Exploration of Quantum Transport Phenomena via Engineering Emergent Magnetic Fields in Topological Magnets (Springer Theses) 981167292X, 9789811672927

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Springer Theses Recognizing Outstanding Ph.D. Research

Yukako Fujishiro

Exploration of Quantum Transport Phenomena via Engineering Emergent Magnetic Fields in Topological Magnets

Springer Theses Recognizing Outstanding Ph.D. Research

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Yukako Fujishiro

Exploration of Quantum Transport Phenomena via Engineering Emergent Magnetic Fields in Topological Magnets Doctoral Thesis accepted by The University of Tokyo, Tokyo, Japan

Author Dr. Yukako Fujishiro Center for Emergent Matter Science RIKEN Wako, Saitama, Japan

Supervisor Prof. Dr. Yoshinori Tokura Center for Emergent Matter Science RIKEN Wako, Saitama, Japan Department of Applied Physics The University of Tokyo Tokyo, Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-16-7292-7 ISBN 978-981-16-7293-4 (eBook) https://doi.org/10.1007/978-981-16-7293-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Supervisor’s Foreword

The concept of topology now forms one of the central schemes in modern condensed matter physics. Intensive research on quantum materials has unveiled that the geometrical phase of electron wave function leads to novel electromagnetic responses through the generation of gauge field or so-called emergent magnetic field. The representative examples include nanoscale spin whirls such as skyrmions or hedgehogs generating unusual Hall responses, as well as the quantum anomalous Hall effect in magnetic topological insulators. These exotic phenomena arising from the topological nature of magnetic/electronic structures in solids are expected to be exploited for low-energy consumption electronic devices in the near future. This doctoral dissertation by Dr. Yukako Fujishiro presents her study on emergent transport properties in topological magnets, especially focusing on the dense topological spin singularities, spin hedgehogs, in chiral magnets as well as the magnetic Weyl semimetal system. By combining state-of-the-art crystal growth techniques, high magnetic field transport measurements and neutron scattering experiments, she has found a new topological spin texture in the course of the transition between hedgehog and skyrmion lattices, gigantic thermoelectric and anomalous Hall effect due to unconventional electron scattering and also succeeded in making thin flakes of magnetic Weyl semimetal towards the realization of high-temperature quantum anomalous Hall effect. These results not only enrich our understanding on the role of topology in real/momentum spaces, but also offer new guiding principles for designing giant emergent responses to produce novel functionalities. Tokyo, Japan July 2021

Prof. Dr. Yoshinori Tokura

v

Parts of this thesis have been published in the following journal articles: 1.

2.

3.

4.

5.

Large magneto-thermopower in MnGe with topological spin texture Y. Fujishiro*, N. Kanazawa*, T. Shimojima, A. Nakamura, K. Ishizaka, T. Koretsune, R. Arita, A. Miyake, H. Miramura, K. Akiba, M. Tokunaga, J. Shiogai, S. Kimura, S. Awaji, A. Tsukazaki, A. Kikkawa, Y. Taguchi, and Y. Tokura Nature Communications 9, 408 (2018). (*equal contributions) Topological transitions among skyrmion- and hedgehog-lattice states in cubic chiral magnets Y. Fujishiro, N. Kanazawa, T. Nakajima, X. Z. Yu, K. Ohishi, Y. Kawamura, K. Kakurai, T. Arima, H. Mitamura, A. Miyake, K. Akiba, M. Tokunaga, A. Matsuo, K. Kindo, T. Koretsune, R. Arita and Y. Tokura Nature Communications 10, 1059 (2019). Engineering skyrmions and emergent monopoles in topological spin crystals Y. Fujishiro, N. Kanazawa, and Y. Tokura Applied Physics Letters 116, 090501 (2020). Topological Kagome magnet Co3 Sn2 S2 thin flakes with high electron mobility and large anomalous Hall effect M. Tanaka*, Y. Fujishiro*, M. Mogi, Y. Kaneko, T. Yokosawa, N. kanazawa, S. Minami, T. Koretsune, R. Arita, S. Tarucha, M. Yamamoto and Y. Tokura Nano Letters 20, 10, 7476 (2020). (*equal contributions) Giant anomalous Hall effect from spin-chirality scattering in a chiral magnet Y. Fujishiro*, N. Kanazawa*, R. Kurihara, H. Ishizuka, T. Hori, F. S. Yasin, X. Z. Yu, A. Tsukazaki, M. Ichikawa, M. Kawasaki, N. Nagaosa, M. Tokunaga and Y. Tokura Nature Communications 12, 317 (2021). (*equal contributions)

