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Magnetic skyrmion is a non-trivial topological object which originates due to the interplay among various types of interactions such as magneto-static, anisotropy, dipolar, exchange interaction, and Dzyaloshinskii-Moriya interaction (DMI). In magnetic thin films, skyrmions originate through the competition between interfacial DMI and exchange interaction&lt;sup&gt;1&lt;/sup&gt;. Patterned nanostructures provides a pathway to predictably stabilize and thus allow understanding their behavior in presence of different important parameters in a more controlled way (ref.). Moreover, due to the potential application of skyrmions in spintronic devices such as race-track memory etc., several studies of their creation, stability, and dynamics in nano wires have been reported&lt;sup&gt;1&lt;/sup&gt;. Skyrmions in nanostructures can be dynamically controlled by magnetic field gradiant&lt;sup&gt;3&lt;/sup&gt; and spin-polarized current&lt;sup&gt;1&lt;/sup&gt;. In case of high-speed skyrmion-based magnetic storage devices, large spin-polarized current density is required to control skyrmion dynamics. This in turn causes Joule heating and skyrmion Hall effect&lt;sup&gt;4&lt;/sup&gt; which limits the efficiency of skyrmion-based devices. These issues can be resolved by employing magnon spin current as a tool to control skyrmion propagation.&lt;br/&gt;However, very few studies have so far reported dynamics of skyrmions in presence of temperature gradient in a nanotrack&lt;sup&gt;5&lt;/sup&gt;. L. Kong &lt;i&gt;et al.&lt;/i&gt; reported the systematic investigation of the dynamics of Bloch skyrmions in presence of the temperature gradient. Skyrmions are found to move from region of lower to higher temperature. Moreover, the velocity of skyrmions is found to increase with the increase of thermal gradient&lt;sup&gt;5&lt;/sup&gt;. Magnon spin current generated due to thermal gradient diffuses from hot to the cold region. This magnon spin current applies a negative torque on the skyrmion resulting in movement of skyrmion cold to hot region&lt;sup&gt;4&lt;/sup&gt;. This study also reports that the skyrmion Hall component is absent or very small with the magnon spin current. A few other studies report the role of temperature on the current induced dynamics which demonstrate the velocity of skyrmion for the constant current density increases with the temperature&lt;sup&gt; 7&lt;/sup&gt;.&lt;br/&gt;Although the dynamical behavior of Bloch skyrmions in presence of a temperature gradient has been reported in literature, there is no comprehensive studies performed to understand the dynamics of Neel skyrmion in presence of temperature gradient. In this work, we performed micromagnetic simulations to investigate the dynamics of Néel skyrmions in a race track of Co/Pt in presence of the temperature gradient. We considered a Co/Pt track with dimensions of 100 nm x40 nm x 1nm. Individual skyrmions were stabilized due to the i-DMI of ~3 mJ/m&lt;sup&gt;2 &lt;/sup&gt;in the nanotrack and thereafter, their dynamics in presence of a temperature gradient&lt;sup&gt;7,8&lt;/sup&gt;  was studied. Our studies reveal that similar to Bloch skyrmions, the temperature gradient plays a significant role on the velocity of Neel skyrmions in the nanotrack. The skyrmions move from cold to the hotter region and the velocity increases as the temperature gradient is increased (Fig. 1(b)). Furthermore, we have also investigated the role of Gilbert damping constant on the skyrmion velocity in presence of thermal gradient on the track. Our results show that the velocity is inversely proportional to the values damping constant. The details of these results are discussed in this work.

**References:**

&lt;sup&gt;1 &lt;/sup&gt;J. Iwasaki, &lt;i&gt;et al.&lt;/i&gt; Nat. Nanotechnol. 8, 742 (2013).&lt;br/&gt;&lt;sup&gt;2 &lt;/sup&gt;A. Casiraghi, &lt;i&gt;et al.&lt;/i&gt; Nat. Commun Phys. 2, 145 (2019).&lt;br/&gt;&lt;sup&gt;3. &lt;/sup&gt;J. J. Liang &lt;i&gt;et al.&lt;/i&gt; &lt;i&gt;New J. Phys.&lt;/i&gt; 20, 053037 (2018).&lt;br/&gt;&lt;sup&gt;4 &lt;/sup&gt;G. Chen, &lt;i&gt;et al&lt;/i&gt;. Nat. Phys 13, 112–113 (2017).&lt;br/&gt;&lt;sup&gt;5 &lt;/sup&gt;L. Kong &lt;i&gt;et al.&lt;/i&gt; Phys. Rev. Lett. 111, 067203, (2013).&lt;br/&gt;&lt;sup&gt;6&lt;/sup&gt; M T Islam &lt;i&gt;et al.&lt;/i&gt; J. Phys.: Condens. Matter 31, 455701 (2019).&lt;br/&gt;&lt;sup&gt;7 &lt;/sup&gt;Litzius, K. &lt;i&gt;et al.&lt;/i&gt; &lt;i&gt;Nat Electron&lt;/i&gt; 3, 30–36 (2020).&lt;br/&gt;&lt;sup&gt;8 &lt;/sup&gt;Everschor, K. &lt;i&gt;et al. &lt;/i&gt;Phys. Rev. B 86, 054432 (2012).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/33fa4a9ffb9f74e344b2669dae0c9fd4.jpg)
Fig.1.(a) Position of a skyrmion in a nanotrack in presence of temperature gradient at three different times. (b) The velocity of Skyrmion as a function of the temperature gradient in K/nm.



