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Current-driven domain wall dynamics in circular cross-section nanowires is a key phenomenon to be studied with a view to designing three-dimensional magnetic memory devices. The information stored in this type of device would be encoded by magnetic domains separated by magnetic domain walls. Two types of domain walls exist in magnetically soft cylindrical nanowires: the transverse-vortex wall (TVW) and the Bloch-point wall (BPW). The latter has a distinct topology and exhibits azimuthal curling of magnetic moments (also called circulation) around a Bloch point – a micromagnetic singularity. This micromagnetic object is expected to prevent the usual Walker breakdown, and thus enable high domain wall speed. The Walker breakdown is expected to not occur in the case of the Bloch-point wall, as it would require a too large energy of dipolar origin with a head-on magnetization configuration along all three directions.&lt;br/&gt;We showed recently [1] that the previously overlooked Oersted field associated with an electric current is a key in experiments to stabilize the BPW and reach speed above 600m&lt;i&gt;/&lt;/i&gt;s with spin-transfer. Here we use micromagnetic simulations, analytical modeling and topological arguments to understand in detail and quantitatively the underlying phenomena, in particular the switching from negative to positive circulation of the BPW, with respect to the applied current [2]. The switching of the azimuthal circulation of the BPW to match that of the OEersted field occurs above a threshold current which we quantify as a function of geometry and material parameters. The key result is a 1&lt;i&gt;/R&lt;/i&gt;&lt;sup&gt;3&lt;/sup&gt; dependence of the switching threshold, with &lt;i&gt;R &lt;/i&gt;the wire radius, with the effect of the OErsted field becoming predominant for wire radius above typically 30nm. To the contrary, the speed of walls remains largely determined by spin-transfer alone, in a below-Walker regime. Thanks to a generalized micromagnetic scaling of lengths and densities of current, the present result can be applied to wires made of any magnetically soft material. We also highlight the complexity of BPW transformation involving the nucleation and/or annihilation of the topological objects at the surface (vortex/antivortex pairs) and in the volume (Bloch points) [Fig.1].

**References:**

[1] M. Schöbitz, A. de Riz, S. Martin &lt;i&gt;et al.&lt;/i&gt;, Phys. Rev. Lett. 123, 217201 (2019)&lt;br/&gt;[2] A. De Riz, J. Hurst, M. Schöbitz, C. Thirion, J. Bachmann, J. C. Toussaint, O. Fruchart, D. Gusakova, arXiv:2011.14869 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/639b0dd3d6b4627fa77ee50fcccf7ce4.jpg)
Figure 1: Snapshots illustrating the switching mechanism of BPW circulation in a permalloy wire under applied current. Top row: unrolled maps of radial magnetization, wire axis along z, periphery along φ. Bottom row: same surface maps of radial magnetization seen from inside of the wire as a 3D view. A, V and B labels highlight surface vortex and antivortex as well as Bloch point.



INTERMAG 2021

 Mechanism of current-assisted Bloch-point wall stabilization for ultra fast dynamics

Current-driven domain wall dynamics in circular cross-section nanowires is a key phenomenon to be studied with a view to designing three-dimensional magnetic memory devices. The information stored in this type of device would be encoded by magnetic domains separated by magnetic domain walls. Two types of domain walls exist in magnetically soft cylindrical nanowires: the transverse-vortex wall (TVW) and the Bloch-point wall (BPW). The latter has a distinct topology and exhibits azimuthal curling of magnetic moments (also called circulation) around a Bloch point – a micromagnetic singularity. This micromagnetic object is expected to prevent the usual Walker breakdown, and thus enable high domain wall speed. The Walker breakdown is expected to not occur in the case of the Bloch-point wall, as it would require a too large energy of dipolar origin with a head-on magnetization configuration along all three directions. We showed recently [1] that the previously overlooked Oersted field associated with an electric current is a key in experiments to stabilize the BPW and reach speed above 600m/s with spin-transfer. Here we use micromagnetic simulations, analytical modeling and topological arguments to understand in detail and quantitatively the underlying phenomena, in particular the switching from negative to positive circulation of the BPW, with respect to the applied current [2]. The switching of the azimuthal circulation of the BPW to match that of the OEersted field occurs above a threshold current which we quantify as a function of geometry and material parameters. The key result is a 1/R3 dependence of the switching threshold, with R the wire radius, with the effect of the OErsted field becoming predominant for wire radius above typically 30nm. To the contrary, the speed of walls remains largely determined by spin-transfer alone, in a below-Walker regime. Thanks to a generalized micromagnetic scaling of lengths and densities of current, the present result can be applied to wires made of any magnetically soft material. We also highlight the complexity of BPW transformation involving the nucleation and/or annihilation of the topological objects at the surface (vortex/antivortex pairs) and in the volume (Bloch points) [Fig.1].