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Acknowledgments

First of all, I would like to express my sincerest gratitude to my advisor Prof. Yoshinori Tokura for offering the world’s best research environment and enlightening guidance throughout this work. My sincere thanks also go to Dr. Naoya Kanazawa for his mentorship. None of my works would have been possible without his generous support. I would like to appreciate especially the following people for their collaboration: Prof. Naoto Nagaosa, Prof. Hiro Ishizuka, Prof. Ryotaro Arita, Prof. Takashi Koretsune, and Dr. Susumu Minami, Prof. Takahisa Aima, Prof. Taro Nakajima, Prof. Kazuhisa Kakurai, Dr. Kazuki Oishi, Dr. Yukihiko Kawamura, Prof. Yasujiro Taguchi, Dr. Akiko Kikkawa, Prof. Masashi Kawasaki, Prof. Masazaku Ichikawa, Prof. Kyoko Ishizaka, Prof. Xiuzhen Yu and Dr. Fehmi Sami Yasin, Prof. Masashi Tokunaga, Dr. Hiroyuki Mitamura, Dr. Atsushi Miyake, Dr. Ryosuke Kurihara, Dr. Kazuto Akiba, Prof. Koichi Kindo, Dr. Akira Matsuo, Prof. Atsushi Tsukazaki, Dr. Junichi Shiogai, Prof. Shojiro Kimura, Prof. Satoru Awaji, Prof. Seigo Tarucha, Prof. Michihisa Yamamoto, Dr. Miuko Tanaka and Dr. Masataka Mogi. I am grateful to Prof. Takahisa Arima, Prof. Tsuyoshi Kimura, Prof. Yukitoshi Motome, Prof. Eiji Saitoh and Prof. Max Hirschberger for their insightful comments and recommendations on this thesis. I also thank all members of the Tokura group in the University of Tokyo and of RIKEN Center for emergent matter science (CEMS) for their kind encouragement and experimental help through my research. Finally and most importantly, I am so grateful to my family for their love and support. Family is my forever source of encouragement and happiness. July 2021

Yukako Fujishiro

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Berry Phase and Emergent Electromagnetism . . . . . . . . . . . . . . . . . . . . 1.3 Berry Phase Engineering in Real Space . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Berry Phase in Real-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Definition of Topology in Spin Structures . . . . . . . . . . . . . . . . . 1.3.3 Formation of Multiple-q State . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Emergent Electromagnetism and THE . . . . . . . . . . . . . . . . . . . 1.4 Berry Phase Engineering in Momentum Space . . . . . . . . . . . . . . . . . . . 1.4.1 Momentum-Space Berry Phase and AHE . . . . . . . . . . . . . . . . . 1.4.2 Gapped Topological Phase and QAHE . . . . . . . . . . . . . . . . . . . 1.4.3 Gapless Topological Phase–Dirac and Weyl Semimetals . . . . 1.4.4 Towards High-Temperature QAHE . . . . . . . . . . . . . . . . . . . . . . 1.5 Purpose of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 4 5 7 9 12 12 14 15 18 19 21

2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Bulk Crystal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 MnSi1−x Gex Poly Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Co3 Sn2 S2 Single Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Magnetization and Electrical Transport Measurement . . . . . . . . . . . . . 2.3 Thermoelectric Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 High-magnetic-field Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Magnetization and Resistivity Measurements . . . . . . . . . . . . . 2.4.2 Thermoelectric Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Neutron Scattering Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Small- and Wide-angle Neutron Scattering Experiment . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 25 25 25 26 27 27 27 29 29 30

