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Magnetic skyrmions are topologically protected chiral spin textures stabilized by the Dzyaloshinskii–Moriya interactions (DMI) in systems lacking inversion symmetry. The attractive features of skyrmions, such as ultra-small size, solitonic nature, and easy mobility with small electrical currents, make them promising as information-carrying bits in low power high-density memory and logic applications [1, 2]. In a skyrmion-based boolean racetrack memory, information can be encoded by the presence (bit “1”) and absence (bit “0”) of skyrmions at a particular position in the track. For an unconventional use of skyrmion racetrack, such as for native temporal memory [1], the information is encoded into the spatial coordinates of the skyrmions, which then can be translated into the timing information needed for race logic operations. The thermal stability of skyrmions is a critical issue for both of these applications, as a randomly displaced skyrmion can alter the sequence of the “0&quot; and “1&quot; bits in a Boolean memory application. Similarly, for race logic applications, the displacement of skyrmions would change the spatial coordinates of the skyrmion and hence the encoded analog timings. For reliable information extraction, it is essential to guarantee the positional stability of skyrmions for a certain amount of time. For example, for a long-term memory application it would require positional stability of years, while for cache memory, hours would be sufficient. In an ideal racetrack, skyrmions are susceptible to thermal fluctuations and exhibit Brownian motion that leads to the diffusive displacement of skyrmions [3]. Moreover, skyrmions show inertia driven drift shortly after a current pulse is removed, rather than stopping immediately. One way to control such undesirable motion is by engineering confinement barriers such as point defects with a different anisotropy, or notches etched into the racetrack (missing materials), which ensure the pinning of skyrmions. If the required unpinning current is too large, it will destabilize the skyrmions, which can cause annihilation of the skyrmions. In addition, high currents have been shown to randomly nucleate unwanted skyrmions as well [4]. This would mean that the energy barrier height needs to be optimized in such a way that it can hold the skyrmions for a long enough time and yet requires a modest current to unpin them. Using micromagnetic simulations [5], we present a quantitative analysis of racetrack geometries with different types of notches to calculate the minimum energy barriers associated with them. To calculate the energy barrier, we use the string method to find the minimum energy path (MEP) for the skyrmion to go over the notch [6]. We vary material parameters, specifically, the DMI, anisotropy ratio between the pinning sites and the rest of the magnetic material, and the geometry of the notches, in order to get the optimal barrier height. Furthermore, we investigate the effects of topology in the unpinning process by comparing the unpinning behavior of a ferrimagnetic vs a ferromagnetic skyrmion. We find a range of energy barriers (up to ~ 45 &lt;i&gt;k&lt;/i&gt;&lt;sub&gt;B&lt;/sub&gt;T) that can provide a long enough positional lifetime (years) of skyrmions for long-term memory applications while requiring a moderate amount of current (~ 10&lt;sup&gt;10&lt;/sup&gt; - 10&lt;sup&gt;11&lt;/sup&gt; A/m&lt;sup&gt;2&lt;/sup&gt;) to move the skyrmions. Our results open up possibilities to design practical skyrmion-based racetrack geometries for spintronic applications.

**References:**

[1]. H. Vakili, M. N. Sakib, and A. W. Ghosh, IEEE J. Explor. Solid-State Comput. Devices Circuits, 1 (2020).&lt;br/&gt;[2]. H. Vakili, Y. Xie, and A. W. Ghosh, Phys. Rev. B, 102, 174420 (2020).&lt;br/&gt;[3]. L. Zhao, Z. Wang, and W. Jiang, Phys. Rev. Lett., 125, 027206 (2020).&lt;br/&gt;[4]. F. Büttner, I. Lemesh, and G. S. D. Beach, Nat. Nanotechnol., 12, 1040-1044 (2017).&lt;br/&gt;[5]. A. Vansteenkiste, J. Leliaert, and B. V. Waeyenberge, AIP Adv., 4, 107133 (2014).&lt;br/&gt;[6]. W. E, W. Ren, and E. Vanden-Eijnden, J. Chem. Phys., 126, 164103 (2007)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/910b0f915b5edec2cfcf6b8a9c88ef0b.jpg)
Fig. 1: a) The racetrack geometry with four identical semicircular notches (radius 100 nm) separated by 200 nm from each other. The space between notches would be the places where the skyrmions would be positioned. b) Calculated minimum energy path for skyrmion passing a notch. The used material parameters are M&lt;sub&gt;s&lt;/sub&gt;=100 kA/m, A&lt;sub&gt;ex&lt;/sub&gt;=12 pJ/m, K&lt;sub&gt;u&lt;/sub&gt; = 50 kJ/m&lt;sup&gt;3&lt;/sup&gt;, D=0.80 mJ/m&lt;sup&gt;2&lt;/sup&gt;, 200 nm racetrack width, and 5 nm thickness. A barrier height of ~ 45 &lt;i&gt;k&lt;/i&gt;&lt;sub&gt;B&lt;/sub&gt;T for a skyrmion with a radius ~ 30 nm is found that can give a positional lifetime of ~ years for the skyrmion, which can be driven with a moderate current. The minimum unpinning current is calculated to be a 4 ns current pulse of 1.4x10&lt;sup&gt;11&lt;/sup&gt; A/m&lt;sup&gt;2&lt;/sup&gt;. Skyrmions can be localized in the racetrack with lifetime long enough even for long term memory, the racetrack will be digitized.

![](https://assets.underline.io/uploads/markdown_image/1/image/ebc6662cc96e93903380732a218f7ec9.jpg)
Fig. 2: The energy barrier (red) and the corresponding minimum unpinning current (black) for a racetrack having a semicircular notch (radius 75 nm) as a function of skyrmion size. The used material parameters are M&lt;sub&gt;s&lt;/sub&gt;=100 kA/m, A&lt;sub&gt;ex&lt;/sub&gt;=12 pJ/m, K&lt;sub&gt;u&lt;/sub&gt; = 50 kJ/m&lt;sup&gt;3&lt;/sup&gt;, D=0.65 - 0.90 mJ/m&lt;sup&gt;2&lt;/sup&gt;, 200 nm racetrack width, and 3 nm thickness. It shows low unpinning current with large enough energy barrier. The energy barrier increases faster with radius than the unpinning current which would help to get a large enough barrier and small enough unpinning current.

