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Cylindrical magnetic nanowires (CNWs) are promising candidates for the building blocks of 3D information technologies such as shift registers, magnetic recording, spintronics and logic gates1-4. Spin-polarized-current is an energy efficient way to excite magnetization dynamics in planar nanostructures. However, in CNWs this research is doing its first steps5. Magnetization dynamics in a CNW of Permalloy, 100 nm diameter and 1 µm length, is investigated under simultaneous application of electric current and magnetic field by micromagnetic simulations. The magnetization reversal process initiates with the creation of open vortex structures with different rotation senses at the nanowire ends. We conclude that the electric current by itself enlarges or reduces the length of these vortex structures according to the rotational sense of the associated Oersted field. Large enough current densities produce a vortex structure which covers the whole nanowire surface. At the same time, the magnetization in the very core of the nanowire remains the same, i.e. no complete magnetization reversal is possible in the absence of external magnetic field. The simultaneous action of the applied electric current and magnetic field allows the complete control of the vortex structures in terms of setting the polarity and vorticity. The resulting diagram of magnetic states obtained after the application of field and electric current showing the values required for the vorticity and axial magnetization switching is presented in Figure 1. This control is essential for future information technologies based on 3D vertical nanostructures. The presented state diagram will become useful for future experiments on current-induced domain wall dynamics in CNWs.References: 1. S. Parkin and S.-H. Yang, Nature Nanotechnology., Vol. 10, pp. 195 (2015). 2. F. Nasirpouri, et al., Scientific Reports., Vol. 9, pp. 9010 (2019). 3. S. Ruiz-Gómez et al., Scientific Reports., Vol. 8, pp. 16695 (2018). 4. J. Garcia et al., Journal of Magnetism and Magnetic Materials., Vol. 383, pp. 88 (2015). 5. M. Schöbitz et al., Physical Review Letters Vol. 123, pp. 217201 (2019). 

MMM 2020

Control of Vortex Structures in Individual Cylindrical Magnetic Nanowires by Simultaneous Action of Applied Magnetic Field and an Electric Current

The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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technical paper

