Magnetic Skyrmions and Their Applications 9780128208151, 9780128209332

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Magnetic Skyrmions and Their Applications

Woodhead Publishing Series in Electronic and Optical Materials

Magnetic Skyrmions and Their Applications Edited by

Giovanni Finocchio Christos Panagopoulos

An imprint of Elsevier

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2021 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-820815-1 ISBN: 978-0-12-820933-2 For information on all Woodhead publications visit our website at https://www.elsevier.com/books-and-journals

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Contributors

Gabriele Bonanno Department of Engineering, University of Messina, Messina, Italy Felix B€ uttner Helmholtz-Zentrum Berlin, Hahn-Meitner-Platz 1, Berlin, Germany Mario Carpentieri Department of Electrical and Information Engineering, Politecnico of Bari, Bari, Italy Xing Chen Fert Beijing Research Institute, BDBC, and School of Microelectronics, Beihang University, Beijing, People’s Republic of China Oksana Chubykalo-Fesenko Instituto de Ciencia de Materiales de Madrid, CSIC, Madrid, Spain Giuseppina D’Aguı` Department of Engineering, University of Messina, Messina, Italy Konstantin Denisov Ioffe Institute, St. Petersburg, Russian Federation Motohiko Ezawa Department of Applied Physics, The University of Tokyo, Tokyo, Japan Peter Fischer Lawrence Berkeley National Laboratory, Berkeley, CA, United States Hans J. Hug Empa, Swiss Federal Laboratories for Materials Science and Technology, D€ ubendorf; Department of Physics, University of Basel, Basel, Switzerland Wang Kang Fert Beijing Research Institute, BDBC, and School of Microelectronics, Beihang University, Beijing, People’s Republic of China Mathias Kl€ aui Institute of Physics, Johannes Gutenberg University Mainz, Mainz, Germany William Legrand Unite Mixte de Physique, CNRS, Thales, Universite Paris-Saclay, Palaiseau, France

x

Contributors

Na Lei Fert Beijing Research Institute, BDBC, and School of Microelectronics, Beihang University, Beijing, People’s Republic of China Andrey O. Leonov Chirality Research Center, Hiroshima University; Department of Chemistry, Faculty of Science, Hiroshima University Kagamiyama, HigashiHiroshima, Hiroshima, Japan; IFW Dresden, Dresden, Germany Sai Li Fert Beijing Research Institute, BDBC, and School of Microelectronics, Beihang University, Beijing, People’s Republic of China Kai Litzius Massachusetts Institute of Technology, Department of Materials Science and Engineering, Cambridge, MA, United States Xiaoxi Liu Department of Electrical and Computer Engineering, Shinshu University, Nagano, Japan Jacques Miltat Laboratoire de Physique des Solides, Universite Paris-Saclay, CNRS, Orsay, France Catherine Pappas Faculty of Applied Sciences, Delft University of Technology, Delft, The Netherlands Stanislas Rohart Laboratoire de Physique des Solides, Universite Paris-Saclay, CNRS, Orsay, France Sujoy Roy Lawrence Berkeley National Laboratory, Berkeley, CA, United States Igor Rozhansky Ioffe Institute, St. Petersburg, Russian Federation Luis Sa´nchez-Tejerina Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences; Department of Biomedical, Dental, Morphological and Functional Imaging Sciences, University of Messina, Messina, Italy Laichuan Shen School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, People’s Republic of China Andre Thiaville Laboratoire de Physique des Solides, Universite Paris-Saclay, CNRS, Orsay, France Riccardo Tomasello Institute of Applied and Computational Mathematics, Heraklion-Crete, Greece Oleg A. Tretiakov School of Physics, The University of New South Wales, Sydney, NSW, Australia

Contributors

xi

Seonghoon Woo IBM Thomas J. Watson Research Center, Yorktown Heights, NY, United States Jing Xia School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, People’s Republic of China Xichao Zhang School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, People’s Republic of China Xueying Zhang Fert Beijing Research Institute, BDBC, and School of Microelectronics, Beihang University, Beijing, People’s Republic of China Weisheng Zhao Fert Beijing Research Institute, BDBC, and School of Microelectronics, Beihang University, Beijing, People’s Republic of China Yan Zhou School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, People’s Republic of China Daoqian Zhu Fert Beijing Research Institute, BDBC, and School of Microelectronics, Beihang University, Beijing, People’s Republic of China Roberto Zivieri Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Messina, Italy

Preface

This book discusses fundamental concepts and research activities in the rapidly growing field of magnetic skyrmions. These are particle-like objects that are topologically stable, highly mobile, and have the smallest magnetic configurations, making them promising for technological applications, including spintronics and neuromorphic computing. Paradoxically, skyrmions were expected to be short-time excitations, quickly collapsing into point or linear singularities. However, the seminal work of A.N. Bogdanov and D.N. Yablonsky in 1989 indicated skyrmions may exist as long-living metastable configurations in low-symmetry condensed matter systems with broken mirror symmetry. For example, in noncentrosymmetric magnetic materials, the underlying crystallographic handedness imposes a unique stabilization mechanism (Dzyaloshinskii-Moriya interaction) for two- and three-dimensional localized states. Hence, material classes expected to host skyrmions may include noncentrosymmetric ferro- and antiferromagnets and multiferroics where the topological field is the magnetization. A similar stabilization mechanism for chiral skyrmions may apply in liquid crystals and ferroelectrics. Expectedly, the scientific and technological relevance of chiral skyrmions is fuelling research in novel classes of materials. Magnetic skyrmions have been reported in a plethora of materials, including bulk ferromagnets, ferrimagnets, and multiferroics. Furthermore, one can engineer a structure that cannot be inverted. For example, by breaking symmetry through the interface between two different materials. An interface between a ferromagnet and a strong spin-orbit metal gives rise to the necessary Dzyaloshinskii-Moriya interaction, even if both metals have inversion-symmetric lattices. This book discusses the relevant fundamental concepts and results in this exciting field of research. Equal weight is dedicated on the realm of technological applications. The chapters start with a description of topologically stable structures in one-, two-, and three-dimensional space. The Thiele equation is derived from a fundamental micromagnetic description of statics and dynamics. A pure mathematical description of topology is included later for a rigorous definition of topological and metric spaces and compactness. Realization of magnetic skyrmions in condensed matter physics propelled the development of numerous materials architectures and advances in experimental methods of characterization. Materials hosting magnetic skyrmions now range from bulk single crystals to synthetic architectures. High-resolution magnetic imaging has been at the center of skyrmion research and development. A comprehensive

xiv

Preface

description provided in this book helps the reader appreciate techniques that continue to make pioneering contributions in this field of research. Intelligent device configurations employed to engineer skyrmion states in materials that would otherwise not accommodate such spin structures are discussed. In addition, exotic structures characterized by anisotropic Dzyaloshinskii-Moriya interaction and Ruderman-Kittel-Kasuya-Yosida interaction to host antiskyrmions and synthetic antiferromagnetic skyrmions are discussed. For instance, skyrmion stabilization and detection in antiferromagnets is at an early stage, facing many challenges ahead. Theoretical foundations however encourage further material development and characterization for the stabilization of skyrmions in antiferromagnetic structures. The community is increasingly active in this direction. Ferrimagnets represent another promising material platform for skyrmionics. Recent advances in imaging, writing, deleting, and electrical detection of ferrimagnetic skyrmions demonstrate their promising impact toward device technology. An account on the statics and dynamics of magnetic skyrmions gives a glimpse in the key attributes of their technological potential. A detailed description of dynamics in the presence of thermal fluctuations is discussed in the light of recent proof of concept of skyrmion-based devices for probabilistic computing. Attention is given to the definition of the skyrmion configurational entropy based on the concept of Boltzmann order function, useful for the theoretical understanding of phase transitions in materials hosting skyrmions. A direction highly relevant for applications is electrical manipulation. A strategy to nucleate skyrmions with electric currents is discussed, considering nucleation via spatial inhomogeneity, local injection of spin-transfer torque, and voltage-controlled magnetic anisotropy. Although the goal is the integration of magnetic tunnel junctions in multilayers hosting skyrmions, the topological Hall effect has proven an effective approach for electrical detection. Analysis of the topological Hall effect in the presence of magnetic skyrmions provides an up to date account of research and discusses pressing challenges. The discussion on the dynamics of skyrmions addresses also the so-called skyrmion Hall angle. Oblique trajectories followed by the skyrmion when moved by spin-orbit torques is a limiting factor in applications such as the racetrack memory. This has motivated the research community further to explore ferrimagnetic and antiferromagnetic materials as energy-efficient skyrmion hosts. Skyrmion-based devices have the potential to store and process information at unprecedentedly small sizes and levels of energy consumption. The presence/absence of a skyrmion could serve as 1/0 in a data bit and multiple skyrmions can aggregate toward multivalued storage devices. The states of such devices can be modulated by an electric current, driving skyrmions in and out of devices in analogy to biological synapses. Researchers have already engineered interfacial skyrmions up to room temperature in magnetic multilayers, making their promise for future technologies more realistic. This offers an opportunity to bring topology into consumer-friendly, lowenergy nanoscale electronics. The ability we have gained to engineer skyrmion-host platforms is propelling new technological opportunities. For example, conventional applications of skyrmions for the realization of memories, logic gates, transistors, and radio frequency circuits such as oscillators. The application of magnetic

Preface

xv

skyrmions in unconventional computing is a tantalizing possibility including proposals for the realization of skyrmion-based memristors. One may also envisage energetically efficient neurons and synapses at device level toward low-power neuromorphics hardware. It has been a pleasure to work with the co-authors and the Elsevier team. Their contributions have made possible the delivery of an inclusive description of the rapidly growing research field of magnetic skyrmions. Giovanni Finocchio Christos Panagopoulos

Acknowledgments

G.F. acknowledges project ThunderSKY, funded by the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under Grant No. 871 and PETASPIN Association. C.P. acknowledges support from the Singapore National Research Foundation (NRF) NRF-Investigatorship (No. NRFNRFI2015-04) and Singapore MOE Academic Research Fund Tier 3 Grant MOE2018-T3-1-002.

Magnetism and topology Andr
e Thiavillea, Jacques Miltata, and Stanislas Roharta a Laboratoire de Physique des Solides, Universit
e Paris-Saclay, CNRS, Orsay, France

1.1

1

Introduction

The application of topology to describe real-space structures in condensed matter has developed over the years, in fact as topology itself was being rigorously constructed. The first structures treated by this approach were the defects, like dislocation lines in crystal lattices, or the lines observed in liquid crystals, at the beginning of the 20th century. The methodology was then extended to structures that contain no defect. We shall keep this historical order for the first half of this chapter, dealing with the static description of magnetic structures (also called magnetic textures). The second half discusses the dynamics of the magnetic structures. In this chapter, mathematical formalism is kept to a minimum. The reader completely unaware of this field of mathematics should refer to textbooks [1, 2] or review papers [3] and, if a deeper mathematical view is desired, to the last chapter of this book. Defects can appear only when there is an underlying order. In condensed matter physics, there are many cases where phases with different types or degree of order appear in sequence, for example, when temperature or pressure is changed. In the Landau theory of phase transitions, order is characterized by the existence of a certain physical quantity called order parameter, which can be a scalar, a complex number, a vector, a tensor, etc. This order parameter is also endowed with a certain number of degrees of freedom, defining a space V called the manifold of internal states. To be specific, let us enumerate the cases for magnetism. The order parameter is the magnetic moment. Depending on the type of magnetism (localized or itinerant), magnetic order (ferromagnetic, ferrimagnetica), and also atomic structure (single crystal, polycrystal, amorphous), the distribution of the magnetic moment on the atomic scale will be very different. However, on the mesoscopic lengthscale where micromagnetics [4, 5] reigns, these atomic subtleties are polished out, leaving only a magnetic moment that is a continuous function of space and time. This moment has a zero thermodynamical average above the Curie temperature, and nonzero one below this temperature, leading to a spontaneous magnetization Ms(T) a sole function of temperature T. The degrees of freedom correspond to the orientation of this moment, and they ! ! ! are described by a unit vector m ð r , tÞ, a function of position r , and time t. Our world being three dimensional (3D), three cases are considered.

a

Note that we do not consider in this chapter the antiferromagnets, for which the order parameter is different.

Magnetic Skyrmions and Their Applications. https://doi.org/10.1016/B978-0-12-820815-1.00012-2 Copyright © 2021 Elsevier Ltd. All rights reserved.

2

l

l

l

Magnetic Skyrmions and Their Applications !

Heisenberg model: Magnetization has n ¼ 3 components as m can take any orientation in 3D, the manifold of internal states is V ¼ 2 , and the unit sphere in three dimensions is described by two variables (e.g., the polar angle θ and the azimuthal angle ϕ), hence the exponent 2. Physically this should be the only case, in general. It is, however, sometimes interesting to consider the two other limiting cases. XY model: This case corresponds to an easy plane, magnetization has n ¼ 2 components, the out-of-plane anisotropy energy being (mathematically) infinite. Physically, it means that other energy terms are much smaller. This can be a description of a soft magnetic thin film, or of a weak ferromagnet (e.g., the orthoferrites) where the Dzyaloshinskii-Moriya interaction confines the ferromagnetic moment to the plane normal to its vector. The manifold of internal states is V ¼ 1 , and the unit circle in the (easy) plane, is described by just one variable (the angle). Ising model: The last case is complementary to the preceding one, with an easy axis of (mathematically) infinite anisotropy, so that magnetization has just n ¼ 1 component. The manifold of internal states is V ¼ 0 , this notation meaning just the two points +1 and 1.

The internal space V plays a capital role in the topological formalism.

1.2

Topological defects in magnetism

In a medium with order, a defect is a point or a collection of points where the order parameter is not defined; one can have a point-like defect, or a defective line, or a defective surface, and so on for space dimensions d that would be larger than 3. The defect is said to be topological when, whatever the continuous modifications exerted on the order parameter distribution over space, it cannot be removed. In short, this defect cannot be cured. The defect dimensionality is called d0 ; its value is d0 ¼ 0 for a point, d0 ¼ 1 for a line, and d0 ¼ 2 for a surface. The dimension of the physical space being denoted d, one has d0 < d. Note that the defect, in this idealization of the order parameter as a continuous function, has no width, it is just a discontinuity of the order parameter. But how can the presence or not of a defect with no size be detected? This can be done with topology. The method is inspired by the Burgers circuit used to detect the presence of dislocation lines in atomic arrangements [6]. The generalization of this method is to consider a closed contour of dimension r in the physical space (i.e., the sphere r , or any continuous deformation of it). To look for a defect of dimension d0 in a space of dimension d, the contour dimension r has to satisfy [7, 8] 0

d + r + 1 ¼ d:

(1.1)

The case of a dislocation line (d0 ¼ 1) in the usual space with d ¼ 3 corresponds to a closed loop r ¼ 1. Similarly, to look for a point defect d0 ¼ 0 in the 3D space (d ¼ 3) requires a sphere-like surface hence r ¼ 2 (Fig. 1.1). Every magnetization texture, by the orientation of the magnetization vector, gives rise to an image of the r-dimensional closed contour on the parameter space V. In mathematical terminology, this is called a mapping. A continuous deformation of

Magnetism and topology

(A)

3

(B)

(C)

Fig. 1.1 Contours to look for topological defects in d ¼ 3 space. The contour dimensionality r is adapted to the defect dimensionality d0 according to Eq. (1.1), as illustrated here for (A) d0 ¼ 2, (B) d0 ¼ 1, and (C) d0 ¼ 0. The defects are drawn in red, and the associated contours in gray.

the magnetization texture leads to a continuous deformation of the image on the parameter space V. The branch of topology called homotopy precisely deals with that. Two images are said to be equivalent (belonging to the same homotopy class) if they can be continuously transformed one into the other. For every space V, there exists an object called the rth homotopy group, denoted π r(V), and each element of the group is a homotopy class. The group has an obvious “zero” (it is called neutral) element, which is the class of the contours whose image can be continuously shrunk to a point. The group is said to be trivial if it consists only of the neutral element. This means that all contours can be continuously contracted to a point. In that case, there can be no topological defect of dimension d0 ¼ d  r  1. Indeed, there will be no obstacle in shrinking, in the physical space, the r-dimensional contour, magnetization will be always defined on the contour and will finally converge to the magnetization vector at the center of the contour when its size reaches zero. The topologically interesting case is the nontrivial one, when the homotopy group has more than one element. In that case, for a contour that has an image on V which belongs to a nonneutral homotopy class, when shrinking it in physical space, at some moment the magnetization orientation has to be undefined: the defect will be met. For the case of magnetism, where V ¼ n1 , with n the number of magnetization components (see Section 1.1), the nontrivial homotopy groups with r < 3 are l

l

l

π 0 ð0 Þ ¼ 2 ≡ f0, 1g: for Ising spins, there exist topological defective surfaces in 3D, topological line defects in two-dimensional (2D), and topological point defects in onedimensional space (1D); π 1 ð1 Þ ¼  ≡ f0, + 1,  1, + 2,  2, …g: for XY spins, there exist topological linear defects in 3D space, and topological point defects in 2D; π 2 ð2 Þ ¼ : for Heisenberg spins, there are topological defects in the form of points in 3D space.

The well-known examples of the second case are the Kosterlitz-Thouless vortices and antivortices (Fig. 1.3), whereas in the third case, it is the Bloch point (Fig. 1.2).

4

Magnetic Skyrmions and Their Applications

(A)

(B)

(C)

(D)

Fig. 1.2 Bloch points for the Heisenberg model. The orientations of the magnetization on a sphere surrounding the Bloch point are schematically drawn. (A) So-called hedgehog Bloch point, with radial magnetization everywhere, covering the unit sphere + 1 time. The variant with opposite magnetization everywhere covers the unit sphere  1 time, so is topologically different. (B) So-called combed Bloch point, with a lower magnetostatic energy [10], which is found in most structures (see Section 1.5). This structure can be derived from (A) by a π/2 rotation of the magnetizations around the vertical axis, hence is topologically equivalent. (C) structure derived from (A) by a π rotation around the vertical axis, so again topologically equivalent to it. (D) Structure which has a S ¼ 1 winding number on the equator, with a p ¼ 1 axial magnetization on the poles, hence also covering + 1 time the sphere. It is also the same as (C), rotating the whole sample by π/2 around a horizontal axis.

Therefore, given that the XY and Ising models are only approximations, the only true topological defect in ferromagnetism is the Bloch point.

1.2.1 The Bloch point The structure and the name were invented by Ernst Feldtkeller in Ref. [9], a pioneering paper for the application of topology to magnetism. It is the first topological magnetic structure studied; it has the defining property that around it every magnetization orientation appears once exactly. Some schematic structures of the magnetization around it are drawn in Fig. 1.2. A surprising property of the Bloch point is that it has a finite energy. In the frame of micromagnetics, even if the exchange energy density diverges with distance r to the center like A/r2 around the Bloch point,b the integral of this density is finite, amounting to 8πAR, R being the radius up to which the Bloch point profile holds. On the other hand, going back to the atomic scale, having a Bloch point amounts to forcing to zero the magnetization in a volume of the size of an atom. Thus, if J is the exchange energy per nearest-neighbor bond, and Z is the number of nearest b

A is the micromagnetic exchange constant, proportional to the exchange energy J and inversely proportional to the lattice constant [5].

Magnetism and topology

5

neighbors, the cost of a Bloch point is about ZJ. For a simple cubic lattice, the two formulae are close for R ¼ a/2. This is the “core” cost of a BP, consisting of exchange only. The magnetostatic cost of a Bloch point was evaluated by D€oring [10]. It was found that the hedgehog Bloch point (Fig. 1.2A), being a monopole, has the highest energy. Conversely, the “combed” Bloch point (Fig. 1.2B) carries much less energy. There is also an anisotropy cost of a Bloch point. Altogether, the insertion energy of a Bloch point in a material with anisotropy (K) dominating magnetostatics is given by a pffiffiffiffiffiffiffiffiffi micromagnetic radius R  A=K [11]. In the framework of atomic micromagnetics, the exchange energy cost as a function of the position of the mathematical center of the Bloch point (in between the atoms) was studied by Reinhardt, who found the best interstitial positions for the Bloch point as well as its saddle point when moving from an atomic cell to another [12]. The friction linked to these saddle points was shown theoretically to be detectable in the motion of so-called Bloch point walls in nanowires [13].

1.2.2 The singular vortex As is apparent from Fig. 1.3, a vortex topological point defect for the XY model (therefore in 2D) can be seen as a cut through the center of a Bloch point, the point topological defect for the Heisenberg model, so in 3D. Regarding the vortex energy, the exchange energy density in the continuous micromagnetic formulation diverges again as A/r2, but now the radial integral also diverges logarithmically [14]. The divergence at small size is removed, in the theory papers, by introducing a finite lower bound for the integration over the radius. This indeed corresponds to the atomic micromagnetic calculation, which gives a finite result.

(A)

(B)

Fig. 1.3 Vortices for the XY model. The orientations of the magnetization around the vortex core, located at the origin, are figured by arrows. (A) Vortex, here in the radial variant. This structure can be continuously converted into the circulating variants (like the velocity field of a vortex in fluid dynamics), with both circulation directions. All these variants are described by the constant C in the relation ϕ ¼ φ + C, with φ the angle of the position vector, and ϕ the angle of the magnetization. (B) Antivortex, described by ϕ ¼ φ (any additional constant can be absorbed by a rotation of the coordinate axes).

6

1.3

Magnetic Skyrmions and Their Applications

Topologically stable structures (topological solitons) in magnetism

In the same pioneering paper by Feldtkeller [9], another topological effect relative to the magnetic structures was introduced. Consider a regular magnetic structure, that is, without any defect, as well as another structure, also without defect. The question is whether or not it is possible to continuously transform one structure into the other (a property called accessibility). This is again a topological question. It is not at all academic, as Feldtkeller was considering the possible obstacles to the magnetization reversal in ferromagnetic rings, used as bits in the ferrite core memory. When it is not possible to transform continuously one structure into another, the structures are said to be topologically stable or, equivalently, they are called topological solitons [15]. The idea behind the latter name is that, similarly to a soliton that keeps its shape while moving, a topological soliton keeps its topology while deforming (moving being one way of deforming). In order to apply homotopy arguments to this situation, we need to realize mappings from r-dimensional spheres r to the order parameter space V. A perfect case is therefore when the sample itself is a sphere, like a film grown on a sphere realizing 2 , or a nanoring realizing 1 . The first case is rare presently, and it could be realized by core-shell magnetic particles with large diameter. The second case has been studied in detail [16], but the magnetic structures (domain walls) are the same as when the nanoring is not closed; they are discussed in Section 1.3.2.3.

1.3.1 Uniform boundary conditions If the physical space is the infinite Euclidian space d , we cannot directly apply homotopy arguments as d is not equivalent to d , even if they have the same number of dimensions. But if, in addition, it is known that magnetization is uniform at infinity, then by a transformation similar to the stereographic projection of the sphere onto the plane, one can transform d to d . In such a situation, we see that all magnetic structures realize mappings of d to the order parameter space V. So they can be classified by the homotopy group π d(V). Let us be more explicit by specifying d, for V ¼ n1 (except the case corresponding to π 0 ð0 Þ ¼ 2 , that is not interesting).

1.3.1.1 Two-dimensional physical space (d52) This case is nontrivial as π 2 ð2 Þ ¼ . This means that the physical space is bidimensional (one can think of a film where nothing changes along the thickness, for example, an ultrathin film with a thickness not much above the magnetic exchange pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi length Λ ¼ 2A=μ0 Ms2 ). So topology tells us that for 3D moments (n  1 ¼ 2) on a plane, which are uniform at infinity (i.e., the structures considered are of finite extent),

Magnetism and topology

7

there exists an infinity of topologically different structures, labeled by the signed integers (called the topological index). Two structures with different index are mutually inaccessible by continuous deformation. Among these structures, only those with index zero can be continuously transformed to the uniform state. The simplest structures are drawn in Fig. 1.4. The structure with index +1 is called a skyrmion, which with index 1 is an antiskyrmion, and the others are higher-order skyrmions. The topological index is easy to calculate in this situation, as it is simply the number of times that the sphere 2 is covered by the mapping. If the plane 2 is described by (x, y) coordinates, we see from geometry that the differential surface covered on the sphere reads

(A)

(B)

(C)

(D)

Fig. 1.4 Skyrmions in a film. The in-plane components of the magnetization are depicted by black arrows, whereas the out-of-plane component is coded in color (blue – down, red – up, white – in-plane). (A) Antiskyrmion with an in-plane winding number S ¼ 1. (B) Nontopological structure with an in-plane winding number S ¼ 0. This structure can be continuously destroyed by a magnetization rotation around the vertical in-plane axis, the angle of rotation being π at the center, π/2 at the wall, and 0 at infinity. (C) A N
eel skyrmion with positive chirality. Note that, by π rotation of the moments around the perpendicular axis, this structure can be continuously transformed into a N
eel skyrmion with negative chirality, as well as into Bloch skyrmions with positive or negative chirality by π/2 rotation. (D) A higher-order skyrmion with an in-plane winding number S ¼ +2.

8

Magnetic Skyrmions and Their Applications

!

!

∂m ∂m  dΩ ¼ ∂x ∂y

! !

 m dxdy:

!

(1.2) !

Indeed, m is the local normal to the sphere, both partial derivatives of m belong to the local tangent plane to the sphere, and the modulus of their vector product is the sine of their angle, which gives the surface of the associated parallelogram. Note that the covered surface can be positive or negative. Altogether, the topological index is the integral of dΩ, divided by 4π which is the surface of the sphere. It is notable that, here, the topological index can be obtained by the space integral of a “topological density.” ! In the case of cylindrical symmetry, namely m ¼ ð sin θ cos ϕ, sin θ sin ϕ, cos θÞ with the spherical angles θ and ϕ (relative to the magnetization at infinity) being sole functions of the radius r and the in-plane angle φ, respectively, one obtains the useful relation Z Ω¼

sin θ

dθ dr dr

Z

dϕ dφ ¼ 4πSp: dφ

(1.3)

The two quantities involved are p ¼ ½ cos θð0Þ  cos θð∞Þ=2, the polarity of the core R 2π of the structure, and S ¼ 0 dϕ dφ dφ=ð2πÞ, the winding number of the planar magnetization component. Note that, if the structure considered does not possess cylindrical symmetry, it can be continuously deformed so as to show it. This formula is therefore general.

1.3.1.2 One-dimensional physical space (d51) This is another nontrivial case, due to the fact that π 1 ð1 Þ ¼ . The physical space is now 1D (a nanowire), the moments are XY (n  1 ¼ 1), and they are identical at both ends, so that the wire can be thought as a closed loop. A physical realization of this situation could be, for example, a nanowire where the moments are constrained to lie in the transverse plane, the magnetic structure being locally a helix. In this case, the topological index is simply the algebraic number of turns of the helix. Another realization of this situation could be an ultrathin film (normal to z) where the moments vary only along one direction x (e.g., a series of parallel domain walls). Then again we could have helices (moments in the (y, z) plane), or also cycloids (moments in the (x, z) plane), these planes being fixed by a strong Dzyaloshinskii-Moriya energy term. The topology is the same for both realizations.

1.3.1.3 Three-dimensional physical space (d53) There exists also π 3 ð2 Þ ¼ . Whereas the previous homotopy groups were rather intuitive, this one is not. It was only reported in 1931 by the mathematician Heinz Hopf [17]. A mapping of 3 to 2 is called a Hopf fibration, the name fiber coming

Magnetism and topology

(A)

9

Fig. 1.5 A hopfion in a vertically magnetized medium. (A) Perspective view. The gray torus figures the locus of the inplane oriented moments (the “wall” of the hopfion). The orientation of magnetization on this surface is depicted by arrows, the loops with the same color drawing the fibers of the structure, i.e., the loci of the points with the same magnetization. These loops are all linked once, this being the topological index of the structure. (B) Cut view through a vertical plane, with the cut of the torus superposed, allowing to see the magnetization inside the torus, which is akin to a reverse domain.

(B)

from the fact that each point on 2 is the image of a line of points in 3 , a fiber. By the same argument as before, 3 is 3 , when moments are identical at infinity, that is, we discuss finite 3D magnetic structures within a uniform background. The structures with index 1 are now called hopfions; they have been observed in hybrid magnetic liquid crystals [18] and predicted in nanostructures of noncentrosymmetric magnets [19] (Fig. 1.5). The topological index was described by Hopf, it is the linking number between any two fibers. It can also be calculated by integration of a density, albeit nonlocal [19a], which is related to the presently much studied Berry curvature [19b].

1.3.2 Nonuniform boundary conditions Nonuniform boundary conditions are frequent. The obvious case is the domain wall, where on both sides magnetization goes to different limits at infinity. Finite samples (like patterned microstructures) are another case, the boundary conditions being imposed by the energetically preferred magnetization orientations at the edges. Two emblematic cases of this are the disk, and the nanostrip. When additional constrains on the order parameter exist at the edge, other mathematical objects exist, called the relative homotopy groups. However, as the emphasis of this chapter is physics rather than mathematics, we shall directly describe the consequences of the boundary conditions, case by case.

10

Magnetic Skyrmions and Their Applications

1.3.2.1 The domain walls !

!

If it is enforced that magnetization goes to m1 at x !∞ and to a different m2 at x ! +∞, then a domain wall has to exist. In topological terms, it means that all paths ! ! in physical space connecting x ¼ ∞ to x ¼ +∞ have to go through m1 and m2 , so that they cannot be contracted to a point: a domain wall has indeed to exist. But we learn nothing by this tautology. The more interesting question is whether all walls satisfying this condition can be continuously transformed one into the other. This depends on the nature of the first homotopy group of the order parameter space π 1(V): if it is trivial then all these walls are topologically equivalent, but if it is not then topologically different walls exist. The latter case occurs for the XY model only (Fig. 1.6).

1.3.2.2 The vortex in a soft magnetic film In a soft magnetic film, the demagnetizing energy dominates the anisotropy term, so that magnetization is confined to the plane of the film. A perfect confinement would mean the XY model, for which we have seen that topological defects (the vortices, antivortices, etc.) exist for d ¼ 2, the present case. As confinement is not perfect in usual samples (finite magnetization, nonzero exchange), these topological defects are regularized by having the core magnetization “escape” in the third dimension, over pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a size proportional to the micromagnetic exchange length Λ ¼ 2A=ðμ0 Ms2 Þ (see Ref. [5, 20] for an analytic calculation of the vortex core profile in the limit of vanishing sample thickness). The question then is: are these structures topologically stable? The answer is Yes, if the magnetization is assumed to stay perfectly in the plane at infinity. In such a case, similar to the plane-to-sphere transformation in Section 1.3, the situation is the same as a finite disk with magnetization assumed to belong to the plane at the disk edge. Let us look at the edge structure first. It realizes a mapping of 1 (the edge) to 1 (the equator on the unit sphere, the locus of the in-plane magnetic moments). So this

(A)

(B)

Fig. 1.6 Topologically stable domain walls in a sample with XY moments. (A) Two N
eel walls with opposite rotation sense, between the same domains, cannot be converted continuously one into the other, if the magnetization in the domains is assumed to be fixed, as the moments are XY. (B) Corresponding paths on the unit circle 1 . For Heisenberg moments, path a2 can be transformed to path a1 by going over a pole, either above or below the plane of the drawing.

Magnetism and topology

11

structure is classified by π 1 ð1 Þ ¼ : it is topologically stable regarding the relevant topological index, which is the winding number of the in-plane magnetization. Thus, from the winding number of the (in-plane) edge magnetic structure, we know if we have a vortex (S ¼ +1), an antivortex (S ¼ 1), nothing (S ¼ 0), or more complex vortices or antivortices (jSj > 1). Physically, as the edge constrain arises from magne! tostatics, the avoidance of surface magnetic charges dictates that m be tangent to the edge. So, for a disk-shape sample, the stabilized winding number is +1. As we have 3D spins inside the disk, the core of the vortex is not singular, but regular with a purely out-of-plane moment mz ≡ p ¼ 1, p being called the polarity of the vortex. From the topology of the edge magnetization, we already know that the vortex cannot be continuously erased into a uniform structure. But could we continuously go from p ¼ +1 to p ¼ 1? The answer is No. Indeed, if this were the case, one would at some moment have the top surface of the film with p ¼ 1, and the bottom one with p ¼ 1. Then mapping the full surface of the disk (the top surface, glued to the bottom surface at the disk edge), which is like 2 , to the parameter space V ¼ 2 would give a full coverage, meaning that the disk contains a Bloch point in its interior: continuity would be broken. We thus see that additional constrains at the sample edge let other topologically stable structures appear. As they cover half of the sphere, the Heisenberg vortex or antivortex are also called merons. Note that this half-integer coverage index of 2 is not quantized: if a perpendicular field is applied to a disk with a vortex, the image of the edge on 2 leaves the equator, increasing or decreasing the coverage depending on the field direction with respect to the core polarity (see Ref. [21] for an experimental demonstration).

1.3.2.3 The soft magnetic nanostrips This case illustrates nicely how the topological objects transform when one progressively lifts the constrains on the magnetization orientation. The topological description provides a general scheme to understand the magnetic structures in the soft nanostrips [22, 23]. It has led to insightful “mechanical” models of the domain walls in these samples [24]. The nanostrip geometry is similar to the disk case, but now the edge is not closed. At an edge, combining the magnetostatic conditions of planar magnetization for the film and tangent magnetization at the edge, one obtains that the lowest energy magnetizations are in 0 . From the fact that π 0 ð0 Þ ¼ 2 , there exists one associated type of topological point defect, which is just a wall in the edge magnetization. These topological defects are regularized by lifting the tangency constrain, that is, by considering, at the edge, V ¼ 1 instead of 0 . Assuming that the magnetization in the two domains separated by the wall is fixed (see Section 1.3.2.1), the topological defects of 0 become topological solitons, in fact merons of π 1 ð1 Þ. Going one step further, by comparing the topological characters of the two edges (which is easy to do, e.g., when, on the left and right of the nanostrip area considered, it is known that magnetization is uniform), one can know if there exists a topological

12

Magnetic Skyrmions and Their Applications

(A)

(B)

(C) Fig. 1.7 Domain walls in soft nanostrips. (A) The vortex wall; (B) the (symmetric) transverse wall; (C) the symmetric transverse wall during its dynamical reversal of the transverse moment. The green points at the edge correspond to topological defects of π 0 ð0 Þ (the edge when tangential magnetization is assumed), which become merons of π 1 ð1 Þ (the edge when planar magnetization is assumed, with values fixed at infinity). The orange points correspond to topological defects of π 1 ð1 Þ (in-plane moments assumed in the sample), which become merons of π 2 ð2 Þ (Heisenberg spins, with in-plane spins far from the core), namely a vortex (A) and an antivortex (C).

defect of π 1 ð1 Þ inside. The typical cases are the vortex wall (with a vortex inside), the transverse wall (no vortex inside), and the transverse wall in the process of reverting (with an antivortex inside), see Fig. 1.7.

1.3.2.4 The vertical Bloch line The vertical Bloch line (VBL) is the second magnetic topological object studied, and it is a topological soliton. Lines in the domain walls of garnet films were intensively studied in the 1970s, in the wake of research about the bubble memories. These samples are single crystals of (at that time) few-micron thickness, epitaxially grown on a single crystal nonmagnetic garnet substrate. As the lattice parameters can be precisely matched, the crystalline defect density is extremely low (1 cm between dislocations, typically). The samples are, however, not perfectly uniform magnetically as the large unit cell of garnets (a cube of 1.3 nm edge) accommodates many substituants, whose concentration can vary from place to place. The magnetic garnets are also ferrimagnetic (in fact, ferrimagnetism was discovered in the garnets), with a tunable low

Magnetism and topology

13

magnetization density. They have a growth-induced perpendicular anisotropy originating from the atomic ordering of the substituants during the growth. The magnetic structure consists of parallel domain walls with a globally disordered structure, with as limiting cases the straight parallel domains array and the bubble lattice. As the sample thickness (few micrometers) is much larger than the domain wall width (a few tenths of a micrometer), in the (111) epitaxy case with small in-plane anisotropy one gets Bloch walls. In the vicinity of the surfaces, they turn to the N
eel orientation due to the stray field of the domains (twisted N
eel-Bloch-N
eel structure [25]). In turn, these walls can contain the two orientations of the Bloch wall, being separated by a line (a “wall in a wall”). This line has its magnetic moment oriented along a N
eel direction, hence the proposition by Feldtkeller to call them N
eel lines. But lines in domain walls had been observed earlier in soft thin films supporting N
eel walls, and called Bloch lines, so that this generic name was kept for all types of lines. As shown in Fig. 1.8, in a garnet film, horizontal Bloch lines (HBL) and vertical Bloch lines (VBL) exist. The HBL is an intermediate state during the dynamics, whereas the VBL is a stable structure. Moreover, in the 1980s, a Bloch line memory was developed, taking advantage of the properties, including the topological ones, of the VBLs [26]. So we concentrate on VBLs here. The topological stability of a VBL within a domain wall is analyzed similarly to the case of the vortex. Magnetization is fixed in the domains far from the wall, as well as in the wall far from the VBL, so that in fact far away (in 2D) from the VBL magnetization is fixed. In the absence of fields transverse to the easy axis, these fixed magnetizations belong to a great circle on 2 , so that the VBL is also a meron of π 2 ð2 Þ.

(A)

(B)

Fig. 1.8 Bloch lines in thin films with perpendicular magnetization. The schematic drawings (perspective view) show (A) a vertical Bloch line and (B) a horizontal Bloch line. The tilt of the domain wall magnetization at the surfaces, from a Bloch to a chiral N
eel orientation, due to the stray field of the domains and that occurs in films thick enough compared to the exchange length [25], is not drawn. The lines can be magnetized in both directions. Only the vertical Bloch line is stable in statics.

14

Magnetic Skyrmions and Their Applications

1.3.3 Role of topological defects for topological solitons The preceding sections have shown relationships between topological defects and topological solitons. These can be seen by comparing the situations described by the same homotopy group. Taking π 2 ð2 Þ as a start, it describes topological point defects in a 3D medium, as well as topological solitons for a 2D medium, both cases being for Heisenberg spins. The relation is simple: in the transformation from a topological soliton to a different one, the breaking of continuity is realized by the presence of a topological defect corresponding to the difference of their topological indices. In this comparison, one goes from a 2D medium to a 3D one either by time stacking the 2D media, or by considering the different slices of its thickness. For example, annihilating a skyrmion requires a Bloch point. Two examples of this are the merging of two skyrmions [27], and the so-called “bobber” structure where a skyrmion exists in the vicinity of one surface of the film only, a Bloch point terminating the structure [28]. Similarly, π 1 ð1 Þ describes topological point defects (vortex, antivortex, etc.) in a 2D medium and for XY spins, as well as topologically stable knots of these spins along a line. Going from a line with one knot to a line without knot requires, similarly, a vortex or an antivortex (see Fig. 1.9).

Fig. 1.9 Structures in soft nanostrips and boundary structures. A soft magnetic strip is drawn,where on the lower edge magnetization is uniform, whereas on the upper edge it makes a 2π turn, one way (A) or the other (B). For XY moments, the four sides of the structures drawn constitute a closed path, on which the moments turn by 2π (A) or +2π (B). Hence, there must be a topological defect of π 1 ð1 Þ inside, the antivortex (A) or the vortex (B). For Heisenberg moments, if we assume that moments far away are in the plane, then the structures are topological solitons (merons) of π 2 ð2 Þ.

(A)

(B)

Magnetism and topology

1.4

15

Effect of topology on magnetization dynamics

Up to now we have discussed the statics of magnetic structures. As we are going to show in this section, the dynamics of magnetic structures is also affected by topology, in some cases. In general, magnetization dynamics in continuous media is governed by the LandauLifshitz-Gilbert (LLG) equation [5, 29, 30], supplemented by additional torque terms in the presence of spin-polarized currents [31, 32]. In this equation, the energy density E !

gives rise to an effective magnetic field H eff defined by a variational equation [29].c Z

Z dE ≡

dE ¼ μ0 Ms

!

!

H eff  d m :

(1.4)

The LLG equation then reads !

!

∂m γ0 ! ! ! ! ∂m ¼ γ 0 H eff  m + α m  ∂t + μ M T : ∂t 0 s

(1.5)

In this equation, γ 0 ¼ μ0kγk is the gyromagnetic factor as adapted to fields, with the negative sign of the electron charge absorbed in the reversal of the vector product, α is !

the dimensionless damping coefficient as introduced by Gilbert [30, 33], and T represents the current-induced torques (spin transfer torques [34], spin-orbit torques [35]), as generally they do not derive from an energy density. As the effective field has contributions from the energy densities of exchange, magnetic anisotropy, applied field, demagnetizing field as well as Dzyaloshinskii-Moriya antisymmetric exchange, etc., this equation is quite complex and in most cases not solvable analytically. Presently many codes, open-source or proprietary, exist for the efficient numerical solution of this integrodifferential equation. But it is always valuable to have approximate analytical solutions in order to understand the physics. For magnetic textures, the Thiele equation [36] is such a tool.

1.4.1 The Thiele equation Suppose that (it is possible to assume that) a structure moves as a solid object, without ! ! deforming. It means that there exists a function m0 ð r Þ such that     ! ! ! ! ! m r , t ¼ m0 r  R ðtÞ , c

(1.6)

Following the use of the magnetic CGS system in the older literature, one continues to use the name of an effective field, hence the μ0 factors appearing at (too) many places. It would be simpler to employ the !

!

effective induction B eff ¼ μ0 H eff , but for homogeneity with the other chapters of this book, we will continue using the effective field.

16

Magnetic Skyrmions and Their Applications !

where R ðtÞ is the position vector of the structure. This allows expressing the time derivative of magnetization !

! ! ! ∂m ¼ ðV  r Þm0 , ∂t !

(1.7)

!

where V ¼ d R =dt is the structure’s velocity vector. We then “solve” the LLG equation in terms of the effective field, and get !

!

H eff ¼

!

1 ! ∂m α ∂m 1 ! ! ! +  m m T +λm , ∂t γ 0 ∂t μ0 Ms γ0

(1.8)

where the last term appears because the component of the effective field along the magnetization cannot be determined by this way. This effective field is used to compute the force acting on the structure, due to the !

!

energy density considered. It is given by mechanics expression F ¼ dE=d R . We can evaluate this derivative using the definition of the effective field Fi ¼ 

∂E ¼ μ0 Ms ∂Ri

Z

!

H eff

!



∂m ¼ μ0 Ms ∂Ri

Z

!

H eff 

!

∂m0 : ∂ri

(1.9)

The next step is to put the expression Eq. (1.8) of the effective field into this expression of the force,d replacing the time derivatives by space derivatives according to the rigid motion assumption (Eq. 1.7). This gives, using the convention of summation on ! ! repeated indices and abbreviating ∂ m =∂ri to ∂i m , Fi ¼

Z Z   μ0 Ms  ! μ Ms  ! ! ! ! m0 Vj ∂j m0  ∂i m + α 0 Vj ∂j m0  ∂i m0 0 γ0 γ0 Z  ! ! ! m0  T  ∂i m0 : +

(1.10)

This can be rewritten, term by term, into the Thiele balance of forces equation [36] !

!

!

!

!

F + F gyro + F dissip + F torques ¼0 :

(1.11)

!

!

The gyrotropic force F gyro and the dissipation force F dissip are proportional to the velocity. The dissipation force can be written Fdissip, i ¼ αDijVj, where the elements !

Dij of the dissipation matrix D appear. The gyrotropic force can be rewritten F gyro ¼ !

!

!

G  V where the components of the gyrovector G are d

Note that the unknown function λ disappears at this moment.

Magnetism and topology

μ Ms Eijk Gk ¼  0 γ0 2

17

Z   ! ! ! ! ∂i m0  ∂j m0  m0 d 3 r ,

(1.12)

where Eijk is the Levi-Civita totally antisymmetric tensor. Comparison with the expression of the surface covered on the 2D sphere of Section 1.3.1.1 allows to see that, for a texture only varying in two dimensions (in a film of thickness h), the gyrovector has only a perpendicular component, that is directly proportional to the π 2 ð2 Þ topological index of the structure: Gz ¼ 

μ0 Ms 4πhNπ2 ð2 Þ : γ0

(1.13)

This relation, together with the Thiele balance of forces equation, constitutes the basis of the link between the topology of magnetic structures and their dynamics. Note that ! it singles out the π 2 ð2 Þ topology. For example, for a pure XY model where m belongs to a plane, the gyrovector is automatically zero: Heisenberg spins are therefore required.

1.4.2 Applications of the Thiele equation The Thiele equation applies to a great variety of situations, much beyond its original scope of describing the steady-state motion of magnetic bubbles in bubble garnet films, as a function of the structure of their walls.

1.4.2.1 Steady-state motion of domain walls Consider a 1D domain wall, that is, magnetization is a sole function of the x coordinate. The gyrovector is zero (this can be seen by analysis, from Eq. 1.13, or from the fact that a curve covers no surface on the unit sphere). Driving the domain wall by an easy axis field H, the steady-state dynamics is simply given by !

!

!

F + F dissip ¼0 . If the wall is a 180 degrees wall, the force per unit domain wall surface under an easy axis field is along the wall normal (x) and reads Fx ¼ 2μ0MsH. The dissipation matrix has only an xx element. The balance of forces thus reads 2μ0MsH ¼ αDxxV. We therefore recover the well-known steady-state dynamics equation V ¼ (γ 0ΔT/α)H, in which instead of the Bloch wall width parameter Δ appears the so-called Thiele domain wall width ΔT given by [37] 2 ¼ ΔT

Z

!

dm dx

!2 dx:

(1.14)

This relation is valid whatever the profile of the domain wall. It explains why, remarkably, vortex walls in nanowires move more slowly than transverse walls under an easy

18

Magnetic Skyrmions and Their Applications

axis field [38, 39], even if their lateral extent is larger. Indeed, the large magnetization gradient around the vortex core leads to a smaller ΔT.

1.4.2.2 Dynamical deflection of topological structures When discussing regular structures, we have seen the π 2 ð2 Þ topological solitons of integer topological index for uniform boundary conditions, and the solitons with noninteger index in the case of nonuniform boundary conditions (in simple terms, the skyrmion on the one hand, the vortex and the VBL on the other hand). Consequently, !

these structures have a nonzero gyrovector. Hence, when moving at a velocity V , the !

!

!

balance of forces includes the gyrotropic force F gyro ¼ G  V , by construction transverse to the velocity. As in the 2D case, the gyrovector is along the film normal, the velocity and the gyrotropic force are in the film plane, and orthogonal. This phenomenon has been known since the invention of the Thiele equation, first in the field of bubble garnet films [36, 37], and then in the field of vortex dynamics [40], under the name of gyrotropic force (to which a gyrotropic deflection is associated). In the context of skyrmions, the effect has been called “skyrmion Hall effect.” The derivation of the Thiele equation shows that the effect is intrinsic to the magnetization dynamics, independent of the stimulus used to drive the texture: it exists in insulators as well as in metals, or when the structure is driven by field or by electrical current. Thus, the description of this effect as due to Newton’s action-reaction law in the case of current-driven motion, because of the electron deflection by the topological Hall effect [41], is misleading. In the case of bubbles, their skew propagation with respect to an applied field gradient has been used to identify the state of the bubble, that is, the number and type of VBLs present in the bubble’s domain wall [25]. The reversal of the gyrotropic force on the VBL as its core magnetization was flipped has been observed by real-time magnetooptical imaging [42]. Similarly, for vortices, the sign of the gyrotropic force has been imaged in real time, as well as its reversal upon flipping the vortex core polarity [43].

1.4.2.3 Topological Brownian motion The Thiele equation can be used to describe the Brownian motion, within the Langevin model where random forces, with an autocorrelation fixed by the fluctuation-dissipation theorem, are used to represent thermal agitation [44]. In the case of magnetism, white-noise random magnetic fields are introduced, which by the integral expression Eq. (1.10) lead to random forces on the structure. Solving the Thiele equation for the velocity vector and performing the statistical average results in a diffusion constant D (leading to average quadratic displacements hX2 ðtÞi ¼ hY 2 ðtÞi ¼ 2Dt after a time t) given by D ¼ kB T

αD G2 + ðαDÞ2

,

(1.15)

Magnetism and topology

19

Fig. 1.10 Skyrmion diffusivity vs. radius and damping. The diffusion constant of a skyrmion, divided by temperature and by atomic numerical factors, is plotted in perspective view as a function of skyrmion radius R (normalized to the domain wall width parameter Δ), and Gilbert damping constant α. The crest obeys Rα ¼ Δ. Above this crest line, the normal behavior where diffusivity decreases as damping increases is observed. Below the crest line, the “topological diffusion” dominates. Adapted from J. Miltat, S. Rohart, A. Thiaville, Phys. Rev. B 97 (2018) 214426.

where D (¼ Dxx ¼ Dyy for a revolution-symmetric structure) is the dissipation matrix diagonal element, and G the gyrovector z component [45]. The relation agrees with the Einstein relation D ¼ kB Tμ with μ ¼ V/F the viscous mobility. This expression shows that, at low damping, the diffusion of topological structures (G 6¼ 0) is completely different from the diffusion of nontopological structures (G ¼ 0), see Fig. 1.10. Recent experiments on skyrmions [46] indicate, however, that, with the present samples quality, this intrinsic diffusion regime is masked by much slower skyrmions trapping and escape processes.

1.5

Topology versus energetic stability

By definition, a topological soliton cannot be erased (a special case of transformation to another topologically different structure) by a continuous deformation of the magnetization. This is sometimes called “topological protection.” But does it mean that

20

Magnetic Skyrmions and Their Applications

this structure cannot be erased at all? The answer is No as we have seen: erasing this soliton requires breaking the continuity by creation of a topological defect (either by having a topological defect enter from one edge of the sample, or by creating a pair of topological defects with opposite topological indices within the sample). We have moreover seen that the Bloch point, the only topological defect of Heisenberg spins, living in 3D space, has a finite energy even in (almost everywhere, mathematically speaking) continuous micromagnetics. So the physical answer is that, in order to erase a topological soliton, a finite energy barrier has to be overcome. And the only physical question left is how large this barrier is: by the Arrhenius law this tells us if the structure is stable on the relevant timescale (the experiment interaction time, the lifetime of a storage device, our life, the lifetime of Earth, etc.). We describe below some cases where this erasure process has been observed, and the associated barrier evaluated.

1.5.1 The collapse of magnetic bubbles In the magnetic bubble memory [25, 47, 48], the information 1/0 is coded by the presence/absence of a cylindrical magnetic domain, called “bubble.” To write and rewrite information, bubbles have therefore to be created and destroyed, at higher than MHz rates. To make things worse, the bubbles used are of the topological type (S ¼ 1 winding number), because these propagate faster as no VBLs slow down their motion (think of the Thiele domain wall width, see Eq. 1.14). Experimentally, one characteristic of a bubble material is the so-called collapse field, above which no bubbles exist (except those pinned at structural defects, which are very rare and optically visible by strain-induced birefringence). Therefore, the “topological protection” has never been mentioned in this context. Moreover, the collapse field can be computed from the energy versus radius profiles, as a function of the bias field, and the value of this field is used to extract the parameters of the sample. Indeed, depending on the bias field, the bubble is stable or metastable with respect to the uniform state, and at the collapse field the local minimum in the profile disappears, at a nonzero radius called the collapse radius. The shape of the energy profile is given by the energies of applied field, demagnetizing field, and domain wall (for the materials used for memories, the bubble radius is much larger than the domain wall width, so that the domain wall can be treated like a mathematical surface, with a surface energy). The excess energy of a bubble at collapse being typically 0.1μ0M2s h3 [49], one sees that as soon as h > 3Λ this energy is larger than the insertion cost of a Bloch point. The bubble collapse is therefore explained by these evaluations. Experiments revealed additionally that bubbles with significantly larger collapse fields, called hard bubbles [25, 50], exist. This was linked to the presence of many VBLs in the wall of the bubble, all with the same rotation sense so that such bubbles are skyrmions with a large topological index (values up to 100 were observed [25, 51]). The larger collapse field results from the progressive annihilation of pairs of VBLs under compression as the bubble shrinks under bias field. A winding VBL pair has a topological index of 1, and it also involves a Bloch point during its annihilation.

Magnetism and topology

21

Note that this observed stability of the VBL pairs has led to the concept of a Bloch line memory [52], in which the writing process of a VBL pair, necessarily discontinuous, was appropriately called Bloch point writing [53].

1.5.2 The vortex core reversal To reverse the core of a vortex in a soft thin film, a Bloch point is required as we have seen. The vortex core reversal in such samples has been the subject of several studies. Statically, the core polarity in NiFe disks has been observed by magnetic force microscopy, various fields being applied ex situ so as to determine a vortex core switching field [54]. The micromagnetic modeling of this experiment [11] has shown that the Bloch point injection has to be made easier by defects; this was checked by using, instead of NiFe, an amorphous CoFeB layer in which structural defects are anticipated to be present. The study of the dynamical switching of the vortex core polarity has revealed a quite interesting behavior. The first experiment [43] used an a.c. in-plane magnetic field to excite the resonant vortex motion in a patterned microstructure, with timeresolved transmission X-ray microscopy in stroboscopic mode for observation. As directly expected from the gyrotropic force, a rotation of the vortex core is obtained, even if the driving field is uniaxial, the rotation sense being given by the sign of the gyrovector which is, as the winding number is S ¼ +1, the same as the polarity of the core. It was also observed that, upon submitting the sample to a sufficiently intense a.c. field pulse, the vortex core polarity could switch. The surprise was that, compared to the static fields of a few 100 mT required to switch the vortex core, much smaller a. c. fields (about 1 mT) were required. The second experiment [55] used an a.c. spin polarized current driving the vortex by spin-transfer torque, to obtain the same polarity switching. The micromagnetic analysis showed that a moving vortex has a distorted profile, with a dip of magnetization perpendicular component appearing next to the core [56, 57]. This dip separates into two parts when it reaches mz ¼ p, one part annihilating with the mz ¼ +p vortex core in a discontinuous process, leaving the other part as a mz ¼ p vortex core.

1.5.3 The skyrmion collapse The annihilation of a skyrmion toward a uniform magnetic structure is equivalent to collapsing an S ¼ 1 bubble. The energetics of a skyrmion is, however, different: whereas the bubble is stabilized by the demagnetizing field, this term is suppressed as the sample thickness decreases, down to one atomic monolayer. Thus, at intermediate thicknesses, both small radius skyrmions and larger radius bubbles can be stabilized [58, 59], with a continuous evolution from one structure to the other as the micromagnetic parameters are varied. This is in full agreement with topology, as both structures are equivalent in this respect. Needless to say, experiments have shown that skyrmions can be created, most directly by localized current [60, 61], and also may annihilate spontaneously [62].

22

Magnetic Skyrmions and Their Applications

In the limiting case with no long-range demagnetizing field and no applied field [63], the skyrmion is always metastable, the energy profile being impacted by a curvature term which was negligible for bubbles. Similarly to bubbles, the stability of a skyrmion requires the calculation of the energy barrier to overcome. Moreover, the prefactor τ0 of the Arrhenius law for the skyrmion lifetime τ (τ ¼ τ0 exp ΔE=ðkB TÞ) needs also to be calculated. Indeed, it has been shown that it is not a constant, but varies rapidly with applied field [64]. Regarding the energy barrier, the situation is more complex than for bubbles as the maximum energy point is more difficult to find. A first approach used atomic-scale micromagnetics for a cobalt monolayer [65], together with the chain of configurations minimization method known as nudged elastic band technique.e As shown in Fig. 1.11, the top of the barrier is attained when the skyrmion reaches an atomic size, hence the necessity of an atomic micromagnetic model. The precise value of the energy barrier is found to depend on all energy terms, even if it duly increases when the Dzyaloshinskii-Moriya exchange term increases, as expected. In fact, based on the Belavin-Polyakov theorem [68] which states that the exchange energy of a structure has a minimum value fixed by the π 2 ð2 Þ topological index, a fair estimation of the energy maximum is this exchange energy limit [58]. The difficulty of the energy barrier evaluation is then transferred to the calculation of the skyrmion energy relative to that of a uniform background. This is exemplified in Fig. 1.12. In the numerical calculation, it is notable that the configuration with the maximum energy is not realized when the skyrmion number is just 1/2. Regarding this 1/2 value, one ought to realize that the definition of the topological index for atomic micromagnetics, which is fundamentally discrete, is problematic. The approach chosen [69] is to keep the geometrical signification of this index, the surface covered on the sphere being evaluated as the sum of the areas of all spherical triangles involved (see Fig. 1.11). The area of a spherical triangle is always defined except when the three summits belong to a plane going through the sphere center. At this mathematical point, the topological index is both 0 and 1 depending on how the area of this triangle is counted, a situation denoted by a 1/2 topological index. Going back to topological arguments, a Bloch point cannot exist in such a sample as it is one atomic layer thick, hence really 2D. Nevertheless, the change from one configuration with topological index 1 to the next with index 0 can be seen as the passage of a Bloch point through the atomic monolayer. The energy barriers obtained by this chain minimization technique were compared to dynamic atomic-scale simulations including temperature-induced fluctuations, in a narrow field window where the skyrmion annihilation is visible. In such conditions, the two determinations of the energy barrier were in agreement. Note finally that, as the skyrmion energy at collapse depends sensitively on exchange, more complex calculations incorporating many-neighbor exchange interactions have predicted nonnegligible changes of this barrier [70]. This might be a path to designing materials with atomic and electronic structures more favorable to skyrmion stability.

e

Note that the initial Ref. [65] contains a methodological error [67], which was corrected afterward [66].

