Magnetic skyrmions are “topologically protected” solitons, that have recently received increased attention due to their potential application as information carriers and for unconventional computing [1]. While the current-driven dynamics has been explored deeply, the theory of temperature gradient-induced dynamics - Skyrmion-Caloritronics [2] - is still at its early stages of development but it is particularly promising due to its low energy consumption.
In this work, we study the effects of thermal gradients on skyrmion motion in different systems (single-layer FM with interfacial Dzyaloshinskii-Moriya interaction (IDMI), multilayer, and synthetic antiferromagnets (SAFs)) inspired by the experimental results of Ref. [2] and from a fundamental point of view. We observe that, when driven by the entropic torque, skyrmions in single layers move from the cold to the hot region with a finite skyrmion Hall angle (Fig. 1), while in multilayers they move in the opposite direction (from hot to cold region) (Fig. 2). The latter is in qualitative agreement with the experimental observations (Fig. 3a in Ref. [2]). We demonstrate that this difference is due to the distinct scaling relations characterizing the two systems, and to the existence of a magnetostatic field gradient linked to the variation of saturation magnetization (MS) which cannot be neglected in magnetic multilayers. The numerical results are corroborated by a generalized Thiele’s equation developed for this scenario. Moreover, we show that the skyrmion Hall angle is completely suppressed in SAFs, similarly to the current-driven skyrmions [3,4].
Our results have fundamental implications in the future development of skyrmionic devices combining thermal gradients and SOTs where the proper temperature dependence of the parameters should be taken into account.
See Ref. [5] for founding statement. 
**References** [1] J. Zázvorka, et al., Nat. Nanotechnol. 14, 658 (2019).

[2] Z. Wang, et al., Nat. Electron. 3, 672 (2020).

[3] X. Zhang, et al., Nat. Commun. 7, 10293 (2016).

[4] R. Tomasello, et al., J. Phys. D. Appl. Phys. 50, 325302 (2017).

[5] This work was supported by the Project No. PRIN 2020LWPKH7 funded by the Italian Ministry of University and Research, and by the PETASPIN association (www.petaspin.com). JB acknowledges support from the Royal Society through a University Research Fellowship. E.R. acknowledges the economical support of the COST Action CA 17123 (MAGNETOFON) within the STSM program. We would like to acknowledge networking support from COST Action No. CA17123 “Ultrafast opto magneto electronics for nondissipative information technology.”

[6] W. Li, et al., Adv. Mater. 31, 1807683 (2019). 

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Skyrmion Caloritronics: a promising Strategy to manipulate Skyrmions

One of the most promising magnetic systems studied in the last years are the 3D nanostructures with curved geometry where novel and interesting magnetization textures and dynamics are involved [1-2]. One of these nanostructures, cylindrical nanowires, present multiple topologically non-trivial magnetization structures such as Bloch-point domain walls, faster and more stable as compared to those in planar nanowires, vortices, and skyrmion tubes [2-3]. 
This talk presents different ways to manipulate the magnetization configuration and the domain wall dynamics in cylindrical nanowires. The control and stabilization of domains and domain walls have been achieved in multi-segmented structures [4] by alternating the magnetic anisotropy or by imprinting notches along the nanostructures length. The domain wall dynamics, under the influence of magnetic field or current pulses, was investigated by XMCD-PEEM and micromagnetic simulations. The results indicate that large current densities induce domain wall nucleation while smaller currents move domain walls preferably against the current direction but also along it. The pinning of domain walls has been observed not only at geometrical constrictions but also induced by the Oersted field which creates chiral domain walls. 
**References:** [1] R. Streubel et al, J. Phys. D: Appl. Phys. 2012, 49, 363001. 
[2] A. Fernández-Pacheco et al., Nat Commun 2017, 8, 15756. 
[3] J.A. Fernandez-Roldan et al., Nanoscale 2018, 10, 5923. 
[4] C. Bran et al., ACS Nano 2020, 14, 12819–12827.

