of the skyrmion velocity and the SkHA. Here, we use the value $\alpha=0.43$. It is extracted from field-driven DW dynamics measurements in a Pt/Co(0.63 nm)/MgO (see Sec. 1.5 within the Supplementary Material). This value is in good agreement with previous DW dynamics experiments in ultrathin Pt/Co/Pt Metaxas et al. (2007) and Pt/Co/AlOx Miron et al. (2011); Pham et al. (2016) films. To explain such large damping values, besides the contributions of surface roughness Min et al. (2010) and spin pumping into the Pt layer Beaujour et al. (2006), the presence of Rashba spin-orbit coupling is also predicted to enhance the damping Kim et al. (2012). This additional contribution can be expected in our stack considering the non-negligible FL-SOT (see Table 1) and can lead to a significant reduction of the DW and skyrmion velocity as well as the SkHA Kim (2015). We note that the damping parameter is measured in a stack with a slightly thinner Co layer, which is required to stabilize larger domains and drive DWs with an external field. In a thicker Co layer, the different interfacial contributions to the damping mentioned above are expected to decrease. The simplest approximation to account for the change in the Co thickness is to assume that $\alpha\sim{}1/t$, leading to $\alpha=0.30$ for the Pt/Co(0.9nm)/MgO film used to study the skyrmion dynamics. Accounting for this correction leads to an enhancement of the calculated skyrmion velocity and of the SkHA $\Theta_{SkH}=$ 45° $\pm$ 8° (see Fig. 2.d and e, red dashed lines) that provides a better agreement with the experimental results, as it will be justified in the next section.

II.3 Micromagnetic simulations

To go beyond the assumptions and limitations of the analytical model, we performed micromagnetic simulations using the parameters listed in Table 1. Both DL-SOT and FL-SOT were taken into account. These torques are given respectively by $\boldsymbol{T}_{DL}=-\gamma_{0}C_{DL}J\boldsymbol{m}\times[(\boldsymbol{\hat{z}}\times\boldsymbol{\hat{j}})\times\boldsymbol{m}]$ and $\boldsymbol{T}_{FL}=-\gamma_{0}C_{FL}J\boldsymbol{m}\times(\boldsymbol{\hat{z}}\times\boldsymbol{\hat{j}})$, with $\gamma_{0}=\mu_{0}\gamma$, $\boldsymbol{m}$ the reduced magnetisation vector, $\boldsymbol{\hat{j}}$ the unit vector in the current direction and $C_{DL}$ (resp. $C_{FL}$) the measured effective field per unit current density associated with the DL-SOT (FL-SOT) Khvalkovskiy et al. (2013). The simulations were carried out using the MICRO3D code Buda et al. (2002). The geometry consists of a $1040\times 560\times 0.9$ nm3 track with a cell size of $3.2\times 3.2\times 0.45$ nm3. An out-of-plane external field $\mu_{0}H_{z}=-5.4$ mT is applied which sets the skyrmion radius to 78.8 nm — that is the experimental one — and thermal fluctuations are neglected. Fig. 3.a and b show snapshots of the skyrmion dynamics during the application of a current with $J=2.9\times{}10^{11}$ A m*-2* and $6.7\times{}10^{11}$ A m*-2* respectively. As indicated by the red arrow, the current is tilted by 45° with respect to the track direction so that the skyrmions move along the track. The simulations reveal that the skyrmion experiences an expansion accompanied by a deformation that becomes more pronounced with increasing driving current. This effect is purely dynamical: when the current is turned off, the skyrmion shrinks back to its original size and recovers its rotational symmetry. In addition, we find that this deformation is independent of the FL-SOT and is therefore only due to the DL-SOT (see Sec. 2.2 within the Supplementary Material). To highlight this effect, we calculated the effective skyrmion size, defined as the total area for which $m_{z}>0$, normalized by the skyrmion area at rest. Fig. 3.c displays its evolution with the current density. It shows that the skyrmion size increases significantly, in contradiction with the rigid-core assumption of the Thiele model. We note that the simulated skyrmion size matches that of the observed skyrmions and it is significantly larger than the one considered in most studies Sampaio et al. (2013); Kim and Yoo (2017); Legrand et al. (2017); Litzius et al. (2017), which could explain why this dynamical effect had not been emphasized before.