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 Positional Stability of Skyrmions via Pinning Sites in a Racetrack Memory

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Magnetic skyrmions are topologically protected chiral spin textures stabilized by the Dzyaloshinskii–Moriya interactions (DMI) in systems lacking inversion symmetry. The attractive features of skyrmions, such as ultra-small size, solitonic nature, and easy mobility with small electrical currents, make them promising as information-carrying bits in low power high-density memory and logic applications [1, 2]. In a skyrmion-based boolean racetrack memory, information can be encoded by the presence (bit “1”) and absence (bit “0”) of skyrmions at a particular position in the track. For an unconventional use of skyrmion racetrack, such as for native temporal memory [1], the information is encoded into the spatial coordinates of the skyrmions, which then can be translated into the timing information needed for race logic operations. The thermal stability of skyrmions is a critical issue for both of these applications, as a randomly displaced skyrmion can alter the sequence of the “0" and “1" bits in a Boolean memory application. Similarly, for race logic applications, the displacement of skyrmions would change the spatial coordinates of the skyrmion and hence the encoded analog timings. For reliable information extraction, it is essential to guarantee the positional stability of skyrmions for a certain amount of time. For example, for a long-term memory application it would require positional stability of years, while for cache memory, hours would be sufficient. In an ideal racetrack, skyrmions are susceptible to thermal fluctuations and exhibit Brownian motion that leads to the diffusive displacement of skyrmions [3]. Moreover, skyrmions show inertia driven drift shortly after a current pulse is removed, rather than stopping immediately. One way to control such undesirable motion is by engineering confinement barriers such as point defects with a different anisotropy, or notches etched into the racetrack (missing materials), which ensure the pinning of skyrmions. If the required unpinning current is too large, it will destabilize the skyrmions, which can cause annihilation of the skyrmions. In addition, high currents have been shown to randomly nucleate unwanted skyrmions as well [4]. This would mean that the energy barrier height needs to be optimized in such a way that it can hold the skyrmions for a long enough time and yet requires a modest current to unpin them. Using micromagnetic simulations [5], we present a quantitative analysis of racetrack geometries with different types of notches to calculate the minimum energy barriers associated with them. To calculate the energy barrier, we use the string method to find the minimum energy path (MEP) for the skyrmion to go over the notch [6]. We vary material parameters, specifically, the DMI, anisotropy ratio between the pinning sites and the rest of the magnetic material, and the geometry of the notches, in order to get the optimal barrier height. Furthermore, we investigate the effects of topology in the unpinning process by comparing the unpinning behavior of a ferrimagnetic vs a ferromagnetic skyrmion. We find a range of energy barriers (up to ~ 45 kBT) that can provide a long enough positional lifetime (years) of skyrmions for long-term memory applications while requiring a moderate amount of current (~ 1010 - 1011 A/m2) to move the skyrmions. Our results open up possibilities to design practical skyrmion-based racetrack geometries for spintronic applications.

**References:**

[1]. H. Vakili, M. N. Sakib, and A. W. Ghosh, IEEE J. Explor. Solid-State Comput. Devices Circuits, 1 (2020). [2]. H. Vakili, Y. Xie, and A. W. Ghosh, Phys. Rev. B, 102, 174420 (2020). [3]. L. Zhao, Z. Wang, and W. Jiang, Phys. Rev. Lett., 125, 027206 (2020). [4]. F. Büttner, I. Lemesh, and G. S. D. Beach, Nat. Nanotechnol., 12, 1040-1044 (2017). [5]. A. Vansteenkiste, J. Leliaert, and B. V. Waeyenberge, AIP Adv., 4, 107133 (2014). [6]. W. E, W. Ren, and E. Vanden-Eijnden, J. Chem. Phys., 126, 164103 (2007)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/910b0f915b5edec2cfcf6b8a9c88ef0b.jpg)
Fig. 1: a) The racetrack geometry with four identical semicircular notches (radius 100 nm) separated by 200 nm from each other. The space between notches would be the places where the skyrmions would be positioned. b) Calculated minimum energy path for skyrmion passing a notch. The used material parameters are Ms=100 kA/m, Aex=12 pJ/m, Ku = 50 kJ/m3, D=0.80 mJ/m2, 200 nm racetrack width, and 5 nm thickness. A barrier height of ~ 45 kBT for a skyrmion with a radius ~ 30 nm is found that can give a positional lifetime of ~ years for the skyrmion, which can be driven with a moderate current. The minimum unpinning current is calculated to be a 4 ns current pulse of 1.4x1011 A/m2. Skyrmions can be localized in the racetrack with lifetime long enough even for long term memory, the racetrack will be digitized.

![](https://assets.underline.io/uploads/markdown_image/1/image/ebc6662cc96e93903380732a218f7ec9.jpg)
Fig. 2: The energy barrier (red) and the corresponding minimum unpinning current (black) for a racetrack having a semicircular notch (radius 75 nm) as a function of skyrmion size. The used material parameters are Ms=100 kA/m, Aex=12 pJ/m, Ku = 50 kJ/m3, D=0.65 - 0.90 mJ/m2, 200 nm racetrack width, and 3 nm thickness. It shows low unpinning current with large enough energy barrier. The energy barrier increases faster with radius than the unpinning current which would help to get a large enough barrier and small enough unpinning current.

technical paper

While for many years magnetic domain walls (DWs) were considered to be Bloch type in ultra-thin magnetic layers with perpendicular anisotropy and particularly in stacks lacking inversion symmetry, the observation of the Dzyaloshinskii Morya interaction (DMI) in these stacks combined with the use of advanced imaging techniques have shown that these DWs were in fact of the Neel type and that they also had a precise chirality. These discoveries have led to significant advances in understanding the mechanisms governing the current induced motion of these DWs and to the reinterpretation of many past experimental results. These discoveries and reinterpretation have also allowed the demonstration of skyrmionic bubbles (Sk) and the development of their current induced dynamics. By analogy with DMI, which is a chirality dependent energy, we have demonstrated the existence of a chirality dependent dissipation: the chiral damping (αC) 1,2. As their chirality is altered, the induced variations of the damping parameter modify the DWs and Sk dynamics. Nevertheless, as this phenomenon shares with DMI the same origins, spin-orbit interaction and symmetry breaking, and as their effects and consequences are often confused, the influence of chiral damping on the dynamics of Sk and chiral DW has remained elusive so far, even though the understanding of this influence is crucial both for the understanding of chiral phenomena in the field of magnetism but also with the aim of developing innovative memory, logic or neuromorphic devices based on these chiral magnetic objects. In this study, we clearly separate the effects of DMI and αC in Sk and DW dynamics and, by focusing on the flow regime, which is both simpler to unambiguously interpret the experimental results and more relevant for the application objectives, we show that the effects of chiral dissipation are ubiquitous: they occur in all possible types of DW and SK motion: i) field-driven DW motion, ii) current induced DW motion, iii) current induced dynamics of skyrmionic bubbles. For each of these experiments, the observed asymmetries cannot be explained by considering either only DMI or only αC. A simple numerical modelling including these two mechanisms and using a single set of parameters makes it possible to reproduce the relevant features of these experimental results (Figure). Finally, these numerical simulations allow us to show the importance of αC for the stability and dynamics of skyrmionic bubbles in the regime of high spin-orbit torques and strong DMI sought for the applications: when they are distorted under the action of an electric current, αC can increase or decrease these distortions. These results show the importance of the chiral damping in the dynamics and stability of DW and Sk and the need to understand and control it in the objectives of applications using these chiral magnetic objects.

