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Ren F, Wang X, Zhang Q, Wang X, Chang L, Zhang Z. Experimental and Theoretical Investigation of External Electric-Field-Induced Crystallization of TKX-50 from Solution by Finite-Temperature String with Order Parameters as Collective Variables for Ionic Crystals. Molecules 2024; 29:1159. [PMID: 38474669 DOI: 10.3390/molecules29051159] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2024] [Revised: 02/26/2024] [Accepted: 02/28/2024] [Indexed: 03/14/2024] Open
Abstract
External electric fields are an effective tool to induce phase transformations. The crystallization of ionic crystals from solution is a common phase transformation. However, understanding of mechanisms is poor at the molecular level. In this work, we carried out an experimental and theoretical investigation of the external electric-field-induced crystallization of TKX-50 from saturated formic acid solution by finite-temperature string (FTS) with order parameters (OPs) as collective variables for ionic crystals. The minimum-free-energy path was sketched by the string method in collective variables. The results show that the K-means clustering algorithm based on Euclidean distance and density weights can be used for enhanced sampling of the OPs in external electric-field-induced crystallization of ionic crystal from solution, which improves the conventional FTS. The crystallization from solution is a process of surface-mediated nucleation. The external electric field can accelerate the evolution of the string and decrease the difference in the potential of mean forces between the crystal and the transition state. Due to the significant change in OPs induced by the external electric field in nucleation, the crystalline quality was enhanced, which explains the experimental results that the external electric field enhanced the density, detonation velocity, and detonation pressure of TKX-50. This work provides an effective way to explore the crystallization of ionic crystals from solution at the molecular level, and it is useful for improving the properties of ionic crystal explosives by using external electric fields.
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Affiliation(s)
- Fude Ren
- School of Chemistry and Chemical Engineering, North University of China, Taiyuan 030051, China
| | - Xiaolei Wang
- School of Chemistry and Chemical Engineering, North University of China, Taiyuan 030051, China
| | - Qing Zhang
- School of Chemistry and Chemical Engineering, North University of China, Taiyuan 030051, China
| | - Xiaojun Wang
- Gansu Yinguang Chemical Industry Group, Baiyin 730900, China
| | - Lingling Chang
- School of Chemistry and Chemical Engineering, North University of China, Taiyuan 030051, China
| | - Zhiteng Zhang
- School of Chemistry and Chemical Engineering, North University of China, Taiyuan 030051, China
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2
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Truc B, Sapozhnik AA, Tengdin P, Viñas Boström E, Schönenberger T, Gargiulo S, Madan I, LaGrange T, Magrez A, Verdozzi C, Rubio A, Rønnow HM, Carbone F. Light-Induced Metastable Hidden Skyrmion Phase in the Mott Insulator Cu 2 OSeO 3. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2023; 35:e2304197. [PMID: 37282751 DOI: 10.1002/adma.202304197] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/05/2023] [Indexed: 06/08/2023]
Abstract
The discovery of a novel long-lived metastable skyrmion phase in the multiferroic insulator Cu2 OSeO3 visualized with Lorentz transmission electron microscopy for magnetic fields below the equilibrium skyrmion pocket is reported. This phase can be accessed by exciting the sample non-adiabatically with near-infrared femtosecond laser pulses and cannot be reached by any conventional field-cooling protocol, referred as a hidden phase. From the strong wavelength dependence of the photocreation process and via spin-dynamics simulations, the magnetoelastic effect is identified as the most likely photocreation mechanism. This effect results in a transient modification of the magnetic free energy landscape extending the equilibrium skyrmion pocket to lower magnetic fields. The evolution of the photoinduced phase is monitored for over 15 min and no decay is found. Because such a time is much longer than the duration of any transient effect induced by a laser pulse in a material, it is assumed that the newly discovered skyrmion state is stable for practical purposes, thus breaking ground for a novel approach to control magnetic state on demand at ultrafast timescales and drastically reducing heat dissipation relevant for next-generation spintronic devices.