**References:**

[1] M. Schöbitz, A. de Riz, S. Martin et al., Phys. Rev. Lett. 123, 217201 (2019) [2] A. De Riz, J. Hurst, M. Schöbitz, C. Thirion, J. Bachmann, J. C. Toussaint, O. Fruchart, D. Gusakova, arXiv:2011.14869 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/639b0dd3d6b4627fa77ee50fcccf7ce4.jpg)
Figure 1: Snapshots illustrating the switching mechanism of BPW circulation in a permalloy wire under applied current. Top row: unrolled maps of radial magnetization, wire axis along z, periphery along φ. Bottom row: same surface maps of radial magnetization seen from inside of the wire as a 3D view. A, V and B labels highlight surface vortex and antivortex as well as Bloch point.

technical paper

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Two-dimensional magnetic (2DM) van der Waals (vdW) materials have attracted significant attention over the last few years, after the first experimental demonstration of ferromagnetism persisting down to the monolayer limit [1]. Since then, new vdW 2DM materials emerge frequently, covering a broad spectrum of diverse properties also found in other 3D counterpart systems – ferro/antiferromagnetism (FM/AFM), metallic/semiconducting behaviour and many non-trivial electronic phases. Out of this expanding pool of 2DM, one category that stands out is FMs with strong out-of-plane anisotropy, which present superior qualities desirable for magnetic random-access memory and other spintronic applications. Fe3GeTe2 (FGT) is a prominent member of this family that features a lot of interesting properties, such as a relatively high, thickness-dependent Tc (150 K – 220 K) that can be tuned to room temperature by doping, high spin-orbit coupling originating from a strongly correlated electronic system, application in spin valves, topological phases and Kondo behaviour, amongst others [2]. FGT is a quasi-2D itinerant ferromagnetic metal, and its structure consists of Fe3Ge layers sandwiched by single-atom layers of Te. Among the FGT layers, Te atoms are bonded together by vdW forces, allowing for exfoliation of the material in ultra-thin flakes with the scotch-tape exfoliation technique. Despite the considerable efforts that have already taken place in order to determine the underlying micromagnetic structure and behaviour of FGT, its highly anisotropic and non-trivial nature has not yet allowed for a consensus in the relevant scientific community. Our work therefore aims to shed light for the case of quasi-bulk FGT, by combining magnetic susceptibility measurements and Raman spectroscopy [3]. Our magnetometry study picks up the main point of controversy around FGT, which is a kink found below the Curie temperature, roughly at 150 K. This possible magnetic transition has been shown to represent a FM to AFM interlayer coupling between the vdW bonded layers [5], however it has been argued that this can be due to different reasons, such as domain wall pinning effects. A temperature dependent Raman study can provide further insight by indirectly detecting a phase change – electronic reconfiguration leads to a qualitative change in the Raman signal, and that is what our results suggest (Fig. 1). The separation between the Raman peaks is a possible sign of this magnetic transition, a signature that often accompanies other 2D material related properties and transitions however rarely related to magnetic properties. Additionally, it has been shown that variations in the sample history, i.e. temperature and applied magnetic field (as well as number of atomic layers), result in different magnetic phases, such as labyrinth and stripe domains, bubble-like lattice and possibly skyrmion phases [4]. Our systematic magnetometry study provides insight regarding the formation of these different magnetic states, following three different magnetization protocols, namely: zero-field cooling (ZFC), field cooling (FC) and field warming (FW) methods (Fig. 2). The magnetometry results are correlated to Raman responses, examined up to 2nd order. Particular interest arises from a prominent feature (a 2nd kink below 150 K) that has not been observed before – suggesting the existence of possibly another magnetic phase. Our results advance the understanding of the underlying mechanisms of Fe3GeTe2, an important 2DM that is often incorporated in spintronic vdW heterostructures and devices.

**References:**

C. Gong, L. Li and X. Zhang, Nature, Vol 546, p. 265-269 (2017) Y. Deng, Y. Yu and Y. Zhang, Nature, Vol 563, p. 94-99 (2018) L. Du, J. Tang and Z. Sun, Adv. Funct. Mat., Vol 29 (2019) H. Wang, C. Wang and J. Li, APL, Vol 116 (2020) J. Yi, H. Zhuang and Z. Gai, 2D Materials, Vol 4 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c55428df00e41337ff9c7e60ce6809c6.jpg)
Fig. 1: Temperature-dependent Raman response: a) 2nd order Raman susceptibility. b) Temperature dependent separation of the peaks @150 and @120 cm-1

![](https://assets.underline.io/uploads/markdown_image/1/image/5131d011fc9a59caf031435ea40dcbe4.jpg)
Fig. 2: Magnetization measurements of quasi-bulk FGT: FC, FW and ZFC protocols obtained under different applied magnetic fields. a) Happl. = 10 mT, b) Happl. = 50 mT, c) Happl. = 100 mT, d) a magnetometry feature (kink) position in temperature under varied applied H field (arrows in a, b, c)

 Correlation of Magnetic Properties of van der Waals Itinerant Magnet Fe3GeTe2 using Magnetometry and Raman Spectroscopy

Magnetic domain walls (DWs) in a Weyl semimetal can induce localized charges and be treated as local axial gauge fields.[1] Dynamics and textures of magnetic DWs may largely alter the electronic behaviors in a Weyl semimetal system.[2] In this work, we imaged the spontaneous magnetization and magnetic susceptibility of a ferromagnetic (FM) Weyl semimetal candidate CeAlSi using scanning SQUID microscopy [3], [4]. We observed the ferromagnetic DWs lined-up with [100] direction (or other degenerate directions) and the coexistence of stable and metastable domain phases, which arise likely due to the magnetoelastic and magnetostriction effects and are highly tunable with small strains. The temperature evolution of the FM domains shows that stable domains shrink while metastable domains grow and eventually collapse to narrow domain-wall-like features as temperature rises towards Tc ~ 8.3K.(fig.1) We applied in-plane external field as the CeAlSi sample was cooled down to ~8K, showing that the pattern of FM domains is strongly correlated with both the amplitude and the orientation of the external field. The area of stable domains increases with field (fig.2) and reaches maximum when the field is parallel to the easy axis of the CeAlSi crystal. Our results suggest these heterogeneous phases can be manipulated and therefore provide a practical way to study the interplay between magnetism and electronic properties in Weyl system.

