Skyrmion Hall Angle: Theory & Experiment

• Skyrmion Hall Angle is defined as the angle between the driving force and the skyrmion’s net velocity, capturing its transverse motion due to the gyrotropic response.
• Experimental and simulation studies show that the angle varies with current density, disorder, internal mode excitations, and temperature, distinguishing pinned, creep, and steady-flow regimes.
• The Hall angle can be actively tuned through substrate symmetry, ferrimagnetic compensation, and spin–orbit interactions, offering new avenues for skyrmion-based device applications.

Searching arXiv for recent and foundational papers on the skyrmion Hall angle. The skyrmion Hall angle (SkH, often written $\theta_{\rm sk}$ or $\Theta_{\rm SkH}$) is the angle between the driving direction and the net drift velocity of a magnetic skyrmion. It quantifies the transverse deflection of skyrmion motion under current, thermal, or other nonequilibrium drives, and is the kinematic signature of the gyrotropic or Magnus response associated with skyrmionic topology. In its most standard form, it is defined by $\theta_{\rm sk}=\arctan(v_y/v_x)$, where $v_x$ and $v_y$ are the longitudinal and transverse velocity components, respectively (Jiang et al., 2016). Although early rigid-skyrmion treatments predict a drive-independent angle fixed by the ratio of gyrotropic and dissipative terms, subsequent experimental, micromagnetic, and particle-based studies established that the observed angle depends strongly on disorder, depinning regime, internal mode excitation, substrate symmetry, torque symmetry, and magnetic compensation, and in some systems it can be suppressed, reversed, quantized, or rendered nearly zero (Reichhardt et al., 2016).

1. Definition and theoretical basis

A magnetic skyrmion is a localized spin texture carrying an integer topological charge,

$$Q=\frac{1}{4\pi}\int \mathbf m\cdot(\partial_x\mathbf m\times \partial_y\mathbf m)\,dx\,dy=\pm1,$$

and the existence of this nontrivial topology is the origin of the emergent gyrotropic response that underlies the skyrmion Hall effect (Jiang et al., 2016). In the conventional rigid-texture approximation, the skyrmion center obeys a Thiele equation in which the gyrovector term generates a transverse component of motion while damping and external torques determine longitudinal drift.

For current-driven motion under spin Hall torque, the steady-state dynamics may be written in the form

$$\mathbf G\times \mathbf v-\alpha\,\mathbf D\cdot \mathbf v+4\pi B_0J_{\rm hm}=0,$$

with $\mathbf G=(0,0,-4\pi Q)$ the gyrovector, $\alpha$ the Gilbert damping, $\mathbf D$ the dissipative tensor, and $\Theta_{\rm SkH}$0 the spin-Hall-torque drive (Jiang et al., 2016). For $\Theta_{\rm SkH}$1, one obtains

$\Theta_{\rm SkH}$2

so that

$\Theta_{\rm SkH}$3

which is independent of drive strength in the rigid-skyrmion limit (Jiang et al., 2016). Closely related particle-based formulations define the intrinsic Hall angle as $\Theta_{\rm SkH}$4, where $\Theta_{\rm SkH}$5 and $\Theta_{\rm SkH}$6 are Magnus and damping coefficients (Reichhardt et al., 2016).

This idealized picture is only the starting point. Later work demonstrated that rigid-body theory is insufficient whenever quenched disorder, plastic flow, substrate periodicity, internal deformations, breathing modes, or topological compensation become important. A central result of the modern literature is therefore that the experimentally observed Hall angle is often a nonequilibrium transport property rather than a simple material constant (Tomasello et al., 2018).