Nucleation and Motion of Domain Walls in Cylindrical Nanowires

As three-dimensional nanomagnetism evolves, novel non-trivial magnetic textures emerge as appealing information carriers for recording and spintronics based on curved nanosystems, and particularly on Cylindrical Nanowires (NWs) [1,2]. One of the most fascinating candidates that is likely to reach the high velocities required for fast recording technologies is the Bloch Point (BP) domain wall (DW). Recently, theoretical evidence indicated that BPs in NWs could reach high velocities close to 2 km/s in the magnonic regime [3]. While the observation of the BP DW in cylindrical NWs is no longer recent [2], scarce numerical studies that combine both current and Oersted field have been published in NWs [4], despite first attempts to measure DW velocities are in progress [5]. In this work we evaluate the dynamics of the BP DW under both current directions in a Ni NW with 100 nm in diameter. We investigate two cases: i) pre-nucleated BP Wall, and ii) the BP DW originated from the transformation of a Vortex-Antivortex DW. The effects of both spin-polarized current and Oersted field are considered. We discuss in detail the role of the chirality of the BP in relation to the Oersted field, also reported previously in precursors of BPs [4]. We show that while the pre-nucleated DW with the same chirality as that of the Oersted field propagates always against the current direction, the BP originated either from the transformation of the BP with the opposite chirality or from the vortex-antivortex DW can either stop the propagation or propagate parallel to the current. We attribute this effect to the role played by the inertia of the BP in its dynamics. Finally, we provide values of the velocities achieved by the BP in the NW as a function of applied current in Fig. 1. We conclude that BPs with vanishing momentum propagate opposite to the current with velocities that may be suppressed by the Oersted field. Importantly for spintronic applications, both momentum and inertial mass play a major role in the dynamics of BPs that has not been envisaged up to know.

![](https://assets.underline.io/uploads/markdown_image/1/image/6ff528b21478a289ec02bc86a4e8eae4.jpg)

The average velocity of a head-to-head BPW driven by a spin-polarized current in a Ni nanowire, far from the ends of the nanowire, as a function of the current density.

Oersted-field- and current- induced Bloch Point domain wall dynamics in cylindrical Ni nanowires

Recent advances on the stabilization and manipulation of chiral magnetic configurations hosted by systems that consist in alternating atomic layers of ferromagnetic and non-magnetic materials hold promise of innovative spintronic devices. The low dimensionality of the systems promotes spin-orbit driven interfacial effects[1] like the antisymmetric Dzyaloshinskii-Moriya interaction (DMI) and interfacial magnetic anisotropy (iMAE), whose relative strengths may be tuned to achieve stable nanometer sized magnetic objects with fixed chirality called magnetic skyrmions[2], which are proposed as information carriers in future spin-orbitronics technologies. While in most of the cases this is obtained by engineering complex multilayers stacks in which interlayer dipolar fields become relevant[3], we consider here a simple epitaxial trilayer in which a ferromagnet (namely Cobalt, Co), with variable thickness, is embedded between a heavy metal and graphene[4]. The latter enhances the perpendicular magnetic anisotropy of the system, promotes a Rashba-type DMI, and can sustain very long spin diffusion length. The work consists in the development of a layer-resolved micromagnetic model capable to account for the interfacial nature of the interactions[5], which leads to macroscopic parameters that depend on the thickness of the ferromagnetic layer. We demonstrate that our model correctly reproduces the experimental magnetization configurations and the spin reorientation transition (SRT) of a gr/Co(tCo)/Pt sample[4]. Next, we study the stabilization of magnetic skyrmions in patterned magnetic nanodots with a diameter of 256 nm. In particular, we analyze the thickness-dependent stability of the skyrmionic state in contrast with a single domain state. Our results demonstrate that, for samples with Co thicknesses larger than 3.6 nm, intrinsic mixed Bloch-Néel (predominantly Bloch-type) skyrmions are stabilized in 256 nm wide dots. For thicknesses larger than ~5 nm, the results show that the geometrical confinement imposed by the nanodot promotes the stabilization of chiral spin textures similar to skyrmions but with complex and fluctuating domain walls.

**References:**

[1] A. Soumyanarayanan, N. Reyren, A. Fert and C. Panagopoulos, Nature, Vol. 539, p. 509-517 (2016) [2] A. Fert, N. Reyren and V. Cros, Nature Reviews Materials, Vol. 2, p. 170301 (2017) [3] W. Legrand, J.-Y. Chauleau, D. Maccariello et al., Science Advances, Vol. 4(7), p. eaat0415 (2018) [4] F. Ajejas, A. Gudín, R. Guerrero et al., Nano Letters, Vol. 18(9), p. 5364-5372 (2018) [5] P. Olleros-Rodríguez, R. Guerrero, J. Camarero et al., Applied Materials and Interfaces, Vol. 12(22), p. 25419-25427 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b0a17000d2dcf3624f3186e212665bbe.jpg)
Fig.1 Sketch of modelled magnetic nanodots at selected cobalt thicknesses. The stabilized spin textures are represented by the arrows at the top of the structures. The color of the arrows represents the out-of-plane component of the magnetization. The color of the surface represents the X-component of the magnetization.