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Contents

3 Topological Transitions Between Skyrmionand Hedgehog-Lattice States in MnSi1−x Ge x . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Generic Aspects of B20-Type Compounds . . . . . . . . . . . . . . . . 3.1.3 Cubic-3q Hedgehog Lattice State in MnGe . . . . . . . . . . . . . . . 3.2 Overview of Topological Magnetic Transitions in MnSi1−x Gex . . . . 3.2.1 Magnetic Period and Direction of q-Vectors . . . . . . . . . . . . . . 3.3 Topological Magnetic Phases in MnSi1−x Gex . . . . . . . . . . . . . . . . . . . . 3.3.1 Formation of SkL States in x ≤ 0.2 . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Formation of HL States in x ≥ 0.4 . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Spin Textures of Tetrahedral-4q Hedgehog Lattice State . . . . 3.4 Topological Hall Effect in MnSi1−x Gex . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Topological Hall Effect in SKL of x = 0.2 . . . . . . . . . . . . . . . . 3.4.2 Topological Hall Effect in HLs of x = 0.6 and 0.8 . . . . . . . . . 3.5 Formation Mechanism of Hedgehog Lattice States . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 31 31 33 34 36 38 38 39 43 45 45 45 47 49 49

4 Giant Magneto-Transport Properties Induced by Spin Fluctuations in MnGe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Large Magneto-Seebeck Effect in Bulk MnGe . . . . . . . . . . . . . . . . . . . 4.2.1 Observation of Field-Induced Large Seebeck Effect . . . . . . . . 4.2.2 Photoemission Spectroscopy and Band Structure Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Specific Heat and Magneto-Resistivity . . . . . . . . . . . . . . . . . . . 4.2.4 High-magnetic-field Measurement . . . . . . . . . . . . . . . . . . . . . . . 4.3 Large Anomalous Hall Effect from Spin-Chirality Skew Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Observation of Giant Hall Response in MnGe Thin Film . . . . 4.3.2 Scaling Relation and Temperature-magnetic-field Profile . . . . 4.3.3 Film-Thickness Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Discussion on Other Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Comparison with Other Materials . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 66 66 70 70 72 75 75

5 Topological Transport Properties of Magnetic Weyl Semimetal Co3 Sn2 S2 Thin Flake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Magnetic Weyl semimetal Co3 Sn2 S2 . . . . . . . . . . . . . . . . . . . . . 5.2 Synthesis of Co3 Sn2 S2 Thin Flakes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 77 78

53 53 56 57 60 60 63

Contents

5.3 Transport Properties of Co3 Sn2 S2 thin flakes . . . . . . . . . . . . . . . . . . . . 5.4 Comparison with Band Structure Calculations . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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81 85 87 88

6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Chapter 1

Introduction

1.1 Preface A key concept of condensed matter physics is emergence: interaction between various degrees of freedom in solids (e.g., charge, spin, lattice, and orbital) gives rise to different phases of matter and macroscopic phenomena [1]. Study and classification of different phases and various phase transitions of materials are the fundamental subjects of this field, which would lead to innovative technologies for the next generation. In this context, the concepts of geometry and topology have opened up a new horizon in exploring quantum materials and their emergent properties. In particular, the quantum transport in solids turned out to be crucially affected by the effective electromagnetic fields, so-called the emergent electromagnetic fields. Unlike the conventional symmetry-broken phases defined by order parameters, topological phases are characterized by nontrivial topological invariants. A well known example in the realspace is the nanometer scale spin whirl, so called skyrmion, which are now observed in many magnetic materials since its first discovery in 2009. Thanks to topological protection, i.e., the fact that the skyrmion number is invariant under any continuous deformations, they show a robust particle-like behavior, which may be exploited for electrically controllable non-volatile memory devices. Furthermore, they generate emergent magnetic field acting on conduction electrons, as defined by the geometric phase of Berry curvature. On the other hand, the importance of band topology had been recognized much earlier, since the discovery of quantum Hall effect in 1980. Again, the idea of Berry phase has succeeded in describing nontrivial transport properties such as the anomalous Hall effect in magnetic materials and the spin Hall effect in semiconductors. More recently, various topological materials including topological insulators, Dirac and Weyl semimetals, topological superconductors have been studied both theoretically and experimentally. Owing to their topological invariants, these topological materials harbour characteristic surface states, which can be exploited for dissipationless electronics or topological quantum computing. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 Y. Fujishiro, Exploration of Quantum Transport Phenomena via Engineering Emergent Magnetic Fields in Topological Magnets, Springer Theses, https://doi.org/10.1007/978-981-16-7293-4_1