Magnetism and topology

23

Fig. 1.11 Annihilation of a skyrmion in a (111) Cobalt monolayer. Calculations were performed using the nudged elastic band technique, with 41 linked configurations, in an atomic micromagnetic code [65, 66]. The DMI energy per Co-Co bond is 1.8 meV. (Top) Variation of total energy and topological number with number of the configuration along the path (the path is shown by its length in configuration space, and the full squares correspond to the images shown in the rest of the figure). (Bottom) Dual representation of chosen configurations along the path, here with numbers 26 and 29 (continued on the next pages with numbers 30–33 and 34–37). For each configuration, the left column shows a real-space view, with the surface height and color displaying the perpendicular mz component (red: positive; blue: negative). The triangular mesh corresponds to the atomic lattice and the nearest-neighbor bonds. The right column depicts the image of the configuration on the unit sphere 2 , as seen from the North direction. The color of the nearest-neighbor bonds expresses the latitude on the sphere (white: North, black: South, blue: Equator). (Continued)

24

Fig. 1.11, Cont’d

Magnetic Skyrmions and Their Applications

Magnetism and topology

Fig. 1.11, Cont’d

25

26

Magnetic Skyrmions and Their Applications

Analytic model Atomistic calculation

Skyrmion energy (meV)

500

250

0

Saddle point energy (meV)

Fig. 1.12 Evaluation of the barrier to annihilate a skyrmion. For a cobalt monoatomic layer, and as a function of the strength of the bond DMI, the analytical model and the atomic micromagnetic model results are compared. (Top) Skyrmion absolute energy, (middle) saddle point absolute energy, and (bottom) resulting energy barrier. One sees that, even if the absolute energies are fairly described, their difference is much poorly obtained by the analytical model.

500

250

0

200

100

0

0

1.4

1.6

Relative error (%)

Barrier energy (meV)

300 250

1.5

1.6 1.7 d (meV)

1.8

Conclusion and perspectives

This chapter has tried to expose all the directions along which topology contributes to our understanding of the magnetic structures, be it in statics or in dynamics. As topology rests on the assumption of continuity and matter is fundamentally discontinuous, it might be tempting to dismiss all topology arguments. However, as the main energy term in ferromagnets is the exchange energy, which promotes continuity, applying topology to magnetic structures is justified. In this usage, the guide is to consider mathematics as a tool, and keep a physical approach. Along this chapter, we have seen that the machinery of topology can be applied with profit in the description of the statics of magnetic textures, and of their transformations, with all the possible homotopy groups for spins with various degrees of freedom being involved. For the intrinsic effect of topology on magnetization dynamics,

Magnetism and topology

27

however, only the full-fledged spin with three components is relevant. This singles out the π 2 ð2 Þ homotopy group, and consequently the magnetic skyrmion, as well as the associated merons, the magnetic vortex, and the vertical Bloch line. As new types of samples are invented, the arguments exposed here can be extended as needed. This applies also to more complex magnetic orders.

Acknowledgments This chapter is based on a tutorial talk delivered at the 61st Annual Conference on Magnetism and Magnetic Materials (New Orleans (USA), 31/10-04/11/2016), for which the presenter (Andr
e Thiaville) is thankful to the conference chairpersons. In addition, A.T. would like to thank F. Pi
echon for discussions, and for communicating Refs. [19a, 19b].

References [1] M. Kl
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[60] N. Romming, C. Hanneken, M. Menzel, J.E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, R. Wiesendanger, Writing and deleting single magnetic skyrmions, Science 341 (6146) (2013) 636. [61] A. Hrabec, J. Sampaio, M. Belmeguenai, I. Gross, R. Weil, S. Ch
erif, A. Stashkevich, V. Jacques, A. Thiaville, S. Rohart, Current-induced skyrmion generation and dynamics in symmetric bilayers, Nat. Commun. 8 (2017) 15765. [62] S. Woo, K. Litzius, B. Kr€uger, M.Y. Im, L. Caretta, K. Richter, M. Mann, A. Krone, R.M. Reeve, M. Weigand, P. Agrawal, I. Lemesh, M.A. Mawass, P. Fischer, M. Kl€aui, G.S.D. Beach, Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets, Nat. Mater. 15 (2016) 501. [63] S. Rohart, A. Thiaville, Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction, Phys. Rev. B 88 (2013) 184422. [64] L. Desplat, D. Suess, J.V. Kim, R.L. Stamps, Thermal stability of metastable magnetic skyrmions: entropic narrowing and significance of internal eigenmodes, Phys. Rev. B 98 (2018) 134407. [65] S. Rohart, J. Miltat, A. Thiaville, Path to collapse for an isolated N
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Materials for skyrmionics Kai Litziusa and Mathias Kla€uib Massachusetts Institute of Technology, Department of Materials Science and Engineering, Cambridge, MA, United States, bInstitute of Physics, Johannes Gutenberg University Mainz, Mainz, Germany

a

2.1

2

Introduction

In recent years, skyrmions have been found in a plethora of different materials. This chapter provides an overview of some of the most common skyrmionics materials that have been used by the community. First, there will be a brief overview of the materials that are capable of hosting stable skyrmions and then a selection of these materials will be discussed in more detail. The primary focus will then be the stabilization of skyrmions at room temperature in systems that show promise for device applications. To understand the differences between the various materials, it is important to emphasize the different types of skyrmions and the microscopic origins of their stability. Fundamentally, skyrmions can be distinguished by the type of their respective domain walls (DWs). When following the magnetization structure of the skyrmion from its core radially to the outside, the magnetic moments can either rotate along this radial direction (Neel type) or perpendicular to it (Bloch type) [1]. This rotation may occur for both of these skyrmion types in righthanded and lefthanded directions, resulting in a magnetization structure that points inward (outward) or clockwise (counterclockwise) for Neel and Bloch skyrmion, respectively. Assuming an uppolarized core, this corresponds to the mentioned right- and lefthanded chiralities. As a result, the simplest skyrmion structures can occur in four different symmetries (Fig. 2.1) [2, 3]. If the material provides a form of symmetry breaking on the atomic level—provided by its chiral crystalline structure or by broken inversion symmetry at interfaces—the resulting skyrmion structure will be governed by the symmetry breaking and will yield usually one of the aforementioned four DW orientations or combinations thereof. Skyrmions can generally be characterized by three parameters: their so-called skyrmion number [4] or topological charge (sometimes also called winding number [5, 6]) Q, their vorticity, and their helicity [7]. These parameters are linked to the generalized definition of a nanoscale skyrmion magnetization field: 1 sinΘðr ÞcosΦðφÞ mðrϕÞ ¼ @ sinΘðr ÞsinΦðφÞ A cosΘðr Þ 0

ΦðφÞ ¼ ωφ + υ, Magnetic Skyrmions and Their Applications. https://doi.org/10.1016/B978-0-12-820815-1.00001-8 Copyright © 2021 Elsevier Ltd. All rights reserved.

32

Magnetic Skyrmions and Their Applications

Fig. 2.1 The spin structure of (from left to right) a right- and lefthanded Neel and a right- and lefthanded Bloch skyrmion. These correspond to the four simplest skyrmion types.

where r is the magnetic moment’s distance from the skyrmion core, ω is the vorticity (+/ 1, see below), and υ is the helicity. The latter is directly related to the four skyrmion states shown in Fig. 2.1 as υ ¼ {0, π} corresponds to right- and lefthanded Neel skyrmions and υ ¼ π2 , 3π 2 to right- and lefthanded Bloch skyrmions (definition with core pointing up). The skyrmion number is usually defined as Q¼

 ð ð  1 ∂m ∂m dx dy  m 4π ∂x ∂y

which describes how many times the vector field m (the magnetization) can be mapped onto a sphere. The sign of Q is directly related to the orientation of the skyrmion core (ω  Q ¼ + 1/  1 for up/down, respectively) [8]. For the simple cases in Fig. 2.1, the skyrmion number is always jQj ¼ 1, however, more complex structures that wrap the sphere multiple times are possible [7, 9, 10]. The skyrmion number is also directly related to the structures’ dynamical behavior. This becomes apparent when looking at the Thiele equation [11] G

dRðtÞ dRðtÞ +Γ ¼ FðRðtÞ, tÞ, dt dt

which is used to predict skyrmion dynamics due to an external drive. Here, F(R(t), t) represents any such external driving forces at time t at position R(t), Γ is a damping term proportional to the Gilbert damping, and G is the gyrocoupling vector. The latter is directly related to the topological charge (in vectorized form) Ð 2 ∂Λ ∂Λ  as G5m d r ∂x  ∂y  Λ via the magnetization density m as proportionality constant [12]. If the material is nonchiral, spin structures like the described skyrmions can still emerge under certain conditions—usually these skyrmions are of the Bloch type and stabilized by dipolar interactions (see below). Note that the distinction between nonchiral and chiral skyrmions for a given material is not always clearly stated in literature. Nonchiral skyrmions should always be distinguished from their counterparts in chiral magnetic systems [8]. For chiral systems with sufficiently strong chiral exchange interaction, only one orientation of the DW is stable. Skyrmions in nonchiral materials can exist with different orientations of the DWs (e.g., clockwise and

Materials for skyrmionics

33

counter-clockwise rotation) within the same host material. This complicates their application in spintronic devices because the orientation of the DW governs the skyrmion’s dynamical response for instance to a spin current [3]. Commonly known examples for the type of bubble skyrmions (large skyrmions with a homogeneous area of magnetization in their center) in a nonchiral system are the magnetic bubbles investigated in the 1980s [13]. Note that the topology of a skyrmion is fully characterized by the three parameters Q, ω, and υ rather than its size so that many different skyrmions (chiral and nonchiral) can exhibit the same topology but vary strongly in size, with larger skyrmions with a plateau region in their center often being referred to as bubble skyrmions (see Section 2.2). Similarly, it is also possible for a skyrmion to exhibit more complex spin textures if its vorticity ω ¼  1. In this case, the skyrmion number will have opposite sign compared with its core polarization and the resulting structure is called an antiskyrmion. The antiskyrmion can be considered the topological antiparticle to the skyrmion and will be discussed in more detail in Section 2.5. The following chapter will introduce materials and requirements for the stabilization of skyrmions in more detail.

2.2

Overview of materials for skyrmion stabilization

Ferromagnec

Paramagnec

Applied magnetic field (OOP)

The number of materials that support skyrmions of varying types ranging from Bloch and Neel to antiskyrmions has been growing steadily in recent years. Among the first materials to be found to host skyrmions were the so-called B20 compounds, i.e., bulk crystals that intrinsically provide the broken inversion symmetry that is necessary to stabilize chiral skyrmions. M€ uhlbauer et al. [14] and Pfleiderer et al. [15] reported their findings of a skyrmion phase pocket in MnSi already in 2009 and 2010, respectively. The advantage of these systems stems from the fact that the skyrmion lattice within the crystal can self-organize into a tightly packed hexagonal skyrmion pattern, which could allow for high storage densities. However, skyrmion lattices in B20 compounds have generally proven to be difficult to manipulate and stabilize beyond their small phase (temperature, field) pockets (see Fig. 2.2), making their application so far rather challenging. Recently, single skyrmion structures in FeGe could be observed much closer to room temperature [16], indicating that more research effort will be

Conical

Helical

Temperature

Fig. 2.2 (A) Schematic B-T-Phase diagram of B20 bulk materials that shows the distinctive “A-phase” (from “anomalous phase”), the skyrmion phase, between the conical and helical orders. Diagrams of such a structure can be found, for example, in MnSi [14] or MnFeSi and MnCoSi [15]. In all cases, the skyrmion phase only occurs at low temperatures and magnetic fields of several 100 mT.

34

Magnetic Skyrmions and Their Applications

(A)

(B)

(C)

Asymmetry when atoms swapped

(D)

(E)

Fig. 2.3 Sketch of the microscopic origin of the DMI for bulk and thin films. (A,B) show the inversion asymmetry in FeGe and MnSi (blue depicts Fe and Mn, respectively), which happens directly inside of the material’s unit cell, leading to the view along the [111] axis shown in (C). The DMI vector points perpendicular to this plane in the [111] direction. (D) and (E) show the DMI vector (black arrow) in a magnetic crystal with impurities (D) and at the interface of a ferromagnetic layer (orange) and a metallic layer with strong spin-orbit coupling (SOC, grey) !

(E). In the latter case, D is perpendicular to the plane, defined by the two ferromagnetic atoms (blue) and the different atom species with strong SOC (red), i.e., pointing along the interface. Because the SOC is much stronger in the bottom layer and thus asymmetric in the triangle, the DMI cannot be compensated and results in a finite net DMI. In the case without a heavy metal layer (D), the DMI generated at impurities averages to zero [17].

necessary to assess the applicability of B20 systems for spintronic devices that usually require temperature stability up to at least 375 K. Another option that allows for the symmetry breaking necessary to host chiral skyrmions is given by thin film multilayer structures, where the breaking of the inversion symmetry and the resulting Dzyaloshinskii–Moriya interaction (DMI) originates from the interfaces between a heavy metal layer and the skyrmion-hosting magnetic layer (see Fig. 2.3). Because such multilayered structures exhibit an immense degree of freedom for the choice of the magnetic, nonmagnetic, and capping layers and additionally provide the possibility to be stacked, these systems have been shown to be extremely versatile. The interfacial DMI in these systems is fundamentally different from the bulk DMI that occurs in B20 compounds: while the B20 crystalline structure promotes a Bloch-type rotation of the DWs in the skyrmion structures, the thin film materials usually (for sufficiently large DMI) promote Neel configurations. In thin films without significant DMI, usually Bloch skyrmions with arbitrary handedness can be observed. The resulting spin structures in the case of interfacial DMI are shown in Fig. 2.4 for high and low DMI. This can be a massive advantage because heavy metal layers like platinum can create spin-orbit torques that can efficiently drive the Neel DW configuration. This topic will be discussed in more detail in Chapters 7 + 8. Prominent examples will be discussed in the following chapters. In addition, a selection of recently investigated skyrmion materials of both B20 and multilayer systems can be found in Table 2.1.

Materials for skyrmionics

35

Fig. 2.4 Illustration of the skyrmion’s topological analogy with a sphere. (A) Sphere with spins on surface pointing radially outward; (C) Combing the spins in one direction; (B) Projecting spins down to the plane—resulting in a Neel skyrmion configuration; (D) Projecting the combed spins down to the plane—resulting in a Bloch skyrmion configuration, which tends to occur with arbitrary sense of rotation in a nonchiral thin film system. Image taken from K. Litzius, Spin-Orbit-Induced Dynamics of Chiral Magnetic Structures. PhD thesis, Johannes Gutenberg-Universit€at Mainz, 2018; C. Pfleiderer, Magnetic order: surfaces get hairy. Nat. Phys. 7 (2011) 673–674, courtesy K. Everschor-Sitte and M. Sitte.

Note that some bulk materials like B20 compounds (e.g., MnSi) provide a Bloch DMI and thus Bloch-type skyrmions [64], whereas other bulk materials like GaV4S8 instead stabilize nonchiral Neel-type skyrmions [5,65]. This is rooted in their crystal structure and along which respective axis the symmetry is broken. In addition, bulk DMI can originate from an inhomogeneous distribution of different elements in the crystal as reported in GdFeCo [66]. Even in the case of a vanishing DMI of either symmetry (neither Neel nor Bloch), skyrmion spin structures can be stabilized purely by dipolar interactions due to a reduction of the resulting stray field energy. The latter is the mechanism that stabilized the magnetic bubbles in the 1980s bubble memory devices [13]. The DMI provides a similar stabilization mechanism by lowering the energy density of DWs σ DW, which is given by sffiffiffiffiffiffiffiffiffiffiffi A  πD σ DW ¼ Ku, eff with A being the exchange constant, Ku,eff the effective anisotropy constant, and D the DMI constant in the material. Therefore, a strong DMI tends to increase the amount of DWs in the material. That is, if the material hosts a skyrmion lattice, the skyrmions in the lattice will become smaller, thus decreasing the periodicity [67]. Note that for a single skyrmion in geometrical confinement, the skyrmion size does not decrease, but increases with DMI [68] as elongating the skyrmion’s DW is in this case usually

Table 2.1 A selection of recently investigated skyrmion materials. These show skyrmions at a temperature of TSky and a DMI of D, leading to spin spirals of period λH and skyrmion diameter dSiSky. Material

Sample

Conduction

Tsky/K

λH/nm

dSiSky/nm

mJ jD j /m 2

Type

Refs.

MnSi MnSi (press.) MnSi Fe1-xCoxSi Fe0.5Co0.5Si FeGe FeGe FeGe Cu2OSeO3 Cu2OSeO3 Co8Zn8Mn4 Co8Zn9Mn3 Co8Zn9Mn3 GaV4S8 Gd3Ru4Al12 Gd2PdSi3 Fe/Ir(111) PdFe/Ir(111) (Ir/Co/Pt)10 (Pt/Co/MgO) Pt/Co/Ta Ta/Pt/[CoB/Ir/Pt]5/Pt Pt/Co/Ru/Pt/Co/Ru/Pt

Bulk Bulk Film (50 nm) Bulk Film (20 nm) Bulk Film (75 nm) Film (15 nm) Bulk Film (100 nm) Bulk Bulk Film (150 nm) Bulk Bulk Bulk Monolayer Bilayer Multilayer Single layer Multilayer Multilayer SAF

Metal Metal Metal Semimetal Semimetal Metal Metal Metal Insulator Insulator Metal Metal Metal Semimetal N.A. Metal N.A. N.A. Metal Metal Metal Metal Metal

28–29.5 5–29 125 > 125 17.7 2.8 2.5 1–20 1–20 30–90 500 480 700 260–300

– – – – 75 N.A. N.A. – – – – – – – – – 300 ≶ 300

344 – 460 300 4200 0 fail to match the measured contrast. Modified from M.A. Marioni, M. Penedo, M. Bacani, J. Schwenk, H.J. Hug, Halbach effect at the nanoscale from chiral spin textures, Nano Lett. 18 (4) (2018) 2263–2267. Copyright 2020. American Chemical Society.

(A)

Δf (Hz)

100 nm

(B)

Hz (mT)

1.0

20

0.5

10

0

100 nm

0

(C)

Hx (mT)

(D)

Hy (mT)

10

10

0

0

-10

–10

Fig. 4.19 (A) MFM image of a skyrmion. (B) z-Component of the magnetic field of the skyrmion obtained from (A) by deconvolution with the response function of the MFM tip. (C and D) x- and y-component of the magnetic field of the skyrmion obtained from the z-component showed in (B). Modified from M.A. Marioni, M. Penedo, M. Bacani, J. Schwenk, H.J. Hug, Halbach effect at the nanoscale from chiral spin textures, Nano Lett. 18 (4) (2018) 2263–2267. Copyright 2020. American Chemical Society.

for example, Hz in a plane above the sample surface (Fig. 4.19B). Hz(x, y) can, for example, be obtained from the deconvolution of the measured MFM image of a skyrmion (Fig. 4.19A). The other field components, for example, Hx (Fig. 4.19C) or Hy (Fig. 4.19D) can then be obtained from Hz. Note that a similar procedure was used to determine all field components from an Hz(x, y) image measured by NV microscopy [48] and Section 4.3.

Mapping the magnetic field of skyrmions and spin spirals by scanning probe microscopy

123

4.2.4.3 Tubular and partial skyrmions A system showing two distinct phases of skyrmions containing two 14.5-nm-thick [Ir (1)/Fe(0.3)/Co(0.6)/Pt(1)]5 multilayers (SK layers) separated by a 3.8-nm-thick ferrimagnetic layer consisting of [(TbGd)(0.2)/Co(0.4)]6/(TbGd)(0.2) has been presented by Mandu et al. [49] (Fig. 4.15). Previous studies [42] demonstrated that the Ir/Fe [50] and Co/Pt [47] interfaces in the Ir/Fe/Co/Pt multilayers exhibit a strong negative DMI supporting skyrmions with a clockwise Neel wall texture at room temperature. The composition and layer thickness of the ferrimagnetic (Fi) layer permit an independent adjustment of its anisotropy and magnetization by the Tb:Gd ratio and rare-earth to Co-layer thickness, respectively. An MFM study of the field evolution is shown in Fig. 4.15B–D. In order to disentangle the MFM contrast arising from skyrmion spin textures existing in individual layers of this complex sample, a series of samples has been fabricated: three consisting of selected parts of the SK/Fi/SK sample, and one in which the Fi layer was replaced by a magnetically inactive Ta layer of the same thickness. Fig. 4.20A–E schematically shows the structures of these samples. Note that in order to quantitatively compare the MFM signals obtained on different samples, the tip-sample distance needs to be kept the same on all samples and controlled precisely (with 100μs. However, the implantation depth of the NV center in the diamond crystal is only one part of total distance between the sample surface and the NV center. For a microscopy application, the diamond crystal containing the NV defect must be integrated into the tip of an AFM. In addition, a confocal microscope and an MW antenna must be combined with the AFM system to measured the spin-dependent PL signal. Two important issues must be simultaneously addressed: first, the photon collection efficiency must be maximized, and second, the AFM must allow a reasonably good control of the diamond apex to surface distance during imaging. An excellent experimental approach to increase the photon collection efficiency has been presented by Maletinsky et al. [56]. Diamond chips with integrated cylindrical nanopillars having diameters of about 200 nm and a lengths of 1 μm are microfabricated from a [001]-oriented diamond crystal containing implanted NV centers. Because of the [001]-orientation, the NV axis orientation is tilted by 54.7 degrees from the nanopillar direction. The nanopillar can then act as a waveguide for the PL light toward the back side of the diamond chip from where the light can be collected with the optical microscope. Typically, the diamond chip with the nanopillar structure is attached to a tuning fork that then acts as an AFM cantilever, and is used for controlling the distance from the nanopillar’s front most point to the surface. This, however, imposes a further limitation: a tuning fork-based AFM is not well suited for scanning larger, that is, micron-sized areas reliably with reasonably short scan times, because the force sensitivity of the tuning fork is limited and its resonance frequency is small. Reliable scanning at a well-controlled tip-sample distance with subnanometer precision as, for example, required for differential imaging techniques or to compare magnetic field data acquired, for example, at different external fields, become challenging. Magnetic stray fields of periodic structures or the different Fourier components of an arbitrary stray field pattern decay exponentially with the distance from the surface of a sample. Hence, a precise and reliable control of the distance between the sensor and the surface of the sample is a mandatory condition to allow a calibration

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measurements, for example, to calibrate the z-position of the NV center within the nanopillar, and then to quantitatively map the stray field of the sample. Note that for a quantitative MFM work, considerable experimental effort has been carried out to obtain an optimized control of the tip-sample distance to make quantitative imaging [18] and differential imaging possible [29]. In spite of these limitations, NV center magnetometry has been extensively used to explore systems with interfacial DMI leading to spin spirals and skyrmions.

4.3.2 Imaging spin spirals and skyrmions One of the key advantages of NV magnetometry is that the NV center is a passive field sensor that unlike an MFM tip does not generate a magnetic field that can influence the magnetic state of the sample. This makes NV magnetometry an ideal technique to map local stray fields emanating from magnetic samples and to particularly assess fluctuations of the micromagnetic state of the sample. Further, once the quantization axis and the exact location of the NV center inside the nanoscale diamond tip and thus the NV center to sample distance zNV have been determined, and under the limitation the field is sufficiently weak, the field component along this axis HnNV(x, y, zNV) can be quantitatively measured. Because the stray field above the sample is a conservative field and thus generated by the negative gradient of a scalar magnetic potential, the other field components perpendicular to this axis can be obtained from the image HnNV(x, y, zNV). Note that the procedure is thus not limited to NV magnetometry methods, but can be applied to field data obtained by all experimental methods capable to image one component of the magnetic stray field (see, e.g., Fig. 4.19). Dussaux et al. [35] studied the local dynamics of topological magnetic defects in FeGe single-crystal samples by both MFM and NV magnetometry. The MFM results are discussed in Section 4.2.2.1 (see also Fig. 4.3). The MFM observation revealed dynamic phase jumps and a relaxation behavior of the helimagnetic order following changes of the applied magnetic field. In order to verify that the observed magnetization dynamics (Fig. 4.3) are intrinsic to the FeGe and not triggered by the stray field of the MFM tip, noninvasive NV magnetometry was applied. For this, nanoscale diamond particles with a nominal diameter of 25 nm and typically one NV center per particle have been dispersed onto the sample surface (Fig. 4.22A) with an areal density sufficiently small such that individual NV centers could be optically resolved. Magnetometry measurements were performed by measuring the difference between the two ESR peaks denoted by the frequencies ω (Fig. 4.22B) and using Eq. (4.8). The temperature was obtained from Eq. (4.10). Fig. 4.22B and C reveals that the field Bk measured by the NV center increases linearly with 1/T for T < TN, that is, for temperatures smaller than the Neel temperature, TN. This is compatible with the formation of a helimagnetical phase with an increasing magnetization that has been imaged by MFM (Fig. 4.3A–C). Fig. 4.22C shows the EPR signal from a different NV center, recorded with decreasing temperature. Below TN, the Zeeman splitting remains roughly constant, but various breakdowns of the Zeemann splitting are observed (see white arrows in Fig. 4.22C). These were attributed to sudden local changes of the orientation of the local magnetization consistent with the tendency of the system

Mapping the magnetic field of skyrmions and spin spirals by scanning probe microscopy

(A)

(B)

+

2.90

Low

2.86

2B||

2.82

– 300

EPR frequency [MHz]

NV fluorescence

(E) 2.90

2.85 285

275

265

0.5 mT

0.5 mT 290 280 Temperature [K] EPR frequency [MHz]

Bulk FeGe

295

T High

10–30 nm

(D)

(C)

T

NV center EPR frequency [MHz]

Nanodiamond

131

280 290 Temperature [K]

3.00

2.80

2.60 0

5

10

15

20

25

Fig. 4.22 (A) Measurement schematic: nanodiamond containing a single NV center is immobilized on the FeGe surface. The local magnetic stray field (dashed red lines) induces Zeeman shifts to electronic spin transitions of the NV center, that is, ms ¼ 0 , ms ¼ 1 that are measured using optically detected EPR. Black arrows indicate the helical spin texture of FeGe. (B) and (C) optically detected EPR spectrum during cooldown and warmup, respectively, revealing the paramagnetic-to-helimagnetic phase transition at TN ¼ 286K. Color coding reflects normalized fluorescence intensity. (D) EPR signal of an NV center showing a splitting below TN. The transient breakdowns in the EPR signal below TN indicate a sudden change of the local magnetization probed by the NV center (white arrows). (E) Temperature evolution and corresponding EPR signal. Transient breakdowns in the EPR signal are detected as long as the temperature changes and vanish completely after the temperature is stabilized. Modified from A. Dussaux, P. Schoenherr, K. Koumpouras, J. Chico, K. Chang, L. Lorenzelli, N. Kanazawa, Y. Tokura, M. Garst, A. Bergman, C.L. Degen, D. Meier, Local dynamics of topological magnetic defects in the itinerant helimagnet FeGe, Nat. Commun. 7 (1) (2016) 12430–12439. Copyright 2020. Springer Nature.

to undergo phase jumps. Fig. 4.22D reveals that these phase jumps become less frequent once the temperature has been kept constant over a longer time. Jenkins et al. [60] also presented a study that makes use of the noninvasiveness of the NV center magnetometry. In their work, microfabricated single-crystalline diamond cantilevers with NV centers close to the apex of the integrated tips were used [56] (Fig. 4.23A) to image skyrmions in a Ta(2)/Co20/Fe60/B20(1)/Pt(0.1)/MgO(2) multilayer system, where the number in the brackets are the layer thicknesses in nm. Fig. 4.23B shows an image of an isolated skyrmion bubble recorded in an external magnetic field of 1 mT acquired in an operation mode that maps the splitting of the two EPR peaks (Fig. 4.23B) (optically detected magnetic resonance [ODMR] contrast), that can be converted to the field component HnNV(x, y, zNV) generated by the skyrmion along the NV center quantization axis nNV. The sample to NV center distance, zNV, was determined to be 58  5nm (Fig. 4.23C). For a more rapid data acquisition, the simpler, contour imaging method [54] is used to characterize the

132

Magnetic Skyrmions and Their Applications

(A)

NV fluorescence

(C)

1

(D)

diamond probe 1.0 0.5 500 nm

4 µm 0

(B)

1.00

0.96

0.92

NV fluorescence ratio

0

(E) 0.50 mT

(F)0.55 mT

(G) 0.65 mT

(J)

500 nm

(K) 400 nm

(G) 0.70 mT

(H) 0.75 mT

(I) 0.80 mT

(L)

0.88 2.83 2.85 2.87 2.89 2.91 Frequency [GHz]

0.0

0.5 1.0 time [s]

1.5

0 -5 -10 -15 0 -5 -10 1.5 1.4 1.3 1.2

NV ODMR [%]

Magnetic skyrmion

NV fluoresc. [kcnts/5 ms]

NV

BNV [mT]

1.5

RF excitation

NV fluorescence

2.0

Fig. 4.23 (A) Diagram of the experimental setup. A single-crystal diamond probe containing a single NV center near the apex of its tip is scanned above the multilayer sample. Simultaneous optical and RF excitation of the NV gives an ESR signal (B), which is used to measure the stray magnetic field at each position in the scan. (B) Example of an NV ESR signal: The NV fluorescence rate decreases when applied microwaves are on resonance with either of the NV’s two spin transitions (ms ¼ 0 ! +1 and ms ¼ 0 !1). The splitting of the two peaks is used to calculate BNV, the magnetic field along the NV center axis. Plotted is the ratio of the NV fluorescence to its off-resonant value. (C) NV magnetic image of a skyrmion bubble with an external magnetic field of 1 mT perpendicular to the film plane. (D) Magnetic contour image with applied microwave frequency equal to 2.870 GHz and with 0.65 mT field perpendicular to the film plane. (E–I) 2.870 GHz contour images of a skyrmion bubble, each labeled by the corresponding Bext applied during that scan. As Bext increases, a section of the domain wall evolves into a bistable configuration, seen most clearly in (F) and (G) with 0.65 and 0.7 mT, respectively. (J) 2.870 GHz contour image of the skyrmion bubble at Bext ¼ 0.92mT. As the field is increased to 0.95 mT, the bubble state becomes unstable and switching between the saturated and bubbles sates is observed as a decrease in the NV ESR contrast. (K) The NV fluorescence collected in 5 ms bins when fixing the NV location at the position indicated by the red dot in (K) and fixing the drive microwave frequency at 2.871 GHz. (L) The NV fluorescence collected in 5 ms bins when fixing the NV location at the position indicated by the red dot in (B) and fixing the drive microwave frequency at 2.871 GHz. A characteristic hopping frequency of 14 Hz is measured over the full 30 s measurement window. Modified from A. Jenkins, M. Pelliccione, G. Yu, X. Ma, X. Li, K.L. Wang, A.C.B. Jayich, Single-spin sensing of domain-wall structure and dynamics in a thin-film skyrmion host, Phys. Rev. Mater. 3 (8) (2019) 083801. Copyright 2020. American Physical Society.

dependence of the magnetic structures on the applied field (Fig. 4.23D–K). Dark contours in the measured image then correspond to locations where the applied microwave frequency is in resonance with the ms ¼ 0 to ms ¼ 1 transitions and are thus (in a first-order approximation) contours of constant magnetic field along the NV center quantization axis. Fig. 4.23D shows a such a magnetic contour image of several skyrmion bubbles recorded with an applied microwave frequency equal to

Mapping the magnetic field of skyrmions and spin spirals by scanning probe microscopy

133

2.870 GHz and with 0.65 mT field perpendicular to the film plane. Fig. 4.23E–J shows a series of contour images acquired in different fields revealing skyrmion bubbles with a shape strongly deviating from the circular shape that would be expected in disorderfree films. The effect of the pinning sites on the skyrmion diameter evolution with applied field becomes particularly apparent in the images shown in Fig. 4.23F and G acquired in fields of 0.65 and 0.70 mT, respectively: contour lines corresponding to both the larger and the smaller diameter are apparent. This can be attributed to the domain wall hopping between two stable positions at a rate much faster than the time required to scan the image. As the field is increased, the contrast of the larger diameter contour progressively decreases while the contrast of the smaller diameter contour increases (compare Fig. 4.23F with H). This behavior arises from the hopping of the domain wall between two stable positions. As the field is increased, the domain wall spends a larger fraction of time at the configuration with the smaller skyrmion diameter. Note that Fig. 4.23I shows only a very weak contrast contour line, which is consistent with the hopping between a state with a skyrmion and the ferromagnetic state. Such a switching between the bubble skyrmion sate and a saturated ferromagnetic state, with a sufficiently slow time scale to be observed at least locally is depicted in Fig. 4.23J–L. A skyrmion bubble that is stable in a field of 0.92 mT (Fig. 4.23J) becomes unstable if the field is raised to 0.95 mT (Fig. 4.23K), and switches stochastically between the skyrmionic and ferromagnetic state. The corresponding telegraph noise of the fluorescence signal measured at the position of the red dot is shown in Fig. 4.23L. Skyrmions with a similar morphology arising from disorder have also been imaged by Gross et al. [61] with NV center microscopy. The studied sample was a symmetric bilayer system with a stack of Pt(5 nm)/FM/Au(3)/FM/Pt(5), where the ferromagnetic layer, FM was a Ni(0.4)/Co(0.7)/Ni(0.4) multilayer. Note that the same sample has also been studied by the same group by MFM [40] (see also Figs. 4.10 and 4.11). Gross et al. [61] used a single NV center embedded close (typically a few tens of nanometers) to the bottom surface of a microfabricated single-crystalline diamond cylinder that is attached to a tuning fork (for details of this setup see Ref. [56]). Micromagnetic simulations revealed the stray field arising from a domain wall 50 nm above the surface of the sample is larger than 20 mT, a field that suppresses and ODMR contrast (see Fig. 4.21D and E). Consequently, all measurements were performed without the application of microwaves, by simply recording the quenching of the PL occurring at the regions of strong fields. As pointed out by the authors, this mode is ideally suited to study the morphology of ferromagnetic textures with high spatial resolution. A tuning fork-based AFM was used to keep the NV center to sample distance at about 50 nm. Fig. 4.24A–D shows contour maps obtained in zero field (A), in 3 mT (B), and in 3 mT after field pulses of 7, and 10 mT showed in (C) and (D), respectively. The initial contrast in Fig. 4.24A is compatible with an up/down domain structure decayed into a single skyrmionic bubble (Fig. 4.24D). For comparison, an MFM image acquired by the same group [40] on the same sample at B ¼ 14mT is showed with the same lateral scale in Fig. 4.24D. Fig. 4.24F and G shows two representative, smaller-scale images of a total of 27 isolated studied skyrmions. Again an MFM image of an isolated skyrmion is shown for comparison in Fig. 4.24H, revealing that both the lateral resolution and signal-to-noise ratio of the MFM used by this group matches that of their

134

Magnetic Skyrmions and Their Applications

(A)

(B)

(C)

(D)

Bz=7mT

(E) MFM

Bz=10mT

500 nm 1

1.2

(F)

Normalized PL

(G)

Normalized PL

100 nm

Occurrence

0 0

(I)

MFM

(J)

100 nm

100 nm

(K)10 5

Normalized PL

(H) MFM

(L)

ds = 268 ± 63 nm

400 600 800 200 Effective skyrmion diameter [nm]

Occurrence

0.8

500 nm

20 Initial diameter

1% disorder ds = 348± 59 nm

No disorder 3 mT

10 0 0

400 600 800 200 Effective skyrmion diameter [nm]

Fig. 4.24 (A–D) Step-by-step formation of isolated skyrmions by applying an external out-ofplane magnetic field Bz. (A) PL-quenching images recorded at zero field and (B) Bz ¼ 3mT. (C), (D) images recorded at Bz ¼ 3mT after applying a 10 s field pulse of (C) 7 mT and (D) 10 mT. The bright PL spots correspond to particles on the sample which serve as position references. (E) MFM image acquired by Hrabec et al. [40] in a field of Bz ¼ 14mT on the same sample for comparison. (F) and (G) PL-quenching images recorded above several isolated skyrmions. The white-dashed contours indicate the PL-quenching ring from which the skyrmion area A is extracted. (H) MFM image acquired by Hrabec et al. [40] in a field of Bz ¼ 14mT on the same sample for comparison. (I, J) Micromagnetic simulations obtained with disorder arising from a 1% variation of the sample thickness. (K, L) Histograms of the skyrmion diameters obtained experimentally and from the micromagnetic calculations, respectively. Note that the simulation performed for a disorder-free sample results in a skyrmion of 860 nm diameter. Modified from I. Gross, W. Akhtar, A. Hrabec, J. Sampaio, L.J. Martı´nez, S. Chouaieb, B.J. Shields, P. Maletinsky, A. Thiaville, S. Rohart, V. Jacques, Skyrmion morphology in ultrathin magnetic films, Phys. Rev. Mater. 2 (2) (2018) 024406. Copyright 2020. American Physical Society.

NV center microscopy, albeit, the stray field of the MFM tip may have perturbed the imaged skyrmion structure. The NV center microscopy data of 27 skyrmions lead to an average skyrmion diameter ds ¼ 268  63nm (Fig. 4.24K). Micromagnetic calculations (Fig. 4.24I and J) were performed to understand the disorder-induced irregular skyrmion shapes. In a disorder-free medium, a skyrmion with a diameter of ds,0 ¼ 860 nm would be expected in a field of 3 mT, considerably larger than the experimental observation. However, in this experiment, a field pulse of 10 mT was applied, which compresses the skyrmion to a smaller diameter, before a relaxation in the field of 3 mT occurs. Fig. 4.24L shows typical results of the skyrmion diameter

Mapping the magnetic field of skyrmions and spin spirals by scanning probe microscopy

135

obtained from the simulations using a sample thickness fluctuation with a 1% relative amplitude. The calculated skyrmion morphology is well comparable to the experimental observations, while the average skyrmions radius ds ¼ 348  59nm is slightly larger. Note that if the same simulations are performed with a 2% and 0.5% relative ½2% ½0:5% ¼ 577  44nm thickness variation, skyrmion radii of ds ¼ 136  55nm and ds were obtained, revealing that the average skyrmion diameter critically depends on the strength of the disorder present in these systems. Work that makes full use of quantitative field measurements by NV microscopy has been presented already many years earlier by Tetienne et al. [62]. In their work, nanoscale diamond crystals containing NV centers were grafted onto the tip of a microfabricated AFM cantilever. Such a setup will not obtain the light collection efficiency as, for example, possible with the diamond nanopillar sensors [56], and the spin decoherence time in nanocrystals will be smaller than those obtained in ultrapure diapffiffiffiffiffiffi mond films [57]. Hence, the sensitivity was only about 10 μB = Hz, which seems poor compared to the best reported values, but is far sufficient for the work performed. Moreover, the use of a cantilever-based AFM setup is highly beneficial for the rapid and robust imaging of the topography even for the case of micropatterned nanowire samples (see Fig. 4.25e and h). This is because such an AFM setup permits the use of established and robust AFM operation modes for tip-sample distance control, such as the intermittent contact mode, typically employed for AFM imaging under ambient conditions. This allows a reproducible mapping of different samples, using different tips, or of one sample set to different micromagnetic states, but also a precise calibration of the location and direction of the NV center above the sample surface, and later its use for imaging an unknown micromagnetic structure. The latter was achieved by mapping the field generated by a current running through a the nanowire structure with the magnetic layer in the saturated state. Once the relevant parameters, such as the NV center to surface distance and the NV center axis have been determined, domain walls in both samples were mapped and compared with micromagnetic simulations. Considering that the thickness of the films is much smaller than the distance above the surface where the field is measured, t ≪ z, the domain wall diameter is smaller than the distance to the surface, ΔDW ≪ z and for a domain wall with an infinite length along the y-direction, the stray field above a Bloch domain wall can be approximated by μ0 Ms t z , π x2 + z2 μ0 Ms t x B? , z ðxÞ   π x2 + z2

B? x ðxÞ 

(4.11)

where the minus sign is for the z-component of the stray field. Note that the indices x, z refer to the component of the field B, whereas the ?-symbol denotes that this field arises from the perpendicular magnetization components (see schematics in Fig. 4.25A). The in-plane magnetization of a domain wall can be described by Mk ðxÞ ¼ Ms = cosh ðx=ΔDW Þ and an angle ψ describing the orientation of the magnetization relative to the x-axis. For a Bloch wall (top schematics in Fig. 4.25B), the angle

Magnetic Skyrmions and Their Applications

DW

(C) 5

Left Néel ccw Néel

4 t

Bx [mT]

x y

(B) Bloch

(E) 500 nm

(F) 500 nm

(G) Zeemann shift [MHz]

(A) z

Bloch

3

Right Néel cw Néel

2 1

-10

(D) 0

Left Néel ccw Néel

2 Bz [mT]

Intermediate

(H)

0 10 z [nm] 200 nm

(I)

200 nm

Bloch

1

Right Néel cw Néel

0

–1

Néel (right-handed, or cw)

–2 –300

0 x [nm]

300

0

20 z [nm]

40

80

40 20 Zeemann shift [MHz]

Left Néel ccw Néel

60

Bloch

40

right Néel cw Néel Data

20

80 60 40 20 Zeemann shift [MHz]

–250 0 250 Position x [nm]

(J) 80 Zeemann shift [MHz]

136

60

40

Data

Left Néel ccw Néel Bloch right Néel cw Néel

–200 –100 0 100 Position x [nm]

Fig. 4.25 (A) Schematic side view of a DW in a perpendicularly magnetized film. (B) Top view of the DW structure in a left-handed Bloch (top panel) or right-handed Neel (bottom panel) configuration, or an intermediate case characterized by the angle ψ. (C) and (D) calculated stray field components Bψx ðxÞ and Bψz ðxÞ, respectively, at a distance z ¼ 120nm above the magnetic layer, with a DW centered at x ¼ 0, Ms ¼ 106A m1, and ΔDW ¼ 20nm. (E) AFM topography image of a 1500-nm-wide nanowire microfabricated from a Ta/CoFeB (1 nm)/MgO thin film sample with a single domain wall at the location of the dashed line. (F) Zeeman shift map recorded at the same location acquired at z ¼ 123  3nm NV center to magnetic layer distance. (G) Cross sections through the DW (see solid line in (F). The markers are the experimental data, while the solid lines are the theoretical predictions for a Bloch (red), a left-handed or ccw Neel (blue), and a right-handed or cw Neel DW (green). The shaded areas show the simulations due to uncertainties in the parameters. The best match between simulation and experimental data are obtained for a Bloch wall. (H) AFM topography image of a 500-nm-wide nanowire microfabricated from a Pt/Co(0.6 nm)/AlOx thin film sample with a single domain wall at the location of the dashed line. (I) Zeeman shift map recorded at the same location. (J) Cross sections through the DW (see solid line in I). The markers are the experimental data, while the solid lines are the theoretical predictions for a Bloch (red), a left-handed or ccw Neel (blue), and a right-handed or cw Neel DW (green). The shaded areas show the simulations due to uncertainties in the parameters. The best match between simulation and experimental data are obtained for a left-handed or ccw Neel wall, which is compatible with the positive DMI of the 0.6-nm Co layer on the Pt layer. Based on J.P. Tetienne, T. Hingant, L.J. Martı´nez, S. Rohart, A. Thiaville, L.H. Diez, K. Garcia, J.P. Adam, J.V. Kim, J.F. Roch, I.M. Miron, G. Gaudin, L. Vila, B. Ocker, D. Ravelosona, V. Jacques, The nature of domain walls in ultrathin ferromagnets revealed by scanning nanomagnetometry, Nat. Commun. 6 (2015) 6733. https://doi.org/10.1038/ncomms7733.

ψ ¼ π/2, whereas for a right or left-handed (clockwise [cw] or counterclockwise [ccw]) Neel wall ψ ¼ 0 or π (see middle and bottom schematics in Fig. 4.25B for an intermediate state and a pure Neel wall, respectively). The spatial variation of the in-plane magnetization can generate magnetic volume charges which generate an additional field Bk above the sample that is given by

Mapping the magnetic field of skyrmions and spin spirals by scanning probe microscopy

1 x2  z2 Bkx ðxÞ  μ0 Ms tΔDW  cos ψ, 2 ðx2 + z2 Þ2 Bkz ðxÞ  μ0 Ms tΔDW

xz ðx2

+ z2 Þ2

 cos ψ:

137

(4.12)

(4.13)

With expressions (4.11)–(4.13), the stray field above a domain wall becomes k Bψx,z ðxÞ ¼ B? x, z ðxÞ + Bx, z ðxÞ, that is, the additional field arising from the volume charges can either amplify or attenuate the stray field arising from the magnetic surface charges generated by the up/down domain pattern, an important effect for chiral magnetization structures such as skyrmions [23]. Local field measurements can thus be used to determine the inner structure of a domain wall. Fig. 4.25F and I shows the experimental data obtained on the nanowire samples fabricated from the Ta/CoFeB(1)/ MgO and Pt/Co(0.6)/AlOx films, respectively. The comparison of the experimental cross-sectional data (black points in Fig. 4.25G) to simulation data obtained from a left-handed or ccw Neel wall (blue line), a Bloch wall (black line), and right-handed or cw Neel wall (green line) reveals that the Bloch wall leads to the best match. For the Pt/Co(0.6)/AlOx nanowire sample, the best match between the experimental data (black points in Fig. 4.25J) was obtained for a left-handed or ccw Neel wall (blue line in Fig. 4.25J), which is compatible with the positive DMI arising from the 0.6-nmthick Co film on the Pt layer. Similar work on a single skyrmion has been reported by Dovzhenko et al. [48] using microfabricated 2 μm diameter disks of a [Pt(3 nm)/Co(1.1 nm)/Ta(4 nm)10 stack on a Ta(3 nm) seed layer] deposited on a flat cleaved quartz tip (Fig. 4.26A and B). The NV center was integrated on a diamond nanopillar fabricated from a high-quality diamond single-crystalline sample with implanted NV center defects [56]. Fig. 4.26C and D shows photocount rate maps acquired in a field of 6.5 and 7.5 mT applied parallel to the NV-axis which is canted by about 54.7 degrees relative to the z-axis. The higher count rates observed inside the disc were attributed to an enhanced collection of photon arising from the reflection from the metal surface, while the variation inside the disk arises from the micromagnetic structure. The stripe-like pattern visible in Fig. 4.26C is reminiscent of the labyrinth domain arrangement expected in such a multilayer structure. When the field is increased by 1 mT, a bubble-like feature can be identified in Fig. 4.26D. Fig. 4.26E shows a smaller-scale image of the stray field component along the NV center axis, obtained at 9.5 mT external field again applied parallel to the NV center axis. As already discussed in Section 4.2.4.2 and demonstrated for MFM data in Fig. 4.18 acquired on a Pt(10)/ Co(0.6)/Pt(1)/[Ir(1)/Co(0.6)/Pt(1)]5/Pt(3) multilayer sample, all other stray field components can easily be obtained from a suitable-sized image of one specific component of the stray field, because the stray field above the sample is generated by a magnetic scalar potential. Dovzhenko et al. [48] could thus obtain maps of Bz(x, y, zNV) and Bx(x, y, zNV) shown in Fig. 4.26F and G from measured BnNV(x, y, zNV) data, respectively. Fitting the stray field of candidate magnetization structures, for example,

Magnetic Skyrmions and Their Applications 6

(E)

4

Diamond with NV pillar

2

6.0

CPW

0 5.5

scanned x,y,z

(B)

1000 nm

(D)

5.0

105 counts per s

-2 Quartz tip

-4 500 nm

4 2 0

4.5

-2 -4 1000 nm

4.0

500 nm

-6

|| [111] N

C V

x C C

-6 6

(G)

z

(F) Stray field [mT]

(C)

(A)

(H) Stray field [mT]

138

500 nm

Fig. 4.26 (A) Sketch of the measurement configuration. A quartz tip with patterned magnetic discs is brought into contact with the diamond nanopillar. The quartz tip and the diamond are mounted on separate stacks of piezo-based positioners and scanners, enabling subnanometer movement along all the three xyz-axes. The inset shows an electron microscopy image of a typical diamond nanopillar which is about 1.5 μm high. (B) False-colored electron microscopic image of a representative quartz tip, where 10 repetitions of a sputtered Pt (3 nm)/Co(1.1 nm)/Ta(4 nm) stack (red) are defined via electron beam lithography and subsequent lift-off. (C, D) NV photoluminescence recorded at 6.5 mT (D) and 7.5 mT (E) external bias field, respectively. Higher counts are observed above the magnetic disc due to reflection from the metallic surface. Within the disc boundary, areas with lower counts correspond to large stray magnetic fields perpendicular to the NV axis. (E) 2D map of the stray field projection Bk on the NV axis (see also F). The measurement was performed at a bias field of Bk, ext ¼ 9.5mT applied along the [111] diamond axis. (F) Sketch of the coordination geometry of a nitrogen-vacancy defect in diamond, illustrating the direction parallel to the quantization axis relative to the Cartesian reference frame of the setup (x, z). Carbon, nitrogen, and vacancy sites are labelled C, N, and V, respectively. The z-axis is orthogonal to the diamond surface. (G) and (H) Reconstructed components of the stray field along the z- and x-directions, respectively. Modified from Y. Dovzhenko, F. Casola, S. Schlotter, T.X. Zhou, F. B€ uttner, R.L. Walsworth, G.S.D. Beach, A. Yacoby, Magnetostatic twists in room-temperature skyrmions explored by nitrogen-vacancy center spin texture reconstruction, Nat. Commun. 9 (1) (2018) 2712. Copyright 2020. Springer Nature.

with different types of walls (Bloch, left- or right-handed Neel or even mixed-type walls) to the measured stray field component, then allows the determination of the wall type and different parameters defining it. Such a quantitative analysis, however, requires data with a sufficiently high signalto-noise ratio that can be acquired under reproducible measurement conditions that can be sustained over different measurements, performed in different external magnetic fields, and during the calibration of the field sensor, that is, an NV center or an MFM tip. Although not being the core interest of most scientists, these issues should be addressed with greater care in the future, before the full potential of the employed experimental methods can be fully harvested.

Mapping the magnetic field of skyrmions and spin spirals by scanning probe microscopy

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Acknowledgments I would like to express my sincere thanks to my collaborators, particularly to Dr. Andrada-Oana Mandru, but also to Dr. Oguz Yildirim and Mrs. Yaoxuan Feng, for their detailed reading, comments, and suggestions. Their help was especially valuable for completing the formidable task to review the large number of publications concerning scanning probe microscopy studies of systems with DMI and to condense this material into a concise book chapter summarizing a viable part of the work performed with SPM on skyrmionic system to date.

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Scanning probe microscopy methods for imaging skyrmions and spin spirals with atomic resolution

5

Hans J. Huga,b a Empa, Swiss Federal Laboratories for Materials Science and Technology, D€ubendorf, Switzerland, bDepartment of Physics, University of Basel, Basel, Switzerland

5.1

Introduction

Since its invention, the scanning tunneling microscope (STM) [1, 2] has become an established surface science tool. In an STM, a metallic tip is brought into a close proximity to a conducting sample. At a sufficiently small tip-sample distance, typically below 1 nm, a tunnel current can flow, which depends on the applied bias, U, on the electronic states of the tip and sample, and exponentially on the tip-sample distance. The tunnel current decays by about one order of magnitude for an increase ˚ . This rapid decay ultimately permits to image surof the tip-sample distance by 1 A faces with atomic resolution, in spite of the radii of several tens of nanometers typical for STM tips, because most of the tunneling current flows through the apex atom. An STM can be operated in an imaging mode where the tip-sample distance is adjusted by a feedback such that the measured tunnel current remains constant. Alternatively, the tip can be scanned at constant average height, or with a slow distance feedback, and the variation of the tunneling current arising from the local topography or spatial variations of the local density of states (DOS) can be mapped, provided that the topography is sufficiently small to avoid a tip-sample crash. To explore the electronic states, the dependence of the tunneling current I on the sample bias U can be dI on the sample bias U at a selected tip posiexplored. For this, either the dependence dU dI tion Rt, or its dependence dU on the tip position Rt at a selected bias U are recorded to either locally map the electronic states or acquire a spectroscopic image of the sample. Using magnetic tips, the current can become spin-polarized, and thus probe local spinpolarized states with atomic-scale resolution. First, spin-polarized scanning tunneling microscopy (SP-STM) experiments have been reported by Wiesendanger et al. [3, 4]. Sections 5.2.1 and 5.2.1.1–5.2.1.3 review magnetic contrast formation in STM, while Sections 5.2.2.1 and 5.2.2.2 are devoted to the imaging of spin spirals and skyrmions and their manipulation. With the invention of the atomic force microscope (AFM) [5] or more generally the scanning force microscope (SFM), a scanning probe microscopy tool to image Magnetic Skyrmions and Their Applications. https://doi.org/10.1016/B978-0-12-820815-1.00015-8 Copyright © 2021 Elsevier Ltd. All rights reserved.

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Magnetic Skyrmions and Their Applications

insulating sample surfaces with highest lateral resolution became available. In their publication, Binnig et al. [5] presented scanlines acquired on an Al2O3 surface displaying features having a width of about 3 nm. Images showing structures with atomic-scale periodicities were presented a few years later by various groups [6–9], but it was soon recognized that the images show atomic periodicity. However, atomic-scale defects or unit cell steps with atomic extension perpendicular to the step edge were never observed. First images with true atomic resolution were obtained almost a decade after the invention of the AFM by Giessibl [10], Kitamura and Iwatsuki [11], and by the Morita group by Ueyama et al. [12] and Sugawara et al. [13]. For this, the tip is brought into close vicinity of the sample surface such that short-range forces arising from incipient chemical bonds between the tip apex atom and surface atoms occur (see Ref. [14] for a review of the earlier work on AFM with atomic resolution). Since 1995, various semiconducting, metallic, and insulating samples have been imaged with atomic resolution. More recently, the controlled functionalization of the tip, either by a CO molecule [15] or by an O atom [16], has become a popular technique to image organic molecules on surfaces. In case the tip is covered with a magnetic material, the instrument becomes sensitive to magnetic stray field emanating from the sample surface (see Chapter 4). However, if the apex atom of an AFM tip coated with a ferromagnetic or antiferromagnetic material is approached sufficiently close to the surface of a magnetic sample, the interatomic chemical bonding energy depends on the relative spin orientation of the tip apex and surface atom. Consequently, a magnetic exchange force can be measured and in the best case a magnetic image with atomic resolution can be obtained. The concept of magnetic exchange force microscopy (MExFM) was first demonstrated by Kaiser et al. [17]. Section 5.3.1 gives a short introduction into the MExFM concepts, while Section 5.3.2 reviews the work performed with MExFM techniques to assess spin spirals and skyrmions. Out of all experimental methods used for skyrmion imaging, SP-STM and MExFM offer the highest spatial resolution (down to the atomic scale). However, in case of the SP-STM, a magnetic contrast can occur only if the electrons can tunnel into the magnetic layer of the sample. Similarly, the MExFM provides a magnetic contrast only if a spin-dependent exchange force between the tip apex atom and atoms on the surface of the sample occurs. Hence, both methods can only be applied, if the magnetic layer is accessible at the surface of the sample. Also, disentangling magnetic and topographical contrast becomes challenging on samples with a roughness on the scale of a few nanometers, which is typical for polycrystalline samples. This then excludes samples where the magnetic layers are covered by other layers, for example, for oxidation protection or by electrodes to introduce electrical currents or perform a readout of the spin texture. Furthermore, almost all reported SP-STM and MExFM experiments on systems with skyrmions have been carried out at low temperatures. These are presumably required to obtain the stability needed to perform such measurements, but also because the thermal stability of atomic-scale spin textures is limited [18]. Thus, imaging technically relevant samples by SP-STM or MExFM remains challenging and has not yet been reported, apart from a nanoskyrmion lattice observed in a Si-wafer-based multilayer system consisting of an Fe ML deposited onto a single-crystalline Ir/YSZ/Si (111) substrate inside the SP-STM system [19].

Scanning tunneling microscopy methods

145

However, in contrast to SPM methods mapping the stray field, SP-STM allows to map the topography and the electronic surface states of a sample with atomic resolution. Moreover, it can also provide spectroscopic information, and different contrast mechanisms can be advantageously combined to obtain local information on spinorbit coupling and the local degree of the spin noncollinearity. SP-STM methods hence provide data sets with atomic-scale lateral resolution allowing a direct comparison with results from ab initio calculations and other spin-resolved atomic-scale modeling work. With this, SP-STM and MExFM methods are ideally suited to study fundamental phenomena in otherwise well-defined single-crystalline systems.