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 Thermal gradient driven dynamics of Neel skyrmions in a nanoracetrack

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Magnetic skyrmion is a non-trivial topological object which originates due to the interplay among various types of interactions such as magneto-static, anisotropy, dipolar, exchange interaction, and Dzyaloshinskii-Moriya interaction (DMI). In magnetic thin films, skyrmions originate through the competition between interfacial DMI and exchange interaction1. Patterned nanostructures provides a pathway to predictably stabilize and thus allow understanding their behavior in presence of different important parameters in a more controlled way (ref.). Moreover, due to the potential application of skyrmions in spintronic devices such as race-track memory etc., several studies of their creation, stability, and dynamics in nano wires have been reported1. Skyrmions in nanostructures can be dynamically controlled by magnetic field gradiant3 and spin-polarized current1. In case of high-speed skyrmion-based magnetic storage devices, large spin-polarized current density is required to control skyrmion dynamics. This in turn causes Joule heating and skyrmion Hall effect4 which limits the efficiency of skyrmion-based devices. These issues can be resolved by employing magnon spin current as a tool to control skyrmion propagation. However, very few studies have so far reported dynamics of skyrmions in presence of temperature gradient in a nanotrack5. L. Kong et al. reported the systematic investigation of the dynamics of Bloch skyrmions in presence of the temperature gradient. Skyrmions are found to move from region of lower to higher temperature. Moreover, the velocity of skyrmions is found to increase with the increase of thermal gradient5. Magnon spin current generated due to thermal gradient diffuses from hot to the cold region. This magnon spin current applies a negative torque on the skyrmion resulting in movement of skyrmion cold to hot region4. This study also reports that the skyrmion Hall component is absent or very small with the magnon spin current. A few other studies report the role of temperature on the current induced dynamics which demonstrate the velocity of skyrmion for the constant current density increases with the temperature 7. Although the dynamical behavior of Bloch skyrmions in presence of a temperature gradient has been reported in literature, there is no comprehensive studies performed to understand the dynamics of Neel skyrmion in presence of temperature gradient. In this work, we performed micromagnetic simulations to investigate the dynamics of Néel skyrmions in a race track of Co/Pt in presence of the temperature gradient. We considered a Co/Pt track with dimensions of 100 nm x40 nm x 1nm. Individual skyrmions were stabilized due to the i-DMI of ~3 mJ/m2 in the nanotrack and thereafter, their dynamics in presence of a temperature gradient7,8 was studied. Our studies reveal that similar to Bloch skyrmions, the temperature gradient plays a significant role on the velocity of Neel skyrmions in the nanotrack. The skyrmions move from cold to the hotter region and the velocity increases as the temperature gradient is increased (Fig. 1(b)). Furthermore, we have also investigated the role of Gilbert damping constant on the skyrmion velocity in presence of thermal gradient on the track. Our results show that the velocity is inversely proportional to the values damping constant. The details of these results are discussed in this work.

**References:**

1 J. Iwasaki, et al. Nat. Nanotechnol. 8, 742 (2013). 2 A. Casiraghi, et al. Nat. Commun Phys. 2, 145 (2019). 3. J. J. Liang et al. New J. Phys. 20, 053037 (2018). 4 G. Chen, et al. Nat. Phys 13, 112–113 (2017). 5 L. Kong et al. Phys. Rev. Lett. 111, 067203, (2013). 6 M T Islam et al. J. Phys.: Condens. Matter 31, 455701 (2019). 7 Litzius, K. et al. Nat Electron 3, 30–36 (2020). 8 Everschor, K. et al. Phys. Rev. B 86, 054432 (2012).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/33fa4a9ffb9f74e344b2669dae0c9fd4.jpg)
Fig.1.(a) Position of a skyrmion in a nanotrack in presence of temperature gradient at three different times. (b) The velocity of Skyrmion as a function of the temperature gradient in K/nm.

poster

Magnetic skyrmions are localized chiral whirling of magnetization that hold great promise as nanoscale information carrier. Skyrmion racetrack memory and logic devices have recently been proposed where trains of skyrmions in tracks are manipulated by electrical current [1]. These devices require the controlled nucleation and current driven motion of skyrmions in narrow tracks. In particular, the skyrmion Hall effect (i.e. the skyrmion motion towards the track edge), is a critical issue as it can lead to the skyrmion annilation [2–5]. Here, we report on the controlled nucleation and current induced guiding of magnetic skyrmions along tracks defined by local He+ ion irradiation on an ultrathin sputtered magnetic film at room temperature. A He+ focused ion beam was used to define 150 nm wide tracks (red areas, Fig.1a) within a sputtered ultrathin Pt/Co/MgO film patterned into 3-µm-wide tracks [5,6]. Magnetometry and Brillouin Light Scattering measurements show that the He+ ion irradiation results in a decrease of the perpendicular magnetic anisotropy as well as the Dzyaloshinskii-Moriya interaction. Under a small perpendicular magnetic field, this leads to the nucleation of skyrmion racetrack (Fig.1b), whose size and density can be tuned by the field amplitude. Experiments of current induced skyrmion motion, show that the skyrmions move along the He+ ion defined tracks (Fig.2 a-f) for two different angles of the current injection. The track acts as a local potential well that guides the skyrmion trajectory, which allows to suppress the skyrmion Hall effect and also it helps in gaining the velocity along the track line. These results are in line with micromagnetic simulations using experimentally derived magnetic and transport parameters (Fig.2 g-h). A deeper insight into the dynamics of the moving skyrmions using micromagnetic simulations show that the steady-state velocity of the skyrmions can be enhanced by tuning the magnetic damping parameter and the height of the ion-defined potential well. These results open a new path to nucleate and guide skyrmions in racetrack memory and logic devices.

**References:**

[1] A. Fert, V. Cros and J. Sampaio, Nat. Nanotechnol. 8, 152 (2013). [2] J. Sampaio, V. Cros, S. Rohart et al., Nat. Nanotechnol. 8, 839 (2013). [3] W. Jiang, X. Zhang, G. Yu et al., Nat. Phys. 13, 162 (2017). [4] S. Woo, K. Litzius, B. Kruger et al., Nat. Mater. 15, 501 (2016). [5] R. Juge, S-G Je, D-S Chaves et al., Phys. Rev. Appl. 12, 044007 (2019). [6] O. Boulle, J. Vogel, H. Yang et al., Nat. Nanotechnol. 11, 449 (2016).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6f27c627202cb6ca06126bc76e792a1e.jpg)
Fig. 1: a) Bottom: SEM image of a 3-µm-wide Pt/Co/MgO track. The red lines indicate the irradiated patterns (6000×150 nm2). Top: AFM image after irradiation (Dose = 6 ion/nm2). b) MFM images showing the skyrmions along the track for μ0H = 19 mT and 24 mT (scale bar is 500 nm for both).