2. Experimental observation and measurement protocols

Direct real-space observation of the skyrmion Hall effect was reported in Ta (5 nm)/Co$\Theta_{\rm SkH}$7Fe$\Theta_{\rm SkH}$8B$\Theta_{\rm SkH}$9 (1.1 nm)/TaO$\theta_{\rm sk}=\arctan(v_y/v_x)$0 (3 nm) trilayers patterned into $\theta_{\rm sk}=\arctan(v_y/v_x)$1 Hall bars and imaged by differential polar magneto-optical Kerr effect microscopy at room temperature (Jiang et al., 2016). In that study, bipolar current pulses of $\theta_{\rm sk}=\arctan(v_y/v_x)$2 with densities up to $\theta_{\rm sk}=\arctan(v_y/v_x)$3 drove $\theta_{\rm sk}=\arctan(v_y/v_x)$4 skyrmions, and the displacement components $\theta_{\rm sk}=\arctan(v_y/v_x)$5 were tracked pulse by pulse to extract average velocities

$\theta_{\rm sk}=\arctan(v_y/v_x)$6

with $\theta_{\rm sk}=\arctan(v_y/v_x)$7 pulses used to suppress stochastic creep (Jiang et al., 2016).

Three transport regimes were identified as a function of electron current density. For $\theta_{\rm sk}=\arctan(v_y/v_x)$8, the skyrmions remained pinned. For $\theta_{\rm sk}=\arctan(v_y/v_x)$9, stochastic creep occurred with $v_x$0 and hence $v_x$1. Above $v_x$2, a steady-flow regime emerged with a clear transverse component (Jiang et al., 2016). At the highest accessible current density, $v_x$3, the measured values were $v_x$4, $v_x$5, and $v_x$6; no saturation was reached, suggesting a larger angle at higher current while still below $v_x$7 (Jiang et al., 2016).

Subsequent high-resolution imaging in chiral magnetic multilayers using scanning transmission X-ray microscopy showed that in a plastic-flow regime the average Hall angle can become effectively diameter-independent. In Ta(3.2 nm)/Pt(2.7 nm)/[Pt(0.6)/Co$v_x$8B$v_x$9(0.8)/Ir(0.4)]$v_y$0/Ta(2.2 nm) wires, over $v_y$1 moving skyrmions with diameters spanning $v_y$2 to $v_y$3, an average velocity of $v_y$4 and mean Hall angle of $v_y$5 were measured, with no observable $v_y$6 dependence expected from the clean rigid-Thiele estimate (Zeissler et al., 2019). That study identified the local energy landscape, rather than diameter alone, as the dominant control parameter in plastic flow.

A different experimental protocol was used in ferrimagnets to infer the Hall angle from transverse elongation of pinned bubble domains. In GdFeCo/Pt, one side of a bubble was pinned, and a uniform current caused the free side to elongate at an angle relative to the current axis. The measured acute angle changed sign with bubble polarity and crossed through zero at the angular-momentum compensation temperature $v_y$7, where the net spin density vanishes (Hirata et al., 2018). This established a direct connection between the Hall response and the temperature-dependent ferrimagnetic spin density.

3. Disorder, depinning, creep, and flow

Disorder-induced suppression near depinning is one of the defining features of the observed skyrmion Hall angle. Particle-based simulations of driven skyrmions in quenched disorder showed that the ratio

$v_y$8

is zero at depinning and increases with drive before saturating to the disorder-free value $v_y$9 at high drive (Reichhardt et al., 2016). In the plastic-flow regime,

$$Q=\frac{1}{4\pi}\int \mathbf m\cdot(\partial_x\mathbf m\times \partial_y\mathbf m)\,dx\,dy=\pm1,$$0

where the slope depends on disorder strength and intrinsic Hall ratio (Reichhardt et al., 2016). For small intrinsic angles, $$Q=\frac{1}{4\pi}\int \mathbf m\cdot(\partial_x\mathbf m\times \partial_y\mathbf m)\,dx\,dy=\pm1,$$1, so the Hall angle itself grows nearly linearly with drive, in agreement with imaging experiments (Reichhardt et al., 2016).