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1 Introduction

Thanks to the intensive research efforts, respective characteristics of each topological phase are now well understood. However, investigations and demonstrations of topological phase transitions and their critical phenomena are scarce. A naive consideration can readily predict the emergence of exotic phases and phenomena at such nontrivial phase transitions which are characterized by the change of topological invariants; along with a dramatic change of the underlying spin textures, electronic bands, and so the emergent magnetic field. Novel electromagnetic responses which can never appear in a single topological phase may also be expected, owing to the interplay and competition between various degrees of freedom around the transition points. This thesis presents the discovery of new topological magnetic phase and electronic transport properties arising from topological phase transitions. First, we investigate the transitions between topologically non-trivial spin textures, i.e., skyrmionand hedgehog-lattice states, through controlling chemical pressure in new chiral magnets MnSi1−x Gex . Second, we study giant electronic transport properties around the topological phase transition between hedgehog lattice state and ferromagnetic state in MnGe. Through high-magnetic-field measurements, we demonstrate the presence of residual magnetic fluctuations in the nominally ferromagnetic region, leading to the unusual electron scattering. Lastly, we study transport properties in thin flakes of magnetic Weyl semimetal Co3 Sn2 S2 , which provides a promising arena for exploring the high-temperature quantum anomalous Hall effect through controlling the dimensionality of electronic bands. In this chapter, we first give a brief introduction about emergent magnetic field and in particular how they are connected to the geometry of spin textures and electronic bands. We then briefly summarize theoretical and experimental backgrounds of topological phases in real and momentum spaces.

1.2 Berry Phase and Emergent Electromagnetism Berry phase is a fundamental concept of quantum physics, which explains the effect of phase factors of geometrical origin on physical properties [2, 3]. Consider the Hamiltonian and its eigenstate which depends on a set of parameters R = (X, Y, . . . ); H (R)φn (R) = E n (R)φn (R).

(1.1)

For simplicity, we assume that there is no energy degeneracy, and parameters R change with time adiabatically, meaning that the eigenstate at any time t has the same index n. Substituting a wave function |ψn (t) = eiγn (t) |φn (R(t)) (γn (t) being = H (R(t))ψ(t) a phase factor) into a time dependent Schroedinger equation i ∂ψ(t) ∂t gives

1.2 Berry Phase and Emergent Electromagnetism

3

  dγn (t) d = i φn (R(t))| |φn (R(t)) dt dt = i φn (R(t))|∇R |φn (R(t)) · ≡ An (R) ·

dR(t) dt

dR(t) . dt

(1.2)

Calculating the time integral gives the Berry phase as t γn (t) − γn (0) =

dt An (R(t)) ·

dR(t) = dt

 dR · An (R). c

0

It is worth noting that the phase factor γn (t) is no longer a function of time but depends on a line integral in a parameter space which can be arbitrarily chosen, indicating the generality of the concept. If the path is a closed curve, it can be transformed by using Stokes theorem as follows;  γn (t) − γn (0) =

 dR · An (R) =

c

 (∇R × An (R)) · d S ≡

S

Bn (R) · d S. (1.3) S

Here An (R) and Bn (R) are called “Berry connection” and “Berry curvature”, respectively, which are associated with “vector potential” and “magnetic field” in the parameter space, respectively. Therefore, the essence of the Berry phase is to introduce an electromagnetic-field-like gauge field in arbitrary parameter space, thus offering a variety of physical counterparts. Importantly, Berry phase Bn (R) modifies the semiclassical equation of motion for the slowly varying wave packet [4] and hence leads to various novel electromagnetic responses; dr(t) ∂n (k) dk(t) = + Bn (r) × , (1.4) dt ∂k dt 

dr(t) dr(t) dk(t) = −eE − e × Bext −  × Bn (r). dt dt dt

(1.5)

