# Harnessing Skyrmion Hall Effect by Thickness Gradients in Wedge-Shaped Samples of Cubic Helimagnets

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## Abstract

**:**

## 1. Introduction

## 2. Phenomenological Model and Equations

## 3. The Simplified Phase Diagram of States in Thin-Layered Systems

#### 3.1. Helicoid–Cone Phase Transition

#### 3.2. Overlap of Surface Twists in Isolated Skyrmions within Thin-Film Helimagnets

#### 3.3. The Properties of ISs in Ultrathin Films

## 4. Static and Dynamic Properties of ISs in Wedge-Shaped Geometries

#### 4.1. Internal Structure of Edge States in Wedge Geometries

#### 4.2. Skyrmion “Pit Stops”

#### 4.3. Current-Driven Dynamics of ISs in Wedge-Shaped Geometries

#### 4.4. Comment on the Skyrmion “Pit Stops” in Truncated Wedges

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bogdanov, A.N.; Yablonsky, D.A. Theormodynamically stable vortices in magnetically ordered crystals. Mixed state of magnetics. Zh. Eksp. Teor. Fiz.
**1989**, 95, 178, [Sov. Phys. JETP 1989, 68, 101].. [Google Scholar] - Nagaosa, N.; Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol.
**2013**, 8, 899. [Google Scholar] [CrossRef] [PubMed] - Moriya, T. Anisotropic Superexchange Interaction and Weak Ferromagnetism. Phys. Rev.
**1960**, 120, 91. [Google Scholar] [CrossRef] [Green Version] - Dzyaloshinskii, I.E. Theory of helicoidal structures in antiferromagnets. I. nonmetals. J. Sov. Phys. JETP-USSR
**1964**, 19, 960. [Google Scholar] - Dzyaloshinskii, I.E. The Theory of Helicoidal Structures in Antiferromagnets. II. Metals. J. Sov. Phys. JETP-USSR
**1965**, 20, 223. [Google Scholar] - Hobart, R.H. On the Instability of a Class of Unitary Field Models. Proc. Phys. Soc. Lond.
**1963**, 82, 201. [Google Scholar] [CrossRef] - Derrick, G.H. Comments on Nonlinear Wave Equations as Models for Elementary Particles. J. Math. Phys.
**1964**, 5, 1252. [Google Scholar] [CrossRef] [Green Version] - Mühlbauer, S.; Binz, B.; Jonietz, F.; Pfleiderer, C.; Rosch, A.; Neubauer, A.; Georgii, R.; Böni, P. Skyrmion lattice in a chiral magnet. Science
**2009**, 323, 915. [Google Scholar] [CrossRef] [Green Version] - Wilhelm, H.; Leonov, A.O.; Roessler, U.K.; Burger, P.; Hardy, F.; Meingast, C.; Gruner, M.E.; Schnelle, W.; Schmidt, M.; Baenitz, M. Scaling Study and Thermodynamic Properties of the cubic Helimagnet FeGe. Phys. Rev. B
**2016**, 94, 144424. [Google Scholar] [CrossRef] [Green Version] - Bogdanov, A.; Hubert, A. Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Magn. Mater.
**1994**, 138, 255. [Google Scholar] [CrossRef] - Bogdanov, A.; Hubert, A. The stability of vortex-like structures in uniaxial ferromagnets. J. Magn. Magn. Mater.
**1999**, 195, 182. [Google Scholar] [CrossRef] - Wiesendanger, R. Nanoscale magnetic skyrmions in metallic films and multilayers: A new twist for spintronics. Nat. Rev. Mater.
**2016**, 1, 16044. [Google Scholar] [CrossRef] [Green Version] - Lebech, B.; Bernhard, J.; Freltoft, T. Magnetic structures of cubic FeGe studied by small-angle neutron scattering. J. Phys. Condens. Matter
**1989**, 1, 6105. [Google Scholar] [CrossRef] - Beille, J.; Voiron, J.; Roth, M. Long period helimagnetism in the cubic B20 Fe
_{x}Co_{1-x}Si and Co_{x}Mn_{1-x}Si alloys. Sol. State Commun.**1983**, 47, 399. [Google Scholar] [CrossRef] - Yu, X.Z.; Onose, Y.; Kanazawa, N.; Park, J.H.; Han, J.H.; Matsui, Y.; Nagaosa, N.; Tokura, Y. Real-space observation of a two-dimensional skyrmion crystal. Nature