![](https://assets.underline.io/uploads/markdown_image/1/image/c47e18a098b8934d1bdb1686c59d170e.jpg)
Fig. 2: Sequence of XMCD-PEEM images showing the current-driven motion of skyrmions in He+-ion-irradiated tracks (red shaded area) at an angle (a-c) 0° and (d-f) 22.5° with respect to the current direction (J = 6.6×1010 A.m-2, pulse width 11 ns, μ0H = 10 mT). (g) Micromagnetic simulations showing the current driven skyrmion motion in a 150-nm-wide irradiated track (region between the dashed line). The current flows along y (J = 6.6×1010 A.m-2) and applied OOP field μ0H = 33 mT. (h) Skyrmion trajectories recorded at different current densities.

 Ion-irradiated Skyrmion Racetracks for current induced Skyrmion guiding at Room Temperature

The magnetic spin textures with non-trivial topology has been an important topic in the fields of magnetism and spintronics for the last few years [1]. For example, the magnetic skyrmion is a promising topological spin texture, which exists in perpendicularly magnetized systems and can be used as a spintronic information carrier. On the other hand, magnetic bimeron is a topological counterpart of skyrmions in in-plane magnets, which can also be used to carry information. In this talk, we will show the static properties of skyrmions and bimerons with different topological structures in a frustrated ferromagnetic monolayer [2], where the topological spin textures are characterized by the vorticity Qv and helicity η. We will also talk about the dynamics of frustrated skyrmions and bimerons driven by the spin-orbit torques, which depend on the strength of the damping-like and field-like torques. We find that these particle-like topological spin textures with Qv = ±1 can be driven into linear or elliptical motion, which depends on the direction of spin polarization. We numerically reveal the damping dependence of the skyrmion/bimeron Hall angle driven by the damping-like torque. Besides, we will show that the isolated bimeron with Qv = ±1 can be driven into self-rotation by the damping-like torque at certain conditions. The rotation frequency is proportional to the driving current density. In addition, we will computationally demonstrate the possibility of creating a bimeron state with a higher or lower topological charge by the current-driven collision and merging of bimeron states with different Qv. Finally, we will give an outlook on other possible topological spin textures in frustrated magnetic systems, such as the skyrmionium and bimeronium. We believe the results presented in this talk could be useful for understanding the topological spin textures physics in frustrated magnetic systems.

**References:**

[1] Nat. Rev. Phys. 2, 492 (2020); J. Phys. D: Appl. Phys. 53, 363001 (2020); J. Phys.: Condens. Matter 32, 143001 (2020); J. Appl. Phys. 124, 240901 (2018); Nat. Rev. Mats. 2, 17031 (2017); Adv. Mater. 29, 1603227 (2017); Phys. Rep. 704, 1 (2017); J. Phys. D: Appl. Phys. 49, 423001 (2016); Proc. IEEE 104, 2040 (2016); Nat. Rev. Mats. 1, 16044 (2016); J. Phys.: Condens. Matter 27, 503001 (2015); Nat. Nanotech. 8, 899 (2013). [2] Phys. Rev. B 101, 144435 (2020); Appl. Phys. Lett. 117, 012403 (2020); Phys. Rev. Appl. 11, 044046 (2019); Nat. Commun. 8, 1717 (2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7f6af7fa51c79a152ccf8514a3846a13.jpg)
Fig. 1. (a) Illustration of a skyrmion with Qv = 1. (b) Illustration of a bimermon with Qv = 1. (c) Top view of simulated static bimeron solutions with different Qv and η. (d) Current-induced collision and merging of an isolated bimeron and a cluster-like bimeron state.

 Static Structures and Dynamics of Frustrated Topological Spin Textures

Magnetic skyrmions are a peculiar kind of chiral magnetic textures stabilized by chiral interactions like the Dzyaloshinskii-Moriya interaction (DMI) [1]. Their topological properties offer interesting robustness qualities but, as a counterpart, add a complexity to their dynamics. Since their first observation in a bulk chiral magnet in 2009 [2], rapid experimental advances demonstrated their stability at room temperature without any magnetic field in ferromagnet/heavy metal bilayers and multi-stacks [3]. The experimental demonstrations of smaller and smaller skyrmions at room temperature (down to few 10s of nm) suggest that they are the perfect candidates for information storage and logic devices. Before building such devices, we first need to master skyrmions' static and dynamical properties, namely the nucleation and stabilization of small skyrmions that can move at high speed inside tracks. One significant issue with skyrmion dynamics is the topological deflection. Also known as Skyrmion Hall effect, it is caused by the topological charge of skyrmion and consists in a deflecting force proportional and perpendicular to the skyrmion velocity. Deflection can lead to the expulsion of skyrmions from the tracks that contain them, and thus imposes a limit to the skyrmion velocity. Experiments and calculations found that the deflection angle varies with skyrmion velocity and diameter probably due to its interaction with pinning disorder [4]. It has been suggested that antiferromagnetically-coupled systems, such as antiferromagnets, synthetic antiferromagnets (SAFs) or ferrimagnets, may be used to overcome these issues. The low net magnetization of these systems reduces the size of the skyrmions, and their low angular momentum density should suppress the troublesome topological deflection. Ferrimagnetic Rare-Earth/Transition Metal alloys possess two anti-parallel sublattices whose relative magnetic moments can be changed by temperature or alloy composition. Ferrimagnets share properties from ferromagnets and antiferromagnets (at the magnetic compensation) with which they present many similarities. They are perfect systems to study the advantages of antiferromagnetically-coupled materials, while preserving the effects that have already been used to stabilize and drive skyrmions in ferromagnets. Recent results on SAF showed the promising advantages of using multi-lattices coupled antiferromagnetically [5], and in 2018 a huge breakthrough was achieved in Rare-Earth/Transition Metal ferrimagnets where the smallest skyrmions were observed (10-30nm) [6] as well as a reduced topological deflection [7]. Using co-evaporation in Ultra-High-Vacuum [8], we grew ferrimagnetic thin film of Ta(1nm)/Pt(5nm)/Gd0.3Co0.7(5nm)/Ta(5nm) with an amorphous structure that decreases pinning of skyrmions. By optimizing the material stack and the process conditions, we were able to strengthen the interfacial DMI while keeping a balanced interfacial anisotropy to obtain out of plane magnetized samples (around room temperature) that can host skyrmions. We then observed the skyrmions by Magneto-Kerr Microscopy with sizes smaller than the technique’s resolution limit (around 500 nm) (Fig. 1). After saturating the sample, we were able to nucleate skyrmions in the tracks and to tune their overall density by applying magnetic fields. Skyrmions were also nucleated on the edges of the track by nanosecond-long current pulses. Thanks to the tunability of the magnetic properties of the ferrimagnet, we studied the sample in a range of magnetic field and temperature (0–40 mT, and 295–320 K). We observed different regimes: saturated magnetic textures, skyrmions with different densities, and stripes textures. Skyrmions were observed between ~295-310 K. The field necessary for nucleating skyrmions increased with temperature (8 mT at 295 K, 15 mT at 305 K), probably due to a decreasing Ms (as the compensation temperature of this sample is above the tested range). We also observed their dynamics under the application of nanosecond current pulses. The current going through the adjacent Pt layer generates a spin orbit torque (SOT) due to the spin Hall effect, which drives the skyrmions. We observed speeds higher than 150 m/s (for ~10 GA/m2) in a denser skyrmion regime and up to 40 m/s in the most diluted regime (Fig. 2), limited for higher current density by edge nucleations. The speed seems to follow a linear dependence with current. In the conditions of higher density, the velocity was higher and the depinning current lower, probably due to a change of the skyrmion diameter. We also studied the evolution of the deflection angle of the skyrmions, which increased from 40° at low current to finally saturate around 60°. This work shows that ferrimagnetic skyrmions can be nucleated, stabilized and propagated by SOT in amorphous rare earth/transition metal thin films. In these results, we stabilized skyrmions far away from the magnetic compensation of the sample (also far away from the angular compensation). We have since prepared new samples with accessible magnetic and angular compensation in order to study the dynamical properties of our sub-micron skyrmions in these very peculiar regimes.