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Affiliation(s)
- Benoit Truc
- Laboratory for Ultrafast Microscopy and Electron Scattering, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
| | - Alexey A Sapozhnik
- Laboratory for Ultrafast Microscopy and Electron Scattering, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
| | - Phoebe Tengdin
- Laboratory for Ultrafast Microscopy and Electron Scattering, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
| | - Emil Viñas Boström
- Max Planck Institute for the Structure and Dynamics of Matter, 22761, Hamburg, Germany
| | - Thomas Schönenberger
- Laboratory for Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
| | - Simone Gargiulo
- Laboratory for Ultrafast Microscopy and Electron Scattering, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
| | - Ivan Madan
- Laboratory for Ultrafast Microscopy and Electron Scattering, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
| | - Thomas LaGrange
- Laboratory for Ultrafast Microscopy and Electron Scattering, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
| | - Arnaud Magrez
- Crystal Growth Facility, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
| | - Claudio Verdozzi
- Division of Mathematical Physics and ETSF, Lund University, Lund, 223 63, Sweden
| | - Angel Rubio
- Max Planck Institute for the Structure and Dynamics of Matter, 22761, Hamburg, Germany
- Center for Computational Quantum Physics (CCQ), The Flatiron Institute, New York, 10010, USA
| | - Henrik M Rønnow
- Laboratory for Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
| | - Fabrizio Carbone
- Laboratory for Ultrafast Microscopy and Electron Scattering, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
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3
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Huang P, Cantoni M, Magrez A, Carbone F, Rønnow HM. Electric field writing and erasing of skyrmions in magnetoelectric Cu 2OSeO 3 with an ultralow energy barrier. NANOSCALE 2022; 14:16655-16660. [PMID: 36330779 DOI: 10.1039/d2nr04399h] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/16/2023]
Abstract
Skyrmions are chiral magnetic textures with non-trivial topology, and due to their unique properties they are widely considered as promising information carriers in novel magnetic storage applications. While electric field writing/erasing and manipulation of skyrmions have been recently achieved, quantitative insights into the energetics of those phenomena remain scarce. Here, we report our in situ electric field writing/erasing of skyrmions in magnetoelectric helimagnet Cu2OSeO3 utilizing real-space and real-time Lorentz transmission electron macroscopy. Through the quantitavie analysis on our massive video data, we obtained a linear dependence of the number of skyrmions on the amplitude of the applied electric field, from which a local energy barried to write/erase skyrmions is estimated to be per skyrmion. Such an ultralow energy barrier implies the potential of precise control of skyrmions in future spintronics applications.
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Affiliation(s)
- Ping Huang
- State Key Laboratory for Mechanical Behavior of Materials, Xi'an Jiaotong University, , CN-710049 Xi'an, China.
- Laboratory for Quantum Magnetism (LQM), Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
- Laboratory for Ultrafast Microscopy and Electron Scattering (LUMES), Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Marco Cantoni
- Centre Interdisciplinaire de Microscopie Électronique (CIME), École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Arnaud Magrez
- Crystal Growth Facility, Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Fabrizio Carbone
- Laboratory for Ultrafast Microscopy and Electron Scattering (LUMES), Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Henrik M Rønnow
- Laboratory for Quantum Magnetism (LQM), Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
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4
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Lee O, Sahliger J, Aqeel A, Khan S, Seki S, Kurebayashi H, Back CH. Tunable gigahertz dynamics of low-temperature skyrmion lattice in a chiral magnet. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 34:095801. [PMID: 34844226 DOI: 10.1088/1361-648x/ac3e1c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2021] [Accepted: 11/29/2021] [Indexed: 06/13/2023]
Abstract
Recently, it has been shown that the chiral magnetic insulator Cu2OSeO3hosts skyrmions in two separated pockets in temperature and magnetic field phase space. It has also been shown that the predominant stabilization mechanism for the low-temperature skyrmion (LTS) phase is via the crystalline anisotropy, opposed to temperature fluctuations that stabilize the well-established high-temperature skyrmion (HTS) phase. Here, we report on a detailed study of LTS generation by field cycling, probed by GHz spin dynamics in Cu2OSeO3. LTSs are populated via a field cycling protocol with the static magnetic field applied parallel to the ⟨100⟩ crystalline direction of plate and cuboid-shaped bulk crystals. By analyzing temperature-dependent broadband spectroscopy data, clear evidence of LTS excitations with clockwise (CW), counterclockwise (CCW), and breathing mode (BR) character at temperatures belowT= 40 K are shown. We find that the mode intensities can be tuned with the number of field-cycles below the saturation field. By tracking the resonance frequencies, we are able to map out the field-cycle-generated LTS phase diagram, from which we conclude that the LTS phase is distinctly separated from the high-temperature counterpart. We also study the mode hybridization between the dark CW and the BR modes as a function of temperature. By using two Cu2OSeO3crystals with different shapes and therefore different demagnetization factors, together with numerical calculations, we unambiguously show that the magnetocrystalline anisotropy plays a central role for the mode hybridization.
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Affiliation(s)
- Oscar Lee
- London Centre for Nanotechnology, University College London, United Kingdom
| | - Jan Sahliger
- Physik-Department, Technische Universität München, D-85748 Garching, Germany
| | - Aisha Aqeel
- Physik-Department, Technische Universität München, D-85748 Garching, Germany
| | - Safe Khan
- London Centre for Nanotechnology, University College London, United Kingdom
| | - Shinichiro Seki
- Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo, 113-8656, Japan
| | | | - Christian H Back
- Physik-Department, Technische Universität München, D-85748 Garching, Germany
- Munich Center for Quantum Science and Technology (MCQST), D-80799 München, Germany
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5
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Leonov AO. Surface anchoring as a control parameter for shaping skyrmion or toron properties in thin layers of chiral nematic liquid crystals and noncentrosymmetric magnets. Phys Rev E 2021; 104:044701. [PMID: 34781482 DOI: 10.1103/physreve.104.044701] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2021] [Accepted: 10/01/2021] [Indexed: 11/07/2022]
Abstract
Existence of topological localized states (skyrmions and torons) and the mechanism of their condensation into modulated states are the ruling principles of condensed matter systems, such as chiral nematic liquid crystals (CLCs) and chiral magnets (ChM). In bulk helimagnets, skyrmions are rendered into thermodynamically stable hexagonal skyrmion lattice due to the combined effect of a magnetic field and, e.g., small anisotropic contributions. In thin glass cells of CLCs, skyrmions are formed in response to the geometrical frustration and field coupling effects. By numerical modeling, I undertake a systematic study of skyrmion or toron properties in thin layers of CLCs and ChMs with competing surface-induced and bulk anisotropies. The conical phase with a variable polar angle serves as a suitable background, which shapes skyrmion internal structure, guides the nucleation processes, and substantializes the skyrmion-skyrmion interaction. I show that the hexagonal lattice of torons can be stabilized in a vast region of the constructed phase diagram for both easy-axis bulk and surface anisotropies. A topologically trivial droplet is shown to form as a domain boundary between two cone states with different rotational fashion, which underpins its stability. The findings provide a recipe for controllably creating skyrmions and torons, possessing the features on demand for potential applications.
