## Introduction

Magnetic skyrmions (skyrmions hereafter) are topologically robust, localized whirls of magnetization1,2,3,4—a simulated image of an isolated magnetic skyrmion is shown in Fig. 1a. The topology of a skyrmion gives rise to a Magnus term in its equation of motion5. As a result, skyrmions are generally driven at an angle to the direction of applied forces; the relationship between driving force, induced velocity and the skyrmion Hall angle is indicated in Fig. 1b. The stability and mobility of skyrmions have inspired the design and manufacture of a multitude of skyrmion-based devices; despite this, the widespread adoption of magnetic skyrmions in computational schemes is far from reality6,7,8,9. For the purposes of device manufacturing, the magnitude of the skyrmion Hall angle must be accurately obtained on a material-by-material basis. This is necessary as the skyrmion Hall angle completely dictates the qualitative dynamical properties of skyrmions in a system, and as variations in, e.g., material quality, doping and surface treatment, can alter the many parameters that control the magnitude of the skyrmion Hall angle (such as density of pinning sites, damping and anisotropies) in a manner that is difficult to predict.

For instance, skyrmions for which the Magnus force is large compared to other terms in their equation of motion have been shown to be much less strongly affected by pinning potentials; in this case, the skyrmion Hall angle typically approaches 90°5. Furthermore, skyrmions with these properties are expected to enjoy significantly reduced depinning thresholds10,11, and can be used in device architectures that are incompatible with typical force–velocity relationships12.

While in many respects a skyrmion Hall angle near to 90° is beneficial, this is not always the case. In device schematics which require skyrmions to be driven along a straight line, it would be desirable for skyrmions to not veer from their path. Non-zero skyrmion Hall angles deflect skyrmions towards the boundary of the structure where they can be trapped or destroyed13. This led to a surge of recent interest in skyrmions in antiferromagnetic, synthetic antiferromagnetic and compensated ferrimagnetic materials, which have a skyrmion Hall angle of 0°14,15,16.

This difference in qualitative behaviour makes the quantitative determination of the skyrmion Hall angle an important topic; one which has been well tackled for the case of skyrmions in magnetic heterostructures16,17,18,19,20,21. In these systems, skyrmions tend to be sparse, and their controlled nucleation has been well studied22. This makes the isolation and manipulation of individual skyrmions possible, allowing the skyrmion Hall angle to be obtained by tracing the real space motion of skyrmions under the influence of a driving force21. Such techniques are inapplicable to the study of the skyrmion lattice state, where individual skyrmions cannot readily be labelled or tracked in real space, or where skyrmions are too small to be resolved with the required time resolution.

Here, we report a study of the signatures of lattice dynamics in reciprocal space, showing that the motion of dislocations through a sheared skyrmion lattice forces the lattice to reorient along its direction of motion. The shear applied to the skyrmion lattice also leads to a build-up of strain, which when combined with the lattice reorientation, allows for the unambiguous measurement of the skyrmion Hall angle in reciprocal space.

## Results

### The high-temperature skyrmion system FeGe

A particularly interesting chiral magnet is the widely studied FeGe, an itinerant helimagnet which hosts sub-100 nm diameter skyrmions near room-temperature23. Despite the wealth of research carried out on this material, measurements of the skyrmion Hall angle in FeGe have remained elusive. To experimentally measure the skyrmion Hall angle in FeGe, we set up a shear force with the stray magnetic field from a current-carrying wire, studying the reciprocal space dynamics using resonant elastic X-ray scattering (REXS)24.

### The effect of shear stress on two-dimensional crystals

To determine the skyrmion Hall angle from reciprocal space measurements, it is necessary to understand the signature of real space motion in the structure factor. For homogeneous drives, this has been studied in detail, though none of the rich dynamic phases explored in the literature allow for the unambiguous determination of the skyrmion Hall angle from the structure factor alone10,25. To move beyond this limitation, it is lucrative to borrow results from the study of crystals under strain. In this case, the properties of a crystal are dictated entirely by its defects and dislocations26—an excellent classical example of this is the shear stress required to plastically deform a crystal. In 1926, it was calculated by Frenkel that the shear stress required for plastic deformation of a lattice should be on the order of its shear modulus27. In reality, experimental values are 4–8 orders of magnitude smaller28. The reason for this stark discrepancy is the existence of dislocations in a real lattice. The shear stress required to plastically deform a real lattice is instead on the order of the Peierls–Nabarro stress, i.e., the shear stress required to displace a dislocation29.