The physical mechanism identified in that work is the side-jump suppression induced by pinning. Pinned skyrmions force moving ones to swerve in such a way that the transverse component is reduced at low drives, making $$Q=\frac{1}{4\pi}\int \mathbf m\cdot(\partial_x\mathbf m\times \partial_y\mathbf m)\,dx\,dy=\pm1,$$2 vanish at depinning and recover only gradually as the system dynamically reorders (Reichhardt et al., 2016). In weak pinning, where the lattice depins elastically rather than plastically, the dependence becomes nonlinear,

$$Q=\frac{1}{4\pi}\int \mathbf m\cdot(\partial_x\mathbf m\times \partial_y\mathbf m)\,dx\,dy=\pm1,$$3

with nonuniversal exponent $$Q=\frac{1}{4\pi}\int \mathbf m\cdot(\partial_x\mathbf m\times \partial_y\mathbf m)\,dx\,dy=\pm1,$$4; one example reported is $$Q=\frac{1}{4\pi}\int \mathbf m\cdot(\partial_x\mathbf m\times \partial_y\mathbf m)\,dx\,dy=\pm1,$$5 for $$Q=\frac{1}{4\pi}\int \mathbf m\cdot(\partial_x\mathbf m\times \partial_y\mathbf m)\,dx\,dy=\pm1,$$6 at $$Q=\frac{1}{4\pi}\int \mathbf m\cdot(\partial_x\mathbf m\times \partial_y\mathbf m)\,dx\,dy=\pm1,$$7 (Reichhardt et al., 2016).

Finite temperature extends this picture by introducing a creep regime below the zero-temperature threshold. In driven skyrmion crystals with quenched disorder and Langevin noise, thermally activated motion consists of intermittent hops between pinned states, but the long-time-averaged transverse velocity remains negligible because skyrmions have time to relax into force-balance points inside pinning sites (Reichhardt et al., 2018). Accordingly, the creep regime is characterized by $$Q=\frac{1}{4\pi}\int \mathbf m\cdot(\partial_x\mathbf m\times \partial_y\mathbf m)\,dx\,dy=\pm1,$$8 even when there is finite longitudinal motion (Reichhardt et al., 2018). Only above a temperature- and drive-dependent crossover to viscous flow does the Hall angle become finite and then increase with drive or temperature toward the intrinsic limit (Reichhardt et al., 2018).

This pinned–creep–flow sequence is now a standard organizing principle. The room-temperature experiment reporting direct observation of the skyrmion Hall effect found precisely such a progression from pinning to creep with vanishing angle, and then to steady flow with increasing transverse deflection (Jiang et al., 2016). A plausible implication is that reported discrepancies between nominally similar materials often reflect different positions in this dynamical phase space rather than inconsistent intrinsic parameters.

4. Deviations from the rigid-skyrmion picture

A major development in the literature is the recognition that current dependence can arise even in ideal samples if internal modes are active. Micromagnetic work on perpendicular ferromagnets showed that thermal modes, especially skyrmion breathing modes, invalidate the simplest rigid approximation and can produce a current-dependent Hall angle through the combined action of field-like and damping-like torques (Tomasello et al., 2018). In that framework, the effective force from the field-like torque acquires a dependence on breathing amplitude $$Q=\frac{1}{4\pi}\int \mathbf m\cdot(\partial_x\mathbf m\times \partial_y\mathbf m)\,dx\,dy=\pm1,$$9, so that

$$\mathbf G\times \mathbf v-\alpha\,\mathbf D\cdot \mathbf v+4\pi B_0J_{\rm hm}=0,$$0

instead of remaining constant (Tomasello et al., 2018). Micromagnetic simulations then showed that strong breathing can reduce the low-current angle and produce gradual saturation only at higher current densities (Tomasello et al., 2018).

Shape effects provide another route beyond the circular rigid model. For elliptical ferromagnetic skyrmions stabilized by anisotropic Dzyaloshinskii–Moriya interaction, the dissipative tensor becomes anisotropic, $$\mathbf G\times \mathbf v-\alpha\,\mathbf D\cdot \mathbf v+4\pi B_0J_{\rm hm}=0,$$1, and the Hall-angle ratio depends explicitly on the ellipse axes (Xia et al., 2019). For the specific analytic profile used there,

$$\mathbf G\times \mathbf v-\alpha\,\mathbf D\cdot \mathbf v+4\pi B_0J_{\rm hm}=0,$$2