The structural inversion symmetry plays an important role in low-dimensional nanomagnets, due to its strong influence on magnetic and electrical properties. It can lead to the appearance of chiral effects, such as the topological Hall effect [1], or to the formation of chiral noncollinear magnetic textures, as skyrmions [2] and chiral domain walls (DWs) [3]. These chiral structures can be the key components for realizing novel concepts for magnonics [4], antiferromagnetic spintronics [5], spin-orbitronics [6], and oxitronics [7]. So far, the main chiral symmetry breaking effect considered as being the origin for the presence of chiral noncollinear magnetic textures is the intrinsic Dzyaloshinskii-Moriya interaction (DMI) [8,9], which appears in certain magnetic crystals in which the unit cell lacks inversion symmetry, such as the gyrotropic magnetic crystals, or appear typically in ultrathin films or bilayers due to the inversion symmetry breaking on the film interface [3]. At present, tailoring of DMI is done by optimizing materials, either doping a bulk single crystal or adjusting interface properties of thin films and multilayers. A viable alternative to the conventional material screening approach can be the exploration of the interplay between geometry and topology. This interplay is of fundamental interest throughout many disciplines in condensed matter physics, including thin layers of superconductors [10] and superfluids [11], nematic liquid crystals [12], cell membranes [13], semiconductors [14]. In the emergent field of curvilinear magnetism chiral effects are associated to the geometrically broken inversion symmetries [15]. Those appear in curvilinear architectures of even conventional materials. There are numerous exciting theoretical predictions of exchange- and magnetostatically-driven curvature effects, which do not rely on any specific modification of the intrinsic magnetic properties, but allow to create non-collinear magnetic textures in a controlled manner by tailoring local curvatures and shapes [16,17]. Until now the predicted chiral effects due to curvatures remained a neat theoretical abstraction. Very recently, we provided the very first experimental confirmation of the existence of the curvature-induced chiral interaction with exchange origin in a conventional soft ferromagnetic material. It is experimentally explored the theoretical predictions, that the magnetisation reversal of flat parabolic stripes shows a two step process [18,19]. At the first switching event, a domain wall pinned by the curvature induced exchange-driven DMI is expelled leading to a magnetisation state homogeneous along the parabola's long axis. Measuring the depinning field enables to quantify the effective exchange-driven DMI interaction constant. The magnitude of the effect can be tuned by the parabola's curvature. It is found that the strength of the exchange-induced DMI interaction for the experimentally realised geometries is remarkably strong, namely ~0.4 mJ/m2, compared the surface induced DMI. The presented study legitimates the predictive power of full-scale micromagnetic simulations to design the properties of ferromagnets through their geometry, thus stabilising chiral textures. We explore these curvilinear magnetic thin films for the realization of novel artificial magnetoelectric materials based on curvilinear helimagnets embedded in piezoelectric matrix [20], to enable the geometrical tuning of the magnetochirality in curvilinear 1D architectures [21], tailoring of magnetic states in flat nanospirals [22] and as components of shapeable magnetoelectronics for interactive wearables [23].References: [1] N. Nagaosa, et al., Nature Nanotech. 8, 899 (2013) [2] U. K. Rößler, et al., Nature 442, 797 (2006) [3] A. Fert, et al., Nature Rev. Mat. 2, 17031 (2017) [4] A. V. Chumak, et al., Nature Physics 11, 453 (2015) [5] T. Jungwirth, et al., Nature Nanotech. 11, 231 (2016) [6] I. M. Miron, et al., Nature 476, 189 (2011) [7] V. Garcia, et al., Nature 460, 81 (2009) [8] I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958). [9] T. Moriya, Phys. Rev. Lett. 4, 228 (1960). [10] J. Tempere, et al., Phys. Rev. B 79, 134516 (2009) [11] H. Kuratsuji, Phys. Rev. E 85, 031150 (2012) [12] T. Lopez-Leon, et al., Nature Physics 7, 391 (2011) [13] H. T. McMahon, et al., Nature 438, 590 (2005) [14] C. Ortix, Phys. Rev. B 91, 245412 (2015) [15] Y. Gaididei, et al., Phys. Rev. Lett. 112, 257203 (2014) [16] J. A. Otálora, et al., Phys. Rev. Lett. 117, 227203 (2016) [17] V. P. Kravchuk, et al., Phys. Rev. Lett. 120, 067201 (2018) [18] O. Volkov et al., PRL 123, 077201 (2019). [19] O. Volkov et al., PSS-RRL 13, 1800309 (2019). [20] O. Volkov et al., J. Phys. D: Appl. Phys. 52, 345001 (2019). [21] O. Volkov et al., Scientific Reports 8, 866 (2018). [22] M. Nord, et al., Small 1904738 (2019). [23] J. Ge, et al., Nature Comm. 10, 4405 (2019).

Curvature-driven Chiral Effects in Nanomagnetism INVITED

The discovery of chiral spin textures such as domain walls, skyrmions, merons and bobbers, with their associated non-trivial topology and superior spintronic properties, is one of the most exciting landmarks in nanomagnetism and spintronics in recent years. Until now, these spin textures have been mostly realised by exploiting symmetry-breaking effects in non-centrosymmetric bulk crystals and synthetic ultra-thin film heterostructures. An emerging alternative route to the generation of chiral spin interactions is geometry [1]. In thin films, for example, curvature can provide the breaking of inversion-symmetry necessary for the generation of localized DMI-like spin interactions [2]. In this work, we employ state of the art 3D nanofabrication [3] to demonstrate the generation of complex chiral spin states using only geometrical effects, with no special material requirements. To do so, we fabricate an artificial magnetic double-helix system (Figure 1a) and perform magnetic X-ray microscopy [4]. This nanostructure is formed by two strongly-interacting strands (Figure 2) with dimensions comparable to fundamental magnetic lengths. By designing geometrical chirality, a wide variety of complex chiral 3D states is imprinted into the internal magnetization vector field. This includes helical magnetic domain walls (Figure 1b), vortex tubes and topologically-induced point-like structures [4]. In this presentation we will describe in detail how the topological mismatch caused by matching a left-handed and a right-handed double helix (See * in Figure 1a) leads to the generation of a localized Néel defect and the formation of an asymmetric vortex state [4]. These results showcase the power of using geometry for the localized generation of complex spin states, opening new paths for the study of non-trivial spin textures.References: [1] Streubel, R. et al. Journal of Physics D: Applied Physics 49.36 (2016): 363001. [2] Kravchuk, V. P., et al. Physical review letters 120.6 (2018): 067201. [3] Skoric, L. et al. Nano Letters 20.1 (2019): 184-191. [4] Sanz-Hernández, D., et al. arXiv:2001.07130 [cond-mat.mes-hall]