Micromagnetic Study of Stable Skyrmions in Dot-Patterned Graphene-Based Magnetic Trilayers

The relaxation of the magnetisation in magnetic materials is of great importance from both an applied and fundamental point of view. Conventional modelling of magnetisation dynamics employs the Landau-Lifshitz-Gilbert (LLG) equation [1], where the damping is included phenomenologically by the Gilbert damping term. This describes the coupling of the magnetic modes (given primarily by the atomic spin) to the non-magnetic modes (lattice vibrations and electron orbits) which are assumed to be in equilibrium. Recent studies of the dynamics induced by THz laser pulses has highlighted the necessity of understanding magnetisation relaxation beyond this assumption. In reality the spin and lattice dynamics mutually influence one another, hence it is necessary to employ a unified model of molecular and spin dynamics [2,3,4], Spin-Lattice dynamics (SLD). In the present work we employ a model where the transfer of energy and angular momentum between the lattice and the spin system (Fig. 1) is realised by the pseudo-dipolar coupling [3,4], which arises from the spin-orbit interaction and can be parameterised by magneto-elastic experiments. The spin system exchange parameters are taken from ab-initio parameterisation for BCC iron [2], while for the phonon system interactions potential we compare the Harmonic and Morse potentials. Our results in Fig. 1 show that equilibration of both sub-systems can be obtained on a sub-nanosecond timescale in both the microcanonical and canonical ensembles, the relaxation time being governed by the energy initially deposited in the system. We also observe that the equilibrium magnetisation is independent of the thermostat and by coupling the spin system only to the lattice vibrations the magnetisation temperature dependence can be reproduced without the need of a phenomenological spin damping [3]. Our model allows the evaluation of the effective magnon-phonon damping (Fig. 2) which agrees well with the values measured in magnetic insulators where they depend predominantly on magnon-phonon coupling rather than on electronic effects. The functional form of the damping variation is quadratic, in accordance with the form of the pseudo-dipolar coupling term. The magnon-phonon damping has been also calculated for an alternate form of the coupling, where the pseudo-dipolar coupling is replaced by an on-site form, i.e a Néel-like anisotropy term. The latter leads to much smaller damping values, as shown in Ref. [3], suggesting that the magnon-phonon damping can clearly have complex behaviour depending on the properties of the system, especially the coupling term. Based on this remark, no universal behaviour of damping as a function of temperature can be deduced for spin lattice models. The model developed in this work [3] opens the possibility to describe the distinct dynamics of spins and phonons, necessary for the understanding of ultrafast magnetisation dynamics experiments and the subsequent angular momentum transfer between the two subsystems. We have demonstrated that the model works well in the absence of a phenomenological Gilbert damping, which consists mainly of electronic contributions, hence the SLD model can be employed to study magnetic insulators, such as YIG, where the principal contribution to damping is via magnon-phonon interactions.

**References:**

 [1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [2] P.-W. Ma et al., Phys. Rev. B, vol. 78, no. 2, p. 024434, Jul. 2008. [3] M. Strungaru et al. arXiv preprint arXiv:2010.00642 (2020) (accepted in Phys. Rev. B) [4] M. Aßmann and U. Nowak, J. Magn. Magn. Mater 6 469 (2019): 217-223.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/0415f635b3cb2e3e3855832e043401d0.jpg)
 
Fig.1: NVE (microcanonical – top panel) and NVT ( canonical – bottom panel) simulations for a 10 × 10 × 10 unit cell BCC Fe system. The spin system is initialised at a spin temperature TS = 0K corresponding to a saturated magnetic state in the z direction, while the lattice velocities are initialised by a Boltzmann distribution at T L = 1200 K (for NVE) and T L = 600 K (for NVT). In both cases we obtain equilibration of the two subsystems under 1ns.

![](https://assets.underline.io/uploads/markdown_image/1/image/6755da1be71fc906a6b75c124d5d162a.jpg)
 
Fig.2 Damping parameter extracted from fitting the z component of the magnetisation for two different choices of potential: HP- Harmonic Potential (green open squares) and MP-Morse Potential (black open circles) as a function of temperature - Panel a) and as a function of the coupling strength – Panel b); The effective damping in Panel a) is calculated for a constant coupling strength of C = 0.5, while in Panel b) the damping is calculated for a constant temperature of T = 300K. The black and green lines represent the average damping parameter obtained from the simulations using the Morse and the Harmonic Potentials, respectively.

