Soft actuators are required to deform, fold, or unfold in order to interact with their surroundings.[1] One strategy to achieve the movement of mechanically soft systems is
the use of magnetic fields for untethered actuation. Flexible magnetic composites have been demonstrated as functional grippers, rollers, and walkers upon applied external magnetic fields.
[2] Being controlled with electromagnetic coils, magnetic actuators can profit from high-speed actuation to quickly respond to their environment.[3]
The development of an appropriate sensory tracking system for soft actuators is a research topic with open challenges. Light, conformal, and mechanically imperceptible sensing systems
are to be developed to be compatible with soft actuators. In this work, we present flexible magnetosensitive electronic skin which relies on thin-film magnetic field sensors to enable
onboard folding control of origami-like soft actuators. The flexible electronic skin consists of thin film giant magnetoresistive GMR and Hall effect sensors that are used to measure the
magnetization state of the actuator, the applied magnetic fields, and the completeness of the bending process. The resulting intelligent material is mechanically designed to fold even when
the flexible magnetic skin is attached to the soft actuator.[4]
The magnetic origami actuators are made of thin DiAPLEX foils, a shape memory polymer, with embedded NdFeB particles. Such composite can be used to achieve magnetically induced
motion using magnetic fields and a directed light source to increase locally the temperature. We experimentally found the best thickness (60 μm) and concentration (40 NdFeB wt%)
parameters to achieve magnetic folds.[4]
The integration of magnetic soft actuators with e-skins allowed the self-guided assembly of the origami foils. We showed two case studies showing that this approach is useful for
assembling boxes and boat-like origami shapes. This integration process was monitored, followed, and controlled by the output of the laminated sensors. [4] 

**References:** 

[1] A. Miriyev, K. Stack, H. Lipson, et al. Nat. Commun., Vol. 8, p.596 (2017) 
[2] S. Wu, W. Hu, Q. Ze, et al. Multifunct. Mater. Vol. 3, p.042003 (2020) 
[3] X. Wang, G. Mao, G. Jin, et al. Commun. Matter. Vol. 1, p.67 (2020) 
[4] M. Ha, G.S. Cañón Bermúdez, J.A.-C- Liu, et al. Adv. Mater. Vol. 33, p.2008751 (2021)

Magnetically Sensitive Electronic Skins for Supervised Folding of Origami Actuators

The interplay between geometry and topology of the order parameter is crucial properties in soft and condensed matter physics, including cell membranes [1], nematic crystals [2,3], superfluids [4], semiconductors [5], ferromagnets [6] and superconductors [7]. Until recently, in the case of magentism, the influence of the geometry on the magnetization vector fields was addressed primarily by the design of the sample boundaries, aiming to tailor anisotropy of the samples. With the development of novel fabrication techniques allowing to realize complex 3D architectures, not only boundary effects, but also local curvatures can be addressed rigorously for the case of ferromagnets and antiferromagnets. It is shown that curvature governs the appearance of geometry-induced chiral and anisotropic responses [6-8]. Here we provide experimental confirmations of the existence of local and non-local curvature-induced chiral interactions of the exchange and magnetostatic origin in conventional soft ferromagnetic materials. Namely, we will present the experimental validation of the appearance of exchange-driven Dzyaloshinskii-Moriya interaction interaction (DMI, local effect) for the case of conventional achiral yet geometrically curved magnetic materials [9,10]. This curvature induced DMI is predicted to stabilize skyrmions [11] and skyrmionium states [12]. Furthermore, we will address the impact of nonlocal magnetostatic interaction on the properties of curvilinear ferromagnets, which enables the stabilization of topological magnetic textures [13,14], realization of high-speed magnetic racetracks [15] and curvature-induced asymmetric spin-wave dispersions in nanotubes [16]. Furthermore, symmetry analysis demonstrates the possibility to generate a fundamentally new chiral symmetry breaking effect, which is essentially nonlocal [13]. Thus, geometric curvature of thin films and nanowires is envisioned as a toolbox to create artificial chiral nanostructures from achiral magnetic materials.

Local and Nonlocal Curvature induced Chiral Effects in Nanomagnetism

The development of functional printable materials enables the production of electronic components in unconventional materials and different form factors[1]. Here, we
show magnetically sensitive inks/pastes based on magnetoresistive powder that can be printed via stencil, dispenser, or screen printing. We employ bismuth microparticles[2] as well as
[Co/Cu], [Py/Cu],[3] and permalloy [4] flakes showing giant, anisotropic, or non-saturating large magnetoresistance, respectively.
We demonstrate that magnetic field sensors based on various types of magnetoresistive flakes can be printed onto rigid, flexible, and deformable substrates. We employed blockcopolymers
and elastic binders to enable mechanical resilience of these sensors: they can withstand bending down to 16 μm, 100% of stretching, and bending for hundreds of cycles
without losing functionality.[3]
Additionally, by automatizing the dispenser printing process of bismuth-based pastes, we demonstrate the production of fully printed magnetic field sensors. The use of a micro-optically
optimized high-power diode laser array provided a versatile approach for selective sintering of sensors over flexible foils in areas exceeding several square centimeters. In this work we
experimentally confirm that such sensors retain their non-saturating magnetoresistive performance (MR = 146%) in high field conditions, allowing operation above 5 T. [2]
Our printed magnetic field sensors can be used to create interfaces that are responsive to magnetic fields through remote human input. Being flexible, they can be laminated on the skin, or
stuck onto any object, from a desk to a house wall. We demonstrate the capabilities of printed magnetic field sensors to work as an interface to navigate through digital maps, as input
panels for smart home applications, and as interactive wallpapers. 