As an ideal skyrmion lattice state is two-dimensional (extending as tubes in depth30,31), the only form of dislocation present in the lattice are edge dislocations, which typically manifest themselves in hexagonal 2D materials as 5–7 defects32. These 5–7 defects can glide in the direction of their Burgers vector for very little energy28. The allowed Burgers vectors for 5–7 defects are real primitive lattice vectors, confining defect glides to three special axes. If a shear stress is applied whose direction is not parallel to one of the three axes, motion is possible only by combining glides and climbs. An example visualization of such a defect in a skyrmion lattice, including its Burgers vector, is shown in Fig. 2a.

The climbing motion of 5–7 defects is much more costly than glides, as each climb changes the macroscopic shape of the crystal28. If the direction of a shear force is at an angle θ from one of the glide axes, in order to plastically deform the crystal, there must be $$\sin \theta$$ defect climbs for every $$\cos \theta$$ defect glides. As such, the system must spend an energy $${E}_{{\rm{Climb}}}\sin \theta +{E}_{{\rm{Glide}}}\cos \theta$$ to propagate a defect when relieving strain, where EClimbEGlide. This creates an effective energy cost $${E}_{{\rm{Defect}}}=K\sin \theta$$ when a 2D crystal is misoriented by an angle θ to the direction of an applied shear, where K is a constant of proportionality that is proportional to the energy cost of an individual defect climb (which is proportional to the number of particles in the crystal) and the magnitude of the applied shear. To minimize this energy contribution, it is overwhelmingly preferential for a 2D crystal to rotate until it is aligned along the direction of an applied shear. In real space, this would correspond to a shear force acting vertically in Fig. 2a, in which case strain can be easily relieved by propagation of the pictured dislocation along its Burgers vector.

As such, 2D crystals make two key responses to shear forces which are observable in their scattering structure factor. Firstly, the crystal responds by reorienting itself so that the crystal’s real lattice vectors are collinear with the shear direction. Secondly, the shear introduces a uniaxial distortion of the structure factor peaks which indicates the direction of motion; this distortion is visible in Fig. 2b. Such distortions are known to occur in sheared colloidal suspensions, and have been recently studied in skyrmion systems33,34.

### Shearing a skyrmion lattice with a magnetic field gradient

To apply a shear stress to a skyrmion lattice, the sample was immersed in a non-uniform magnetic field. Assuming that the magnetic field varies slowly over the length scale of a single skyrmion, the Zeeman energy of a skyrmion at position $$\vec{R}=\left(x,y,z\right)$$ can be written as

$${E}_{{\rm{Zeeman}}}(\vec{R})=-\vec{B}(\vec{R})\cdot {I}_{{\rm{Sk}}},$$
(1)

where we defined $${I}_{{\rm{Sk}}}={\int}_{{A}_{{\rm{Sk}}}}{\vec{m}}_{{\rm{Sk}}}(\vec{r}-\vec{R})dA$$. Here, $${\vec{m}}_{{\rm{Sk}}}$$ is the magnetization configuration of a skyrmion and the integral is taken over the area ASk bounded by the skyrmion’s radius. In the measurements, a constant magnetic field $${\vec{B}}_{0}={B}_{0}\hat{z}$$ was applied by permanent magnets, on top of which a perturbative non-uniform magnetic field was generated from a single turn of wire. This gives rise to a field profile which can be written as

$${B}_{z}(y)={B}_{0}+{B}_{1}/y,$$
(2)

where the wire lies along the x-axis in the Cartesian coordinate system (cf. Oersted wire in Fig. 3). This leads to a force Fy = ISkB1/y2 acting on the skyrmion lattice.

The steady-state motion of a skyrmion under the influence of a driving force is well described by the Thiele equation, given by35

$${\vec{F}}_{M}-\alpha \vec{v}={\vec{F}}_{{\rm{ext}}}={I}_{{\rm{Sk}}}{B}_{1}\hat{y}/{y}^{2},$$
(3)

where α is the damping coefficient, $$\vec{v}$$ is the velocity of the skyrmion, and $${\vec{F}}_{{\rm{ext}}}$$ is the external force that acts on the skyrmion. The Magnus force $${\vec{F}}_{M}$$ can be written as $${\vec{F}}_{M}=\vec{G}\times \vec{v}$$ and it acts perpendicularly to the direction of the applied velocity, where $$\vec{G}=\pm 2\pi N\hat{z}$$ is the gyrovector. Here, the topological winding number N is the number of times the skyrmion moments wrap the unit sphere. The angle from the y-axis at which skyrmions are driven is given by

$${\theta }_{{\rm{SkH}}}=\arctan (G/\alpha ).$$
(4)

Figure 3 shows velocity fields obtained from solutions of Eq. (3) as a function of θSkH. These velocity fields shear the skyrmion lattice along its direction of motion. The direction of the shear can be inferred from the structure factor, and the direction of the force is known; the skyrmion Hall angle is then the angle between the shear and the applied force.