Creating and Interfacing Chiral Spin Textures Using 3D Geometries INVITED

Authors: F. Ajejas, W. Legrand, Y. Sassi, S. Collin, A. Vecchiola, K. Bouzehouane, N. Reyren, V. Cros, A. Fert, Unite Mixte de Physique CNRS/Thales, Palaiseau, Île-de-France, FRANCE|S. Pizzini, Institut NEEL, Grenoble, Auvergne-Rhône-Alpes , FRANCE|

Abstract Body: Magnetic multilayers (MML) with large perpendicular magnetic anisotropy (PMA) and Dzyaloshinskii-Moriya interaction (DMI) have attracted great attention in recent years owing to the possibility of stabilizing non-collinear magnetic textures such as chiral domain walls (DW) [1], spin spirals [2] or skyrmions [3-4] with Neeel chiral spin rotation. The latter are promising candidates as carriers of information for next-generation race-track magnetic memories or logic devices. To this aim, the accurate experimental determination of DMI is an important challenge allowing the engineering of future devices with tailored properties. The determination of the effective DMI amplitude D in MML is still a current subject of research, in particular for metallic multilayers with a high number of repetitions. In this study, we have performed a series of measurements to determine DMI values by two different techniques: domain size periodicity (λ) [5] of aligned-stripe domains after in-plane (IP) demagnetization by magnetic force microscopy (MFM), and asymmetric expansion of domains in the presence of an in-plane magnetic field, by Kerr microscopy [6-7]. For this purpose, we selected different materials to build asymmetric trilayers, with the general structure is [Pt/Co/M]xN with M = Ru, Ni, Pd, Al, Al/Ta and N = 2, 3, 4, 5, 6 the number of repetitions. We observe a correlation between Ds = D/t (t is the Co thickness), the interfacial DMI obtained in the previous methods and intrinsic material parameters such as atomic number (Z), Pauling electronegativity (χ) [8] work function (Wf) [9] of Co/M interfaces. We find a clear linear behaviour between Ds and Wf. This correlation points to Rashba-like interfacial fields, leading to the modulation of the effective interfacial DMI. French ANR grant TOPSky TOPSKY (ANR-17-CE24-0025) and DARPA TEE program grant (MIPR#HR0011831554) and EU grant SKYTOP (H2020 FET Proactive 824123) are acknowledged for their financial support.References: [1] A. Thiaville et al. EuroPhys. Lett, 100, 57002 (2012) [2] P. Ferriani et al. Phys. Rev. Lett. 101, 27201 (2008). [3] A. Fert, N. Reyren, V. Cros, Nat. Rev. Mat. 2, 17031 (2017). [4] C. Moreau-Luchaire et al. Nature Nanotechnology 11, 444 (2016) [5] I. Lemesh et al. Physical Review B 95, 174423 (2017) [6] F. Ajejas et al. Appl. Phys. Lett. 111, 202402 (2017); [7] T. Ha Pham et al EPL 113 67001 (2016) [8] H. Jia et al. Phys. Rev. M 4, 024405 (2020) [9] Y.-K. Park et al. NPG Asia Materials 10, 995 (2018)

GMW Best Poster Award 1 - Experimental correlation of Interfacial Dzyaloshinskii-Moriya interaction amplitude and Work Function in magnetic multilayers

Authors:  A. Semiannikova, V. Irkhin, Y.A. Perevozchikova, P. Korenistov, V. Marchenkov, M N Mikheev Institute of Metal Physics of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Sverdlovskaâ, RUSSIAN FEDERATION|V. Marchenkov, Ural Federal University named after the first President of Russia B N Yeltsin, Ekaterinburg, Sverdlovskaa, RUSSIAN FEDERATION|