**References:**

 [1] – A. Fert et al., Nature Nanotechnology, Vol 8, p 152–156 (2013) [2] – X. Z. Yu et al. Nature, Vol 465, p 901–904 (2010) [3] – A. Hrabec et al., Nature Com. Vol 8, 15765 (2017) [4] - Joo-Von Kim et al. Appl. Phys. Lett., Vol 110, 132404 (2017) [5] – W. Legrand et al., Nature Materials, Vol 19, p 34–42 (2020) [6] – L. Caretta et al., Nature Nanotechnology, Vol 13, p 1154–1160 (2018) [7] – S. Woo et al. Nature Communications Vol 9, 959 (2018) [8] – E. Haltz et al. Physical Rev. Materials Vol 2, 104410 (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4deb0696a0fd43f90ed46a6cc224f887.jpg)
Fig.1 : Magneto-Kerr Microscopy image of skyrmions in GdCo micro-structured sample.

![](https://assets.underline.io/uploads/markdown_image/1/image/25864383d3eb7d0bd0916c7804a27862.jpg)
Fig. 2 : SOT-driven velocity at room temperature in diluted and denser regime.

 Spin Orbit Torque Dynamics of Magnetic Skyrmions in GdCo Ferrimagnetic Thin-Films

Typically, stabilizing small, robust skyrmions requires a strong Dzyaloshinskii-Moriya Interaction (DMI). Accurately measuring the strength of DMI in prospective skyrmion host materials is therefore crucial to realizing proposed skyrmion-based devices for racetrack memory and neuromorphic computing.[1–2] In the creep regime, the velocity of a domain wall is governed by an Arrhenius law and is inversely proportional to its elastic energy.[3] Early work on the impact of DMI on domain walls showed that the resting energy of a wall is modulated by the DMI's effective field and that the presence of an in-plane field results in asymmetric domain growth due to the (mis)alignment of the two fields.[4] Subsequent studies have proposed creep models that reflect the intricacies of the relevant velocity and elastic energy scales.[5–6] As such, the sign and strength of the Rashba DMI found in thin film systems has been deduced from the asymmetric creep expansion of domains when an in-plane field is present; wall segments with core magnetizations parallel to the in-plane field have reduced energies, resulting in enhanced velocities—and vice-versa.[5–6] However, Brock et al., recently reported experimental results that deviate remarkedly from the above-mentioned creep models. Observations made in [Co(0.7 nm)/Ni(0.5 nm)/Pt(0.7 nm)]N&gt;3 films include highly asymmetric dendritic domain growth perpendicular to the in-plane field for low fields and sharp variations of the growth direction with the strength of the in-plane field.[7–8] While one study focused on the apparent preference of Bloch walls of a specific handedness and attributed most of the phenomena to interlayer DMI [8], further investigation of the findings in [7] revealed that the trends that clearly defy the symmetries expected from an intrinsically favored wall chirality with Hz.[7] In this work, we address these peculiarities by considering the impact of the driving field on the domain wall magnetization, dispersive stiffness, and creep velocity. We show that even in the creep regime, the core magnetization of a domain wall, as per the Landau-Lifshitz-Gilbert equation, can be driven to a steady-state equilibrium that deviates significantly from the static case. We find that for material parameters comparable to those reported in [7], a bubble domain in the presence of an in-plane field can exhibit sharp discontinuities in its steady-state wall magnetization profile. Similar to the findings reported in [10], such a point represents a sudden reorientation of the magnetizations along the domain wall due to the nuanced competition between the effective DMI field, the domain wall anisotropy field, the in-plane field, the driving field and Gilbert damping. Consequently, the domain wall energy and stiffness landscapes can be equally complex, highly sensitive to the in-plane field and significantly different from the static case. Incorporating a steady-state treatment of domain wall profiles into the dispersive stiffness framework developed in [5], we compute the steady-state stiffness and creep velocity profiles of a an expanding domain for the scenarios reported in [7]. As seen in Figure 2., our modelling of the N=3 film shows remarkable agreement with the observed dendritic growth directionalities as a function of the strength of the in-plane field: first propagating uniformly for very weak fields, then at 90° from the field direction, then gradually reducing to ≈20°, and then abruptly reversing direction to ≈200°. Moreover, the model reproduces the curious observation that that the propagation direction is constant for both driving field directions; switching the domain polarization switches the directions of both the DMI and applied field torques, leaving the Bloch chirality at steady state unchanged. Comparing the magnetization profiles to the propagation directions and stiffness values reveals that segments with the lowest elastic energy (and thus the highest velocity) occur close to reorientation positions in the magnetic profile. The sudden emergence of such points is also responsible for the drastic changes in the growth direction. In light of these insights, we discuss the impact of steady-state profiles on DMI values obtained via bubble expansion experiments. Finally, we consider how our steady-state stiffness model could be used in tandem with these growth experiments to more easily determine the value of Gilbert damping in thin films.