**References:**

1. Jué, E. et al., Nat. Materials, 15, 272–277 (2016). 2. Akosa, C. A., Miron, I. M., Gaudin, G. & Manchon, A. Phys. Rev. B, 93, 1–5 (2016).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/344f02225ea9513667ef104df38bf7ea.jpg)
a. Measured DW velocity vs. Hip (red for down/up DW, blue for up/down DW) at constant HZ= -82 mT. b. When only DMI is considered, numerical simulations cannot reproduce the crossing of curves: the up-down DW velocity is always higher than that of down-up DW for positive in-plane fields. c. When DMI and αC are taken into account, experimental results are well reproduced by numerical simulations.

 Effect of Chiral Damping on the dynamics of chiral domain walls and skyrmions INVITED

Recently, an ever-growing interest of the community of solid-state physics has been captured by the promising potentials of the magnetic skyrmions [1]. The skyrmion chiral property is generally driven by an additional anti-symmetric term in the exchange energy, namely the Dzyaloshinskii-Moriya interaction (DMI), induced by the lack of structural inversion symmetry and the spin-orbit coupling. As they have a high stability, small size and they can be moved by very small currents densities, skyrmions are considered as the ultimate magnetic storage bit for a new generation of devices of ultra-dense information storage and low power consuming spintronic logic and neuromorphic computing devices [2]. One of the major advantages of using interfacial DMI in thin films is that several materials parameters, that define not only the existence of skyrmions but also their dimensions, can be experimentally tuned, such as the magnetic anisotropy, or the DMI strength, which can be efficiently tailored. Here, we show a novel method to control interfacial effect in thin films with DMI using He irradiation. We compare SQUID, Ferromagnetic resonance (FMR), Magnetic force microscopy (MFM) as well as Brillouin light scattering (BLS) results on Ta(4.7 nm)/[Pt(4nm)/Co(1.4nm)/Ta(1.9nm)]20 stacks as a function of He irradiation dose to study the behaviour of the magnetic properties and interfacial effect of the stacks, such as magnetic anisotropy, intrinsic damping, effective magnetization, exchange stiffness as well as DMI. Our results show clear evidence of i) tailoring interface ii) different magnetic properties in thin film with different irridiation dose. These results can open novel way to manipulate skyrmions, as well as control of interface of spin-orbit torque devices and spin injection into a thin film, for example using spin pumping. Figure 1 (a) shows typical magnetization measurements while sweeping the external magnetic field in the direction perpendicular (out-of-plane) and parallel (in-plane) to the substrate at room temperature. Figure 1 (b) shows anisotropy energy density K and anisotropy field Hk with different Helium irridiation dose IR. As IR is increased further, a significant IR dependent diminishment of K and Hk occurs that is explained by the tailoring interface of the samples via IR. Figure 2 (a) show typical two-dimensional color plots of microwave absorption as a function of microwave frequency f and perpendicular magnetic field B. We determined the effective magnetization Meff from out-of-plane fits of the resonance Hres as different frequency using frait equation [3]. We show that Meff decrease with increasing IR (Fig. 2 (c)). The linear dependence of FMR linewidth ΔH on frequency f (Fig. 2(b)) allows the computation of the Gilbert-type damping constant α [4]. We see the clear enhancement of α with IR (Fig. 2(d)), suggesting of high rates of spin injection in the non-magnetic layers by spin pumping. Funding from the EMPIR project TOPS “Metrology for topological spin structures” (#17FUN08)) is acknowledged.

**References:**

[1] A. Fert et al. Nat. Nanotech. 8, 152 (2013)] [2] J. Grollier et al. Nat. Electronics, 3, 360–370(2020)) [3] S Klingler et al. J. Phys. D: Appl. Phys. 48 015001 (2015) [4] J.-C. Rojas-Sanchez, M. Cubukcu et al. Phys. Rev. B. 88, 064403 (2013)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/f89f26dd8da5338caf4117406cbcea1c.jpg)
Figure 1. SQUID measurements. (a) Normalized magnetization M/Ms versus applied magnetic field B plots for out-of-plane and in-plane configurations for sample with He irridation dose IR=8.1014 cm-2. (b) Anisotropy energy density K and anisotropy field Hk plots as a function of He irridation dose IR.

![](https://assets.underline.io/uploads/markdown_image/1/image/b4ac438e8762f4a8c4d6fb0e3d228d84.jpg)
Figure 2. FMR measurements. (a) Two-dimensional colour plots of microwave absorption as a function of microwave frequency f and magnetic field B for the out-of-plane configuration for sample with IR=8.1014 cm-2. We can clearly identify two modes; n=0 FMR uniform mode and n=1 spin wave mode. The solid line is fit to used to deduce the Meff. (b) FMR linewidth ΔH as a function of frequency f for the out-of-plane configuration for sample with IR=8.1014 cm-2. The solid line is fit to used to deduce the Gilbert damping constant α. Meff (c) and damping α (d) as a function of IR.