5.2

Scanning tunneling microscopy methods

5.2.1 Scanning tunneling microscopy contrast formation If a magnetic sample is studied with an STM using a spin-polarized tip, a magnetic contrast can arise since the tip and sample density of states are spin-dependent (Fig. 5.1A) [20, 21]. To date, tips providing a spin polarization are typically prepared by depositing a few monolayers (MLs) of a ferromagnetic or antiferromagnetic materials onto an etched tungsten tip that has been heated up to 2200 K in UHV to form a semisphere with a radius of about 500 nm. Coating the W tip with 3–5 MLs of Fe results in a tip with an in-plane magnetization, while W tips coated with 7–9 MLs Gd or 10–15 MLs of Gd90Fe10, or
35 MLs of Cr show an out-of-plane magnetization [20, 22]. Note that the Cr-coated tip is antiferromagnetically ordered, which minimizes the stray field of the tip. Alternatively, bulk antiferromagnetic Cr tips have been used [23]. The magnetic moment direction of the tip apex atom of a bulk Cr tip is along an arbitrary direction, but Schlenhoff et al. [23] showed that it can be changed by

GMR / TMR

e−

(A)

TAMR

e−

e−

(B)

NCMR

e−

e−

e−

(C)

Fig. 5.1 Magnetoresistive effects in tunneling junctions. (A) Sketch of the tunneling magnetoresistance (TMR) effect, in which two magnetic electrodes are separated by an insulator or vacuum. (B) The tunneling anisotropic magneto resistance (TAMR) effect does not require a magnetic sensor electrode (tip). The effect arises from the intrinsic spin-orbit coupling (SOC) within the magnetic layer giving a different conductance for an out-of-plane and in-plane magnetization of the sample. (C) The noncollinear magnetoresistance (NCMR) effect arises from a sample with a noncollinear spin texture giving rise to a contrast if the degree of the noncollinearity changes. Modified from C. Hanneken, F. Otte, A. Kubetzka, B. Dupe, N. Romming, K. von Bergmann, R. Wiesendanger, S. Heinze, Electrical detection of magnetic skyrmions by tunnelling noncollinear magnetoresistance, Nat. Nanotechnol. 10 (12) (2015) 1039–1042. Copyright 2020. Springer Nature.

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voltage pulses or controlled collisions of the tip with the sample surface. Because of the antiferromagnetic order, the magnetic moment direction is robust toward an applied external field, making such Cr tips ideal for field-dependent studies. A magnetic contrast can also arise from a local tunneling anisotropic magnetoresistance (TAMR) [24–27], because the band structure can become dependent on the local magnetization direction through spin-orbit coupling (SOC) (Fig. 5.1B). A change of the local differential tunnel conductance can further arise from the noncollinearity of spin textures in the sample (noncollinear magnetoresistance [NCMR] effect), which can induce a mixing between the spin channels and consequently result in a change of the local electronic state [28, 29] (Fig. 5.1C). Using single-crystalline model systems, for example, consisting of a one or two monolayers of a (magnetic) transition metal elements, or an PdFe bilayer deposited on suitable single-crystalline substrates, SP-STM delivers rich data sets that allow a detailed comparison with micromagnetic and spin-resolved DFT calculations and with this deep insight into the complex and fundamental physics of magnetic samples. With this, SP-STM methods are ideally suited to study fundamental phenomena in otherwise well-defined single-crystalline systems.

5.2.1.1 Contrast formation in SP-STM According to Bardeen [30], the tunnel current becomes IðRt ,UÞ ¼

i  2 4πe X h s f ðEμ  EF Þ  f ðEtν  EF Þ  δðEtν  Esμ  eUÞMνμ ðRt , UÞ , ħ μ, ν (5.1)

where U is the bias potential applied to the sample, Rt is the position of the tip above the sample, f(E) and EF are the Fermi function and energy, respectively, and Esμ and Etν are the energies of the sample and tip states, respectively. Note that a factor of 2 has been introduced to account for the two possible spin states of the electrons. Tersoff and Hamann [31] assumed that the tip wave function can be described by an s-wave function. The tunneling matrix element Mνμ(Rt, U) can then be written as Mνμ ðRt , UÞ ¼ hψ tν jUjψ sμ i ¼ 

2πCħ2 s ψ , κm μ

(5.2)

where C is a normalization constant, κ ¼ ħ1 ð2mϕÞ1=2 is the minimum inverse decay length of the wave functions in vacuum, and ϕ is the work function. Wortmann et al. [32] extended this description to a spin-polarized case. For this, the coefficient 4πe ħ in Eq. (5.1) needs to be divided by 2, and two-component spinors 

ψ sμ, ",#

 ψ sμ" , ¼ ψ sμ#

(5.3)

Scanning tunneling microscopy methods

Tip

147

Tip

Sample

Tip

Sample

Sample

(A)

(B)

Density of states

(C)

Density of states

Fig. 5.2 (A) Schematics of a spin-polarized tip above a magnetic sample in a field μ0H. (B and C) Electronic states for spin-up and spin-down electrons for a parallel and antiparallel arrangement of mt with ms, respectively. Based on S.-H. Phark, D. Sander, Spin-polarized scanning tunneling microscopy with quantitative insights into magnetic probes, Nano Converg. (2017) 1–17. Copyright 2020. Springer Nature.

have to be used for the wave functions of the sample and tip (Fig. 5.2). Assuming spin conserving tunneling across the vacuum barrier, that is, ignoring spin-flip processes, for example, due to the spin-orbit interaction or defects, the potential Ut used in Eq. (5.2) is diagonal in spin space. Assuming that the spin-up n"t and spin-down n#t tip DOS are constant in energy but different in size to describe the magnetization of the tip as mt ¼ ðn"t  n#t ÞetM , Wortmann et al. [32] described the tunneling current as the sum of a spin-independent (I0) and a spin-dependent (IP) part as IðRt ,U, θÞ ¼ I0 ðRt ,UÞ + IP ðRt , UÞ ¼

i 4π 3 C2 ħ3 e h  ~ n ðR , UÞ + m m ðR , UÞ , n t s t t t s κ 2 m2

(5.4) (5.5)



where m s ðRt , UÞ is the energy integral of the local magnetization density of states ms(Rt, U) given by 

Z

m s ðRt , UÞ ¼

gðEÞms ðRt ,EÞ dE

(5.6)

and ms ðRt ,EÞ ¼

X μ

s δðEμ  EÞψ s{ μ ðRt Þσψ μ ðRt Þ,

(5.7)

where g(E) ¼ f(E EF)  f(E + eU EF). The integrated local density of states (ILDOS) of the sample n~s ðRt , UÞ is defined analogously. The spin-polarized part of the tunneling current in Eq. (5.5) depends on the relative directions of the magnetic moments of the tip apex atom mt and the sample ms. Hence, with an adjustment of, for example, the tip magnetic moment direction,

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Magnetic Skyrmions and Their Applications

different components of the sample magnetization can be addressed. In principle, the tip moment can be adjusted by the applied magnetic field, but this field may also influence the magnetic state of the sample. This procedure can thus only be applied to situations where the magnetic state of the sample remains unaffected by the applied field (see, e.g., Bode et al. [33]). Alternatively, bulk Cr tips can be used for which the magnetic moment direction of the apex atom can be changed by voltage pulses or by bringing the tip into a gentle contact with the sample surface [23]. After such a process, the tip magnetic moment direction is, however, in an arbitrary state and needs to be recalibrated by imaging a known or previously determined magnetic moment direction. A major reported advantage of bulk Cr tips is that once the magnetic moment is set, it remains unaffected by applied magnetic fields that can thus be used to change the magnetic state of the sample. Alternatively, the in-plane orientation of the sample magnetic moments can be deduced from images of identical magnetic objects acquired in different rotational domains resulting from the substrate symmetry. Images acquired on equivalent magnetic features in the different rotational domains then show a distinct appearance which is equivalent to data that would be obtained by imaging one of the features with different tip in-plane magnetic moments rotated by the rotational angle between the rotational domains. Separating topographical and magnetic contrast is one of the main objectives in SPSTM experiments. One approach is to operate the SP-STM in the spectroscopic mode that is typically carried out by simultaneously recording a constant current topography  dI  , using a lock-in amplifier driving the sample and differential conductivity data, dU U bias at UðtÞ ¼ Umod + sin ωmod t. The recorded topography image is the vertical adjustment Δz(rk, U, θ) of the tip above the sample at the position rk ¼ (xt, yt) required to keep the tunnel current constant. It arises from the energy integrated quantities n~s and   m s . n~s and with it I0 always increase with U whereas m s and IP remains constant, such that the constant current image is typically dominated by the sample topography. The differential conductivity  dI  ∝nt ns ðRt , EF + eUÞ + mt ms ðRt , EF + eUÞ dU U

(5.8)

is mapped at a bias U than can be chosen advantageously to maximize the ratio of jmsj over ns to obtain a strong magnetic contrast. In spite of this, the first SP-STM revealing atomic resolution (Fig. 5.3B) was obtained in a constant current mode with a current set point of 40 nA and a bias of 3 mV: Heinze et al. [34] used nonpolarized and spin-polarized STM tips to image a single-Mn monolayer grown pseudomorphically on a W(110) substrate (inset in Fig. 5.3A). With a nonmagnetic STM tip an atomic resolution image showing the pseudomorphic growth of the Mn on the W(110) substrate is obtained (Fig. 5.3A). Using a Fe-coated W-tip having an in-plane magnetic moment orientation, the theoretically predicted c(2  2) AFM superstructure (Fig. 5.3B) was imaged. As discussed earlier, the spin-polarized part of the tunneling current in

Scanning tunneling microscopy methods

149

5 nm

Mt

Experiment

Experiment

B 0.2 nm

[001]

Co

5

Theory

8

Theory

[110]

(A)

ms

o ato

ith C

Pt w

Spin-pol. dI/dU

(B)

(C)

Fig. 5.3 (A and B) 2  2 nm2 constant-current STM images of one monolayer of Mn on W(110) imaged with an unpolarized W-tip and spin-polarized Fe-coated W-tip, respectively. While panel (A) recorded with the nonpolarized tip reveals the Mn atomic lattice, the data shown in panel (B) acquired shows the c(2  2) antiferromagnetic ground state. (C) shows data obtained on a partial Co layer on a Pt(111) substrate recorded at 0.3 K. The STM topograph is colorcoded spectroscopic dI/dU data. Modified from R. Wiesendanger, Spin mapping at the nanoscale and atomic scale, Rev. Mod. Phys. 81 (4) (2009) 1495–1550. Copyright 2020. American Physical Society.

Eq. (5.5) is a small part of the total current and thus expected to generate only a very small magnetic contrast in a constant current STM image. For the c(2  2) AFM superstructure image here, the situation is, however, different [32]. The constant current STM image Δz(Rk, U, θ) is determined by the change ΔI of the tunneling current that is given by ΔIðRk , U,θÞ ¼

X n6¼0

n

ΔIGnk ðz, U, θÞeiGk Rk ,

(5.9)

where Gnk denote the reciprocal lattice vectors parallel to the surface of the sample. Eq. (5.9) holds for both, the unpolarized part I0 as well as for the spin-polarized part Ip of the tunneling current given by Eq. (5.5). The expansion coefficients decay exponentially with the tip-sample distance z and with increasing lengths of the reciprocal lattice vectors Gnk . The STM image is thus dominated by the smallest nonvanishing reciprocal lattice vector: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΔIG1 ðz,U, θÞ∝ e k

2z

2m jE + eUjðG1k =2Þ2 ħ2 F :

(5.10)

The reduced translational symmetry of the c(2  2) AFM superstructure leads to a smaller reciprocal lattice vector compared to that arising from the atomic Mn lattice, and thus to a slower decay of the corresponding part of the tunneling current with distance z from the surface of the sample. Consequently, the constant current STM image with a spin-polarized tip is dominated by the spin-polarized part Ip of the tunneling current rather than by the more rapidly decaying unpolarized part I0 arising from the smaller atomic lattice.

150

Magnetic Skyrmions and Their Applications

Generally, SP-STM with atomic resolution of nonperiodic structures is, however, best performed using a spectroscopic imaging mode at a well-selected bias to maximize the spin-polarized term in Eq. (5.8). An example measured by Meier et al. [35] is shown in Fig. 5.3C. The image shows a 3d-representation of data acquired on a submonolayer of Co deposited on Pt(111). The Co forms a magnetic nanowire along the Pt(111) step edges with a magnetization that is either up or down, yellow or red arrow in Fig. 5.3C, respectively. The STM topograph recorded at I ¼ 0.8nA and U ¼ 0.3V is dI data acquired simultaneously using a bias modulation of 20 mV. color coded with dU Apart from the magnetic nanowires, single Co-atoms with a magnetization along the external upfield are visible on the Pt-terraces. The magnetic moment of these isolated Co-atoms could be imaged as a function of the applied field, documenting the atomic resolution of spin textures.

5.2.1.2 Contrast formation by TAMR in STM The contrast formation in SP-STM has been reviewed in the last section. A magnetic contrast can, however, also occur from tunneling anisotropic magneto-resistance effects as shown in Fig. 5.1B. A first observation of such a contrast has been presented by Bode et al. [24] for 1.75 monolayer (ML) of Fe on a W(110) surface. The Fe grows in a step-flow mode such that the step edges of the W substrate are decorated with Fe nanostripes of alternating monolayer and double layer thicknesses. A topography image is shown in Fig. 5.4A.

(A)

(B)

(C)

(D)

Fig. 5.4 STM data of 1.75 ML Fe on W(110). All images are 250  250 nm2 in size. (A) Constant-current topography image. (B) Scanning tunneling spectroscopic (STS) image recorded simultaneously with (A) at U ¼ 0.7V with an Gd-coated W-tip revealing an up/down domain pattern. (C) STS image recorded at U ¼ 0.7V (at a different location on the sample) with an Fe-coated W-tip revealing the in-plane domain walls with a magnetization parallel and antiparallel to the ½110-direction. (D) STS image recorded at U ¼ 0.05V with a nonpolarized W-tip revealing a TAMR domain wall contrast. Modified from M. Bode, S. Heinze, A. Kubetzka, O. Pietzsch, X. Nie, G. Bihlmayer, S. Bl€ ugel, R. Wiesendanger, Magnetization-direction-dependent local electronic structure probed by scanning tunneling spectroscopy, Phys. Rev. Lett. 89 (23) (2002). Copyright 2020. American Physical Society.

Scanning tunneling microscopy methods

151

26 nm

The spectroscopic image recorded simultaneously at U ¼ 0.7V with a Gd-coated W tip with a perpendicular magnetization shown in Fig. 5.4B shows the pattern of up/ down domains appearing in the 2 ML thick Fe nanostripes. Fig. 5.4C shows the spectroscopic image (on a different area of the same) obtained with a Fe-coated W-tip that has an in-plane magnetization. Consequently, the domain walls pointing toward or away from the step edges become visible with a bright and dark contrast. While the contrast in Fig. 5.4B and C arises from local changes of the orientation of mt relative to ms, the domain-wall contrast observed with a nonpolarized (uncoated) W-tip shown in Fig. 5.4D arises from the tunneling anisotropic magneto-resistance (TAMR) contrast. If SOC is absent, the sample wave functions in the tunneling matrix element given in Eq. (5.2) are diagonalized by a majority ψ sμ" and a minority ψ sμ# spinor. Bode showed that the presence of SOC can be treated by means of first-order perturbation theory [24]. The change in the wave function of state ψ kkμ of the state jkkμi is proportional to the expectation value hHSOi ¼ hσkkμjlsjσ 0 kkμ0 i of the spin Hamiltonian. The orbital part of the matrix element (hlsi) depends on the magnetization direction and can mix majority and minority states as well as orbitals of the same spin channel. Thus, in a system with SOC, the electronic band structure depends on the axis of the magnetization orientation and hence changes the local density of states at specific bias voltages. TAMR contrast can also be used to image magnetic structures on a much larger length scale. This has been demonstrated by Herve et al. [36] who imaged 10 ML of Co deposited onto Ru(0001) postannealed at 450°C using I ¼ 1nA and U ¼ 330mV (see Fig. 5.5). TAMR at the single-atom limit was demonstrated by Neel et al. [27] who showed that the spin direction of single Co atoms adsorbed on a 2 ML Fe film on W(110) can be detected.

30

(D)

10 0 0.0

(B)

0.5

1.0 1.5 Distance [µm]

150 nm

30

20

2.0

(C)

[1120]

dI/dU [a.u.]

m

n 500

0 nm

(A)

300 nm

2.2 2.0 1.8

(E)

0

50

100 150 Distance [nm]

200

Fig. 5.5 (A) Large-scale topographic STM image of 10 ML of Co deposited on Ru(0001). (B) Line scan across the structure showing flat island surfaces and the general tilt of the Ru substrate. (C) Map of the dI/dU signal showing a domain wall contrast (I ¼ 1nA, U ¼ 330 mV, ΔUrms ¼ 30 mV). (D) Zoomed image and (E) line scan of the area marked by a green-dotted box of (C). Modified from M. Herve, T. Balashov, A. Ernst, W. Wulfhekel, Large tunneling anisotropic magnetoresistance mediated by surface states, Phys. Rev. B 97 (22) (2018). Copyright 2020. American Physical Society.

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Magnetic Skyrmions and Their Applications

5.2.1.3 Contrast formation by NCMR in STM NCMR contrast can solely occur in the presence of noncollinear spin structures. Nanoscale skyrmions are therefore ideal candidate spin textures to give rise to such a contrast. Although, chronologically, the first images of skyrmions by STM were obtained by mapping local variations of the spin polarization [37] in a system consisting of 1 ML of Fe on Ir(111) (see Fig. 5.9), in this section that is devoted to a discussion of the NCMR contrast mechanism, scanning tunneling spectroscopy images of skyrmions using STM tips without spin polarization reported by Hanneken et al. [28] are discussed first. The experiment was carried out on a PdFe atomic bilayer deposited onto an Ir(111) single-crystalline substrate. Skyrmions in this system have previously been studied by SP-STM [37]. The skyrmions in this system are Neel-type skyrmions with clockwise chirality as depicted schematically in Fig. 5.6A. The skyrmion imaging was performed at T ¼ 8K in a field of 1.8 T, with I ¼ 1nA and at a sample bias of 0.7 V. Under these conditions, the skyrmions show a reduced differential conductivity and thus appear as dark spots in Fig. 5.6B where the STM topograph has been color coded with dI -STM spectroscopy image recorded at U ¼ 0.7 V. See also zoomed image of two the dU skyrmions recorded at B ¼ 2.5 T, U ¼ +0.7 V, I ¼ 1 nA, and T ¼ 4 K, shown in Fig. 5.6C. STS data recorded at sample biases from 0 to 1 V at the center of the skyrmion (red circles in Fig. 5.6D) are significantly different than data recorded at locations showing the ferromagnetic background (black circles in Fig. 5.6D). In order to relate the observed contrast to the noncollinearity of the spin structures, skyrmions were imaged in magnetic fields varied from 1 to 3 T, leading to a shrinking of the skyrmion radius. Fig. 5.6E shows the polar angles of the of the skyrmion profiles for different magnetic fields as obtained by previous SP-STM measurements [38] for different applied fields. From these spin profiles, the angle αc between the spin at the skyrmion center and the neighboring spins can be calculated. In the previous work [38], αc was found to depend linearly on the applied field (see inset in Fig. 5.6E). The effect of the local magnetic noncollinearity on the electronic properties could then be corroborated from the linear dependence of the ΔE (the difference in energy of the dI peaks characteristic for the skyrmion) on the angle αc (see inset in Fig. 5.6F). The two dU dI latter observation also becomes apparent from Fig. 5.6G that shows laterally resolved dU images obtained in fields of 1, 1.75, and 2.5 T. The maximum of noncollinearity moves from the rim of the skyrmion to its center with increasing magnetic field, in agreement with the skyrmion profiles in Fig. 5.6E. DFT calculations showed that for noncollinear spin structures a mixing between the spin-up and spin-down channels occurs that results in a change of the band structure and consequently of the LDOS [39].

5.2.2 Imaging spin-spiral states and skyrmions by STM Providing atomic resolution and magnetic contrast arising from different effects such as spin polarization, TAMR and NCMR, the STM is an ideal tool to explore complex spin textures at the atomic scale.

Scanning tunneling microscopy methods

30 nm

Pd Fe Ir(111) 40

0 0

(E)

6 4 2 0

1 2 3 Distance [nm]

2.3 1.2 -3.5 d [nm] 3.5

1.2

3 2 1 FM Skyrmion center

0 2.3

0 0.2 0.4 0.6 0.8 1 Sample bias [V]

4

(F)

0 0

3.8 40 [°] Sk [T] FM 1.00 1.75 2.50

0 0.2 0.4 0.6 0.8 1 Sample bias [V]

-1.00 T

-1.75 T

-2.50 T

dI/dUb [a.u.]

8

ΔE [V]

dI/dUb [a.u.]

0 0.5 B [T] 3.5 Field [T]

Polar angle [°]

1.0 1.5 2.0 2.5 3.0

Fe

(D)

5 nm

0.2

[°]

180

(C)

dI/dUb [a.u.]

(B)

dI/dU [a.u.]

(A)

153

1.8

0.5 nm

(G)

Fig. 5.6 (A) Sketch of a magnetic skyrmion; cones represent the magnetization direction. dI signal. The yellow areas indicate PdFe (B) STM constant-current image, color coded with the dU dI and red circular features are the skyrmions. (C) dU map of two skyrmions: the inset presents a dI profile along the arrow. (D) dU tunnel spectra in the center of a skyrmion (red) and outside the skyrmion in the ferromagnetic (FM) background (black) (B ¼ 2.5 T, T ¼ 4 K; stabilization parameters, U ¼ 1 V, I ¼ 1 nA). (E) Skyrmion profiles for different magnetic field values, plotted as polar angle of the magnetization versus distance from the skyrmion center. Inset: Evolution of the angle between a central spin of a skyrmion and its neighbors, αc, with the dI external magnetic field B. (F) dU tunnel spectra measured with a W tip at the center (Sk) and outside (FM) of an individual skyrmion at different magnetic field values (T ¼ 8 K; stabilization parameters U ¼ 0.3 V, I ¼ 0.2 nA). Inset: Evolution of the energy shift of the high-energy peak with respect to the FM state, ΔE, as a function of the angle between spins in the center of the skyrmion, αc (see the inset in a for the relation between αc and B). (F) Corresponding dI laterally resolved dU maps (U ¼ +0.7 V, I ¼ 1 nA, T ¼ 8 K). (G) dI/dU images obtained in fields of 1, 1.75, and 2T, showing the evolution of the skyrmion NCMR contrast on the fielddependent diameter of the skyrmion. Modified from C. Hanneken, F. Otte, A. Kubetzka, B. Dupe, N. Romming, K. von Bergmann, R. Wiesendanger, S. Heinze, Electrical detection of magnetic skyrmions by tunnelling noncollinear magnetoresistance, Nat. Nanotechnol. 10 (12) (2015) 1039–1042. Copyright 2020, Springer Nature.

5.2.2.1 Exploring the spin texture by STM The first observation of a chiral magnetization structure on a surface was performed by SP-STM on 1 ML of Mn on W(110) [33]. Earlier SP-STM work performed on this system has already revealed the antiferromagnetic ground state of the Mn spins in this system [34] (see Fig. 5.3B). Fig. 5.7A shows a constant current SP-STM image recorded at I ¼ 15nA and U ¼ 3mV with a Cr-coated tip that is sensitive to the in-plane magnetization. Compatible with earlier work [34], periodic stripes running along the [001] direction, with an interstripe distance of (0.47
0.03)nm matching the surface lattice constant along ½110 direction are visible. Apart from this signal arising from the antiferromagnetic

154

Magnetic Skyrmions and Their Applications

(A)

_ [110]

(C)

W W Mn

(D)

[001]

Mn

8

Mn

4 0 _ [110]

(E)

[001]

10 nm

(B)

8 4 0 _ [110]

2 nm

[001]

(F)

8

8

4

4

0

0 0

5 10 Lateral displacement [nm]

15

SDW

h-SS

c-SS

0

15 5 10 Lateral displacement [nm]

Fig. 5.7 (A) Topography of 0.77 atomic layers of Mn on W(110). (B) High-resolution constantcurrent image of the Mn monolayer taken with a Cr-coated tip (tunneling parameters: I ¼ 15 nA, U ¼ 13 mV). The stripes along the [001] direction are caused by spin-polarized tunneling between the magnetic tip and the sample. The averaged line section reveals a magnetic corrugation with a nominal periodicity of 0.448 nm and a long-wavelength modulation. Comparison with a sine wave (red), expected for perfect AFM order, reveals a phase shift of π between adjacent antinodes. In addition, there is an offset modulation (blue line), which is attributed to a varying electronic structure owing to spin-orbit coupling. (C) Schematic view of the considered spin structures: a spin-density wave (SDW), a helical spin spiral (h-SS), and a left-handed cycloidal spin spiral (c-SS). (D–F) Constant current images at I ¼ 2nA and U ¼ 30mV acquired with a ferromagnetically coated tip in external fields of 0 T (D), 1 T (E), and 2 T (F). As sketched in the insets, the external field rotates the tip magnetization from in-plane (A) to out-of-plane (C), shifting the position of maximum spin contrast. This proves that the Mn layer does not exhibit a spin-density wave but rather a spin spiral rotating in a plane orthogonal to the surface. Modified from M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch, S. Bl€ugel, R. Wiesendanger, Chiral magnetic order at surfaces driven by inversion asymmetry, Nature 447 (7141) (2007) 190–193. Copyright 2020. Springer Nature.

ground state, the magnetic amplitude is modulated with a period of about 6 nm. This modulation is, however, not symmetric, that is, superimposed on a constant offset current I0 (Eqs. 5.4, 5.5) but arises from a long spatial wavelength modulation of the I0 (blue line in Fig. 5.7B). Such a modulation of the nonpolarized part of the tunnel current must arise from a TAMR contrast (Section 5.2.1.2) and thus from spatial variations of antiferromagnetic spin axis. This long-wavelength modulation can arise from two fundamentally different spin structures: a spin-density wave (SDW in Fig. 5.7C) characterized by a sinusoidal modulation of the magnetic moment size, or from a spin spiral consisting of a directional rotation of magnetic moments of approximately constant magnitude as approximately constant magnitude as indicated

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155

by the h-SS and c-SS spin textures in Fig. 5.7C. To disentangle the possible spin structures, an Fe-coated W-tip was used in zero external field and in strong fields applied perpendicular to the surface of the sample or the bottom of the spherical tip apex. Such fields will rotate the magnetization of the tip from the in-plane direction (preferred direction in zero field) to a direction fully perpendicular in Bz ¼ 2T. Hence, the direction of the magnetization of the tip mt can be adjusted to modify the spin-polarized part of the current in Eq. (5.5). Note that the antiferromagnetic ground state of the sample does not have a net magnetization, and thus remains unaffected by an applied field. While for the case of the spin-density wave, the long-wavelength modulation is not be affected by a change of the tip magnetization from an in-plane to an out-ofplane direction, the phase between the long-wavelength modulation and the atomic-scale modulation should shift for the case of the spin spirals. The experimental results (Fig. 5.7D–F) confirm that such a field-dependent phase shift exists, which thus rule out the spin-density wave case. Because the SP-STM instrumentation used in the work of Bode et al. [33] did not permit to rotate the magnetization of the tip in the plane, it was not possible to experimentally test whether the observed spin spiral is helical or cycloidal. However, symmetry arguments permitted the conclusion that the spin spiral is cycloidal. Moreover, density functional calculations revealed that a left-handed (counterclockwise) cycloidal spin spiral with a wavelength of about 8 nm occurs in this system. Meckler et al. [40] have later used an SP-STM instrument equipped with a triple-axis vector field magnet [41], which permitted to orient the tip magnetization in the film plane but also perpendicular to the plane to show that an Fe double layer on W(110) forms an inhomogeneous right-rotating cycloidal spin spiral. The type and chirality of a spin spiral can be determined by changing the magnetic moment direction of the tip, which in case of tips coated with a ferromagnetic layer [33, 40] can be achieved with a typically strong magnetic field of the order of some Teslas. An alternative method to assess the spin structure has been demonstrated by Herve et al. [42], who have advantageously used TMR and TAMR contrast contributions obtained at different sample biases to characterize the type of a spin-spiral occurring in a Co monolayer on a single-crystalline Ru(0001) substrate. Fig. 5.8A shows a dI/ dU map recorded at I ¼ 10nA, U ¼ 320mV, and Urms ¼ 50 mV with a nonmagnetic W tip, and a representative cross-section. Because of the nonmagnetic tip, the contrast arises from TAMR. Fig. 5.8B shows the same system (in a different area) but recorded with a Cr-coated W tip having an out-of-plane magnetization, as indicated in the inset. Note that the TAMR contrast changes if the local magnetization direction rotates by 90 degrees but remains unchanged by two different collinear directions. Hence, two maxima and two minima are observed for each spin-spiral period, while the TMR contrast is direction sensitive and thus reveals one maximum and one minimum per spinspiral period. This explains the halved periodicity of the TAMR cross-section compared to cross-section obtained from spin-polarized measurements. In order to disentangle contrast contributions arising from TAMR from those generated by TMR, the bias dependence of these contributions was studied. The bias dependence of the TAMR contrast observed with a nonmagnetic W tip is shown as the red curve in

Magnetic Skyrmions and Their Applications

(C)

(B) [1120]

[1120]

ML BL

TMR [%]

(A)

40

6

20

3

0

0

-20

-3

-40

-6 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 U [V]

100 nm

2.9 2.8

0

50 100 Position [nm]

150

dI/dU [a.u.]

dI/dU [a.u.]

dI/dU [a.u.]

(D)

3.0

TAMR [%]

156

3.7 3.6 3.5 3.4 0

50

100 150 200 Position [nm]

2.2 2.1 2.0 1.9 2.8 2.7 2.6 2.5 0

U = –500 mV

Fit

U = –400 mV

20

40 60 Position [nm]

80

Fig. 5.8 (A) Top: dI/dU map of Co atomic monolayer (ML) and bilayer (BL) islands on Ru(0001) acquired with a nonspin-polarized W tip (at I ¼ 10 nA, U ¼ 320 mV, and ΔUrms ¼ 50mV) revealing a periodic stripe pattern through TAMR effect. Bottom: Line profile of dI/dU plotted along the red arrow in the image and corresponding sketch of the magnetization. (B) Top: spin-polarized dI/dU map taken with a Cr-coated W tip revealing the periodic magnetic stripe pattern through the TMR effect (at I ¼ 1 nA, U ¼ 300 mV, ΔUrms ¼ 30 mV). The tip spin polarization was out-of-plane. Bottom: profile of dI/dU plotted along the green arrow in the image and corresponding sketch of the magnetization. (C) voltage dependency of the TMR (green curve, ΔUrms ¼ 20 mV) and TAMR (red curve, ΔUrms ¼ 10 mV). (D) Experimental dI/dU profiles and fits. Modified from M. Herve, B. Dupe, R. Lopes, M. B€ottcher, M.D. Martins, T. Balashov, L. Gerhard, J. Sinova, W. Wulfhekel, Stabilizing spin spirals and isolated skyrmions at low magnetic field exploiting vanishing magnetic anisotropy, Nat. Commun. 9 (1) (2018) 1–8. Copyright 2020. Springer Nature.

Fig. 5.8C. A significant contrast occurs solely in the range between 400 and 170 mV, and a change of sign is observed at 300 mV. In addition, the bias dependence of the TMR contrast was analyzed (green curve in Fig. 5.8C). For this, the contrast difference between the maximum and minimum signal recorded with a Cr-coated W tip with an out-of-plane magnetization was measured. Note that when measuring the contrast difference between areas with opposite but collinear magnetic moment direction, a TAMR contrast does not occur, and the measured signal arises solely from TMR, that is, from parallel and antiparallel moment orientation of the magnetic moment of the tip and the sample. The TMR contrast is significant for bias voltages between 800 and 50 mV. As shown in Fig. 5.8C, at 500 mV (see vertical blue line in Fig. 5.8C), the TAMR is negligible (1%). Consequently, the dI/dU profile plotted along the axis perpendicular to the stripes shows a sinusoidal behavior (blue curve in Fig. 5.8C) that can be well fitted by a simple sine function (red curve) having a periodicity of 37 nm. However, at 400 mV (see vertical gray line in Fig. 5.8C), the TAMR (3%) and TMR (10%) are of comparable amplitude. The dI/dU profile then shows a Sawtooth-like shape, because both TAMR and TMR are present. This combination of both signals

Scanning tunneling microscopy methods

157

thus contains information about the direction of the tip magnetization. The mixed TMR and TAMR signals shown in Fig. 5.8D were the fitted by     dI 2π 4π ðxÞ ¼ A1 sin x + ϕ1 + A2 sin x + ϕ2 , dU λ λ

(5.11)

where the first and second terms correspond to the TMR and TAMR signal, respectively. A wavelength λ ¼ 37nm was found and from the phase information the tip spin polarization was deduced to be 47 degrees away from the normal direction. A nonchiral spin structure would show arbitrary changes of the rotational sense, which would lead to reversals of the Sawtooth profile. As this was not observed, the authors concluded that the spin spiral has a unique rotational sense as expected in the presence of DMI. The full potential of the spatial resolution of the SP-STM becomes apparent from the work of Heinze et al. [37], who have imaged a two-dimensional (2D) atomic-scale skyrmion lattice in a system consisting of 1 ML of Fe deposited on an Ir(111) single-crystalline surface. The pseudomorphic growth of Fe on the Ir (111) substrate becomes apparent from Fig. 5.9A. The SP-STM image obtained with a 2 T field perpendicular to the sample surface shown in Fig. 5.9B surprisingly shows a square magnetic superstructure with a size of about 1 nm by 1 nm that is incommensurate with the atomic lattice. SP-STM data, acquired in zero field and hence with an in-plane tip magnetization, are shown in the overview image (Fig. 5.9C) that is sufficiently large to contain all three possible rotational domains arising from a combination of a square magnetic structure and a hexagonal atomic lattice. While the direction of the in-plane tip magnetization was not explicitly controlled here, its direction is the same for all three domains and hence catches the different in-plane sample magnetization components of the rotational domains with respect to a unique axis of the magnetic unit cell. The excellent agreement of the experimental data (Fig. 5.9D–F) with simulated SP-STM images (insets in Fig. 5.9D–F) and also the possibility to direct the in-plane magnetization axis of the tip along the crystallographic directions allowed the conclusion that the observed magnetic structure is compatible with a nanoscale skyrmion lattice (and not a vortex lattice). To disentangle contrast arising from the spin-polarized current from that generated by spin-orbit interaction, additional STM measurements were performed with nonmagnetic tips. Because of the small signal size of the TAMR contrast, the atomic structure is visible together with the magnetic contrast arising from the spin-orbit coupling. These data allowed the conclusion that the nanoskyrmion lattice is incommensurable to the underlaying atomic lattice. To understand the microscopic origin of the nanoskyrmion lattice, an extended Heisenberg model including a four-spin interaction term apart from the terms describing the normal Heisenberg exchange and the DMI was proposed. The four-spin interaction was found to be the driving force for the observed square symmetry of the spin texture, while the DMI interaction further reduces the energy by the formation of skyrmions with a unique rotational sense.

158 (A)

Magnetic Skyrmions and Their Applications (B)

(C)

(D)

(E)

F

D E

(F)

Fig. 5.9 (A) Atomic-resolution STM image of the pseudomorphic hexagonal Fe layer at an Ir step edge. Upper inset: The FFT. Lower inset: A side view of the system (tunnel parameters U ¼ +5 mV, I ¼ 30 nA). (B) SP-STM image of the Fe ML on Ir(111) with a magnetic tip sensitive to the out-of-plane component of magnetization (Fe-coated W tip, B ¼ +2 T along the tip axis, U ¼ +50 mV, I ¼ 0.5 nA): bright (dark) spots indicate areas with magnetization parallel (antiparallel) to the tip magnetization. Upper inset: simulated SP-STM image of the nanoskyrmion with out-of-plane magnetic tip. Lower inset: FT of the experimental SP-STM image shown in the two-dimensional Brillouin zone. (C) Three-dimensional representation of a sample area with all three possible rotational magnetic domains measured with a tip sensitive to the in-plane component of magnetization as shown (Fe-coated W tip, U ¼ +5 mV, I ¼ 0.2 nA, sketched tip magnetization axis inferred from comparison to simulated SP-STM images). (D–F) Closer views of the three rotational domains indicated by squares in (C) displayed with a z-scale of 23 pm; the tip magnetization is indicated by the arrows. Insets: simulations of SP-STM measurement of the nanoskyrmion with this tip magnetization. Modified from S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, S. Bl€ugel, Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions, Nat. Phys. 7 (9) (2011) 1–6. Copyright 2020. Springer Nature.

A spin-spiral and a more conventional skyrmion phase was observed by Romming et al. [43] in a PdFe bilayer on Ir(111). The SP-STM data acquired at zero field with a bulk Cr spin-polarized tip revealed a spin-spiral magnetic ground state with a period of 6–7 nm (Fig. 5.10A and B). When a field of 1 T is applied perpendicular to the sample surface, skyrmions coexist with spin-spiral domains (Fig. 5.10D), whereas at 1.4 T a slightly disordered hexagonal skyrmion lattice is observed (Fig. 5.10E). Higher fields finally lead to a homogeneous ferromagnetic state (not shown). In later work, Romming et al. [38] presented a field-dependent study of the skyrmion size and shape. Again bulk Cr tips were used which permit to change the magnetic moment direction of the apex atom by voltage pulses or by gently touching the sample surface [23]. Moreover, the magnetic moment direction remains unaffected by an applied field, making such tips ideal for field-dependent studies. Fig. 5.11A shows a schematical view of the clockwise spin texture of a skyrmion in a RhFe bilayer deposited on the Ir(111) single-crystalline surface. An SP-STM image recorded with a tip sensitive to the out-of-plane direction is shown in Fig. 5.11B. In a down field, the skyrmion core magnetization is up, generating a positive Δz (blue)

159

b)

c)

d)

B = +1.0 T

-60 pm

a)

+55 pm

Scanning tunneling microscopy methods

50 nm

e)

B = +1.4 T

Fig. 5.10 SP-STM data obtained at T ¼ 8K on a PdFe bilayer on an Ir(111) single-crystalline substrate as a function of the applied field. (A) Schematics of the zero-field spin-spiral state and (B) corresponding SP-STM data measured at B ¼ 0 T. (C) Schematics of the skyrmion lattice state. (D and E) SP-STM data obtained at B ¼ 1 T and B ¼ 1.4 T, showing a mixed spin-spiral skyrmion state and a skyrmion lattice, respectively. Modified from N. Romming, C. Hanneken, M. Menzel, J.E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, R. Wiesendanger, Writing and deleting single magnetic skyrmions, Science 341 (6146) (2013) 636–639. Copyright 2020. Science.

(A)

(B)

(C)

(D)

B

B

B = -3.25 T

B = +3.25 T

B 20 nm

Fig. 5.11 (A) Sketch of the experimental setup of a spin-polarized STM tip probing a magnetic skyrmion. (B) Constant-current SP-STM image measured at U ¼ +200mV, I ¼ 1nA, B ¼ 1.5T at T ¼ 2.2K with out-of-plane sensitive magnetic tip (each blue circular entity is a skyrmion). (C and D) Magnetic contrast of two contrast of two skyrmions obtained with a tip with a magnetic moment direction of the apex atom pointing to the left (see mt arrows) for an external field pointing in the down and up direction, respectively. The data were recorded at U ¼ +250mV, I ¼ 1nA, B ¼ 3.25T at T ¼ 4.2K. Modified from N. Romming, A. Kubetzka, C. Hanneken, K. von Bergmann, R. Wiesendanger, Field-dependent size and shape of single magnetic skyrmions, Phys. Rev. Lett. 114 (17) (2015) 177203-5. Copyright 2020. American Physical Society.

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Magnetic Skyrmions and Their Applications

contrast for a tip with an up magnetization. SP-STM images obtained with a tip with an in-plane magnetization for a field along the down and up direction are shown in Fig. 5.11C and D, respectively. As Fig. 5.11A shows, a skyrmion with core magnetization that is up (appearing in a down field) will have a magnetic moments pointing to the left/right directions on the left/right sides of the skyrmion core. For tip with magnetic moment of the apex atom pointing to the left (as indicated by the arrows in Fig. 5.11C and D), this results in a positive (blue) and negative (red) SP-STM Δz contrast on the left and right side of the skyrmion core (blue/red lobe). If the external field is reversed, that is, applied along the up direction, the skyrmion core magnetization points down and with it the direction of the magnetic moments to the left and right side of the skyrmion core also reverse, resulting in the reversed red/blue-lobe contrast visible in Fig. 5.11D. The dependence of the size and shape of the skyrmions on the applied field can be obtained from field-dependent SP-STM data shown in Fig. 5.12A–D for fields of 1.15, 1.5, 2.02, and 2.97 T, respectively. Note the color scale in Fig. 5.12A–D refers to the differential conductance ΔdI/dU, measured at U ¼ 20mV, I ¼ 3nA, and Umod ¼ 2.4mV. Since there is no exact analytical expression to describe skyrmion profiles, Romming et al. [38] approximated the skyrmion wall profile by a standard 360 degrees domain wall profile as 8X    ρ
c > > arcsin tanh + π for B  z > 0 > < +, w=2 θðρ,c, wÞ ¼   X  > ρ
c > > for B  z < 0, arcsin tanh : +, w=2

(5.12)

where θ defines the polar angle of the magnetization at position ρ, and c and w define the position and width of two overlapping 180 degrees domain walls, respectively. Because of the axial symmetry of the skyrmion spin texture, the skyrmion can then be described by 0

1  sin ½θðρ, c,wÞ  x=ρ S ðx, yÞ ¼ @  sin ½θðρ, c,wÞ  y=ρ A, cos ½θðρ,c, wÞ

!

(5.13)

pffiffiffiffiffiffiffiffiffiffiffiffiffi where ρ ¼ x2 + y2 is the radial distance from the center of the skyrmion located at the origin. Fits of this model to the field-dependent profiles of the skyrmion highlighted by the dotted box in Fig. 5.12A then yields values for the independent parameters c and w that fully describe the skyrmion spin profile according to Eq. (5.13). The values obtained for the different fields are plotted (as red points) in Fig. 5.12E. In the framework of micromagnetic continuum theory, the skyrmion energy can be described as

Scanning tunneling microscopy methods

(A)

161

(F)

(B)

0

B = -1.15T

B = -2.97T

-90 B = -1.15T

B = -1.50T

(C)

B -1.15T -1.50T -2.02T -2.97T

(D) -180

0

(G) B = -2.97T

Skyrmion parameters d, c, w [nm]

(E) 4

180 w 90

3

0 0

d/2 d 2 Dx [nm]

2

4

3

4

5

Fit with Eq. (18)

Sim. data

B = -1.15T

1.0 B = -1.40T

B = 2.02T

0.0 -2

2

Experimental data

1 -1

1

A = 2 pJ/m |D| = 3.9 mJ/m2 K = 2.5 MJ/m3 Ms = 1.1 MA/m

20 nm ΔdI/dU [arb. units)

B = -2.02T

2 nm

-3 B [T]

-6

B = 2.97T

-4

-2

0

2

4

6

Fig. 5.12 (A–D) SP-STM differential conductance maps with in-plane magnetized tip (U ¼ 20mV, Umod ¼ 2.4mV, I ¼ 3nA, T ¼ 4.2K). (E) The size and shape of the skyrmion indicated by the box in (A) is evaluated by a fit with Eq. (5.13) as a function of the magnetic field. The inset shows the geometrical meaning of c, w, and of d, which is numerically calculated; the dashed blue line is a fit to d with 1/(B  B0). Solid black lines are obtained theoretically for the fitted set of material parameters A, D, and K of the energy functional given by Eq. (5.14). (F) Spin textures of the skyrmion as described by Eq. (5.12) for different fields. The inset shows visualization of spins with atomic distance as parametrized by Eq. (5.13). (G) Comparison of experimental and simulated data across an individual skyrmion (see inset boxes) for different field values, and fits with Eq. (5.12). The left and right insets show SP-STM experimental data from (A) to (D) and micromagnetic simulations based on the derived material parameters, respectively. Modified from N. Romming, A. Kubetzka, C. Hanneken, K. von Bergmann, R. Wiesendanger, Field-dependent size and shape of single magnetic skyrmions, Phys. Rev. Lett. 114 (17) (2015) 177203-5. Copyright 2020. American Physical Society.

Z

∞

EðA, D, KÞ ¼ 2πt

( " A

0

dθ dρ

2

#   sin 2 θ dθ sin θ cos θ + + 2 +D ρ dρ ρ )

(5.14)

K cos θ  Bz Ms cos θ ρdρ, 2

where the exchange stiffness A, the DMI constant D, the uniaxial effective anisotropy constant K, and the saturation magnetization Ms are the material-dependent parameters, Bz is the external out-of-plane magnetic field, and t is the film thickness. Using an

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Magnetic Skyrmions and Their Applications

Ms  1.1MA m1, Romming et al. minimized the energy functional for each set of A, D, K, Bz with respect to c and w. The obtained theoretical curves c(Bz) and w(Bz) were then fitted to the experimentally obtained values with A, D, and K as fitting parameters. The solid black lines in Fig. 5.12E are the calculated values of d, c, and w for A ¼ (2.0
0.4) pJ m1, D ¼ (3.9
0.2) mJ m2, and K ¼ (2.5
0.2) MJ m3 as a function of the applied magnetic field. Fig. 5.12F then shows the skyrmion profiles obtained from this analysis for different applied fields. The magnetic field dependence of skyrmions and particularly their annihilation at a critical field was later studied by Mougel et al. [44] in the Co/Ru(0001) system. It is noteworthy that Ru is a 4d element and thus leads to an SOC that is considerably reduced compared, for example, to those obtained for magnetic 3d transition metal elements on substrates consisting of 5d elements, for example, the DMI of a Pd/Fe bilayer on Ir(111) is about 6 times stronger than that of the Co/Ru(0001) system. Nevertheless, chiral magnetic states were observed in the latter system [42]. The magnetic ground state of Co/Ru(0001) in zero external magnetic field is a spin spiral with a unique rotational sense defined by the sign of the DMI and a period of 37 nm. This state transforms into a skyrmion state under the application of moderately strong magnetic fields of a few hundreds of mT. To explore the annihilation of the skyrmions and determine the required critical magnetic field, Mougel et al. [44] relied on the TAMR contrast, obtained with nonmagnetic tips that are not affected by the applied magnetic field, and measured at a very low temperature T  30mK using a home-built STM operated in a dilution refrigerator [45]. As expected for TAMR contrast the skyrmions appear as ring-like objects. In Fig. 5.13, the measured skyrmion radius is plotted versus the applied field (red points) and compared to the 1/B dependence predicated by analytical models [46, 47]. The experimental results were compared with atomistic simulations using E ¼

X

Jij Mi Mj 

ij



X ij



X

Dij ðMi  Mj Þ

ij

Ms Mi B 

X

κM2i,z

i

(5.15)

X μ M2 Mi Mj rij2  ðMi rij ÞMj ðrij Þ 0 s , 4π rij5 ij

where Jij is the magnetic exchange interaction, Dij is the Dzyaloshinskii-Moriya vector, κ is the magnetocrystalline anisotropy, and B is the external magnetic field. Using parameters J1 ¼ 13.1mV, D1 ¼ 0.2meV, κ ¼ 13 μeV, and Ms ¼ 1.8 μB per atom obtained from DFT calculations [42], the radius of the skyrmion without the dipole-dipole interaction was calculated as a function of the field (see blue points in Fig. 5.13). Compared to the experiment, the skyrmion radius was found to decrease more steeply with increasing fields. At lower fields, the experimentally observed

Scanning tunneling microscopy methods

163

Atomistic simulation Round skyrmion deformed skyrmions Short axis Long axis

20

Collapse region

Skyrmion radius [nm]

25

15 10 5 Be 0

0

100 200 300 400 500 600 700 800 900 Magnetic field [mT]

Fig. 5.13 Theoretical (blue-filled circles) values of the dependence of the skyrmion radius on the applied magnetic field in Co/Ru(0001) together with the experimental values for round skyrmions B > Be  230 mT as illustrated in the right inset. At B < Be, the skyrmions are elongated, and both the radii along the long (orange) and short (green) axes are plotted as illustrated in the left inset. The black-dashed curve is a fit to r/B. At fields in the shaded green area, annihilation of skyrmions was experimentally observed. The insets are differential conductance maps (white level proportional to dI/dV of 130  130nm2, V ¼ 350mV, I ¼ 2nA, modulation: U ¼ 60 mV). Left inset: B ¼ 0 mT and right inset: B ¼ 400 mT. Modified from L. Mougel, P.M. Buhl, R. Nemoto, T. Balashov, M. Herve, J. Skolaut, T.K. Yamada, B. Dupe, W. Wulfhekel, Instability of skyrmions in magnetic fields, Appl. Phys. Lett. 116 (26) (2020) 262406. Copyright 2020. AIP Publishing.

smaller skyrmion radius was attributed to the more dense packing of the skyrmions. The atomistic simulations predict a collapse field of Bc ¼ 820
10 mT which is in good agreement with the experimental value. A first systematic study of the thermal stability of nanoscale skyrmions in the Pd/ Fe/Ir(111) system was presented recently by Lindner et al. [18]. Skyrmions are found at temperatures up to 80 K and in external magnetic fields between 1 and 2.5 T. Compared to the system of Fe/Ir(111) without Pd capping, the critical temperature is increased by a factor 3. The system consisting of Fe on Ir(111) was later revisited by von Bergmann et al. [48], who deposited slightly less than 1 ML of Fe. Fig. 5.14A reveals that both fcc and hcp Fe-islands coexist. SP-STM data recorded at 0 and 2 T reveal that the magnetic structure of these polymorphs is very different: while at zero field an approximately 10 nm sized hcp-island does not show any magnetic structure, the square nanoskyrmion lattice previously observed by Heinze et al. [37] is again observed in the fcc island (Fig. 5.14B). At 2 T, the state with the net-up magnetization would be favored such that a hexagonal skyrmion lattice appears (Fig. 5.14C) also in the

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Magnetic Skyrmions and Their Applications

(E)

(A)

(B)

(F)

(D)

(C)

Fig. 5.14 (A) 600nm  600 nm 3D topography image of Fe on Ir(111). At room temperature, the Fe monolayer grows pseudomorphic as hcp or fcc monolayer high islands and fcc monolayer stripes at the Ir step edges; on top of most islands double-layer high patches can be observed. (B– F) SP-STM constant current images of Fe islands. (B) A hcp island without magnetic contrast and an fcc stripe with magnetic contrast (square nanoskyrmion lattice) at B ¼ 0 T. (C) Same as in (B) but in an applied magnetic field of B ¼ +2 T, exhibiting a hexagonal magnetic superstructure in the hcp island. (D) Larger hcp island showing the hexagonal magnetic state at B ¼ 0 T. (E) A hcp island with double-layer Fe in the center showing different magnetic signals at B ¼ 0 T, see magnified views at the bottom; the histograms show the height distribution of the two magnetic domains. (F) Same island as in (E) at B ¼ +2 T showing mostly the hexagonal magnetic superstructure with dark dots. All data were recorded at T ¼ 7.5K̇ , U ¼ +50 mV, I ¼ 0.5 nA. Modified from K. von Bergmann, M. Menzel, A. Kubetzka, R. Wiesendanger, Influence of the local atom configuration on a hexagonal skyrmion lattice, Nano Lett. 15 (5) (2015) 3280–3285. Copyright 2020. American Chemical Society.

small hcp island, while the square nanoskyrmion lattice in the fcc-island remains unchanged. A larger fcc-island, however, shows a hexagonal skyrmion lattice also in zero field (Fig. 5.14D). Experimental data obtained at 0 T on an hcp-island with a double Fe layer in the center reveals the coexistence of three different magnetic structures (Fig. 5.14E). While in the top-left a hexagonal skyrmion lattice appearing as an array of bright dots occurs, a hexagonal array of dark dots is observed at the topright, and no magnetic contrast is detected at the bottom of the island. The distinct appearance of these two different hexagonal arrays suggests that the magnetic structures appearing in these islands may have a net-down and net-up magnetization. Possibly, for smaller islands, rapid transitions between the two states do occur such that on average no magnetic contrast is observed (Fig. 5.14B). An applied magnetic field would then favor the hexagonal state with a net-magnetization parallel to the applied field, in case of Fig. 5.14F the skyrmion state appearing as a hexagonal array of dark spots. The hexagonal skyrmion lattice was found to be commensurable with the atomic lattice (12 atoms per skyrmion). Comparing TAMR data (recorded with a nonmagnetic tip) to simulated TAMR data allowed the conclusion that the magnetic ground state is a hollow-state with threefold symmetry.

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165

Such a subtle dependence of the magnetic state on the stacking sequence was later also observed by Romming et al. [49], who found that higher-order exchange interactions depend sensitively on the stacking sequence: For fcc-Rh/Fe/Ir(111), frustrated exchange interactions are dominant and lead to the formation of a spin spiral ground state with a period of about 1.5 nm. For hcp-Rh/Fe/Ir(111), higher-order exchange interactions favor an up-up-down-down (""##) state. Apart from the stacking, the local magnetic state was also found to depend critically on the size of the magnetic islands as, for example, discussed in the work of von Bergmann et al. [48] (compare Fig. 5.14C with D) or also in the work of Hagemeister et al. [50], who elaborated on the orientation of the skyrmion lattice relative to the island edges. Confined geometries can also induce new skyrmion-like magnetic states characterized by extra spin rotations such as the target state. Cortes-Ortun˜o et al. [51] have imaged confined magnetic states in small islands consisting of Pd/Fe and Pd2/Fe on Ir(111). SP-STM performed with a bulk Cr tip showed (Fig. 5.15A–C) a (A)

(D)

(B)

(E)

(C)

(G)

(I)

(H)

(J)

(F)

Fig. 5.15 (A–C) SP-STM conductance maps of the magnetic states observed on a roughly hexagonal hcp-Pd island on fcc-Fe/Ir(111) during a magnetic field sweep from 0 T (A), to 1.61 T (B), to 2.5 T (C). (D–F) Discrete spin simulations, which are obtained by relaxation of the experimentally found magnetic state using the using the experimentally determined material parameters; the calculated topological charge Q is indicated. (G) Line profiles of the dI/ dU signal near the boundary of a Pd/Fe island, see red-dotted rectangle in the STM image, measured at positive (solid lines) and negative (dashed lines) magnetic fields. The tip magnetization direction mt is predominantly in-plane and was obtained from 2D fits to the skyrmions. Dotted lines refer to the respective difference between the line profiles taken at positive and negative fields (scale on the right axis). (H) Experimentally derived in-plane mx components at the boundary for the two magnetic field values are shown as dotted lines together with the ones from the simulations as solid lines. (I and J) Magnetic states of Pd islands on an extended Pd/Fe film on Ir(111) imaged at B ¼ +3 T. STM constant-current images with the NCMR contrast. (I) and (J) were obtained with U ¼ 50mV, I ¼ 1nA, T ¼ 4.2K, after sweeping to 3 T from 0 and 7 T, respectively. Modified from D. Cortes-Ortu no, N. Romming, M. Beg, K. von Bergmann, A. Kubetzka, O. Hovorka, H. Fangohr, R. Wiesendanger, Nanoscale magnetic skyrmions and target states in confined geometries, Phys. Rev. B 99 (21) (2019). Copyright 2020. American Physical Society.

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spin-spiral ground state at B ¼ 0 T, the coexistence of skyrmions with spin spirals at B ¼ 1.61 T, and skyrmions at B ¼ 2.5 T. These experimental results could be well reproduced with discrete spin simulations (Fig. 5.15D–F). Confined geometries can also induced canted edge states as for example explored by MFM in larger microfabricated multilayer structures by Cubukcu et al. [52] (see Fig. 4.9). Such canted edge states were also observed in the much smaller PdFe islands by SP-STM. Fig. 5.15G dI SP-STM contrast across the edge of such an island. The local mx magshows the dU netization component derived from this experimental data shown in Fig. 5.15G shows a significant canting of the magnetic moment along the x-direction with the last 2 nm toward the island edge. Such a canting of the spins toward the island edge induces significant changes of the topology, that is, leads to a noninteger topological charge Q of the magnetic state of the island. The topological number measures the number of times that the spin-orientation wraps around a unit sphere, and hence is equal to the number of skyrmions in an infinite sample, where every skyrmion contributes with Q ¼ 1. Simulations have also predicted target skyrmions (skyrmions with two spin rotations) in confined geometries. While these were not found in the PdFe islands, spin spirals and skyrmion target states appeared in the Pd2Fe islands (Fig. 5.15H and I). The signal observed in Fig. 5.15H and I was attributed to arise from an NMCR, that is, from differences in the local noncollinearity of the magnetic moments (see Section 5.2.1.3).