To quantify the dynamics of a noncircular skyrmion, we define its position as that of its barycenter. The obtained skyrmion velocity and SkHA are plotted as blue dots in Fig. 4.a and b respectively. Remarkably, the simulations are in very good agreement with the simple prediction of the analytical model (red line). This highlights the relevance of discussing our experimental results in the light of the Thiele model. In addition, it emphasizes that the effect of the FL-SOT, which is not taken into account in the model, is not significant in our system. Surprisingly, the DL-SOT-induced expansion and/or deformation does not significantly alter the skyrmion dynamics in the range of current density considered, in spite of the large increase in the skyrmion size (Fig. 3.c). Therefore, this effect cannot account for the evolution of the skyrmion velocity and SkHA observed experimentally (Fig. 2). Furthermore, mechanisms involving a sizeable FL-SOT have been put forward to explain the drive-dependence of the SkHA of slightly deformed skyrmions Litzius et al. (2017) or breathing skyrmions Tomasello et al. (2018). However, our micromagnetic simulations including both SOTs with realistic relative amplitudes (up to $C_{FL}=C_{DL}$) show no significant effect of the FL-SOT (see Sec. 2.2 within the Supplementary Material). Note finally that the Oersted field may also impact the measured SkHA as it generates a force on the skyrmion perpendicular to the current direction. For skyrmions with a core magnetised along $+\boldsymbol{\hat{z}}$, which corresponds to the condition of most of our experiments, this force is opposite to the gyrotropic force Hrabec et al. (2017). Thus, besides being of negligible amplitude, it may only reduce the measured SkHA and therefore cannot explain the pronounced drive-dependence observed experimentally (Fig. 2.e).

Another explanation to account for our experimental results is the presence of pinning sites in the material Reichhardt et al. (2015b); Reichhardt and Reichhardt (2016); Kim and Yoo (2017); Legrand et al. (2017). To include such an effect, we introduce disorder, modeled by a distribution of grains with different anisotropies Voto et al. (2016); Garcia-Sanchez et al. (2016). The average grain size is 30 nm, a value in line with previous estimations in similar systems Legrand et al. (2017); Zeissler et al. (2017); Gross et al. (2018), and the anisotropy is varied randomly between $\left(1\pm 0.05\right)K_{u}$. This approach was shown to provide a good qualitative agreement with experimental observations of the skyrmions statics under applied field Zeissler et al. (2017); Juge et al. (2018); Gross et al. (2018); Kim and Yoo (2017); Legrand et al. (2017). Fig. 3.d displays the motion of a skyrmion for a given disorder configuration and for $J=6.7\times{}10^{11}$ A m*-2*. It reveals that the skyrmion experiences an additional dynamical deformation due to the local variation of anisotropy that creates minima for the DW energy. This makes it difficult to draw a distinction between the two contributions, namely that of the DL-SOT and that of the disorder. The skyrmion at rest is also distorted, since it relaxes on an inhomogeneous energy landscape, explaining the (static) deformation of some skyrmions in Fig. 1 Zeissler et al. (2017); Gross et al. (2018); Juge et al. (2018). Fig. 3.e and f display the skyrmion trajectories recorded for different grain distributions and for $J=2.9\times{}10^{11}$ A m*-2* and $J=6.7\times{}10^{11}$ A m*-2*, respectively. The blue dotted lines represent the trajectories in the absence of disorder, which deviate slightly from linearity due to the skyrmion deformation. The first plot reveals a stochastic dynamics, with a large dispersion in the direction of motion. In some cases, the skyrmions are stopped on a strong pinning site, preventing further motion. For larger current densities, the number of such events decreases and the trajectory approaches that of the ideal case. This is emphasized in Fig. 4.a, which shows the skyrmion velocity as a function of the current density in the ideal case (blue dots) and the disordered case (black stars).