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Affiliation(s)
- Andrey O Leonov
- Chirality Research Center, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan; Department of Chemistry, Faculty of Science, Hiroshima University Kagamiyama, Higashi Hiroshima, Hiroshima 739-8526, Japan; and IFW Dresden, Postfach 270016, D-01171 Dresden, Germany
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6
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Real-space observations of 60-nm skyrmion dynamics in an insulating magnet under low heat flow. Nat Commun 2021; 12:5079. [PMID: 34426575 PMCID: PMC8382761 DOI: 10.1038/s41467-021-25291-2] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2021] [Accepted: 08/03/2021] [Indexed: 11/25/2022] Open
Abstract
Thermal-current induced electron and spin dynamics in solids –dubbed “caloritronics”– have generated widespread interest in both fundamental physics and spintronics applications. Here, we examine the dynamics of nanometric topological spin textures, skyrmions driven by a temperature gradient ∇T or heat flow, that are evaluated through in-situ real-space observations in an insulating helimagnet Cu2OSeO3. We observe increases of the skyrmion velocity and the Hall angle with increasing ∇T above a critical value of ~ 13 mK/mm, which is two orders of magnitude lower than the ∇T required to drive ferromagnetic domain walls. A comparable magnitude of ∇T is also observed to move the domain walls between a skyrmion domain and the non-topological conical-spin domain from cold to hot regions. Our results demonstrate the efficient manipulation of skyrmions by temperature gradients, a promising step towards energy-efficient “green” spintronics. Skyrmions are a type of topological spin texture that great potential across a wide variety of technological applications. Here, Yu et al. study the thermally driven motion of Skyrmions and find a minimum temperature gradient for the motion of skyrmions two orders of magnitude smaller than for domain walls.
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7
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Abstract
Skyrmion, a concept originally proposed in particle physics half a century ago, can now find the most fertile field for its applicability, that is, the magnetic skyrmion realized in helimagnetic materials. The spin swirling vortex-like texture of the magnetic skyrmion can define the particle nature by topology; that is, all the constituent spin moments within the two-dimensional sheet wrap the sphere just one time. Such a topological nature of the magnetic skyrmion can lead to extraordinary metastability via topological protection and the driven motion with low electric-current excitation, which may promise future application to spintronics. The skyrmions in the magnetic materials frequently show up as the crystal lattice form, e.g., hexagonal lattice, but sometimes as isolated or independent particles. These skyrmions in magnets were initially found in acentric magnets, such as chiral, polar, and bilayered magnets endowed with antisymmetric spin exchange interaction, while the skyrmion host materials have been explored in a broader family of compounds including centrosymmetric magnets. This review describes the materials science and materials chemistry of magnetic skyrmions using the classification scheme of the skyrmion forming microscopic mechanisms. The emergent phenomena and functions mediated by skyrmions are described, including the generation of emergent magnetic and electric field by statics and dynamics of skrymions and the inherent magnetoelectric effect. The other important magnetic topological defects in two or three dimensions, such as biskyrmions, antiskyrmions, merons, and hedgehogs, are also reviewed in light of their interplay with the skyrmions.
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Affiliation(s)
- Yoshinori Tokura
- Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan.,RIKEN Center for Emergent Matter Science (CEMS), Wako, 351-0198, Japan.,Tokyo College, University of Tokyo, Tokyo 113-8656, Japan
| | - Naoya Kanazawa
- Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
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8
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Das S, Hong Z, Stoica VA, Gonçalves MAP, Shao YT, Parsonnet E, Marksz EJ, Saremi S, McCarter MR, Reynoso A, Long CJ, Hagerstrom AM, Meyers D, Ravi V, Prasad B, Zhou H, Zhang Z, Wen H, Gómez-Ortiz F, García-Fernández P, Bokor J, Íñiguez J, Freeland JW, Orloff ND, Junquera J, Chen LQ, Salahuddin S, Muller DA, Martin LW, Ramesh R. Local negative permittivity and topological phase transition in polar skyrmions. NATURE MATERIALS 2021; 20:194-201. [PMID: 33046856 DOI: 10.1038/s41563-020-00818-y] [Citation(s) in RCA: 31] [Impact Index Per Article: 10.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/14/2019] [Accepted: 09/01/2020] [Indexed: 06/11/2023]
Abstract
Topological solitons such as magnetic skyrmions have drawn attention as stable quasi-particle-like objects. The recent discovery of polar vortices and skyrmions in ferroelectric oxide superlattices has opened up new vistas to explore topology, emergent phenomena and approaches for manipulating such features with electric fields. Using macroscopic dielectric measurements, coupled with direct scanning convergent beam electron diffraction imaging on the atomic scale, theoretical phase-field simulations and second-principles calculations, we demonstrate that polar skyrmions in (PbTiO3)n/(SrTiO3)n superlattices are distinguished by a sheath of negative permittivity at the periphery of each skyrmion. This enhances the effective dielectric permittivity compared with the individual SrTiO3 and PbTiO3 layers. Moreover, the response of these topologically protected structures to electric field and temperature shows a reversible phase transition from the skyrmion state to a trivial uniform ferroelectric state, accompanied by large tunability of the dielectric permittivity. Pulsed switching measurements show a time-dependent evolution and recovery of the skyrmion state (and macroscopic dielectric response). The interrelationship between topological and dielectric properties presents an opportunity to simultaneously manipulate both by a single, and easily controlled, stimulus, the applied electric field.