5.2.2.2 Manipulation of the spin texture by STM The magnetic ground state of a thin film system with DMI depends on both the system’s temperature and the applied external field. Generally, the zero-field ground state is the spin-spiral state that develops into a skyrmion state when a suitably strong external field is applied and that ultimately transforms into a uniformly magnetized ferromagnetic state in larger fields. At lower temperatures, the system can, however, be locked in a metastable state, permitting local transitions between the ferromagnetic and the skyrmion state, for example, the writing and annihilation of individual skyrmions by local injection of a current from an STM tip. Such experiments were first carried out by Romming et al. [43] again in a system consisting of a RhFe bilayer deposited onto an Ir(111) single-crystalline substrate (see Fig. 5.16). By adjusting the applied field B such that the energy of a skyrmion and that of a homogeneously magnetized state are degenerate, skyrmions could be written and deleted by local electron injection. Fig. 5.16A shows a constant-current SP-STM image of a sample area with four defects, each of them hosting a skyrmion. The data were acquired at U ¼ +250mV, I ¼ 1nA, and B ¼ +3.23T, at a temperature T ¼ 4.2K using a tip sensitive to an in-plane magnetic component. The difference SP-STM images shown in Fig. 5.16B–E then show the individual annihilation of all skyrmions achieved by a local injection of electrons at a higher bias potential of +750mV (while the STM feedback loop was switched off). Using the same process, the skyrmions could be individually rewritten (Fig. 5.16G–J). In order to disentangle the possible mechanisms that may trigger the observed transitions between the skyrmion and FM states, the switching rate was studied as a

Scanning tunneling microscopy methods

167 +8 pm

5 nm

G

(G)

2.5 2.0 1.5 1.0 0.5

(K)

4.0

Time [s] 10 1 0 0

Time [s] 10 0 0 20 40 60 80 100 120 140 160 180 200 Current I [nA]

(I)

(J)

3.5 3.0

dI/dU [nor.]

1 0 0

(E) J

2.5 2.0 1.5 1.0 0.5

(L)

dI/dU [nor.]

3.0

(D) I

(H)

Switching rate [Hz]

3.5

dI/dU [nor.]

Switching rate [Hz]

4.0

dI/dU [nor.]

(F)

(C) H

1

0 0 1

Time [s] 10

0 0

Time [s]

10

0 0 100 200 300 400 500 600 700 800 900 Bias U [mV]

Skyrmion probability [%]

(B)

(A) F

90 80 80 60 60 40 40 20 10 0 3.5

(M)

-8 pm

Skyrmionic state dominant +6 00 mV

-6 00 m

V

FM state dominant 3.6 3.7

3.8 3.9 4.0 4.1 4.2 Field B [T]

Fig. 5.16 (A) Constant-current image of a sample region with four defects, each hosting a skyrmion marked by a circle imaged at B ¼ 3.25T with a magnetically in-plane sensitive tip (the other tunneling parameters are given in the text). (B–E) Sequence of difference SP-STM images [with respect to (F)] showing the selective erasing of all four skyrmions using local voltage sweeps (feedback loop switched off while bias voltage was increased to 750mV). (F) The sample area without skyrmions (constant-current image) and (G) to (J) their successive rewriting (difference images). (K) Dependence of the switching rate on the current measured with U ¼ 650mV, and B ¼ 2.9T. (L) Dependence of the switching rate on the applied bias measured at I ¼ 300nA and B ¼ 2.7T. (M) The probability of the skyrmion state decreases with the applied field. About 100 mT are needed to compensate for different current directions at U ¼
600 and I ¼ 100nA. Modified from N. Romming, C. Hanneken, M. Menzel, J.E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, R. Wiesendanger, Writing and deleting single magnetic skyrmions, Science 341 (6146) (2013) 636–639. Copyright 2020. Science.

function of the applied bias voltage, current and applied magnetic field strength and direction. While the dependence on the current was found to be linear (Fig. 5.16K) suggesting a constant switching probability per injected electron, the switching rate was found to depend strongly on the applied bias (Fig. 5.16L): a significant switching rate was observed only for biases above jUj > 300mV. No dependence of the switching rate on the power, that is, product P ¼ U  I was found, ruling out switching arising from a local temperature increase from the injected electrons. The strong dependence on the bias reveals that the energy of the injected electrons jeUj is the dominant factor determining the switching rate. As expected, the switching rate was found to decrease when the applied magnetic field is increased such that the ferromagnetic state becomes energetically more favorable (Fig. 5.16L). Interestingly, the switching-rate versus field curves for positive and negative fields are shifted relative to each other by ΔB  100mT (Fig. 5.16M). This indicates that spin torque transfer (STT) effects play a role and thus provide a method to control the directionality of the switching process.

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Magnetic Skyrmions and Their Applications

Switching between a skyrmion and ferromagnetic state by the electric field between the tip and the sample was demonstrated by Hsu et al. [53] in a system consisting of 3 ML of Fe on Ir(111). An overview STM image is shown in Fig. 5.17A. The 3 ML Fe layer shows a three different orientational domains of dislocation lines having a periodicity between 4 and 9 nm to at least partially reduce the strain of the Fe growing on the Ir(111) surface. A SP-STM dI/dU image recorded with a bulk Cr tip sensitive to an out-of-plane magnetization at B ¼ 0T, U ¼ 0.7V, I ¼ 1nA, and T ¼ 7.8K is shown in Fig. 5.17B. The magnetic ground state is a spin-spiral state with a magnetic period of about 3.8 nm similar to that observed for the Fe double layer (D)

(A)

e

[112]

Fe-TL Fe-TL g

Fe-DL

Fe-TL

Fe-TL f

30 nm

(E)

(F)

(G)

(I)

(J)

5 nm Fe-DL

(H) 30 nm

(C)

B = +2.5 T

10 nm

dI/dU [a.u.]

B = +0 T

0.51

(B)

(K)

(L)

0.40

Fig. 5.17 (A) SP-STM constant-current image of about 2.7 ML of Fe on Ir(111) shows a sample topography. Both the Fe triple layer (Fe-TL) and the Fe double layer (Fe-DL) are reconstructed because of uniaxial strain relief. (B) Spin-resolved dI/dU map of the area indicated by the rectangle in (A) at B ¼ 0 T, the magnetic ground state of the Fe-TL is a spin spiral with zigzag wave fronts and a periodicity is about 3.8 nm. (C) Same as (B) but in B ¼ +2.5 T. (B) and (C) have been recorded with an out-of-plane magnetization direction of the tip. (D) Perspective view of an SP-STM constant-current image of about 2.5 ML of Fe on Ir(111) measured at B ¼ +2.5 T. (E–G) Spin-resolved dI/dU maps of each of the three possible rotational domains of the reconstructed Fe triple layer at B ¼ +2.5 T. One skyrmion is highlighted by the dotted lines; the image areas are indicated by dashed squares in (D); the tip magnetization direction of the tip apex atom is in-plane and indicated by the arrows. (H–J) Spin-resolved dI/dU maps of one representative magnetic object of each rotational domain, rotated to have the dislocation lines vertical (image area is 7.5  3.5 nm2), and SP-STM simulations (bottom) of the spin structure displayed in (H)–(J). (K) Proposed spin structure of the magnetic objects of the Fe triple layer, derived from the in-plane magnetic contrasts. (L) Spin structure of an axially symmetric magnetic skyrmion for comparison. Based on P.-J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von Bergmann, R. Wiesendanger, Electric-field-driven switching of individual magnetic skyrmions, Nat. Nanotechnol. 12 (2017) 123–126. https://doi.org/10.1038/nnano.2016.234.

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169

on the same substrate. The wave fronts show a zigzag shape with a periodicity corresponding to that of the dislocation lines. In a field of 2.5 T, the dark lines shrink in length and bean-like black objects, such as the one highlighted by the white-dashed ellipse in Fig. 5.17C, appear. In order to study the local spin texture of these objects, a tip with an in-plane magnetic moment was used. In principle, the in-plane magnetic configuration could be explored by recording several images with different in-plane directions of the magnetic moment of the tip. Here the existence of three rotational domains (Fig. 5.17D) provides a more convenient experimental strategy: from images of the same type of bean-like object (Fig. 5.17E–G) taken in the three orientational domains using the same tip with a fixed in-plane magnetic moment direction, the local in-plane spin direction can be deduced. The bean-like object shows a unique rotational sense of the magnetization wrapping around the unit sphere, and thus has the topology of a skyrmion. Fig. 5.18A shows that the bean-shaped skyrmions can be erased one-by-one by positioning the tip above the skyrmion and ramping the bias from the U ¼ +0.3V used for imaging at a current I ¼ 0.5nA to a negative sample bias of U ¼ 3V. Ramping to a positive bias of U ¼ +3V allows the rewriting of the skyrmions one-by-one (Fig. 5.18B). Because the bias direction determines the type of switching process (erasure or writing), Joule heating can be ruled out as a process. Instead, the nature of the switching process must be governed either by a spin-transfer torque (SOT) arising from the spin-polarized current or be caused directly by the electric field which could modify the local occupation of the electronic states around the Fermi level, possibly inducing a change of the local anisotropy or DMI or also caused by an electric field induced a structural modification

Fig. 5.18 (A) Sketch of the experimental setup for skyrmion annihilation and a series of successive SP-STM constant-current images of the same Fe triple layer area showing a one-byone potential-driven annihilation of the skyrmions. The imaging conditions were U ¼ +0.3V, I ¼ 0.5nA, T ¼ 7.8K, B ¼ +2.5 T. For the skyrmion annihilation the bias is ramped to  3V. (B) Sketch of the experimental setup for skyrmion nucleation and a series of successive SP-STM constant-current images of the same Fe triple layer area showing a one-by-one potential-driven nucleation of the skyrmions (for nucleation the bias is ramped + 3V). Based on P.-J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von Bergmann, R. Wiesendanger, Electric-field-driven switching of individual magnetic skyrmions, Nat. Nanotechnol. 12 (2017) 123–126. https://doi.org/10.1038/nnano.2016.234.

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Magnetic Skyrmions and Their Applications

of the magnetic film. In order to disentangle an SOT effect from a switching process governed solely by the electric field, nonspin-polarized W tips were used. The beanshaped skyrmions can could still be imaged because of the NCMR effect, and reproducibly be erased and written by choosing the appropriate direction of the electric field. The authors concluded that for an electric field pointing toward the sample surface (negative sample bias as shown in Fig. 5.18A), an inward relaxation of the Fe atomic nuclei occurs which leads to an increase of the magnetic exchange energy and, consequently, to a preference of the ferromagnetic state, that is, to an annihilation of a skyrmion.

5.3

Magnetic exchange force microscopy

5.3.1 Contrast mechanism Atomic force microscopy measures the force acting on the tip that is brought into close vicinity of the sample surface. Typically, an AFM or SFM is operated in a dynamic mode, where shifts of the cantilever resonance frequency, Δf, arising from the z-derivative of the z-component of the force weighted over the tip-sample distance range covered by the oscillation of the cantilever tip [14], are measured. It has been demonstrated that AFM can obtain (true) atomic resolution provided, that the tip apex atom approaches the surface atoms to a distance sufficiently small that short range forces, arising from a chemical interaction between the tip apex atom and individual surface atoms, can occur (see Ref. [14] for a review of the earlier work on AFM with atomic resolution). On magnetic samples and when using a magnetic tip material, the interatomic chemical bonding energy also depends on the relative spin orientation of the two interacting atoms. Consequently, the force between tip apex atom having a fixed spin direction arising from a spin-up surface atom is expected to be different from that of a spin-down surface atom. The concept of magnetic exchange force microscopy (MExFM) was first demonstrated by Kaiser et al. [17] who used an Fe-coated cantilever tip in a field of 5 T to image rows of antiferromagnetically ordered Ni spins of opposite orientation on the surface of a NiO(001) single crystal (Fig. 5.19). In a later work, Schmidt et al. [54] also resolved the antiferromagnetic structure of an Fe monolayer on W(001). While Kaiser et al. [17] and Schmidt et al. [54] both relied on an AFM equipped with a microfabricated cantilever, MExFM contrast on the NiO (001) surface has also been demonstrated by tuning fork AFM [55]. Note that chemical interaction forces (including the spin-dependent part) are much larger than typical magnetic dipole forces measured by MFM such that the inferior force or force gradient sensitivity of the tuning forks compared to that of cantilevers is not relevant. On the contrary, when operated with small oscillation amplitudes comparable to the range of the interatomic forces, tuning fork-based AFMs can easily obtain a signal-to-noise ratio that can surpass that of cantilever AFM operated with nanometer-sized oscillation amplitudes [55–57]. Moreover, the high spring constant of a tuning fork and the mesoscopic metal tip attached to it, considerably facilitated measurements that

Scanning tunneling microscopy methods

171

(B)

(C) Corrugation [pm]

(A)

B=5T

2 0

[010] [100]

Nickel

1 nm

(D)

0

0.2 0.4 0.6 0.8 Distance along [100] direction [nm]

0

0.4 0.6 0.8 0.2 Distance along [100] direction [nm]

(E) Corrugation [pm]

(1 11 )

(001)

4

Oxygen

[001]

4 2 0

[010] [010]

[100]

[100]

1 nm

Fig. 5.19 (A) Concept of MExFM on the insulating antiferromagnetic NiO(001) single crystal. Ni magnetic moments (spins) within {111} planes are ordered ferromagnetically and point in h211i directions. Because neighboring {111} planes are aligned antiferromagnetically, Ni spins of opposite orientations (red and green) alternate along h110i directions on the (001) surface. (B) Raw topography data recorded at T ¼ 7.6K, with a frequency shift kept constant at  22Hz and unit cell-averaged data of the chemical unit cell (inset at the top right). A cantilever with a spring constant of 34N m1 and a free resonance frequency f0 ¼ 159kHz was used with an oscillation amplitude A ¼ 6.65nm at U ¼ 1.2V. (C) Line section of the spatially averaged magnetic unit cell along the [001] direction the purely chemical apparent height difference between nickel and oxygen in a obtained at a relatively large tip-sample distance can be measured to be about 4.5 pm. (D) Raw data as measured at Δf ¼ 23.4Hz, that is, with a smaller tip-sample distance showing the magnetic unit cell and unit cell averaged data of the chemical unit cell (inset at the top right). (E) Cross-section: an additional apparent height difference between nickel atoms of opposite spin orientations due to the magnetic exchange interaction with the spin of the iron tip is about 1.5 pm. Based on U. Kaiser, A. Schwarz, R. Wiesendanger, Magnetic exchange force microscopy with atomic resolution, Nature 446 (2007) 522–525. https://doi.org/10.1038/nature05617.

simultaneously acquire tunnel current and frequency shift signals, either by operation at constant tunnel current (STM-type feedback scheme) or at constant frequency shift (AFM-type feedback scheme). It has been pointed out by Hauptmann et al. [57] that force-based detection of magnetic structures by MExFM can be advantageous compared to current-based measurements by SP-STM: MExFM can address magnetic insulators [17, 55]. Further, force-based magnetic detection can be combined with simultaneous spin-polarized current measurements (SPEX) [56] providing additional information facilitating the disentanglement of magnetic and topographic contrast contributions [58]. Moreover, magnetic exchange force spectroscopy (MExFS) can directly quantify exchange forces [56, 59], which, combined with first-principles calculations, allows for determining the interplay of various exchange mechanisms [60] as well as the role of chemical functionalization of the tip [61, 62].

172

Magnetic Skyrmions and Their Applications

5.3.2 Imaging spin-spiral states and skyrmion by MExFM A first application of MExFM for imaging skyrmions was presented by Grenz et al. [63] using a cantilever-type AFM, and just 2 months later by Hauptmann et al. [56] applying a combination of SP-STM and MExFM (abbreviated SPEX) with a tuningfork-based AFM. The studied system was an Fe monolayer on Ir(111) previously studied by SP-STM that shows an incommensurate square nanoskyrmion lattice at zero field (Fig. 5.9) arising from a four-spin interaction and interfacial DMI [37]. Fig. 5.20A shows an overview noncontact AFM topography image of the Fe/Ir (111) sample with a coverage of 0.7 ML of Fe measured with a frequency shift kept constant at 12Hz using a cantilever with a stiffness kc ¼ 147.1N m1 with an oscillation amplitude A ¼ 2.3nm, at a sample bias of U ¼ +0.1V applied to minimize the electrostatic tip-sample force. Fig. 5.20B shows a frequency shift (Δf) atomic resolution image resolution image recorded with a cantilever with a stiffness kc ¼ 148.3N m1 with an oscillation amplitude A ¼ 1.2nm, and at a sample bias of U ¼ +0.9 V, acquired by scanning at constant height parallel to the surface. To make the cantilever tip magnetically sensitive the super-sharp Si tips were coated by a Ti adhesion layer of several nanometers, followed by an Fe layer of similar thickness. For magnetic imaging, the tip was again scanned at constant height, and a field of 4 T was used to obtain an out-of-plane magnetization of the tip apex. The data shown in Fig. 5.20C and the Fourier-filtered image (Fig. 5.20D) reveal both the hexagonal atomic structure and the incommensurate nanoskyrmion square lattice. In contrast to STM, MExFM facilitates the determination of the relation between the atomic and magnetic structure, because both of these data set can be simultaneously measured.

(A)

(B)

(C)

(D)

Ir ML Fe

998 pm

0] 0 pm

[11

ML Fe

1 nm -315 Hz

-345 Hz

-232 Hz

1> Δ/2, so that both spin subbands are populated with electrons. Taking into account the time inversion symmetry, the asymmetric scattering rates J ss0 ðθ,ηÞ introduced in Eq. (9.41) can be expressed in the form: J "" ðθ, ηÞ ¼ ηΓ 1 ðθÞ + ΠðθÞ, J ## ðθ,ηÞ ¼ ηΓ 1 ðθÞ  ΠðθÞ, J "# ðθ, ηÞ ¼ J #" ðθ,ηÞ ¼ ηΓ 2 ðθÞ,

(9.42)

where Γ 1, 2(θ) and Π(θ) themselves have no dependence on the background polarization sign η. This representation is convenient for treating the topological charge effect and SHE independently. Indeed, the terms ηΓ 1, 2 describe the asymmetric scattering in the same transverse direction determined by η and independent of the incident electron spin state. These terms, therefore, lead to the charge Hall effect. On the contrary, the term Π describes the scattering of spin-up and spin-down electrons in the opposite transverse directions, independent of η. This process leads to the SHE, it is absent for spin-flip channels. For a magnetic skyrmion, both Γ 1, 2 and Π change their signs with the sign change of the skyrmion vorticity ϰ !ϰ. As we have discussed in the previous sections, THE can be due to SHE in the adiabatic case or due to charge Hall effect in the weak-coupling regime. Which of the two contributions dominate, strongly depends on whether the spin-flip processes are activated or not. When the spin splitting does not exceed the Fermi energy EF ≫ Δ/2 so that both electron spin projections are available, the rate of the spin-flip scattering is controlled by the adiabatic parameter λa defined in Eq. (9.1). In the case of λa  1, the spin-flip processes are effective and give rise to the spin-independent asymmetric scattering described by Γ 1, 2, this is the weak-coupling case. In the opposite case of large adiabatic parameter λa ≫ 1, the spin flip processes are suppressed and the emerging Hall effect is spin dependent. The adiabatic contribution to the asymmetric scattering response is, therefore, described by Π. Let us assume that the considered 2D sample contains spatially localized chiral spin textures having the same vorticity ϰ, the background polarization is η ¼ +1. We consider the dilute regime, when the scattering rate on spin textures is much smaller than that on nonmagnetic impurities. Solving the system (9.39) for fs ðpÞ, we express the topological Hall resistivity ρTyx as a sum of the two contributions [36]: ρTyx ¼ ρc + ρa ,

Z

2π

ρc ¼ R0 ðϕ0 nsk Þ 0

Z

2π

ðΓ 1 + Γ 2 Þsin θdθ, ρa ¼ R0 Ps ðϕ0 nsk Þ

Π sin θdθ,

0

(9.43) where R0 ¼ (nec)1. The term ρc describes the charge transverse current (charge Hall effect) originating from asymmetric scattering due to spin-independent terms Γ 1, 2 (see Eq. 9.42). The term ρa describes the transverse spin current (SHE) driven by the spin-dependent

Topological Hall effect

305

contribution to the asymmetric scattering Π (Eq. 9.42). The spin current does not lead to a charge separation unless there is unequal number of spin-up and spin-down carriers in the system. Therefore, this contribution to the Hall resistivity is proportional to the carrier spin polarization Ps ¼ (n" n#)/(n" + n#) ¼ Δ/2EF. In Eq. (9.43), the notation n ¼ n" + n# stands for the total electron sheet density. Finally, for the complete description of the Hall response, the scattering rates Γ 1, 2 and Π should be calculated explicitly. This can be done numerically using the phasefunction method as described in Refs. [21, 37].

9.4.2 Scattering on a chiral spin texture in an intermediate case The crossover from spin-independent to spin-dependent scattering on a magnetic skyrmion is driven by the adiabatic parameter. The latter depends on the exchange spin splitting Δ, electron wavevector k, and the skyrmion size a. The details of the crossover depend on which of these parameters are varied. Before considering this crossover from a general point of view, let us illustrate evolution of a scattering cross-section asymmetry with increasing of the adiabatic parameter. Let us separate the differential scattering cross section from an incident state β into an outgoing state α into symmetric and antisymmetric parts: dσ αβ ¼ Gαβ ðθÞ + Σ αβ ðθÞ, dθ

(9.44)

where Gαβ(θ) ¼ Gαβ(θ) is symmetric and Σ αβ(θ) ¼ Σ αβ(θ) is antisymmetric with respect to the scattering angle θ. The details of the crossover appear to be different depending on whether Δ is varied keeping ka constant or the skyrmion size a is varied keeping the exchange strength fixed. This is because the wave parameter ka also influences the scattering together with the adiabatic parameter. For example, ka ≪ 1 in analogous to Rayleigh scattering as the scatterer size is smaller than the electron wavelength, the scattering goes into a broad range of scattering angles. The case ka 1 corresponds to Mie scattering, the scattering occurs mostly into a narrow range of small angles. The evolution of the antisymmetric part of the differential cross-section Σ αβ(θ) with the skyrmion size is shown in Fig. 9.5. Only spin-conserving channels are shown for the purpose of clarity. The first and the last frames in Fig. 9.5A and F correspond to the limiting cases. At a small λa and ka (Fig. 9.5A), both spin-up and spin-down electrons scatter into the same half-plane (for background polarization η ¼ +1, it is the right half-plane). As λa is increased further, the crossover begins with more and more terms of the Born series contributing to the T-matrix and breaking down the sign compensation for the opposite spins. At the same time, the angular dependence evolves as the wave parameter ka changes. Note that with a background magnetization, the Fermi wavevector is different for spin-up an spin-down electrons. The crossover ends with an adiabatic regime characterized by the asymmetry for spin-up and spin-down electrons (Fig. 9.5E and F). The scattering asymmetry for η-parallel channel (spin-up for η ¼ +1) is the same at the opposite sides of the crossover: spin-up electron is mostly

0.5

1

1.5

−0.5

2

2.5

θ 3

↓↓

0

0.5

1

1.5

2

2.5

θ 3

↑↑

−0.5

1

Σαβ (θ)

Σαβ (θ)

0

λa = 1.4

×10−2

Σαβ (θ)

λa = 0.6

×10−4

0

λa = 3.5

×10−2

↑↑ 0.5

1

↑↑

1.5

2

2.5

3

↓↓

−1

−1

θ

↓↓

−1

−2

(A)

= 0.3, ka = 2

Δ 2E

(B)

= 0.3, ka = 4.6

Σαβ (θ)

0.05 0

↓↓ 0.5

1

1.5

θ 2

2.5

3

(D)

0.5

1

θ 1.5

2

2.5

3

0

−0.1

−0.05 −0.1

0

0.5

↓↓

0.1

↑↑ Δ 2E

−0.2

= 0.3, ka = 13

(E)

−0.5

↑↑ Δ 2E

= 0.3, ka = 17.6

Fig. 9.5 Evolution of Σ "", Σ ## (in units of a/2) tuned by skyrmion size a.

= 0.3, ka = 11.7

λa = 7.2

λa = 5.3 Σαβ (θ)

λa = 3.9

Δ 2E

(C)

Σαβ (θ)

Δ 2E

(F)

↓↓ θ 0.5

1

1.5

2

2.5

3

↑↑ Δ 2E

= 0.3, ka = 24

λa = 0.6

× 10−4

λa = 3.9 θ

0.5

1

1.5

2

−2

Δ 2E

Σαβ (θ)

Δ 2E

1.5

2

= 0.16 ka = 24

λa = 7.2 ↓↓

−0.5

1

↑↑

(B)

0.5

0

0.5

θ

−2

−0.1

= 0.025 ka = 24

(C)

0

−5 × 10

↑↑ ↓↓

−4

(A)

Σαβ (θ)

Σαβ (θ)

0

↓↓

5× 10−2

2

0.5

θ 1

1.5

2

↑↑ Δ 2E

= 0.3 ka = 24

Fig. 9.6 Evolution of Σ "", Σ ## (in units of a/2) tuned by an exchange coupling Δ.

scattered into the right half-plane as in the weak-coupling regime. For η-antiparallel channel (spin-down for η ¼ +1), the sign of the asymmetry is changed; spin-down electron in Fig. 9.5F is scattered into different (left) half-plane than for small λa (Fig. 9.5A). Although probably more difficult from experimental point of view, the transition from weak to adiabatic regime can be also tuned by the exchange constant keeping ka ¼ const. The crossover in the same range of λa tuned by the exchange strength Δ is shown in Fig. 9.6. Note that this type of the crossover is possible only at ka ≫ 1. If, on the opposite ka ≪ 1, then λa ¼ 1 corresponds to Δ ≫ EF, where EF is the Fermi energy, but then the spin-down electrons with a nonzero kinetic energy do not exist and so spin and charge Hall currents coincide. At ka ≫ 1, a number of angular harmonics contribute to the scattering already in the weak-coupling case, so the whole evolution of the asymmetry occurs within the range of the small angles close to the forward scattering.

9.4.3 Interplay between different THE regimes Let us summarize most important features that arise from the competition between spin-dependent and spin-independent terms in the topological Hall response.

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In Fig. 9.7 we present the dependence of the THE resistivity calculated from Eq. (9.43) on the skyrmion diameter for the crossover regime, that is, when the adiabatic parameter is neither small nor large. In this figure, we also demonstrate the dependence of ρc, ρa. The exact calculation reveals that the spin-independent part of THE remains relevant and even dominates the whole amplitude of ρTH up to λa  5. As a result of the competition between ρc and ρa, the magnitude of the Hall current depends nonmonotonically on the skyrmion size. The same nonmonotonic behavior can be also driven by the variation of the Fermi energy. The second important feature of the crossover regime concerns the role of the topological charge in the Hall resistivity. The Hall resistivity becomes quantized when the adiabatic parameter is large and the electron motion inside skyrmion is described by the Berry curvature. On the contrary, Eq. (9.33) for the asymmetric scattering rates in the weak-coupling regime represents no explicit dependence on the magnetization winding number. Therefore, both topologically charged and trivial chiral spin textures are able to produce the Hall response. As soon as we associate the features of the weakcoupling regime with ρc contribution to ρTH , we expect the similar behavior even at higher values of the adiabatic parameter. Fig. 9.8 demonstrates the dependence of ρTH on the spin texture diameter for three different skyrmion configurations, namely (Λ1(r) ¼ π(1  2r/a), Λ2 ðrÞ ¼ π sin 2 ½ðπ=2Þð1 + 2r=aÞ) correspond to magnetic skyrmions, while (Λ3(r) ¼ (2r/a)π(1  2r/a)) has zero topological charge. It is clearly seen from Fig. 9.8 that all three configurations exhibit nonzero ρTH featured by nonmonotonic dependence on the spin texture size. Naturally, the Hall response driven by chiral ordering of the magnetization can be induced by the topologically trivial configurations of the magnetization as well as by topologically charged. It is interesting to see how the Hall response becomes dependent on the topological charge when increasing the skyrmion size, and, hence, the adiabatic parameter. In Fig. 9.9, we show the continuation of the dependence of ρTH on the texture size for the same Λ1, 2, 3 spin textures at larger λa. It is evident that the Hall signal for the topologically trivial configuration goes down to zero, while both charged configurations Λ1, 2 are featured by the saturation of ρTH [38]. The limiting value at λa ≫ 1 naturally coincides with the result of the adiabatic theory: ρTH ! R0 Ps ðϕ0 nsk ÞQ. Fig. 9.7 The dependence of ρTyx on chiral texture diameter ka for different texture profiles in the region of crossover. The parameters Ps ¼ 0.4, nsk ¼ 2  1011 cm2.

Topological Hall effect

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Fig. 9.8 The dependence of ρTyx on chiral texture diameter ka for different texture profiles in the adiabatic region. The parameters Ps ¼ 0.4, nsk ¼ 2  1011 cm2. The saturation magnitude Ps(ϕ0nsk)  3.3 T.

9.4.4 General character of THE and beyond skyrmions While THE is often associated with the geometric Berry phase in skyrmion crystals [8–10], or regular and disordered arrays of magnetic skyrmions, this effect is obviously more general. As we have discussed in this chapter, when the skyrmion size is reduced, one should be careful estimating the magnitude of THE solely from the topological charge. Indeed, the mean-field Berry-phase approach is valid only when the criteria of the adiabaticity is strictly satisfied. Let us estimate the characteristic values of λa for some of the skyrmion systems experimentally studied to date. Magnetic skyrmions realized in multilayers stack systems, such as in Refs. [31, 32, 39], and on another material platform, see Refs. [29, 30], and also Ref. [33], have spatial size typically ranging from 40 to 100 nm. For an estimate, we take a ¼ 50 nm, Δ ¼ 0.6 eV, EF ¼ 5 eV, the effective in-plane mass m ¼ m0 and obtain λa  30. Therefore, the estimates based on the mean-field adiabatic approach are definitely correct for these material platforms. An example of substantially different system is Ta/FeCoB/TaOx structure with skyrmionic bubbles of 1 μm size [40]. The electron transport in such systems is also described by the adiabatic theory. Fig. 9.9 The dependence of ρTyx on chiral texture diameter ka for different texture profiles in the adiabatic region. The parameters Ps ¼ 0.4, nsk ¼ 2  1011 cm2. The saturation magnitude Ps(ϕ0nsk)  3.3 T.

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The decrease of λa down to the order of unity leading to the nonadiabatic transport regime is expected for nanometer-size chiral magnetic textures in metallic systems with typical ferromagnets such as Co or Fe. In some of the recent experimental studies a few-nanometer-size skyrmions were successfully stabilized [35]. Taking a ¼ 5 nm, one gets λa  3 suggesting that the system is in the vicinity of the crossover from adiabatic to weak-coupling regime. Adjusting the skyrmions size with an external magnetic field [38] would allow one to trace the transition between charge and spin Hall regimes of THE in such systems. Let us note that the electron mean free path (MFP) can also contribute to THE properties. In particular, if MFP is smaller than the skyrmion size, the electron can spend more time inside the skyrmion, thus, effectively increasing the adiabatic parameter as suggested in Refs. [27, 41]. Alternatively, the nonadiabatic scenario of THE can be achieved in the DMS. The existence of chiral spin textures in DMS with spin-orbit interaction has been suggested [42, 43], in particular, on the basis of chiral magnetic polaron [44]. The solid advantage of DMS is that both the Fermi energy and the exchange interaction strength can be tuned, allowing to control the adiabatic parameter in a wide range. For example, taking a n-type Cd1xMnxTe-based quantum well (electron effective mass m ¼ 0.11 m0, the exchange spin splitting depends on Mn fraction as x  220 meV) with a chiral spin texture radius equal to the donor-bound state Bohr radius of 3 nm, for x ¼ 0.08 and the electron sheet density n1 ¼ 5  1011 cm2 we get λa  2.5. By decreasing the sheet density down to n2 ¼ 1  1011 cm2 or decreasing Mn fraction down to x ¼ 0.02, the adiabatic parameter can be adjusted to λa  6 and λa  0.8, respectively. The weak-coupling regime of THE (which can be also referred as chiral Hall effect) is not sensitive to the global topological properties of the magnetization field and, therefore, should manifest itself in various systems, where the magnetization experiences local chiral fluctuations of a various origin. Indeed, any chiral spatial profile of magnetization and any noncoplanar spins arrangement would lead to the chiralitydriven exchange skew scattering of electrons. The chiral spin textures should not be regarded as something very exotic. For instance, fluctuations of the magnetization in the presence of spin-orbit coupling become chiral and can give rise to the Hall effect [23, 45]. Also, an electrostatic disorder in conducting systems with broken timereversal symmetry universally leads to chiral Friedel oscillations of the electron gas spin density, which are skyrmion-like textures [46]. This phenomenon has a universal character and it is expected in a variety of experimentally studied systems, such as DMS [47, 48], thin films of ferromagnets [49, 50], Bi2Se3 doped by magnetic impurities [51, 52], magnetic topological insulators [53], or due to the proximity effect with magnetic insulators or ferromagnets [54]. So far, we have discussed THE and the spin chirality in real space. However, the appearance of magnetic skyrmions and similar spin vortices is often associated with Dzyaloshinskii-Moriya interaction and, hence, SOC, which favors noncollinear spins alignment. SOC involved in skyrmions formation can also influence the free carriers spectra introducing Berry curvature in momentum space. When both exchange interaction and SOC have a comparable influence on electron motion, one expects transport effects driven by the mixed real-momentum-space Berry curvature. This scenario is discussed as an important ingredient needed to describe the transport in

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skyrmion-based systems [55]. In particular, the mixed real-momentum-space Berry curvature allows the asymmetric response due to a domain wall as described in Ref. [56], which is generally absent via the topological Hall scenario described by the real-space Berry curvature. Apart from magnetic skyrmions, there are other topologically nontrivial spin textures, which are promising for THE experiments and further development of the theory of THE. For instance, THE is possible for antiferromagnetic skyrmions [57]. One can also expect the similar transport features, at least in the adiabatic regime, for merons [58], which possess fractional topological charge, or antiskyrmions, observed recently [59]. More complex textures like hopfions have been also considered from the point of view of THE. THE is not expected for hopfions in the adiabatic regime due to compensation of the Berry phases from different parts of the hopfion [60]. However, for the weak-coupling regime, the problem of THE for hopfions and other complex spin textures have not been studied so far giving enough room for further developments.

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Skyrmions in ferrimagnets

10

Xichao Zhanga, Yan Zhoua, and Seonghoon Woob a School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, People’s Republic of China, bIBM Thomas J. Watson Research Center, Yorktown Heights, NY, United States

10.1

Ferrimagnetic skyrmions: Topological properties and materials

Topologically nontrivial skyrmions were theoretically predicted to exist in magnetic materials by Bogdanov and Yablonskii in 1989 [1], where the skyrmions are localized noncollinear spin textures stabilized by the asymmetric exchange interaction, that is, the Dzyaloshinskii-Moriya interaction (DMI) [2–7]. The first experimental observation of magnetic skyrmions was achieved in 2009 [8], which revealed the existence of skyrmions in the form of lattice array in noncentrosymmetric ferromagnet MnSi with bulk DMIs. Since then, magnetic skyrmions have been the focus of magnetism and spintronics research as they are widely considered to be useful for future information storage and processing applications [7, 9–15]. So far, magnetic skyrmions have been observed and even manipulated in different material systems, including a number of chiral magnets with bulk DMIs and ferromagnetic multilayers with interface-induced DMIs. Most recently, theoretical works suggested that skyrmions can also be metastable solutions in antiferromagnets [16–19], and several experimental works have reported the existence of skyrmions in uncompensated antiferromagnets, that is, ferrimagnets [20–22]. Especially, due to the fundamental differences in underlying lattice structure and topological structure, antiferromagnetic and ferrimagnetic skyrmions could demonstrate novel dynamic properties that are different to their counterparts in conventional ferromagnetic materials. In this chapter, we first introduce the concept of skyrmions in antiferromagnetic and ferrimagnetic systems, and then, we focus on the imaging, writing, deleting, and electrical detection of ferrimagnetic skyrmions. We also review and discuss the intriguing dynamics of ferrimagnetic skyrmions and the associated skyrmion Hall effect in ferrimagnets. This chapter will end with a brief review and outlook on potential applications of ferrimagnetic skyrmions. The spin configuration of a ground-state skyrmion in ferromagnetic materials is given in Fig. 10.1A, where the skyrmion carries an integer topological charge of Q ¼ +1. The topological charge Q in ferromagnetic system is defined as Ð Q ¼  m  (∂xm × ∂ym)dxdy/4π, where m being the reduced magnetization. Fig. 10.1B shows a ground-state skyrmion in ferromagnetic materials carrying a topological charge of Q ¼  1. For the antiferromagnetic system, generally one could consider that there are two antiferromagnetically exchange-coupled sublattices, where Magnetic Skyrmions and Their Applications. https://doi.org/10.1016/B978-0-12-820815-1.00002-X Copyright © 2021 Elsevier Ltd. All rights reserved.

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Fig. 10.1 Illustration of skyrmions in different material systems. (A) A ferromagnetic skyrmion with a topological charge of +1. (B) A ferromagnetic skyrmion with a topological charge of 1. (C) An antiferromagnetic skyrmion. (D) A ferrimagnetic skyrmion. (E) A synthetic antiferromagnetic skyrmion. (F) A synthetic ferrimagnetic skyrmion. Note that the direction and size of an arrow denote the magnetization direction and relative magnitude, respectively.

all spins are fully compensated, thus leading to a zero net magnetization. In such an antiferromagnetic system, a skyrmion can be seen as a structure consisting of two sublattice skyrmions, which carry opposite topological charges (i.e., Q ¼ +1 and Q ¼  1) and are antiferromagnetically exchange-coupled rigorously. The schematic of an antiferromagnetic skyrmion is given in Fig. 10.1C. Under the magnetization-based definiÐ tion of topological charge, i.e., Q ¼  m  (∂xm × ∂ym)dxdy/4π, the topological charge of an antiferromagnetic skyrmion is equal to zero. However, it should be noted that the topological charge in antiferromagnetic system can also be defined based on the configuration of antiferromagnetic order parameter n ¼Ð (m1  m2)/2 (i.e., the N
eel vector). In such a case, the topological charge Q ¼  n  (∂xn × ∂yn)dxdy/4π and the topological charge of a ground-state antiferromagnetic skyrmion is thus equal to +1 or 1, indicating it is still a skyrmionic object with nonvanishing topological characteristics and a counterpart of skyrmions in ferromagnets. If the two opposite sublattices are uncompensated, as shown in Fig. 10.1D, the system is known as the ferrimagnet and the given skyrmion state is referred to as a ferrimagnetic skyrmion. Namely, the ferrimagnetic and antiferromagnetic skyrmions share the same configuration of reduced magnetization (i.e., spin), whereas the ferrimagnetic skyrmion has a nonzero net magnetization. On the other hand, it is worth mentioning that magnetic skyrmions can be formed in both synthetic antiferromagnets [23–28] and synthetic ferrimagnets [25, 29]. Fig. 10.1E and F show the schematics of a bilayer synthetic antiferromagnetic skyrmion and a bilayer synthetic ferrimagnetic skyrmion, where the skyrmions in top and bottom ferromagnetic layers are

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exchange-coupled together in an antiferromagnetic manner, through the RudermanKittel-Kasuya-Yosida (RKKY) mechanism [24, 30–32]. Due to the nontrivial topological structure, a ferromagnetic skyrmion demonstrates few topology-dependent dynamic behaviors. For example, as shown in Fig. 10.2A, a current-driven ferromagnetic skyrmion may move at an angle with respect to the direction of the driving current, of which the phenomenon is referred to as the skyrmion Hall effect [33–35]. The skyrmion Hall effect is given by the quantum-mechanical origin, called Berry phase [36]. When conduction electrons flow through noncolinear magnetic textures, the electrons can be strongly coupled to the local spin field and thus acquire a Berry phase, which then acts as an effective electromagnetic field, therefore termed as an “emergent” field [37]. In detail, an emergent magnetic field arises as electrons flow through topological spin textures (i.e., skyrmions), which then leads to an additional voltage signal in the Hall measurements induced by the topology of skyrmions, therefore called as the topological Hall effect [38]. Hence, phenomenologically, the skyrmion Hall effect could result in the accumulation of ferromagnetic skyrmions at sample edges, which could lead to the destruction of skyrmions when they are forced to touch the sample edge by large currents [24]. Thus, from the point of view of practical applications, it is necessary to avoid or reduce the skyrmion Hall effect for applications based on the in-line motion of skyrmions, such as the racetrack-type memory [9, 24, 31, 39–42], where skyrmions carrying binary information should safely move along a straight line on the nanotrack with zero or minimal interaction with the edges. Because the skyrmion Hall effect is caused by the topology-related Magnus force acting on the skyrmion texture, a ferrimagnetic skyrmion can have a reduced skyrmion Hall effect due to its two sublattice structures, where the Magnus forces acting on two sublattice skyrmions are perpendicular to the velocity but pointing along opposite directions. As the magnitude of the sublattice Magnus force is also proportional to

Fig. 10.2 Illustration of the skyrmion Hall effect. (A) The skyrmion Hall effect of a ferromagnetic skyrmion. (B) The reduction of skyrmion Hall effect of a ferrimagnetic skyrmion due to the partial cancellation of sublattice Magnus forces. (C) The elimination of skyrmion Hall effect of an antiferromagnetic skyrmion due to the cancellation of sublattice Magnus forces.

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the sublattice magnetization saturation (i.e., the net spin density [43]), the net Magnus force acting on a ferrimagnetic skyrmion could be much smaller than that of a ferromagnetic skyrmion driven by the same current provided that the magnetization magnitude of ferromagnet is much larger than the net magnetization magnitude of ferrimagnet, as illustrated in Fig. 10.2B. Therefore, a ferrimagnetic skyrmion could reach a higher speed in a nanotrack and is more robust than a ferromagnetic one due to the smaller transverse shift caused by the reduced skyrmion Hall effect. Moreover, as we will discuss in Section 10.3, the effective net magnetization of ferrimagnets can be controlled by controlling either material composition or temperatures, and remarkably, at a specific temperature or composition where the net spin density vanishes, the ferrimagnet could behave as an antiferromagnet, leading to the zero skyrmion Hall angle [43, 44]. Indeed, for the perfect antiferromagnetic system, as shown in Fig. 10.2C, the Magnus forces acting on the two sublattice skyrmions can exactly cancel each other out, leading to a zero net Magnus force and thus zero skyrmion Hall effect. Therefore, antiferromagnetic skyrmions are also promising for high-speed in-line motion in narrow channels. We will discuss more about the current-induced dynamics of ferrimagnetic skyrmions in Section 10.3, and the dynamics of antiferromagnetic skyrmions will be discussed in detail in Chapter 10. Regarding the materials hosting ferrimagnetic skyrmions, here we limited our discussion to the two-dimensional or quasi-two-dimensional case. That is, the variation of skyrmion texture in the thickness direction can be reasonably ignored, whereas sometimes such magnetic texture variation in film thickness direction cannot be ignored due to strong emergence of dipolar field and its interplay with chiral textures driven by DMIs [45–47]. Generally, one will need to have ferrimagnetic thin films or multilayers with bulk or interfacial DMIs. Most notable approach was using the bulk alloy of 4f rare-earth and 3d transition-metal ferromagnet alloy, e.g., Gd-Co or Tb-Fe, as was heavily studied in other field of study—laser-induced ultrafast magnetization dynamics and all-optical switching [48–51]. In these materials, magnetic moments in 4f rare-earth and 3d ferromagnet elements couple antiferromagnetically, whereas the relative compensation of their magnetic moments varies as a function of temperature and material composition as briefly discussed above. Based on such ferrimagnetic alloy, one can induce strong DMIs by engineering interfaces, for example, by establishing asymmetric heterostructure where a ferrimagnet is sandwiched between high spin-orbit coupling heavy metal and oxides, where the heavy metal (e.g., Pt or Ta) provides sufficient DMIs to stabilize chiral textures, including skyrmions, as was shown in GdFeCo multilayers [20, 22]. Recently, the bulk form of DMI is also observed in the same ferrimagnetic materials, GdFeCo alloy [52], induced by nonequivalent element distribution within the film. It is worth mentioning that skyrmion-like bubbles can exist in ferrimagnetic thin films even in the absence of DMIs [53], which share the same topological and chiral nature but usually have a larger size in diameter. More theoretical and experimental works about skyrmions in ferrimagnets will be reviewed in the next section and will be linked to the functions of spintronic information storage and processing devices.

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10.2

319

Imaging, writing, deleting, and electrical detection of ferrimagnetic skyrmions

In this section, we review recent theoretical and experimental findings and advances in imaging, detection, writing, and deleting of skyrmions in ferrimagnets. In analogy to ferromagnetic skyrmion-based spintronic applications, the most essential task for nonvolatile and low-energy information storage and processing is to realize the writing (i.e., creation), deleting (i.e., annihilation), and electrical detection of skyrmion states in a reliable and reproducible manner. Indeed, to investigate the static and dynamic properties of a ferrimagnetic skyrmion, it is also vital to realize effectively imaging of skyrmions in ferrimagnets. In 2009, Ogasawara et al. [54] experimentally realized the creation of a skyrmionlike bubble by utilizing the laser-induced magnetization reversal in a TbFeCo thin film under a small out-of-plane magnetic field. The diameter of the laser-induced reversal bubble area in the TbFeCo thin film ranges from 400 to 1500 nm, which can be controlled by the pump pulse energy. Note that the skyrmion-like bubble can have the same topology of a nanoscale skyrmion; however, due to the absence of DMI, the topological charge of a skyrmion-like bubble may not be fixed, i.e., achiral, and the size of a skyrmion-like bubble is usually much larger than that of a nanoscale skyrmion (e.g., 10–100 nm) owing to the effect of dipole-dipole interaction. In 2013, Finazzi et al. [55] experimentally realized the laser-induced writing of a nanoscale skyrmion (about 400 nm in diameter) in a thin TbFeCo film, where no external magnetic field is required to stabilize the skyrmion and other skyrmionic textures. TbFeCo is a promising ferrimagnetic material to host skyrmion-like spin textures as a thin film of TbFeCo grown on a heavy metal layer can have a good perpendicular magnetic anisotropy [56], where DMIs can also be induced at TbFeCo-heavy metal interfaces as we noted above [57]. With the support of perpendicular magnetic anisotropy and DMI, isolated chiral skyrmions can be stabilized even in the absence of an out-of-plane external magnetic field. In 2018, Woo et al. [20] realized in room-temperature experiments the creation of nanoscale skyrmions by external magnetic field in a ferrimagnetic multilayer [Pt/ GdFeCo/MgO]20, where the created ferrimagnetic skyrmion has a diameter down to 200 nm. To observe the skyrmion in ferrimagnetic GdFeCo alloy layer, where Gd and FeCo sublattices are antiferromagnetically exchange-coupled, Woo et al. used the element-specific scanning transmission X-ray microscopy (STXM), which gives opposite contrast at Fe-absorption and Gd-absorption edges, revealing the antiferromagnetic spin ordering within the ferrimagnetic GdFeCo alloy. In the same year, Woo et al. [22] also reported the deterministic electrical writing and deleting of a single isolated skyrmion in a ferrimagnetic multilayer [Pt/GdFeCo/MgO]20 at room temperature, where the writing and deleting of isolated skyrmions are realized by applying electric pulses with two different unique pulse profiles, as shown in Fig. 10.3A and B. The electric current pulses injected into the ferrimagnetic multilayer will induce

Fig. 10.3 Imaging, writing, deleting, and electrical detection of skyrmions in ferrimagnets. (A and B) Current-induced writing and deleting of ferrimagnetic skyrmions in a nanotrack. (C) Experimental confirmation of the controlled skyrmion nucleation near a constriction. (D) Ferrimagnetic skyrmions in a ferrimagnetic Pt/GdCo/TaO film. (A and B) From S. Woo, K.M. Song, X. Zhang, M. Ezawa, Y. Zhou, X. Liu, M. Weigand, S. Finizio, J. Raabe, M.-C. Park, K.-Y. Lee, J.W. Choi, B.-C. Min, H.C. Koo, J. Chang, Deterministic creation and deletion of a single magnetic skyrmion observed by direct time-resolved X-ray microscopy, Nat. Electron. 1 (2018) 288. © 2018 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. (C) From F. B€ uttner, I. Lemesh, M. Schneider, B. Pfau, C. M. G€unther, P. Hessing, J. Geilhufe, L. Caretta, D. Engel, B. Kr€ uger, J. Viefhaus, S. Eisebitt, G.S.D. Beach, Field-free deterministic ultrafast creation of magnetic skyrmions by spin–orbit torques, Nat. Nanotechnol. 12 (2017) 1040. © 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. (D) From L. Caretta, M. Mann, F. B€uttner, K. Ueda, B. Pfau, C.M. G€ unther, P. Hessing, A. Churikova, C. Klose, M. Schneider, D. Engel, C. Marcus, D. Bono, K. Bagschik, S. Eisebitt, G.S.D. Beach, Fast current-driven domain walls and small skyrmions in a compensated ferrimagnet, Nat. Nanotechnol. 13 (2018) 1154. © 2018, Springer Nature.

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spin-orbit torques due to the spin Hall effect in the Pt layers. Isolated skyrmions can be created in the ferrimagnetic multilayer nanotrack by a writing pulse and then deleted by a deleting pulse in a deterministic and reproducible manner. The ultrafast processes of skyrmion writing and deleting are imaged by using the stroboscopic pump-probe X-ray measurements, that is, a time-resolved nanoscale X-ray imaging technique. Using spin dynamics simulations, Woo et al. further revealed the microscopic mechanism of the topological fluctuations that occur during the writing and deleting of skyrmions in ferrimagnetic GdFeCo films. As shown in Fig. 10.3C, similar observation was also reported by B€ uttner et al. [58], where the authors demonstrated that single isolated skyrmions can be generated and annihilated at custom-defined positions in a magnetic racetrack deterministically, using the nanoseconds pulses leading to the generation and application of nanoseconds-long spin-orbit torques. Caretta et al. [21] also experimentally created, using current pulse injections, roomtemperature-stable skyrmions in ferrimagnetic Pt/GdCo/TaOx films with sizeable DMIs in 2018, where ferrimagnetic skyrmions were imaged using X-ray holography at room temperature. As shown in Fig. 10.3D, Caretta et al. found that their ferrimagnetic skyrmions range in size with a mean diameter of 23 nm and minimum observed diameters approaching 10 nm, which is significantly smaller than that of ferromagnetic skyrmions and therefore highly appealing for practical applications. On the other hand, Streubel et al. [59] observed chiral ferrimagnetic spin textures in amorphous GdCo films utilizing Lorentz microscopy and X-ray magnetic circular dichroism spectroscopy, which is also a similar ferrimagnetic platform hosting skyrmions. It is noteworthy that the N
eel-type chiral nature of domain walls has also been observed in ferrimagnetic oxide heterostructures by Avci et al. [60] in 2019, broadening the class of materials where ferrimagnetic skyrmions could be stabilized and manipulated. Ferrimagnetic skyrmions can also be imaged by using the magnetic force microscopy (MFM). For example, in 2019, Branda˜o et al. [61] observed skyrmions in a nanostructured ferrimagnetic Pt/CoGd/Pt multilayer discs at zero magnetic field and room temperature, where a single isolated ferrimagnetic skyrmion with a diameter of about 70 nm was obtained. In 2020, by using the MFM technique, Mandru et al. found two distinct skyrmion phases in a hybrid ferromagnet/ferrimagnet/ferromagnet multilayer [47], which can coexist at certain out-of-plane magnetic field and can be controlled by adjusting the magnetic field. From the point view of practical applications, the electrical detection of skyrmions is a prerequisite for future information storage and processing devices based on skyrmions. The skyrmions in ferromagnetic materials can be electrically detected by measuring the topological Hall effect [38, 62–65], and it is also possible to use the same method to detect skyrmions in ferrimagnets. Indeed, in 2019, Yu et al. [66] reported an extraordinary Hall signal near the magnetization compensation point in ferrimagnetic CoTb single layer, which may originate from the noncollinear skyrmion spin texture stabilized by DMIs. However, as the magnitude of electrical signals from topological Hall effect is significantly smaller than the required values for applications, further exploration on the electrical detection scheme may be necessary to provide effective solutions.

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10.3

Magnetic Skyrmions and Their Applications

Current-induced dynamics of ferrimagnetic skyrmions

Similar to ferromagnetic skyrmions, skyrmions in ferrimagnetic materials can also be driven into motion and manipulated by electric currents [20, 22, 67]. Specifically, the current-induced dynamics of a ferrimagnetic skyrmion could be induced by either the spin-transfer torques [68] or the spin-orbit torques [20, 44]. In 2016, two theoretical reports [16, 17] suggested that skyrmions can be driven into motion by spin currents in antiferromagnetic system, which shows no skyrmion Hall effect and is thus promising for spintronic applications based on in-line motion of skyrmions. Since then, the current-induced motion of antiferromagnetic skyrmions and the current-induced motion of skyrmions in uncompensated ferrimagnets have aroused great interest of many researchers in the field of chiral magnetism and spintronics. In 2017, Kim et al. [43, 69] theoretically studied the current-driven dynamics of a single isolated skyrmion in a ferrimagnetic nanotrack. Such a ferrimagnetic nanotrack can be made of rare-earth transition-metal ferrimagnetic thin-film alloys, which can reach the angular momentum compensation point by varying the temperature or chemical composition. Kim et al. [43, 69] found that in a nanotrack with varying net angular momentum density, the skyrmion can move strictly along the line of the angular moment compensation points due to the absence of Lorentz force, which is a force perpendicular to the skyrmion velocity and is proportional to the net angular momentum density. This work suggested that the line of the angular momentum compensation points (i.e., the line of points where net angular momentum density equals zero) can be used as a self-focusing racetrack for ferrimagnetic skyrmions. In 2018, the current-driven motion of ferrimagnetic skyrmions was realized for the first time by Woo et al. [20] in ferrimagnetic GdFeCo films, where they also demonstrated the inhibition of the skyrmion Hall effect in ferrimagnetic system. In the work by Woo et al. [20], the dynamics of ferrimagnetic skyrmions is mainly driven by the damping-like spin-orbit torque, which is generated by the spin Hall effect in the heavy metal Pt layer attached to the ferrimagnetic GdFeCo layer. As shown in Fig. 10.4A, Woo et al. found that a train of skyrmions propagates along the ferrimagnetic nanotrack driven by the spin torque, and that their alignment and trajectory show a finite angle with respect to the current flow direction, which is the hallmark of the skyrmion Hall effect [33–35]. The driving current pulse amplitude-dependent skyrmion velocity and its skyrmion Hall angle are given in Fig. 10.4B and C, respectively. It can be seen that the skyrmion velocity increases linearly with pulse amplitudes, and the maximum velocity approaches 50 m s1 at a driving current density of 3.55  1011 A m2. At the same time, a reduced skyrmion Hall angle, up to about 20 degrees, is observed, which is far lower than the skyrmion Hall angles (>30 degrees) observed for ferromagnetic skyrmions in Ta/CoFeB/MgO and Pt/CoFeB/MgO structures [34, 35]. Compared with ferromagnetic skyrmions, the reduced skyrmion Hall effect of ferrimagnetic skyrmions means that ferrimagnetic skyrmions can be driven into high-speed motion with much smaller skyrmion Hall angle (i.e., transverse shift),

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Fig. 10.4 Current-driven behavior of ferrimagnetic skyrmions and their velocity and skyrmion Hall effect in Pt/GdFeCo/MgO multilayer.(A) Sequential STXM images showing the responses of multiple skyrmions after injecting unipolar current pulses along the track. (B) Experimental and simulated average skyrmion velocity versus current density. (C) Experimental and simulated average skyrmion Hall angle versus current density. From S. Woo, K.M. Song, X. Zhang, Y. Zhou, M. Ezawa, X. Liu, S. Finizio, J. Raabe, N.J. Lee, S.-I. Kim, S.-Y. Park, Y. Kim, J.-Y. Kim, D. Lee, O. Lee, J.W. Choi, B.-C. Min, H.C. Koo, J. Chang, Current-driven dynamics and inhibition of the skyrmion hall effect of ferrimagnetic skyrmions in GdFeCo films, Nat. Commun. 9 (2018) 959. Creative Commons CC BY.

and thus are more suitable for in-line motion in narrow nanotracks, such as the racetrack memory geometry. On the other hand, Woo et al. [20] also found that the skyrmion Hall angle of ferrimagnetic skyrmions increases monotonically at low current densities and saturates at high current densities, as shown in Fig. 10.4C. Such a phenomenon has also been observed for the skyrmion Hall effect in ferromagnets [34]. The reason may be that the skyrmion dynamics is dominated by pinning effects at low driving forces, which is

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similar to the creep motion of ferromagnetic skyrmions in low-current-density regime caused by the pinning potential [34, 70]. As discussed in Section 10.1, the skyrmion Hall effect is a detrimental effect for high-speed on-demand skyrmion motion in narrow nanotracks, which may result in the destruction of skyrmions at sample edges. Although the ferrimagnetic skyrmions have reduced skyrmion Hall effect and thus have better mobility and reliability, their nonvanishing angles could still result in similar issues such as skyrmion annihilation in a longer distance scale. Unlikely, the skyrmion Hall effect can be totally suppressed in antiferromagnets, and ferrimagnets can also become antiferromagnets by compensating two opposite magnetic sublattices, which can be achieved by either temperature or relative composition variations. Indeed, in 2019, Hirata et al. [44] experimentally studied the temperature-dependent skyrmion Hall angle in ferrimagnetic GdFeCo films. As shown in Fig. 10.5, Hirata et al. estimated the skyrmion Hall effect from

Fig. 10.5 Current-driven elongation of magnetic bubble as a function of temperature in ferrimagnetic GdFeCo/Pt films. MOKE images at T ¼ 343 (A), T ¼ 283 K (B), and T ¼ 253 K (C). (D) Elongation angle θ as a function of T for each magnetization state. (A–C) From Y. Hirata, D.-H. Kim, S.K. Kim, D.-K. Lee, S.-H. Oh, D.-Y. Kim, T. Nishimura, T. Okuno, Y. Futakawa, H. Yoshikawa, A. Tsukamoto, Y. Tserkovnyak, Y. Shiota, T. Moriyama, S.-B. Choe, K.-J. Lee, T. Ono, Vanishing skyrmion hall effect at the angular momentum compensation temperature of a ferrimagnet, Nat. Nanotechnol. 14 (2019) 232. © 2019, Springer Nature.