The disorder introduces different regimes in the skyrmion dynamics: at low driving current, the average velocity, calculated from the total displacement after a computation time of 8 ns, is close to zero. This pinned regime is highlighted by the shaded areas in Fig. 4. At higher current density, the force due to the current becomes large compared to the pinning force and the velocity increases and finally reaches the disorder-free velocity, defining the flow regime. This behavior is analogous to that of current-driven DWs Miron et al. (2011) and points out that care should be taken when interpreting the regime of the skyrmion dynamics. In the pinned regime, a reduction in the average SkHA is observed. Upon increasing the driving force, the SkHA increases monotonically and converges to the disorder-free SkHA in the flow regime. These results can be understood as follows: the pinning can be viewed as a force acting opposite to the ideal skyrmion motion direction Iwasaki et al. (2013), thus lowering its velocity. The gyrotropic response to this force will make the skyrmion deviate from its initial trajectory in a direction transverse to this pinning force, according to the second term of Eq. (1). This effect, referred to as extrinsic SkHE Kim and Yoo (2017), prevails at low current density and becomes negligible as the force due to the current becomes sufficiently large as compared to the pinning force.

These results are in excellent agreement with our observations and strongly suggest that the observed drive-dependence of the skyrmion velocity and that of the SkHA is largely due to pinning. Note that only the nonshaded areas in Fig. 4 are to be compared with the experimental results in Fig. 2. Although the critical current densities and skyrmion velocities for which the depinning and flow regimes are reached depend on the choice of disorder parameters Kim and Yoo (2017); Legrand et al. (2017); Salimath et al. (2019), this approach constitutes a good qualitative description of the pinning effect on the current-driven skyrmion dynamics. Qualitatively, the micromagnetic simulations reveal that the disorder-free velocity and SkHA constitute upper bounds for the velocity and SkHA in disordered systems. This supports the aforementioned assumption of a lower damping in our film, which would increase the disorder-free velocity and the SkHA (Fig. 2, red dashed lines), leading to an excellent agreement with the simulations. Therefore, it confirms that the observed skyrmion dynamics are in a depinning regime and this suggests that the flow regime is reached for the largest current densities.

III Conclusions

We have studied the current-driven dynamics of small magnetic skyrmions in an ultrathin Pt/Co/MgO film at room temperature. We observe that isolated magnetic skyrmions exhibit fast motion (100 m s*-1*) and a drive-dependent SkHE. Supported by a detailed characterization of the film properties and the SOTs governing the skyrmion dynamics, we show that these observations can be well accounted for by Thiele’s model at high current density, which is particularly relevant in an ultrathin film hosting skyrmions with a well defined chirality. Micromagnetic simulations including the measured DL-SOT and FL-SOT as well as material inhomogeneities reproduce the different regimes of the drive-dependent skyrmion velocity and SkHA. This, in turn, allows us to rule out the impact of the FL-SOT on the skyrmion dynamics and to identify pinning as the dominant effect responsible for the drive dependence of the SkHE. Our results shed light on the current-driven skyrmion dynamics in ultrathin films, which paves the way for future experimental investigations for the purpose of developing low-power skyrmion-based applications.

ACKNOWLEDGEMENT

We thank Nikita Strelkov for his contribution to the simulations and Jordi Prat for his support at the CIRCE beam line. We acknowledge the help from Olivier Fruchart and Simon Le Denmat for performing magnetic force microscopy with low-moment tips. The authors acknowledge the support of the Agence Nationale de la Recherche, Project ANR-17-CE24-0045 (SKYLOGIC), and the support of the DARPA TEE program through Grant No. MIPR HR0011831554. The XMCD-PEEM experiments were performed at (1) the CIRCE beam line at the ALBA synchrotron Aballe et al. (2015); Foerster et al. (2016), with CALIPSOplus funding (Grant No. 730872), (2) the Nanospectroscopy beam line at the Elettra synchrotron Menteş et al. (2014), with funding from the European Community’s Horizon 2020 Framework Programme under Grant Agreement 730872, and (3) the the beam line I06 at Diamond Light Source under proposal number SI-I4035.