 Tailoring interfacial effect in thin films with DMI by He irradiation

Chiral magnets hosting non-collinear spin textures such as skyrmion lattices (SkL) [1] provide novel functionality in magnonics because of the formation of bandgaps in the magnon band structure and asymmetric magnon dispersion relations [2]. Multiple SkL phases were observed in the bulk chiral magnet Cu2OSeO3, e.g. high temperature SkL near TC, low temperature SkLs when the magnetic field was applied along an easy axis and meta-stable SkL when Cu2OSeO3 was quenched by fast cooling down [3-5]. Their dynamic modes are hard to be discriminated in large bulk samples because they coexist with conical and helical phases. Conventionally detected spin excitations (magnons) show complex spectra with a multitude of resonances due to multiple domains of different phases and boundary conditions. It is challenging to understand the one-to-one correspondence between an individual SkL phase and its magnon spectra. Here we report the local spectroscopy of magnons by scanning Brillouin light scattering (BLS) microscopy on Cu2OSeO3 in SkL phases. Thanks to the high sensitivity of BLS, we resolved the dynamic behavior of all reported SkL phases. Clockwise, counterclockwise and breathing modes of hexagonal SkL were detected by BLS proving simultaneous thermal excitation of all three modes which exhibit different polarization characteristics. Using the magnon modes, we were able to locate SkL phases and investigate the stabilization mechanism of non-collinear multi-domains. Our observations of locally excited thermal magnons in Cu2OSeO3 deepen the understanding of the SkL dynamics in chiral magnet. They pave the way for further design of magnonic devices based on chiral magnets. We acknowledge the finical support from Swiss National Science Foundation (SNSF) via Sinergia Network NanoSkyrmionics CRSII5 171003.

**References:**

[1] S. Mühlbauer, B. Binz and F. Jonietz, etc. Science, 323, 915 (2009). [2] M. Garst, J. Waizner and D. Grundler, J. Phys. D. Appl. Phys., 50, 293002 (2017). [3] A. Chacon, L. Heinen, M. Halder, etc. Nat. Phys. 14, 936-941 (2018). [4] F. Qian, L.J. Bannenberg, H. Wilhelm, etc. Sci. Adv. 4, eaat7323 (2018). [5] R. Takagi, Y. Yamasaki, T. Yokouchi, etc. Nat. Commun., 11, 5685 (2020).

**Images:**

 Spin Dynamics of Skyrmion Lattices in a Chiral Magnet Resolved by Micro-Focus Brillouin Light Scattering



The use of spin waves (SWs) as data carriers in spintronic and magnonic logic devices offers operation at low power consumption, free of Joule heating. Nevertheless, the controlled emission and propagation of SWs in magnetic materials remains a significant challenge. Here, we propose that skyrmion–antiskyrmion bilayers form topological charge dipoles and act as efficient sub-100 nm SW emitters when excited by in-plane ac magnetic fields. The propagating SWs have a preferred radiation direction, with clear dipole signatures in their radiation pattern, suggesting that the bilayer forms a SW antenna. Bilayers with the same topological charge radiate SWs with spiral and antispiral spatial profiles, enlarging the class of SW patterns. We demonstrate that the characteristics of the emitted SWs are linked to the topology of the source, allowing for full control of the SW features, including their amplitude, preferred direction of propagation, and wavelength.

**References:**

[1] S. A. Diaz, T. Hirosawa, D. Loss, and C. Psaroudaki, Nano Letters 20, 6556 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/227a77e02f052fbbadd7732d22309280.jpg)
Skyrmion-antiskyrmion bilayers form topological charge dipoles and act as efficient sub-100 nm spin wave emitters when excited by in-plane ac magnetic fields.



 Spin Wave Radiation by a Topological Charge Dipole

Magnetic skyrmions are chiral whirling textures of magnetization with a non-trivial topology. Their nanometric size and particle-like behavior has engendered huge interest for their potential applications in memory and logic devices. The observation of magnetic skyrmions at room temperature has triggered extensive research to decipher their various static and dynamic properties which is crucial for their eventual implementation in spintronic devices [1]. In the case of ultra-thin multilayer films consisting of heavy metal_(HM)/ferromagnet_(FM)/insulator_(I), skyrmions are primarily stabilized by interfacial Dzyaloshinskii-Moriya interaction (iDMI) in combination with perpendicular anisotropy, dipolar, exchange and Zeeman energies, where each of these contributions can be finely tuned [2]. The non-trivial skyrmion topology and its emergent electrodynamics is at the heart of phenomena like the skyrmion Hall effect and the topological Hall effect. Another interesting consequence of the skyrmions topology is reflected in their unique spectral signatures given by the gyrotropic rotational modes [3] (clockwise and counter-clockwise) and the breathing mode [4], when excited by an in-plane or an out-of-plane rf magnetic field. These unique dynamics open up new prospects for skyrmion-based microwave detectors [5] and nano-oscillators [6]. However, the experimental observation of skyrmion resonance dynamics remains limited. So far, skyrmion eigen modes have been mainly observed at low temperature in bulk systems [7]. Only few studies have been carried out in thin films, but with zero iDMI [8]. The experimental observation of the excitation modes of homochiral skyrmions stabilized by iDMI at room temperature remains elusive and challenging owing to usually elevated damping parameter and material inhomogeneities in HM/FM/I systems. Here, we study the magnetization dynamics in a [Pt/CoFeB/AlOx]×20 multilayer deposited by sputtering. First, the system is optimized by tuning the Pt and CoFeB thicknesses to host magnetic skyrmions at room temperature, along with a minimized damping parameter α ~ 0.02, measured by ferromagnetic resonance (FMR). The quasi-static magnetic domain configuration is observed by magnetic force microscopy (MFM) where on sweeping an out-of-plane (OP) magnetic field from saturation to zero, random skyrmion nucleation occurs, forming a lattice structure which then breaks into a mixture of skyrmions and stripes at lower fields; finally transforming into labyrinthine domains as shown in Fig. 1. The dynamic response of the system is measured by Vector Network Analyser (VNA) FMR with an rf field applied in-plane and varied over a frequency (f) range of 0.1-20 GHz, and an OP dc magnetic field swept from -0.55 T to +0.55 T. The frequency-field dispersions indicated in Fig. 2 (a) show several resonant modes corresponding to the domain configurations observed by MFM at the respective applied fields. At fields above the saturation, the well-known Kittel mode is observed, pertaining to uniform precession. Below saturation, distinct modes arise in the resonance spectrum with both positive and negative dispersions at low (f&lt;2GHz), medium (2&lt;f&lt;7GHz) and high frequencies (f&gt;7). The low frequency mode is observed on increasing the field towards saturation as can be seen on comparing Fig. 2 (a) and (b). Micromagnetic simulations further give an insight into the mechanisms involved in the subsequent dynamic behavior of the skyrmions. We observe that the low frequency mode (LFM) is due to the excitations localized at the skyrmion edges, corresponding to the breathing dynamics. The medium frequency mode (MM), on the other hand, arises due to the localized excitations in the inter-skyrmion region, i.e. the non-reversed magnetic zone, and has a negative dispersion. The origin of the high frequency mode (HFM) lies in the excitations across the skyrmion lattice and corresponds to the spin waves travelling across both the reversed and non-reversed regions. These latter modes (MM and HFM) are hence attributed to the magnonic cystal modes of the skyrmion lattice. This observation opens up new possibilities of creating and designing dynamically modulable skyrmion-based magnonic crystals at room temperature.