 Magnon-phonon damping calculations in the spin-lattice dynamics model

Antiferromagnetic spintronic devices have the potential to greatly outperform their current ferromagnetic counterparts due to their robustness to external fields, ultrafast dynamics and potential for high data density storage. Mn2Au is one of the most promising materials for antiferromagnetic spintronics due to its very high Néel temperature, moderate in-plane anisotropy and layered two sublattice spin structure [1]. Here we present atomistic spin simulations that allow us to estimate the temperature dependent magnetic properties of Mn2Au single crystals. We introduce a thermodynamically consistent spin Hamiltonian guided by previous ab-initiocalculations [3], which includes easy plane and rotational anisotropy terms. Temperature scaling of the anisotropy parameters is crucial for the performance optimization of antiferromagnetic devices, from the scale of magnetic domains to stability under thermal activation [2]. However, scaling in antiferromagnet has been barely studied due to the inherent difficulties to measure antiferromagnetic order parameter, n. We find a scaling of the easy plane anisotropy with the sublattice magnetization of n2.32. This is consistent with its two-ion origin [4], whereas the rotational anisotropy presents a n10dependence, consistent with the Callen-Callen scaling in ferromagnets of a 4thorder anisotropy. The crystal structure is shown in Fig. 1 showing two distinct Mn sites with a 180 degree relative orientation. We obtain quantitative agreement with experimental measurements of the temperature dependence of the order parameter by using a thermal-bath temperature rescaling, which has been demonstrated to work in ferromagnets. Our model provides a critical temperature (Neel) of 1256 K from the ab initiovalues of the exchange interactions. Due to the presence of four easy directions, the domain wall structure of 90 degrees and 180 degrees are stable in Mn2Au. We find a narrow domain wall width of 23.13 +/- 0.085 nm at zero temperature in both cases as shown in Fig. 2. Applying thermal spin fluctuations yields a domain wall width that increases with temperature, up to 25 +/- 2 at 300K and to 53 +/- 18 nm at 800K, matching well with experimental results of Sapozhnik et al[5]. In the case of a 180 degree wall the two separate 90 degree walls are weakly bound and separate at high temperatures. Further analysis of the domain wall width could provide information about the scaling of the exchange stiffness in antiferromagnets, a pivotal parameter for the construction of hierarchical multiscale models in magnetism, which are missing for antiferromagnets. When an electrical current is applied to the system along the [110]direction a spin orbit torque (SOT) is generated by the flow of electrons through the system interacting with the Au atoms. These SOTs cause the domain wall to move. For a Neel anti-damping SOT field of 0.05 T we find a domain wall velocity of 250 m/s at 0K, which is different to the usually fast dynamics displayed by anti-ferromagnetic materials.This is because the SOT applied is precessional in nature and both sublattices have precessional motion in the same direction. Applying a field-like SOT yields the usual fast motion of the domain walls. The atomistic spin model [6] used here to compute thermal properties of Mn2Au provide new insights on the temperature dependent magnetic properties – magnetic order and anisotropies and domain wall widths in Mn2Au. The understanding of the macroscopic parameters is necessary for the developments of multiscale models of Mn2Au that could greatly help to the development of novel antiferromagnetic spintronic devices.

**References:**

[1] V. Barthem et al Nature comminucations 4 (2013) [2] L Rózsa, S Selzer, T Birk, U Atxitia, U Nowak Physical Review B 100, 064422 (2019) [3] A. B. Shick et al. Phys. Rev. B 81, 212409 (2010) [4] R.F.L. Evans et al. Phys. Rev. B 102, 020412(2020) [5] A. A. Sapozhnik, Phys. Rev. B 97, 134429 (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c38d47266d2475804a86e9579cb756b5.jpg)
Magnetic unit cell of Mn2Au. The Mn atoms (atoms with arrows) form two anti-parallel sublattices aligned along the [110] and [-1-10] crystal directions. The gold coloured atoms represent Gold.

![](https://assets.underline.io/uploads/markdown_image/1/image/22f100ca89efd90f63a1ac9868813590.jpg)
90 degree and 180 degree domain wall profiles.