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the angle between the electric current direction and the bubble elongation direction. It is found that the skyrmion Hall effect vanishes at the angular momentum compensation temperature where the net spin density vanishes, which is in line with previous theoretical prediction [43]. The fact that the skyrmion Hall angle can be adjusted by temperature and relative compositions in ferrimagnets provides a potential route to build future spintronic logic devices in which the dynamics of ferrimagnetic skyrmions are controlled by both material engineering and thermal gradient.

10.4

Potential applications of ferrimagnetic skyrmions

As discussed in Section 10.2, ferrimagnetic skyrmions can be created, deleted, and detected by electric means, which serves as the foundation for spintronic applications based on ferrimagnetic skyrmions. As the ferrimagnetic skyrmions can be driven into high-speed motion with significantly reduced skyrmion Hall effect as well as sizes appropriate for modern electronics [20, 21, 44], it is desired to use them as a type of nonvolatile information carriers in information storage and processing devices, e.g., the racetrack-type memory, where the information write-in and read-out operations are based on the current-induced motion of skyrmions in nanotracks, as shown in Fig. 10.6A. As the skyrmion Hall effect of skyrmions is reduced and can become zero in ferrimagnets, the current-induced motion of ferrimagnetic skyrmions can reach a higher speed without being destroyed at the edge of devices. Therefore, the racetrack memory devices based on ferrimagnetic skyrmions may have a higher information processing speed. In addition, the packing density of nanotrack components in a racetrack memory with multiple ferromagnetic nanotracks is usually limited by the stray fields. Namely, the magnetization dynamics in a nanotrack can be affected by the stray fields from adjacent nanotracks, which may lead to undesired motion of domain walls [71] or skyrmions [72]. Hence, it is necessary to maintain certain spacing between adjacent ferromagnetic nanotracks in a racetrack memory device to avoid significant influence of the stray fields. For this reason, as the partially compensated magnetization in ferrimagnets could lead to smaller stray fields compared with ferromagnets, it is envisioned that the racetrack memory devices based on ferrimagnetic materials could have a smaller nanotrack spacing and thus a higher packing density than conventional ferromagnetic racetrack memories. In addition to information storage devices, ferrimagnetic skyrmions can also be used to build novel bio-inspired spintronic devices. For example, in 2020, Song et al. [67] experimentally demonstrated the electrical creation, motion, detection, and deletion of magnetic skyrmions in a single device made of ferrimagnetic GdFeCo multilayers, which can emulate the functions of an artificial synapse for neuromorphic computing. Fig. 10.6B shows the ferrimagnetic skyrmion-based artificial synapse, where the potentiation and depression are successfully realized. The potentiation and depression are essential behaviors of a biological synapse determining the synaptic weight and are vital for the learning function in neuromorphic applications [73, 74]. In the ferrimagnetic skyrmion-based artificial synapse studied by Song et al. [57], the

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Fig. 10.6 Potential applications based on skyrmions in ferrimagnets. (A) Concept of a racetrack memory based on ferrimagnetic skyrmions, where the binary information is encoded by the absence and presence of ferrimagnetic skyrmions. (B) Concept of an artificial synapse based on ferrimagnetic skyrmions. In the left schematic, the two-dimensional N
eel-type skyrmions in thin films in the experiments are mapped onto spheres and are shown in three-dimensional space. The red and blue arrows represent magnetic moments pointing in the + z and z directions within skyrmions, respectively. The right panel shows the measured Hall resistivity change and calculated skyrmion number as a function of injected pulse number. Note that the red and blue symbols and colored areas correspond to resistivity changes (left axis) during potentiation and depression, respectively. Green symbols are used to indicate the number of skyrmions (corresponding to the right axis). Insets: electrical pulses indicating the direction of the charge current pulse, opposite to the direction of electron flow. Error bars represent the standard deviation of the resistivity measurements at each state. From K.M. Song, J.-S. Jeong, B. Pan, X. Zhang, J. Xia, S. Cha, T.-E. Park, K. Kim, S. Finizio, J. Raabe, J. Chang, Y. Zhou, W. Zhao, W. Kang, H. Ju, S. Woo, Skyrmion-based artificial synapses for neuromorphic computing, Nat. Electron. 3 (2020) 148. © 2020, Springer Nature.

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linear variations of synaptic weights during potentiation and depression are realized through the electric current-controlled creation and annihilation of ferrimagnetic skyrmions as the synaptic weights (i.e., the measured Hall resistivity) are proportional to the number of skyrmions. Namely, the accumulation and dissipation of ferrimagnetic skyrmions in the device lead to decrease and increase of resistance, respectively, and therefore result in multiple distinguishable resistivity states in analog fashion. Song et al. [67] also simulated the response of an artificial neural network based on ferrimagnetic skyrmion synapses to implement a supervised pattern recognition task, where a high learning accuracy of 89% was reached, comparable to about 93% of ideal cases. Because the dynamics of ferrimagnetic skyrmions depend on not only the skyrmion topology and driving current but also the temperature, ferrimagnetic skyrmions have the potential to perform more complex functions compared with conventional ferromagnetic skyrmions. For example, as discussed several times above, the skyrmion Hall angle can be adjusted by varying temperature and can even be eliminated at the angular momentum compensation temperature [44]. Therefore, it is envisioned that ferrimagnetic skyrmions can be used to build future advanced information computing devices, where the dynamics of ferrimagnetic skyrmions are controlled by both electric current and applied thermal gradient.

10.5

Summary

In this chapter, we have introduced the concept of ferrimagnetic skyrmions and reviewed recent theoretical and experimental works about the dynamics of skyrmions in ferrimagnets. In particular, we have focused on recent progress on the imaging, writing, deleting, and detection of ferrimagnetic skyrmions. We have also shown that ferrimagnetic skyrmions can be used as information carries in racetrack-type memories and can also be utilized for unconventional devices, e.g., to mimic the functions of an artificial synapse for neuromorphic computing. The ferrimagnetic skyrmions have the intermediate properties between ferromagnetic and antiferromagnetic skyrmions. The current-induced motion of ferrimagnetic skyrmions shows certain but much reduced skyrmion Hall effect compared with that of ferromagnetic skyrmions, which means ferrimagnetic skyrmions can be driven into high-speed motion with much smaller transverse shift, guaranteeing a better mobility and reliability of skyrmions in narrow channels. The ferrimagnetic skyrmions also have small nonzero net magnetization due to uncompensated sublattices, which makes all detection methods used for ferromagnetic skyrmions also available for detecting ferrimagnetic skyrmions. Most importantly, the nature of ferrimagnetic materials (thus ferrimagnetic skyrmions) can be engineered between ferromagnet-like and antiferromagnet-like, by controlling materials and their environment. Therefore, ferrimagnetic skyrmions could serve as promising nonvolatile information carriers in future memory and logic computing devices, which combine the essential merits of both ferromagnetic and antiferromagnetic ones.

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Acknowledgments S.W. acknowledges the support from IBM Research and management support from G. Hu and D. Worledge. S.W. also acknowledges the support from his previous institution, Korea Institute of Science and Technology (KIST), where he conducted most of his original researches on ferrimagnetic skyrmions. X.Z. acknowledges the support by the National Natural Science Foundation of China (Grant No. 12004320), and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110713). X.Z. also acknowledges the fruitful discussions with X. Liu and M. Ezawa. Y.Z. acknowledges the support by Guangdong Special Support Project (Grant No. 2019BT02X030), Shenzhen Peacock Group Plan (Grant No. KQTD20180413181702403), Pearl River Recruitment Program of Talents (Grant No. 2017GC010293), and National Natural Science Foundation of China (Grant Nos. 11974298 and 61961136006).

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Skyrmions in antiferromagnets

11

Oleg A. Tretiakov School of Physics, The University of New South Wales, Sydney, NSW, Australia

11.1

Introduction

Skyrmions are topological textures, which can be formed in different magnetic materials [1]. Recently, they became a very active area of research in spintronics due to their potential in information encoding, transfer [2, 3] and computing [4]. Feldtkeller [5] and Thiele [6] discussed similar topological defects earlier in magnetism, while Belavin and Polyakov [7] were the ones who introduced skyrmions [8, 9] in the context of the two-dimensional (2D) Heisenberg model. Skyrmions have some advantages over other proposed storage technologies, such as domain wall registers, because of the low currents required to move them due to the ability to move past pinning sites [10]. Implementing skyrmion devices in ferromagnetic materials involves issues in common with other spintronic concepts, such as the sensitivity to stray fields. Skyrmions in ferromagnets also possess a further complication in that they experience a skyrmion Hall effect, that is, a Magnus force perpendicular to the applied current, thus making it difficult to move them strictly along the current [11, 12]. By contrast, antiferromagnets are not sensitive to stray fields. Also, it is found that skyrmions in antiferromagnets move always in a straight line along the current, which is very distinct from skyrmion motion in ferromagnets. In a two-sublattice antiferromagnet, the skyrmion forms as a pair of strongly coupled topological objects, one belonging to each sublattice. As it will be shown in more detail below, the opposing topological charge for the magnetization of each sublattice causes an exact cancelation of the Magnus force, hence there is no skyrmion Hall effect [13, 14]. The longitudinal velocity is also found to strongly depend on the material parameters (α, β) and as a result the antiferromagnetic (AFM) skyrmion can reach high velocities of the order of kilometer per second [13–15]. Moreover, the thermal properties of AFM skyrmions are found to be rather different from their ferromagnetic counterparts [13].

11.2

Antiferromagnetic materials for skyrmions

In many modern material systems, such as in Pt/CoFeB/MgO, Ir/Co/Pt, Co/Pd, and Pt/ Co/MgO, ferromagnetic skyrmions are stabilized by the Dzyaloshinskii-Moriya interaction (DMI) [16, 17], which is induced by the breaking of inversion symmetry in magnetic materials. Moreover, it has been shown that in ferromagnets the skyrmions can be stabilized also by four-spin interactions, frustrated exchange interactions [18, 19], dipolar interactions, etc. [1]. However, in the following, we will focus on the DMI, since it is the most relevant stabilization mechanism in realistic AFMs. Magnetic Skyrmions and Their Applications. https://doi.org/10.1016/B978-0-12-820815-1.00009-2 Copyright © 2021 Elsevier Ltd. All rights reserved.

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A compelling reason to study skyrmions in antiferromagnets apart from the insensitivity to stray fields, is that the Dzyaloshinskii-Moriya interaction (DMI), which is essential for the formation of individual skyrmions, is more commonly found in antiferromagnetic (AFM) materials than ferromagnetic (FM) materials. Most recent experimental results on FM skyrmions rely in the presence of an interfacial DMI to stabilize skyrmions; however, bulk DMI is more prevalent in AFMs [16, 17]. AFMs are also considerably more abundant in nature than ferromagnets, although metallic AFMs are not so common but examples include FePt3 and Mn2Au. Skyrmions can form in different systems where there is a competition between the DMI and another energy contribution, for example, the Zeeman energy from an applied field or a uniaxial anisotropy [20]. Here, the last option is studied by necessity because of the antiferromagnets insensitivity to applied fields and the lack of a significant demagnetizing field precludes these mechanisms from forming a skyrmion [21]. We also focus on individual skyrmions, rather than a skyrmion lattice, as the ability to move and manipulate individual bits of information is more relevant to the suggested technological applications [2]. In terms of the required DMI, the stabilization of antiferromagnetic skyrmions is not problematic [13, 22]. Two skyrmions with mutually reversed spins (i.e., two skyrmions with opposite polarity and a helicity difference of π) are energetically favored by the same type of DMI. Furthermore, both subskyrmions need to be coupled antiferromagnetically quite strongly, to accomplish the antiparallel alignment of the corresponding magnetic moments. And indeed, very recently, the bilayer-type antiferromagnetic skyrmions have been observed in synthetic antiferromagnets [23, 24] at room temperature. In Ref. [23], the small stray fields resulting from the bilayer setup have been detected using magnetic force microscopy (MFM). In Ref. [24], the authors explain a method to prepare synthetic antiferromagnets with a tunable net moment. While they can achieve a completely compensated system, they deliberately prepared also a system with a small net moment to be able to perform magneto-optical Kerr effect (MOKE) measurements.

11.3

Statics of antiferromagnetic skyrmion

In the continuous limit, a single magnetic skyrmion would be stable due to the topological protection, that is, it cannot be transformed into a uniform magnetic state continuously, even though this state is the ground state of the system and, therefore, has the lower energy. However, since all real materials are not continuous but consist of lattice of magnetic moments, this topological protection of single skyrmions is not strict [25] and they have finite lifetimes in real antiferromagnets [22]. Considering the G-type antiferromagnet, formed by a three-dimensional (3D) chessboard-like pattern, the AFM skyrmion forms in much the same way as an FM skyrmion, by introducing a topological defect, reversing the A and B sublattices within a small area and allowing the system to relax. The DMI prevents the metastable domain from reversing. The 3D spin structure of the AFM skyrmion is shown in Fig. 11.1. It is analogous to the hedgehog skyrmion state of an FM but with one of

Skyrmions in antiferromagnets

335

Fig. 11.1 The 3D spin texture of the G-type AFM skyrmion. The core is represented not by a single spin but rather by a compensated structure combining the two sublattices of the AFM. Reproduced with permission from J. Barker, O.A. Tretiakov, Static and dynamical properties of antiferromagnetic skyrmions in the presence of applied current and temperature, Phys. Rev. Lett. 116 (2016) 147203.

the sublattices inverted. Hence, the topological defect exists in the N
eel field and the magnetization is nearly zero everywhere. At the center of the AFM skyrmion, neither sublattice dominates, but instead there is a compensation of opposite spins around the true center of the skyrmion. At zero temperature, the skyrmion radius and radial profile for a given DMI are the same in both FM and AFM skyrmions for the magnetization and N
eel parameters, respectively. AFM skyrmions can be understood as the combination of two skyrmions with mutually reversed spins. Therefore, they are characterized by a vanishing topological charge for the magnetization. However, in the present case, the subskyrmions are not spatially separated but intertwined. For this reason, the magnetization density vanishes locally and the N
eel order parameter, the main order parameter for antiferromagnets, can be considered instead of the magnetization. Calculating the topological charge with this parameter gives
1. Thus, the antiferromagnetic skyrmions are still skyrmions from the viewpoint of topology, but exhibit different dynamics compared to ferromagnetic skyrmions. These antiferromagnetic dynamics can also be described by the Thiele equation but by using the N
eel order parameter [26]. For antiferromagnetic skyrmions in a single layer (not the case of a synthetic antiferromagnet bilayer), the detection is another problem. Both, the magnetization and topological charge density of the magnetization, are compensated globally and locally. These antiferromagnetic skyrmions would, therefore, appear invisible for the real-space techniques, such as MFM or LTEM. Furthermore, anomalous and topological Hall signatures do not exist. Luckily, a different hallmark has been predicted: the topological spin Hall effect [27–29]. The resulting signal is the analog of the conventional spin Hall effect, but originates in the noncollinearity of the spin texture. The topological spin Hall effect can most easily be comprehended if one assumes two electronically uncoupled subskyrmions (the results hold also for the coupled case): due to the opposite spin alignment, the emergent fields of the two subskyrmions are oriented oppositely. This leads to a transverse deflection of the electrons in the opposite

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directions. The spins of the two species of electrons align with their respective texture and can, therefore, be considered as “spin-up” and “spin-down” states, again due to the opposite spin alignment.

11.4

Dynamics of antiferromagnetic skyrmion

The dynamic properties of single antiferromagnetic skyrmions have first been predicted in antiferromagnets [13, 30] and synthetic antiferromagnetic bilayers [14], and shortly after, they have been extended to two-sublattice antiferromagnetic skyrmion crystals [28]. Let us first review the AFM skyrmion dynamics due to an applied current at zero temperature. It is convenient to compare the AFM skyrmion dynamics using Thiele’s equation [6, 10, 13, 31–33] with the one for the FM skyrmions. Then next we will consider more complicated thermal effects. Coupling to the current assumes that the electrons of up and down spin are transported predominantly through their corresponding magnetization sublattice [34]. For the G-type AFM, this is reasonable, but for other AFMs the transport of the electronic current through the AFM may be different. From this assumption, the spatial derivative !!

r M is calculated for the magnetization of each sublattice, rather than the net local magnetization which is almost zero. Comparing the AFM dynamics with those of an FM skyrmion (where the only change in material parameters is in the sign of the exchange interaction), highlights two main intrinsic differences in the dynamics resulting from the AFM characteristics. First, the AFM skyrmion always has zero transverse velocity v?, relative to the current. In the FM, this is only true for the highly symmetric case of α ¼ β, where α is the Gilbert damping constant and β is the nonadiabatic spin-transfer torque parameter. In an FM, the transverse velocity is due to the Magnus force acting on the skyrmion and the direction (
^ y) is determined by the winding number of the skyrmion: Q

ðkÞ

!

Z ¼ !

d2 r ðkÞ Eij Eαβγ mαðkÞ ∂i mβ ∂j mγðkÞ , 8π

(11.1) !

!

where m ðkÞ ð r Þ is the unit vector parallel to the local magnetization M ðkÞ ð r Þ and k ¼ 1, 2 label the sublattices in the AFM case. The AFM skyrmion is essentially composed of two topological objects with opposite winding numbers (Q(k) ¼
1), which are strongly coupled through the AFM exchange interaction. Both sublattices generate a Magnus force, but there is a perfect cancelation (Fig. 11.2C) thus resulting in no v?. As a result, the AFM skyrmions move strictly along the current. One can also directly define the winding number for the AFM order parameter (N
eel field) ! ! ! ! ! n ð r , tÞ ¼ mð1Þ ð r , tÞ  m ð2Þ ð r ,tÞ and thus show that AFM skyrmions are topologically nontrivial textures with the AFM topological charge
1. The second notable difference between AFM and FM skyrmions is that the longitudinal velocity in the AFM can greatly exceed the FM skyrmion drift velocity, which is always close to the electron drift velocity (v  200 m s1 for current j ¼ 200 m s1 and α, β ≪ 1). For low α or high β, AFM skyrmions can move at kilometer per second

Skyrmions in antiferromagnets

337

(A)

–

(B) Current

Drag

(C) Fig. 11.2 Current-induced dynamics of the AFM skyrmion. (A) Along the current component of the skyrmion velocity is shown for different combinations values of spin-transfer torques α and β. The inset demonstrates that the mass term is small and the skyrmion reaches terminal velocity within 2 ps. (B) The transverse components of the AFM skyrmion velocity from the same simulations demonstrate that there is no transverse motion. (C) The AFM skyrmion is composed of two FM skyrmions (of opposite topological charge) on two sublattices. Thus, the magnus forces act in the opposite directions and there is no overall skyrmion Hall effect for the AFM skyrmion (i.e., no transverse motion). Reproduced with permission from J. Barker, O.A. Tretiakov, Static and dynamical properties of antiferromagnetic skyrmions in the presence of applied current and temperature, Phys. Rev. Lett. 116 (2016) 147203.

while remaining stable (Fig. 11.2A). Recent theoretical studies of AFM dynamics give an insight into this [26, 35]. The dynamics can be studied based on the generalized Thiele’s equations [26], which describe the motion of a spin texture in terms of collective coordinates bj Mij b€j + αΓ ij b_ j ¼ Fi ,

(11.2)

which correspond to the soft modes of the skyrmion. Here, Mij is the mass tensor, αΓ ij characterizes viscous friction and is related to damping in the AFM, and finally Fi is the generalized force due to the current. The mass term contributes to the dynamics only on short time scales (up to 2 ps), as shown in the inset of Fig. 11.2A. The AFM

338

Magnetic Skyrmions and Their Applications

skyrmion reaches its terminal velocity quickly, which is understood more intuitively by switching back to the two-sublattice description and writing the Thiele’s equations for magnetizations m(1, 2)(r, t). It is then clear that the Magnus forces for m(1)(r, t) and m(2)(r, t), G^z  j with the gyrocoupling constant G ¼ 4πQ(k), cancel each other (Fig. 11.2C) and the remaining generalized drag force, F ¼ Γβj, leads to the AFM skyrmion velocity β vk ¼ j, α

(11.3)

which is only along the direction of the current. The velocity is plotted as lines in Fig. 11.2A, showing an excellent agreement with the simulations. Thermal properties. It is also important to understand the thermal properties of AFM skyrmions and compare them to those of FM skyrmions [36]. On a macroscopic level, skyrmions diffuse due to the thermal perturbations of the magnetic moments. We also consider the temperature dependence of the macroscopic material parameters, such as anisotropy and exchange stiffness, and find this can lead to a change in the balance of the competing energy terms in skyrmionic systems. The skyrmion spin texture is also subjected to deformations due to internal dynamics, which are stimulated by thermally induced spin waves. Using Langevin Landau-Lifshitz-Gilbert simulations, we have studied these thermal effects for the AFM skyrmion and compared it with that of the FM skyrmion. Simulating the Brownian motion of a single skyrmion (Fig. 11.3), one finds the AFM skyrmion to be diffusive [13], meaning the mean square displacement Fig. 11.3 (A) Brownian motion trajectories of the AFM Skyrmion for Gilbert damping constants α ¼ 0.1 and 0.01 are depicted at temperature T/Tc ¼ 0.25. (B) The diffusion coefficient of the AFM Skyrmion as a function of Gilbert damping α. The points are evaluated using the mean-squared displacements hr2i shown in the inset. The solid line is D ¼ λkB T=ð2αsσÞ. Reproduced with permission from J. Barker, O.A. Tretiakov, Static and dynamical properties of antiferromagnetic skyrmions in the presence of applied current and temperature, Phys. Rev. Lett. 116 (2016) 147203.

(A)

(B)

Skyrmions in antiferromagnets

339

hr2i∝ t, as was shown for the FM skyrmion [36]. However, the diffusion coefficient of the AFM skyrmion is greater than that of the FM and D ∝ 1/α as a direct consequence of the absence of a net Magnus force, which plays the dominant role in the FM diffusion, where D ∝ α [36]. In Ref. [37], the diffusion coefficient of AFM textures in one-dimensional case was shown to be D ¼ λkB T=ð2αsσÞ, that is, inversely proporpffiffiffiffiffiffiffiffiffi tional to the Gilbert damping α, where s ¼ ħS=a3 , λ ¼ A=K is the exchange length, and σ is the cross-sectional area of the domain wall – in this case the circumferential area of the skyrmion, 2πaRs(T). The absence of the Magnus force allows this expression to be generalized to 2D diffusion, comparing it with the numerical simulations (see solid line in Fig. 11.3) one finds an excellent agreement. The demonstrated increased thermal mobility of the AFM skyrmions may be useful when moving these spin textures with heat gradients. However, it also may pose challenges for more conventional current-driven motion as the thermal perturbations may cause excessive randomness in their motion. At the same time, if one would consider the AFMs of not G-type (different symmetry), it thus may be possible that these AFMs could contain the benefit of high current-driven velocity but with a lower diffusion constant.

11.5

Bimerons in antiferromagnets

Magnetic bimeron is a topologically nontrivial spin texture carrying an integer topological charge, which can be regarded as the counterpart of skyrmion in easy-plane magnets. The controllable creation and manipulation of bimerons are crucial for practical applications based on topological spin textures [38]. In Ref. [39], the dynamics of an antiferromagnetic bimeron driven by a spin current have been investigated. Numerical simulations demonstrated that the spin current can create an isolated bimeron in the AFM thin film via the damping-like spin-torque, see Fig. 11.4. The spin current can also effectively drive the antiferromagnetic bimeron without showing a transverse drift. Furthermore, the steady motion speed of an antiferromagnetic bimeron was analytically derived. It was found that the alternating-current-induced motion of the AFM bimeron can be described by the Duffing equation due to the presence of the nonlinear boundary-induced force. In Ref. [39], the dynamics of an isolated AFM bimeron induced by spin currents have been studied and the results demonstrated the inertial nature of the AFM bimeron dynamics. It was shown that a spin current can create an isolated bimeron in the AFM film, and also drive the AFM bimeron at a speed of a few kilometers per second. Based on the Thiele approach, the steady motion speed was derived, which was shown to be in a good agreement with the simulation results. Furthermore, it was found that the AFM bimerons can be used as the Duffing oscillator and chaos can be achieved in ac current-driven systems. These results may provide useful guidelines for building spintronic devices based on AFM bimerons.

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Magnetic Skyrmions and Their Applications

t = 0.00 ns

t = 0.02 ns

t = 0.01 ns

t = 0.04 ns

–

t = 0.05 ns

(E)

(B)

(F)

t = 0.06 ns

(C)

(D)

t = 0.07 ns

(G)

t = 0.20 ns

(H)

Q

1.0

100

Q j

0.5 0.0 0.00

(I)

50 0

0.05

0.10

0.15

j (MA cm–2)

(A)

0.20

t (ns)

Fig. 11.4 (A–H) The evolution of the N
eel order parameter induced by a spin-polarized current with the polarization vector p ¼ ez, the colors represent the out-of-plane components of the N
eel vector. Only the damping-like spin-orbit torque (SOT) is taken into account. (I) The topological charge Q and injected current density j as a function of time t. In the simulations, the current of j ¼ 100 MA cm2 is injected in the central circular region with a diameter of 30 nm (see green lines in A–D); the other parameters were adopted from Barker and Tretiakov [13]. Reproduced with permission from L. Shen, J. Xia, X. Zhang, M. Ezawa, O.A. Tretiakov, X. Liu, G. Zhao, Y. Zhou, Current-induced dynamics and chaos of antiferromagnetic bimerons, Phys. Rev. Lett. 124 (3) (2020) 037202.

11.6

Potential applications of antiferromagnetic skyrmions

In this section, we will discuss future potential applications of skyrmions and their topological analogues – bimerons – in antiferromagnets. One of these applications is related to nano-oscillators, which can be used, for example, as microwave signal generators, where the signal is generated by the precession of uniform magnetization or gyrotropic motion of a magnetic vortex [40] or skyrmion [41]. Skyrmion-based spin-torque nano-oscillators are potential next-generation microwave signal generators. However, ferromagnetic skyrmion-based spin-torque nanooscillators cannot reach high oscillation frequencies. In Ref. [42], it was proposed to use the circular motion of an antiferromagnetic skyrmion to create the oscillation

Skyrmions in antiferromagnets

341

Fig. 11.5 The motion of FM and AFM skyrmions is depicted in a nanodisk. The current directions in (B) and (D) are opposite to (A) and (C). Reproduced with permission from L. Shen, J. Xia, G. Zhao, X. Zhang, M. Ezawa, O.A. Tretiakov, X. Liu, Y. Zhou, Spin torque nano-oscillators based on antiferromagnetic skyrmions, Appl. Phys. Lett. 114 (4) (2019) 042402.

signal in order to overcome this obstacle. The micromagnetic simulations have demonstrated that the antiferromagnetic skyrmion-based spin-torque nano-oscillators can produce high frequencies (tens of gigahertz). In Fig. 11.5, the comparison of FM and AFM skyrmions motion in a nanodisk is presented. In Fig. 11.5A and B, the trajectories of the AFM skyrmion are shown driven by positive and negative currents, respectively, while in Fig. 11.5C and D, the same trajectories are shown for the FM skyrmion. Furthermore, the speed of the circular motion for an antiferromagnetic skyrmion in a nanodisk was analytically derived based on the Thiele’s equation approach [6, 10, 13, 31, 33], which agrees well with the results of numerical simulations. These results are useful for the understanding of the inertial dynamics of an antiferromagnetic skyrmion in a nanodisk and the development of future skyrmion-based spin-torque nano-oscillators. It was proposed to build a novel spin-torque nano-oscillator based on the AFM skyrmion, where the boundary-induced force acts as the centripetal force to maintain the AFM skyrmion in a circular motion. The numerical simulations show that the AFM skyrmion can move at speeds of a few kilometers per second and such a nano-oscillator gives relatively high frequencies (tens of gigahertz). Furthermore, the analytically derived velocity becomes larger with the increase of the applied current and skyrmion size or reduction of the damping that is in good agreement with the numerical simulations. These results provide a promising route to modulate the frequency in a wide range for future skyrmion-based nano-oscillators. For antiferromagnetic skyrmion-based logic [43] or racetrack [13] applications, a controlled generation process is needed. Conventional skyrmions, for example, have

342

Magnetic Skyrmions and Their Applications

been generated by directed, deterministic approaches such as spin-torques or magnetic fields (see reviews [1, 44–50]). These approaches are, however, difficult to utilize for antiferromagnetic skyrmions, since all vectorial quantities would have to act either only on one of the two subskyrmions, so that the other generates automatically due to the strong antiferromagnetic coupling, or they would have to act on both subskyrmions with an opposite sign. Both is hardly feasible, since magnetic fields for example cannot change their sign on the length scale of the lattice constant. Stochastic processes, like the generation of nano-objects at defects or from the confinement (as shown for conventional skyrmions [51, 52]), appear to be more advantageous in this regard. The stabilization of antiferromagnetic skyrmions becomes even more challenging when an antiferromagnetic skyrmion crystal shall be stabilized. By analogy with skyrmion crystals, a stabilizing magnetic field is inevitable. It has to be oriented along
z for the two subsystems, respectively. This problem may be circumvented by growing a potential antiferromagnetic skyrmion host on top of a collinear antiferromagnet with the same crystal structure at the interface. By doing so, a staggered magnetic field is mimicked by the exchange interaction at the interface [28].

11.7

Conclusions and outlook

Ultrasmall topological spin textures can serve as bits of information and manipulating them by electric or thermal currents is one of the main challenges in the field of spintronics. Ferromagnetic skyrmions in the recent years attracted enormous interest due to their small sizes and ability to avoid pinning sites while moved by electric currents. Nevertheless, ferromagnetic skyrmions still affected by stray fields and the transverse to the current dynamics are causing difficulties in employing them in spintronic applications. The AFM skyrmions and bimerons overcome these disadvantages, being insensitive to stray fields and having the dynamics to be strictly along the current (no Hall effect) while also being faster compared to their ferromagnetic analogues. This makes AFM skyrmions and bimerons to be ideal candidates for future information carriers. Moreover, synthetic antiferromagnetic skyrmions have been recently observed and their current-driven motion has been realized [24]. Furthermore, the topological spin Hall effect may play a crucial role in observing these AFM topological spin textures despite their completely compensated magnetization. Even in antiferromagnetic insulators, the skyrmions were predicted and can potentially be moved by an electrically created anisotropy gradient [53] or thermal gradients. In addition, some thermal properties, such as the AFM skyrmion diffusion constant, may serve as an advantage over ferromagnetic skyrmions when AFM skyrmions are driven by thermal currents. As a final remark, the considerations of skyrmions in antiferromagnets with more than two sublattices have been generalized recently, see for example, Monte Carlo simulations of three-sublattice skyrmion crystals in Refs. [54, 55]. However, these topological textures do not possess the advantages of two-sublattice antiferromagnetic skyrmions because their topological charge for the magnetization is nonzero.

Skyrmions in antiferromagnets

343

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Multiple skyrmionic states and oblique spirals in bulk cubic helimagnets

12

Andrey O. Leonova,b,c and Catherine Pappasd a Chirality Research Center, Hiroshima University, Higashi-Hiroshima, Hiroshima, Japan, b Department of Chemistry, Faculty of Science, Hiroshima University Kagamiyama, HigashiHiroshima, Hiroshima, Japan, cIFW Dresden, Dresden, Germany, dFaculty of Applied Sciences, Delft University of Technology, Delft, The Netherlands

12.1

Introduction

Chiral magnetic skyrmions – topological defects with a complex noncoplanar magnetic structure [1, 2] – were first spotted in bulk cubic helimagnets such as MnSi [3] and Cu2OSeO3 [4]. In these systems, skyrmions form hexagonal lattices in a small pocket of the temperature-magnetic field-phase diagram just below the transition temperature TC, the so-called A-phase. The quest for stabilization mechanisms shows that, within an isotropic phenomenological model, the skyrmion lattice (SkL) is always a metastable solution. However, the energetic difference to its main competitor, the conical phase, is weak and reduces to a minimum for those magnetic fields that stabilize the A-phase [5, 6]. As a consequence, weak interactions such as the softening of the magnetization modulus [7, 8], dipolar interactions, fluctuations [3, 9], etc. may modify the energetic landscape and eventually stabilize the SkL in the A-phase pocket. Due to this subtle energetic balance, the boundaries of the A-phase can be changed substantially by relatively small external stimuli, such as pressure [10] or electric fields [11–14]. Theoretical models that are based on the Dzyaloshinskii theory [15] and include anisotropy (Section 12.2), however, show that SkLs should be stabilized over a wide range of magnetic fields and temperatures well beyond the A-phase also in bulk cubic helimagnets [1, 2, 5, 6, 16, 17]. In particular, cubic anisotropy (CA) may stabilize skyrmions even for low temperatures (LTs) for specific directions of the applied magnetic field [5, 18]. Recently a new impetus to these theoretical considerations was provided by the reported LT-SkL in the bulk insulating cubic helimagnet Cu2OSeO3 [19–21]. In this chapter, we show that, besides the stabilization of skyrmions, which is addressed in Section 12.4, CA also leads to many other remarkable phenomena. In particular, it accounts for the reorientation processes of oblique spiral states that occur not only at the low critical field Hc1, discussed in Section 12.3.1, but also near the

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Magnetic Skyrmions and Their Applications

transition into the saturated state Hc2, addressed in Section 12.3.2. Furthermore, as discussed in Section 12.3.3, CA may stabilize various elliptically distorted spiral states and induce the transitions between them. Although these transitions occur without any reorientation of wave vectors and with almost negligible period change, they have clear marks in the magnetization and susceptibility curves, waiting to be identified experimentally. In these sections, we tacitly imply that the theoretical results for easy h100i, or h111i, axes of CA may explain the behavior of the bulk helimagnets Cu2OSeO3, or MnSi, respectively.

12.2

Phenomenological theory of modulated states in chiral helimagnets

12.2.1 The general micromagnetic energy functional Within the phenomenological theory introduced by Dzyaloshinskii [15], the magnetic energy density of a noncentrosymmetric ferromagnet with spatially dependent magnetization M can be written as W ðMÞ ¼ A

X ∂mj 2 i, j

∂xi

+ D wD ðMÞ  M  H + Wa ðmÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(12.1)

W0 ðMÞ

where A and D are coefficients of exchange and Dzyaloshinskii-Moriya (DM) interactions; H is an applied magnetic field; and xi is the Cartesian components of the spatial variable. The function wD is composed of Lifshitz invariants ðkÞ Li, j ¼ Mi ∂Mj =∂xk  Mj ∂Mi =∂xk – energy terms involving first derivatives of the magnetization with respect to the spatial coordinates. In this chapter, all calculations have been done for cubic helimagnets with wD ¼ M  rotM although the results may be applied for magnets with other symmetry classes [1] including different combinations of Lifshitz invariants. Dzyaloshinskii’s phenomenological approach is a main theoretical tool to analyze and interpret experimental results on chiral magnets. During the last three decades of intensive investigations of chiral modulations in different classes of noncentrosymmetric magnetic systems, a huge empirical material has been organized and systematized within the framework of this theory (see, e.g., the review of Ref. [22] and the bibliography in Ref. [16]). The Dzyaloshinskii interaction functional, Eq. (12.1), plays in chiral magnetism a similar role as the Frank functional in liquid crystals [23] or the Ginzburg-Landau functionals in superconductivity [24, 25]. The functional W0(M) includes only the basic interactions essential to stabilize skyrmion and helical states and specifies their most general features attributed to all chiral ferromagnets.

Multiple skyrmionic states and oblique spirals in bulk cubic helimagnets

349

12.2.2 One-dimensional spiral modulations within the isotropic theory The DM interactions wD play a crucial role in destabilizing the homogeneous ferromagnetic arrangement and twisting it into a helix. At zero magnetic field such helices are single-harmonic modes forming the global minimum of the functional W0(M) [15], M ¼ Ms ½n1 cos ðq  rÞ + n2 sin ðq  rÞ, where n1, n2 are the unit vectors in the plane of the magnetization rotation orthogonal to the wave vector q ¼ n3/(2LD) (n1?n2; n1?q; n2?q). LD is proportional to the ratio of the counter-acting exchange and Dzyaloshinskii constants and introduces a fundamental length characterizing a magnitude of chiral modulations in noncentrosymmetric magnets, LD ¼ A/D. The helical modulations have a fixed rotation sense determined by the sign of DM constant D and are continuously degenerate with respect to any specific propagation directions in space. An applied magnetic field lifts the degeneracy of the helices and stabilizes two types of one-dimensional modulations: helicoids (with propagation vector perpendicular to an applied magnetic field) and cones (with propagation direction along the magnetic field).

12.2.2.1 Helicoids For helicoids, analytical solutions for the polar angle θ(x) of the magnetization written in spherical coordinates, M ¼ Ms ð sin θðxÞ cos ψ, sin θðxÞ sinψ, cosθðxÞÞ, are derived by solving the pendulum equation Ad 2 θ=dx2  H cos θ ¼ 0. Such solutions are expressed as a set of elliptical functions [15] and describe a gradual expansion of the helicoid period with increased magnetic field (see, e.g., the set of angular profiles θ(x) in Fig. 4.1. of Ref. [5]). At sufficiently high magnetic fields, above the critical field HH, the helicoid infinitely expands and transforms into a system of isolated noninteracting 2π-domain walls (solitons) separating domains with the magnetization along the applied field [2, 15]. Recently, such an expansion of the chiral magnetic soliton lattice has been observed by Lorenz microscopy and small-angle electron diffraction in Cr1/3NbS2 [26]. The nondimensional value of this critical field is hH ¼ HH/HD ¼ π 2/8 ¼ 0.30843, with HD ¼ D2/AM. The azimuthal angle ψ, on the contrary, is fixed by the different forms of the Lifshitz invariants. In particular, ψ ¼ π/2 for Bloch helicoids in cubic helimagnets and ψ ¼ 0 for N
eel helicoids or cycloids in lacunar spinels with Cnv symmetry [27].

12.2.2.2 Cones The conical state combines properties of the homogeneous state and the flat spiral as a compromise between the Zeeman and DM energies. Thus, a conical spiral retains its single-harmonic character with ψ ¼ z/2LD and cos θ ¼ 2jHj=HD . The magnetization component along the applied field has a fixed value M? ¼ M cos θ ¼ 2MH=HD and the magnetization vector M rotates within a cone surface. The critical value hd ¼ 0.5 marks the saturation field of the conical phase. For the functional W0(M), the conical phase is the global minimum in the whole range of the applied fields where modulated states are stabilized (0 < h < hd). In this approach, helicoids and SkLs may occur only as metastable states.

350

Magnetic Skyrmions and Their Applications

12.3

Rich behavior of spiral states in helimagnets with cubic anisotropy

The functional Wa(m) of Eq. (12.1) includes short-range anisotropic terms: Wa ðmÞ ¼ Kc ðm2x m2y + m2x m2z + m2y m2z Þ  Ku ðm  aÞ2

(12.2)

where Kc and Ku are coefficients of cubic and uniaxial magnetic anisotropies, correspondingly; a is the unit vector along the easy axis. In the following, we will operate with the vector field of the magnetization m(r) ¼ M(r)/M under the constraint jmj ¼ 1. The effects imposed by the coupling between angular (twisting) and longitudinal modulations arising near the ordering temperature of noncentrosymmetric ferromagnets are described in Refs. [5, 8]. The energy density is measured in nondimensional units, w(m) ¼ W(M)/HDM. Generically, there are only small energy differences between the various modulated states. On the other hand, weaker energy contributions to Wa(m) impose distortions on the solutions of the model of Eq. (12.1), which reflect crystallographic symmetry and the specific values of magnetic interactions in particular chiral magnets. It is essential to acknowledge that these weaker interactions play a crucial role in determining the stability limits of the different modulated states. Historically, the effect of uniaxial anisotropy (UA) was addressed first [17] and was shown to be an essential ingredient for understanding the skyrmion stability in thin-film magnets with induced UA [28] and with stress-dependent limits of skyrmion stability in bulk helimagnets [29, 30]. UA along the propagation direction of the cones suppresses the conical phase for values of the anisotropy coefficient Ku, which are much smaller than for the skyrmion and helical phases, an effect that opens the door to the stabilization of skyrmions. On the other hand, UA does not alter the ideal singleharmonic type of the magnetization rotation in the conical spiral, but just leads to the gradual closing of the cone. CA, on the contrary, can either increase the energy of the conical phase or decrease it depending on the mutual arrangement of easy anisotropy axes and propagation direction of the cone. As a matter of fact, one should first implement a simple analysis of the homogeneous states in a system with CA in an applied magnetic field as described, for example, in Ref. [31] and depicted in Fig. 12.1. The manifold of energy extrema provides a hint on the distortions of the conical state as the rotation of the magnetization occurs within a complex energetic landscape generated by CA. The direction of the wave vector q itself is also specified by competing Zeeman and CA energies. These considerations explain the main features of the phase diagrams of bulk cubic helimagnets such as MnSi and Cu2OSeO3, at the critical field Hc1, where oblique spirals align along the field, and at Hc2 where the conical phase undergoes the phase transition to the homogeneous state.

q

z

3

q

z

h || [110]

3

z

h || [110]

q h || [001] 3

z q 3 h || [111]

q

2

H || [001]

q

q

1

h

h y q

h

x

y 2

x

q

w –0.250

0.13 0.14 q

1

0.15

–0.24 q

w

q

4

q

q

1

(D)

3

4

kc 0

x

q

(B)

1

(A)

h

q 2 y

q

(F)

(G)

q

(H)

1

h || [110] –0.4

0

a

q

2

0.4 0.8

(I)

Fig. 12.1 Reorientation of the spirals into the conical state at the critical field hc1, for kc > 0. When hjj[110] (A), (B), (E), (F), the spiral states with q1jj[100] and q2jj[010] cant toward the field, as illustrated by the black line in (H), with a subsequent FOPT into q4jj[110] (B). The spiral q3jj[001] is metastable with respect to q1,2 but may persist up to higher magnetic fields, due to the large energy barrier, as shown in (F). Both transitions are reversible with decreasing magnetic field. For hjj[001] (C), (G), the global energy minimum is immediately reached at q3, which is along the field. The other spiral states are metastable solutions and they are not recovered, thus they do not exhibit hc1, with decreasing magnetic field. For hjj[111] (D), domains of the three spiral states with qijjh001i coexist. The angle α with respect to the field slightly varies (red line in (H)) till the jump into the conical state occurs. In (E–G), the energy density is plotted versus the angle α, in radians, for qi varying in the corresponding planes. (I) Reorientation of the spirals into the conical state for kc < 0 and different directions of the field. q-vectors of spirals are shown by blue arrows and point along h111i crystallographic axes (see text for details). Modified from F. Qian, L.J. Bannenberg, H. Wilhelm, G. Chaboussant, L.M. DeBeer-Schmitt, M.P. Schmidt, A. Aqeel, T.T.M. Palstra, E.H. Bruck, A. J.E. Lefering, C. Pappas, M. Mostovoy, A.O. Leonov, New magnetic phase of the chiral skyrmion material Cu2OSeO3, Sci. Adv. 4 (2018) eaat7323.

352

Magnetic Skyrmions and Their Applications

12.3.1 The helical-to-conical-phase transition at the critical field Hc1 In this section, we present results on spiral reorientation processes, which we obtained by exact numerical minimization of the energy functional, given by Eq. (12.1), including the CA Kc in Eq. (12.2). For this purpose, we consider that the dimensionless anisotropy constant, kc ¼ KcA/D2, is fixed to the value 0.1, and that the magnetic field H is applied along the high-symmetry directions [110], [111], and [001]. Since CA with kc > 0 favors easy axes of h001i type, in zero field one has domains of spiral states with the corresponding directions of the q-vectors. Fig. 12.1A–D shows a schematic layout of the q-vectors in stable and metastable spiral states with respect to the field and the crystallographic directions, whereas Fig. 12.1E–G depicts the energy densities of skewed spiral states depending on the q-vector direction.

12.3.1.1 Hk[110] Fig. 12.1A shows the three spiral domains with the wave vectors q1, q2, and q3 at low magnetic fields. The zero-field degeneracy of the spiral states is lifted by the magnetic field, which favors the wave vectors q1 and q2 in the (001)-plane. Fig. 12.1E shows the energy density in the field range h ¼ 0.130.16 as a function of α, the angle between q1 and the field direction. The first-order phase transition (FOPT) between the spiral states with the wave vectors q1 and q2 and the conical state with the wave vector q4kH (Fig. 12.1B) occurs at h  0.15. The spiral flip is preceded by a gradual rotation of q1 and q2 toward the field as shown by the black line in Fig. 12.1H. On the other hand, the wave vector of the metastable helical spiral state with q3k[001] remains perpendicular to the field (Fig. 12.1B), up to the critical field h  0.125, where the energy minima become equal (Fig. 12.1F) and the spiral q3 flips into the conical state. Although this critical field is lower than that for spirals q1, 2, the value of the energy barrier between q3 and q4, is higher than between q1,2 and q4, which may shift this transition toward higher field values. Thus, for Hk[110], the helical-to-conical transition is a two-step, not a simple one-step, process. This ð1Þ ð2Þ explains the two Hc1 lines, Hc1 and Hc1 , in the phase diagram of Cu2OSeO3, determined experimentally by combining neutron diffraction and magnetization measurements [19]. With decreasing magnetic field the transitions at both critical fields should occur in a reproducible manner, possibly with some hysteresis.

12.3.1.2 Hk[001] In this configuration only the spiral with q3k[001], represented by blue cones in Fig. 12.1C, is stable whereas the spirals with q1k[100] and q2k[010], represented by red springs, become metastable. Fig. 12.1G shows the dependence of the spiral energy on the angle α, between q and the [001] axis, for q varying in the (010) plane. At low fields, the wave vectors of the metastable spirals remain parallel to [100] and [010]. However, these states become unstable at h  0.22, that is, at the magnetic field,

Multiple skyrmionic states and oblique spirals in bulk cubic helimagnets

353

where the energy minimum corresponding to these spirals disappears. With decreasing magnetic field only one spiral (or conical) domain with q3jj[001] survives in zero field, that is, the other spiral domains along [100] and [010] are not recovered.

12.3.1.3 Hk[111] In this case, all three equivalent spiral states with qkh001i jump by FOPT to the field direction. As shown by the red line in Fig. 12.1H, this jump is preceded by a slight modification of the angle α.

12.3.1.4 Negative values of kc For kc < 0, the reorientation processes of spirals can be analyzed in the same way. When the field is applied along [001], there are four equivalent h111i easy directions for q vectors. The transition from a multidomain helical phase to a single domain conical phase is a reproducible FOPT, similar to the one described in Section 12.3.1.3. The spirals first gradually rotate toward the field and subsequently jump into the conical state. When the field is applied along [111], there are three equivalent h111i directions and one direction (along the field) which is energetically more favorable, a situation identical to that considered in Fig. 12.1C. The transition from a multidomain helical phase to a single-domain conical phase is sharp and of first order. When decreasing the magnetic field from a single-domain conical phase, no multidomain state is stabilized. When the field is applied along [110], there are two sets of two equivalent h111i directions at an angle of either α  35.3 degrees or α ¼ 90 degrees with respect to the field. Thus, the transition from a multidomain helical phase to a single-domain conical phase is a two-step reproducible process with jumps of stable and metastable domains and is conceptually the same as the one illustrated in Fig. 12.1A.

12.3.2 The conical-to-homogeneous-phase transition at the critical field Hc2 In this section, we consider the solutions for the conical phase and the nature of the transition to the homogenous phase at Hc2. As it will become evident later, if the field is applied along the easy axis, [001] or [111] depending on the sign of kc, this transition is of first order [32]. On the contrary, if the field points to hard anisotropy axes, the minima of CA gradually align along the field and underlie a second-order phase transition (SOPT) at Hc2. In the following, we carry out a comprehensive analysis of the magnetization rotation in the conical phase by considering the three-dimensional polar plots of CA energy shown in Fig. 12.2D–G as their topology determines the nature of the phase transition. For kc > 0 and h ¼ 0, the minima and maxima of CA are along the easy h001i, green circles in Fig. 12.2D, and hard h111i, red circles in Fig. 12.2D, axes, respectively. At h ¼ 0, the magnetization rotates in the (001) plane and while rotating, it leaves one energy minimum, corresponding to one h001i direction and gets into another one

0.3 0

A BC 0.2 kc = 0.1

–0.2

(A)

0

0.2

0.4

h

1.0 0.8

1.0 0.8 0.6 0.4 0.2 0

(B)

B 0.2 kc = 0.1

0

h = 0.2

h=0

0.5

0.4

w

0.4 0.2

0.4

q

0.6

H || [001]

2

q

1

0.4

q

kc = –0.1

q 4

3

0.2

0.3 h

–0.3 –0.5

0

(C)

0.5 h = 0.2

h=0

h = 0.4



1.0

h

1.5 h = 0.4



H || [001]

(D) h=0



Hard axes h = 0.2

k c = 0.5

h = 0.4

h=0

h = 0.2

kc = –0.5 h = 0.4







(F)

H || [001] Hard axes

(E)

H || [111]

Hard axes

kc = –0.5 (G)

Hard axes H || [111]

k c = 0.5

Fig. 12.2 Conical phase and phase transition into the homogeneous state at the critical field hc2. For kc > 0 and hjj[001] (A), (B), (D), as well as for kc < 0 and hjj [111] (F), the transition at hc2 is of first order. This is seen at the energy density difference between the conical and homogenous phases, depicted in (A), and the magnetization curves shown in (B). The jumps in (B) occur at the field value where the energy density difference in (A) becomes positive. The first-order character of the phase transition is stipulated by the topology of polar plots for CA (D), (F): in both cases, the hard anisotropy axes between the field direction and the trajectory of the magnetization rotation (shown by the black curves in (D)) exclude a smooth alignment of the magnetization along the magnetic field, as it would occur in an isotropic case. For kc < 0 and hjj[001] (C), (E), as well as for kc > 0 and hjj[111] (G), the mutual arrangement of easy and hard anisotropy axes, shown in (E) and (G), underlies the SOPT. The magnetization curves are shown in (E). The solid lines show jumps of wave vectors along the field at hc1. The conical phases shown by the dotted lines have nonzero magnetization even in zero field, but their energies are higher as compared with the oblique spiral states. Modified from A. O. Leonov, C. Pappas, I. Kezsmarki, Field and anisotropy driven transformations of spin spirals in cubic skyrmion hosts, Phys. Rev. Res. 2 (2020) 043386.

Multiple skyrmionic states and oblique spirals in bulk cubic helimagnets

355

along another h001i direction, through a saddle point between the hard h111i axes. The resulting trace of the magnetization is shown by the thick black line in Fig. 12.2D. In an applied magnetic field along h001i, the magnetization is forced to rotate in the environment of hard h111i axes, see Fig. 12.2D for h ¼ 0.2, 0.4, and this underlies the subsequent FOPT into the homogeneous state, evidenced by, for example, the jumps in the magnetization curves shown in Fig. 12.2B. Fig. 12.2A depicts the field dependence of the difference between the energy density of the conical spiral and that of the homogeneous state. The point B for kc ¼ 0.1, in particular, corresponds to the FOPT, which occurs when the energy difference between the two phases is zero. Logically a similar situation occurs for hjj[111] and kc < 0, as seen from the topology of polar plots in Fig. 12.2F. In this case, the three hard anisotropy axes h001i distort the conical phase and these distortions underlie the FOPT, seen as a jump, to the homogeneous state [32]. For kc < 0 and h ¼ 0 (Fig. 12.2E), the maxima of CA energy correspond to the hard h001i directions and the minima are along the easy h111i axes. The angle between these easy directions and a field hjj[001] is 70.6 degrees. Thus, in the conical phase, the magnetization rotates in such a way as to sweep the easy directions h111i and even in zero field it has a nonzero component along the field, as illustrated by the curves in Fig. 12.2C. However, this low-field conical phase is a metastable solution, overshadowed by the oblique spiral states considered in Section 12.3.1. In an applied magnetic field, the global minima of CA gradually approach the field direction and thus display an SOPT to the homogeneous state (Fig. 12.2C and E). The same arguments are also valid for hjj[111] and kc > 0. In this configuration, the field direction is at an angle of 54.7 degrees to the easy h001i axes but at an angle of 70.6 degrees to the hard h111i axes. The corresponding topology of polar plots, depicted in Fig. 12.2G, shows a gradual alignment of the magnetization along the hard direction [111] with increasing magnetic field, resulting to an SOPT at Hc2.

12.3.3 Phase transition between two elliptically distorted conical states for Hjj[110] For Hjj[110] independently on the sign of kc, the q-vector of the conical state is surrounded by two hard and two easy anisotropy axes h111i and h100i. This rosette of anisotropy axes leads to a characteristic elliptical distortion of the conical state, the characteristics of which moreover change depending on whether the magnetization rotation occurs in the exterior or the interior of the rosette. In particular for kc < 0 and low field values, the trajectory of the rotating magnetization is elongated in-plane along the hard h100i axes, as illustrated in Fig. 12.3A and B. With increasing magnetic field, the magnetization approaches the rosette, penetrates the rosette’s interior, and changes its elliptical distortion being now elongated along the easy h111i axes (see Fig. 12.3C and D). The q-vector during this process remains along [110]. The ratio of the major to the minor axes of the ellipse is plotted in Fig. 12.3E in dependence on the field for different values of kc. Since the rotating magnetization tries to avoid

[001]

(110)

[111]

[111]

[111] [111]

[111]

[100]

[111] [111]

[111]

[010] (001)

[001]

(110)

(B)

h [110]

[111]

(110) h = 0.20 [111]

[010] [100]

(001)

[110]

[110]

[111]

[111]

kc = –0.3

) (110

(A)

h [110]

)

(110) h = 0.38

(C)

(D)

(110

1.2

0.8

ax

0.6 0.4

(E)

0.6 7

–0.2 –0.3

0.4

–0.4

0.2

0

0.2

h

0.4

0.6

–0.3

1.1

–0.2 –0.1 0

0

–0.5

0

0

(F)

0.2

h

–0.4

1.2 l/L D

–0.1 ay

kc = 0 –0.2

dmh/dh

ax / ay

1.0

–0.4 0.8 mh

kc = 0

h

0.4

0.6

–0.2

kc = 0

1.0

0.6

(G)

0

0.2

h

0.4

0.6

Fig. 12.3 hjj[110]. Trajectories of the magnetization rotation in the conical phase and in the presence of CA depicted by the black curves for kc < 0 (A)–(D). The anisotropy axes are represented by the red straight lines for the h100i axes, and the blue lines for the h111i axes. The corresponding projections of the magnetization traces on the planes (110), shown in (B), (C), reveal the change of ellipticity, which is also characterized by the ratio of ellipse half-axes in (E). The case of kc < 0 is accompanied by magnetization jumps and peaks in the magnetic susceptibility, as shown in (F) and its inset. Similar fingerprints of the ellipticity change are not detected for kc > 0. For small anisotropy constants, the periodicity changes only by a few percent (G). Modified from A. O. Leonov, C. Pappas, I. Kezsmarki, Field and anisotropy driven transformations of spin spirals in cubic skyrmion hosts, Phys. Rev. Res. 2 (2020) 043386.

Multiple skyrmionic states and oblique spirals in bulk cubic helimagnets

357

the two hard axes [100] and [010], this results in a jump of the magnetization from the rosette’s exterior to its interior (Fig. 12.3F) also indicated by a λ-shaped spike of dmh/ dh (inset). Interestingly, this phase transition between two different conical states has almost no effect on the periodicity of the spiral state, which, for low values of CA, changes just by a few percent, less than 4% for kc ¼ 0.2, as shown in Fig. 12.3G. It is therefore not surprising that this effect has not been detected so far, for example, by neutron scattering experiments.