**References:**

[1] A. Fert et al., Nat. Rev. Mater. 2, 17031 (2017) [2] W. Legrand et al., Sci Adv 4, 7, (2018) [3] M. Garst et al. J. Phys. D: Appl. Phys. 50, 293002 (2017) [4] J.-V. Kim, et al., Phys. Rev. B 90, 064410 (2014) [5] G. Finocchio et al., Appl. Phys. Lett. 107, 262401 (2015) [6] F. Garcia-Sanchez et al., New J. Phys. 18, 075011 (2016) [7] Onose et al. Phys. Rev. Lett. 109, 037603 (2012) [8] S. A. Montoya et al. Phys. Rev. B 95, 224405 (2017)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/85f55608a7a181b99a46e77d5eb4df88.jpg)
Fig. 1. Magnetic domain configurations: randomly distributed skyrmions, skyrmion lattice, skyrmion-stripe mix, and labyrinthine domains, measured by MFM under the application of OP magnetic field swept from negative to positive values. The OP hysteresis curve in blue is plotted below. The solid blue curve corresponds to the OP field sweep from negative to positive (NtoP) values and the dashed curve from positive to negative (PtoN). The corresponding skyrmion density (red spheres) determined from the MFM images is also plotted. A dense skyrmion lattice is seen to be stable between 0.18T and 0.32T. The dashed grey line indicates the possible extrapolation of the observed skyrmion density at higher field values in accordance with the OP hysteresis curve (NtoP).

![](https://assets.underline.io/uploads/markdown_image/1/image/d3c6b3fe1ca9c66501cb406ed6122b5f.jpg)
Fig. 2. (a) Frequency-field dispersion map (with dashed curves at HFM as a guide to the eye) measured by VNA-FMR by scanning over a frequency range of 0.1 GHz to 20 GHz with an applied OP field swept from -0.55 T to +0.55 T. (b) and (c) Zoom of the areas indicated by white squares in (a).

 Ferromagnetic resonance of skyrmions in thin film multilayers

Magnetic skyrmions are magnetic quasi-particles with a number of interesting properties for possible future applications in, memory storage devices, or non-conventional computing [1-7]. We have shown that in low energetic landscapes of thin film magnetic multilayers, skyrmions show Brownian-like diffusion [7], which can be influenced by magnetic fields, temperature, and the magnetic properties of the pinning site [8,9]. While the diffusive dynamics have experimentally been studied in the continuous film [7,8,10], the device application requires a fabrication process in general, which means that skyrmions move rather in confined geometry. Here we show that the diffusive dynamics of skyrmions can be significantly modified for the geometry of the confined system and the skyrmion density. It is experimentally shown that the critical factor to unravel the underlying physics is the commensurability effect which is corroborated with Molecular dynamics simulations, as well as the pinning effect. The combination of experiment and analysis enables us to reveal the intriguing behavior that pinning sites for magnetic skyrmion depends on the magnetic field strength of the order of μT. Our findings could pave the way to control the thermal dynamics of skyrmions and the pinning effect to realize novel skyrmionic device applications. Stable skyrmions are nucleated by applying a magnetic out-of-plane (OOP) field while pulsing once an in-plane field [10]. We study the skyrmion diffusion in confined geometries varying both the symmetry of the geometry as well as the number of skyrmions from a sparse population to a fully lattice-like situation [11]. Basic geometries like triangles and circles are used. By measuring the mean squared displacement (MSD) of the trajectories of the skyrmions in the structures, the diffusion coefficient is calculated. Figure 1 shows the thermal behavior of skyrmions as a function of the skyrmion density in triangular geometry [11]. As can be seen in Fig. 1(a), The number of skyrmions can be precisely controlled by varying the OOP field to allow us to investigate the skyrmion density dependence of the skyrmion diffusion in confined geometries. Figure 1(b) presents the time-averaged configurations over 9600 Kerr images, where some configurations exhibit blur skyrmions which move with arbitrary time scale. It is shown that the diffusion qualitatively depends on the commensurability of the skyrmion number. Also, we find that the states for skyrmion numbers of 1, 3, 6, and 10 commensurate with the geometry and lead to a saturation of the MSDs as a function of time. The MSDs for an incommensurate state exhibit a non-monotonic relation and do not saturate over the experimental timescale. As shown in Fig. 1(c), molecular simulations agree qualitatively with the experimental findings that corroborating the basic dependence of skyrmion numbers on the motion in triangular confinement [11]. The quantitative difference from the simulation could be related to the pinning effect on the magnetic skyrmion. To explore the behavior closely, we investigate the magnetic field dependence of the skyrmion diffusion under a simplified environment. Figure 2 shows the OOP magnetic field dependence of histograms of skyrmion center positions in the circular geometry. For eliminating the commensurability effect, and the symmetry breaking of magnetic potential, we nucleate a single skyrmion in a circular geometry. We can see not only the modification of the depth of the pinning potential but also the pinning site transition with a small magnetic field of the order of μT, where the size of the skyrmion is modified. More intriguingly, the behavior shows hysteresis to a magnetic field, on the other hand, the reproducibility confirmed after applying a large magnetic field of 8 T suggests that the pinning sites themselves are not linked with the magnetic origin of pinned spins. Thus, the possible origin could arise from the topological spin texture itself. In conclusion, we reveal that the skyrmion diffusion is governed by the commensurability in confined geometry, as well as the magnetic field-dependent skyrmion pinning. The results show that for any potential skyrmionic device based on thin magnetic layers the numbers of skyrmions in the structured geometry as well as the geometry itself influence not only the arrangement but also the thermal diffusive motion of the skyrmions. Also, the pinning effect remains significant however, our findings indicate that magnetic skyrmions could be a new useful tool for experimentally exploring the pinning effect, which is a critical factor for any magnetic objects, and the possibility that the pinning sites are able to be tailored even after film deposition and fabrication process.