with $$\mathbf G\times \mathbf v-\alpha\,\mathbf D\cdot \mathbf v+4\pi B_0J_{\rm hm}=0,$$3 and $$\mathbf G\times \mathbf v-\alpha\,\mathbf D\cdot \mathbf v+4\pi B_0J_{\rm hm}=0,$$4 the semi-axes along $$\mathbf G\times \mathbf v-\alpha\,\mathbf D\cdot \mathbf v+4\pi B_0J_{\rm hm}=0,$$5 and $$\mathbf G\times \mathbf v-\alpha\,\mathbf D\cdot \mathbf v+4\pi B_0J_{\rm hm}=0,$$6 (Xia et al., 2019). Elongation along the current direction reduces the Hall angle, whereas elongation transverse to the current increases it (Xia et al., 2019).

The question of size dependence has also been revisited. One analytic treatment based on a single-length-scale skyrmion profile concluded that, for a given profile shape, the Hall angle is independent of both current density and the length scale controlling skyrmion radius, with

$$\mathbf G\times \mathbf v-\alpha\,\mathbf D\cdot \mathbf v+4\pi B_0J_{\rm hm}=0,$$7

where $$\mathbf G\times \mathbf v-\alpha\,\mathbf D\cdot \mathbf v+4\pi B_0J_{\rm hm}=0,$$8 is a profile-dependent pure number (Bera et al., 2020). This stands in contrast to earlier large-skyrmion estimates such as $$\mathbf G\times \mathbf v-\alpha\,\mathbf D\cdot \mathbf v+4\pi B_0J_{\rm hm}=0,$$9, which imply $\mathbf G=(0,0,-4\pi Q)$0 (Zeissler et al., 2019). The experimental observation of diameter independence in plastic flow does not directly validate the clean single-scale theory, but it does show that apparent size scaling can be masked or overridden by disorder-dominated dynamics (Zeissler et al., 2019).

A further line of work examines additional transverse viscous terms generated by magnon fluctuations in insulating chiral magnets. In that case, a DMI-derived Hall viscosity modifies the transverse response, leading to analytic Hall-angle expressions dependent on skyrmion shape, size, and the ratio $\mathbf G=(0,0,-4\pi Q)$1, while being roughly independent of Gilbert damping (Kim, 20 Oct 2025). Since that work postdates the earlier transport literature, it should be viewed as an extension rather than a replacement of the conventional Magnus–damping framework.

5. Substrate symmetry, quantization, reversal, and absolute transverse mobility

On periodic substrates, the skyrmion Hall angle acquires a much richer structure than in random disorder. For a skyrmion moving over a two-dimensional periodic obstacle array under dc and ac drives, the combination of Magnus dynamics, substrate periodicity, and phase locking generates plateaus of quantized Hall angles (Vizarim et al., 2020). When the skyrmion translates by $\mathbf G=(0,0,-4\pi Q)$2 lattice constants during one ac period $\mathbf G=(0,0,-4\pi Q)$3,

$\mathbf G=(0,0,-4\pi Q)$4

and therefore

$\mathbf G=(0,0,-4\pi Q)$5

with observed locked values including $\mathbf G=(0,0,-4\pi Q)$6, $\mathbf G=(0,0,-4\pi Q)$7, $\mathbf G=(0,0,-4\pi Q)$8, $\mathbf G=(0,0,-4\pi Q)$9, and $\alpha$0 (Vizarim et al., 2020). Between these plateaus, motion can be chaotic and the Hall angle fluctuates strongly (Vizarim et al., 2020).

Under biharmonic ac forcing on a periodic substrate, sign reversals are possible. When the ac amplitudes differ in the two directions, elliptical rather than circular ac orbits break additional spatiotemporal symmetries, and ratchet-like rectification can produce alternating sign intervals of $\alpha$1, hence multiple reversals of $\alpha$2 as the dc drive is varied (Vizarim et al., 2020). The same work reported phases of absolute transverse mobility, where $\alpha$3 and $\alpha$4, giving $\alpha$5 despite nonzero damping (Vizarim et al., 2020).