**References:**

[1] Y. Araki, A. Yoshida, and K. Nomura, Phys Rev B, vol. 98, no. 4, p. 045302, Jul. 2018 [2] J. D. Hannukainen, Y. Ferreiros, A. Cortijo, and J. H. Bardarson, Phys. Rev. B, vol. 102, no. 24, p. 241401, Dec. 2020 [3] H.-Y. Yang et al., ArXiv200607943 Cond-Mat, Jun. 2020 [4] B. Xu, J. Franklin, H.-Y. Yang, F. Tafti, and I. Sochnikov, ArXiv201106168 Cond-Mat, Nov. 2020

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/03e8ecc3bef65d96e91883ee81c1252f.jpg)
Figure 1. The temperature evolution of the magnetic domains in the CeAlSi crystal. Upper row: dc magnetizations in flux units. Middle and lower rows are the in-phase component and the out-of-phase component of the ac susceptibility. Dark green and dark marine areas in the in-phase and out-of-phase components, respectively, are stable in-plane domains. Light green and light marine areas in the in-phase and out-of-phase components, respectively, are metastable domains. The stable domains shrink, while the metastable grow towards the transition, and eventually collapse to narrow features reminiscent of the domain boundaries roughly lined up with the crystallographic <1 0 0> direction. The dc magnetization contrast reduces towards the transition, while the amplitude of the in-phase and out-of-phase signals grow. The metastable regions show enhanced signals. Immediately above the transition temperature, all signals collapse to small values, and no domains persist.

![](https://assets.underline.io/uploads/markdown_image/1/image/954b2477fcd8048e89881b7176b39e5d.jpg)
Figure 2. The growth of domains with magnetic field in a CeAlSi crystal that was as-cooled through the transition at ~8K. The stable and metastable domains in the in-phase and out-of-phase components are represented the same way as in fig 1.

 Direct observation of tunable magnetic domains in noncentrosymmetric ferromagnetic Weyl semimetal CeAlSi

The surface states of three-dimensional topological insulators (TIs) exhibit a large spin-orbit coupling leading to spin-momentum locking, the electron spin being locked perpendicularly to the electron momentum. This property is promising for spintronic applications, as the Rashba Edelstein effect can create, from a charge current at the conducting surface of a TI, a spin accumulation. Reciprocally, the injection of a spin current creates a net charge current by inverse Edelstein effect. Recently, we have demonstrated a large spin-to-charge conversion efficiency in strained mercury telluride (HgTe) TI at room temperature [1], with conversion rates one order of magnitude higher compared to conventional materials. Achieving the mirror charge-to-spin conversion would be of great interest for memory and logic applications, with two effects in particular that could be exploited. First, the spin-orbit torques (SOT), in which the spin accumulation induces a torque on the magnetization [2], which can be used as a switching mechanism in a magnetoresistive random access memories (SOT-MRAM). Second, the unidirectional spin-hall magnetoresistance (USMR), in which depending on the relative directions between the spin accumulation and the magnetization, the system holds high or low resistance state [3]. While conventional SOT devices suffer from a three-terminal geometry, this USMR effect has potential to be used as a new reading mechanism for developing a novel type of SOT-MRAM with a two-terminal geometry and smaller footprint. Here we report unidirectional spin-Hall magnetoresistance in HgTe topological insulator-ferromagnet heterostructures up to room temperature. Results are shown from a 3 µm wide Hall bar device made of a NiFe ferromagnetic layer with in-plane magnetization on top of a HgTe layer with a thin HgCdTe insertion to protect the TI surface state (Figure 1). Harmonic measurements of the longitudinal and Hall resistance are performed to investigate both USMR and spin-orbit torque effects. A change in the nonlinear magnetoresistance upon switching the magnetization from +y to –y direction is demonstrated (Figure 1b), corresponding to the signature of the USMR driven by Eldelstein effect. Anomalous Nernst and spin Seebeck effects are measured from the nonlinear magnetoresistance in the x direction, showing negligible contributions. The results show USMR efficiency of 2.6 ppm A cm-1 at 10K, three times higher than in ferromagnet-normal bilayer and on the same order of low temperature Bismuth Selenide TI. USMR signal is seen to increase linearly with the current density, confirming the unidirectional nature of USMR in HgTe (Figure 2). Experimental results at room temperature further show comparable USMR efficiency, bringing HgTe compatible for room temperature applications. Ongoing material engineering is discussed to improve further the USMR efficiency on HgTe system, the high efficiency observed in this first demonstration constituting an important step towards SOT-MRAM with two-terminal geometry at room temperature.