**References:** 

[1] A. Kamyshny, S. Magdassi. Chem. Soc. Rev. Vol. 48, p.1712 (2019) 
[2] E.S. Oliveros-Mata, G.S. Cañón Bermúdez, M. Ha, et al. Appl. Phys. A Vol. 127, p.280 (2021) 
[3] M. Ha, G.S. Cañón Bermúdez, T. Kosub, et al. Adv. Mater. Vol. 33, p.2005521 (2021) 
[4] E.S. Oliveros-Mata, C. Voigt, G.S. Cañón Bermúdez, et al. Adv. Mater. Technol. p.2200227 (2022)

Detecting Magnetic Fields with Printed Magnetoresistive Sensors on Rigid and Flexible Substrates

Ultrathin asymmetrically sandwiched ferromagnetic films support fast moving chiral domain walls (DWs) and skyrmions [1,2]. This paves the way to the realization of prospective racetrack memory concept, the performance of which is determined by the static and dynamic micromagnetic parameters [3]. The necessity of having strong Dzyaloshinskii-Moriya interactions (DMI) and perpendicular magnetic anisotropy requires the utilization of ultrathin (~1 nm) layers, which compromize structural quality and substantially enhance the damping for non-collinear magnetic textures. 
Here, we present the experimental and theoretical analysis of ultrathin Co films with asymmetric interfaces //CrOx/Co/Pt and estimation of their micromagnetic parameters based on the analysis of the temperature dependence of magnetization as well as imaging of the DW morphology in stripes. We show that in the frame of magnon thermodynamics the best fit to the magnetometry data up to room temperature is obtained within a quasi-2D model, accounting for the lowest transversal magnons [4]. The developed approach provides access to the exchange constant in asymmetric stacks, which is found to be about 1 order of magnitude smaller compared to the bulk Co. The experimentally observed tilt of magnetic DWs in stripes in statics can be explained based on two models: (I) A unidirectional tilt could appear in equilibrium as a result of the competition between the DMI and additional in-plane easy-axis anisotropy, which breaks the symmetry of the magnetic texture and introduce tilts [5]. (II) A static DW tilt could appear due to the spatial variation of magnetic parameters, which introduce pinning centers for moving tilted DWs driven by magnetic field and can fix them at remanence [6]. We found that the second model is in line with the experimental observations and allows to determine self-consistently the DW damping parameter and DMI constant for the particular layer stack. The DW damping is found to be about 0.1 and explained by the enhanced longitudinal relaxation mechanism. The latter is shown to be much stronger than the standard transversal relaxation and can be even stronger than the spin pumping contribution for the case of ultrathin magnetic films [7]. 
**References:** [1] N. Nagaosa and Y. Tokura, “Topological properties and dynamics of magnetic skyrmions”, Nat. Nanotechnol. 8, 899 (2013). 
[2] A. Fert, N. Reyren, and V. Cros, “Magnetic skyrmions: advances in physics and potential applications”, Nat. Rev. Mater. 2, 17031 (2017). 
[3] C. Garg, S.-H. Yang, T. Phung, A. Pushp and S. S. P. Parkin, “Dramatic influence of curvature of nanowire on chiral domain wall velocity”, Sci. Adv. 3, e1602804 (2017). 
[4] I. A. Yastremsky, O. M. Volkov, M. Kopte, T. Kosub, S. Stienen, K. Lenz, J. Lindner, J. Fassbender, B. A. Ivanov and D. Makarov, “Thermodynamics and Exchange Stiffness of Asymmetrically Sandwiched Ultrathin Ferromagnetic Films with Perpendicular Anisotropy”, Phys. Rev. Appl. 12, 064038 (2019). 
[5] O. V. Pylypovskyi, V. P. Kravchuk, O. M. Volkov, J. Fassbender, D. D. Sheka and D. Makarov, “Unidirectional tilt of domain walls in equilibrium in biaxial stripes with Dzyaloshinskii–Moriya interaction”, J. Phys. D: Appl. Phys. 53, 395003 (2020). 
[6] O. M. Volkov, F. Kronast, C. Abert, E. Se. Oliveros Mata, T. Kosub, P. Makushko, D. Erb, O. V. Pylypovskyi, M.-A. Mawass, D. Sheka, S. Zhou, J. Fassbender and D. Makarov, “Domain-Wall Damping in Ultrathin Nanostripes with Dzyaloshinskii-Moriya Interaction”, Phys. Rev. Appl. 15, 034038 (2021). 
[7] I. A. Yastremsky, J. Fassbender, B. A. Ivanov, and D. Makarov, “Enhanced Longitudinal Relaxation of Magnetic Solitons in Ultrathin Films”, Phys. Rev. Appl. 17, L061002 (2022).

Quantification of Micromagnetic Parameters in Ultrathin Asymmetrically Sandwiched Magnetic Thin Films

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)

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 Micromagnetic description of symmetry-breaking effects in curvilinear ferromagnetic shells

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Magnetically Sensitive Electronic Skins for Supervised Folding of Origami Actuators

Local and Nonlocal Curvature induced Chiral Effects in Nanomagnetism

Detecting Magnetic Fields with Printed Magnetoresistive Sensors on Rigid and Flexible Substrates

Quantification of Micromagnetic Parameters in Ultrathin Asymmetrically Sandwiched Magnetic Thin Films

Micromagnetic description of symmetry-breaking effects in curvilinear ferromagnetic shells

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