**2010**, 465, 901. [Google Scholar] [CrossRef] - Foster, D.; Kind, C.; Ackerman, P.J.; Tai, J.-S.B.; Dennis, M.R.; Smalyukh, I.I. Two-dimensional skyrmion bags in liquid crystals and ferromagnets. Nat. Phys.
**2019**, 15, 655. [Google Scholar] [CrossRef] [Green Version] - Birch, M.T.; Cortes-Ortuno, D.; Khanh, N.D.; Seki, S.; Stefancic, A.; Balakrishnan, G.; Tokura, Y.; Hatton, P.D. Topological defect-mediated skyrmion annihilation in three dimensions. Commun. Phys.
**2021**, 4, 175. [Google Scholar] [CrossRef] - Wilhelm, H.; Baenitz, M.; Schmidt, M.; Roessler, U.K.; Leonov, A.A.; Bogdanov, A.N. Precursor phenomena at the magnetic ordering of the cubic helimagnet FeGe. Phys. Rev. Lett.
**2011**, 107, 127203. [Google Scholar] [CrossRef] - Pappas, C.; Lelievre-Berna, E.; Falus, P.; Bentley, P.M.; Moskvin, E.; Grigoriev, S.; Fouquet, P.; Farago, B. Chiral Paramagnetic Skyrmion-like Phase in MnSi. Phys. Rev. Lett.
**2009**, 102, 197202. [Google Scholar] [CrossRef] [Green Version] - Yu, X.Z.; Kanazawa, N.; Onose, Y.; Kimoto, K.; Zhang, W.Z.; Ishiwata, S.; Matsui, Y.; Tokura, Y. Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe. Nat. Mater.
**2011**, 10, 106. [Google Scholar] [CrossRef] - Rybakov, F.N.; Borisov, A.B.; Bogdanov, A.N. Three-dimensional skyrmion states in thin films of cubic helimagnets. Phys. Rev. B
**2013**, 87, 094424. [Google Scholar] [CrossRef] [Green Version] - Leonov, A.O.; Togawa, Y.; Monchesky, T.L.; Bogdanov, A.N.; Kishine, J.; Kousaka, Y.; Miyagawa, M.; Koyama, T.; Akimitsu, J.; Koyama, T.; et al. Chiral surface twists and skyrmion stability in nanolayers of cubic helimagnets. Phys. Rev. Lett.
**2016**, 117, 087202. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rybakov, F.N.; Borisov, A.B.; Bluegel, S.; Kiselev, N.S. New spiral state and skyrmion lattice in 3D model of chiral magnets. N. J. Phys.
**2016**, 18, 045002. [Google Scholar] [CrossRef] - Cortes-Ortuno, D.; Wang, W.; Beg, M.; Pepper, R.A.; Bisotti, M.-A.; Carey, R.; Vousden, M.; Kluyver, T.; Hovorka, O.; Fangohr, H. Thermal stability and topological protection of skyrmions in nanotracks. Sci. Rep.
**2017**, 7, 1. [Google Scholar] [CrossRef] [Green Version] - Schulz, T.; Ritz, R.; Bauer, A.; Halder, M.; Wagner, M.; Franz, C.; Pfleiderer, C.; Everschor, K.; Garst, M.; Rosch, A. Emergent electrodynamics of skyrmions in a chiral magnet. Nat. Phys.
**2012**, 8, 301–304. [Google Scholar] [CrossRef] [Green Version] - Jonietz, F.; Mühlbauer, S.; Pfleiderer, C.; Neubauer, A.; Münzer, W.; Bauer, A.; Adams, T.; Georgii, R.; Böni, P.; Duine, R.A.; et al. Spin Transfer Torques in MnSi at Ultralow Current Densities. Science
**2010**, 330, 1648–1651. [Google Scholar] [CrossRef] [Green Version] - Hsu, P.