Related studies of dc-plus-ac driving over square obstacle arrays identified the coexistence of Shapiro steps and directional locking. Depending on whether the ac drive is parallel or perpendicular to the dc drive, the Hall angle may remain constant on steps, increase or decrease with increasing drive, overshoot the intrinsic Hall angle, or even show sign reversals at small drives (Vizarim et al., 2020). In particular, a transverse ac drive can yield large overshoots such as a first nonzero locked state at $\alpha$6 for an intrinsic angle of only about $\alpha$7 (Vizarim et al., 2020).

Obstacle-lattice symmetry also determines privileged locking angles. For triangular and honeycomb arrays, the principal robust locking directions are $\alpha$8 and $\alpha$9, corresponding to low-collision channels selected by the substrate symmetry (Vizarim et al., 2021). The locking windows shift with obstacle density and Magnus-to-damping ratio, and in triangular arrays the $\mathbf D$0 and $\mathbf D$1 plateaus can be mutually exclusive in $\mathbf D$2 space (Vizarim et al., 2021). This suggests that patterned obstacle landscapes can function as trajectory selectors or sorters.

Commensuration with a background skyrmion lattice adds another control parameter. For a driven skyrmion moving through a pinned skyrmion background, integer filling fractions can produce symmetry-aligned channeling with $\mathbf D$3, while incommensurate fillings yield finite Hall angles and more disordered motion (Reichhardt et al., 2022). Under commensurate conditions, multistep depinning and narrow-band velocity noise appear; under incommensurate conditions, depinning is single-step and the velocity noise is broad-band (Reichhardt et al., 2022).

6. Suppression, cancellation, and nonstandard Hall responses

Because transverse deflection can drive skyrmions into device edges, a substantial part of the literature is devoted to suppressing or nullifying the Hall angle. One route is ferrimagnetic compensation. In ferrimagnets, the gyrotropic vector is proportional to the net areal spin density $\mathbf D$4,

$\mathbf D$5

so that

$\mathbf D$6

At the angular-momentum compensation temperature $\mathbf D$7, where $\mathbf D$8, the Hall angle vanishes exactly (Hirata et al., 2018). This was confirmed experimentally in GdFeCo/Pt by observing a sign change of the elongation angle above and below $\mathbf D$9 and a zero crossing near the compensation point (Hirata et al., 2018).

A second route uses graded magnetization in synthetic ferrimagnets. Micromagnetic simulations of synthetic ferrimagnetic skyrmions with a linear saturation-magnetization gradient $\Theta_{\rm SkH}$00 showed that the gradient changes the normalized radius $\Theta_{\rm SkH}$01, thereby modifying the dissipative tensor and the Hall angle dynamically (Bo et al., 2024). In the reported Gd/Co-based system, the average Hall angle changed almost linearly with $\Theta_{\rm SkH}$02 and, for $\Theta_{\rm SkH}$03, was reduced to $\Theta_{\rm SkH}$04 from a zero-gradient value of about $\Theta_{\rm SkH}$05 (Bo et al., 2024). This result supports the broader idea of graded-index skyrmionics.

Gate-controlled spin–orbit coupling offers an electronic cancellation mechanism. In a Thiele treatment including both spin-transfer torque and Rashba spin–orbit torque, the Hall angle becomes

$\Theta_{\rm SkH}$06

where $\Theta_{\rm SkH}$07 is the inverse spin-orbit length and can be tuned by gate voltage (Plettenberg et al., 2019). Straight motion occurs when the numerator vanishes, giving a critical $\Theta_{\rm SkH}$08 that cancels transverse drift (Plettenberg et al., 2019). This mechanism was proposed for electrically steered racetrack operation.

Hybrid DMI provides another symmetry-based method. In systems containing both interfacial and bulk DMI, a global spin rotation maps the magnetic energy to a purely interfacial-DMI problem but rotates the effective current direction by an angle $\Theta_{\rm SkH}$09 (Kim et al., 2018). The resulting Hall angles for the two skyrmion polarities become

$\Theta_{\rm SkH}$10

so the response is asymmetric in polarity and one polarity can be tuned to zero Hall angle by satisfying $\Theta_{\rm SkH}$11 (Kim et al., 2018). This is a particularly direct example of symmetry breaking converting the usually symmetric $\Theta_{\rm SkH}$12 response into an asymmetric one.