**References:**

1. R. Tomasello, E. Martinez, R. Zivieri et al., Scientific Reports., Vol. 4, (2014) 2. G. Bourianoff, D. Pinna, M. Sitte and K. Everschor-Sitte, AIP Advances., Vol. 8, p. 055602 (2018) 3. P. J. Metaxas, J. P. Jamet, A. Mougin, Physical Review Letters., Vol. 99, p. 217208 (2007) 4. S. Je, D. Kim, S. Yoo et. al., Physical Review B., Vol. 88, p. 214401 (2013) 5. J.P. Pellegren, D. Lau, and V. Sokalski. Physical Review Letters., Vol. 119, p. 027203 (2017) 6. D. Lau, J. P. Pellegren, H. T. Nembach et al., Physical Review B., Vol. 98, p. 184410 (2018) 7. J. Brock, R. Medapalli and E.E. Fullerton, Bulletin of the American Physical Society. (2020) 8. S.D. Pollard, J. A. Garlow, K. Kim,et al., Physical Review Letters., Vol. 125, p. 227203 (2020) 9. L. Sánchez-Tejerinaa, O. Alejosa, and E. Martínez, Journal of Magnetism and Magnetic Materials., Vol. 423, p. 405–410 (2017)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/5c391c3ecd610921b55255792a947aef.jpg)
Simulated static (blue/dark) and steady-state (yellow/light) equilibrium wall magnetization profiles for an expanding bubble domain similar to those observed in [7].

![](https://assets.underline.io/uploads/markdown_image/1/image/8a760a928ff1a5a294b48a522d4d7bce.jpg)
(a) Conventions used for defining the growth direction of a domain in the presence of an in-plane field. (b) Experimental (markers) and semi-analytically computed (lines) plots of domain growth direction against in-plane field for the [Co(0.7 nm)/Ni(0.5 nm)/Pt(0.7 nm)]3 sample referenced in [7].

 Impacts of Steady-State Domain Wall Configuration on Domain Wall Stiffness and Directional Domain Propagation in the Creep Regime

The spintronics devices have been considered as promising candidates for future artificial neural networks by mimicking the functionality of biological neurons with nonvolatile control, which can enable effective reduction of energy consumption and heat generation during the information processing and transmission. Among them, voltage-controlled magnetic skyrmions have attracted special attentions because that they will satisfy the demands of well-controlled, high efficient, and energy-saving for future skyrmion-based neuron device applications. In this work, we propose a compact leaky-integrate-fire (LIF) spiking neuron device by deploying the voltage-driven skyrmion dynamics in a multiferroic nanodisk structure. The relationship between the skyrmion dynamics and the characteristics of voltage induced piezostrains have been investigated, where the skyrmion radius can be effectively modulated by the applied piezostrain pulses. Similar to the biological neuron, the proposed skyrmionic neuron will accumulate a membrane potential as skyrmion radius varying by inputting the continuous piezostrain spikes, and the skyrmion radius will return to the initial state in the absence of stimulus. Therefore, this skyrmion radius based membrane potential will reach a definite threshold value by the strain stimuli and then reset by removing the stimuli. The skyrmion size variation can be transformed to resistance detection signal by the magnetoresistance effect, which means that the skyrmionic neuron could fire an output resistance spike with controlling the input strain pulses. Such LIF neuronal functionality and behaviors of the proposed skyrmionic neuron device are elucidated through the comprehensive micromagnetic and theoretical studies. Our results provide a new route for realizing voltage-controllable skyrmionic neuron in future energy-efficient and tunable neuromorphic computing hardware.

**References:**

1. P. A. Merolla, J. V. Arthur, and D. S. Modha, Science, Vol. 345, p. 668 (2014). 2. X. Liang, G. Zhao, and Y. Zhou, Applied Physics Letters, Vol. 116, p. 122402 (2020). 3. X. Chen, Y. Zhou, and W. Zhao, Nanoscale, Vol. 10, p. 6139 (2018).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/624573e1b7b93a13d5c0eb82c5b43516.jpg)
(a) Schematic of the proposed spiking neuron and demonstration of the LIF behaviors of the skyrmion-based artificial neuron. (b) Skyrmion radius varying with the spiking strain signals under the homogeneous square wave with the frequency of 1.09 GHz. (c) The creation of LIF functionality based on the Resistance-Strain characteristic by magnetoresistance effect of the skyrmionic neuron device.

 Voltage-controllable Magnetic Skyrmion Dynamics for Spiking Neuron Device Applications

Magnetic skyrmions (Sks) and antiskyrmions (aSks) are topologically protected non-trivial spin textures1, 2. Those objects can be used as magnetic bits in spintronics devices3, 4. However, the presence of the skyrmion Hall effect in the Sk always hinders its movement along the straight path, and the transverse defection of it will cause a problem on reading data in devices. The unwanted Hall effect can be eliminated in aSk host systems2, 5 such that it can be moved in a straight path along the current direction6. Therefore, it is highly desired to prepare nano-stripes along specific crystallographic directions of the aSk host compound. Here we stabilize a single-chain of aSks in nano-stripes of Mn1.4Pt0.9Pd0.1Sn prepared along different crystallographic directions within the tetragonal basal plane7. We observe the enhanced stability of aSks compared to bulk systems without providing in-plane magnetic field components. Additionally, the presence of dipole-dipole interactions with the anisotropic Dzyaloshinskii-Moriya exchange interaction (DMI) in the Heusler compound Mn1.4Pt0.9Pd0.1Sn allows the stability of distinct spin textures called elliptical Bloch skyrmions7,8. The co-existence of elliptical skyrmions (eSks) and aSks is found in nano-stripes at room temperature7. The simultaneous existence of different spin textures in a racetrack has great advantages. It is not needed to write the distances between the magnetic bits anymore, unlike the racetrack of only single type magnetic objects. Our results provide a gateway for the remaining steps towards realizing an advanced version of racetrack storage devices based on magnetic aSks and eSks.