-J.; Kubetzka, A.; Finco, A.; Romming, N.; von Bergmann, K.; Wiesendanger, R. Electric-field-driven switching of individual magnetic skyrmions. Nat. Nanotechnol.
**2017**, 12, 123–126. [Google Scholar] [CrossRef] - Kang, W.; Huang, Y.; Zheng, C.; Lv, W.; Lei, N.; Zhang, Y.; Zhang, X.; Zhou, Y.; Zhao, W. Voltage Controlled Magnetic Skyrmion Motion for Racetrack Memory. Sci. Rep.
**2016**, 6, 23164. [Google Scholar] [CrossRef] [Green Version] - Fert, A.; Cros, V.; Sampaio, J. Skyrmions on the track. Nat. Nanotechnol.
**2013**, 8, 152. [Google Scholar] [CrossRef] - Toscano, D.; Mendonca, J.; Miranda, A.; de Araujo, C.; Sato, F.; Coura, P.; Leonel, S. Suppression of the skyrmion Hall effect in planar nanomagnets by the magnetic properties engineering: Skyrmion transport on nanotracks with magnetic strips. J. Magn. Magn. Mater.
**2020**, 504, 166655. [Google Scholar] [CrossRef] [Green Version] - Gobel, B.; Mook, A.; Henk, J.; Mertig, I. Overcoming the speed limit in skyrmion racetrack devices by suppressing the skyrmion Hall effect. Phys. Rev. B
**2019**, 99, 020405(R). [Google Scholar] [CrossRef] [Green Version] - Kolesnikov, A.G.; Stebliy, M.E.; Samardak, A.S.; Ognev, A.V. Skyrmionium—High velocity without the skyrmion Hall effect. Sci. Rep.
**2018**, 8, 16966. [Google Scholar] [CrossRef] [Green Version] - Leonov, A.O.; Ler, U.K.R.; Mostovoy, M. Target-skyrmions and skyrmion clusters in nanowires of chiral magnets. EPJ Web Conf.
**2014**, 75, 05002. [Google Scholar] [CrossRef] - Barker, J.; Tretiakov, O.A. Static and Dynamical Properties of Antiferromagnetic Skyrmions in the Presence of Applied Current and Temperature. Phys. Rev. Lett.
**2016**, 116, 147203. [Google Scholar] [CrossRef] [Green Version] - Mukai, N.; Leonov, A.O. Skyrmion and meron ordering in quasi-two-dimensional chiral magnets. Phys. Rev. B
**2022**, 106, 224428. [Google Scholar] [CrossRef] - Bak, P.; Jensen, M.H. Theory of helical magnetic structures and phase transitions in MnSi and FeGe. J. Phys. C Solid State Phys.
**1980**, 13, L881. [Google Scholar] [CrossRef] - Vansteenkiste, A.; Leliaert, J.; Dvornik, M.; Helsen, M.; Garcia-Sanchez, F.; Waeyenberge, B.V. The design and verification of MuMax3. AIP Adv.
**2014**, 4, 107133. [Google Scholar] [CrossRef] - Leliaert, J.; Gypens, P.; Milosevic, M.V.; Waeyenberge, B.V.; Mulkers, J. Coupling of the skyrmion velocity to its breathing mode in periodically notched nanotracks. J. Phys. D Appl. Phys.
**2019**, 52, 024003. [Google Scholar] [CrossRef] [Green Version] - Nakanishi, O.; Yanase, A.; Hasegawa, A.; Kataoka, M. The origin of the helical spin density wave in MnSi. Solid State Comm.
**1980**, 35, 995. [Google Scholar] [CrossRef] - Leonov, A.O.; Pappas, C. Reorientation processes of tilted skyrmion and spiral states in a bulk cubic helimagnet Cu
_{2}OSeO_{3}. Front. Phys.**2023**, 11, 1105784. [Google Scholar] [CrossRef] - Leonov, A.O. Skyrmion clusters and chains in bulk and thin-layered cubic helimagnets. Phys. Rev. B