Thermally driven transport can also strongly suppress the angle. In Co/Pt bilayer nanoracetracks under a thermal gradient, micromagnetic simulations found motion toward the hotter region with a nearly vanishing Hall angle over a specific material window; the reported suppressed regime corresponds to $\Theta_{\rm SkH}$13, $\Theta_{\rm SkH}$14, and $\Theta_{\rm SkH}$15, with $\Theta_{\rm SkH}$16 and in some cases below $\Theta_{\rm SkH}$17 (Kumar et al., 8 Oct 2025). This suggests that the effective force decomposition under thermal gradients differs qualitatively from ordinary current-driven Magnus-dominated motion.

7. Extensions, misconceptions, and device implications

A common misconception is that the skyrmion Hall effect is determined solely by topological charge and therefore vanishes automatically whenever the net topological charge is zero. This is only partially correct. It holds for certain compensated ferrimagnetic settings, where the net spin density and hence the fictitious magnetic field vanish (Hirata et al., 2018), but it fails more generally for spin–orbit-torque-driven topologically trivial textures such as skyrmioniums or synthetic antiferromagnetic skyrmions. In those systems, the Hall angle can depend directly on helicity rather than net topological charge (Msiska et al., 2021). For pure damping-like spin–orbit torque, the derived result is $\Theta_{\rm SkH}$18, where $\Theta_{\rm SkH}$19 is the helicity (Msiska et al., 2021). A plausible implication is that “topological triviality” does not by itself guarantee straight racetrack motion unless the torque symmetry and helicity are also controlled.

An even stronger challenge to the conventional charge-based viewpoint appears in altermagnets. There, skyrmions can have zero net magnetization and zero skyrmion charge, yet still exhibit a Hall effect mediated by a magnetic quadrupole and a hidden Aharonov–Casher-type gauge field (Jin et al., 2024). In the steady state driven by spin-transfer torque, one finds

$\Theta_{\rm SkH}$20

and

$\Theta_{\rm SkH}$21

where the sign flips under $\Theta_{\rm SkH}$22 through $\Theta_{\rm SkH}$23 (Jin et al., 2024). This demonstrates that neutral, compensated textures can still support Hall transport when a different internal multipolar gauge structure replaces the usual topological gyrovector.

From a device perspective, the skyrmion Hall angle is both a liability and a resource. It is a liability in narrow racetracks because transverse drift promotes edge annihilation and data loss, motivating compensation schemes, synthetic ferrimagnets, gate control, graded magnetization, and anisotropy engineering (Bo et al., 2024). It is a resource in patterned or multichannel geometries because the angle can be used for topological sorting, controlled edge accumulation or ejection, ratchet transport, directional selection, and logic or routing operations that rely on sideways motion rather than suppressing it (Jiang et al., 2016). Under combined dc and ac drives on periodic substrates, quantized Hall-angle states and absolute transverse mobility suggest reconfigurable transport primitives, while obstacle-density engineering in triangular and honeycomb arrays suggests trajectory programming through patterned landscapes (Vizarim et al., 2021).

The modern understanding of the skyrmion Hall angle is therefore not a single formula but a hierarchy of regimes. In clean rigid motion it is set by the gyrotropic-to-dissipative ratio; near depinning it is suppressed by disorder; in periodic media it can lock to symmetry angles; in ferrimagnets it can vanish by compensation; in hybrid or spin–orbit-coupled systems it can be steered electronically or made asymmetric; and in topologically compensated or altermagnetic systems it can persist through helicity or hidden gauge fields even when conventional expectations predict its absence (Jin et al., 2024). This suggests that the skyrmion Hall angle is best regarded as a diagnostic of the full driven quasiparticle dynamics, not merely of skyrmion topology in isolation.