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Affiliation(s)
- S Das
- Department of Materials Science and Engineering, University of California, Berkeley, CA, USA.
- Department of Physics, University of California, Berkeley, CA, USA.
| | - Z Hong
- Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA, USA
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA
| | - V A Stoica
- Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA, USA
| | - M A P Gonçalves
- Materials Research and Technology Department, Luxembourg Institute of Science and Technology (LIST), Esch/Alzette, Luxemburg
- Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Universidad de Cantabria, Cantabria Campus Internacional, Santander, Spain
- Physics and Materials Science Research Unit, University of Luxembourg, Belvaux, Luxembourg
| | - Y T Shao
- School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA
| | - E Parsonnet
- Department of Physics, University of California, Berkeley, CA, USA
| | - E J Marksz
- National Institute of Standards and Technology, Boulder, CO, USA
| | - S Saremi
- Department of Materials Science and Engineering, University of California, Berkeley, CA, USA
| | - M R McCarter
- Department of Physics, University of California, Berkeley, CA, USA
| | - A Reynoso
- Department of Physics, University of California, Berkeley, CA, USA
| | - C J Long
- National Institute of Standards and Technology, Boulder, CO, USA
| | - A M Hagerstrom
- National Institute of Standards and Technology, Boulder, CO, USA
| | - D Meyers
- Department of Materials Science and Engineering, University of California, Berkeley, CA, USA
| | - V Ravi
- Department of Materials Science and Engineering, University of California, Berkeley, CA, USA
| | - B Prasad
- Department of Materials Science and Engineering, University of California, Berkeley, CA, USA
| | - H Zhou
- Advanced Photon Source, Argonne National Laboratory, Argonne, IL, USA
| | - Z Zhang
- Advanced Photon Source, Argonne National Laboratory, Argonne, IL, USA
| | - H Wen
- Advanced Photon Source, Argonne National Laboratory, Argonne, IL, USA
| | - F Gómez-Ortiz
- Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Universidad de Cantabria, Cantabria Campus Internacional, Santander, Spain
| | - P García-Fernández
- Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Universidad de Cantabria, Cantabria Campus Internacional, Santander, Spain
| | - J Bokor
- Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA
| | - J Íñiguez
- Materials Research and Technology Department, Luxembourg Institute of Science and Technology (LIST), Esch/Alzette, Luxemburg
- Physics and Materials Science Research Unit, University of Luxembourg, Belvaux, Luxembourg
| | - J W Freeland
- Advanced Photon Source, Argonne National Laboratory, Argonne, IL, USA
| | - N D Orloff
- National Institute of Standards and Technology, Boulder, CO, USA
| | - J Junquera
- Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Universidad de Cantabria, Cantabria Campus Internacional, Santander, Spain
| | - L Q Chen
- Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA, USA
| | - S Salahuddin
- Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA
| | - D A Muller
- School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA
- Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA
| | - L W Martin
- Department of Materials Science and Engineering, University of California, Berkeley, CA, USA
- Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA
| | - R Ramesh
- Department of Materials Science and Engineering, University of California, Berkeley, CA, USA.
- Department of Physics, University of California, Berkeley, CA, USA.
- Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA.
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9
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Aya S, Araoka F. Kinetics of motile solitons in nematic liquid crystals. Nat Commun 2020; 11:3248. [PMID: 32591526 PMCID: PMC7319993 DOI: 10.1038/s41467-020-16864-8] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2019] [Accepted: 05/26/2020] [Indexed: 11/09/2022] Open
Abstract
The generation of spatially localized, soliton-like hydrodynamic disturbances in microscale fluidic systems is an intriguing challenge. Herein, we introduce nonequilibrium solitons in nematic liquid crystals stimulated by an electric field. These dynamic solitons are robust as long as the electric field is maintained. Interestingly, their kinetic behaviours depend on the field condition-Tuning of the amplitude and frequency of the applied electric field alters the solitons to self-assemble into lattice ordering like physical particles or to command them to various dynamic states. Our key property to the realisation is the electrohydrodynamic instability due to the coupling between the fluid elasticity and the background convection. This paper describes a new mechanism for realising dynamic solitons in fluid systems on the basis of the electrohydrodynamic phenomena.