**2022**, 105, 094404. [Google Scholar] [CrossRef] - Loudon, J.C.; Leonov, A.O.; Bogdanov, A.N.; Hatnean, M.C.; Balakrishnan, G. Direct observation of attractive skyrmions and skyrmion clusters in the cubic helimagnet Cu2OSeO3. Phys. Rev. B
**2018**, 97, 134403. [Google Scholar] [CrossRef] [Green Version] - Du, H.; Zhao, X.; Rybakov, F.N.; Borisov, A.B.; Wang, S.; Tang, J.; Jin, C.; Wang, C.; Wei, W.; Kiselev, N.S.; et al. Interaction of Individual Skyrmions in a Nanostructured Cubic Chiral Magnet. Phys. Rev. Lett.
**2018**, 120, 197203. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Togawa, Y.; Koyama, T.; Takayanagi, K.; Mori, S.; Kousaka, Y.; Akimitsu, J.; Nishihara, S.; Inoue, K.; Ovchinnikov, A.S.; Kishine, J. Chiral Magnetic Soliton Lattice on a Chiral Helimagnet. Phys. Rev. Lett.
**2012**, 108, 107202. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lilley, B.A. Energies and widths of domain boundaries in ferromagnetics. Phil. Mag.
**1950**, 41, 792–813. [Google Scholar] [CrossRef] - Iwasaki, J.; Mochizuki, M.; Nagaosa, N. Current-induced skyrmion dynamics in constricted geometries. Nat. Nanotech.
**2013**, 8, 742. [Google Scholar] [CrossRef] [Green Version] - Leonov, A.O.; Mostovoy, M. Edge states and skyrmion dynamics in nanostripes of frustrated magnets. Nat. Commun.
**2017**, 8, 14394. [Google Scholar] [CrossRef] [Green Version] - Wang, B.; Wu, P.; Salguero, N.B.; Zheng, Q.; Yan, J.; Randeria, M.; McComb, D.W. Stimulated Nucleation of Skyrmions in a Centrosymmetric Magnet. ACS Nano
**2021**, 15, 13495. [Google Scholar] [CrossRef] - Zhang, X.; Zhao, G.P.; Fangohr, H.; Liu, J.P.; Xia, W.X.; Xia, J.; Morvan, F.J. Skyrmion-skyrmion and skyrmion-edge repulsions in skyrmion-based racetrack memory. Sci. Rep.
**2015**, 5, 7643. [Google Scholar] [CrossRef] [Green Version] - Rohart, S.; Thiaville, A. Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction. Phys. Rev. B
**2013**, 88, 184422. [Google Scholar] [CrossRef] [Green Version] - Meynell, S.; Wilson, M.N.; Fritzsche, H.; Bogdanov, A.N.; Monchesky, T.L. Surface twist instabilities and skyrmion states in chiral ferromagnets. Phys. Rev. B
**2014**, 90, 014406. [Google Scholar] [CrossRef] [Green Version] - Lin, S.Z.; Reichhardt, C.; Batista, C.D.; Saxena, A. Particle model for skyrmions in metallic chiral magnets: Dynamics, pinning, and creep. Phys. Rev. B
**2013**, 87, 214419. [Google Scholar] [CrossRef] [Green Version] - Thiele, A.A. Steady-State Motion of Magnetic Domains. Phys. Rev. Lett.
**1973**, 30, 230. [Google Scholar] [CrossRef] - Brearton, R.; Burn, D.M.; Haghighirad, A.A.; Laan, G.v.; Hesjedal, T. Three-dimensional structure of magnetic skyrmions. Phys. Rev. B
**2022**, 106, 214404. [Google Scholar] [CrossRef] - Navau, C.; Del-Valle, N.; Sanchez, A. Analytical trajectories of skyrmions in confined geometries: Skyrmionic racetracks and nano-oscillators. Phys. Rev. B
**2016**, 94, 181104. [Google Scholar] [CrossRef] - Iwasaki, J.; Koshibae, W.; Nagaosa, N. Colossal Spin Transfer Torque Effect on Skyrmion along the Edge. Nano Lett.
**2014**, 14, 4432. [Google Scholar] [CrossRef] - Leonov, A.O. Surface anchoring as a control parameter for shaping skyrmion or toron properties in thin layers of chiral nematic liquid crystals and noncentrosymmetric magnets. Phys. Rev. E
**2021**, 104, 044701. [Google Scholar] [CrossRef]