**References:**

[1] A. Fert et al., Nat. Nanotechnol.8, 152 (2013). [2] X. Zhang et al., Sci. Rep. 5, 7643 (2015). [3] X. Zhang et al., Sci. Rep. 5, 9400 (2015). [4] F. Garcia-Sanchez et al., New J. Phys. 18, 075011 (2016). [5] S. Zhang et al., New J. Phys. 17, 023061 (2015). [6] D. Prychynenko et al., Phys. Rev. Appl. 9, 014034 (2018). [7] J. Zázvorka et al., Nat. Nanotechnol. 14, 658 (2019). [8] T. Nozaki et al., Appl. Phys. Lett. 114, 012402 (2019). [9] N. Kerber et al., preprint arXiv:2004.07976 (2020). [10] J. Zázvorka et al., Adv. Funct. Mater. 30, 2004037 (2020). [11] C. Song et al., preprint arXiv:2009.03995 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3767ae61748fcafeb8f7fd6ebb2d7005.jpg)
Fig. 1 Thermal behavior of skyrmions as a function of the skyrmion density in triangular geometry. (a) The initial state of the skyrmion configurations. (b) The time-averaged configurations over 9600 Kerr images. (c) Simulations of the histograms of skyrmion center positions for configurations of 1 to 10 skyrmions inside the triangle containing positions from 1 million independent frames.

![](https://assets.underline.io/uploads/markdown_image/1/image/4f7ce9fe1971db64496cd0232ff4f441.jpg)
Fig. 2 The OOP magnetic field dependence of histograms of skyrmion center positions at 341.5 K in the circular geometry, where we plot and integrate the position where skyrmion exists at each frame in a 60 minutes video (57600 frames) on a two-dimensional map. A brighter contrast means that the skyrmion dwell time is longer at these points corresponding to an energy minimum.

 Skyrmion diffusion in a confined system

Magnetic skyrmions are topologically-protected magnetization textures characterized by a non-trivial topology [1]. They have been observed in bulk materials [2] and asymmetric multilayers [3,4] in presence of a finite Dzyaloshinskii-Moriya interaction (DMI). The manipulation (nucleation, shifting, and detection) of skyrmions usually occurs via electrical currents [3], however, thermal effects have been also used to reshuffle skyrmions for low-power unconventional applications [5]. Recently, thermal gradients have been applied to magnetic multilayers where thermally-generated skyrmions unidirectionally diffuse from hot regions to cold regions [6]. This observation has been explained through the combination of repulsive forces between skyrmions, thermal spin-orbit torques, magnonic spin torques as well as entropic forces. Here, inspired by Ref. [6], we study, by means of micromagnetic simulations, the effect of thermal gradients on skyrmion motion from a fundamental point of view. We have previously demonstrated [7] that temperature induces a variation of the magnetic parameters (exchange, interfacial DMI, perpendicular anisotropy constants) which reduce with temperature following atomistically-computed scaling relations. The set of scaled values of those parameters can be used to deterministically simulate the effect of thermal fluctuations instead of a stochastic thermal field [7]. With this in mind, we consider a linear thermal gradient applied along the x-axis (Fig. 1) with the hot and cold regions placed on the left and right side, respectively. We firstly study a 1-repetition ferromagnetic sample with a number of cells Ncx=300, Ncy=300, and Ncz=1 where a Néel skyrmion is initially placed at the center of the sample. We analyze the effect of a linear gradient of only one parameter at time. The skyrmion exhibits two velocity components that follow the generalized Thiele equation [8], where the component parallel to the gradient direction depends on the parameter varied. In particular, the perpendicular anisotropy and exchange gradients move the skyrmion from the cold to the hot region, therefore to the region where those parameters are smaller. On the contrary, the saturation magnetization (magnetostatic field) and the DMI shift the skyrmion from the hot to the cold region, thus towards the region where those parameters are larger. Afterward, we study the effect of the gradients all at once, observing an overall skyrmion motion from the cold to the hot region (Fig. 1). Furthermore, we study the effect of the skyrmion profile on the thermal gradient-driven skyrmion motion. In magnetic multilayers, the skyrmion is characterized by a thickness-dependent profile which gives rise to the so-called hybrid skyrmion [4]. Therefore, we study the effect of the thermal gradients on a single hybrid skyrmion as a function of the number of ferromagnetic repetitions. Fig.1 shows a comparison of the skyrmion trajectories, where a dependence on the number of repetitions is observed. Our results can be useful for a deeper understanding of the effect of thermal gradients and therefore for the design of low-power thermally-driven skyrmion applications. This work was supported by the project “ThunderSKY,” funded by the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under grant agreement No. 871.

**References:**

[1] G. Finocchio et al., J. Phys. D. Appl. Phys. 49, 423001 (2016). [2] X. Z. Yu et al., Nature 465, 901 (2010). [3] S. Woo et al., Nat. Mater. 15, 501 (2016). [4] W. Li et al., Adv. Mater. 1807683, 1 (2019). [5] J. Zázvorka et al., Nat. Nanotech. 14, 658 (2019). [6] Z. Wang et al., Nat. Electron. 3, 672 (2020). [7] R. Tomasello et al. Phys. Rev. B 97, 060402 (2018). [8] R. Tomasello et al. Phys. Rev. B 98, 024421 (2018).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e7af58671e85206fb87cfe24d058a58f.jpg)
Figure 1: skyrmion trajectory under a thermal gradient (gradients of the parameters all at once) in a magnetic multilayer for different ferromagnetic repetitions, as obtained by micromagnetic simulations. The hot and cold regions are located on the left and right side, respectively.