12.3.4 Spiral flip near the saturation field Hc2 Remarkably, spiral reorientation processes akin to those evidenced at hc1 may also occur near the transition to the homogenous state at hc2. As the conical phase, coaligned with the field, is considered to be the most favorable state that benefits from the Zeeman interaction, such discontinuous spiral reorientation processes are less expected at these high magnetic fields. Nevertheless, they can be induced by relatively moderate values of the CA constant, as evidenced in Fig. 12.4. These jumps of the spiral q-vectors depend both on the direction of the applied magnetic field and the value of kc. In the following, we show theoretically that versatile tilted spiral states may originate from the interplay between Zeeman and CA interactions, which is generic to chiral magnets. This interplay is, therefore, an important factor for understanding the properties of specific cubic helimagnets under magnetic field. As an illustrative example, we consider the case of hjj[110] and kc < 0 (Fig. 12.4). As shown in Fig. 12.4E, the q-vector undergoes an FOPT into a tilted spiral state with an almost constant tilt angle. The specific field range, where this tilted spiral phase occurs, depends on kc. The sketches in Fig. 12.4A–D are instructive to deduce the nature of this tilted spiral. Indeed, in an increasing magnetic field, the magnetization of the conical phase sweeps only two h111i easy anisotropy axes (Fig. 12.4A and B). In the tilted state, as shown in Fig. 12.4C and D, the magnetization tries to embrace all four easy h111i axes at the expense of Zeeman energy. Thus, the maximum tilt angle reached at higher kc values is π/4. The mentioned phase transition is accompanied by the characteristic jumps in the magnetization curve shown in the inset of Fig. 12.4F. However, such a reorientation phase transition requires a threshold value of kc. For kc ¼ 0.1 not even a hint of the metastable tilted spiral becomes apparent in the angle-dependent energy density plot, similar to the one shown in Fig. 12.4F. Conceptually the same reorientation process may occur also for hjj[111]. The spiral canting in this case is stipulated by the same increasing influence of the easy h111i anisotropy axes in the (110) plane. In the limiting case, qjj[110].

0.8 [001]

[001] h



h

(001)

[010]

[100]

a 0.4

(001)

(001)

)

)

0

(001)

0

(E)

[001]

[001] q h

0.2

(010

(C)

kc = –0.2

[010]

[100]

(010

(A)

–0.3

0.6

h = 0.22 kc = –0.3

–0.4

h || [110]

h || [110]

h

h || [110] –0.34

0.1

0.2

h

0.3

0.4

1

mh

h || [110]

Helicoid q w

[100]

[100]

0

0

0.2 h 0.4

kc = –0.3 –0.35

(B)

(010)

(D)

Cone

(F)

h = 0.2

Canted spiral

(010) 0

0.4

0.8

1.2

a

Fig. 12.4 Spiral reorientation near the saturation field hc2, for hjj[110] and kc < 0. As seen from the sketches (A), (B), the magnetization in the conical state embraces two easy axes of CA and is favored by the Zeeman energy. In the canted spiral state (C), (D), the magnetization attempts to embrace all four easy h111i axes, at the expense of the Zeeman energy. This is an FOPT as seen from the energy density (F), for q varied in the plane (010), and the jumps in the magnetization curves (inset of (F)). The deflection angle α of the wave vector q with respect to the field, shown in (E), depends on the value of kc and tends to π/4, thus to qk[100] in the present configuration.

Multiple skyrmionic states and oblique spirals in bulk cubic helimagnets

12.4

359

Skyrmion stability by means of cubic anisotropy

12.4.1 Skyrmion solutions in the presence of cubic anisotropy Alongside with the drastic influence on spiral states, CA significantly distorts the skyrmion states. In particular, the symmetry of the skyrmion cores reflects the underlying energy landscape of CA (Fig. 12.5A). Moreover, versatile SkLs oblique with respect to the field can be found (see Section 12.4.4). In the search for the SkL with the lowest energy density, the direction of the SkL axis n has been offset with respect to the field and easy anisotropy axes. In this section, we focus on the case with kc > 0 and hjj[001], which enhances skyrmion stability due to strong cone deformations (Fig. 12.5E). The equilibrium position of SkL was found to be codirectional with the applied magnetic field, njjh. For the considered case, two energetically close SkL minima occur in the transversal (001) plane. The first minimum corresponds to the easy h100i cubic axes pointing along the diagonals of the hexagonal SkL and along their apothems. In the second minimum, the hexagonal lattice is rotated by the angle π/4. The cores of skyrmions in both states become square shaped with the tendency either to elongate or to shorten along particular directions (see two panels in Fig. 12.5A). For larger anisotropy values, the easy cubic axes along the diagonals may induce an elliptical instability of the SkL similar to that of isolated skyrmions [2, 33] and trigger the phase transition into the helical state. Thus only the second SkL minimum is preserved. At the phase diagram of Fig. 12.5E, the line v-w separates the two SkL phases.

12.4.2 Stabilization mechanisms of skyrmions In order to investigate the stabilization mechanisms of skyrmions within the isotropic model of Eq. (12.1), we consider the energy difference, Δw ¼ wsk  wc, between the equilibrium energy density of the SkL, wsk, and that of the conical phase, wc, as a function of the reduced magnetic field h. The result is given by the red curve in Fig. 12.5B. Δw decreases in the field range 00.2 and reaches its minimum at the critical field value h* ¼ 0.2, which is accompanied by the decrease of the skyrmion periodicity l, as shown in Fig. 12.5C. Remarkably, the densest packing of skyrmions at h* exhibits negligibly small higher harmonics of the SkL Fourier transform [34], and the average SkL mz component equals that of the conical state [5]. With increasing magnetic field for h > h*, the antiparallel magnetization in the cell core is gradually suppressed. This reduces the energetic advantage of the “double-twist” and the skyrmion energy density increases. Nevertheless, at h*, the point with the minimal energy difference with respect to the conical phase, additional small energy contributions can stabilize SkLs. The effect of such a stabilization mechanism is highlighted in Fig. 12.5B, which shows the evolution of Δw with increasing values of kc in Eq. (12.2): the minimum of Δw shifts toward higher field values and decreases substantially, with increasing kc. Remarkably, for kc ≳0:04, Δw even changes sign and the SKL becomes thermodynamically stable in a small region of magnetic fields.

h||[001] kc > 0

kc > 0 1

h||[001]

0.5 a

kc < 0

mz

0.4 i

h||[001] kcc > 0

A FM

h

0

0.3 b

-1





0.1

h||[001]

0.07 0.05

1.3 h||[001]

0.03 0.01

1.2 0.01

0.01

kc = 0

h*

0.03 0.05

0

0.07

(B)

B

c d j f 1.2

Helicoid

e Cone

kc = 0

1.4

0.02

0

C D

SkL v

0.2

(A)

g w

0.1

0.2

h

0.3

0.4

(C) 0

0.1

0.2

(E) 0 h 0.2

0.4

0.8

Cone

0.4 0.1

kc

h||[001] kc < 0 Helicoid

A

0.01

(D)

0.3 0.1

0.02

0 0

h

0

kc = 0

h||[111] 0.1

0.2

0.3 h 0.4

0

(F)

0

0.5

1.0

kc

Fig. 12.5 (A) Contour plots for mz-component of the magnetization in unit cells of hexagonal SkLs for both signs of CA constant kc. SkLs are oriented perpendicular to the field. The cores of skyrmions become distorted. Easy axes of CA are shown as white arrows. For kc > 0 and hjj[001] (first two panels in (A), kc ¼ 0.3, h ¼ 0.2), two stable SkL states rotated by the angle π/4 with respect to each other are realized. (B) Differences between the energy density for SkLs and the conical phases plotted as a function of the applied magnetic field hjj[001] and CA ranging from 0 (red) to 0.07 (green-dotted line). For kc ≳0:04, for example, for kc ¼ 0.05, SkL is stabilized in the vicinity of h*, and thus undergoes an FOPT to the conical phase at both boundaries of the formed stability pocket. For higher values of kc, however, the stability region shifts toward higher fields, and at the upper field boundary, for example, for kc ¼ 0.07, the SkL transforms into the homogeneous state by an FOPT. The field dependence of the SkL period ‘ is shown in (C) and is interrupted at the transition line to the homogeneous state for all considered values of kc. For kc ¼ 0, ‘ diverges to infinity (red line). For hjj[111], (D) the tendency is quite the opposite: the SkL becomes even more energetically unfavorable, and the minimum of Δw shifts toward lower fields (black-dotted line). Moreover at lower fields, the spiral states become energetically advantageous as compared with the conical phase (solid black line). The phase diagrams for kc > 0 and kc < 0 are shown in (E) and (F), respectively. The magnetic field h is applied along [001] and the regions of the thermodynamical stability are colored red (skyrmions), blue (cones), and green (helicoids). The detailed description of the phase transitions between the modulated phases is discussed in details in the text. Modified from L.J. Bannenberg, H. Wilhelm, R. Cubitt, A. Labh, M. Schmidt, E. Lelievre-Berna, C. Pappas, M. Mostovoy, A.O. Leonov, Multiple low-temperature skyrmionic states in a bulk chiral magnet, NPJ Quantum Mater. 4 (2019) 11.

Multiple skyrmionic states and oblique spirals in bulk cubic helimagnets

361

For other directions of the field and kc > 0, the SkL still remains a metastable solution. In particular, for hjj[111], the magnetization must rotate in the environment of the easy cubic axes h100i, which makes the conical state even more favorable. The values of Δw, shown by the solid black line in Fig. 12.5D, correspond to an energy difference between the SkL and the thermodynamically stable spiral state, which is a helicoid with qjjh100i to the left of the point A and a conical state with qjjh111i to the right. A metastable conical state is shown by the dotted line and indicates not only an increase in energy as compared with the isotropic case (red line), but the shift of Δw into the region of smaller fields with increasing kc. For kc < 0 the same reasoning is applicable after interchanging the field directions [111] and [001], respectively.

12.4.3 Phase diagrams of states in the presence of magnetocrystalline cubic anisotropy The phase diagrams of states for both signs of CA constant kc and the field applied along [001] are plotted in Fig. 12.5E and F. These phase diagrams have been built based only on the comparison of the energies of the representative modulated phases.

12.4.3.1 Phase diagram of states for kc > 0, hjj[001] Cones as modulated states with negative energy relative to the homogeneous state exist below the line a-A-g-C-d in Fig. 12.5E. At this line the cones flip into the saturated state by an FOPT as described in Section 12.3.2. Nevertheless, above this line, cones may still exist as states with positive energy. Only within the region filled with the blue color, the conical phase is thermodynamically stable. At the lines A-B and B-f the cones transform discontinuously into skyrmions (red-colored area) and helicoids (green-colored region), respectively. The dotted lines mark phase transitions between metastable states: e-B is the line of the FOPT between skyrmions and helicoids and for kc ¼ 0 the point “e” corresponds to this transition in the isotropic case. The greendotted line B-g, where the point (g) has the coordinates (0.269, 0.253), stands for the transition between metastable cones and helicoids in the region where skyrmions are thermodynamically stable. The line B-D (red-dotted line) is the line of the FOPT between skyrmions and cones in the region of stability of helicoids. As it is seen from the phase diagram (Fig. 12.5E), skyrmion states are thermodynamically stable within a curvilinear triangle (A-B-C) with vertices (A) ¼ (0.047, 0.379), (B) ¼ (0.293, 0.264), and (C) ¼ (0.613, 0.203). At the line B-C they transform into helicoids, and at i-A-C-D-j – into the homogeneous state with the magnetization along the field. The point (D) ¼ (0.651, 0.176) is the intersection of the lability lines for cones (bluedashed lines) and skyrmions. At the point C, the lability lines for skyrmions and helicoids cross each other. For kc > kc(D) helicoids and cones can exist for much larger values of the applied magnetic field than skyrmions, and skyrmions undergo an elongation into helicoids in this region. The solutions for helicoids exist below the line b-g-C-c where they turn into the homogeneous state. In the green-shaded region of the phase diagram, helicoids are

362

Magnetic Skyrmions and Their Applications

the thermodynamically stable states. Due to the strong influence of CA on the conical phase, helicoids can exist in higher fields than cones for kc > kc(g). Conceptually, this is the same phenomenon as the one described in Section 12.3.4.

12.4.3.2 Phase diagram of states for kc < 0, hjj[001] For kc < 0 and hjj[001], only one-dimensional chiral modulations are present in the phase diagram as thermodynamically stable states (see Fig. 12.5F). At zero field, helicoids with qjjh111i have lower energy than the cones, as explained in Section 12.3.1. In an applied magnetic field, these helicoids transform into the conical state by an FOPT. In this case, skyrmions are the only metastable solutions. The dashed line in Fig. 12.5F indicates the hidden transition between the metastable SkL and the metastable conical state, whereas the dotted line the transition between the metastable SkL and helicoidal states.

12.4.3.3 General remarks on the stabilization of skyrmion states in the presence of cubic anisotropy The phase diagrams shown in Fig. 12.5E and F lead to some qualitative skyrmion stabilization scenarios in the presence of CA. (i) As it was shown in Section 12.3.2, CA effectively suppresses the conical phase for kc > 0 and hjj[001]. The same effect may be achieved for hjjh111i and kc < 0. The phase diagram in this case looks qualitatively similar to that in Fig. 12.5E, but with slightly different coordinates for all critical points. Therefore, the suppression of the conical phase depends on the sign of the CA constant kc: for kc > 0 the field must be applied along h100i, for kc < 0 – along h111i. The CA kc must be larger than some threshold value corresponding to the point A in Fig. 12.5E. For kc < 0 and hjj[001] or kc > 0 and hjjh111i, the spiral states are the thermodynamically stable states of the system (see Fig. 12.5F), because in these configurations the magnetization sweeps the global minima of the energy functional in Eq. (12.2). Under these conditions, SkL can occur only as a metastable state. (ii) At the same time, the CA constant kc must not be larger than the critical value kc(C). Otherwise, the skyrmions tend to elongate into spirals. Such an instability of skyrmions is related to the easy anisotropy axes h100i in the plane of the SkL, see Fig. 12.5A.

12.4.4 Oblique skyrmions due to the cubic anisotropy In the same way, as it was considered for spirals, the spatial orientation of SkLs is specified by the competing effect of easy anisotropy axes and applied magnetic field. For example in zero magnetic field and kc > 0, SkLs occupy the crystallographic {001} planes with easy h001i axes (Fig. 12.6A and E). In an applied magnetic field hjj[110] (Fig. 12.6B) or hjj[111] (Fig. 12.6F), these SkLs smoothly rotate and orient perpendicular to the field. Fig. 12.6D and H shows the angle ϕ between the field and the normal n to the SkL plane for both field directions. The equilibrium angle ϕ is obtained by the energy minimization procedure, in which we smoothly rotate n in

n

[100] x

q1 z [001]

(A)

[100] x

(B)

q1

n || [100] –0.2110

h=0 SkL energy density, w

z [001]

n || [110] n || [010] h= 0 k c = 0.10

–0.2130 –0.2140 –0.2312 h = 0.1 –0.2316 [010] y –0.2320 [100] q2 –0.2630 h = 0.2 x –0.2640 H (E) –0.2650 –0.8 –0.4 0 f 0.4 0.8 (C) 0.8 n || [100] 0.7 0.6 f 0.5 0.4 [010] 0.3 y [100] 0.2 q2 0.1 n || [110] x 0 H 0 0.05 0.10 0.15 h (F)

(D)

z

[001] h = 0

–0.210 –0.211 w –0.212 –0.213 H –0.214 [010] –0.224 y –0.226 w –0.228 –0.230 [001] z

(G) H

h= 0 n || [001]

n || [110]

–1.2 –0.6 f 0 0.6 h= 0.1 k c = 0.10

–1.2 –0.6 f 0

0.6

0

f

n || [111]

[010] –0.5 y n || [001] –1.0

(H)

0

0.1

0.2 h 0.3

Fig. 12.6 Schematic representations of hexagonal SkLs occupying the (001) planes (A), (E) for CA with easy h001i axes. In an applied magnetic field hjj[110] (B) (hjj[111] (F)) the SkLs smoothly rotate and then jump perpendicular to the field, that is, into the plane (110) (B) ((111) (F)). The field dependence of the angle ϕ between the skyrmion axes and the field is shown in (D) and (H). The SkL energy density, given in (C) and (G) for kc ¼ 0.1 in dependence on the field, is minimized with respect to the direction of n.

364

Magnetic Skyrmions and Their Applications

the plane (001) (Fig. 12.6C) and in the plane ð110Þ (Fig. 12.6G), correspondingly. Fig. 12.6G, in particular, exhibits two minima for the SkL orientation: the global minimum in the plane (001) and the local one in the plane (110). In an increasing magnetic field, the latter local minimum shifts to the position njjh, and the SkL undergoes a smooth reorientation as also shown by the continuous curve ϕ(h) in Fig. 12.6H.

12.5

Conclusions

Our theoretical findings provide a comprehensive overview of cubic-anisotropydriven effects, which lead to deviations from the universal behavior of modulated states at the magnetic phase diagrams reported for bulk cubic helimagnets [35]. In fact, the reorientation processes of spirals do not only boil down to the phenomenon of the canonical critical field Hc1 (Section 12.3.1) observed since the first experiments, for example, in MnSi [36–38]. We show that the q-vectors of spirals may undergo FOPT between different oblique states when it is least expected, that is, near the critical field Hc2 (Section 12.3.4) [19]. Moreover, we show that the effect of anisotropy is more subtle than assumed so far, as it can lead to an elliptical distortion of spirals. This unexpected effect may change drastically the observed behavior and lead to surprising experimental results even without any reorientation of the spiral wave vectors (Section 12.3.3). Remarkably, the “well-being” of skyrmions also rests on the stability of spiral states. Indeed, in order to stabilize SkLs (Section 12.4.2), one should orient the field along the easy axes of the CA, and thus ensure that anisotropy modifies the energetic balance in favor of the skyrmion phase (Section 12.3.2). To summarize, our theoretical findings imply that modern experiments on bulk cubic helimagnets would benefit if done alongside with theoretical predictions. First of all, the experimental setups must be carefully devised and scan the whole q-space to spot multiple oblique skyrmion and spiral states. This is of crucial importance in the case of competing, for example, exchange and cubic anisotropies as observed in Refs. [19, 21]. Moreover, it is important that the timescale of the experiments is adjusted according to the phenomena to be probed. Furthermore, experimental findings that seem at first sight of secondary importance such as the high-order peaks in SANS patterns, which are usually neglected, may deliver the paramount signature of subtle processes, like the elliptical distortions of the spiral states, and explain the stabilization of skyrmions at LTs and high magnetic fields, thus outside the A-phase pocket. These LT thermodynamically stable skyrmions may not necessarily be an extended lattice. They could represent skyrmion clusters surrounded by the conical state, which may also be brought in relation with the FOPT into the homogenous state as considered in Ref. [39].

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[2] A. Bogdanov, A. Hubert, Thermodynamically stable magnetic vortex states in magnetic crystals, J. Magn. Magn. Mater. 138 (1994) 255 (195 (1999) 182). [3] S. M€uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, P. B€oni, Skyrmion lattice in a chiral magnet, Science 323 (2009) 915. [4] S. Seki, X.Z. Yu, S. Ishiwata, Y. Tokura, Observation of skyrmions in a multiferroic material, Science 336 (2012) 198. [5] A.A. Leonov, Twisted, Localized, and Modulated States Described in the Phenomenological Theory of Chiral and Nanoscale Ferromagnets (Ph.D. Thesis), Technical University Dresden, 2012. [6] M.N. Wilson, A.B. Butenko, A.N. Bogdanov, T.L. Monchesky, Chiral skyrmions in cubic helimagnet films: the role of uniaxial anisotropy, Phys. Rev. B 89 (2014) 094411. [7] H. Wilhelm, M. Baenitz, M. Schmidt, U.K. Roessler, A.A. Leonov, A.N. Bogdanov, Precursor phenomena at the magnetic ordering of the cubic helimagnet FeGe, Phys. Rev. Lett. 107 (2011) 127203. [8] A.O. Leonov, A.N. Bogdanov, Crossover of skyrmion and helical modulations in noncentrosymmetric ferromagnets, New J. Phys. 20 (2018) 043017. [9] S. Buhrandt, L. Fritz, Skyrmion lattice phase in three-dimensional chiral magnets from Monte Carlo simulations, Phys. Rev. B 88 (19) (2013) 195137. [10] I. Levatic, P. Popcevic, V. Surija, A. Kruchkov, H. Berger, A. Magrez, J.S. White, H.M. Ronnow, I. Zivkovic, Dramatic pressure-driven enhancement of bulk skyrmion stability, Sci. Rep. 6 (2016) 21347. [11] Y. Okamura, F. Kagawa, S. Seki, Y. Tokura, Transition to and from the skyrmion lattice phase by electric fields in a magnetoelectric compound, Nat. Commun. 7 (2016) 12669. [12] J.S. White, K. Prsa, P. Huang, A.A. Omrani, I. Zivkovic, M. Bartkowiak, H. Berger, A. Magrez, J.L. Gavilano, G. Nagy, J. Zang, H.M. Ronnow, Electric-field-induced skyrmion distortion and giant lattice rotation in the magnetoelectric insulator Cu2OSeO3, Phys. Rev. Lett. 113 (2014) 107203. [13] A.J. Kruchkov, J.S. White, M. Bartkowiak, I. Zivkovic, A. Magrez, H.M. Ronnow, Direct electric field control of the skyrmion phase in a magnetoelectric insulator, Sci. Rep. 8 (2018) 10466. [14] J.S. White, I. Zivkovic, A.J. Kruchkov, M. Bartkowiak, A. Magrez, H.M. Ronnow, Electric-field-driven topological phase switching and skyrmion-lattice metastability in magnetoelectric Cu2OSeO3, Phys. Rev. Appl. 10 (2018) 014021. [15] I.E. Dzyaloshinskii, Theory of helicoidal structures in antiferromagnets. I. Nonmetals, J. Sov. Phys. JETP-USSR 19 (1964) 960. [16] U.K. Roessler, A.N. Bogdanov, C. Pfleiderer, Spontaneous skyrmion ground states in magnetic metals, Nature 442 (2006) 797. [17] A.B. Butenko, A.A. Leonov, U.K. R€oßler, A.N. Bogdanov, Stabilization of skyrmion textures by uniaxial distortions in noncentrosymmetric cubic helimagnets, Phys. Rev. B 82 (2010) 052403. [18] A. Leonov, Chiral skyrmion states in non-centrosymmetric magnets, arXiv:1406.2177 (2014). [19] F. Qian, L.J. Bannenberg, H. Wilhelm, G. Chaboussant, L.M. DeBeer-Schmitt, M.P. Schmidt, A. Aqeel, T.T.M. Palstra, E.H. Bruck, A.J.E. Lefering, C. Pappas, M. Mostovoy, A.O. Leonov, New magnetic phase of the chiral skyrmion material Cu2OSeO3, Sci. Adv. 4 (2018) eaat7323. [20] A. Chacon, L. Heinen, M. Halder, A. Bauer, W. Simeth, S. M€ uhlbauer, H. Berger, M. Garst, A. Rosch, C. Pfleiderer, Observation of two independent skyrmion phases in a chiral magnetic material, Nat. Phys. 14 (2018) 936.

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Conventional applications of skyrmions

13

Xichao Zhanga, Jing Xiaa, Laichuan Shena, Motohiko Ezawab, Xiaoxi Liuc, and Yan Zhoua a School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, People’s Republic of China, bDepartment of Applied Physics, The University of Tokyo, Tokyo, Japan, cDepartment of Electrical and Computer Engineering, Shinshu University, Nagano, Japan

13.1

Introduction

The existence of skyrmions in magnetic systems was theoretically proposed by Bogdanov and Yablonskii in 1989 [1] and has been experimentally realized by M€ uhlbauer et al. in 2009 [2]. Since 2009, magnetic skyrmions have been extensively studied due to their intriguing physics and significant potential in magnetic and spintronic applications [3–10]. The intriguing physics of magnetic skyrmions, including both static and dynamic properties, has been introduced and discussed in previous chapters. In this chapter, we focus on possible conventional applications based on the manipulation of magnetic skyrmions. The concept of conventional skyrmion-based applications means the use of magnetic skyrmions in traditional magnetic and spintronic devices, which have been proposed for magnetic domain walls or magnetic vortices. Replacing those by magnetic skyrmions may improve the device performance and device energy efficiency, and therefore, could lead to new and better designs of devices. For example, Parkin et al. proposed and designed the domain wall-based racetrack-type memory in 2008 [11], which is a current-driven information storage device in which information are stored by a series of magnetic domains with opposite polarizations. In 2013, Fert et al. suggested using magnetic skyrmions to carry information in racetrack-type memories [3] due to several intrinsic properties of skyrmions that are beneficial for the current-driven operation, such as the low depinning current threshold. In Section 13.2, we will review skyrmion-based racetrack memory device concepts, including both single layer and multilayer systems. In addition to information storage, information computing based on binary logic, i.e., Boolean computing, is also a vital task in conventional magnetic and spintronic applications. Both theoretical and experimental works have demonstrated that magnetic domain walls or vortices can be used in logic computing [12]. Indeed, magnetic skyrmions can provide alternative approaches for logic computing, which have been proposed in several theoretical works [13–16]. In particular, skyrmion-based logic computing could be combined with information storage in principle [17], which may lead to an in-memory computing architecture. We will introduce and review Magnetic Skyrmions and Their Applications. https://doi.org/10.1016/B978-0-12-820815-1.00013-4 Copyright © 2021 Elsevier Ltd. All rights reserved.

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Magnetic Skyrmions and Their Applications

the concept of using skyrmions for conventional logic computing in Section 13.3. On the other hand, in Sections 13.4 and 13.5, we will review skyrmion-based transistorlike functional devices and skyrmion-based nano-oscillator devices, respectively. These device concepts are also inspired and developed based on existing technologies in the field, however, they may provide new features to spintronic applications due to unique topology-dependent dynamics of skyrmions. As discussed in Chapter 6, the dynamics of skyrmions can be precisely controlled by electric current owing to the spin-transfer torques and spin-orbit torques [18–38]. It is worth mentioning that skyrmions can also be manipulated by other methods, including electric field [39–53], spin wave [54–56], chiral laser [57–63], and so on. As an example, we will review and discuss in Section 13.6 the electric-field control of skyrmion for future applications. This chapter will be concluded with a brief summary (cf. Section 13.7) on proposed concepts of conventional skyrmion-based applications and possible skyrmion-based applications that can be further studied in the following years.

13.2

Skyrmion-based racetrack memory

In this section, we first introduce the concept of racetrack-type memory and then focus on the magnetic skyrmion-based racetrack-type devices, where the driving force is provided by spin currents or spin waves. The racetrack-type memory is a type of nonvolatile information storage devices driven by electric currents, where binary information is represented by magnetic domains with opposite polarizations in ferromagnetic racetracks. For example, magnetic domains magnetized along the positive and negative out-of-plane can be used to represent binary digits “1” and “0,” respectively. In contrast to commercial hard disk drives, the racetrack memory has no mechanical moving parts. The magnetic domains and domain walls carrying information can be shifted in the racetrack by direct injection of electric current into the racetrack [11], where the driving force is provided by spin-transfer torques or spinorbit torques [18, 19]. Then, the binary digital information can be written to and read out from the ferromagnetic racetrack by using magnetic tunnel junction element connected to the racetrack. Such a device concept was proposed and experimentally demonstrated in 2008 by an IBM team led by Stuart Parkin [11]. Nowadays, the team has developed different generations of racetrack memory with improved performance [64], and meanwhile, many studies have also contributed to the improvement of racetrack memory [3–9, 65–67]. As shown in Fig. 13.1, in the first generation of racetrack memory, the magnetic domains are in-plane magnetized and driven by spin-transfer torques, whereas in recent implementation of racetrack memory, the synthetic antiferromagnetic structure was used where magnetic domains are magnetized in an out-of-plane manner without significant fringing fields and driven by a giant exchange coupling torque [68]. Obviously, the performance and reliability of racetrack memory 4.0 have been much improved due to the ingenious design. In the following, we will see that similar design

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Fig. 13.1 Schematic illustrations of magnetic racetrack memories based on domain walls. (A) Racetrack memory 1.0. (B) Racetrack memory 2.0. (C) Racetrack memory 3.0. (D) Racetrack memory 4.0. (E) Three-dimensional (3D) arrays of racetrack memories. From S. Parkin, S. H. Yang, Memory on the racetrack, Nat. Nanotechnol. 10 (3) (2015) 195– 198. © 2015 Macmillan Publishers Limited.

concept can also be applied to skyrmion-based racetrack memory and will bring additional benefits. Although the racetrack memory has been studied and significantly upgraded for more than a decade, some development challenges still remain and must be overcome before any racetrack memory product can go into market and reach its full potential. For example, the operation of a racetrack memory relies on the current-induced motion of domain walls, and thus it is important to reduce the threshold current density for domain wall motion while ensuring domain walls can be shifted in high velocities. On the other hand, the increase of information data storage density is also a vital task for the development of a competitive racetrack memory. Note that Parkin et al. have shown that racetrack memory can be arranged in a three-dimensional array [64], which offers a solution to increase the storage density by increasing the packing density. However, the direct increase of magnetic bit density in each ferromagnetic racetrack is still important. An alternative approach to improve the racetrack memory is using magnetic skyrmions instead of traditional magnetic domains (or domain walls) to carry binary information in ferromagnetic racetracks [3–9]. In 2013, Sampaio et al. [24] and Iwasaki et al. [23] independently carried out theoretical and simulation studies on current-driven motion of isolated skyrmions in ferromagnetic racetracks. In particular, Fert et al. suggested that the intrinsic properties of magnetic skyrmions may help overcome some challenges in racetrack memory devices [3], such as avoiding the detrimental effects of defects and impurities in materials. Because the magnetic skyrmion is quasiparticle-like nanoscale object, which has a rigid shape and

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topology-dependent dynamics, it could be easier to overcome the pinning and thus leading to lower critical depinning current compared with magnetic domain walls [3, 23, 69–71]. The possible minimal size of a single isolated skyrmion down to a few nanometers also provides a great potential to fabricate a skyrmion-based racetrack memory with ultrahigh storage density. Owing to these potential advantages of replacing domain walls by skyrmions in racetrack memory, since 2013, magnetic skyrmion has been widely regarded as a promising candidate to be used as information carrier in future solid-state information storage devices, which may provide features including but not limited to ultrahigh storage density and ultralow energy consumption. As shown in Fig. 13.2A, in the initially proposed and most widely studied prototype of skyrmion-based racetrack memory, the binary information digits “1” and “0” are represented by the presence and absence of isolated skyrmions [3, 24, 72]. Namely, a sequence of binary data is coded in a chain of isolated skyrmions with variable spacings. Theoretically, in the absence of pinning effect and thermal effect, all skyrmions in a racetrack will move together in a simultaneous manner, so that the binary information can be written into or read out from the racetrack by a sensor placed on a certain location of the racetrack (see Fig. 13.2). The driving force can be provided by spin-transfer torques or spin-orbit torques, which are generated by electric currents injected into the ferromagnetic racetrack itself or its heavy metal substrate. Specifically, for the skyrmion-based racetrack memory made of a single ferromagnetic layer, Tomasello et al. numerically demonstrated four different scenarios for the design in 2014 [73], as shown in Fig. 13.2B. The performances of both Bloch-type and Neeltype skyrmions driven by either spin-transfer torques or spin-orbit torques in a singlelayer racetrack are simulated and analyzed by Tomasello and coworkers. It was found that the racetrack containing Neel-type skyrmions driven by spin-orbit torques has the best performance in terms of the current-velocity relation, reliability, and fabrication flexibility. Therefore, it is desired to fabricate racetrack memory in a ferromagnetheavy metal structure, where the ferromagnet-heavy metal interface could provide interfacial Dzyaloshinskii-Moriya interaction [74] to stabilize Neel-type skyrmions, and the heavy metal subtract can provide spin-orbit torques to drive skyrmions due to the spin Hall effect. Indeed, the skyrmion-based racetrack memory is not limited to the single ferromagnetic structure. Similar to the domain wall-based racetrack memory 4.0 (see Fig. 13.1), synthetic antiferromagnetic structures can also be used to host skyrmions and thus lead to an upgraded skyrmion-based racetrack memory design, which was found to be more suitable for the in-line motion and transport of skyrmions on the track [75, 76]. In the synthetic antiferromagnetic bilayer structure, two ferromagnetic layers with identical thickness are exchange-coupled via an antiferromagnetic interfacial coupling, which could be realized by the Ruderman-Kittel-Kasuya-Yosida interaction [68, 75, 77]. In such a structure, as shown in Fig. 13.3, the magnetic skyrmions in top and bottom ferromagnetic layers have opposite skyrmion numbers, and therefore, the antiferromagnetically exchange-coupled skyrmion pair has a zero skyrmion number. Due to the zero total skyrmion number, the magnetic skyrmion in the synthetic antiferromagnetic bilayer racetrack shows no skyrmion Hall effect [75] and can be displaced along the

Fig. 13.2 Schematic illustrations of skyrmion-based racetrack memory. (A) Skyrmion-based racetrack memory. (B) Four different scenarios for the design of a skyrmion racetrack memory. (A) From X. Zhang, G.P. Zhao, H. Fangohr, J.P. Liu, W.X. Xia, J. Xia, et al., Skyrmion-skyrmion and skyrmion-edge repulsions on the skyrmionbased racetrack memory, Sci. Rep. 5 (2015) 7643. CC BY. (B) From R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Carpentieri, G. Finocchio, A strategy for the design of skyrmion racetrack memories, Sci. Rep. 4 (2014) 6784.

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Fig. 13.3 Schematic of synthetic antiferromagnetic bilayer skyrmions. (A) Synthetic antiferromagnetic bilayer skyrmion. (B) Simulated current-induced motion of a synthetic antiferromagnetic bilayer skyrmion in the racetrack. From X. Zhang, Y. Zhou, M. Ezawa, Magnetic bilayer-skyrmions without skyrmion hall effect, Nat. Commun. 7 (2016) 10293. CC BY.

driving current direction without strong interaction with racetrack edges. Note that the skyrmion Hall effect is an intrinsic nature of current-driven skyrmions [78], which depends on the topological structures of skyrmions and has been experimentally observed [79, 80]. As a result, the magnetic skyrmion in the synthetic antiferromagnetic bilayer racetrack can be safely driven into high-speed motion [77] and thus promises for reliable ultrafast operations of information carriers. The idea of building the skyrmion-based racetrack memory based on the synthetic antiferromagnetic bilayer structure to avoid detrimental effect of the skyrmion Hall effect was first proposed by Zhang et al. in 2016 [75]. The skyrmion-based synthetic antiferromagnetic racetrack was also studied in several theoretical and computational works [76, 77, 81]. In particular, the synthetic antiferromagnetic skyrmions have been experimentally realized by Legrand et al. [82] and Dohi et al. [83] in two independent works published very recently. It should be noted that the synthetic antiferromagnetic skyrmion shows no skyrmion Hall effect only when it consists of even ferromagnetic layers with identical thickness, as theoretically suggested by Zhang et al. in 2016 [75]. Here, it is worth mentioning that the skyrmion Hall effect can also be eliminated in antiferromagnets [84] or be reduced in ferrimagnets [66], of which two types of magnetic materials are also possible for building skyrmion-based racetrack memory. There are different ways to encode binary bits “0” and “1” in magnetic skyrmionbased racetrack memory devices. As mentioned above, the straightforward way is to use the skyrmion state and ferromagnetic state in a single ferromagnetic racetrack to represent binary digit “1” and “0,” respectively, as shown in Fig. 13.4A. Namely, the information is encoded by the spacing between neighboring skyrmions. However, such a method may not be reliable for real materials with certain defects and

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Fig. 13.4 Possible methods of encoding binary bits “0” and “1” in magnetic skyrmion-based devices. (A) Using a pair of skyrmion and ferromagnetic states. (B) Using a pair of skyrmions with opposite polarities. (C) Using a pair of Neel-type and Bloch-type skyrmions. (D) Using a pair of skyrmion and antiskyrmion. (E) Using a pair of synthetic antiferromagnetic bilayer skyrmions with opposite top-layer and bottom-layer skyrmion polarities. (F) Using a pair of skyrmion tube and magnetic bobber.

impurities, especially at room temperature. The reason is that the pinning effect induced by defects or impurities as well as the random walk of skyrmions induced by thermal effect may lead to the change of the spacing between two adjacent skyrmions and thus may result in the distortion or loss of information. Hence, different information encoding methods have been proposed for the skyrmion-based racetrack memory. For example, using a two-lane racetrack instead of a single-lane racetrack scheme, where the information is encoded in the lane index of each skyrmion [65]. Such as method is more robust against pinning effect—and thermal effect—induced disturbance as the information carried by skyrmions is unaffected by the spacing between skyrmions. Moreover, it is possible to use a pair of skyrmions with opposite polarities to represent binary data bits “1” and “0” in the two-lane racetrack, as shown in Fig. 13.4B. Similarly, by fabricating a two-lane racetrack based on two ferromagnetic materials with different types of Dzyaloshinskii-Moriya interactions, it would be possible to encode information in pairs of Neel-type and Bloch-type skyrmions or in pairs of skyrmion and antiskyrmion, as shown in Fig. 13.4C and D. On the other hand, in the synthetic antiferromagnetic bilayer racetrack, with either single-lane or twolane design, binary data bits “1” and “0” can be represented by synthetic antiferromagnetic bilayer skyrmions with two different spin coupling configurations, as illustrated in Fig. 13.4E. Recently, it was found that in three-dimensional racetracks where both

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skyrmion tubes and magnetic bobbers can be stabilized [85], binary data bits “1” and “0” can be carried by skyrmion tube and magnetic bobber, respectively, as shown in Fig. 13.4F. Most recently, the research on skyrmion-based racetrack memory has also been largely concerned with how to incorporate the skyrmion-based racetrack into fully functional electronic circuits [86], where the skyrmion-based racetrack memory can perform multiple fundamental tasks such as the information update, deletion, and insertion. We note that latest information about the skyrmion-based racetrack memory, from the point of view of electronics, was reported in several comparative studies on domain wall-based and skyrmion-based racetrack memories [77, 87].

13.3

Skyrmion-based logic computing gates

In the previous section, we showed that both domain walls and skyrmions can be displaced in magnetic racetracks driven by electric currents, where magnetic domains or skyrmions can be used as binary information carriers for data storage. In this section, we show that it is also possible to perform information computing based on the manipulation of magnetic skyrmions. The information computing based on classical binary logic—Boolean logic—is an important and necessary function of any conventional spintronic information processing device. Many theoretical and experimental works have shown that basic Boolean logic functions can be carried out by the manipulation of magnetic domain walls [88]. For example, in 2005, Allwood et al. [88] experimentally demonstrated that four basic logic operations, including logical NOT, logical AND, signal fan-out, and signal cross-over, can be realized by manipulating domain walls in simple geometric designs via a global applied rotating magnetic field. In 2014, Zhou and Ezawa [26] theoretically predicted the reversible conversion between isolated skyrmions and domain wall pairs (see Fig. 13.5A), which opens a way to connect traditional domain walls with topological skyrmions. With the ability to mutually convert skyrmions and domain walls in one magnetic system, existing technologies used in domain wall logic can be directly transferred to perform skyrmion logic computing. Inspired by this fact, in 2015, Zhang et al. [13] proposed the first skyrmion-based logic computing concept and computationally demonstrated that skyrmions can be used to perform basic logical OR and logical AND operations. As shown in Fig. 13.6, by combining the wide-narrow nanotrack junctions with a Y-shape fan-out junction structure, it is possible to realize the current-driven duplication and merging of isolated skyrmions based on the domain wall-skyrmion mutual conversion and domain wall fan-out mechanisms [13]. Namely, to duplicate a single isolated skyrmion into two isolated skyrmions, an isolated skyrmion is first converted to a domain wall pair, then the domain wall pair is duplicated by the fan-out junction, and finally, the created two domain wall pairs are converted to two isolated skyrmions through wide-narrow nanotrack junctions driven by the spin-transfer torque. Similarly, by reversing the direction of injected driving current, one could merge two isolated skyrmions into one single isolated skyrmion in the same geometric design. Note that the driving force can be either spin-transfer or spin-orbit torques, as demonstrated

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Fig. 13.5 Conversion between magnetic skyrmions and magnetic domain walls. (A) Simulated reversible conversion between isolated skyrmions and domain wall pairs. (B) Experimental demonstration of conversion between domain walls and skyrmions. (A) From Y. Zhou, M. Ezawa, A reversible conversion between a skyrmion and a domain-wall pair in junction geometry, Nat. Commun. 5 (2014) 4652. © 2014 Macmillan Publishers Limited. (B) From W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M.B. Jungfleisch, F.Y. Fradin, et al., Blowing magnetic skyrmion bubbles, Science 349 (6245) (2015) 283–286, with permission from AAAS.

in a recent experiment by Jiang et al. [27] where skyrmions are created by blowing domain walls by spin-orbit torques (see Fig. 13.5B). Based on the basic duplication and merging processes of isolated skyrmions, Zhang et al. [13] further proposed and computationally demonstrated the logical OR and logical AND operations, as shown in Fig. 13.7, where the geometric designs of the logical OR gate and logical AND gate have a delicate difference on the width of the Y-shape junction. In the proposed skyrmion-based logic computing scheme, both the logical OR and AND gates have two input branches and one output branch, and the binary data bits “1” and “0” are represented by the presence of a skyrmion and absence of a skyrmion, respectively. It should be noted that the computing operation of “0” + “0” ¼ “0” is trivial in such a scheme, which means that when there is no input, there is no output. However, the skyrmion computing operations of “0” + “1” ¼ “1”, “0” + “1” ¼ “1”, and “1” + “1” ¼ “1” can be realized in the geometric design with a narrow Y-shape junction, whereas the skyrmion computing operations of

Fig. 13.6 Duplication and merging of magnetic skyrmions. (A) Duplication of an isolated skyrmion into two isolated skyrmions. (B) Merging of two isolated skyrmions into an isolated skyrmion. From X. Zhang, M. Ezawa, Y. Zhou, Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions, Sci. Rep. 5 (2015) 9400. CC BY.

Fig. 13.7 Skyrmion-based logic gates. (A) Skyrmion-based logical OR gate. (B) Skyrmion-based logical AND gate. From X. Zhang, M. Ezawa, Y. Zhou, Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions, Sci. Rep. 5 (2015) 9400. CC BY.

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“0” + “1” ¼ “0”, “0” + “1” ¼ “0”, and “1” + “1” ¼ “1” can be realized in the geometric design with a wide Y-shape junction. Therefore, the logical OR and logical AND operations can be implemented in the geometric designs with narrow and wide Yshape junctions, respectively. For example, in the logical OR gate, “1” + “0” means that there is a single isolated skyrmion in input A and no skyrmion in input B. By applying an electric current, which produces spin-transfer torques, the isolated skyrmion in input branch A moves toward the output side. The isolated skyrmion is first converted into a pair of domain walls, which is then converted back to an isolated skyrmion via the narrow Y-shape junction. The isolated skyrmion finally moves into the output branch, leading to the implementation of the “1” + “0” ¼ “1” operation. It is worth mentioning that the proposed logic gates can be directly connected to magnetic racetracks, and thus, it is envisioned that an in-memory computing architecture can be constructed by combining the skyrmion-based racetrack memory and skyrmion-based logic gates. On the other hand, a lot of theoretical and computational works on skyrmion-based logic computing have also emerged soon after the proposal of first skyrmion-based logical OR gate and logical AND gate. Zhang et al. [14] designed the skyrmionbased logical NIMP, XOR, and IMP gates in 2015. In 2016, Xing et al. [15] computationally demonstrated the logical NAND and NOR gates. In 2018, Luo et al. [16] computationally demonstrated a series of skyrmion-based logical operations, including AND, OR, NOT, NAND, NOR, XOR, and XNOR in the ferromagnetic system. In 2019, Liang et al. [89] extended the hosting system of skyrmion-based logic computing from ferromagnets to antiferromagnets and computationally demonstrated the basic Boolean logical operations of AND, OR, NOT, NAND, and NOR. Here, it should be mentioned that magnetic skyrmions can also be employed to perform unconventional and non-Boolean computing, such as the probabilistic computing [90] and reservoir computing [91–93], which will be discussed in next chapter.

13.4

Skyrmion-based nano-oscillator devices

In this section, we introduce the concept of using skyrmions in spin-torque nanooscillator (STNO) devices and review recent advances in skyrmion-based STNO devices. Because the spin-transfer torque was predicted independently by Slonczewski [94] and Berger [95], it has attracted a great deal of attention due to its important role in modern spintronic devices, such as the racetrack-type memory [11] and STNO [96] In particular, the arrays of STNOs can be used to perform the neuromorphic computing. [97–99] Skyrmion-based STNO, which was proposed in recent years, is currently a hot topic, because it can excite microwave signals with small linewidth and is expected to improve the output power. In 2015, Zhang et al. [96] first demonstrated that in nanocontact oscillators (see Fig. 13.8A), a uniform current can excite the ferromagnetic skyrmion to move in a circular motion, so that an oscillating signal can be acquired by detecting the skyrmion position and using the magnetoresistance effect. In 2016, Garcia-Sanchez et al. [100] proposed an alternative skyrmion-based STNO, which

Fig. 13.8 The model for spin-torque nano-oscillator (STNO) devices based on the circular motion of magnetic skyrmions. (A) Nanocontact oscillators, where the current flows through the point-contact electrodes; (B) Nanopillar oscillators, where the fixed layer with a magnetic vortex configuration is used to generate the spin-polarized current with a vortex-like polarization. (A) From S.F. Zhang, J.B. Wang, Q. Zheng, Q.Y. Zhu, X.Y. Liu, S.J. Chen, et al., Current-induced magnetic skyrmions oscillator, New J. Phys. 17 (2) (2015) 023061. © 2015 IOP Publishing Ltd. and Deutsche Physikalische Gesellschaft. (B) From L. Shen, J. Xia, G. Zhao, X. Zhang, M. Ezawa, O.A. Tretiakov, et al., Spin torque nano-oscillators based on antiferromagnetic skyrmions, Appl. Phys. Lett. 114 (4) (2019) 042402, with the permission of AIP Publishing.

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requires a fixed layer with a vortex magnetic configuration to generate the spinpolarized current with a vortex-like polarization (see Fig. 13.8B). On the other hand, due to the presence of a large Magnus force, the fast-moving ferromagnetic skyrmion will be destroyed at the nanodisk edge, so that the oscillation frequencies of the ferromagnetic skyrmion-based STNOs are low (about 1 GHz). To improve the oscillation frequency, putting multiple skyrmions in one STNO was adopted. [96, 101, 102] In 2019, Feng et al. proposed a modified nanodisk structure, in which the nanodisk edge is enhanced by applying high perpendicular magnetic anisotropy and showed that the oscillation frequency can be increased by 75% [103]. In addition, modifying the profile of Dzyaloshinskii-Moriya interaction can also lead to the increase in oscillation frequency. [104] In addition, in 2019, Shen et al. [105] proposed to use the circular motion of an antiferromagnetic skyrmion to create the oscillation signal. Such a nano-oscillator can produce high frequencies (tens of gigahertz), because the antiferromagnetic skyrmion obeys the inertial dynamics and it has no net Magnus force. Fig. 13.9 shows the comparison of ferromagnetic and antiferromagnetic skyrmion-based STNOs. It can be seen that the ferromagnetic

Fig. 13.9 The comparison of ferromagnetic and antiferromagnetic skyrmions in the nanodisk. The trajectory for an antiferromagnetic skyrmion driven by positive (A) and negative (B) currents. The trajectory for a ferromagnetic skyrmion driven by positive (C) and negative (D) currents. From L. Shen, J. Xia, G. Zhao, X. Zhang, M. Ezawa, O.A. Tretiakov, et al., Spin torque nanooscillators based on antiferromagnetic skyrmions, Appl. Phys. Lett. 114 (4) (2019) 042402, with the permission of AIP Publishing.

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skyrmion moves toward the nanodisk center when the positive current is applied, whereas for the negative current, the skyrmion is destroyed at the nanodisk edge. Under the action of the same currents, the antiferromagnetic skyrmion moves steadily in the nanodisk, which is independent of the sign of the applied current.

13.5

Skyrmion-based transistor-like functional devices

The concept of skyrmion-based transistor-like functional devices was inspired by the idea of controlling the current-driven transport of skyrmions in a channel by a gate voltage, which relies on the phenomenon of voltage-controlled magnetic anisotropy (VCMA) [106, 107]. Like the semiconductor transistor that usually has three terminals, where a small voltage or current applied to a pair of the terminals can control a large current through another pair of terminals, a small voltage applied in the skyrmion-based transistor-like device can control the motion of skyrmions driven by a large current. Such an idea of controlling skyrmion motion by a gate voltage, mimicking a transistor-like function, was independently proposed by Zhang et al. [53] and Upadhyaya et al. [46] in 2015. Both the works [46, 53] theoretically and numerically demonstrated that the current-induced skyrmion motion can be manipulated by the electric-field-controlled magnetic anisotropy (see Fig. 13.10A). In particular, Zhang et al. [53] demonstrated that a gate voltage can be used to switch on/off a skyrmion transport circuit, whereas Upadhyaya et al. [46] demonstrated transistor-like and multiplexer-like spintronic devices controlling skyrmion current by defining electric-field gates. As shown in Fig. 13.10B, in the ferromagnetic racetrack where skyrmions are displaced by a driving current applied along the racetrack, the perpendicular magnetic anisotropy in the voltage-gated region is controlled by the applied electric field due to the charge accumulations. There are two working states when the driving current is applied. For the ON state, the gate voltage is turned off and the skyrmion dynamics is only driven by the spin-polarized current, which means skyrmions can be delivered from the left terminal of the racetrack-type device to the right terminal (or vice versa). For the OFF state, the gate voltage is turned on and induces an energy barrier in the voltage-gated region by changing the perpendicular magnetic anisotropy, which prevents current-driven skyrmions going through the voltage-gated region. As shown in Fig. 13.10C, in a wing-shaped nanowire junction, the multiplexer-like function can also be implemented by controlling the skyrmion transport channel based on the same gate voltage method. In the skyrmion-based transistor-like functional device, the driving force can be provided by either spin-polarized current [13] or spin wave [108]. The antiferromagnetic skyrmion-based transistor-like device controlled by strain manipulation was also proposed by Zhao et al. in 2018 [109]. It is noteworthy that the transistor-like function can be used in skyrmion-based logic computing, as computationally demonstrated by Liang et al. [89] in 2019.

Fig. 13.10 Design of the skyrmion-based transistor-like functional device. (A) Schematic view (xz-plane) of the spin-polarized current-driven prototype of the skyrmion transistor. (B) Schematic view (xy-plane) of three states of the skyrmion transistor: initial, off, and on. (C) Micromagnetic simulations showing the multiplexer-like function. (A and B) From X. Zhang, Y. Zhou, M. Ezawa, G.P. Zhao, W. Zhao, Magnetic skyrmion transistor: skyrmion motion in a voltage-gated nanotrack, Sci. Rep. 5 (2015) 11369. CC BY. (C) From P. Upadhyaya, G. Yu, P.K. Amiri, K.L. Wang, Electric-field guiding of magnetic skyrmions, Phys. Rev. B 92 (13) (2015) 134411. ©2015 American Physical Society.

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13.6

383

Electric field control of skyrmion for future applications

In the previous section, we have shown that electric field can be used to control the motion of skyrmions in a transistor-like functional device owing to the VCMA effect. The electric field is promising for future spintronic applications because it produces zero or negligible Joule heating and could result in a low-energy consumption [43, 47–49, 110–114]. Here, we continue to discuss the effect of pure electric field on the skyrmion dynamics. Magnetic skyrmions can be also controlled by harnessing the magnetoelectric coupling effect in magnetoelectric materials. For example, Mochizuki and Watanabe [44] theoretically proposed in 2015 that isolated magnetic skyrmions can be created by locally applying a pure electric field in a multiferroic thin film. In 2018, the transition between the skyrmion lattice phase and conical phase controlled by the electric field was experimentally realized in magnetic insulator Cu2OSeO3 [43]. On the other hand, it is also possible to control or manipulate magnetic skyrmions in common ferromagnetic materials by a pure electric field, usually based on the voltage-induced magnetic anisotropy change, or the so-called VCMA effect [106, 107]. In 2017, by using a scanning tunneling microscope, Hsu et al. demonstrated that a local electric field can be used to induce reversible transition between a skyrmion and the ferromagnetic state in the Fe triple layer on Ir(111) [47]. The experiments carried out by Hsu et al. suggest that one can write and delete magnetic skyrmions by using a local electric field, which relies on the electric-field-induced change of magnetic anisotropy. Indeed, in 2017, Schott et al. [48] experimentally realized the creation and annihilation of skyrmion bubbles controlled by electric field in a Pt/Co/oxide trilayer at room temperature, as shown in Fig. 13.11A. The electric field control of the skyrmion chirality was also experimentally demonstrated by Srivastava et al. in 2018 [49], as shown in Fig. 13.11B. Such a phenomenon was realized through the electric-field-induced change of Dzyaloshinskii-Moriya interaction. In 2019, Ma et al. [50] experimentally demonstrated in a Pt/CoNi/Pt/CoNi/Pt multilayer, which has a slope in thickness, that skyrmion bubbles can be created and driven into motion by applying an increasing electric field, as shown in Fig. 13.11C. In contrast to the skyrmions created in flat thin films or multilayers, in such a wedge-shaped asymmetric multilayer, the skyrmion bubbles will be annihilated when the electric field is removed. On the other hand, in addition to create, delete, and drive skyrmions using electric field, it is also possible to manipulate the Brownian motion of skyrmions or skyrmion bubbles by applying a pure electric field, which has been experimentally demonstrated by Nozaki et al. [52] in a W/FeB/ Ir/MgO multilayer structure. In the future, it is envisioned that the pure electric field will be used in skyrmion-based spintronic applications, where creation, deleting, and motion of skyrmions are induced and controlled by pure electric fields. In addition, as discussed in Section 13.5, electric field can also be used to assist the current-driven operations of magnetic skyrmions, leading to the implementation of skyrmion-based devices that can perform advanced information processing, such as the skyrmion-based transistor-like devices.

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Fig. 13.11 Electric field control of skyrmions. (A) Creation and annihilation of skyrmion bubbles controlled by electric field. (B) Electric field control of the skyrmion chirality. (C) Electric-field-induced creation and motion of skyrmion bubbles in a wedge-shaped multilayer. (A) From M. Schott, A. Bernand-Mantel, L. Ranno, S. Pizzini, J. Vogel, H. Bea, et al., The skyrmion switch: turning magnetic skyrmion bubbles on and off with an electric field, Nano Lett. 17 (5) (2017) 3006–3012. © 2017 American Chemical Society. (B) From T. Srivastava, M. Schott, R. Juge, V. Krizˇa´kova´, M. Belmeguenai, Y. Roussigne, et al., Large-voltage tuning of dzyaloshinskii–moriya interactions: a route toward dynamic control of skyrmion chirality, Nano Lett. 18 (8) (2018) 4871–4877. © 2018 American Chemical Society. (C) From C. Ma, X. Zhang, J. Xia, M. Ezawa, W. Jiang, T. Ono, et al., Electric field-induced creation and directional motion of domain walls and skyrmion bubbles, Nano Lett. 19 (1) (2019) 353–361. © 2018 American Chemical Society.

13.7

Summary

In this chapter, we have introduced and reviewed several typical examples of conventional skyrmion-based applications, including skyrmion-based racetrack memory, skyrmion-based logic computing gates, and skyrmion-based nano-oscillator, and skyrmion-based transistor-like device. We also reviewed recent progress on the electric field control of magnetic skyrmions, which may lead to more energy efficient spintronic devices based on skyrmions. We have shown that skyrmions can be used to carry binary information in both traditional ferromagnetic and a more advanced synthetic antiferromagnetic racetrack-type memories, where skyrmions can be driven into motion by either spin-transfer or spin-orbit torques. In particular, in the synthetic antiferromagnetic bilayer structure, skyrmions can strictly move along the driving

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force direction without showing the skyrmion Hall effect. Note that the skyrmion Hall effect may lead to the destruction of skyrmions at racetrack edges and thus may limit the maximum speed of skyrmions. Due to the nanoscale size and topological nature of skyrmions, the racetrack memory based on perfect compact skyrmions may have increased data storage density and low critical depinning current density. Also, we have shown that there are different methods to encode binary information in the skyrmion-based racetrack memory, for example, a pair of skyrmions with opposite polarities can be used as a basic binary bit carrier in the two-lane racetrack memory, which is a more reliable encoding method compared with using a skyrmion and the ferromagnetic state. In addition, the applications of magnetic skyrmions in the logic computing devices and nano-oscillators have been overviewed. We have shown that skyrmions are versatile objects that can be used for both information storage and computing and signal generation. Both the skyrmion-based logic gates and nanooscillators can be made of ferromagnetic and antiferromagnetic materials. Notably, the antiferromagnetic skyrmion-based nano-oscillator promises a better performance. In addition, we have reviewed the skyrmion-based transistor-like functional devices, in which skyrmions are driven by the electric current but controlled by the electric field. The electric field is a promising method to control or manipulate the dynamics of magnetic skyrmions, which produces zero Joule heating and consumes lower energy. Hence, we provided an overview of recent trends and advances in creating, driving, and manipulating skyrmions and skyrmion bubbles using pure electric field. All theoretical and experimental works on the skyrmion-based applications demonstrate a great potential that skyrmions can be used in future electronic and spintronic circuits and lead to novel devices with improved performance and efficiency.