**References:**

1. Mühlbauer et. al., Science 323, 915 (2009). 2. Nayak et. al., Nature 548, 561 (2017). 3. Parkin et. al., Science 320, 190 (2008). 4. Fert et. al., Nat. Nanotechnol. 8, 152 (2013). 5. Jena et. al., Nano Lett. 20, 59 (2019). 6. Huang et. al., Phys. Rev. B 96, 144412 (2017). 7. Jena et. al., Sci. Adv. 6, 49, eabc 0723 (2020). 8. Jena et. al., Nat. Commun. 11, 1115 (2020).

**Images:**

 Evolution and competition between chiral spin textures in nano-stripes with D2d symmetry

1. Introduction As quasiparticle-like swirling topological spin textures in nanostructures1, magnetic skyrmions are expected to be widely used in future spintronic applications2. The creation and transmission of an isolated skyrmion in magnetic films are important for skyrmionic applications, which utilize skyrmions as information carriers3. An exemplary application is the racetrack-type storage memory4. Recently, it has been reported that spin-polarized current, spin wave, magnetic field, and temperature gradients, can effectively modulate the dynamics of skyrmions5. As the most widely used method, the magnetic field has been used to manipulate magnetic spin textures in studies6. Slonczewski and Berger first proposed theoretically spin transfer torque (STT)7 can be induced when spin polarized current passes through the magnetic layer/non-magnetic layer. STT is an effect in which the orientation of a magnetic layer in a spin valve can be modified by applying a spin-polarized current. On the other hand, in magnetic multilayered structures, metallic ferromagnets in contact with heavy metals exhibit strong effects due to spin-orbit coupling. Such spin-orbit coupling effects give rise to spin-orbit torque (SOT)8 on the ferromagnetic (FM) materials. In this work, we demonstrate that a skyrmion guide on the nanotrack under both in-plane magnetic field and spin-transfer torque or spin-orbit torques driven. In applying both in-plane magnetic field and STT conditions, we focus on spatially sinusoidal-varying Gilbert damping coefficients and non-adiabatic STT coefficients with in-plane magnetic field. For in-plane magnetic field and SOT conditions, we discuss the effect on damping-like toque, field-like torque with in-plane magnetic field. The results provide a guide for designing and developing the skyrmion transporting nanotrack channel in spintronic devices based on the manipulation of skyrmions. 2. Simulation Methods The skyrmion dynamics guide is based on an ultrathin FM nanotrack structure. All simulations are processed by using the Object Oriented Micromagnetic Framework (OOMMF). The time-dependent magnetization dynamics are governed by the LLG equation including STT (adiabatic and non-adiabatic) and SOT (field-like and damping-like). Note that we consider the effect of both in-plane magnetic field and STT or SOT driven. In the first part, the magnetization dynamics is based on the Landau-Lifshitz-Gilbert (LLG) equation augmented with in-plane magnetic field, adiabatic and non-adiabatic STT: dm/dt=-|γ|m×Heff+α(m×dm/dt)+u(m×dm/dt×m)-βu(m×dm/dt) where m represents the reduced magnetization, M/MS. γ and α are the gyromagnetic ratios and Gilbert damping constant, respectively. β is the strength of the non-adiabatic STT coefficient and u represents the adiabatic STT coefficient. The Gilbert damping coefficient α and the non-adiabatic STT coefficient β are defined as functions of the longitudinal coordinate x as follows: α(x)=αamp{1+sin[2π(x/λα)]} β(x)=βamp{1+sin[2π(x/λβ)-θ]} where αamp is the amplitude of α function and βamp represents the amplitude of β function, respectively. λβ and θ denote the wavelength and phase of β function. In the second part, the magnetization dynamics is based on the Landau-Lifshitz-Gilbert (LLG) equation augmented with in-plane magnetic field, and spin-orbit torque (SOT): dm/dt=-|γ|m×Heff+α(m×dm/dt)+ζDLT m×(mp×m)-ζFLT m×mp where ζSOT, ζDLT, and ζFLT stand for the spin-orbit torque, damping-like torque and field-like torque, respectively. 3. Results and Discussion The effects of varying damping coefficient α and non-adiabatic coefficient β on velocity in both x and y direction, and skyrmion hall angle φ are illustrated in Fig. 1, by fixing external magnetic field as 800 mT. We set αamp and βamp from 0.1 to 0.5 in our simulation to obtain the discipline, by fixing βamp and αamp in Fig. 1(a)-(c) and Fig. 1(d)-(f), respectively. The contour lines in Fig. 1(g)-(i) result from the combinations of αamp and βamp. The results show that the damping coefficient α exhibits declining effect on velocity, while non-adiabatic STT coefficient β exhibits increasing influence. The polarization rate is fixed at 0.4 and the value of field-like torque is modified to change the ratio of K=ζFLT/ζDLT . Applying both magnetic field and SOT to drive skyrmion simultaneously at magnetic field H = 800 mT. The ratio of field-like torque and damping-like torque K and damping-like torque ζDLT both indicates increasing effect on skyrmion velocity, as shown in Fig. 2(a)-(b). However, damping-like torque ζDLT provides more contribution on skyrmion motion.

**References:**

1. X. Zhang, Y. Zhou, K. M. Song, T.-E. Park, J. Xia, M. Ezawa, X. Liu, W. Zhao, G. Zhao and S. Woo, J. Phys. Condens. Matter 32 (14), 143001 (2020). 2. A. Fert, N. Reyren and V. Cros, Nat. Rev. Mater. 2 (7), 1-15 (2017). 3. W. Kang, X. Chen, D. Zhu, S. Li, Y. Huang, Y. Zhang and W. Zhao, presented at the 2018 Design, Automation & Test in Europe Conference & Exhibition (DATE) (unpublished). 4. C. Song, C. Jin, J. Wang, H. Xia, J. Wang and Q. Liu, Appl. Phys. Lett. 111 (19), 192413 (2017). 5. C. Schütte, J. Iwasaki, A. Rosch and N. Nagaosa, Phys. Rev. B 90 (17), 174434 (2014). 6. J. Müller, New J. Phys. 19 (2), 025002 (2017). 7. M. D. Stiles and A. Zangwill, Phys. Rev. B 66 (1), 014407 (2002). 8. Y. Li, K. W. Edmonds, X. Liu, H. Zheng and K. Wang, Adv. Quantum Technol. 2 (1-2), 1800052 (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/fca6ea5a3e90fff1a8d05d79a4845325.jpg)
Figure 1. Both magnetic field and STT driven skyrmion are applied to study the effect of damping coefficient α on velocity in both x and y direction, and skyrmion hall angle φ: (a) The Vx as a function of α at a fixed β = 0.2-0.5. (b) The Vy as a function of α at a fixed β = 0.2-0.5. (c) The φ as a function of α at a fixed β = 0.2-0.5. The effect of non-adiabatic β on velocity in both x and y direction, and skyrmion hall angle φ. (d) The Vx as a function of β at a fixed α = 0.2-0.5. (e) The Vx as a function of β at a fixed α = 0.2-0.5. (f) The Vx as a function of β at a fixed α = 0.2-0.5. (g) Vx, (h) Vy, and (i) skyrmion hall angle φ at magnetic field H = 800 mT.