**References:**

[1] P. Noel et al., Phys. Rev. Lett. 120, 167201 (2018). [2] I. M. Miron et al., Nature 476, 189–193 (2011). [3] C. O. Avci et al., Nature Physics 11, 570–575 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/f2e1172201ca3e61e64fb3ee032e8b46.jpg)
Sample layout and nonlinear magnetoresistance (a) Structure of the sample and measurement geometry. (b) Second harmonic longitudinal resistance as a function of the magnetic field along the y direction. (c) Second harmonic Hall resistance as a function of the magnetic field along the x direction. All measurement were performed with a current density of j2D = 3.3 A cm-1 at a temperature of 10K.

![](https://assets.underline.io/uploads/markdown_image/1/image/0aec57feaef1307ba3166b3fc3a3daa8.jpg)
XY angular dependence and extraction of USMR efficiency. (a) Angular dependence of the second harmonic longitudinal resistance when the magnetic field is rotated in the xy plane. (b) Current dependence of the USMR resistance at 10K. Inset shows USMR as a function of the temperature.

 ferromagnet heterostructures

Topological insulators (TI) are gaining attention from a technological point of view due to their foreseen highly efficient capability to control adjacent magnetic media through spin to charge (S2C) conversion [1]. However, in order to make a decisive step toward technology transfer, it is necessary to develop fabrication methods suitable to guarantee their large-scale production. We developed a Metal Organic Chemical Vapour Deposition technique (MOCVD) process to grow epitaxial-quality antimony telluride (Sb2Te3) topological insulator (TI) on 4” Si(111) substrates [2]. When compared to granular-Sb2Te3 grown on SiO2 [3], the epitaxial-Sb2Te3 on top of Si(111), shows improved magnetoconductance (MC) performances, especially upon proper annealing, as described in [2]. Figure 1 displays the MC signal emerging from the epitaxial (annealed) Sb2Te3 on Si(111), at 5.5 K. Clearly, the MC displays a non-parabolic shape, which we attribute to weak antilocalization. In the framework of the Hikami-Larkin-Nagaoka (HLN) model [3], we extract the parameters α (being connected to the number of conducting channels), and coherence length (lφ), as depicted in the inset of Figure 1. The values of α and lφ for the annealed sample are 0.3 and 55 nm, respectively, thus indicating a 2D-type of conduction due to the existence of topologically-protected surface states (TSS). The as-deposited sample shows a much smaller MC signal, and the extracted α and lφ are comparable with those previously obtained for granular-Sb2Te3 on SiO2 [3]. Following the MC studies, both the as-deposited and epitaxial Sb2Te3 samples have been covered with evaporated 57Fe(5nm) and subsequently in situ capped with Au(5nm) to prevent Fe oxidation. To investigate the potential role of the Sb2Te3 preparation on S2C efficiency, terahertz (THz) emission spectroscopy has been conducted on both samples. The Fe layer is saturated by an external in-plane magnetic field. The multilayer system is excited using femtosecond laser pulses, which launch a longitudinal spin-polarized electron current. Through S2C conversion, the spin current is converted into a transverse charge current concomitantly ejecting a THz pulse which we record in the time domain via electro-optic (EO) detection (see Figure 2) [4,5]. Our results show a considerable impact of Sb2Te3 annealing on the THz signal amplitude suggesting a S2C conversion contribution from the Fe/Sb2Te3 interface. Conversion-electron Mössbauer spectroscopy (CEMS) has been conducted on the same samples studied by THz emission spectroscopy. The Fe/as-deposited-Sb2Te3 spectrum resembles those previously reported for the Fe/granular-Sb2Te3 system [6], and CEMS reports that a large fraction of Fe atoms (≤60%) coordinates paramagnetically with Te in a “FeTe” type of bonding. Surprisingly, this fraction collapses to ≤10% at the interface with annealed (epitaxial) Sb2Te3, with a CEM-spectrum mostly dominated by pure Fe, reflecting a remarkable improvement of the chemical-structural-magnetic quality of the Fe/annealed- Sb2Te3 interface. We propose the suppression of the “FeTe”-type of bonding at the interface as being the main source for the enhancement of the S2C conversion as observed by THz emission spectroscopy. To conclude, the annealed epitaxial Sb2Te3 develops TSS-connected conduction as shown by MC (Figure 1), which turns out in an enhancement of the THz signal in Fe/Sb2Te3 heterostructures as due to S2C conversion (Figure 2). By means of CEMS, the evolution of the S2C efficiency with the Sb2Te3 annealing is attributed to a remarkable suppression of intermixing occurring at the Fe/Sb2Te3 interface, when Fe is deposited in direct contact on top of annealed and epitaxial Sb2Te3.

**References:**

[1] A. Soumyanarayanan, N. Reyren, A. Fert et al., Nature 539, 509 (2016) [2] M. Rimoldi, R. Cecchini, C. Wiemer et al., RSC Advances 10(34), 19936 (2020) [3] R. Cecchini, R. Mantovan, C. Wiemer et al., Phys. Status Solidi RRL 12, 1800155 (2018) [4] T. Seifert, S. Jaiswal, U. Martens et al., Nat. Photonics, 10, 483 (2016) [5] T. Seifert, S. Jaiswal, J. Barker et al., Nat Commun 9, 2899 (2018). [6] E. Longo, C. Wiemer, R. Cecchini et al., Adv. Mater. Interfaces 7, 2000 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6fdf4a13fdf66c175dd0d951784e14b2.jpg)
Fig.1. Magnetoconductance signal of the epitaxial (annealed) Sb2Te3 on Si(111), at 5.5 K. The inset shows the fit in the framework of the Hikami-Larkin-Nagaoka model, from which the α and lφ parameters are extracted.