Acknowledgments X.Z. acknowledges the support by the National Natural Science Foundation of China (Grant No. 12004320), and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110713). M.E. acknowledges the support from the Grants-in-Aid for Scientific Research from JSPS KAKENHI (Grant Nos. JP18H03676, JP17K05490 and JP15H05854) and the support from Core Research for Evolutional Science and Technology, JST (Grant Nos. JPMJCR16F1 and JPMJCR1874). X.L. acknowledges the support by the Grants-in-Aid for Scientific Research from JSPS KAKENHI (Grant Nos. JP17K19074, 26600041, and 22360122). Y.Z. acknowledges the support by Guangdong Special Support Project (Grant No. 2019BT02X030), Shenzhen Peacock Group Plan (Grant No. KQTD20180413181702403), Pearl River Recruitment Program of Talents (Grant No. 2017GC010293), and National Natural Science Foundation of China (Grant Nos. 11974298 and 61961136006).

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14

Wang Kang, Sai Li, Xing Chen, Daoqian Zhu, Xueying Zhang, Na Lei, and Weisheng Zhao Fert Beijing Research Institute, BDBC, and School of Microelectronics, Beihang University, Beijing, People’s Republic of China

With the rapid research and development of skyrmionics in both theory and experiment, in particular the successful demonstration of the electrical functions, e.g., generation, motion, detection, and manipulation, a variety of skyrmionic devices and applications have been proposed recently, such as racetrack memory, neuromorphic computing, probabilistic computing, and reservoir computing (RC). This chapter focuses on introducing unconventional applications of skyrmions through exploiting the unique characteristics that may be inaccessible to conventional electronic devices. These applications could bring large density storage and high energy-efficient computing.

14.1

Skyrmion-based memristors and neuromorphic computing

14.1.1 Introduction to memristor and neuromorphic computing In conventional computing systems based on the von Neumann architecture, memory and processor are spatially separated and much of energy is consumed for transferring data between the memory and the processor, resulting in “von Neumann bottleneck” [1], which limits the processing performance and energy efficiency. Inspired by the biological nervous system, neuromorphic computing has attracted considerable attention and has shown great potential in replacing or complementing conventional computing platforms in the big-data era. In the human brains, the biological neural network is composed of a large number of neurons massively connected by an even larger number of synapses, schematically shown in Fig. 14.1. Synapses have the ability to reconfigure the strength with which they connect two neurons according to the past electrical activity of these neurons, which is called the synaptic plasticity [3]. This phenomenon allows memories to be formed and stored, and neural networks can learn and adapt to a changing environment. In the past few years, artificial neural networks have attracted worldwide attention and attempted to mimic the behaviors of biological synapses and neurons and to realize the cognition-and perception-related tasks with high energy efficiency. Magnetic Skyrmions and Their Applications. https://doi.org/10.1016/B978-0-12-820815-1.00004-3 Copyright © 2021 Elsevier Ltd. All rights reserved.

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Biological neural network

I1

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Synapse Neuron Fig. 14.1 Illustration of a biological neural network composed of a large number of neurons massively connected by an even larger number of synapses [2]. Copyright 2017, IOP Publishing Group.

In recent decades, a variety of nano-electronic devices have been developed to emulate synapses, including phase change memories, spintronic memories, and memristors/resistive memories. Among them, memristors are considered as the best solution to realize such plastic nano-synapses. Memristor is a type of fundamental passive circuit element that complements resistor, capacitor, and inductor [4]. It was first proposed by Leon Chua in 1971 [5]. The memristor maintains a relationship between the time integrals of current and voltage across a two-terminal element. In fact, its resistance depends on the charge that flowed through the circuit. When current flows in one direction, the resistance increases; in contrast, when the current flows in the opposite direction, the resistance decreases [6]. This property is similar to the behavior of synapse and the application of memristors in neuromorphic computing has been widely studied since its discovery. However, it is not until 2008 that the first experimental evidence involving a TiO2/TiO2
x memristor was observed [7]. Until now, most memristors are built using simple capacitor like structures consisting of electrodes and a switching layer. Memristive switching layers are generally comprised of dielectrics and the dielectrics are mainly binary oxide materials [8]. However, a device that fulfils all the following requirements, such as nanoscale size, long write/erase endurance, long retention, nanosecond switching speed, and low programming energy, has not been found [9]. In particular, the relatively short endurance of traditional memristors hinders them to be applied on neural networks with “on-chip” training. The emergence of magnetic skyrmions provides a new opportunity for the application of spintronic devices in neuromorphic computing. Recently, varieties of

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skyrmion-based devices and prototypes, including skyrmionic memristor, devoted to neuromorphic computing, have been proposed.

14.1.2 Application of skyrmionic memristor in neuromorphic computing In a biological neuro system, the information communication between neurons and synapses is realized by neurotransmitters. This particle-like substance can be released after an action potential has reached the terminal of one neuron (referred to as presynapse). Then, the neurotransmitter crosses the synaptic gap and attaches to the postsynapse, either potentiate or depress the receiving neurons depending on the type of neurotransmitter. Analogously, skyrmions are particle-like spin textures with topological stability that carry spin angular momentums. They can be generated in one region of devices, moved by a sequence of pulses or “spikes.” When receipted by the target region, skyrmion can change the local state, for example, changing the topologic number or the local magnetization, thereafter changing the spin-related properties of the device such as the tunneling magnetoresistance (MR). In this way, the synaptic weight of the device can be strengthened/weakened by positive/negative stimuli, mimicking the potentiation/depression process of a biological synapse [10]. Therefore, skyrmion can indeed function as information carriers like neurotransmitters in artificial neural networks. Our group has proposed a skyrmion-based device with synaptic plasticity [10], as shown in Fig. 14.2. The main body of the device is a ferromagnetic (FM) layer/heavy metal (HM) nanotrack with perpendicular anisotropy (PMA) and interfacial-induced Dzyaloshinskii-Moriya interactions (DMIs), which serves as the holder and transport channel of skyrmions. A gated barrier at the center of the nanotrack is designed to separate the entire nanotrack into two regions, referred to as the pre- and postsynapse regions. The postsynapse region is covered by a magnetic tunnel junction (MTJ), via which the change of the number of skyrmions in the postsynapse can be transformed into the resistance signal of the device. A high tunnel MR is highly preferred for highsensitivity skyrmion detection [11]. Fig. 14.2C illustrates the operation principle of the device. During the learning phase, a bidirectional charge current flows through the HM, injecting a vertical spin current into the FM layer from terminal A to terminal B (or vice versa) and driving skyrmions into (or out of) the postsynapse region to increase (or decrease) the synaptic weight, mimicking the potentiation/depression process of a biological synapse. In the spike transmission mode of operation, the spike produced by a preneuron is modulated by the weight (MR) of the synaptic device, which generates a postsynaptic spike current from terminal C to terminal B. Both short-term plasticity and long-term potentiation functionalities have been demonstrated for a spiking time-dependent plasticity scheme based on this device. Recently, an experiment has demonstrated such an electrically operating skyrmion-based artificial synaptic device for neuromorphic computing, and controlled current-induced creation, motion, detection, and deletion of skyrmions in ferrimagnetic multilayers has been be harnessed at room temperature to imitate the behaviors of biological synapses [12].

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Fig. 14.2 Schematic of (A) a biological synapse and (B) our proposed skyrmionic synaptic device. To mimic a neuromodulator, a bidirectional learning stimulus flowing through the heavy metal from terminal A to terminal B (or vice versa) drives skyrmions into (or out of) the postsynapse region to increase (or decrease) the synaptic weight, as shown in (C), mimicking the potentiation/depression process of a biological synapse 3. A detection device at terminal C measures the weight of the synaptic device via the magnetoresistance effect, which also modulates the post-synaptic spike current (depending on the weight of the synaptic device) [10]. Copyright 2017, IOP Publishing Group.

In addition to emulating the behavior of a synapse, skyrmionic devices can also be designed to mimic the behavior of neurons. In this field, the leaky-integrate-fire (LIF) model has been widely adopted for emulating a spiking neuron, which can be briefly described as follows: when a neuron receives spikes from other connected neurons, the membrane potential of the neuron will accumulate with a leaky-integrate process. Once the membrane potential reaches a definite threshold value, the neuron will fire an output spike and then reset. Such a LIF neuronal behavior can be practically described by the skyrmion motion dynamics on a nanotrack under a driving electric current with the moving distance of the skyrmion indicating the membrane potential of biological neurons. We have proposed such a skyrmion-based artificial spiking neuron device [2], the schematic of which is given in Fig. 14.3. The primary component of the device is an FM/HM-layered racetrack with interfacial PMA and DMI, which can hold skyrmions. The thickness of the FM layer is tuned so that a linear PMA along the nanotrack is obtained. Although a constant DMI value is considered in this work, a skyrmion generation unit is located at one of the ends and a skyrmion detection unit is placed at a location with a given threshold distance on the nanotrack. If one or several preneurons spike, the accumulated spike current drives the skyrmion motion along the nanotrack because of the spin-orbit torque (SOT). On the other hand, the energy barrier of the nanotrack gradually grows because of the linear increase of the PMA energy. If the amplitude or (and) the frequency of the accumulated spike

Unconventional applications of skyrmions

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Fig. 14.3 (A) Schematic of the proposed skyrmion-based artificial neuron device; (B) The front cross-section view; (C) The linear increase of the PMA value along the nanotrack [2]. Copyright 2017, IOP Publishing Group.

current is large enough to overcome the gradually increased PMA energy, the skyrmion will move forward; otherwise, the skyrmion will move backward. When the skyrmion moves into the detection area and the neuron “fires” an output spike, the neuron is reset and the skyrmion is driven backward to the origin of the nanotrack with a reset current. This current-driven skyrmion motion and reset behavior is analogous to the LIF neuronal model. A similar prototype of the skyrmionic neuron device to mimic the LIF functionality has also been proposed based on a wedge-shaped nanotrack [13]. In this device, the state of the back and forward behavior of skyrmion is determined by the competition between the current-induced force and the edgeinduced repulsive force. Except for emulating the behavior of synapse or neuron using the displacement of skyrmions, another strategy to realize neural functionality is creating memristive devices relying on the generation and annihilation of skyrmions, for example, the combination of skyrmions and MTJs, as shown in Fig. 14.4. In fact, the free layer of a perpendicular MTJ can serve as the holder of skyrmions. The general composite of the tunneling barrier layer/free layer/capping layer of an MTJ is MgO/FM/HM, a structure that can generate considerable DMIs. Our research has proved that the DMI in an MgO/FM/HM structure can be tuned through the thickness of the MgO [15]. In particular, the strength of DMIs is sensitive to the thickness of MgO at the value around 0.8 nm, which is a commonly adopted thickness for the tunneling layer of MTJ. The generation and annihilation of skyrmions can be achieved by the spin-transfer torque (STT) produced by the current through the MTJ. The change

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Pre-neuron

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CL FL BL SPL RL HL

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Fig. 14.4 Analogy between the biological synapse and the electronic ternary synapse based on the skyrmionic MTJ device [14]. Copyright 2019, IEEE Publishing Group.

of skyrmion number in the free layer of the MTJ can be read out by the tunneling MR. Thus, the combination of skyrmions and MTJs can create a device with multilevel resistance and the resistance can be increased or decreased via the current flowing through the junction. This kind of device has the characteristics of memristors and can be used for neuromorphic computing. Our previous works have theoretically studied the creation of skyrmions in MTJs [13]. It is found that the properties of skyrmions in the free layer of MTJs is related to the strength of DMIs and can be changed by the spin-current pulse. In particular, the stray field from the pinned layer can be used to assist the generation and stabilization of the skyrmions. Benefiting from this, the operation current of the skyrmionic MTJ can be reduced to a low value. Thereafter, our work proved that this device can also be used in the implementation of the ternary neural networks [14]. Circuit-level simulations for image recognition tasks were conducted in this study. The recognition rate can reach up to 99% with 5% device variation and average power consumption of 29.23 μW. Recently, some other interesting results have been published on skyrmionic memristors or skyrmionic neuromorphic computing. For example, Luo et al. proposed a voltage-controlled skyrmion memristor based on a multiferroic heterostructure, which is announced to be more energy efficient [16]. Under electric-field-modulated magnetic anisotropy via remnant strain, continuously tunable resistance can be obtained due to the skyrmion size modulation in the FM layer. The readout resistance values can be efficiently modulated by applying a voltage as well as the assistance of STT. Yu et al. proposed a skyrmion-based nanodevice composed of a synthetic antiferromagnet and a piezoelectric substrate. In this device, the interlayer anti-FM coupling can be manipulated under a weak electric field, giving rise to a continuous transition between a large skyrmion bubble and a small skyrmion, thus inducing the variation of the resistance of an MTJ that can mimic the potentiation/depression of a synapse and the LIF function of a neuron. The cost of energy consumption is as low as 0.3 fJ per operation [17]. According to the above studies, the energy consumption per spike based on the skyrmionic devices is far less than that of silicon-based spiking neurons. An improvement in energy consumption by two orders of magnitude as compared with 45-nm

Unconventional applications of skyrmions

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technology complementary metal oxide semiconductor implementation have been proved [18]. Even compared with other spintronic neuromorphic devices, e.g., domain wall (DW)-based devices, skyrmionic devices are still much more energetically efficient, considering the existence of defects in the devices [19,20].

14.2

Skyrmion for stochastic computing

This section introduces the skyrmion-based stochastic computing (SC) devices. Recent studies show that thermally induced skyrmion motion dynamics provides a new dimension for unconventional computing applications, as they have been demonstrated to be useful for probabilistic/SC tasks. The thermal-induced random walk of skyrmion motion around room temperature has recently been reported theoretically and experimentally [21–24], and great focus is given to the use of skyrmions as basic elements in the design of SC devices. Instead of using numerical values encoded as conventional binary format, SC processes random bit-streams, whose information contains in the probability of appearing a 1 or 0 in a random binary sequence. For example, a bit-stream containing 25% 1s and 75% 0s represents the number P ¼ .25, known as “P” value, denoting that the probability of observing a 1 at an arbitrary bit position is p. The main attraction of SC is that it enables low-cost implementations of arithmetic operations using standard logic elements. For example, a multiplication of two input bit-streams p1 and p2 can be performed by a stochastic circuit consisting of a single AND gate, where the output is p1  p2 (see Fig. 14.5A). Another outstanding feature of SC is its inherent error tolerance. With a single bit changed in a long bit-stream, the value of stochastic number only alters a little, whereas it could result in huge error if it is represented in the conventional format [25, 26]. It should be noted that it is important to have the two bitstreams suitably uncorrelated or independent in SC. An extreme case here is shown in Fig. 14.5B, where two identical bit-streams denoting the value p applied to an AND gate will result in an output of p, rather than the correct one p2. Therefore, it is crucial to be able to reshuffle the input signals to keep them uncorrelated before doing logic operations. However, in conventional circuits, reshuffling signals produced by combining a pseudorandom number generator with a shift register requires long-term memory and consumes a lot of energy. In this context, an alternative reshuffler design as the first low-energy, compact device proposal of its kind, the skyrmion reshuffle, was devised by Pinna et al. in 2018 to overcome this obstacle [25]. They proposed that by prefixing such devices to each gate in a circuit, inconvenient correlations can be washed out effectively (see Fig. 14.5C). Fig. 14.6. shows the working concept of the skyrmion reshuffle device, which consists of two circular chambers with input-output conduit tracks capable of ushering skyrmions into and out of the chambers. The net drift of skyrmions is achieved by a static current flowing across the entire structure from one conduit to the other. The up-and-down (binary) states of the bit-stream are used to select which chamber to inject the generated skyrmions into. Skyrmions are generated at a constant rate onto the input conduit of one of the two chambers (up and down) depending on the input signal’s state. As a consequence, among the total population of N generated

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(A)

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(C) Fig. 14.5 AND gate used as multiplication operation: (A) The P value of the output is the exact product of input signals. (B) The output result is not the expected value because there are correlations in the two input signals. (C) The output value is an approximate result of the exact one if the input signals pass through a reshuffler [25]. Copyright 2018, American Physical Society.

Skyrmion

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Fig. 14.6 The skyrmion reshuffle device consisting of two magnetic chambers. The input signals are injected to the selected chamber depending on the state of the input signal. Skyrmions in the chambers are driven by a constant current to the exit, where the output order is different from the input signals because of the thermal diffusion [25]. Copyright 2018, American Physical Society.

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skyrmions, the populations in the up chamber and the below chamber are Nup ¼ pN and Ndown ¼ (1
p)N, respectively. To ensure the p value of the output from the chamber is identical to that of the input signal, whenever an outgoing skyrmion is read out from the up (down) chamber, the outgoing signal into an up (down) state is switched until all of the skyrmions are read out from the chambers. Through this brilliant design of the complementary structure, where the interplay between skyrmion particle stability and thermal diffusivity is fully leveraged for efficient reshuffling of input signals, this skyrmion reshuffle could help tackle the long-standing problem in the field of SC, where computations defined over stochastic signals require to be uncorrelated. Based on this innovative exploration for efficient SC, the thermally excited skyrmion diffusion was then experimentally observed and was further used to construct a signal reshuffler device as part of a skyrmion-based SC by Za´zvorka et al. in 2019 [27]. Fig. 14.7 shows an example of several typical skyrmion trajectories without any external excitations. By evaluating the mean squared displacement (MSD) of the measured skyrmions, it is concluded that skyrmions exhibit diffusive dynamics, which is relevant at room temperature on macroscopic timescales. Fig. 14.8 presents the skyrmion reshuffle device operation. The proof-of-concept reshuffler device successfully demonstrates the creation of uncorrelated signal streams by means of thermal skyrmion motion dynamics, which lays the foundation for skyrmion-based system to implement SC. Another important issue regarding SC is efficient generation of true random numbers. The conventional circuits-based, e.g., oscillator sampling or directly noise amplifying true random number generators (TRNGs) usually requires a large area and has a large power consumption, whereas the quantum physics-based, e.g., photoelectric effect is more difficult to implement and usually requires additional postprocessing circuitry. To address this problem, Yao et al. proposed a TRNG based on continuous skyrmion thermal Brownian motion in a confined geometry [28].

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Fig. 14.7 Trajectories of selected skyrmions at 296 K. The inset shows the time-averaged MSD (black line) and the linear fit of the data (red dashed line) [27]. Copyright 2019, Nature Publishing Group.

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Fig. 14.8 Skyrmion reshuffler device. (A) Reshuffler operation with skyrmion nucleation by a direct current. The input signal is constructed as a time frame in which the skyrmion crosses the blue threshold line, whereas the output is determined on crossing the orange line. (B) The corresponding input (blue) and reshuffled output signals (orange) [27]. Copyright 2019, Nature Publishing Group.

(A)

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Fig. 14.9 (A) The schematic structure of the skyrmion-based TRNG: the ferromagnetic layer is used for skyrmion Brownian motion and the two MTJs are used to detect the skyrmion. (B) The top view of the FM chamber. (C) The schematic of the comparator. It is assigned a bit of “0” if the skyrmion topological is on the left; otherwise, a bit of “1” is indicated [28].

Unconventional applications of skyrmions

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Fig. 14.10 Random bit sequence generated by skyrmion Brownian motion. (A) The motion trajectory of the skyrmion in the chamber. (B) A random bit sequence obtained by detecting the relative position of skyrmion in the x-axis. (C) Outputs of selected 50 bits [28].

The structure consists of a rectangular region with two semicircular regions, as show in Fig. 14.9. Under the influence of the thermal fluctuations, the skyrmion exhibits random motion dynamics, the output bits are then assigned depending on the relative position of skyrmion inside the chamber, which could be determined through the differential voltage of the two detection MTJs. The readout sequence, displayed in Fig. 14.10, is proven to be nonrepeatable and unpredictable, which is evaluated by the National Institute of Standards and Technology suite. The authors further implemented a probability-adjustable TRNG, in which a desired ratio of “0” and “1” can be acquired by adding an anisotropy gradient through voltage-controlled magnetic anisotropy (VCMA) effect. This work provides a new perspective to implement efficient TRNG with small area and low power consumption for SC.

14.3

Skyrmion for reservoir computing

In this section, we first address the concept of RC to develop a baseline for understanding the need to exploit spintronic device, especially skyrmionic device, as a physical implementation. Then, we introduce the recent studies of skyrmionic device designs and their applications to RC. Finally, we explore the prospects of using dynamic process of skyrmion to build reservoir, highlighting the potential to solve spatial-temporal problems.

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14.3.1 Principle of RC Owing to the improvement of the computing capacity of central processing units and graphics processing units, deep learning has been investigated explosively by many scientific communities and commercial companies to solve classification, recognition, and prediction tasks with the rapid development of algorithms. However, the energy efficiency is far bigger than that of a human brain, limited by the hardware. As we know, a human brain requires power consumption of only 20 W, far less than the 250-W expense of a standard computer for recognition tasks [29]. Therefore, bioinspired computing, rooting in the emulation of the biological neural structure and brain operation, has captured great attention in both academic and industry communities [30]. Among diverse bio-inspired computing paradigms, RC has recently been introduced for its unique characteristics and simple implementation, especially applicable to solve the temporal/sequential data problems in the real-word tasks [32]. RC is mainly derived from two recurrent neural network (RNN) models independently: echo state networks [33] and liquid-state machines [34]. The most important characteristics of RC is that a randomly created medium called reservoir could transform the input data into spatiotemporal patterns in a high-dimensional feature space, then be readout by a pattern analysis. As shown in Fig. 14.11, the input weights (Win) and the weights within the reservoir (W, namely, the connectivity structure of the reservoir) are fixed, instead of trained, only the readout weights (Wout) connecting the reservoir and the output layer need to be trained with a simple learning algorithm such as linear regression [31].

(A)

(B) Fig. 14.11 Comparative concepts of (A) RNN and (B) RC [31]. Copyright 2018, American Physical Society.

Unconventional applications of skyrmions

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The reduced training load enables RC to outperform classical fully trained RNNs in lots of tasks within flexible task-dependent reservoir adaptation. The advantages of RC compared with other neural networks include [35]: (1) The notable modeling accuracy in the identification, prediction, and classification of nonlinear system, e.g., in predicting chaotic dynamics [36]; (2) The modeling capacity allows RC to generally apply in continuous-time, continuous-value real-time systems in possession of time and value resolution [37, 38]; (3) The biological plausibility is reflected in that the interconnected nodes of reservoir resemble the architectural and dynamical properties of brains [39, 40]; (4) Extensibility. RC provides a solution to the main challenge of neural network research in ensuring that previously learned models could be extended by fully utilizing previously learned results rather than impairing or destroying them. The separate output weights make it simple that new items can be appended to the established reservoir as new output units [41].

14.3.2 Skyrmionic RC To process temporal information, the reservoir operates as an intermediate to map sequential inputs into a new high-dimensional space within a short-term memory [42]. It is obvious that the internal topology of reservoir has a dramatic impact on the efficiency, say, the complexity of learning algorithm of training on output weights. Although randomly creating a fixed reservoir is simple and is considered the key benefit of RC when it is first proposed, altering the reservoir design to improve performance is more in the spotlight when addressing a given modeling task [43]. Recently, the research focus is emphasized on understanding the role of reservoir characteristics on task performance and exploring suitable reservoir design ranging from electronics, photonics, mechanics, biology to spintronics. Several studies have demonstrated the physical implementation of RC with nanodevices, such as memristors, atomic switches, and spintronic devices. The special outpouring of concern for the physical RC is fast information processing speed and low learning cost, which can be realized by the intrinsic physical phenomena of the device. Among different technologies, spintronic devices are the most promising candidates for the reason that the harness of electron’s spin can decrease or even eliminate the Joule heating [44–49]. At the time of writing, a plenty of spintronic reservoir devices has been proposed to implement RC, such as spin-torque nano-oscillators-based RC devices [45, 46], MTJ-based RC devices [31], spin wave-based RC devices [47], and skyrmionbased RC devices [48, 49]. Here, we focus on the last one. Magnetic skyrmion is extremely suitable for RC design as a spiking node served in the reservoir layer. Prychynenko et al. first proposed a skyrmionic RC structure, a twoterminal device with skyrmion embedded in magnetic films as shown in Fig. 14.12, where the pinning-induced nonlinear voltage characteristics of skyrmion originating from anisotropic MR generates a nonlinear resistive effect [48]. As shown in Fig. 14.12A, the skyrmion is initially pinned at the center of the device, then a voltage is applied between the two contacts, which are embedded on opposite sides of the film and lead to a characteristic current flow calculated based on the anisotropic magnetoresistance (AMR). The output is to read out the resistance

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(A)

(B)

Fig. 14.12 (A) Schematic of the skyrmion-based RC device. A voltage U is applied between two contacts of the ferromagnetic thin film. A potential pinning center is located in the center of the film; (B) Resistance of a pinned skyrmion as a function of an applied voltage visualizing the nonlinear behavior of the device [48]. Copyright 2018, American Physical Society.

varying with the shape, type, relative size, and position of the skyrmion. Because of the combination of pinning force and the STT force, the skyrmion will be deformed or stretched when the applied voltage is smaller than the critical depinning threshold. Therefore, the self-consistent current j [U, m] can be calculated according to the AMR effect, where the current density through the conductivity tensor σ[m] could be written as, j ½U, m ¼
σ ½m  E½U , where E is the electric field produced by the input voltage. It drives the skyrmion to approach one of the contacts and eventually disappear. The nonlinear current-voltage characteristics can be described by the equation dI/dU ¼ G[U, m] ¼ R
1[U, m], where G(R) represents the differential conductivity (resistance) function, depending on the voltage U and the magnetic configuration m. The relative resistance of the RC device as a function of the applied voltage is given in Fig. 14.12B, fully displaying its nonlinear current-voltage features. It is worth mentioning that the random thermal fluctuations may even help enhance the nonlinear signals, which could turn the temperature from a negative effect on traditional device, e.g., racetrack memory or logic gate, to a positive effect on the performance of the RC systems. The resistance rises when the skyrmion nears the contact, enabling the current density being largest there, as can be seen in Fig. 14.13. Afterward, the skyrmion will be absorbed, thus the resistance becomes the same value as the outof-plane FM configuration without a skyrmion. It should be stressed that the specific type of MR (AMR or noncollinear magnetoresistance) or the structure type of skyrmion (Neel or Bloch) is not so crucial in this RC device design and the analysis results concerning the RC function could be grafted when the functional relationship between the current and voltage for the particular type of MR is nonlinear. Skyrmion-based reservoir could offer some potential advantages: (1) small size, because skyrmions have been reported to be able to downscale to even 1 nm; (2) low power consumption, roughly 1 μW is obtained in the report; (3) low complexity, many internal degrees of freedom of skyrmion system is due to its interaction with currents and magnons. (4) high tunability, the current forces

Unconventional applications of skyrmions

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Fig. 14.13 Relative resistance dependence (R
R0)/R0 on time for Neel-type and Bloch-type skyrmions at a fixed voltage [48]. Copyright 2018, American Physical Society.

competing with the applied bias fields could make the skyrmion-based RC owe more adjustability. (5) homeostatic operation, additional thermal effects and bias fields can be applied to maintain homeostatic operating points. Another skyrmion-based RC study expanding on Ref. [48] proposes to use a two-terminal nanotrack device embedded with skyrmion fabrics, detected by resulting AMR-mediated current flow [49]. The more complex magnetic texture, compared with a single Neel or Bloch skyrmion, could generate nonlinear current flow responding to different applied voltages, reminiscent of the recursive connectivity demanded by the reservoir network. One advantage of this device is that the rich characteristics of the skyrmion fabrics can be tuned by optimizing the material properties or applying external field, making more nonlinearity of the reservoir’s dynamics for RC implementation.

14.4

Skyrmions for Boolean logic computing gates

This section introduces the skyrmion-based Boolean logic computing gates by using the skyrmion dynamics with unique behaviors, which are different from conventional semiconductor-based Boolean logic gates and may be exploited for unconventional computing applications. The work done by Zhang et al. [50], which proposed to implement Boolean logic gates using skyrmion-DW conversion, has stimulated the optimization of the skyrmion-based logic gates and the exploration of other forms of skyrmion-based unconventional computing. In 2018, Luo et al. presented by micromagnetic simulations an all-skyrmion logic gate that can realize complete Boolean logic operations, such as AND, OR, and XOR [51]. Interestingly, the conversion between skyrmions and DWs is not necessary in their scheme, which may largely simplify the design

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of skyrmionics logic gates and reduce the power consumption and area footprint. In contrast, several effects that are usually supposed to be suppressed, especially in the design of skyrmion racetrack memory, such as the skyrmion Hall effect and the interactions between skyrmions, play a critical role during the logic operations. Fig. 14.14A–D shows the implementation process of their skyrmionic AND/OR logic gates upon different inputs. The skyrmion trajectories are indicated by the white dashed lines. If there is only one skyrmion injected from the bottom input terminal, as shown in Fig. 14.14B, the skyrmion driven by SOT from the HM (not shown) underneath the FM layer, will move toward the top output port because of the skyrmion Hall effect. If this skyrmion is generated at another terminal, as shown in Fig. 14.14C, again, it will eventually enter the same port under the joint effect of the skyrmion Hall effect and the repulsion from the upper edge. Note that the notch at the upper edge is used to modulate the skyrmion motion. It seems obvious in this case but becomes critical when there are two input skyrmions. As can be seen from Fig. 14.14D, these two skyrmions meet at the cross. As a consequence, the skyrmion coming from the bottom input terminal will be pushed to the bottom output port due to the repulsion force, which is strong enough after the introduction of the notch, from the other skyrmion. With the VCMA effect, reconfigurable logic gates can be further achieved. Because skyrmions can be trapped by energy barriers formed at the VCMA region

Fig. 14.14 (A–D) The implementation process of a skyrmionic AND/OR logic gate upon different inputs. (E) The schematic of a reconfigurable logic gate using VCMA effect and the corresponding output table [51]. Copyright 2018, American Chemical Society.

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when a proper voltage is applied, the voltage actually functions as a switch, offering extra degree of freedom to control the logic functionality. As shown in Fig. 14.14E, two VCMA voltages, Vk1 and Vk2, are introduced in the AND/OR logic gate, thus determining whether the output is an AND or OR operation of the inputs. Another voltage VM, which is applied at the anti-FM layer of the detection MTJ, is used to switch the magnetization of the top FM layer. Therefore, the detection MTJ, together with VM, realizes a controllable NOT gate. With these control voltages, AND/NAND and OR/NOR can be achieved in this reconfigurable logic gate. This concept of all-skyrmion logic gates was also adopted by Chauwin et al. [52]. The way they implemented AND/OR gates was nearly the same as that of Luo et al., except for the different geometry used. In addition, they also designed a NOT/COPY gate, as presented in Fig. 14.15, by virtue of the skyrmion Hall effect and skyrmionskyrmion interaction. A skyrmion is always provided in the control input terminal, whose trajectory will be altered by the presence of another skyrmion from the terminal IN. Therefore, the output ports COPY1 and COPY2 can obtain a duplication of the input signal while the middle port operates as a NOT output. Based on the proposed elementary skyrmionic logic gates, Chauwin et al. further demonstrated a cascaded one-bit full adder. Fig. 14.16 exhibits the timing sequence and the corresponding skyrmion motion trajectories. An SOT current (JSOT) with density 0.5  1011 A/m2 is applied during the logic operations, whereas a 150-ps-wide SOT pulses with an amplitude of 2  1011 A/m2 and a period of 5 ns is used for signal synchronization. The synchronization is achieved by placing suitable notches along the track, as shown in Fig. 14.16B. Skyrmions will be impeded by the notches at JSOT ¼ 0.5  1011 A/m2 because of the repulsion force, but can safely escape from the traps at JSOT ¼ 2  1011

Fig. 14.15 (A and B) The implementation of a skyrmion NOT/COPY logic gate upon different inputs. The skyrmion trajectories are indicated by the black lines. The scale bar is 40 nm [52]. Copyright 2019, American Physical Society.

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Magnetic Skyrmions and Their Applications

J(1011 A/m2)

2 1.5 1 0.5 0

(A)

z 60 nm

A=1

y

t = 11.8 ns

t = 7.1 ns

t = 3.1 ns 0 t = 0.1ns

5

t (ns)

10

15 t = 11.8 ns

t = 3.1 ns

x AÅBÅCin

Cin = 1

B=1

AÅB

SUM

=1

CinÙAÅB B=1

Cin = 1

Cout = 1

A=1 t = 0.1ns

AÙB

(AÙB)Ú[CinÙ(AÅB)] t = 7.1 ns

(B) Fig. 14.16 (A) The timing sequence and (B) the skyrmion trajectories during the one-bit full adder simulation. The scale bar is 60 nm [52]. Copyright 2019, American Physical Society.

A/m2, thanks to their topological stability. At t ¼ 11.8 ns, the target output A  B  Cin is obtained, as denoted by the pink dashed box. Meanwhile, because their proposed logic gate is a conservative system, i.e., no skyrmions vanish during the process, byproducts like Cin ^ AB and (A ^ B) _ [Cin ^ (A  B)] can be expected and even used as the inputs for the computing of a next step. With the clocked structure, pipelined full adder was further demonstrated in their work. It is also worth noting that the authors claim for many times that their skyrmion logic gates are reversible, which may provide new perspectives in skyrmion-based quantum computing. As discussed in Section 14.2, SC can implement multiplier and adder with just simple Boolean logic gates. By using a skyrmionic AND gate as a stochastic multiplier and a skyrmion multiplexer as a stochastic adder, Zhang et al. investigated the possibility of implementing SC by skyrmionic logic devices [53]. Because the skyrmionic AND gate proposed by Luo et al. and Chauwin et al. requires precise control of skyrmion collision at the center cross, it may fail as the pervasive pinning sites can modify the skyrmion trajectories. Fortunately, recent studies have suggested that the skyrmion Hall angle is relevant to skyrmion velocity [54, 55]. Zhang et al. proposed another scheme that utilizes VCMA effect to design dynamic energy landscape, thus guiding the motion of skyrmions [53]. Fig. 14.17A is the top view of the proposed AND/OR gate with an HM layer underneath the FM layer to provide SOT current. Two VCMA regions are set in the device and their timing sequences are displayed in Fig. 14.17B. As shown in Fig. 14.17C and D, if only one skyrmion is injected from

Unconventional applications of skyrmions

411

In1

And_out

120 nm

10 nm

In0

Vg2

Vg1

Or_out

50 nm 10 nm

FM layer 200 nm

(A) t = 0 ns

10 MA·cm -²

JSOT

6 MA·cm -²

Vg1

2V 0V

Vg2

2V 0V

JSOT

3 ns 2 ns 3 ns

(B) t = 0 ns

t = 0 ns

mz +1 0 -1

(C)

t = 1 ns

t = 1 ns

t = 2 ns

t = 5 ns

t = 5 ns

t = 5 ns

t = 8 ns

t = 8 ns

t = 8 ns

(D)

(E)

Fig. 14.17 (A) The schematic of the proposed skyrmionic AND/OR gate. The yellow regions, Vg1 and Vg2, are used to dynamically modify the energy landscape using VCMA effect. (B) The timing sequence of the skyrmionic AND/OR gate. (C–E) The detailed implementation of the AND/OR gate under different inputs [53].

either input terminal In0 or In1, this skyrmion will finally enter Or_out port because a voltage of +2 V is applied on the region Vg1, hence preventing the skyrmion from entering the And_out port. When there are two input skyrmions, as shown in Fig. 14.17E, the first skyrmion from the input terminal will enter the Or_out port and stop near the Vg2 region, whereas the next skyrmion will be impeded by both the first skyrmion and Vg1 region. Then, at t ¼ 3 ns, the voltage on the Vg1 region is removed, thus the second skyrmion enters the And_out port and also stops near the Vg2 region at the upper track. After Vg2 is removed at t ¼ 5 ns, the two skyrmions export from the two output terminals. Therefore, by virtue of the VCMA effect, AND/OR gate can be realized in this device. Because of the introduction of VCMAcontrolled gates, the peripheral circuits will inevitably become more complicated. The design of a concise and robust skyrmion logic gate is still a pending question. Based on the two-input skyrmion AND gate, the computation of two 3-bit stochastic numbers,

412

Magnetic Skyrmions and Their Applications 20 MA·cm -²

In1

In0

And_out

J SOT

10 MA·cm -² 4.5 MA·cm -²

0,1,1,0,1,0,1,0 (P1=4/8)

0,0,1,0,1,0,1,0 (Pout=3/8) Vg1

2V 0V

1,0,1,1,1,0,1,1 (P0=6/8)

Vg2

2V 0V 3 ns 0.6 ns

(C) In1 In0

11 ns

3 ns 2 ns

19.6 ns One Cycle

AND OR

(B) t = 0 ns

120 nm

t = 10 ns t = 35 ns t = 168 ns

(C) Fig. 14.18 (A) The schematic of the computation of 4/8 and 6/8 using a skyrmionics AND gate. (B) The device structure and (C) the timing sequence of one clock cycle to implement the computation. (D) Snapshots of the magnetization configuration of the device at selected times [53].

i.e., the length of the bit stream is 8, was designed and validated using micromagnetic simulation. Fig. 14.18A shows the schematic of the computation of 4/8 and 6/8 with a two-input AND gate, which will be implemented in the structure shown in Fig. 14.18B. Periodic notches are set along the tracks for synchronization and to divide the tracks into eight-bit regions denoted by the yellow box. The corresponding timing sequence of one clock cycle is displayed in Fig. 14.18C. In every circle, an SOT current pulse of 20 MA cm
2 is applied to drive skyrmions to the region of the next bit. After eight clock cycles, there are three skyrmions on the upper track of the device, as shown in Fig. 14.18D, which represents a probability of 3/8 and is the result of 4/8  6/8.

14.5

Challenges and perspectives

Thanks to the outstanding features such as nonvolatility, topological stability, nanoscale size, low driving current, fast speed, and rich dynamics, skyrmion shows great potential for unconventional computing applications and a variety of devices has been proposed. However, a number of challenges have to be addressed before real applications. Most of the devices/prototypes as discussed in this chapter are conceptually demonstrated based on simulations. Few of them have been experimentally validated owing to the extraordinary requirements for the materials and nanofabrication techniques [56–58]. More efforts deserve to be paid on both theoretical and experimental studies to develop skyrmionic devices and to promote the application of skyrmions on unconventional computing.

Unconventional applications of skyrmions

413

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Further reading L. Sai, K. Wang, Z. Xichao, N. Tianxiao, Z. Yan, L.W. Kang, Z. Weisheng, Magnetic skyrmions for unconventional computing, Mater. Horiz. (2020), https://doi.org/10.1039/ D0MH01603A. In press.

Introduction to topology Gabriele Bonanno and Giuseppina D’Aguı` Department of Engineering, University of Messina, Messina, Italy

15

This chapter is dedicated to review the theoretical fundamental concepts of topology. The interest in topology is growing in several fields, including physics, economics, chemistry, engineering, and other mathematical and scientific fields. The main motivation is that by using topological concepts one can categorize and count objects using qualitative information as opposed to exact values. For instance, topology enables us to handle qualitative laws and determine qualitative, but demonstrable, chaotic behavior in dynamical systems. As discussed in the previous chapters, the topology has an important role in understanding both the static and dynamic properties of magnetic skyrmions. In detail, the chapter starts from the definition and main theorems of topological and metric spaces up to driving the reader to the concept of winding number and its applications.

15.1

Topological spaces

A space is a set whose elements are called points. Providing a space X with a topological structure means knowing, for each subset, which are its interior, exterior, and boundary points. The main rules that must be satisfied are: 1. 2. 3. 4. 5.

Each point of X is interior of X. If x is internal of A, then x  A. If x is interior of A and A
B, then x is interior of B. If x is interior of both A and B, then it is also interior of A \ B. If A˚ denotes the set of interior points of A, then every point of A˚ is interior of A˚.

The previous five conditions are the axiomatic definition of a topological structure; however, it is now considered standard to define such a structure in terms of the open family, and this is what we will do to define topological spaces. The notion of topological space is general and from a mathematical point of view makes sense to focus on the classification aspects of topological spaces. Let’s discuss qualitatively a simple example, we have two spaces and we have to demonstrate if they are topologically equivalent. The classical way to proceed is to evaluate the cardinalities and then one can introduce some properties of the topological spaces that are invariant for homeomorphism (this concept will be discussed extensively ahead in the chapter) for checking the topological equivalence. For example, the properties of being metrizable or of Hausdorff have such a characteristic. In some cases, the invariance for homeomorphism is clear from the definitions; in other cases, it is not obvious and theoretical work is required to claim such a conclusion. However, the theory Magnetic Skyrmions and Their Applications. https://doi.org/10.1016/B978-0-12-820815-1.00006-7 Copyright © 2021 Elsevier Ltd. All rights reserved.

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already developed for topological spaces allows us to save time and check by using theorems the topological equivalence in a fashionable way. The first step is to define open sets without reference to distance, which provides the abstract notion of “nearness.” Definition 15.1. A topology T on a set X is a collection of subsets such that (1) ∅, XT (2) The union of arbitrarily many sets in T is in T . (3) The intersection of finitely many sets in T also lies in T .

Definition 15.2. The pairing of a set and a topology, ðX, T Þ, is a topological space. The subsets of X that lie in T are called the open sets in ðX, T Þ. Example 15.1. Take a three-point set X ¼ {a, b, c}. There are 23 possible subsets of X. ∅ and X must be in T , and there are six other subsets to be added in. One topology could be for instance T ¼ f∅, X, fa, bg, fcgg. Example 15.2. The trivial topology is composed just with X and the empty set. The discrete topology consists of all possible subsets of X. Definition 15.3. Let T and T 0 be two topologies on X, we say l

l

l

T 0 is finer than T if T
T 0 T 0 is coarser than T if T 0
T The two topologies are said to be comparable if one is finer than the other. Otherwise, they are incomparable.

Example 15.3. X ¼ fa, b, c, d g,T ¼ f∅, X, fag, fbg, fa, bg, fb, cg, fa, b, cgg. Finer sets include: l

l

The discrete topology Adding {c} and {a, c}

Coarser sets include: l

l

The trivial topology {∅, X, {a}}

Incomparable: l

Anything with {d}, e.g., {∅, X, {d}, {b, c, d}}

Remark 15.1. In general, the trivial topology is coarser than all topologies on X, and the discrete topology is always finer. Definition 15.4. A neighborhood of x is an open set U such that  U. Proposition 15.1. A set A is open in a topological space X if and only if for all points x 08 A, there is a neighborhood U containing x.

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Definition 15.5. For a set X, a basis for topology T on X is a collection B of subsets of X, called basis elements, such that (1) For all x  X, x lies in some basis element; (2) If B1 \ B2, then there exists a B3 containing x such that B3
B1 \ B2.

Theorem 15.1. A basis generates a topology T on X. The open sets are precisely the subsets U
X such that for all x  U there is a basis element B
U containing x. Proof. (1) ∅ is satisfied vacuously. XT because every x  X must lie in a basis element

B
X. S (2) For arbitrary unions, let U ¼ α Uα . Take x  U. Then, x lies in some Uα, which is open. By definition, there is some basis element B
Uα such that x  B and B
Uα
U. Thus, U is open. T (3) Last, for finite intersections, let ¼ ni¼1 Vi , where each Vi is open. Take x  V. Then, x  V1, which is open. So, there exists a basis element B1
V1 containing x. This is true for each Vi, 1  i  n; namely, there exists a basis element Bi
Vi containing x. By definition, there is a basis element C2
B1
B2 such that x  C2. And x  B3 \ C2; so, there exists a C3
B3 \ C2 such that x  C3. And so forth. Eventually, there will be some T T Cn
B3 \ C2 \ ⋯ \ Cn1 such that x  Cn. Note that Cn
ni¼1 Bi
ni¼1 Vi ¼ V; so, V is open. Hence, T is a topology.

□ Definition 15.6. A closed set A is the complement of an open set U : A ¼ X  U. Example 15.4. l

l

l

Closed disks ℝ X and ∅ in any space X

Proposition 15.2. The complement of an open set is closed. Example 15.5. In any topological space X, ∅ and X are both closed and open. In ℝ, the only sets that are both open and closed are ∅ and ℝ. Theorem 15.2. Let X be a topological space. Then, (1) ∅ and X are closed (2) Arbitrary intersections of closed sets are closed (3) Finite unions of closed sets are closed.

Definition 15.7. A set X is Hausdorff if for all x, y  X there exist disjoint open sets U and V such that x  U and y  V. Definition 15.8. For A
X, the interior of A, denoted A˚ of Int(A), is the union of all open sets contained in A.

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Definition 15.9. The closure of A, denoted A or Cl(A), is the intersection of all closed sets containing A. Facts: (1) Int(A) is open (2) Cl(A) is closed (3) Int(A)
A
Cl(A)

Proposition 15.3. For any topological space X, (i) (ii) (iii) (iv) (v)

If U is open and
A, then U
Int(A). If C is closed and C  A, then C  Cl(A). If A
B, then Int(A)
Int(B) and Cl(A)
Cl(B). A is open ⟺A ¼ Int(A). A is closed ⟺ A ¼ Cl(A).

Proof. (i) and (ii) follow directly from the definitions of interior and closure. (iii) Note that Int(A)
A
B, so Int(A) is an open set contained in B. Then, by (i), Int(A)
Int(B). Likewise, A
B
Cl(B). Thus, B contains A and is closed so B  A. (iv) If A is open, then A
A. So, by (i), A
Int(A). And clearly, Int(A)
A, so we see that Int(A) ¼ A. For the other direction, if A ¼ Int(A), then Int(A) is an open set, which means that A is open too. (v) is provided in a similar fashion. □

Definition 15.10. A set A
X is dense if A ¼ X. Theorem 15.3. Let y  A
X. Then, (1) y  Int(A)⟺ there is an open set U with y  U
A. (2) y  A⟺ every open set containing y intersects A.

Theorem 15.4. Let A, B
X. Then, (1) (2) (3) (4)

Int(X  A) ¼ X  Cl(A) Cl(X  A) ¼ X  Int(A) Int(A) [ Int(B)
Int(A [ B) but they are not equal in general Int(A) \ Int(B) ¼ Int(A \ B).

In topology, limit points are defined using open neighborhoods. Definition 15.11. A limit point x of A
X has the property that every neighborhood U of x intersects A somewhere other than x. The set of limit points of a set A will be denoted A0 . Proposition 15.4. Every limit point of A is in the closure of A. Theorem 15.5. B ¼ B [ B0 . Proof. B
B for any set B, and B0
B, so B [ B0
B. Now, take x  B. Then, every open neighborhood U of x must intersect B at some point y. If x  B, x is in the union already, so assume x 62 B. Because y  B, x 6¼ y, thus U intersects B at a point other □ than x. So, x is a limit point. Hence, x  B [ B0 ¼)B ¼ B [ B0 .

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Corollary. B is closed if and only if B contains all of its limit points. Proof. Assume B is closed, by which B ¼ B. Then, by Theorem 15.5, B ¼ B [ B0 , which means B0
B. Now, assume B0
B. Then, B [ B0 ¼ B and by the same theo□ rem, B ¼ B [ B0 ¼ B, so B is closed. Definition 15.12. In a topological space X, a sequence (xn) is convergent if there exists a limit x  X such that for all open neighborhoods U containing x, there exists an N > 0 such that if n  N, then xn  U. Theorem 15.6. If X is Hausdorff, then every convergent sequence has a unique limit. Proof. Let (xn) converge to x  X. Consider a point y  X that is distinct from x. Because X is Hausdorff, there are disjoint neighborhoodsUx and Uy containing x and y, respectively. Because (xn) ! x, there is some N  ℕ such that for all n  N, xn  Ux. But for all n  N, xn 62 Uy and thus (xn) cannot converge to y. Because y □ was any point in X other than x, the limit (xn) ! x is unique. Definition 15.13. Let A
X. Then, the boundary of ¼ AnIntðAÞ. Theorem 15.7. x  δ A if and only if every neighborhood of x intersects both A and A  X. Proposition 15.5. For A
X, (1) (2) (3) (4) (5) (6) (7)

δA is closed δA ¼ A \ X  A δA \ Int(A) ¼ ∅ δ [ IntðAÞ ¼ A δA
A ⟺ A is closed δA \ A ¼ ∅ ⟺ A is open δA ¼ ∅ ⟺ A is both open and closed.

Definition 15.14. Let Y
X where X is a topological space. Define the subspace topology on Y by T Y ¼ fU \ Yj U isopenin Xg. Then, ðY, T Y Þ is itself a topological space. Claim. T Y is a topology on Y. Proof. (1) Because ∅ is open in X and ∅ \ Y ¼ ∅, ∅ is open in Y. And because X \ Y ¼ Y, Y is open inSY. (2) Let V ¼ α Vα where each Vα is open in Y. So, each Vα ¼ Uα \ Y where Uα is open in X. S  S Thus, V ¼ α ðUα \ Y Þ ¼ α Uα \ Y, and the arbitrary union of open sets in X is open, S so α Uα ¼ U is open, which implies U \ Y is open in Y. T (3) Let W ¼ ni¼1 Wi where each Wi is open in Y. Then, each Wi ¼ Ui \ Y for some Ui open in X. T  T T Then, W ¼ ni¼1 Wi ¼ ni¼1 ðUi \ Y Þ ¼ ni¼1 Ui \ Y, and because finite intersections of Tn open sets in X are open, i¼1 Ui ¼ U is open in X. Thus, U \ Y is open in Y. We conclude that T Y is indeed a topology on Y.

□ Proposition 15.6. Let Y
X have the subspace topology and let U
Y. Then,

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(1) U is open in X ¼) U is open in Y. (2) If Y is open in X, then U open in Y implies U open in X.

Proposition 15.7. Let Y
X. If X is Hausdorff, then Y is Hausdorff. Suppose we have two topological spaces X and Y and cross them: X  Y. If U is open in X and V is open in Y, we want U  V (see Fig. 15.1) to be open in X  Y. However, not all open sets in X  Y will be a product like U  V. Definition 15.15. The basis for the product topology on X  Y is B ¼ fU  Vj U open in X, V open in Y g. Proposition 15.8. B is a basis for the product topology X  Y. Example 15.6. S1  [0, 1] is a cylinder (see Fig. 15.2). Example 15.7. S1  S1 ¼ T2 is a torus (see Fig. 15.3). Definition 15.16. Let X be a topological space and let p : X ! Q be a surjective map onto set Q. The quotient topology T p on Q is given by: l

l

l

V
Q is open in Q ⟺ p1(V) is open in X Q is then called a quotient space p is called a quotient map between topological spaces.

Fig. 15.1 Product space.

Fig. 15.2 A cylinder.

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Fig. 15.3 A torus.

Claim. T p is a topology. Definition 15.17. For topological spaces X and Y, let f : X ! Y be a well-defined function. Then, f is continuous if for every open set, U
Y, f1(U) is open in X. Remark 15.2. For metric spaces, the ε-δ definition of continuity is equivalent to the topological definition. Example 15.8. The identity function id : X ! X defined by id(x) ¼ x is continuous for any topological space X. Example 15.9. The constant map Ck : X ! Y defined by Ck(x) ¼ k for some k  Y is continuous.  X, k  U ð U Þ ¼ To see this, take any U
Y, which is open. Then, C1 k ∅, k 62 U: Example 15.10. Quotient maps are continuous. Definition 15.18. An open map is a function that maps open sets to open sets. Proposition 15.9. Let T 1
T 2 and suppose f : ð X, T 1 Þ ! Y is continuous. Then, f : ð X, T 2 Þ ! Y is continuous as well. Proposition 15.10. Let T 1
T 2 and suppose g : X ! ð Y, T 2 Þ is continuous. Then, g : X ! ð Y, T 1 Þ is continuous as well. Theorem 15.8. Given f : X ! Y, the following are equivalent: (1) (2) (3) (4)

For all open sets, U
Y, f1(U)is open in X (definition of continuity). Given a basis B on Y, for all BB, f1(B) is open in X. For all closed sets, C
Y, f1(C) is closed in X.   For all subsets, A
X, f A
f ðAÞ.

Theorem 15.9. If f : X ! Y and g : Y ! Z are continuous functions, then h ¼ g ∘ f : X ! Z is continuous. Theorem 15.10. Continuous functions map limits of sequences to limits of sequences. In other words, if (an)!x, then (f (an)! f(x)).

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Definition 15.19. Let f : X ! Y be a bijection between topological spaces. If both f and f 1are continuous maps, then f is a homeomorphism, and X and Y are homeomorphic. Remark 15.3. Homeomorphisms are point-wise and open-set bijections. Also note that f( f 1(U)) ¼ U and f 1 ( f(V)) ¼ V if and only if f is a bijection. Some other useful properties of homeomorphism are: (1) (2) (3) (4)

id : X ! X is homeomorphism. If f : X ! Y is a homeomorphism, then f1 : Y ! X is also a homeomorphism. Compositions of homeomorphisms are homeomorphisms. The property of being homeomorphic is an equivalence relation on topological spaces, so we can separate all topological spaces into their respective homeomorphism classes.

Definition 15.20. A topological property is a property of topological spaces that is invariant across a homeomorphism class. Example 15.11. f : ℝ ! ℝ defined by f(x) ¼ x3. Then, f 1(x) ¼ x1/3, which is bijective and continuous. Thus, f is a homeomorphism. Definition 15.21. An embedding is a function f : X ! Y such that X ffi f(X). Theorem 15.11. Hausdorff is a topological property. Proof. Let f : X ! Y be homeomorphism. Note that f1 : Y ! X is also a homeomorphism, so it suffices to show the forward direction. Suppose X is Hausdorff. Take y1, y2  Y such that y1 6¼ y2. Then, these pull back bijectively to f1(y1) ¼ x1 and f1 (y2) ¼ x2. Because X is Hausdorff, there exist disjoint, open sets U1, U2
X such that xi  Ui for i ¼ 1, 2. Let V1 ¼ f(U1) and V2 ¼ f(U2). Then, because f is a homeomorphism, V1, V2
Y are open, and y1  V1 and y2  V2. Take z  V1 \ V2. Then, f1 (z)  U1 \ U2, but this intersection is empty. Hence, V1 \ V2 is empty as well, so Y is Hausdorff. □

15.2

Metric spaces

One of the most common and useful types of topological space is the so-called metric space. Metric spaces are topological spaces that result from having a means for measuring distance between points in the underlying set. Definition 15.22. A metric d on a topological space X is a function d : X  X ! ℝ satisfying the following for all x, y, z  X: (1) Positivity: d(x, y)  0 and d(x, y) ¼ 0 ⟺ x ¼ y (2) Symmetry: d(x, y) ¼ d(y, x) (3) Triangle inequality: d(x, z)  d(x, y) + d(y, z).

15.12. The Euclidean qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dðx, yÞ ¼ ðx1  y1 Þ + ðx2  y2 Þ2 .

Example

metric

on

R 2:

for

x,yℝ 2 ,

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Fig. 15.4 Taxicab metric example.

 Example 15.13. The Euclidean metric on Rn: for x,yℝn , dðx, yÞ ¼

n P

1=2 ðx i  yi Þ2

.

i¼1

Example 15.14. Other metrics on ℝn l

The taxicab metric:d1 ðx, yÞ ¼ |x1  y1 | + … + |xn  yn | (see Fig. 15.4) d1 ðð0, 1Þ, ð4, 4ÞÞ ¼ 4 + 3 ¼ 7:

l

l

The sup metric: d∞ ðx, yÞ ¼ sup1ⅈ_ n jxi  yi j In the above diagram, d∞((1, 1), (5, 3)) ¼ 4. For an integer 1  p  ∞, the p-metric: dp ðx, yÞ ¼



n P

1=p ðx i  y i Þp

i¼1

The Euclidean and sup metrics are special cases for p ¼ 2 and p ¼ ∞, respectively. Remark 15.4. Any norm on a vector space induces a metric: let k k denote a norm on a vector space X. Then, define d(x, y) ¼ jj x  y jj. Note that a norm possesses: l

l

l

Nonnegativity Scalability The triangle inequality.

Note that every metric space is a topological space, but the converse is not true in general. Definition 15.23. For any metric space X, we define the open and closed metric balls Bd and Bd by Bd ðx, εÞ ¼ fy j d ðx, yÞ < εg Bd ðx, εÞ ¼ f y j d ðx, yÞ  ε g: Proposition 15.11. The collection B ¼ f Bd ðx, εÞ j x  X, ε > 0 g is a basis for a topology on the metric space (X, d), called the metric topology. Example 15.15. Function spaces.

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Function spaces are an important type of topological space. For an interval [a, b]
ℝ, we define C[a, b] ¼ { f j f is continuous on [a, b]}. Consider two metrics on C[a, b]: (1) d ð f , gÞ ¼ max x½a, b jf ðxÞ  gðxÞj: Note that the Extreme Value Theorem says that continuous functions on closed, bounded intervals attain their maximum value(s), so we can write this metric using max instead of sup. Ðb (2) pð f , gÞ ¼ jf ðxÞ  gðxÞjdx. This is in a sense the measure of the area between the two a

functions.