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Affiliation(s)
- Satoshi Aya
- South China Advanced Institute for Soft Matter Science and Technology (AISMST), School of Molecular Science and Engineering, South China University of Technology, Guangzhou, People's Republic of China.
| | - Fumito Araoka
- Physicochemical Soft Matter Research Team, RIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan.
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10
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Zhang X, Zhou Y, Mee Song K, Park TE, Xia J, Ezawa M, Liu X, Zhao W, Zhao G, Woo S. Skyrmion-electronics: writing, deleting, reading and processing magnetic skyrmions toward spintronic applications. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:143001. [PMID: 31689688 DOI: 10.1088/1361-648x/ab5488] [Citation(s) in RCA: 59] [Impact Index Per Article: 14.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
The field of magnetic skyrmions has been actively investigated across a wide range of topics during the last decades. In this topical review, we mainly review and discuss key results and findings in skyrmion research since the first experimental observation of magnetic skyrmions in 2009. We particularly focus on the theoretical, computational and experimental findings and advances that are directly relevant to the spintronic applications based on magnetic skyrmions, i.e. their writing, deleting, reading and processing driven by magnetic field, electric current and thermal energy. We then review several potential applications including information storage, logic computing gates and non-conventional devices such as neuromorphic computing devices. Finally, we discuss possible future research directions on magnetic skyrmions, which also cover rich topics on other topological textures such as antiskyrmions and bimerons in antiferromagnets and frustrated magnets.
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Affiliation(s)
- Xichao Zhang
- School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, People's Republic of China
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11
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Pöllath S, Aqeel A, Bauer A, Luo C, Ryll H, Radu F, Pfleiderer C, Woltersdorf G, Back CH. Ferromagnetic Resonance with Magnetic Phase Selectivity by Means of Resonant Elastic X-Ray Scattering on a Chiral Magnet. PHYSICAL REVIEW LETTERS 2019; 123:167201. [PMID: 31702336 DOI: 10.1103/physrevlett.123.167201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2019] [Revised: 07/24/2019] [Indexed: 06/10/2023]
Abstract
Cubic chiral magnets, such as Cu_{2}OSeO_{3}, exhibit a variety of noncollinear spin textures, including a trigonal lattice of spin whirls, the so-called skyrmions. Using magnetic resonant elastic x-ray scattering (REXS) on a crystalline Bragg peak and its magnetic satellites while exciting the sample with magnetic fields at gigahertz frequencies, we probe the ferromagnetic resonance (FMR) modes of these spin textures by means of the scattered intensity. Most notably, the three eigenmodes of the skyrmion lattice are detected with large sensitivity. As this novel technique, which we label REXS FMR, is carried out at distinct positions in reciprocal space, it allows us to distinguish contributions originating from different magnetic states, providing information on the precise character, weight, and mode mixing as a prerequisite of tailored excitations for applications.
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Affiliation(s)
- S Pöllath
- Institut für Experimentelle Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - A Aqeel
- Physik-Department, Technische Universität München, D-85748 Garching, Germany
| | - A Bauer
- Physik-Department, Technische Universität München, D-85748 Garching, Germany
| | - C Luo
- Physik-Department, Technische Universität München, D-85748 Garching, Germany
- Helmholtz-Zentrum Berlin für Materialien and Energie, D-12489 Berlin, Germany
| | - H Ryll
- Helmholtz-Zentrum Berlin für Materialien and Energie, D-12489 Berlin, Germany
| | - F Radu
- Helmholtz-Zentrum Berlin für Materialien and Energie, D-12489 Berlin, Germany
| | - C Pfleiderer
- Physik-Department, Technische Universität München, D-85748 Garching, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, D-80799 München, Germany
| | - G Woltersdorf
- Institut für Physik, Universität Halle-Wittenberg, D-06120 Halle (Saale), Germany
| | - C H Back
- Institut für Experimentelle Physik, Universität Regensburg, D-93040 Regensburg, Germany
- Physik-Department, Technische Universität München, D-85748 Garching, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, D-80799 München, Germany
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12
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Abstract
Abstract
In this article, we focus on (1) type-II multiferroics driven by spiral spin orderings and (2) magnetoelectric couplings in multiferroic skyrmion-hosting materials. We present both phenomenological understanding and microscopic mechanisms for spiral spin state, which is one of the essential starting points for type-II multiferroics and magnetic skyrmions. Two distinct mechanisms of spiral spin states (frustration and Dzyaloshinskii–Moriya [DM] interaction) are discussed in the context of the lattice symmetry. We also discuss the spin-induced ferroelectricity on the basis of the symmetry and microscopic atomic configurations. We compare two well-known microscopic models: the generalized inverse DM mechanism and the metal-ligand d-p hybridization mechanism. As a test for these models, we summarize the multiferroic properties of a family of triangular-lattice antiferromagnets. We also give a brief review of the magnetic skyrmions. Three types of known skyrmion-hosting materials with multiferroicity are discussed from the view point of crystal structure, magnetism, and origins of the magnetoelectric couplings. For exploration of new skyrmion-hosting materials, we also discuss the theoretical models for stabilizing skyrmions by magnetic frustration in centrosymmetric system. Several basic ideas for material design are given, which are successfully demonstrated by the recent experimental evidences for the skyrmion formation in centrosymmetric frustrated magnets.