![](https://assets.underline.io/uploads/markdown_image/1/image/a97ebfc4c249fdb79705013e0f677e76.jpg)
Fig.2. THz emission signals of the Au/Fe/as-deposited-Sb2Te3 (black curve) and Au/Fe/epitaxial-Sb2Te3 (red curve) heterostructures.

 Evolution of the THz emission signal as a function of the interface quality between Fe and large-area topological insulator Sb2Te3 on Si

Atomically thin, magnetic materials have gained a lot of attention since 2017, when the first 2D ferromagnet was reported [1]. This breakthrough has triggered research on 2D magnetic materials [2]. They are not only important from a fundamental point of view -to understand the theory of magnetism in reduced dimensions-, but also for technological applications. However, probing the magnetic order of the 2D systems by conventional magnetic experimental setups is very challenging. On the other hand, it is well known, that even in the single layer limit, semiconducting two-dimensional materials strongly absorb light. Therefore, optical spectroscopy is a good method for their characterization. Here we present the results of the comprehensive theoretical investigation of the binding energy of the 2D magnetic crystal MnPS3 based on the density functional theory and versatile formalism of the Bethe-Salpeter equation. The MnPS3 is one important example from the large family of transition metal phosphorus trisulfide (MPS3) [3]. We demonstrate that the large binding energy of the exciton equal to more than 1 eV and obtained for monolayer of MnPS3 [4], is two times greater than in the case of transition metal dichalcogenides. We also highlight the role of the magnetic orderings in electronic and optical properties. In particular, the strong impact of the magnetic ordering on the binding energy of excitons in the monolayer limit is demonstrated, as well as its substantial influence on effective mass of the carriers. In addition, we have shown that the magnetic state of the monolayer samples is sensitive to the polarization of light. Aforementioned results indicate the possibility of indirect probing of magnetic properties via optical or transport measurements. The study was accomplished thanks to the funds allotted by the National Science Centre, Poland within the framework of the research project ‘SONATA12’ no. UMO-2016/23/D/ST3/03446. Financial support from theDeutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) under Project-ID 314695032 (SFB1277) is acknowledged by P.E.F.J. and J.F. and un-der Project-ID 317551441 by J.K. Access to computing facilities of TU Dresden ZIH for the project ”TransPheMat”, PL-Grid Polish Infrastructure for Supporting Computational Science in the European Research Space, and of the Interdisciplinary Center of Modeling (ICM), University of Warsaw are gratefully acknowledged.

**References:**

[1] B. Huan et al., Nature, Vol. 546, p. 270-273 (2017) [2] M. Gibertini et al., Nature Nanotechnology, Vol. 14, p. 408 (2019). [3] R. Brec, Solid State Ionics vol. 22, p. 3-30 (1986). [4] M. Birowska, Paulo E. Faria Junior, J. Fabian, J. Kunstmann, http://arxiv.org/abs/2009.05030.

**Images:**

 2D magnetic crystals: Can we probe magnetic order? Theoretical study of vdW layered MnPS3 crystals exhibiting large exciton binding energy.

Curvilinear magnetism is a rapidly developing research area of modern magnetism which deals with curved magnetic wires and curved magnetic films [1]. By extending planar geometries into three dimensional space, there appear new, sometimes, unexpected physical effects, stemming from mutual interplay between the geometrical properties of the underlying substrate and the magnetic order parameter. By tailoring curvature and topology of the conventional magnetic materials there appears a possibility to control material response leading to modification or even launching new functionalities [2, 3]. This is granted by complementary expertise and advances of fundamental researched, materials sciences and technologies. This talk focuses on the peculiarities emerging from geometrically curved magnetic objects, including 3D bent and twisted curved wires and films. The curvilinear geometry manifests itself in emergent anisotropy and chiral interactions. These curvature-induced interactions can be local stemming from the exchange energy [4, 5], but also can be nonlocal due to magnetostatics [6]. As a consequence, family of novel curvature-driven effects emerge, which include magnetochiral effects and topological patterning, resulting in theoretically predicted unlimited domain wall velocities, chirality symmetry breaking etc. Current and future challenges of the curvilinear magnetism will be discussed.

**References:**

[1] E. Y. Vedmedenko, R. K. Kawakami, D. D. Sheka et al, JPhysD, vol. 53 , p.453001 (2020) [2] R. Streubel, P. Fischer, F. Kronast et al, JphysD, vol. 49, p.363001 (2016) [3] P. Fischer P, D. Sanz-Hernández, R. Streubel et al, APL Materials, vol. 8, p. 010701 (2020) [4] Y. Gaididei, V. P. Kravchuk and D. D, Sheka, PRL, vol. 112, p. 257203 (2014) [5] D. D. Sheka, V. P. Kravchuk and Y. Gaididei, JPhysA, vol. 48, p.125202 (2015) [6] D. D. Sheka, O. V. Pylypovskyi, P. Landeros et ak, Communications Physics, vol. 3, p. 128 (2020)