 Temperature-dependent magnetic properties and domain wall width of the antiferromagnet Mn2Au

Skyrmions typically appear in systems with perpendicular magnetic anisotropy (PMA) and Dzyaloshinskii-Moriya interactions (DMI). Bloch skyrmion states can be stabilized in magnetic dots with no need of DMI [1]. However, PMA is typically considered to be necessary. Nanoparticle geometry is an additional important factor for the skyrmion stabilization due to the influence of magnetostatic energy. Therefore, the sample shape (e.g. spherical) can result in stabilization of the skyrmions [2]. Recently, stabilization of the half-hedgehog skyrmions in soft magnetic (Permalloy) hemispherical dots by the MFM tip was demonstrated [3]. Here we conduct micromagnetic simulations in 3D soft magnetic dots with planar (cylindrical) or curved (spherical cap) geometry with neither PMA nor DMI. The geometries of both magnetic elements correspond to the base radius 60-120 nm and the aspect ratio thickness/ radius ranged from 0.4 to 2 and magnetic parameters of permalloy. In a wide range of the dot sizes, we observed the co-existence of multiple magnetic states. Particularly, in the spherical caps we stabilized for the same geometry, the in-plane, the out-of-plane (flower-like), the 3D vortex-like (Bloch skyrmion), the 3D Néel skyrmion –like (see Fig.1) and the Bloch point-like states. The Bloch skyrmion is characterized by a vortex configuration in the basal plane while the Neel-like skyrmon – by the radial vortex configuration and is similar to the half hedgehog structure. The existence of magnetic skyrmions without DMI or PMA in cylindrical nanodots was demonstrated by analytical calculations using the Belavin-Polyakov ansatz [4]. Both Néel and Bloch (vortex-like) skyrmions structures have been proven to be metastable or stable states in some range of dot sizes. The Néel skyrmion always has larger energy than the Bloch skyrmion. Direct micromagnetic simulations confirm the co-existence of both types of skyrmions, although in a smaller range of geometrical parameters than predicted by analytical calculations. In spherical caps, the curved geometry provides an additional factor for the Néel skyrmion stabilization which was obtained for a wide range of geometrical parameters. Typically, the Néel skyrmion has the largest energy of all structures, however its energy decreases as a function of the cap basal diameter and aspect ratio. Consequently, for some geometrical parameters we found a Néel skyrmion having smaller energy than that of the Bloch skyrmion. The spherical geometry also leads to a coupling between the skyrmion polarization and radial magnetization components, so that only the skyrmions with positive Polarity (P) and outward chirality (Q) or negative polarity and inward chirality are possible, see Fig. 1 Finally, we present the state diagram for skyrmions in soft magnetic caps and nanopillars. We have also calculated 2D topological charges and the gyrovector values, both depending on the geometrical parameters.

**References:**

[1] K. Guslienko, K. Y. IEEE Magn. Lett. Vol. 6, p,4000104 (2015) [2] V.P. Kravchuk et al Phys. Rev. B, Vol. 85, p.144433 (2012). [3] E.Berganza et al, Nanoscale, Vol.12, p.18646 (2020) [4] A. A. Belavin and A. M. Polyakov, Pis’ma Zh. Eksp. Teor. Fiz. Vol. 22, p. 503 (1975) .

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c0723cd91b5db4d72e227b49f23c8980.jpg)
Figure.1 (a-b) Chiral (Neel-like) skyrmion magnetic configurations in permalloy caps with diameter 60 nm and height 30 nm colored by the axial-component mz. Left and right figures are cross sections in the XZ plane and in the XY basis plane respectively. P and Q stand for the polarity and the chirality of the skyrmion respectively.

 Skyrmions in 3D soft magnetic nanodots with no Dzyaloshinskii-Moriya interactions

EOC: Skyrmions I

technical paper

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Presentations

Skyrmion Caloritronics: a promising Strategy to manipulate Skyrmions

Nucleation and Motion of Domain Walls in Cylindrical Nanowires

Oersted-field- and current- induced Bloch Point domain wall dynamics in cylindrical Ni nanowires

Micromagnetic Study of Stable Skyrmions in Dot-Patterned Graphene-Based Magnetic Trilayers

Magnon-phonon damping calculations in the spin-lattice dynamics model

Temperature-dependent magnetic properties and domain wall width of the antiferromagnet Mn2Au

Skyrmions in 3D soft magnetic nanodots with no Dzyaloshinskii-Moriya interactions

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