![](https://assets.underline.io/uploads/markdown_image/1/image/338c7353053a46418aa778ff47c803f2.jpg)
Figure 2. Applying both magnetic field and SOT to drive skyrmion at magnetic field H = 800 mT: (a) Vx as the function of the ratio of field-like torque and damping-like torque from 0.1 to 10. (b) Vx as the function of damping-like torque , which indicates the polarization.

 Dynamic property of ferromagnetic skyrmion in an in-plane magnetic field

Topological non-trivial whirling spin textures such as magnetic skyrmions have recently triggered a lot of interest by virtue of their potential nanosized solitonic nature, stability and tunability. All these attributes, if combined with the efficient current-induced motion via spin orbit (SO) torque, make them attractive to build elementary units of digital information for spintronic logic devices and memories. In ultra-thin ferromagnetic films with Perpendicular Magnetic Anisotropy (PMA) sandwiched between a heavy metal with strong SO and an insulator (i.e. HM/FM/I), the chirality of this spin arrangements is ascribable to an antisymmetric exchange, Interfacial Dzyaloshinskii-Moriya Interaction (iDMI) [1,2]. Among the possible topological magnetic quasi-particles, which can be stabilised in such peculiar trilayer systems, a lower dimension (1D) non-trivial magnetic texture, named Domain Wall Skyrmion (DWSk), has been newly reported and experimentally observed using Lorentz Transmission Electron Microscopy [3]. The DWSk is a 1D topological excitation sharing the same integer topological charge, noted Q, as conventional 2D skyrmions (Q = 1). It is describable as a localized 2π rotation of the In-Plane (IP) component of magnetization within a Néel domain wall which stands between two perpendicular magnetic domains. It can be intuitively viewed as the counterpart of a 2π-vertical Bloch lines coming about in Bloch instead of Néel domain walls. Skyrmions with chiral kinks inside the otherwise homochiral domain wall, hereinafter called Double Skyrmion (DSk), have been numerically predicted as a soliton solution [4,5]. In our work, we investigated the stability and the spin-driven dynamics of the DSk in a sample of the type Pt/Co(0.9nm)/MgO [6] using micromagnetic simulations [7]. Interestingly, as shown in Fig.1a, we are able to stabilise simultaneously various spin-textures including skymion (Sk, Q=1), double skyrmion (DSk, Q=2) and domain wall skyrmion (DWSk, Q=1). It is found that such magnetization pattern is robust against both thermal fluctuations and normally distributed anisotropy, for different grain-like region sizes. We also proved the resilience of double skyrmion to material parameters variation (DMI, anisotropy) and applied field. The static analysis is complemented with the investigation of the dynamic regime of motion under electrical current. Fig.1b shows the trajectory of the double skyrmion induced by spin-orbit torque. In the limit of moderate injected current the skyrmion and the double skyrmion behave similarly (Fig.1c). However, depending on the geometry of the injected current the double skyrmion can be annihilated or splits in two skyrmions, opening the path to conceive logical devices.

**References:**

[1] I.E. Dzyaloshinskii Sov. Phys. JETP 5, (1957), J.Phys. Chem Solids, 4, 241 (1958) [2] T. Moriya Phys. Rev Lett., 4, 228, (1960), Phys. Rev. 120, 91, (1960) [3] Li at al., https://arxiv.org/abs/2004.07888 (2020) [4] Kuchkin et al., Phys. Rev. B 102, 144422 (2020) doi.org/10.1103/PhysRevB.102.144422 [5] S.-G. Je Current Applied Physics 21, 175 (2021) doi.org/10.1016/j.cap.2020.10.021 [6] Juge et al., Phys. Rev. Applied 12, 044007 (2019) doi.org/10.1103/PhysRevApplied.12.044007 [7] Vansteenkiste et al. AIP Adv. 4, 107133 (2014). doi/10.1063/1.4899186

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/469f5808fec70a1e12fc572d65ebf2ad.jpg)
Fig. 1: a) Multidomain equilibrium state of 0.5µmx0.5µm ideal Pt/Co(0.9nm)/MgO dot at 0K and no applied magnetic field. The color scale is associated to the perpendicular component of magnetization. b) Sequence of micromagnetic simulation snapshots illustrating the current-induced SOT driven DSk dynamics at 0K with Japp=50x1010 A/m2. The skyrmion Hall angle of DSk is 64°. c) Comparison of DSk and Sk velocities (orange) and effective diameters (black) as a function of injected current density.