### REXS measurements of the skyrmion Hall angle

To use this approach to measure the skyrmion Hall angle experimentally, REXS was performed in transmission on a thin, 10 × 10-μm2-sized FeGe lamella with a measured TC of 273 K31. The presence of stray temperature or magnetic field gradients is known to cause the skyrmion lattice to rotate12,36. To verify that no such unintended external forces were present in our experimental setup, the motion of the peaks in the REXS diffraction pattern was tracked prior to the application of the shearing magnetic field gradient. This motion was found to be very small, and can be seen by comparing Fig. 4a and b. Subsequently, the response of the skyrmion lattice to the application of a shearing magnetic field gradient was measured; the average of the diffraction patterns obtained after this drive is shown in Fig. 4c. An example video of the response of the skyrmion lattice to the shearing magnetic field is included in the Supplementary Material (Supplementary Video 1), as well as a video of the average response (Supplementary Video 2).

While these data make the reorientation of the skyrmion lattice in response to a shear force strikingly clear, the six-fold symmetry of the acquired diffraction pattern made the unambiguous determination of the skyrmion Hall angle impossible. To break this symmetry, sufficient strain must build up in the skyrmion lattice in response to the shear to broaden the diffraction peaks34. Leveraging the reduction in magnetic softness further from TC, the same experiment was performed at 250 K; the results are shown in Fig. 4d. The angle between the labelled direction of motion and the direction of the applied force is the skyrmion Hall angle, and was measured to be 55° ± 2°.

## Discussion

This angle is larger than any previous measurement of the skyrmion Hall angle in magnetic multilayer systems. However, this is still far from the theoretical limit of 90°. The primary damping mechanism for skyrmions moving through a metal occurs via interaction with conduction electrons37. An insulating skyrmion-hosting material (such as Cu2OSeO3) may have a skyrmion Hall angle which is closer to the 90° limit, due to greatly reduced damping. The skyrmion Hall angle in such candidate low-dissipation materials can be measured using the presented technique, unlocking the possibility of developing ultra-high mobility devices which take advantage of this unique motion10,11,12.

Recent theoretical work suggests that the presence of pinning potentials in real materials should make the skyrmion Hall angle dependent on the magnitude of an applied drive, approaching zero in the low drive limit, while the value given in Eq. (4) is expected to correspond to the high drive limit10,11. This hypothesis is supported by experimental evidence, obtained by measuring the skyrmion Hall angle in sputtered magnetic multilayers17. The highly energetic nature of the sputtering technique is expected to be the source of the pinning potentials present in these materials systems17. As the FeGe lamella used in this experiment was cut from a high-quality single crystal31, we anticipate that this drive dependence will be suppressed in our sample, and that the angle measured here should be considered to be a lower bound that is close to the theoretical value. A detailed study of the drive dependence of the skyrmion Hall angle in samples cut from a single crystal would make for an interesting topic of future research.

This technique also opens up the prospect of studying the effect of chemical doping on the skyrmion Hall angle in a precise and systematic manner. The ability to fine tune the response of a skyrmion to an applied force represents a significant milestone in the field of skyrmionics, moving the skyrmion Hall angle from an unknown variable to a free parameter. The direction of motion of skyrmions can be further controlled by exploiting the direction in which external forces are applied. While the skyrmion Hall angle is defined as the angle between an applied force and a skyrmion’s resultant velocity, the direction in which forces are applied can be complicated by the internal structure of skyrmions. In Eq. (3), the external force $${\vec{F}}_{{\rm{ext}}}$$ is set to be equal to the gradient of the Zeeman energy of a skyrmion. When $${\vec{F}}_{{\rm{ext}}}$$ is instead equated to the force due to spin-orbit torque from an adjacent heavy metal layer, the Thiele equation becomes

$${\vec{F}}_{M}-\alpha \vec{v}=kR(\phi ){\vec{j}}_{{\rm{HM}}}$$
(5)