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Affiliation(s)
- Takashi Kurumaji
- Physics , Massachusetts Institute of Technology , Cambridge , MA, USA
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13
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Functional Ferroic Domain Walls for Nanoelectronics. MATERIALS 2019; 12:ma12182927. [PMID: 31510049 PMCID: PMC6766344 DOI: 10.3390/ma12182927] [Citation(s) in RCA: 34] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/20/2019] [Revised: 09/05/2019] [Accepted: 09/06/2019] [Indexed: 11/17/2022]
Abstract
A prominent challenge towards novel nanoelectronic technologies is to understand and control materials functionalities down to the smallest scale. Topological defects in ordered solid-state (multi-)ferroic materials, e.g., domain walls, are a promising gateway towards alternative sustainable technologies. In this article, we review advances in the field of domain walls in ferroic materials with a focus on ferroelectric and multiferroic systems and recent developments in prototype nanoelectronic devices.
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14
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Qian F, Bannenberg LJ, Wilhelm H, Chaboussant G, Debeer-Schmitt LM, Schmidt MP, Aqeel A, Palstra TTM, Brück E, Lefering AJE, Pappas C, Mostovoy M, Leonov AO. New magnetic phase of the chiral skyrmion material Cu 2OSeO 3. SCIENCE ADVANCES 2018; 4:eaat7323. [PMID: 30255145 PMCID: PMC6155131 DOI: 10.1126/sciadv.aat7323] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 04/03/2018] [Accepted: 08/08/2018] [Indexed: 06/08/2023]
Abstract
The lack of inversion symmetry in the crystal lattice of magnetic materials gives rise to complex noncollinear spin orders through interactions of a relativistic nature, resulting in interesting physical phenomena, such as emergent electromagnetism. Studies of cubic chiral magnets revealed a universal magnetic phase diagram composed of helical spiral, conical spiral, and skyrmion crystal phases. We report a remarkable deviation from this universal behavior. By combining neutron diffraction with magnetization measurements, we observe a new multidomain state in Cu2OSeO3. Just below the upper critical field at which the conical spiral state disappears, the spiral wave vector rotates away from the magnetic field direction. This transition gives rise to large magnetic fluctuations. We clarify the physical origin of the new state and discuss its multiferroic properties.
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Affiliation(s)
- Fengjiao Qian
- Faculty of Applied Sciences, Delft University of Technology, Mekelweg 15, 2629JB Delft, Netherlands
| | - Lars J. Bannenberg
- Faculty of Applied Sciences, Delft University of Technology, Mekelweg 15, 2629JB Delft, Netherlands
| | - Heribert Wilhelm
- Diamond Light Source Ltd., Chilton, Didcot, Oxfordshire OX11 0DE, UK
| | | | | | - Marcus P. Schmidt
- Max Planck Institute for Chemical Physics of Solids, Nöthnitzer-Straße 40, 01187 Dresden, Germany
| | - Aisha Aqeel
- Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, Netherlands
| | - Thomas T. M. Palstra
- Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, Netherlands
| | - Ekkes Brück
- Faculty of Applied Sciences, Delft University of Technology, Mekelweg 15, 2629JB Delft, Netherlands
| | - Anton J. E. Lefering
- Faculty of Applied Sciences, Delft University of Technology, Mekelweg 15, 2629JB Delft, Netherlands
| | - Catherine Pappas
- Faculty of Applied Sciences, Delft University of Technology, Mekelweg 15, 2629JB Delft, Netherlands
| | - Maxim Mostovoy
- Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, Netherlands
| | - Andrey O. Leonov
- Department of Chemistry, Hiroshima University, 1-3-1, Kagamiyama, Higashi-Hiroshima 739-8526, Japan
- Chiral Research Center, Hiroshima University, 1-3-1, Kagamiyma, Higashi Hiroshima, Hiroshima 739-8526, Japan
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15
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Huang P, Cantoni M, Kruchkov A, Rajeswari J, Magrez A, Carbone F, Rønnow HM. In Situ Electric Field Skyrmion Creation in Magnetoelectric Cu 2OSeO 3. NANO LETTERS 2018; 18:5167-5171. [PMID: 30040904 DOI: 10.1021/acs.nanolett.8b02097] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Exploiting additional degrees of freedom in solid-state materials may be the most-promising solution when approaching the quantum limit of Moore's law for the conventional electronic industry. Recently discovered topologically nontrivial spin textures, skyrmions, are outstanding among such possibilities. However, the controlled creation of skyrmions, especially by electric means, remains a pivotal challenge in technological applications. Here, we report that skyrmions can be created locally via electric field in the magnetoelectric helimagnet Cu2OSeO3. Using Lorentz transmission electron microscopy, we successfully write skyrmions in situ from a helical-spin background. Our discovery is highly coveted because it implies that skyrmionics can be integrated into modern field effect transistor based electronic technology, in which very low energy dissipation can be achieved and, hence, realize a large step forward toward its practical applications.