**Figure 1.**(color online) The simplified phase diagram of states. (

**a**) The phase diagram for model (3) is constructed in reduced variables for the film thickness $\nu =T/4\pi {L}_{D}$ and the applied magnetic field $h=H/{H}_{D}$. Filled areas indicate the regions of stability for the helicoid (blue; the wave vector is perpendicular to the field), cone (green; the $\mathbf{q}$-vector is along the field), and the ferromagnetic state. The point A is a critical point, in which the phase transition between cones and helicoids changes its type from the first- to the second-order with decreasing layer thickness $\nu $: at the line $A-b$ the first-order phase transition occurs whereas at the line $a-A$ − the second-order one. At the line $A-c$ the period of the spiral state expands to infinity. The line $d-B-e$ is the corresponding process of SkL expansion. In the red-shaded area below the line $d-B-f$, the eigen-energy of isolated skyrmions becomes negative with respect to the surrounding homogeneous or conical phase. Along the line C–D, the eigen-energy of skyrmions exhibits minimum for a fixed field value and varying thickness $\nu $. (

**b**) Magnetic structure of a helicoid with an expanding period for the field value corresponding to the line a–A. Comparison with the internal structure of a cone in (

**c**) indicates appearance of conical domains with some phase shift formed within the helicoid and thus the second-order nature of their phase transition (see text for details).

**Figure 2.**(color online) Overlap of chiral surface twists. (

**a**) Energy density $\epsilon \left(z\right)$ within isolated skyrmions plotted across the film thickness for different ratios $\nu /\lambda $. The energy is plotted after integrating over the x and y coordinates with respect to the energy of the homogeneous state ${\epsilon}_{FM}=0.5$. The negative energy near the surfaces is gained due to the additional surface twists of the magnetization. Starting and finishing points of each line correspond to the energy density at the surfaces. In (

**b**), the energy density was integrated only with respect to the y coordinate and plotted as a color plot on the plane $xz$. Such an energy distribution exhibits the energy excess in the middle of the layer as well as near the confining surfaces, which may be the underlying reason of toron and bobber formation. Overlap of surface twists in (

**a**) results in the thickness-dependent minimum of the total energy of isolated skyrmions (

**c**), which also corresponds to the line C–D at the phase diagram in Figure 1a. To avoid any deformation of the skyrmion shape by the conical phase or an oblique spiral state as depicted in (

**d**), we concentrate on the field values $h\ge 0.5$ when skyrmions are embraced by the homogeneous state.

**Figure 3.**(color online) Variation of skyrmion shapes with decreasing layer thickness $\nu $. The sizes of ISs (

**a**) are defined according to the Lilley rule (

**b**) by using the lines tangential to the inflection points of the magnetization profiles at the surface and in the middle of the layer. It gives the dependencies ${R}_{L}^{\mathrm{surf}}\left(\nu \right)$ (dotted red line) and ${R}_{L}^{\mathrm{middle}}\left(\nu \right)$ (solid blue line in (

**a**)) with the crossing point ${\nu}_{0}$ between them. Profiles ${R}_{L}\left(z\right)$ across the film interface shown as red lines in (

**c**) reveal the skyrmion appearances as convex and/or concave barrels. In addition, we color in blue the skyrmion core with the magnetization rotation from ${m}_{z}=-1$ to 0. Schematic representation of skyrmions with the two characteristic profiles and the corresponding distribution of the magnetization field are depicted in (

**d**). Skyrmion helicity $\gamma $ is shown to sweep quite a broad angular range for small film thicknesses until it saturates at the value $\approx {47}^{\circ}$ (

**e**). Interestingly, the helicity value was found to vary across the skyrmion center (

**f**) as clearly deduced from the zoomed image (

**h**). In (

**g**), we plot the helicity profiles across the layer defined under condition ${m}_{z}=0$ at each coordinate z. While the profiles for thicker films have an interval with constant helicity, $\gamma =\pi /2$, which is inherent for bulk cubic helimagnets with the Bloch fashion of the magnetization rotation, in thinner films, the magnetization continuously rotates from the upper to the lower surface (as shown by dotted lines in (

**g**)).

**Figure 4.**(color online) Static properties of isolated skyrmions in wedge-shape nanostructures. (

**a**) Schematic representation of an isolated skyrmion in a wedge with the tilt angle $\alpha $. The distance d of the skyrmion from the sharp edge is measured with respect to the skyrmion center. The corresponding thickness is defined as $\nu =dtan\alpha $. The current I is applied along the y axis, and the magnetic field is parallel to z. As a result, skyrmions are expected to move along the current with a small deviation along x. (

**b**) The internal structure of so-called edge states arising due to the open boundary conditions. First two panels in (