 Static and dynamic behaviour of double skyrmions in Pt/Co/MgO trilayer

Chiral interactions between magnetic moments in ferromagnetic systems with broken inversion symmetry has attracted lots of attention in the recent years. Dzyaloshinskii-Moriya interaction[1] (DMI) present in these systems leads to an ordering of the spins that forms various exotic spin configurations such as chiral domain walls[2] and skyrmion[3]. DMI is an anti-exchange interaction which prefers to cant the neighbouring spins orthogonal to each other that forms whirling structures such as skyrmions. A magnetic skyrmion is a topological object having particle-spin like configuration, which can be stabilized in ferromagnetic systems with DMI. Recently magnetic skyrmions are proposed as promising candidates for non-volatile memory and logic applications due to their nanoscale size, high stability, high density and small driving currents required for their motion. Besides memory applications, skyrmions also hold great potential in the emerging area of magnonics where collective spin excitations (magnons) are used as a medium of information processing. In addition, observation of skyrmions on low Gilbert damping materials may allow the use of the skyrmions for manipulations of magnon transport. Recently skyrmion generation at an antidot is demonstrated using micromagnetic simulations[4]. In this work, we present the various routes for obtaining giant tunability in microwave properties based on skyrmion dynamic via micromagnetic simulations using Mumax3. We have investigated microwave properties of magnetic skyrmions in a circular nanodisk having antidot at the centre. An in-plane ac magnetic field excites gyrotropic dynamics with a clockwise and a counter-clockwise mode. We have systematically investigated skyrmion resonant modes as a function of DMI and perpendicular magnetic anisotropy (K). Large variation of resonant modes as a function of small out-of-plane bias magnetic field has been demonstrated. Simulated microwave spectrum of this Néel type skyrmion in an antidot nanodisk has been shown in figure 1(a) and it is compared with an antidot nanodisk having uniform OOP magnetization without any skyrmions. It is to be noted that there is no external bias field applied for these results in fig 1(a). Microwave responses are found in 13-22 GHz frequency range. The most prominent modes are found at 13 GHz and 17 GHz for the disk with a skyrmion and no skyrmion, respectively. Therefore, the modes at 13.1 GHz can be associated to the skyrmion resonances whereas the other higher frequency modes arise from the bulk magnetization (OOP) of the disk. Next, we demonstrate the tunability of the skyrmion resonance modes by applying a small bias field (Hext) along the OOP direction of the nanodisk with a single skyrmion. We have varied the polarity and the magnitude of the OOP field. The results are shown in fig 1(b) where the microwave resonance spectra are shown at different external fields: 0 to 35 mT. A relatively large frequency shift of 3.3 GHz has been observed for a change of 35 mT bias field. To further explore the functionalities, we have investigated skyrmion stabilization with various combinations of the parameters, DMI and K. Néel skyrmion is found to be stabilized at K = 0.25×106 J/m3 and DMI = 1.8×10-3 J/m2. Next, we have varied the K from 0.22 to 0.34 (×106) J/m3 for a fixed value of DMI = 1.8 mJ/m2 and we have varied the parameter DMI from 1.6 mJ/m2 to 1.9 mJ/m2, for a fixed value of K = 0.25×106 J/m3 shown in fig 2(a & b) respectively. Furthermore, a broad microwave tunability of 9.8 GHz has been shown with a variation of magnetic anisotropy of 1×105 J/m3. The manifestation of inertial mass of the skyrmion, its dependence on the resonance responses and the size of the skyrmion are explained by using Thiele’s equation. The inertial mass of the skyrmion is estimated to be in the order of 10-23 kg. We believe these results will have potential implications in the area of skyrmion based nano-scale reconfigurable microwave devices.

**References:**

[1] Dzyaloshinsky 1958 A Thermodynamic Theory of Gadolinium Molybdate J. phys.chem.solids 46 763–72 [2] DeJong M D and Livesey K L 2017 Domain walls in finite-width nanowires with interfacial Dzyaloshinskii-Moriya interaction Phys. Rev. B 95 054424 [3] Fert A, Reyren N and Cros V 2017 Magnetic skyrmions: Advances in physics and potential applications Nat. Rev. Mater. 2 17031 [4] Kumar Behera A, Murapaka C, Mallick S, Bhusan Singh B and Bedanta S 2021 Skyrmion racetrack memory with an antidot J. Phys. D. Appl. Phys. 54

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b61b65d775f8963b8b27c1292b1525e5.jpg)
Fig 1. (a) Dynamic responses of a disk with a skyrmion and without a skyrmion in an antidot and (b) Skyrmion resonance modes for different OOP bias magnetic field ranging from 0-35 mT.

![](https://assets.underline.io/uploads/markdown_image/1/image/4ce77cfb5ab580ce27e4122c11df437d.jpg)
Fig 2. (a) Variations of the gyrotropic modes of skyrmion as a function of K for a fixed DMI = 1.8 mJ/m2 and (b) variations of the gyrotropic modes of skyrmion as a function of DMI for a fixed K = 0.25×106 J/m3.

 Tunable microwave properties of a skyrmion in an antidot nanodisk structure

Magnetic skyrmions are topologically protected spin texture with diameters in the nanometer length scale. Thus, skymions possess profound applications for next-generation spintronics, such as racetrack memory and logic gates. In order to be used in devices, it is necessary to consider how to manipulate magnetic skyrmions. By reading literature，we learned that skyrmion has two driving methods, one is current drive and the other is voltage-controlled magnetic anisotropy (VCMA) drive. The method of using the potential well generated by the VCMA can induce the motion of the skyrmion, which can not only avoid the Joule heating caused by the current-induced methods, but also simplify the existing complex structure. Therefore, studying the movement properties of skyrmion induced by potential wells generated by VCMA alone is necessary. First we study the possibility of the potential well induced skyrmion movement within a certain distance. Then, the relevant factors that affect the distance of skyrmion induction, such as the D value of DMI, the perpendicular magnetic anisotropy (PMA) constant Ku, the external magnetic field H, and the relation between the size of skyrmion and the inductive distance, are studied. The simulation result is shown in Fig.1. All the parameters that cause the radius of skyrmion to change will affect the maximum sensing distance of skyrmion. Finally, the continuous motion scheme of skyrmion induced only through the potential well generated by VCMA explored in Fig.2. In the picture, the green area represent the voltage-controlled potential well and 1, 2 are two sets of potential energy wells driven by inverted square wave voltage signals. In this way, setting a spaced potential well with alternating voltage control on a long track could enable the skyrmion to achieve continuous motion on the track. Overall, the method of using the potential well generated by the VCMA can realize the drive control of skyrmion continuous movement. This work provide guidance for the design of low energy consumption voltage-driven skyrmion driving method and a new concept for track memory.

**References:**

[1] Wang, X. et al. Efficient skyrmion transport mediated by a voltage controlled magnetic anisotropy gradient. Nanoscale 10, 733-740, doi:10.1039/c7nr06482a (2018). [2]Zhang, X., Ezawa, M. & Zhou, Y. Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions. Scientific Reports 5, doi:10.1038/srep09400 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/cc2100a2c7bf12d613fce3df0996541b.jpg)
Fig.1

![](https://assets.underline.io/uploads/markdown_image/1/image/6a2127e6f4db009ab5c521b6e9a3b34b.jpg)
Fig.2

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Tunable microwave properties of a skyrmion in an antidot nanodisk structure