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Affiliation(s)
| | - Marco Cantoni
- Centre Interdisciplinaire de Microscopie Électronique (CIME) , École Polytechnique Fédérale de Lausanne (EPFL) , CH-1015 Lausanne , Switzerland
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16
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Kruchkov AJ, White JS, Bartkowiak M, Živković I, Magrez A, Rønnow HM. Direct electric field control of the skyrmion phase in a magnetoelectric insulator. Sci Rep 2018; 8:10466. [PMID: 29992965 PMCID: PMC6041276 DOI: 10.1038/s41598-018-27882-4] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2018] [Accepted: 05/25/2018] [Indexed: 11/09/2022] Open
Abstract
Magnetic skyrmions are topologically protected spin-whirls currently considered as promising for use in ultra-dense memory devices. Towards achieving this goal, exploration of the skyrmion phase response and under external stimuli is urgently required. Here we show experimentally, and explain theoretically, that in the magnetoelectric insulator Cu2OSeO3 the skyrmion phase can expand and shrink significantly depending on the polarity of a moderate applied electric field (few V/μm). The theory we develop incorporates fluctuations around the mean-field that clarifies precisely how the electric field provides direct control over the free energy difference between the skyrmion and the surrounding conical phase. The quantitative agreement between theory and experiment provides a solid foundation for the development of skyrmionic applications based on magnetoelectric coupling.
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Affiliation(s)
- A J Kruchkov
- Department of Physics, Harvard University, Cambridge, MA, 02138, USA.
- Laboratory for Quantum Magnetism (LQM), Insititute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland.
| | - J S White
- Laboratory for Neutron Scattering and Imaging (LNS), Paul Scherrer Institut (PSI), CH-5232, Villigen, Switzerland
| | - M Bartkowiak
- Laboratory for Scientific Developments and Novel Materials (LDM), Paul Scherrer Institut (PSI), CH-5232, Villigen, Switzerland
| | - I Živković
- Laboratory for Quantum Magnetism (LQM), Insititute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland
| | - A Magrez
- Crystal Growth Facility, Insititute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland
| | - H M Rønnow
- Laboratory for Quantum Magnetism (LQM), Insititute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland
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17
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Hou Z, Zhang Q, Xu G, Gong C, Ding B, Wang Y, Li H, Liu E, Xu F, Zhang H, Yao Y, Wu G, Zhang XX, Wang W. Creation of Single Chain of Nanoscale Skyrmion Bubbles with Record-High Temperature Stability in a Geometrically Confined Nanostripe. NANO LETTERS 2018; 18:1274-1279. [PMID: 29299928 DOI: 10.1021/acs.nanolett.7b04900] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Nanoscale topologically nontrivial spin textures, such as magnetic skyrmions, have been identified as promising candidates for the transport and storage of information for spintronic applications, notably magnetic racetrack memory devices. The design and realization of a single skyrmion chain at room temperature (RT) and above in the low-dimensional nanostructures are of great importance for future practical applications. Here, we report the creation of a single skyrmion bubble chain in a geometrically confined Fe3Sn2 nanostripe with a width comparable to the featured size of a skyrmion bubble. Systematic investigations on the thermal stability have revealed that the single chain of skyrmion bubbles can keep stable at temperatures varying from RT up to a record-high temperature of 630 K. This extreme stability can be ascribed to the weak temperature-dependent magnetic anisotropy and the formation of edge states at the boundaries of the nanostripes. The realization of the highly stable skyrmion bubble chain in a geometrically confined nanostructure is a very important step toward the application of skyrmion-based spintronic devices.
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Affiliation(s)
- Zhipeng Hou
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences , Beijing 100190, China
- Physical Science and Engineering, King Abdullah University of Science and Technology (KAUST) , Thuwal 23955-6900, Saudi Arabia
| | - Qiang Zhang
- Physical Science and Engineering, King Abdullah University of Science and Technology (KAUST) , Thuwal 23955-6900, Saudi Arabia
| | - Guizhou Xu
- School of Materials Science and Engineering, Nanjing University of Science and Technology , Nanjing 210094, China
| | - Chen Gong
- Physical Science and Engineering, King Abdullah University of Science and Technology (KAUST) , Thuwal 23955-6900, Saudi Arabia
| | - Bei Ding
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences , Beijing 100190, China
| | - Yue Wang
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences , Beijing 100190, China
| | - Hang Li
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences , Beijing 100190, China
| | - Enke Liu
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences , Beijing 100190, China
| | - Feng Xu
- School of Materials Science and Engineering, Nanjing University of Science and Technology , Nanjing 210094, China
| | - Hongwei Zhang
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences , Beijing 100190, China
| | - Yuan Yao
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences , Beijing 100190, China
| | - Guangheng Wu
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences , Beijing 100190, China
| | - Xi-Xiang Zhang
- Physical Science and Engineering, King Abdullah University of Science and Technology (KAUST) , Thuwal 23955-6900, Saudi Arabia
| | - Wenhong Wang
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences , Beijing 100190, China
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18
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Wild J, Meier TNG, Pöllath S, Kronseder M, Bauer A, Chacon A, Halder M, Schowalter M, Rosenauer A, Zweck J, Müller J, Rosch A, Pfleiderer C, Back CH. Entropy-limited topological protection of skyrmions. SCIENCE ADVANCES 2017; 3:e1701704. [PMID: 28975152 PMCID: PMC5621974 DOI: 10.1126/sciadv.1701704] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2017] [Accepted: 09/07/2017] [Indexed: 05/27/2023]
Abstract
Magnetic skyrmions are topologically protected whirls that decay through singular magnetic configurations known as Bloch points. We used Lorentz transmission electron microscopy to infer the energetics associated with the topological decay of magnetic skyrmions far from equilibrium in the chiral magnet Fe1-x Co x Si. We observed that the lifetime τ of the skyrmions depends exponentially on temperature, [Formula: see text]. The prefactor τ0 of this Arrhenius law changes by more than 30 orders of magnitude for small changes of the magnetic field, reflecting a substantial reduction of the lifetime of skyrmions by entropic effects and, thus, an extreme case of enthalpy-entropy compensation. Such compensation effects, being well known across many different scientific disciplines, affect topological transitions and, thus, topological protection on an unprecedented level.
