France



Magnetic skyrmions (Sks) and antiskyrmions (aSks) are topologically protected non-trivial spin textures&lt;sup&gt;1, 2&lt;/sup&gt;. Those objects can be used as magnetic bits in spintronics devices&lt;sup&gt;3, 4&lt;/sup&gt;. 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 systems&lt;sup&gt;2, 5 &lt;/sup&gt;such that it can be moved in a straight path along the current direction&lt;sup&gt;6&lt;/sup&gt;. 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 Mn&lt;sub&gt;1.4&lt;/sub&gt;Pt&lt;sub&gt;0.9&lt;/sub&gt;Pd&lt;sub&gt;0.1&lt;/sub&gt;Sn prepared along different crystallographic directions within the tetragonal basal plane&lt;sup&gt;7&lt;/sup&gt;. 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 Mn&lt;sub&gt;1.4&lt;/sub&gt;Pt&lt;sub&gt;0.9&lt;/sub&gt;Pd&lt;sub&gt;0.1&lt;/sub&gt;Sn allows the stability of distinct spin textures called elliptical Bloch skyrmions&lt;sup&gt;7,8&lt;/sup&gt;. The co-existence of elliptical skyrmions (eSks) and aSks is found in nano-stripes at room temperature&lt;sup&gt;7&lt;/sup&gt;. 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., &lt;i&gt;Science&lt;/i&gt; 323, 915 (2009).  2. Nayak et. al., &lt;i&gt;Nature&lt;/i&gt; 548, 561 (2017).&lt;br/&gt;3. Parkin et. al., &lt;i&gt;Science&lt;/i&gt; 320, 190 (2008).  4. Fert et. al., &lt;i&gt;Nat. Nanotechnol.&lt;/i&gt; 8, 152 (2013).&lt;br/&gt;5. Jena et. al., &lt;i&gt;Nano Lett.&lt;/i&gt; 20, 59 (2019).  6. Huang et. al., &lt;i&gt;Phys. Rev. B&lt;/i&gt; 96, 144412 (2017).&lt;br/&gt;7. Jena et. al., &lt;i&gt;Sci. Adv. &lt;/i&gt;6, 49, eabc 0723 (2020). 8. Jena et. al., &lt;i&gt;Nat. &lt;/i&gt;&lt;i&gt;Commun.&lt;/i&gt; 11, 1115 (2020).

**Images:**





INTERMAG 2021

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

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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:**

poster

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 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.

 Going Beyond Skyrmions: Alternative Magnetic Nano-Objects for Spintronics

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

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.

 Positional Stability of Skyrmions via Pinning Sites in a Racetrack Memory

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].

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