**b**) indicate the rotation of the magnetization towards the upper surface and within the sharp end. The third panel indicates the negative energy density related to these twists. Isolated skyrmions placed within the wedge develop an energy minimum of the edge–IS interaction, which, according to (

**c**) is located at about the same skyrmion elevation $\nu $, and according to (

**d**), moves closer to the edge. The energy minimum persists only in the angular range $\alpha \in [0,{22}^{\circ}]$. For larger tilts, the equilibrium “pit stop” position of skyrmions near the edge disappears, and skyrmions, left “alone” within the wedge, would slide down and annihilate. The inset shows the energy density in the wedge, including an IS. A ring of negative energy density known to form around the skyrmion [10] rests upon the part of the negative energy associated with the edge states within the sharp wedge end.

**Figure 5.**(color online) Current-driven skyrmion dynamics. (

**a**) Trajectories of moving isolated skyrmions for different current values indicating both x (i.e., d) and y coordinates of a skyrmion center. After a short time period, when the skyrmion actually “climbs” the hill, the skyrmion track becomes essentially a straight line parallel to the sharp edge. In (

**b**), we plot the same trajectories as the time-dependent curves $d\left(t\right)$. Panel (

**c**) summarizes the information about the skyrmion trajectories; it exhibits the largest skyrmion distance from the edge (left axis) as well as the skyrmion elevation (right axis). The inset shows corresponding velocity along y. The current-driven skyrmion motion bears an essentially non-linear character. At smaller magnitudes of the current density, the skyrmion trajectories are almost equidistant, in accordance with the current increment as shown by the yellow lines in (

**d**). Larger current densities, however, drive skyrmions to higher elevations, a process which first results in larger separations between skyrmion tracks and eventually culminates in an unbound skyrmion escape up to the upper wedge boundary at the critical current, ${I}_{cr}\approx 7.6\times {10}^{11}$ A/m${}^{2}$. Remarkably, the decreasing current density may induce backward skyrmion movement down the “hill” as shown by the gray dashed lines in (

**a**,

**b**). During these processes, we first calibrated the skyrmion motion for ${I}_{1}=5\times {10}^{11}$ A/m${}^{2}$. Then, we switched the current to ${I}_{2}=7\times {10}^{11}$ A/m${}^{2}$. As soon as the skyrmion stabilized to its equilibrium motion, we dropped the current back to $5\times {10}^{11}$ A/m${}^{2}$, which returned the skyrmion exactly to its initial trajectory.

**Figure 6.**(color online) Static properties of isolated skyrmions in truncated wedge-geometries and thin films. (

**a**) Edge-skyrmion interaction potentials in thin films with different thicknesses $\nu $. Note that the curves saturate for the configurations with the ISs located at large distances from the edges. The internal structure of the edge states in thin films (

**b**) differs from the essentially one-dimensional edge states in samples with the infinite lateral boundary (

**c**). Nevertheless, such a 2D magnetization pattern also underlies the edge-skyrmion repulsion. The edge-skyrmion interaction potential in truncated wedges (

**d**) loses its minimum when going from the sample $\u266f1$ with the thinner lower end to the sample $\u266f2$ with the thicker edge. The internal structure of the edge states in truncated wedges (

**e**) is similar to those in thin films. However, such edge states are concluded to be less efficient in maintaining skyrmions within the racetracks.

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**MDPI and ACS Style**

Shigenaga, T.; Leonov, A.O.
Harnessing Skyrmion Hall Effect by Thickness Gradients in Wedge-Shaped Samples of Cubic Helimagnets. *Nanomaterials* **2023**, *13*, 2073.
https://doi.org/10.3390/nano13142073

**AMA Style**

Shigenaga T, Leonov AO.
Harnessing Skyrmion Hall Effect by Thickness Gradients in Wedge-Shaped Samples of Cubic Helimagnets. *Nanomaterials*. 2023; 13(14):2073.
https://doi.org/10.3390/nano13142073

**Chicago/Turabian Style**

Shigenaga, Takayuki, and Andrey O. Leonov.
2023. "Harnessing Skyrmion Hall Effect by Thickness Gradients in Wedge-Shaped Samples of Cubic Helimagnets" *Nanomaterials* 13, no. 14: 2073.
https://doi.org/10.3390/nano13142073