**Images:**

 Curvilinear Magnetism INVITED

The behaviour of any physical system is governed by the order parameter, determined by the geometry of the physical space of the object, namely their dimensionality and curvature. Usually, the effects of curvature are described using local interactions only, e.g. local spin-orbit- or curvature-induced Rashba and Dzyaloshinskii-Moriya interactions (DMI). In the specific case of ferromagnetism, until recently, there was no analytical framework, which was treating curvature effects stemming from local [1] and non-local [2] interactions on the same footing. The lack of a proper theoretical foundation impedes the description of essential micromagnetic textures like magnetic domains, skyrmion-bubbles and vortices. Here, we present a micromagnetic theory of curvilinear ferromagnetic shells, which allows to describe the geometry-driven effects stemming from exchange and magnetostatics within the same framework [3]. A general description of magnetic curvilinear shells can be done using tangential derivatives of the unit magnetization vector. Tangential derivatives are represented by the covariant derivatives of in-surface components and the regular derivative of the normal magnetization component, normalized by the square root of the corresponding metric tensor coefficient. This allows to separate the explicit effects of curvature and spurious effects of the reference frame. The shape of a given thin shell can be determined by two principal curvatures k1 and k2, which are functions of coordinate. The respective classification of curvilinear surfaces operates with (i) developable surfaces, where one of the principal curvatures equals to zero; (ii) minimal ones, where the mean curvature k1 + k2 = 0; and (iii) the general case. The local geometry-driven energy contributions are represented by the DMI and anisotropy, whose coefficients are determined by powers of the principal curvatures. This allows to cancel the influence of one of the DMI terms for the developable surfaces for any magnetic texture. The magnetostatic interaction is a source of new chiral effects, which are essentially non-local in contrast to the conventional DMI. The physical origin is the non-zero mean curvature of a shell and the non-equivalence between the top and bottom surfaces of the shell. We demonstrate that the analysis of non-local effects in curvilinear thin shells can become more straightforward when introducing three magnetostatic charges. In this respect, in contrast to the classical approach by Brown [4], we split a conventional volume magnetostatic charge into two terms: (i) magnetostatic charge, governed by the tangent to the sample's surface, and (ii) geometrical charge, given by the normal component of magnetization and the mean curvature. In addition to the shape anisotropy (local effect), there appear four additional non-local terms, determined by the surface curvature. Three of them are zero for any magnetic texture in shells with the geometry of minimal surfaces. The fourth term is determined by the non-equivalence of the top and bottom surfaces of the shell and becomes zero only for the special symmetries of magnetic textures. The discovered non-local magnetochiral effects introduce handedness in an intrinsically achiral material and enables the design of magneto-electric and ferro-toroidic responses. This will stimulate to rethink the origin of chiral effects in different systems, e.g. in fundamentally appealing and technologically relevant skyrmionic systems, and further theoretical investigations in the field of curvilinear magnetism as well as experimental validation of these theoretical predictions. These developments will pave the way towards new device ideas relying on curvature effects in magnetic nanostructures. The impact of effects predicted in this work goes well beyond the magnetism community. Our description of the vector field behaviour can be applied to different emergent field of studies of curvature effects. The prospective applications include curved superconductors [5], twisted graphene bilayers [6], flexible ferroelectrics [7], curved liquid crystals [8].

**References:**

[1] Yu. Gaididei, V. P. Kravchuk, D. D. Sheka, Phys. Rev. Lett., 112, 257203 (2014); D. D. Sheka, V. P. Kravchuk, Yu. Gaididei, J. Phys. A: Math. Theor., 48, 125202 (2015); O. V. Pylypovskyi, V. P. Kravchuk, D. D. Sheka et al, Phys. Rev. Lett., 114, 197204 (2015); V. P. Kravchuk, D. D. Sheka, A. Kakay et al, Phys. Rev. Lett., 120, 067201 (2018) [2] P. Landeros, A. S. Nunez, J. Appl. Phys. Vol. 108, p. 033917 (2010); J. A. Otalora, M. Yan, H. Schultheiss et al, Phys. Rev. Lett., 117, 227203 (2016); J. A. Otalora, M. Yan, H. Schultheiss et al, Phys. Rev. B, 95, 184415 (2017) [3] D. D. Sheka, O. V. Pylypovskyi, P. Landeros et al., Comm. Phys. 3, 128 (2020) [4] W. F. Brown Jr. Micromagnetics (Wiley, New York, 1963) [5] V. Vitelly, A. M. Turner, Phys. Rev. Lett., 93, 215301 (2004) [6] W. Yan, W.-Y. He, Z.-D. Chu et al, Nat. Comm., 4, 2159 (2013) [7] M. Owczarek, K. A. Hujsak, D. P. Ferris et al, Nat. Comm., 7, 13108 (2016) [8] G. Napoli, L. Vergori, Phys. Rev. Lett., 108, 207803 (2012)

**Images:**

 Micromagnetic description of symmetry-breaking effects in curvilinear ferromagnetic shells