where k is a constant of proportionality, R(ϕ) is the rotation matrix in the plane spanned by $${\vec{F}}_{M}$$ and $$\vec{v}$$, ϕ is the helicity angle of a skyrmion and $${\vec{j}}_{{\rm{HM}}}$$ is the current density in the heavy metal layer7. As Bloch-type skyrmions have a helicity of ±π/2, the force due to spin-orbit torque is normal to $${\vec{j}}_{{\rm{HM}}}$$; in this case, the direction of the skyrmion’s induced velocity is parallel to $${\vec{j}}_{{\rm{HM}}}$$ only when the skyrmion Hall angle is 90°. A detailed discussion of the relationship between current, force and drive direction for Néel- and Bloch-type skyrmions can be found in ref. 7.

In conclusion, we have introduced a novel technique for the straightforward measurement of the skyrmion Hall angle in skyrmion lattice systems. The success of the technique demonstrates that even the properties of a crystal consisting of topologically wound magnetic moments are completely determined by its defects and dislocations, mirroring the discovery of the Peierls–Nabarro stress. We have used this technique to measure the skyrmion Hall angle in a thin lamella of FeGe, finding an angle of 55° ± 2°. This angle far exceeds previous measurements of the skyrmion Hall angle in magnetic multilayer systems, providing verification for previous theoretical work38. This giant skyrmion Hall angle opens the door to ultra-low dissipation devices, while the technique allows for the methodical study of the skyrmion Hall angle in the skyrmion lattice state.

## Methods

### Resonant elastic X-ray scattering

The REXS experiment was carried out in the transmission geometry on a 400-nm-thick, 10 × 10 μm2-sized FeGe lamella using a CCD camera31. To protect the camera from overexposure in this geometry, a horizontal beamstop was used to block the direct X-ray beam. Nevertheless, the Airy rings which arise from optical diffraction of the beam are extremely bright. These, alongside other contributions from the direct beam, have been masked in all figures to emphasize the magnetic diffraction signal.

### Magnetic gradient field

To provide the perturbative field, a single turn of 600-μm-diameter Kapton-insulated Cu wire was suspended ~1 mm from the sample, with the core of the wire approximately lying in the plane of the sample surface. An out-of-plane magnetic field gradient of ~1 mT mm−1 was mapped out with a Hall probe, obtained by driving 11 A through the wire. The wire was thermally anchored on points, each ~ 10 mm away on both from the sample, and the experiment was performed under ultrahigh vacuum conditions, which minimizes the effects of a perturbative temperature gradient.

Entering the skyrmion phase by field sweeping from the helical state, a roughly randomly oriented skyrmion lattice was obtained. Due to the nature of the small 10 × 10 μm2 FeGe samples (prepared by focused-ion-beam milling) there were preferential orientations for the skyrmion lattice in the absence of a driving force, which could be the result of, e.g., a strong shape anisotropy. As a result of this energetic anisotropy, we observed minor natural reorientation of the skyrmion lattice for 1–2 min upon entering the skyrmion pocket in the absence of any external drive. In order to completely rule out any effects due to this reorientation, we found that waiting for 15 min was sufficient. The field gradient was applied after this 15 min pause and the lattice reorientation was measured for a further 15 min. This experiment was performed 15 times at 268 K on a sample of FeGe with a measured TC of 273 K: four times with an out-of-plane magnetic field of 55 mT, four times at 50 mT, four times at 45 mT, and three times at 60 mT. The reorientation in response to the application of the shearing Oersted field was found to be independent of the out-of-plane magnetic field and was extremely significant in all experiments. The average overall 15 final states is shown in Fig. 4c.

The same experiment was then repeated 24 times at 250 K, eight times each at external magnetic field value of 65, 60, and 55 mT. An average over the final frames obtained after application of the magnetic field gradient for 15 min at 250 K is given in Fig. 4d, with the anisotropic peak broadening indicating the rough direction of motion.

The skyrmion Hall angle was calculated from the data set acquired just below TC at 268 K, as shown in Fig. 4c. Measuring at this temperature has the advantage that the same information can be obtained in a shorter period of time, as compared to measuring at a lower temperature. As such, the strain induced anisotropy measured in the 250 K experiment was used to infer the approximate direction of motion, while the precise measurement of the skyrmion Hall angle was made using the diffraction spots imaged in Fig. 4c. The quoted uncertainty in the measured value is half of the angular width of the average overall 15 final peak orientations imaged in Fig. 4c.