 Micromagnetic study of thermal gradient-driven skyrmion motion in magnetic multilayers



The second-generation of Artificial intelligence (AI) has experienced intense growth due to its excellence in big-data applications. However, state-of-the-art AI lacks the advanced cognitive abilities seen in the brain, including the abilities to adapt to the environment, to understand context and to build coherent representations of the outside world. In addition, the implementation of second-generation AI with conventional CMOS uses large amounts of energy and area [1]. Spintronic nanodevices are particularly attractive for neuromorphic implementation of AI due to their small footprint, high endurance and low power consumption [2]. While these devices can implement ultra-low-power AI [3], they have yet to mimic the more rich, adaptive and computationally powerful behavior seen in the brain which forms the basis for advanced cognition. Here, we show the proof-of concept of a self-adaptive artificial neuron that utilizes adaptive spintronic materials that alter their structure and properties in response to external stimuli, in order to realize advanced cognitive abilities such as context-awareness and feature binding. The self-adaptive neuron is constructed from an artificial skyrmion lattice with five skyrmions hosted in a bilayer of thulium iron garnet (TmIG) and platinum (Pt). We report micromagnetic simulation results that show the neuron, when excited by an oscillating magnetic field, produces spin waves originating from skyrmion oscillations with a multi-frequency spectrum. The spectrum consists of four distinct resonant modes, identified as gyration, breathing, and hybridizations of both. Since each resonant frequency can represent one bit of information, the multi-frequent spectrum enables a large basis of information representation. Crucially, we show that both the amplitude and frequency of these resonant modes can be modulated by re-arranging the skyrmions in the lattice. This ability of the lattice to regulate its oscillatory dynamics in response to external input mimics a critical neural property called neuromodulation which plays a key role in advanced cognitive processes, including information transfer, decision-making, memory, object representation, consciousness etc. As a result of neuromodulation, the self-adaptive neuron demonstrates bio-plausible properties such as bursting and cross-frequency coupling [7], as well as advanced cognitive processes, namely context-awareness and feature binding. Context-awareness allows the neuron to make complex, high-level decisions while accounting for multiple factors to develop a deeper understanding of its circumstance. To implement context-awareness, the neuron receives a human spoken command and contextual information about the color of a box. Integrating both inputs, the neuron decides to open the box if and only if the human being gives the command to open and the box is not red in color. In addition, feature binding allows the neuron to correctly combine different segments of information (say, ‘red’ color and ‘circular’ shape) and construct a coherent percept (‘red circle’). To implement feature binding, a network of neurons receives visual information about two objects and processes information about their color and shape separately. The network then correctly binds together the two features (color and shape) to obtain coherent percepts of the original object. These results realize context-aware AI and the fusion of information from different sensory streams, which can have significant impact on human-machine collaboration, bio-medicine, smart energy, advanced manufacturing, agriculture and education. Adaptive materials can thus help realize neuromorphic circuits with advanced cognitive abilities for third generation AI, one that understands and adapts to a complex and ever-changing environment, learns without supervision from small datasets and better collaborates with human beings.

**References:**

[1] Cisco, Cisco Global Cloud Index: Forecast and methodology 2016–2021,  1513879861264127, (2018). [2] S. Singh, A. Sarma, N. Jao et al., ACM/IEEE 47th Annual International Symposium on Computer Architecture (ISCA), Valencia, Spain (2020). [3] P. A. Merolla, et al., Science, vol. 345, pp. 668-673 (2014). [4] E. Marder, Advances in Neural Information Processing Systems 6 (NIPS) (1993). [5]         L. Gao, P.-Y. Chen and S. Yu, Appl. Phys. Lett., vol. 111, p. 103503, (2017). [6] T. Jackson, S. Pagliarini and L. Pileggi, 28 nm CMOS, IEEE International Conference on Rebooting Computing (2018). [7]      R. F. Helfrich and R. T. Knight, Trends in Cognitive Sciences, vol. 20, p. 916, (2016).

**Images:**





 Advanced cognitive computing using adaptive spintronic materials INVITED

Recently, magnetic skyrmions have been employed as information carriers via the digital encoding of bits of information to perform information storage and computing (see e.g. [1]). In this work, it is proposed a simple scheme to use Néel skyrmions hosted in ferromagnetic materials as information entropy carriers. To show that, we start from the definition of entropy noting that the Boltzmann’s configurational entropy of magnetic skyrmions [2] has a complete identification with Jaynes’s information entropy [3], the continuous limit of Shannon’s entropy. The information entropy can be regarded as the average value of the information (or information content) produced by a stochastic source of data measuring the unpredictability of the state. The more the probability is lower, the more the information content is higher and the more the information entropy is higher. In the 1D case Jaynes’s continuous information entropy takes the form [4]: ∫X p(x) log2(p(x)/m(x)) dx (1) where p(x) is the probability density function referred to the continuous variable x, the function m(x), called the “invariant measure”, makes part of the definition of the limit of the density of a set of N discrete points {xi} as N →∞ and X = { x1, x2, .., xN } is a random variable.The key message contained in Jaynes’s information entropy is the following: a low-probability event carries more information than a high probability event. In other words, if an event is less probable, due to its high degree of unpredictability, is more interesting than a high-probability event. If this concept is applied to the configurational statistical entropy of a skyrmions population, that is defined as S = −kB &lt;lnf0&gt; (kB is the Boltzmann constant) in terms of the statistical average &lt;...&gt; of lnf0 with f0 the Gaussian probability distribution, the information is contained in lnf0. As for the case of Jaynes’s information entropy, the amount of information carried out by each of the different events is a random variable represented by the skyrmion size at a given T and instant of time whose expectation value is the information entropy determined at each temperature. Let us suppose having a distribution of skyrmion diameters obtained taking pictures of the skyrmions at different instants of time having the same average energy. The event (message) occurring at the instant of time t1 when the skyrmion area Askymp corresponds to the most probable skyrmion diameter Dskymp carries less information than all the events occurring at different instants of time corresponding to skyrmion diameters Dsky such that Asky ≠ Askymp with Asky the generic skyrmion area. The more Dsky is far away from Dskymp, the less is its probability and the more the corresponding event carries information. In other words, the more Asky differs from Askymp, the more the event is unpredictable and the more it encodes bits of information. It has been found that the effect of temperature is the broadening of the distributions of the skyrmion diameters at every external bias field Hext [2]. Hence, with increasing temperature the interval of diameters belonging to the distribution increases leading to a higher number of events carrying more information and to a higher entropy that encapsulate more information. On the other hand, the effect of Hext is opposite: at fixed temperature there is the narrowing of the distributions of diameters with increasing Hext leading to a lower number of events carrying more information and to a lower entropy encapsulating less information. Hence, via the analogy between Jaynes’s entropy and the configurational thermodynamic entropy, the following conclusions can be drawn: (1) room temperature magnetic skyrmions carry more information entropy than the ones at low temperature and (2) at fixed T, in the region of metastability, magnetic skyrmions that are not perturbed by Hext encapsulate more information entropy than the ones subjected to Hext. Owing to the above considerations, a simple entropy data communication system based on the employment of ferromagnetic materials hosting magnetic skyrmions at a given T may be suggested and employed in future spintronic devices. Data communication systems consists of: (1) a data source, (2) a communication channel and (3) a receiver. The receiver should interpret the data that come from the source collecting them. The proposed data communication system is schematically shown in Fig.1. The data source of entropy information can be regarded as a thermal bath at a given T in thermodynamic equilibrium with the ferromagnetic material (e.g., cylindrical dot) hosting the magnetic skyrmion. The thermal bath excites the thermal breathing mode that acts as a source of entropy. The communication channel can be thought of the ferromagnetic material itself hosting the magnetic skyrmion able to collect the entropy information bits (message) coming from the magnetic skyrmion. The receiver could be a special device able to identify and collect the bits of entropy. It can be ideally supposed that the amount of entropy is less than the capacity of the communication channel leading to the communication and to the storage of all the data to the receiver. In this way, magnetic skyrmions may be employed as novel temperature and magnetic field dependent information entropy carriers via the transmission of bits of entropy.