We investigate the effects of the competition between exchange (J) and dipolar (D) interactions on the magnetization reversal mechanisms of ferromagnetic nanotubes. Using atomistic Monte Carlo simulations for a model with Heisenberg spins on a cylindrical surface, we study how the competition between both interactions influences the low temperature magnetic order of the nanotubes as well as the thermal-driven transitions involved, showing that helical (H) states become stable for a range of intermediate values of γ=D/J and establishing phase diagrams for the stability of FM, H and vortex states as a function of the radius R and length L of the nanotube, see Fig. 1 [1]. We also study the reversal behavior under application of a magnetic field, showing that for which helical (H) states are energetically favorable at zero applied field, we show that the hysteresis loops can occur in four different classes that are combinations of two reversal modes with well-differentiated coercivities with probabilities that depend on R and L. This variety in the reversal modes is found to be linked to the metastability of the H states during the reversal that induces different paths followed along the energy landscape as the field is changed. We further demonstrate that reversal by either of the two modes can be induced by tailoring the nanotube initial state so circular states with equal or contrary chirality are formed at the ends (Fig. 2), thus achieving low or high coercive fields at will without changing γ. Finally, the results of additional micromagnetic simulations performed on tubes with a similar aspect ratio show that dual switching modes and its tailoring can also be observed in tubes of microscopic dimensions [2]. Acknowledgements: Work supported by Spanish MINECO (PGC2018-097789-B-I00), Catalan DURSI (2017SGR0598) and EU FEDER funds (Una manera de hacer Europa), Colciencias Beca de Doctorados Nacionales, ConvocatoriaNo. 727, Project No. CODIUdeA 2017-16253, also CSUC for supercomputer facilities.

**References:**

[1] H. D. Salinas, J. Restrepo, Òscar Iglesias, Sci. Rep. 8, 10275 (2018). [2] H. D. Salinas, J. Restrepo, Òscar Iglesias, Phys. Rev. B 101, 054419 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4c95db4a71866b9df94625a04b88cb42.jpg)
Fig. 1: Typical phase diagram.

![](https://assets.underline.io/uploads/markdown_image/1/image/7cb117a7f970c0aa3b067c1bece94d70.jpg)
Fig. 2: Snapshots of spin configurations during a reversal mode.

 Magnetic phase diagrams and helicity control of reversal modes in ferromagnetic nanotubes

Recently, it was demonstrated that confined helimagnetic nanostructures, composed of grains of different chirality (and consequently, a different sign of Dzyaloshinskii-Moriya energy constant), can host stable Bloch points [1]. In addition, it was shown that Bloch points undergo hysteretic behaviour and that their type can be changed using an external magnetic field. In this work, using Ubermag [2] micromagnetic simulations with OOMMF [3] as a computational backend, we explore if Bloch point magnetisation states can be manipulated using spin-polarized currents in nano-disk and nano-strip FeGe samples. We start by showing that a zero-field stable Bloch point emerges not only in thin-film disk samples [1] but also in an extensive range of planar geometries, including nanostrips. We then explore whether Bloch point's type can be switched using spin-polarized current pulses. We simulate a FeGe [4] nanodisk with 150 nm diameter, 20 nm thickness of the bottom layer with D&lt;0 and 10 nm thickness of the top layer with D&gt;0. We begin by relaxing a uniform +z magnetisation state, which results in a head-to-head Bloch point. To a relaxed head-to-head Bloch point, we apply a spin-polarised current pulse with current density J and duration T, polarised in the negative z-direction. We vary current density J and pulse duration T and observe whether the type of Bloch point changes. In Fig. 1 we show a diagram depicting J-T regions for which Bloch point type switching occurs from head-to-head to tail-to-tail. In the second part of our work, we simulate nanostrips with 450 nm length and 90 nm width. Similar to nanodisks previously simulated, the thickness of the bottom layer with negative Dzyaloshinskii-Moriya constant is 20 nm and the thickness of the top layer with positive DM constant is 10 nm. We apply current polarised in -y-direction and track both x and y positions of the Bloch point. In Fig. 2, we show how Bloch point moves in the x-direction for different current densities J. Our findings demonstrate that Bloch points can be manipulated using spin-polarized currents. Using current pulses polarised in -z-direction, we were able to switch the Bloch point's type, whereas, with currents polarised in the negative y-direction, we managed to move Bloch point in the x-direction. Our results, in addition to being of interest to fundamental physics, bring Bloch points one step closer to being potential candidates for future data storage and information processing devices.

**References:**

[1] M. Beg et al. Stable and manipulable Bloch point. Scientific Reports 5, 17137 (2015). [2] M. Beg et al. User interfaces for computational science: A domain-specific language for OOMMF embedded in Python. AIP Advances, 7, 056025 (2017). [3] M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, Interag. Rep. NISTIR 6376, Natl. Inst. Stand. Technol. Gaithersburg, MD, 1999. [4] M. Beg et al. Ground state search, hysteretic behaviour, and reversal mechanism of skyrmionic textures in confined helimagnetic nanostructures. Scientific Reports, 5, 17137 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/640e10a6e4df4a9d468372576f020ff1.jpg)
Switching diagram of a head-to-head Bloch point in a 150 nm FeGe diameter disk using current pulse polarised in negative z-direction, with current density J and pulse duration T.

![](https://assets.underline.io/uploads/markdown_image/1/image/55c492a4be70dde428d04bd6e85f4c49.jpg)
Bloch point's x-coordinate as a function of time for different current densities J polarised in the negative y-direction.