Proposition 15.12. Metric spaces are Hausdorff. Proof. Let (X, d) be a metric space. Take x, y X where x 6¼ y and   let l ¼ d(x, y). 1 1 Because x 6¼ y, l > 0. Consider the open balls Bd x, 2 and Bd y, 2 . By the triangle inequality, there are disjoints. Hence, X is Hausdorff. □ Corollary. If a space is not Hausdorff, it is a not a metric space (and cannot have a metric). Proposition 15.13. For metric spaces (X, dX) and (Y, dY), a function f : X ! Y is continuous if for all x  X, ε > 0, there is some δ > 0 such that dX(x1, x2) < δ ¼) dy(f(x1), f(x2)) < ε. Definition 15.24. In (X, d), the distance between subsets A
X and B
X is defined as d ðA, BÞ ¼ inf fdða, bÞg aA bB

Definition 15.25. A topological space X is metrizable if there exists a metric d on X inducing its topology. Example 15.16. As mentioned before, any space that is not Hausdorff is not metrizable. Results: (i) Subspaces of metric spaces are metrizable (via the same metric). (ii) Countable products of metric spaces are metrizable. (iii) Some “order” topologies are metrizable, but others are not. For example, the vertical interval topology on ℝ2 is metrizable. (iv) Metrizability is a topological property.

15.3

Connected sets

To the question “How many pieces does the space X ¼ R  {0}is composed?”, everyone intuitively answers immediately “two”. This is possible because we can distinguish two parts of X, {x < 0} and {x > 0}, which in our mind are well suited to

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[0,1] ∪ [2,3]

Fig. 15.5 Disconnected set.

the word “piece” used in the question. Topological structures allow us to define this mathematically with the notions of connected space and connected component, which correspond to the intuitive comparisons of “object all in one piece” and of “piece of cake” (the cake is already cut), respectively. This section introduces the connectedness. Intuitively, a topological space is connected if it is made of only one piece. For example, R is all one piece, whereas [0, 1] [ [2, 3] consists of two pieces. If a space is not connected, it is natural to ask how many pieces there are and to attempt to describe them. A is an example of a set that is disconnected (Fig. 15.5). Definition 15.26. A topological space X is disconnected if there exist nonempty open sets U, V
X such that U \ V ¼ ∅ and U [ V ¼ X. Such open sets U and V are called a separation of X. Definition 15.27. If no such open sets exist, then X is connected. Example 15.17. The discrete topology. For any X with discrete topology (and |X | > 1), X is disconnected. This is because for any nonempty subset U
X, U and X  U are both open. Example 15.18. The trivial topology. X with the trivial topology is connected because X is the only nonempty open set. Theorem 15.12. Xis connected if and only if the only subsets of X that are both open and closed are ∅ and X. Corollary. Let T 1
T 2 be topologies on X. (1) If ðX, T 1 Þ is disconnected, then ðX, T 2 Þ is disconnected. (2) If ðX, T 2 Þ is connected, then ðX, T 1 Þ is connected (by contrapositive).

Definition 15.28. A subset A
X is disconnected in X if there is some open sets U, V
X such that each intersects A, A
U [ V and U \ V is disjoint from A. Theorem 15.13. Let X be a connected space and let f : X ! Y be continuous. Then, f(X) is connected. T Theorem 15.14. Let {Cα} be a collection of connected subsets of X, with α Cα nonempty. T Then, α Cα is connected. Theorem 15.15. Connectedness is a topological property. Theorem 15.16. The n-sphere Sn for n  1 is connected. Remark. If X is connected and p : X ! Q is a quotient map, then Q is connected.

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As a result, the following are all connected spaces: l

l

l

l

l

l

Cylinder Annulus M€obius strip Sphere Sn Klein bottle Torus

Example 15.19. The square [0, 1]2 is connected because [0, 1] is connected and products of connected spaces are connected. Definition 15.29. A path from x to y where x, y  X, is a continuous function f : [0, 1] ! X such that f(0) ¼ x and f(1) ¼ y, see Fig. 15.6. If X is disconnected, paths between points may not exist. Definition 15.30. X is path connected if for every x, y  X there exists a path from x to y. Theorem 15.17. If X is path connected, then X is connected. Theorem 15.18. If f : X ! Y is continuous and surjective and X is path connected, then Y is also path connected. Proof. Take y1, y2  Y. Because f is surjective, there exist x1  f1(y1) and x2  f1 (y2). By path connectedness of X, there exists a path p : [0, 1] ! X such that p   and p(1) ¼ x2. Define p : ½0, 1 ! Y by p ¼ f ∘p. Then, (0) ¼ x1    p ð0Þ ¼ f ðpð0ÞÞ ¼ f ðx1 Þ ¼ y1 and p ð1Þ ¼ f ðpð1ÞÞ ¼ f ðx2 Þ ¼ y2 . Thus, p is a path connecting y1 and y2, so Y is path connected. Note: The hypothesis that f is continuous is required because the composition f ∘ p needs to be continuous. □

Fig. 15.6 Example of path.

Introduction to topology

15.4

429

Compact sets

In this section, we present another property: compactness. This concept is not as intuitive as continuity or connectedness. In ℝn, the compact sets are the closed and bounded sets, but in a general topological space, the compact sets are not as simple to describe. In fact, we need the following definitions. Definition 15.31. A collection O of subsets of X is a cover of X if the union of all sets in O equals X. O is an open cover if all sets in O are open in X. For a subset A
X, O covers A if A is contained in the union of all the sets in O. Definition 15.32. C is a subcover of O if C covers X and C
O. Definition 15.33. A topological space X is compact if every open cover of X has a finite subcover. Example 15.20. ℝ is not compact. Definition 15.34. A subset A
X is compact in X if it is compact in the subspace topology on A inherited from X. Theorem 15.19. Compactness is a topological property. Theorem 15.20. Let A
X be compact and suppose f : X ! Y is continuous. Then, f(A)
Y is compact. Theorem 15.21. The following are properties of compact sets: (1) Finite unions of compact sets are compact. (2) If X is Hausdorff, then arbitrary intersections of compact sets are compact.

Theorem 15.22. If X is Hausdorff and A
X, then A compact ¼) A closed. Proof. Hausdorff implies for each a  A and x  X\A there exist open, disjoint sets Ua, Va
X such that x  Ua and a  Va. Then, {Va}aA is an open cover of A. Because T A is compact, there exists a finite subcover {V1, V2, …, Vn} of {Va}aA. Let U ¼ ni¼1 Ui . S Then, U is open, x  U is disjoint from ni¼1 Vi  A. So, U
X\A. We have thus found an open set containing any point x  X\A. Therefore, X\A is open, proving the theorem. □ Theorem 15.23. Let C be closed and C
D, which is compact in X. Then, C is compact. In other words, closed subsets of compact sets are compact. Theorem 15.24. If X and Y are compact spaces, then X  Y is compact. Example 15.21. The following are compact spaces:

430 l

l

l

l

l

l

Magnetic Skyrmions and Their Applications

The square [0, 1]2 Sphere Torus Klein bottle Projective space ℝP2 M€obius band

Theorem 15.25. (Tychonoff). Products of uncountably many compact sets are compact. Theorem 15.26. [a, b]
ℝ is compact. Theorem 15.27. (Heine-Borel). Let A be a subset of ℝn for finite, positive n. Then, A is compact if and only if A is closed and bounded. Example 15.22. The torus T2is compact. Theorem 15.28. For a metric space (X, d), A compact in X ¼) A closed and bounded. Definition 15.35. A metric space X is complete if every Cauchy sequence in X converges. Theorem 15.29. Every compact metric space is complete. In addition to the topological definition of compactness given in this chapter, there are several other definitions that are more useful in different settings. Definition 15.36. A subset A
X is sequentially compact if every sequence (an) in A has a convergent subsequence (ank). Definition 15.37. A subset A
X is limit point compact if every infinite subset of A has a limit point. Theorem 15.30. Compactness implies sequential compactness and limit point compactness. Moreover, if X is metrizable, then all three are equivalent. In general, however, they are not equivalent.

15.5

Homotopy theory

The fundamental notion of topological equivalence is homeomorphism. However, in topology, we often want to consider two spaces to be equivalent, or have the “same shape,” in a much broader sense than homeomorphism. Two connected and regular subsets of ℝ2, where regular is understood in an intuitive sense, as the opposite of complicated, bizarre, pathological, etc., have equivalent forms if they have the same number of holes. For example, the letters of the alphabet can be seen as a connected union of lines and segments of ℝ2 and they are divided into two classes of equivalence of forms: in fact, the three letters O, P, and A have equivalent shapes because they have only one hole, whereas M, T, and I have equivalent shapes because they have no holes. So, mathematically, having the same type of homotopy will correspond to the intuitive notion of having equivalent forms.

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Definition 15.38. A homotopy is a family of maps ft : X ! Y for t  I ¼ [0, 1] such that the function F : (X  I) ! Y defined by F(x, t) ¼ ft(x) is continuous. Definition 15.39. Two maps g, h : X ! Y are said to be homotopic, denoted g ’ h, if there is a homotopy F : X  I ! Y such that f0 ¼ g and f1 ¼ h. Example 15.23. Let X ¼ I, Y ¼ [0, 1], f(x) ¼ x and g(x) ¼ x. Define ft(x) ¼ x for all t  I. Then, F(x, t) ¼ x is a projection map, so F is continuous and thus a homotopy. This shows that f(x) ’ g(x). Example 15.24. Let X ¼ I, Y ¼ ℝ, f(x) ¼ 0, and g(x) ¼ 4x. Define F(x, t) ¼ 4xt.Then, F(x, 0) ¼ f(x) and F(x, 1) ¼ g(x). Because F is continuous on X  I, this proves that f and g are homotopic. Theorem 15.31. Homotopy is an equivalence relation. Proof. (1) Let F(x, t) ¼ f(x). Then, F is clearly continuous, so f(x) ’ f(x). This shows that homotopy is reflexive. (2) Suppose f(x) ’ g(x). Then, there is a family of functions F(x, t) such that F(x, 0) ¼ f(x) and F(x, 1) ¼ g(x). Let G(x, t) ¼ F(x, 1  t). Then, G(x, 0) ¼ g(x) and G(x, 1) ¼ f(x) by construction. Moreover, G inherits continuity from F, so G is a homotopy. This shows symmetry. (3) Finally, suppose f(x) ’ g(x) and g(x) ’ h(x). Then, there is a map F(x, t) such that F(x, 0) ¼ f(x) and F(x, 1) ¼ g(x) and there is another function G(x, t) such that G(x, 0) ¼ g(x) and G(x, 1) ¼ h(x). Define 8 1 > < Fðx, 2tÞ,0  t  2 H ðx, tÞ ¼ > : Gðx, 2t  1Þ, 1 < t  1 2 Then, H : X  I ! Y is continuous because lim t!1=2 Hðx, tÞ ¼ gðxÞ and H is continuous on each piece. Moreover, H(x, 0) ¼ f(x) and H(x, 1) ¼ h(x), so f(x) ’ h(x). This establishes transitivity, so homotopy is an equivalence relation. In this section, we explore the earliest, and perhaps most important, connection between topological spaces and algebraic structures: the fundamental group. To do so, we first develop the concepts of homotopy type and path composition.

□

Definition 15.40. A map f : X ! Y is called a homotopy equivalence if there exists a map g : Y ! X such that fg is homotopic to identity on Y and gf is homotopic to the identity on X. If there exists such a homotopy equivalence between X and Y, then we say X and Y have the same homotopy type. Proposition 15.14. Topological equivalence (i.e., homeomorphism) implies homotopy equivalence. Proof. Let f : X ! Y be a homeomorphism. Then, f1 exists and is continuous, and for any x  X, y  Y, we have f1 ∘ f(x) ¼ x and f ∘ f1(y) ¼ y. Hence, f is a homotopy equivalence. □ Example 15.25. ℝ is homotopic to itself.

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Magnetic Skyrmions and Their Applications

Example 15.26. ℝ is homotopic to a point. Let X ¼ {0} and Y ¼ ℝ. Define f : X ! Y by f(x) ¼ x and g : Y ! X by g(x) ¼ 0. Then, gf(0) ¼ 0 so gf is the identity on {0}. However, fg(x) ¼ f(0) ¼ 0, which is not the identity on ℝ. But we did show that 0 ’ x, so again f is a homotopy equivalence and {0} and ℝ have the same homotopy type. Theorem 15.32. Let B
ℝn and X be any topological space and suppose f, g : X ! B are maps such that for all x  X, the line segment between f(x) and g(x) lies in B. Define H to be the function that traces out the line segment at a constant rate from t ¼ 0 to 1, i.e., H(x, t) ¼ f(x) + t(g(x)  f(x)). Then, H is a homotopy, called the straight line homotopy. Theorem 15.33. Two functions f : {p} ! Y and g : {p} ! Y are homotopic if Y is path connected. We are commonly interested in paths in X, by which we mean continuous maps τ : I ! X. Definition 15.41. Given two paths τ1,τ2 : I ! X such that τ1(1) ¼ τ2(0), we define their product or path composition τ1τ2 : I ! X by 8 > < τ1 ð2tÞ,

1 0x 2 τ 1 τ 2 ðt Þ ¼ > : τ2 ð2t  1Þ, 1 < 0  1 2 Definition 15.42. An n-manifold (possibly with boundary) is a topological space such that for every point x there is a neighborhood of x that is homeomorphic to a neighborhood of some point in ℝn+ ¼ {(x1, …, xn)  ℝn j xn  0}. Definition 15.43. For a smooth map f : X ! Y where X is a compact manifold and Y is connected, we define the mod 2 intersection number for any closed manifold Z
Y to be the number of points in f1(Z) mod 2, denoted I2(f, Z). When X and Y are both n-manifolds, I2(f, {y}) is the same for all y  Y. In this case, the number is called the mod 2 degree of f, denoted deg2(f). Some useful properties of the mod 2 degree are: l

l

If f : X ! Y and g : X ! Y are homotopic, then deg2(f) ¼ deg2(g). If X ¼ δW and f : X ! Y may be extended to all of W then deg2(f) ¼ 0.

15.6

Winding numbers

This section focuses on the concept of winding number, whose intuitive idea is simple, but less easy is to give the strict introduction. This concept is linked to the concept of skyrmion number already introduced in previous chapters.

Introduction to topology

433

The winding number of a loop γ : I ! ℝ  {P} around a point P is simply the number of times the loop γ winds around the point P. The winding number is important because it gives indices of vector fields and degrees of mappings. We simply define the winding number of γ to be the change in θ (taking polar coordinates on ℝ2 with P at the origin) along γ. The problem is that θ is not a well-defined continuous function on all of ℝ2  P. The angle θ as defined in modern calculus books makes a jump from θ ¼ 2π to θ ¼ 0 on the ray θ(x, y) ¼ 0. We define the map p : ℝ+  ℝ ! ℝ2  P by (recall that ℝ+ ¼ [0, ∞)) pðr, θÞ ¼ P + ðr cos θ, r sin θÞ This map is, of course, just polar coordinates on ℝ2  P centered at P. Let a denote the   beginning point of γ and let a denote any point in p1(a). Then, let γ : I ! ℝ +  ℝ be a      path such that γ ð0Þ ¼a and p γ ðsÞ ¼ γ ðsÞ for all s. Any such path γ will be called a  lift of γ beginning at a . Then, we define the winding number of γ around P to be W ðγ, PÞ ¼

  1    θ γ ð 1Þ  θ γ ð 0Þ 2π

where θ : ℝ+  ℝ ! ℝ is the projection onto the second coordinate. 

Lemma 15.1. Suppose that γ is a loop in ℝ2  P beginning at a and γ is a lift of γ     beginning at a p1 ðaÞ. Then, for any a 0 p1 ðaÞ, there is a lift γ 0 beginning at a 0 such that         θ γ 0 ð 1Þ  θ γ 0 ð 0Þ ¼ θ γ ð 1Þ  θ γ ð 0Þ         Proof. We simply shift γ up or down to get γ . Define γ 0 by γ 0 ðsÞ ¼γ 0 ðsÞ + 0, a 0  a :    Then, γ 0 is a lift of γ because a 0  a is an integer multiple of 2π and the functions cos θ and sin θ are periodic with period 2π. Then, the required equation holds. □ 

Definition 15.44. Let X be a topological space and let p : X ! X be onto map. We 

say that p is a covering map and that X is a covering space for X if for every x  X there is a neighborhood U of x such that p1(U) is the disjoint union of open sets each of which is mapped homeomorphically onto U by p. A neighborhood U as in this definition is called an elementary neighborhood for the covering map. 

If p : X ! X is a covering map that is n to one (each point in X has n preimages), then 

we say that X ! X is an n-sheeted covering. 

Suppose that f : X ! Y is a map that Y is a covering space for Y with covering map 



p : Y ! Y. Then, a function f : X ! Y is called a lift if it satisfies f ¼ p ∘ f.

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Magnetic Skyrmions and Their Applications

Lemma 15.2. (The Path-Lifting Lemma) Suppose γ : I ! X is a pathbeginning at a 



point a  X, p : X ! X is a covering map for X, and a is a point in p1(a). Then, there 





exists a unique lift γ : I !X beginning at a . Corollary 15.1. The winding number from Equation does not depend on the choice of    the point a or the lift γ , and a lift γ always exists. Thus, the winding number is well defined.  We now prove that not only does every path have a unique lift for each point a , but  every homotopy between paths has a unique lift for each a . Lemma 15.3. (The Homotopy-Lifting Lemma) Suppose that γ 0, γ 1 : I ! X are both 

paths beginning at a point a  X and ending at a point b  X, that p : X ! X is a cov ering map for X, and that a is a point in p1(a). Also, suppose that γ0 and γ1 are homo



topic via a homotopy H. Then, there exists a unique lift H : I  I !X of H beginning  at a . Theorem 15.34. Two loops γ 0, γ 1 : I ! ℝ2  P are base point homotopic in ℝ2  P if and only if they have the same winding number around P: γ 0 ’ γ 1 ⟺W ðγ 0 , PÞ ¼ W ðγ 1 , PÞ In particular, if n is the winding number of γ about P, then γ is homotopic to the loop γ n(s) ¼ (cos(nπs), sin(nπs)). 

Proof. If the two loops are base point homotopic, then, for any b p1 ðbÞ, their lifts 

based at b are homotopic by the homotopy lifting lemma. Hence, the lifts end at the same point and W(γ 0, P) ¼ W(γ 1, P).   If W(γ 0, P) ¼ W(γ 1, P), then let γ 0 and γ 1 be lifts of γ 0 and γ 1 both based at the same 

point b p1 ðbÞ. Then, both lifts end at the same point. Hence, the lifts are paths in ℝ+  ℝ that have common endpoints. Because ℝ+  ℝ is convex, the lifts are homo



topic by some H : I  I ! ℝ +  ℝ. Then, γ 0 and γ 1 are homotopic via H ¼ p∘ H .

□

Corollary 15.2. Let γ n denote the path γ n(s) ¼ (cos(nπs), sin(nπs)). Then, any loop γ in ℝ2  (0, 0) based at (1, 0) is path homotopic to a unique loop γ n. Two loops γ 0, γ 1 : I ! X are said to be homotopic as loops if there is a homotopy H : I  I ! X such that H(0, t) ¼ H(1, t) for all t. In this case, the path γ t defined by γ t(s) ¼ H(t, s) is a loop for all t. Corollary 15.3. Two loops γ 0, γ 1 : I ! ℝ2  {0} are homotopic as loops if and only if W(γ 0, 0) ¼ W(γ 1, 0). Proposition 15.15. (Dog on a Leash) Suppose that α and β are closed paths in ℝ2  {P} and that ∣α(s)  β(s) ∣ < ∣α(s)  P ∣ for all s  I. Then, W(α, P) ¼ W(σ, b).

Introduction to topology

435

Suppose that γ is a loop taking an interval I into an open set U in ℝ2. Then, γ can be treated as a map F : C ! U. To be specific, assume that C is the unit circle in the plane and define F by   F ei2πs ¼ γ ðsÞ, for s  ½0, 1

where π : I ! C is the quotient map. The map F is well defined because γ is a loop and hence γ(0) ¼ γ(1). By definition, γ is the composition of the map F with the map taking s to ei2πs. Thus, this composition is continuous and also the map F is continuous. Moreover, each loop γ corresponds exactly to one F and vice versa. For any map F : C ! ℝ2  P, we then define the winding number of F to be W ðF, PÞ ¼ W ðγ, PÞ where γ is the loop satisfying γ ¼ F ∘ π. Proposition 15.16. (Dog on a Leash) Suppose that F and G are maps from a circle C to ℝ2  {P} and that ∣ F(x)  G(x) ∣ < ∣ F(x)  P∣ for all x  C. Then, W(F, P) ¼ W(F, b). Suppose that F is a map from a circle C to another circle C0 , both in the plane. Let P denote the center of C0 and define the degree of F by degðFÞ ¼ W ðF, PÞ Intuitively, the degree F : C ! C0 is just the number of times that F winds C around C0 . As an example, consider the map Fn(z) ¼ zn where x is a point in the unit circle in the complex plane. [The map is equivalent to the map of ℝ2 given in polar coordinates by (1, θ) ⟼ (1, nθ). ] Then, for each n, the path γ n satisfying γ n ¼ Fn ∘ π is just γ n(t) ¼ (cos(n2πt), sin(n2πt)). Thus, from Equation, and we have degðFn Þ ¼ W ðFn , ð0, 0ÞÞ ¼ W ðγ n , ð0, 0ÞÞ ¼ n Lemma 15.4. Two maps F, G : C ! C0 have the same degree if and only if they are homotopic. Moreover, any map F : C ! Cof degree n is homotopic to the map f(z) ¼ zn.

15.7

Applications of windings numbers

In this section, we want to introduce an interesting connection between calculus and topology: the interaction between homotopy of paths and integrals of vector fields along paths. This leads to an intuitively simple definition of the winding number of a path in terms of integral.

436

Magnetic Skyrmions and Their Applications

If A is an open subset of ℝn, then a vector field on A is a function V : A ! ℝn that assigns to each point x  A a vector V(x). In coordinates, we write V as V ðx1 , x2 , …, xn Þ ¼ ðV1 ðx1 , …, xn Þ, V2 ðx1 , …, xn Þ, …, Vn ðx1 , …, xn ÞÞ: A function between open sets in ℝn is said to be continuously differentiable if it has continuous partial derivatives. We will always assume that vector fields are continuously differentiable. If we consider the two-dimensional case V ðx, yÞ ¼ ðV1 ðx, yÞ, V2 ðx, yÞÞ: If γ : I ¼ [0, 1] ! A is a continuously differentiable path in U, defined in coordinates by γ(t) ¼ (x(t), y(t)), then the integral of V along γ is defined to be ð γ

V¼

ð1

V ðγ ðtÞÞ ∙ γ 0 ðtÞdt

(15.1)

0

This integral can be written in coordinates as ð γ

V¼

ð1

½V1 ðxðtÞ, yðtÞÞx0 ðtÞ + V2 ðxðtÞ, yðtÞÞy0 ðtÞ dt

(15.2)

0

This integral has a nice geometric interpretation. First, it is a standard result from calculus that this integral is independent of the parametrization of γ, so we only need to consider the unit-speed parametrization of γ (i.e., we can assume that kγ 0 (t)k ¼ 1). Recall also that the dot product of two vectors a and b, when kak ¼ 1, is the length of the projection of b onto a. Hence, the dot product V(γ(t)) γ 0 (t) is equal to the projection of V onto γ 0 , or, in other words,Ð V(γ(t)) γ 0 (t) is the component of V in the direction of γ 0 . Hence, we can interpret γ V as the total amount of the vector field that is pointing along γ in the direction of increasing t. Example. Let A be a subset of ℝ2, A ¼ ℝ2  {(0, 0)}. Define the vector field V0(x, y) by l

l



y x V0 ðx, yÞ ¼ 2 2 , 2 2 x +y x +y



and define γ by γ(t) ¼ (cos(2πt), sin(2πt) ). Then, the integral of V0 along γ is, ð γ

¼

V¼

ð1  0

ð1

V ðγ ðtÞÞ∙γ 0 ðtÞdt

0

 sin ð2πtÞ cos ð2πtÞ , ð2π sin ð2πtÞ, 2π cos ð2πtÞÞdt cos 2 ð2πtÞ + sin 2 ð2πtÞ cos 2 ð2πtÞ + sin 2 ð2πtÞ

Introduction to topology

¼

ð1

437

ð sin ð2πtÞÞð2π sin ð2πtÞÞ + ð cos ð2πtÞÞð2π cos ð2πtÞÞ dt

0

¼

ð1

2π sin 2 ð2πtÞ + 2π cos 2 ð2πtÞ dt

0

¼

ð1 2π dt 0

¼ 2π

Now, consider the loop γ n defined by γ n(t) ¼ (cos(n2πt), sin(n2πt)). It is easy to verify that ð γn

V0 ¼ n2π:

(15.3)

For a vector field V on an open set A ℝ2, we define the curl of the vector field to be Curl ðV Þ ¼

∂V2 ðx, yÞ ∂V1 ðx, yÞ  ∂x ∂y

It is a straightforward calculation to show that the curl of the vector field V0 above is zero. 

y x Curl ðV ðx, yÞÞ ¼ Curl 2 2 , 2 2 x +y x +y



    ∂ x ∂ y  ¼ ∂x x2 + y2 ∂y x2 + y2 ¼

¼

ðx2 + yr Þ  xð2xÞ ðx 2 + y 2 Þ 2



ðx2 + yr Þ + yð2yÞ

2ðx2 + y2 Þ  2ðx2 + y2 Þ ðx 2 + y 2 Þ2

ðx 2 + y 2 Þ 2 ¼0

A homotopy H between differentiable paths in ℝn is said to be a differentiable homotopy if the map H : I  I ! ℝn is differentiable. Theorem 15.35 is the key connection between path integration of vector fields and differential homotopy. Theorem 15.35. Suppose that A is an open set in ℝn and that γ 0 and γ 1 are differentiable paths in A with common endpoints. Suppose also that V is a vector field on A with Curl(V) ¼ 0. If there exists a differentiable homotopy from γ 0 to γ 1, then

438

Magnetic Skyrmions and Their Applications

ð

ð γ0

V¼

V γ1

There is an important corollary to this theorem. Through this corollary, vector fields and integration provide important information about the topology of paths in ℝ  {(0, 0)}. Corollary 15.4. If n 6¼ m, then the loops γ n and γ m defined above are not homotopic in to the constant map. ℝ2  {(0, 0)}, and γ 0 is the only one that is homotopic Ð 1 Observe from Eq. (15.3) that the integral 2π V is equal to the number of times a 0 γn loop γn goes around the origin. This integral is thus our friend: the winding number. The winding number of a differentiable loop γ : I ! ℝ2 around (0, 0) is ð 1 V0 W ðγ, 0Þ ¼ 2π γ The two differentiable paths in ℝ2  {0} are differentiably homotopic if and only if they have the same winding number. For an arbitrary point P  ℝ2, we can translate the vector field V0 to define the winding number for a loop γ : I ! ℝ2  P. For a point P ¼ (a, b)  ℝ2, define the vector field Vθ, P by Vθ, P ðx, yÞ ¼ Vθ ðx  a, y  bÞ ¼

ðy  bÞ

ð x  aÞ

!

, : ð x  aÞ 2 + ð y  bÞ 2 ð x  aÞ 2 + ð y  bÞ 2

Suppose that γ : I ! ℝ2  P is a differentiable loop. Hence, the winding number of γ around P is W ðγ, PÞ ¼

15.8

ð 1 Vθ, P : 2π γ

(15.4)

Conclusions

In this chapter, we presented a short introduction to the basic idea of general topology. Here, basic concepts in point set topology (topological spaces, continuous maps, metric spaces, constructions of topological spaces, compactness, connectedness) are discussed. Then, an introduction to algebraic topology, such as homotopy and furthermore winding numbers, presented the essential parameters to characterize topologically nontrivial textures in the magnetization field.

Introduction to topology

Further reading C. Adams, R. Franzosa, Introduction to Topology: Pure and Applied, Pearson, 2007. H. Hatcher, Algebraic Topology, Cambridge University Press, 2002. W.F. Basener, Topology and Its Applications, Wiley, 2006. W. Fulton, Algebraic Topology: A First Course, Springer, 1995.

439

Index Note: Page numbers followed by f indicate figures and t indicate tables. A Adiabatic parameter, 290–291, 297, 305, 308 Adiabatic theory of topological Hall effect, 290–296 AHE. See Anomalous Hall effect (AHE) AMR. See Anisotropic magnetoresistance (AMR) Analogous X-ray holography, 75 Analytical model for Neel skyrmion energy calculation, 184–186 ANE. See Anomalous Nernst effect (ANE) Anisotropic DMI and antiskyrmions, 44–46 Anisotropic magnetoresistance (AMR), 405–406 Annihilation of skyrmion, 22, 23–26f Anomalous Hall effect (AHE), 266–268, 289–290 Anomalous Nernst effect (ANE), 270, 278 Antiferromagnetic (AFM) skyrmion, 316–317, 333–334 dynamics of, 336–339 potential applications of, 340–342 statics of, 334–336 thermal properties, 338 vs. ferromagnetic skyrmion, 380–381, 380f Antiferromagnets, 333 bimerons in, 339 Arrhenius law, 182 Atomic force microscope (AFM), 143–144, 170 Atomic-resolution STM image, 152, 158f Atomic structure of the NV defect, 125–127, 126f Average skrymion energy, 193–194 Average skyrmion diameter, 189, 202 B B20 bulk materials, 33–34, 33f B20 single-crystalline systems with bulk DMI, 104–106 Berry-phase and spin Hall effect, 291–293

Bilayer systems, 46 Bimerons in antiferromagnets, 339 Binary digital information, 368 Biological neural network, 393, 394f Biological synapse, 395, 396f Biskyrmions, 61–62, 61f Bloch- and Neel-type skyrmions, 58 Bloch point, 4–5 Boltzmann kinetic equation, 295–296 Boolean computing, 367–368 Boolean logic computing gates, skyrmion for, 407–412, 408–409f, 411f Breathing vs. translation vs. rotation, magnetic skyrmions, 69 Brownian motion of skyrmion, 338–339, 338f Bulk crystals, 33–34, 78–80 C CA. See Cubic anisotropy (CA) CDF. See Cumulative distribution function (CDF) Chiral bobbers, 46–47, 64–66 Chiral Hall effect, 302, 310 Chiral helimagnets, modulated states in micromagnetic energy functional, 348 in one-dimensional spiral modulations, 349 Closed set, 419 Coherent diffractive imaging and ptychography, 74 Coherent X-ray scattering, 84 Compactness, 430 Compact sets, 429–430 Configurational entropy, 182–183, 190–191, 203–205 Confined geometries, 165–166 Conical-to-homogeneous-phase transition at critical field, 353–355, 354f Connectedness, 427–428 Connected sets, 426–428 Continuously differentiable, 436 Contrast formation

442

Contrast formation (Continued) magnetic force microscopy, 100–103 nitrogen-vacancy center microscopy, 125–130 by noncollinear magnetoresistance (NCMR), 152 scanning tunneling microscope, 145–152 by noncollinear magnetoresistance, 152 by tunneling anisotropic magnetoresistance, 150–151 in spin-polarized scanning tunneling microscopy, 146–150, 147f by tunneling anisotropic magnetoresistance, 150–151 Convergent, 421 Cover, 429 Covering map, 433 Covering space, 433 Crystallinity and annealing, 47–48 Cubic anisotropy (CA), 347, 350 oblique skyrmions due to, 362–364 phase diagrams of states, 361–362 stabilization of skyrmion states, 362 Cumulative distribution function (CDF), 210, 210f Current-induced dynamics of AFM skyrmion, 336, 337f of ferrimagnetic skyrmions, 322–325, 323f Current-induced skyrmion lattice generation, 41, 42f Current in-plane (CIP), 256–262 Current perpendicular to the plane (CPP), 256–257, 262–264 Current pulse-assisted nucleation, 112–113, 114f D DDI. See Dipole-dipole interaction (DDI) Defect, 2 Density of states (DOS), 143 Determination of chirality, 121–122 Differentiable homotopy, 437–438 Dilute magnetic semiconductors (DMS), 310 Dipole-dipole interaction (DDI), 162–163 Disconnected set, 427, 427f DMI. See Dzyaloshinskii–Moriya interaction (DMI)

Index

DMS. See Dilute magnetic semiconductors (DMS) Domain walls (DWs), 31 DOS. See Density of states (DOS) Dynamical deflection of topological structures, 18 Dynamical skyrmion and oscillations, 244–247 Dynamics of antiferromagnetic skyrmion, 336–339 Dynamic studies of skyrmions, 68–69 breathing vs. translation vs. rotation, 69 linear motion of skyrmions on racetrack, 68 skyrmion fluctuations, 69 skyrmion hall effect, 68–69 Dzyaloshinskii interaction functional, 348 Dzyaloshinskii–Moriya interaction (DMI), 34, 38, 55, 181, 255, 261–262, 315, 318, 334, 395–397 E Electrical detection of magnetic skyrmions, 266–277 via perpendicular reading schemes, 275–277 via transport magnetoresistive and thermoelectric effects, 266–270 topological Hall effect, 270–275 Electrical detection of skyrmions, 321 Electrical detection via transport magnetoresistive and thermoelectric effects, 266–270, 267f topological Hall effect, 270–275, 272–273f Electrical nucleation of magnetic skyrmions, 256–266 current in-plane (CIP), 256–262 current perpendicular to the plane (CPP), 256–257, 262–264 electrical nucleation without current flow, 264–266 Electrical nucleation without current flow, 264–266 Electric field control of skyrmion, 383, 384f Electron scattering on magnetic skyrmion, 293, 294f Elementary neighborhood, 433 Elliptically distorted spiral states, 347–348

Index

Embedding, 424 Emergent electrodynamics, 274 Energy fluctuations, 194–195 Energy of skyrmion state, 211–214 Energy perpendicular uniform state, 211–214 Entropy density, 198 Exploring spin texture by STM, 153–166 F Ferrimagnetic skyrmions, 315–318, 316f, 325 antiferromagnetic skyrmion vs., 380–381, 380f current-induced dynamics of, 322–325, 323f imaging, writing, deleting, and electrical detection of, 319–321, 320f potential applications of, 325–327, 326f Ferromagnetic multilayers, skyrmion motion in, 240–243 Ferromagnetic resonance (FMR) frequency, 244–245 Ferromagnets, skyrmion motion in, 238–240, 239f First-order phase transition (FOPT), 352–353, 355, 361 FOPT. See First-order phase transition (FOPT) Full-field transmission X-ray microscopy, 73–74 Full width at half maximum (FWHM), 198 Function spaces, 425 G Gaussian distribution, 198 GdFeCo alloy, 319–321 Ginzburg-Landau functionals, 348 H Hall resistivity, 308 Hausdorff, 419 HBL. See Horizontal Bloch lines (HBL) Heisenberg model, 2, 4f Helical-to-conical-phase transition at critical field, 351f, 352–353 Hjj[001], 352–353 Hjj[110], 352 Hjj[111], 353 negative values of kc, 353

443

Helimagnetic domain walls in FeGe, 106, 107f Heusler materials, 39 Homeomorphic, 423 Homeomorphism, 423 Homotopy, 2–3 equivalence, 431 theory, 430–432 type, 431 Hopfions, 66 Horizontal Bloch lines (HBL), 13, 13f Hybrid skyrmions, 241 materials, 43–44 low DMI or hybrid DMI states, 44 stray field effects on the DWs, 43–44 I ID. See Instanton droplet (ID) IEC. See Interlayer exchange coupling (IEC) Imaging micromagnetic state, 104–117 B20 single-crystalline systems with bulk DMI, 104–106 metallic thin film multilayers with interfacial DMI, 110–117 skyrmions in epitaxial oxide heterostructures, 106–110 Imaging spin spirals and skyrmions, 130–138, 131–132f Imaging spin-spiral states and skyrmions by magnetic exchange force microscopy, 172–176 by scanning tunneling microscope, 152–170 exploring spin texture by STM, 153–166 manipulation of spin texture by STM, 166–170 Imaging with electrons, 75–76 Lorentz TEM, 75–76 scanning electron microscopy with polarization analysis, 76 spin-polarized low-energy electron microscopy, 76 Imaging with photons, 70–75 coherent diffractive imaging and ptychography, 74 full-field transmission X-ray microscopy, 73–74

444

Imaging with photons (Continued) scanning transmission X-ray microscopy, 74 X-ray holography, 75 X-ray photoemission electron microscopy, 72–73 Imaging with scanning probes, 76–77 Information computing, 367–368 Instanton droplet (ID), 246 Interface-dominated multilayer materials, 38–43 magnetic states in thin film materials systems, 40–42 skyrmions in ion-irradiated films, 42–43 Interfacial Dzyaloshinskii-Moriya interaction (IDMI), 183, 186 Interlayer exchange coupling (IEC), 243 Interplay between different THE regimes, 307–308 Invariant measure, 224–225 Ion-irradiated skyrmion stabilization, 42, 43f Ising model, 2 Isolated skyrmions, 255–256, 319–321 J Jaynes information entropy, 224–226 K Kelvin probe microscopy (KPFM), 104–105, 105f KPFM. See Kelvin probe microscopy (KPFM) L Landau-Lifshitz-Gilbert (LLG) equation, 15, 183 Large-scale topographic STM image, 151, 151f Leaky-integrate-fire (LIF), 396–397 Lift, 433 Limit point, 420 Limit point compact, 430 Linear motion of skyrmions on racetrack, 68 Linear scaling, 289 Lorentz TEM, 75–76 Low DMI or hybrid DMI states, 44 Low temperatures (LTs), 347 behavior of skyrmion diameter, 213, 214f

Index

M Macrospin model of spin-orbit torque switching, 261–262, 261f Magnetic bimeron, 339 Magnetic bubbles, collapse of, 20–21 Magnetic contrast, 146 Magnetic domains, 368–369 Magnetic exchange force, 144 Magnetic exchange force microscopy (MExFM), 144, 171–172f contrast mechanism, 170–171 imaging spin-spiral states and skyrmion by, 172–176 Magnetic field dependence of skyrmions, 162–163, 163f Magnetic force microscopy (MFM), 99, 104–105, 104–105f, 321 contrast formation, 100–103 imaging the micromagnetic state, 104–117 B20 single-crystalline systems with bulk DMI, 104–106 metallic thin film multilayers with interfacial DMI, 110–117 skyrmions in epitaxial oxide heterostructures, 106–110 manipulation of the micromagnetic state, 117–118 quantitative assessment of complex spin textures, 118–124 determination of the chirality, 121–122 measuring average and local DMI, 119–121, 120f tubular and partial skyrmions, 123–124 Magnetic image, 102–103 Magnetic racetrack memory, 368–369, 369f Magnetic skyrmions, 57, 152, 153f, 255, 315, 369–370, 383, 405 characterization, 255–256 conversion between magnetic domain walls and, 374, 375f duplication and merging of, 374–375, 376f electrical detection, 266–277 magnetoresistive and thermoelectric effects, 266–270, 267f topological Hall effect, 270–275, 272–273f via perpendicular reading schemes, 275–277

Index

electrical nucleation of, 256–266 current in-plane (CIP), 256–262 current perpendicular to the plane (CPP), 256–257, 262–264 electrical nucleation without current flow, 264–266 electron scattering on, 293, 294f static properties of 3D skyrmion diameters distribution (see 3D skyrmion diameters distribution) micromagnetic model, 183–184 Neel skyrmion energy calculation, 184–186 Magnetic states in thin film materials systems, 40–42 Magnetic systems, 56 Magnetic thin film, 80 Magnetic transmission x-ray microscopy (MTXM), 58–61, 60f, 73 Magnetic tunnel junction (MTJ), 395, 397–398 Magnetic X-ray holography, 75 Magnetism topological defects in, 2–5 Bloch point, 4–5 singular vortex, 5 topologically stable structures in, 6–14 nonuniform boundary conditions, 9–13 topological defects for topological solitons, 14 uniform boundary conditions, 6–9 Magnetization dynamics, effect of topology on, 15–19 Thiele equation (see Thiele equation) Magneto-optical (MO) effects, 71 Magnetoresistive and thermoelectric effects, 266–270, 267f Magnetoresistive effects in tunneling junctions, 145–146, 145f Magnus force, 236 Manipulation of micromagnetic state, 117–118 Manipulation of spin texture by STM, 166–170 Materials for hybrid skyrmions, 43–44 low DMI or hybrid DMI states, 44 stray field effects on the DWs, 43–44 Materials for skyrmionics, 33–38

445

Mean free path (MFP), 310 Mean squared displacement for skyrmion, 401 Mean square fluctuation of energy, 195, 215–216, 215f Measuring average and local DMI, 119–121, 120f Memristive switching layers, 394–395 Memristor computing, 393–395 Metallic thin film multilayers with interfacial DMI, 110–117 Metastable skyrmion lattice periodicities, 40–41, 41f Metastable skyrmions, 255–256 Metric, 424 Metric spaces, 424–426 Metric topology, 425 MExFM. See Magnetic exchange force microscopy (MExFM) MFM. See Magnetic force microscopy (MFM) MFP. See Mean free path (MFP) Microcanonical partition function and free energy, 216–217, 217f Micromagnetic energy functional, 348 Micromagnetic model, 183–184 Micromagnetic simulations, 207 Modulated states in chiral helimagnets micromagnetic energy functional, 348 one-dimensional spiral modulations, 349 MO effects. See Magneto-optical (MO) effects Monolayers (MLs), 145–146 MTJ. See Magnetic tunnel junction (MTJ) MTJ device, 248, 248f MTXM. See Magnetic transmission x-ray microscopy (MTXM) N Nano-electronic devices, 394–395 Nanometer-sized skyrmions, 38, 39f NCMR. See Noncollinear magnetoresistance (NCMR) Neel skyrmion energy calculation, 184–186 Neel skyrmion motion, 236–238, 237f Neel-type skyrmions, 370 Neuromorphic computing, 393–395 skyrmionic memristor in, 395–399 Neutron scattering, 84–85

446

Nitrogen-vacancy (NV) center microscopy, 99 contrast formation, 125–130 imaging spin spirals and skyrmions, 130–138, 131–132f N-manifold, 432 Noncollinear magnetoresistance (NCMR), 146, 275 contrast formation by, 152 Nonstatic and dynamics studies, 82 Nontopological droplet (NTD), 246 Nontrivial skyrmions, 315 Nonuniform boundary conditions, 9–13 domain walls, 10 soft magnetic nanostrips, 11–12, 12f vertical Bloch line, 12–13 vortex in soft magnetic film, 10–11 NTD. See Nontopological droplet (NTD) Numerical results for skyrmion diameters distribution, 206–226 2D vs. 3D average diameters and standard deviations, 221, 222f 2D vs. 3D configurational entropies, 221–224, 223f 2D vs. 3D skyrmion diameters distribution, 217–219, 220f 3D model micromagnetic and analytical results, 207–210 energy of perpendicular uniform state, 211–214 energy of skyrmion state, 211–214 Jaynes information entropy, 224–226 mean square fluctuation of energy, 215–216, 215f microcanonical partition function and free energy, 216–217, 217f pressure and equation of state, 217–219 statistical thermodynamic Boltzmann entropy, 224–226 O Oblique skyrmions due to cubic anisotropy, 362–364 One-dimensional spiral modulations, 349 cones, 349 helicoids, 349 Open and closed metric balls, 425 Open cover, 429

Index

Open map, 423 Open sets, 418 Optical holography, 75 P Particle-like substance, 395 Partition function and free energy of skyrmion diameters, 196–197 Path, 428, 428f Path composition, 432 Path connected, 428 Perpendicular anisotropy (PMA), 395–397 Perturbative impulse on skyrmion lattice, 82 Phase diagrams of states, cubic anisotropy, 361–362 Phase transition between two elliptically distorted conical states, 355–357, 356f PMA. See Perpendicular anisotropy (PMA) Polarizer, 245–246 Pressure of skyrmion diameters population, 197–201, 217–219, 218f Product topology, 422 Q Quantitative assessment of complex spin textures, 118–124 determination of the chirality, 121–122 measuring average and local DMI, 119–121, 120f tubular and partial skyrmions, 123–124 Quotient map, 422 Quotient space, 422 Quotient topology, 422 R Racetrack memory devices, 325 Racetrack-type memory, 368–369, 369f RC. See Reservoir computing (RC) Recurrent neural network (RNN), 404, 404f Reservoir computing (RC), 403, 404f principle of, 404–405 skyrmionic, 405–407 Resonant coherent x-ray scattering studies, 82–84, 83f Resonant modes of skyrmions, 247–248 Resonant X-ray scattering, 77–84 bulk crystals, 78–80 nonstatic and dynamics studies, 82

Index

resonant coherent x-ray scattering studies, 82–84, 83f thin films and thinned crystals, 80–81 RNN. See Recurrent neural network (RNN) Ruderman-Kittel-Kasuya-Yosida (RKKY)–DMI, materials for, 47 S Sackur-Tetrode entropy equation, 191, 206 SANS. See Small-angle neutron scattering (SANS) Scanning electron microscopy with polarization analysis (SEMPA), 76 Scanning force microscope (SFM), 143–144 Scanning probe microscopes (SPMs), 99 Scanning transmission X-ray microscopy (STXM), 74 Scanning tunneling microscope (STM), 143 contrast formation, 145–152 by noncollinear magnetoresistance (NCMR), 152 in spin-polarized scanning tunneling microscopy (SP-STM), 146–150, 147f by tunneling anisotropic magnetoresistance (TAMR), 150–151 exploring spin texture by, 153–166 imaging spin-spiral states and skyrmions by, 152–170 manipulation of spin texture by, 166–170 Scattering asymmetry in adiabatic regime, 293–296 Scattering on chiral spin texture in intermediate case, 305–307 Scattering techniques, 77 Second-order phase transition (SOPT), 353 SEMPA. See Scanning electron microscopy with polarization analysis (SEMPA) Sequentially compact, 430 SHE. See Spin-Hall effect (SHE) Sheeted covering, 433 SHMR. See Spin Hall magnetoresistance (SHMR) Singular vortex, 5 Skew scattering due to spin chirality, 297–299 Skyrmion annihilation, 169–170, 169f Skyrmion-based artificial neuron device, 396–397, 397f

447

Skyrmion-based logic computing gates, 374–378 Skyrmion-based logic gates, 375–378, 377f Skyrmion-based nano-oscillator devices, 378–381 Skyrmion-based racetrack memory, 368–374, 371f Skyrmion-based spin-torque nano-oscillators, 340–341 Skyrmion-based transistor-like functional devices, 381–382, 382f Skyrmion Brownian motion, 401–403, 403f Skyrmion collapse, 21–25 Skyrmion configurational entropy, 190–191, 212, 213f Skyrmion diameters distribution, 186–189, 188t, 201–202, 209f 2D model for, 201–206 average skyrmion diameter, 202 configurational entropy, 203–205 low temperature, 205–206 skyrmion diameters distribution, 201–202 standard deviation, 202–203 3D model for, 186–201 average skrymion energy, 193–194 average skyrmion diameter, 189 configurational entropy, 190–191 energy fluctuations, 194–195 equation of state, 197–201 low temperatures, 192–193 partition function and free energy, 196–197 pressure, 197–201 standard deviation, 189 numerical results, 206–226 standard deviation of, 189 Skyrmion dynamics at fundamental time scales, 87–91 Skyrmion energy, 255–256, 256f Skyrmion Hall angle, 236–238 Skyrmion Hall effect, 68–69, 317, 317f, 324–325, 370–372 Skyrmionic memristor in neuromorphic computing, 395–399 Skyrmion lattice (SkL), 33–34, 40–41, 41f, 347 Skyrmion motion, 236 in ferromagnetic multilayers, 240–243

448

Skyrmion motion (Continued) in ferromagnets, 238–240, 239f implications and perspectives, 248–249 in synthetic antiferromagnets (SAFs), 243–244 Skyrmion nucleation by “blowing” of stripe domains, 262, 263f by current perpendicular to plane, 262–264, 264f by gating, 264–266, 265f Skyrmion number, 32 Skyrmion racetrack memory, 239f, 248–249, 255, 256f Skyrmion reshuffler device, 401, 402f Skyrmions, 6–7, 7f, 333 annihilation of, 22, 23–26f antiferromagnetic materials for, 333–334 for Boolean logic computing gates, 407–412, 408–409f, 411f in different material systems, 316f dynamic studies of, 68–69 breathing vs. translation vs. rotation, 69 linear motion of skyrmions on racetrack, 68 skyrmion fluctuations, 69 skyrmion hall effect, 68–69 in epitaxial oxide heterostructures, 106–110 in ferrimagnetic materials, 322 in ferromagnetic materials, 321 in ferromagnets, 333 fluctuations, 69 at fundamental length scales, 86–87 in ion-irradiated films, 42–43 lattices, 62–63 magnetic (see Magnetic skyrmions) mean squared displacement (MSD), 401 for reservoir computing, 403–407, 406f solutions in presence of cubic anisotropy, 359 spin textures in, 55–66, 92–93 three-dimensional skyrmion textures (see Three-dimensional skyrmion textures) two-dimensional skyrmion textures (see Two-dimensional skyrmion textures) stabilization mechanisms of, 359–361

Index

for stochastic computing, 399–403 strings and tubes, 63–64 topological analogy, 34, 35f in various multilayered thin films, 38–39, 40f in weak-coupling regime, 299–302 Skyrmion shifting, 233–244 motion in ferromagnetic multilayers, 240–243 motion in ferromagnets, 238–240, 239f motion in SAFs, 243–244 skyrmion Hall angle, 236–238 spin-transfer torque and spin-Hall effect, 233–234, 234f Thiele’s equation, 234–236 Skyrmions lattice in Cu2OSeO3, 78, 79f Small-angle neutron scattering (SANS), 84–85 SOC. See Spin-orbit coupling (SOC) Soft magnetic nanostrips, 11–12, 12f SOPT. See Second-order phase transition (SOPT) SOT. See Spin-orbit torque (SOT) Space, 417 Spin chirality, 297–299, 302 Spin configurations of ground-state skyrmion, 315–316 for skyrmions, 58, 59f Spin-dependent and nonadiabatic contributions to THE, 302–305 Spin-Hall effect (SHE), 233–234, 234f, 237f, 289 Berry-phase and, 291–293 Spin Hall magnetoresistance (SHMR), 268 Spin-orbit coupling (SOC), 146, 151, 289, 310–311 Spin-orbit torque (SOT), 258–262 skyrmion nucleation by, 261–262, 262f Spin-polarized low-energy electron microscopy (SPLEEM), 76 Spin-polarized scanning tunneling microscopy (SP-STM), 153, 159f constant-current image, 159f, 168–169, 168–169f contrast formation in, 146–150, 147f Fe bilayer islands on Ir, 174–175, 174f of Fe ML on Ir, 152, 158f monolayer and bilayer islands of Fe, 173–174, 173f

Index

Spin structures for a magnetic ChB and an SkT, 64, 65f Spin texture in Bloch-type skyrmion, 58, 60f Spin texture of magnetic vortex, 57, 57f Spin textures in skyrmions, 55–66 three-dimensional skyrmion textures, 63–66 chiral bobbers, 64–66 hopfions, 66 skyrmion strings and tubes, 63–64 two-dimensional skyrmion textures, 57–63 biskyrmions, 61–62, 61f Bloch- and Neel-type skyrmions, 58 skyrmion lattices, 62–63 target skyrmions, 58–61 Spin-torque nanooscillator (STNO) devices, 244–247, 378, 379f, 380–381 Spin-transfer torque (STT), 233–234, 234f, 258–262, 260f Spintronics, 315, 322 Spiral flip near the saturation field, 357–358, 358f Spiral states in helimagnets with cubic anisotropy, 350–358 conical-to-homogeneous-phase transition, 353–355, 354f helical-to-conical-phase transition, 351f, 352–353 phase transition between two elliptically distorted conical states, 355–357, 356f spiral flip near the saturation field, 357–358, 358f SPLEEM. See Spin-polarized low-energy electron microscopy (SPLEEM) SPMs. See Scanning probe microscopes (SPMs) SP-STM. See Spin-polarized scanning tunneling microscopy (SP-STM) Stabilization of skyrmion states, cubic anisotropy, 362 Standard deviation of skyrmion diameters distribution, 189, 202–203 Static properties of magnetic skyrmions 3D skyrmion diameters distribution (see 3D skyrmion diameters distribution) micromagnetic model, 183–184

449

Neel skyrmion energy calculation, 184–186 Static real-space microscopy, 68 Static studies of skyrmions, 67–68 Statistical thermodynamic Boltzmann entropy, 224–226 Steady-state motion of domain walls, 17–18 STM. See Scanning tunneling microscope (STM) STM constant-current image, 152, 153f STNO devices. See Spin-torque nanooscillator (STNO) devices Straight line homotopy, 432 Stray field effects on DWs, 43–44 STT. See Spin-transfer torque (STT) STXM. See Scanning transmission X-ray microscopy (STXM) Subcover, 429 Subspace topology, 421 Synthetic antiferromagnetic bilayer skyrmions, 370–372, 372f Synthetic antiferromagnetic materials for skyrmion, 46, 46f Synthetic antiferromagnetic skyrmion, 243–244, 315–316, 316f Synthetic ferrimagnetic skyrmion, 315–317, 316f T TAMR. See Tunneling anisotropic magnetoresistance (TAMR) Target skyrmions (TSks), 58–61 Taxicab metric, 425, 425f TbFeCo thin film, 319 THE. See Topological Hall effect (THE) Thiele equation, 15–17, 234–236 applications of, 17–19 dynamical deflection of topological structures, 18 steady-state motion of domain walls, 17–18 topological Brownian motion, 18–19 Thin films and thinned crystals, 80–81 Thin film skyrmion materials, 40 Three-dimensional skyrmion textures, 63–66 chiral bobbers, 64–66 hopfions, 66

450

Three-dimensional skyrmion textures (Continued) skyrmion strings and tubes, 63–64 3D model micromagnetic and analytical results, 207–210 3D skyrmion diameters distribution, 186–201 average skrymion energy, 193–194 average skyrmion diameter, 189 configurational entropy, 190–191 energy fluctuations, 194–195 equation of state, 197–201 low temperatures, 192–193 partition function and free energy, 196–197 pressure, 197–201 skyrmion diameters distribution, 186–189, 188t standard deviation, 189 Time-lapsed X-ray scattering, 82 Tip transfer function, 102 TMR effects. See Tunneling magnetoresistance (TMR) effects TNE. See Topological Nernst effect (TNE) Topological Brownian motion, 18–19 Topological defects for topological solitons, 14 Topological droplet (TD), 246 Topological Hall effect (THE), 270–275, 272–273f, 289–290 adiabatic theory of, 290–296 in general case interplay between different THE regimes, 307–308 scattering on chiral spin texture, 305–307 spin-dependent and nonadiabatic contributions, 302–305 general character of, 309–311 weak-coupling regime, 310 skew scattering due to spin chirality, 297–299 skyrmions, 299–302 Topological Nernst effect (TNE), 275 Topological property, 424 Topological soliton, 19–20 Topological spaces, 417–424 Topology, 417 Topology vs. energetic stability, 19–25

Index

collapse of magnetic bubbles, 20–21 skyrmion collapse, 21–25 vortex core reversal, 21 Toward skyrmion applications, 91–92 Trivial topology, 418 TRNGs. See True random number generators (TRNGs) True random number generators (TRNGs), 401–403, 402f TSks. See Target skyrmions (TSks) Tubular and partial skyrmions, 123–124 Tunneling anisotropic magnetoresistance (TAMR), 146, 156f, 275 contrast formation in STM by, 146 Tunneling magnetoresistance (TMR) effects, 155–157 2D model for skyrmion diameters distribution, 201–206 average skyrmion diameter, 202 configurational entropy, 203–205 low temperature, 205–206 skyrmion diameters distribution, 201–202 standard deviation, 202–203 Two-dimensional skyrmion textures, 57–63 biskyrmions, 61–62, 61f Bloch- and Neel-type skyrmions, 58 skyrmion lattices, 62–63 target skyrmions, 58–61 2D vs. 3D average diameters and standard deviations, 221, 222f 2D vs. 3D configurational entropies, 221–224, 223f 2D vs. 3D skyrmion diameters distribution, 217–219, 220f U Uniaxial anisotropy (UA), 350 Uniform boundary conditions, magnetism, 6–9 one-dimensional physical space, 8 three-dimensional physical space, 8–9 two-dimensional physical space, 6–8 V VBL. See Vertical Bloch line (VBL) VCMA. See Voltage-controlled magnetic anisotropy (VCMA)

Index

VCMA effect. See Voltage-controlled magnetic anisotropy (VCMA) effect Vector field, 436 Velocity-current relation for Neel skyrmion, 240–241, 241f Vertical Bloch line (VBL), 12–13, 13f, 261–262 Voltage-controlled magnetic anisotropy (VCMA) effect, 381, 383, 401–403, 408–412 von Neumann bottleneck, 393 Vortex core reversal, 21 Vortex in soft magnetic film, 10–11 W Weak-coupling regime, topological Hall effect in skew scattering due to spin chirality, 297–299

451

skyrmions, 299–302 Winding numbers, 432–435 applications of, 435–438 X XMCD. See X-ray magnetic circular dichroism (XMCD) X-PEEM. See X-ray photoemission electron microscopy (X-PEEM) X-ray Fresnel Zone Plates (FZP), 73 X-ray holography, 75 X-ray magnetic circular dichroism (XMCD), 71 X-ray photoemission electron microscopy (X-PEEM), 72–73 X-ray scattering, 77 XY model, 2