**References:**

[1] J. Müller, New J. Phys., Vol. 19, p. 025002 (2017) [2] R. Zivieri, R. Tomasello, O. Chubykalo-Fesenko et al., Phys. Rev. B, Vol. 99, p. 174440 (2019) [3] E.T. Jaynes, IEEE Trans. Syst. Sci. Cybern., Vol. SSC-4, p. 227 (1968) [4] R. Zivieri, Appl. Sci., Vol. 10, p. 52 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/fca95be2870fa98bcf4cf1be1527b6bc.jpg)
Fig.1. Scheme of the entropy data communication system. FM: ferromagnet, HM: heavy metal.

 Magnetic Skyrmions as Information Entropy Carriers

Magnetic skyrmions have attracted enormous research interest since their discovery a decade ago. Especially the non-trivial real-space topology of these nano-whirls leads to fundamentally interesting and technologically relevant effects – the skyrmion Hall effect of the texture and the topological Hall effect of the electrons. Furthermore, it grants skyrmions in a ferromagnetic surrounding great stability even at small sizes, making skyrmions aspirants to become the carriers of information in the future. Still, the utilization of skyrmions in spintronic devices has not been achieved yet, among other reasons, due to shortcomings in their current-driven motion. In this talk, we present our recent advances in the field of topological spin textures that go beyond skyrmions [1]. The majority of the discussed objects can be considered the combination of multiple skyrmions or the skyrmion analogues in different magnetic surroundings, as well as three-dimensional generalizations. We classify the alternative magnetic quasiparticles – some of them observed experimentally, others theoretical predictions – and present the most relevant and auspicious advantages of this emerging field. A special focus is on magnetic antiskyrmions [2,3,4], bimerons [5], antiferromagnetic skyrmions [6] and hopfions [7]. These objects (shown in Fig. 1) exhibit advantageous emergent electrodynamic effects compared to conventional skyrmions, either due to their lower symmetry or due to a compensated topological charge. As we will show, all four of these objects can be driven parallel to the current, without a skyrmion Hall effect which makes them the ideal bits in data storage devices. Furthermore, we show that the antiferromagnetic skyrmions exhibit a topological version of the spin Hall effect which could help their detection. For bimerons and hopfions we show that the topologically induced Hall response can be distinguished much more easily from the anomalous and conventional Hall effect. This is because their emergent field is perpendicular to the net magnetization and the stabilizing magnetic field; all three quantities are (anti)parallel for skyrmions and most other magnetic nano-objects. Also, we show that some of these objects can even coexist, allowing for an advanced version of the racetrack memory data storage, where a bit sequence could, for example, be encoded by a sequence of skyrmions (‘1’ bit) and antiskyrmions (‘0’ bit); cf. Fig. 2. This concept would be more reliable than conventional racetracks: When the information is encoded by the presence or absence of a single type of topological object, the information is susceptible to thermal motion and interactions between the carriers of information. An unintended change in the bit sequence would lead to a loss of the stored information. Only when the information is encoded by at least two distinct carriers, the bit sequence can become irregular without any harm to the stored data.

**References:**

 [1] B. Göbel, I. Mertig, O. Tretiakov. “Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles”. Physics Reports in press (2020), arXiv preprint: 2005.01390 [2] J. Jena*, B. Göbel*, S. Parkin. et al. “Elliptical Bloch skyrmion chiral twins in an antiskyrmion system”. Nature Communications 11, 1115 (2020). [3] J. Jena*, B. Göbel*, S. Parkin. et al. “ Evolution and competition between chiral spin textures in nanostripes with D2d symmetry”. Science Advances 6, eabc0723 (2020). [4] P. Sivakumar, B. Göbel, S. Parkin. et al. “ Topological Hall Signatures of Two Chiral Spin Textures Hosted in a Single Tetragonal Inverse Heusler Thin Film”. ACS Nano 14, 13463 (2020). [5] B. Göbel, I. Mertig, O. Tretiakov. et al. “Magnetic bimerons as skyrmion analogues in in-plane magnets”. Phys. Rev. B. 99, 060407(R) (2019). [6] B. Göbel, A. Mook, J. Henk, I. Mertig. “Antiferromagnetic skyrmion crystals: Generation, topological Hall, and topological spin Hall effect”. Phys. Rev. B. 96, 060406(R) (2017). [7] B. Göbel, C. Akosa, G. Tatara, I. Mertig. “Topological Hall signatures of magnetic hopfions”. Phys. Rev. Research 2, 013315 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1df6edc367627cf9506b431bbb84b52b.jpg)
Overview of the discussed magnetic nano-objects. (a) Magnetic antiskyrmion, (b) magnetic bimeron, (c) antiferromagnetic skyrmion, (d) magnetic hopfion. The arrows correspond to magnetic moments whose color is determined by their orientation.

![](https://assets.underline.io/uploads/markdown_image/1/image/fa4bb03806ecc363261983a4f2311ee5.jpg)
In Heusler materials the anisotropic DMI stabilizes antiskyrmions (red) and elliptically deformed skyrmions (blue). These coexisting types of topologically distinct objects may encode data as ‘0’ and ‘1’ bits, respectively.

Positional Stability of Skyrmions via Pinning Sites in a Racetrack Memory

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Positional Stability of Skyrmions via Pinning Sites in a Racetrack Memory

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Effect of Chiral Damping on the dynamics of chiral domain walls and skyrmions INVITED

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Sensitivity optimization of spin-torque diode with perpendicular anisotropy through free layer thickness tuning

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