 Spin-Polarised Current Bloch Point Manipulation

Understanding magnetization dynamics in nanosized systems is a key condition not only to realize innovative new applications, but also to further expand our comprehension of the underlying fundamental physics. Only recently has it been possible to image magnetic objects such as domain walls (DWs) with high temporal and spatial resolution and versatile environments or excitations, using, for example time resolved x-ray magnetic circular dichroism with scanning transmission x-ray microscopy (XMCD STXM). Continuous developments of this technique have allowed for a shift of interest from simple systems such as flat nanostrips, that lend themselves more readily to magnetic imaging, to more intricate systems such as three dimensional nanostructures, with added complexity from the volume [1,2]. A textbook case for such an investigation is provided by cylindrical magnetic nanowires (NWs) featuring a novel type of DW, the Bloch-point wall (BPW) [3] which exhibits azimuthal curling of magnetic moments around a Bloch-point on the NW axis. The dynamics of these walls stand out compared to DWs in flat nanostrips due to fascinating theoretical predictions of fast, stable speeds and the controlled emission of spin waves [4]. Recent experiments in NWs indicate that the Œrsted field induced by nanosecond pulses of current plays a key role in stabilising walls exclusively of the BPW type and further switches the azimuthal circulation to be parallel to the field [5]. This allows controlling the wall structure and enables fast DW motion with speeds &gt; 600 m/s and an absence of Walker breakdown. The importance of the Œrsted field in NWs was further studied with simulations to determine a strong radial dependence, scaling as one-over radius cubed, and a complex mechanism of the BPW switching process, involving nucleation and annihilation of pairs of Bloch-points [6]. In the case of longitudinally magnetised domains, the peripheral magnetic moments also tend to align with the Œrsted field with the degree of tilt related to a competition between magnetic exchange and Zeeman energy. It is clear that the Œrsted field has a major influence on magnetization dynamics in NWs, yet few predictions have been confirmed experimentally due to the complexity of such experiments. Further, without time resolved measurements, no quantitative comparisons of eg. tilt angles in magnetization may be made with the simulations. Verifications of these effects are therefore strongly required in order to control the system and employ it in possible future applications. Here, we make use of time resolved XMCD STXM to image the magnetization dynamics in NWs subjected to nanosecond pulses of electrical current. Time resolved images were acquired with a temporal resolution of 200 ps, to image the effect of positive and negative polarity 3 ns duration voltage pulses of chosen amplitude applied to magnetically soft Co30Ni70 NWs with diameters 93, 97 and 101 nm. We first observed the tilting of magnetization towards the applied Œrsted field direction in longitudinally magnetized domains (Fig. 1a) and created a model based on the absorptivity of x-rays and a description of the magnetization in a NW cross section to accurately reproduce the observed magnetic contrast and extract quantitative information. We find that the degree of tilt on the surface of these NW samples is of the order of 18±8° per 1012 A/m2 applied (Fig. 1b), which is in agreement with theoretical predictions from ref. [6]. We then observe the expansion (Fig. 2b,d) and compression (Fig. 2a,c) of the BPW width due to the application of a parallel and antiparallel Œrsted field, respectively. This effect, previously predicted by simulations, occurs when the applied current density is insufficiently strong to induce the BPW circulation switching. The magnetic contrast of the phenomenon is again reproduced with our simple model giving quantitative information about this “breathing” of the BPW. We expect that this work lays the foundation for and provides an incentive to further studying complex magnetisation dynamics in NWs. With increased control of the NW materials, the predicted ultrafast DW motion and associated spin wave emissions [4] may be visualized using this technique. Further, the quantitative analysis provided herein shows the depth of information obtainable with time resolved XMCD STXM and that a direct comparison of the observed dynamics to ie. simulations and theory is possible. This type of direct collaboration will significantly grow the understanding of three dimensional magnetic nanosized systems and enable better control over them.

**References:**

[1] C. Donnelly et al., Time-resolved imaging of three-dimensional nanoscale magnetization dynamics. Nat. Nanotech. 15 p.356-360 (2020) [2] A. Fernández-Pacheco et al., Three-dimensional nanomagnetism. Nat. Comm. 8 p.15756 (2017) [3] S. Da Col et al., Observation of Bloch-point domain walls in cylindrical magnetic nanowires. Phys. Rev. B. 89 p.180405 (2014) [4] R. Wieser et al., Current-driven domain wall motion in cylindrical nanowires. Phys. Rev. B. 82, p.1-7 (2010) [5] M. Schöbitz et al. Fast Domain Wall Motion Governed by Topology and Œrsted Fields in Cylindrical Magnetic Nanowires. Phys. Rev. Lett. 123 p.217201 (2019) [6] A. De Riz et al. Mechanism of current-assisted Bloch-point wall stabilization for ultra fast dynamics. arxiv: 2011.14869v1

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/dad38862b68e88e07ac7345f1feba12e.jpg)
Figure 1: (a) Time resolved imaging of the tilting of magnetization in cylindrical NWs due to an applied Oersted field with white arrows indicating the direction of the Œrsted field. (b) Tilt angle calculated from the magnetic images for three different NW diameters and compared with theoretical predictions.

![](https://assets.underline.io/uploads/markdown_image/1/image/266b1bac51bb6fd30599e47d72aee2f6.jpg)
Figure 2: (a) Time resolved images of the compression (a) and expansion (b) of a Bloch-point DW in cylindrical NWs due to an applied Oersted field. Illustrations of the compressed (c) and expanded (d) DW from (a,b) with the Oersted field direction indicated.

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