Abstract Body: Half-metallic ferromagnets (HMFs) and spin gapless semiconductors (SGSs) are promising materials for spintronics with almost 100% spin polarization of charge carriers. They possess a gap near the Fermi energy for the current carriers with spin down. However, for the opposite spin projection, these materials have a difference: the energy gap is absent in HFMs [1], while it is zero in SGSs. Parameters of the energy gap and, consequently, physical properties can vary quite strongly in different X2YZ Heusler alloys [2, 3, 4], where Y-elements are 3d-transition metals and Z-components are s- and p-elements. Thus, the aim of the work is to follow the changes in the electrical, magnetic and galvanomagnetic properties and to establish their interconnection and basic behavior patterns of the Co2YSi (Y = Ti, V, Cr, Mn, Fe, Co, Ni) and Co2MnZ Heusler alloys (Z = Si, Al, Ge, Sb) in the HMF and/or SGS states. The electrical resistivity was measured in a wide temperature range from 4.2 to 300 K, the field and temperature dependences of magnetic and galvanomagnetic properties being studied too. Significant changes in the values of the coefficients of the normal and anomalous Hall Effect, magnetization, residual resistivity, type, and concentration of current carriers and their mobility were found. At the same time, the correlation between the changes in these electronic and magnetic characteristics depending on the number of valence electrons are observed. The work was performed within the framework of the state assignment of the Ministry of Science and High Education of Russia (the themes “Spin”, No. AAAA-A18-118020290104-2-2 and “Quantum” No. AAAA-A18-118020190095-4) with partial support from the RFBR (project No. 18-02-00739 and 20-32-90065) and the Government of the Russian Federation (Decree No. 211, Contract No. 02.A03.21.0006).References: [1] M.I. Katsnelson, V.Yu. Irkhin, L. Chioncel et al., Rev. Mod. Phys., Vol. 80, p. 315 (2008). [2] V.V. Marchenkov, Yu.A. Perevozchikova, N.I. Kourov et al., J. Magn. Magn. Mat., Vol. 459, p. 211 (2018). [3] Yu.A. Perevozchikova, A.A. Semiannikova, A.N. Domozhirova, P.B. Terentyev, E.B. Marchenkova, E.I. Patrakov, M. Eisterer, P.S. Korenistov and V.V. Marchenkov, Low Temp. Phys., Vol. 45, p. 789 (2019). [4] Yu.A. Perevozchikova, A.A. Semiannikova, P.B. Terentyev, M. Eisterer, P.S. Korenistov and V.V. Marchenkov, Journal of Physics: Conf. Series, Vol. 1389, p. 012110 (2019). 

GMW Best Poster Award 3 - Electroresistivity, magnetic and galvanomagnetic properties of Co-based Heusler alloys in the states of half-metallic ferromagnet and spin gapless semiconductor

Unlike the common expectation that superconducting spintronics and long-range triplet (LRT) proximity effects require complex ferromagnetic multilayers, relying on noncollinear/spiral magnetization or half-metals, we propose a new platform compatible with commercial spintronics. Recently, we have experimentally demonstrated that interfacial spin-orbit coupling (SOC) and symmetry-filtering in all-epitaxial V/MgO/Fe junctions, cooled below the critical temperature of vanadium, provide a thousand-fold increase in tunneling anisotropic magnetoresistance, supporting the LRT formation [1]. Here we report the converse effect: transformation of the magnetocrystalline anisotropy of a Fe(001) layer driven by the superconductivity of an underlying V layer through a spin-orbit coupled MgO interface [2]. We attribute this to an additional contribution to the free energy of the ferromagnet arising from the controlled generation of triplet Cooper pairs, which depends on the relative angle between the exchange field of the ferromagnet and the spin-orbit field. This observation offers the ability to fundamentally tune magnetic anisotropies using superconductivity - a key step in designing future cryogenic magnetic memories.References: [1] I. Martínez et al; Phys. Rev. Appl. 13, 014030 (2020). [2] C. González-Ruano et al; accepted for publication as Rapid Communication in Phys. Rev. B, Cond Mat. ArXiv https://arxiv.org/abs/2006.02118

Superconductivity-induced change in magnetic anisotropy in epitaxial ferromagnet-superconductor hybrids with spin-orbit interaction (B)