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Affiliation(s)
- Johannes Wild
- Institut für Experimentelle Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - Thomas N. G. Meier
- Institut für Experimentelle Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - Simon Pöllath
- Institut für Experimentelle Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - Matthias Kronseder
- Institut für Experimentelle Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - Andreas Bauer
- Physik Department, Technische Universität Bremen, D-85748 Garching, Germany
| | - Alfonso Chacon
- Physik Department, Technische Universität Bremen, D-85748 Garching, Germany
| | - Marco Halder
- Physik Department, Technische Universität Bremen, D-85748 Garching, Germany
| | - Marco Schowalter
- Institut für Festkörperphysik, Universität Bremen, Otto-Hahn-Allee 1, D-28359 Bremen, Germany
| | - Andreas Rosenauer
- Institut für Festkörperphysik, Universität Bremen, Otto-Hahn-Allee 1, D-28359 Bremen, Germany
| | - Josef Zweck
- Institut für Experimentelle Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - Jan Müller
- Institut für Theoretische Physik, Universität zu Köln, D-50937 Köln, Germany
| | - Achim Rosch
- Institut für Theoretische Physik, Universität zu Köln, D-50937 Köln, Germany
| | | | - Christian H. Back
- Institut für Experimentelle Physik, Universität Regensburg, D-93040 Regensburg, Germany
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19
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Kanazawa N, Seki S, Tokura Y. Noncentrosymmetric Magnets Hosting Magnetic Skyrmions. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2017; 29:1603227. [PMID: 28306166 DOI: 10.1002/adma.201603227] [Citation(s) in RCA: 44] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2016] [Revised: 11/30/2016] [Indexed: 06/06/2023]
Abstract
The concept of a skyrmion, which was first introduced by Tony Skyrme in the field of particle physics, has become widespread in condensed matter physics to describe various topological orders. Skyrmions in magnetic materials have recently received particular attention; they represent vortex-like spin structures with the character of nanometric particles and produce fascinating physical properties rooted in their topological nature. Here, a series of noncentrosymmetric ferromagnets hosting skyrmions is reviewed: B20 metals, Cu2 OSeO3 , Co-Zn-Mn alloys, and GaV4 S8 , where Dzyaloshinskii-Moriya interaction plays a key role in the stabilization of skyrmion spin texture. Their topological spin arrangements and consequent emergent electromagnetic fields give rise to striking features in transport and magnetoelectric properties in metals and insulators, such as the topological Hall effect, efficient electric-drive of skyrmions, and multiferroic behavior. Such electric controllability and nanometric particle natures highlight magnetic skyrmions as a potential information carrier for high-density magnetic storage devices with excellent energy efficiency.
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Affiliation(s)
- Naoya Kanazawa
- Department of Applied Physics, University of Tokyo, Tokyo, 113-8656, Japan
| | - Shinichiro Seki
- RIKEN Center for Emergent Matter Science (CEMS), Wako, 351-0198, Japan
| | - Yoshinori Tokura
- Department of Applied Physics, University of Tokyo, Tokyo, 113-8656, Japan
- RIKEN Center for Emergent Matter Science (CEMS), Wako, 351-0198, Japan
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20
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Ozawa R, Hayami S, Motome Y. Zero-Field Skyrmions with a High Topological Number in Itinerant Magnets. PHYSICAL REVIEW LETTERS 2017; 118:147205. [PMID: 28430467 DOI: 10.1103/physrevlett.118.147205] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/27/2016] [Indexed: 06/07/2023]
Abstract
Magnetic Skyrmions are swirling spin textures with topologically protected noncoplanarity. Recently, Skyrmions with the topological number of unity have been extensively studied in both experiment and theory. We here show that a Skyrmion crystal with an unusually high topological number of two is stabilized in itinerant magnets at a zero magnetic field. The results are obtained for a minimal Kondo lattice model on a triangular lattice by an unrestricted large-scale numerical simulation and variational calculations. We find that the topological number can be switched by a magnetic field as 2↔1↔0. The Skyrmion crystals are formed by the superpositions of three spin density waves induced by the Fermi surface effect, and hence, the size of Skyrmions can be controlled by the band structure and electron filling. We also discuss the charge and spin textures of itinerant electrons in the Skyrmion crystals which are directly obtained in our numerical simulations.
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Affiliation(s)
- Ryo Ozawa
- Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
| | - Satoru Hayami
- Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
| | - Yukitoshi Motome
- Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
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