When applied to C2V ferromagnetic/heavy metal interfaces, Moriya’s rules do not deterministically predict the orientation of interatomic DM vectors. Low-symmetry magnetic systems can exhibit both Rashba (DR) and Dresselhaus (DD) forms of the Dzyaloshinskii-Moryia interaction (DMI), as both spin-orbit couplings are generally present [1]. We assert that the spatial variation of the DM vectors—and thus that of the strength and direction of the effective DMI field—is determined by the DMI’s character (i.e., ||DD||/||DR||), as well as the unit cell’s aspect ratio [2]. Overall, the DMI is thus nominally anisotropic and hybridized—potentially stabilizing antiskyrmions and Bloch-Néel (i.e. mixed) skyrmions [2-3]. Ascertaining the existence and tunability of [anti]skyrmions in such systems is of technological interest and would elucidate the nature of DMI [2]. Employing the Wulff construction [3], ξ-vector formalism [4], and an augmented wall energy that accounts for DMI character, this work models the equilibrium shape and wall magnetization profile of bubble domains and skyrmions in thin films with C2v symmetry. We show that for sufficiently large DMI, square-shaped antiskyrmions are stable for certain DM vector orientations while lens- and oval-shaped mixed skyrmions are favored otherwise. A symmetry-breaking applied in-plane field stabilizes domain shapes resembling ovals and Reuleaux triangles. The impact of the unit cell’s aspect ratio, the DM vector’s orientation, and other material parameters on the shape of bubble domains and skyrmions are discussed. Moreover, our results reveal unusual relationships between faceting behavior and wall stiffness under rather reasonable conditions. Elucidating the interplay between the different effective fields present, we explore the validity of these results based on pedal plots, discontinuities in the magnetization profiles, and local extrema in the wall energy.References: U. Güngördü et al., Physical Review B., 93, 064428 (2016) M. Hoffman et al., Nature Communications., 8, 308 (2017) J. P. Pellegren, D. Lau, and V. Sokalski, Physical Review Letters., 119, 027203 (2017) A. A. Wheeler, Journal of Statistical Physics., 95, 1245–1280 (1999)

Energetics, equilibrium shape, and vorticity of skyrmions and antiskyrmions stabilized by the anisotropic Dzyaloshinskii-Moriya interaction in thin films with C2v symmetry

New fascinating phenomena involving coupled spin waves and phonons are observed at the magneto-elastic resonance: magnetic field dependent spin Seebeck effect [1], acoustically-assisted spin pumping [2, 3] and magnetization swiching [4, 5]. In this context, it has been shown that an efficient mean to excite phonons is by surface acoustic waves (SAW). The integration of SAW-FMR (ferromagnetic resonance) mechanism in spintronic and magnonic devices are claimed to operate with very low power. However, most ferromagnetic materials have precession frequencies well above 5 GHz, i.e. above the highest frequency reached by today's SAW technologies. This restricts the research on SAW-FMR to low magnetization and/or low magnetic anisotropy materials, such as (Ga,Mn)As or (GaMn)(As,P), Ni, and YIG. Here, we show that thin films of Fe epitaxially grown on GaAs(001), a spintronic and magnonic compatible magnetoelastic and piezoelectric heterostucture, give the opportunity to obtain SAW-FMR, at room temperature, below 1 GHz despite the high Fe magnetization. This is obtained by exploiting the softening of the magnetic stiffness when the field is applied along a hard axis, at moderate intensity (60 mT). Slight variations of the magnetic field direction, less than 0.1 deg off, increase dramatically magnetization precession frequency leading to off-resonance interaction and suppressing SAW attenuation. The observed phenomena are described by adopting a very simple magnetization dynamics model that catches the physics of the interaction. The observation of resonant, low frequency and highly directional magnetoelastic coupling in Fe suggests that SAW mature technology offers the opportunity to magnonics and spintronics to handle magnetization precession in a remote way, without using any dissipative electron current. [6]References: [1] Takashi Kikkawa, et al., Phys. Rev. Lett. 117, 207203, (2016) [2] M. Weiler, et al. Phys. Rev. Lett. 108, 176601, (2012) [3] Ken ichi Uchida, et al., Solid State Communications 198, 26, (2014) [4] L. Thevenard,et al. Phys. Rev. B 93, 134430, (2016) [5] P. Kuszewski, et al. Journal of Physics: Condensed Matter 30, 244003 (2018) [6] J.-Y. Duquesne, P. Rovillain, et al. Phys. Rev. Appl. 12, 024042 (2019)

Sub-GHz resonant magnetoelastic coupling in epitaxial Fe thin films

Voltage control of the structure, magnetic and electronic properties of materials enables the realization of new types of nonvolatile magnetoelectric memory devices. Here, we investigate the effect of ionic liquid gating on the magnetic properties of self-assembled heterostructure films consisting of nanoscale CoyFe3-yO4 (CFO) spinel pillars embedded in a SrCo1-xFexO3-δ (SCFO) perovskite (P) or brownmillerite (BM) matrix, over a range of compositions x and y. These self-assembled nanocomposites are grown by pulsed laser deposition on SrTiO3 substrates using SrCoO3-δ-SrFeO3-δ, SrCoO3-δ-Fe3O4, or SrFeO3-δ-Co3O4 pairs of targets. We first describe the growth kinetics of these nanocomposites, focusing on how the oxygen pressure affects the distribution of Co and Fe between the two phases and the formation of P vs. BM structure for the matrix. Unusually, at high oxygen pressures, the two-phase structure can grow from the codeposition of SrCoO3-δ and SrFeO3-δ, attributed to depletion of the Sr in the film. The pillar CFO magnetism depends on y, e.g. ferrimagnetic with room temperature saturation magnetization Ms ~300 kA/m for y=1 and paramagnetic for y=0. The matrix SCFO with the BM phase has antiferromagnetic ordering irrespective of x, while the magnetism of the P phase depends strongly on x and can exhibit a room temperature Ms ~100 kA/m for x=0.5 but almost no moment for x=0 and x=1 [1]. With the gating voltage of -3V/+3V, the lattice parameter of matrix SCFO changes ±3% and induces strain transfer around ±1% to the pillar phase (Fig. 1). In particular, for x=0.5 and y=0 (Fig. 2), the negative gating inserts oxygens into the BM phase and changes it into a P phase showing ferromagnetism. Our study clarifies the manipulation of the magnetism of composite materials through an electrochemical process, expanding the applications of magnetoelectric coupling.References: [1] S. Ning, Q. Zhang and C. A. Ross, ACS Nano, https://dx.doi.org/10.1021/acsnano.0c03750 (2020)

Controlling Magnetic Properties of Epitaxial Nanocomposite Thin Films Through Ionic Liquid Gating

The focus of lab-on-a-chip technology centers on small, low power devices that can be integrated on a microscopic scale to perform laboratory functions within one unit or “chip”. In order to achieve this, one has to understand how to transfer fluid at this scale. Due to the low Reynolds number limit, such devices need to break time-reversible symmetry to produce a net flow. There has been a degree of interest and research into using magnetic materials to achieve this. In this work, we propose a new model employing ferromagnetic elements for construction of a practical microfluidic pump. The design is based on a repeated chain of magnetic discs connected together via links with hinges at the centers of the discs, which allow the discs to rotate (Fig 1a). While the model is based on that of ferromagnetic swimmers[1]. The device is surrounded by fluid and is activated by an external AC magnetic field. The magnetic and elastic forces in this device can work together against the fluidic forces to create a time-irreversible movement, which results in a net flow. While magnetisation of all the discs are oriented in the same direction, the relative rotational stiffness of the discs (or hinge stiffness) is varied to produce a lag in the response to the magnetic torque. This collective effect is similar to that of a metachronal wave, which is ultimately responsible for the flow. We found that if both discs had the same rotational stiffness, then no flow was produced. Depending on the chosen frequency, there is an optimal rotation stiffness of the two discs where a maximum flow rate is achieved. We observed as the frequency increases so does the flow, until the frequency is too large for the discs to turn to their maximum angle before the direction of the magnetic field reverses. To conclude, we demonstrated that an elasto-magnetic device can produce a net flow at low Reynolds number. The flow direction can be controlled by altering the frequency of the driving magnetic field or by using different rotational stiffness. These results could lead to new types of ferromagnetic pumping devices.References: [1] A. D. Gilbert, et al., Q. J. Mech. Appl. Math., 64, 239 (2011).

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