Here, Bn (k) and Bn (r) represent the Berry phase in momentum- and real-spaces, respectively. Bn (k) generates the so-called “anomalous velocity” (the second term in Eq. 1.4), giving rise to a Hall current which is essentially dissipationless. Indeed, it is the origin of various important phenomena discovered in condensed matter physics such as ferroelectricity [5, 6], quantum Hall effect [7, 8], intrinsic spin Hall effect [9], and anomalous Hall effect [10]. On the other hand, Bn (r) act as the additional magnetic field besides the external ones (Bext ), leading to the topological Hall effect observed in non-coplanar and topological spin textures [11].

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1 Introduction

1.3 Berry Phase Engineering in Real Space 1.3.1 Berry Phase in Real-Space When conduction electrons flow with their spin direction aligned along the underlying spin texture, they acquire Berry phase which act as the real-space emergent electromagnetic field [11]. In the continuum limit, the emergent magnetic field (bi ) and electric field (ei ) are described as h εi jk n · (∂ j n × ∂k n), 8π e h ei = n · (∂i n × ∂t n), 2π e

bi =

(1.6) (1.7)

where n is a unit vector parallel to the direction of spins and εi jk is the Levi-Civita symbol. Thus, the emergent magnetic field (bi ) is expected to show up in non-coplanar spin structures. If we suppose three spins 1,2, and 3, the electrons acquire Berry phase which corresponds to 1/2 of the solid angle ( ) subtended by these spins, which is so-called the scalar-spin chirality (SSC) given by Si · (S j × Sk ) (Fig. 1.1). This SSC mechanism of the Hall effect first attracted attentions in manganites and pyrochlore ferromagnets with non-coplanar spin excitations or structures [12–14], and gained a renewed interest in the field of topological spin textures [11, 15]. On the other hand, the emergent electric field (ei ) is related to the time-dependent dynamics of spin structures. It was demonstrated through a current-dependent Hall response in a skyrmion lattice [16], as well as by an emergent induction in a helix which paves the way for microscale inductors of quantum mechanical origin [17].

Fig. 1.1 a Schematics of the emergent magnetic field b associated with non-collinear spin structures [1]. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Ref. [1] Copyright 2017, Nature Publishing Group. The solid angle subtended by the three neighbouring spins, which is proportional to the scalar-spin chirality given by Si · (S j × Sk )

1.3 Berry Phase Engineering in Real Space

5

1.3.2 Definition of Topology in Spin Structures Skyrmions and other topological spin textures are other examples with finite SSC. The concept of skymions was originally proposed by Skyrme in the field of high-energy physics in the 1960s [18, 19], while the possibility of finding magnetic skyrmions in acentric magnets was first predicted by Bogdanov et al. in 1989 [20–22]. The topological nature of a spin structure is characterized by the winding number (or so-called the skyrmion number) w=

1 εi jk 8π

 d Sk n(r) · (∂i n(r) × ∂ j n(r)),

(1.8)

S

which counts how many times the direction of local magnetization n = m(r)/|m(r)| wraps the unit sphere (i.e., = 4π for a single skyrmion) (Fig. 1.2a) [11]. The fact that the skyrmion number is invariant under any continuous deformations gives a particle nature of a skyrmion. Indeed, this topological protection manifests itself in aggregation and collapse dynamics [23, 24], thermodynamic metastability [25–28], driven motion of skyrmions with low electric-current excitation [29, 30], all of which may be exploited for future applications to non-volatile memory devices [31–33]. A single skyrmion carries one emergent flux owing to its integer skyrmion number (Fig. 1.2b). There are versatile spin structures experimentally verified so far, which are topologically distinct with each other (Fig. 1.3) [15, 34]. The sign of the emergent magnetic field corresponds to the sign of the skyrmion number. Skyrmions (w = −1) (Fig. 1.3a) most typically emerge in non-centrosymmetric magnets with broken inver-

Fig. 1.2 a Spin arrangement for a single skyrmion (bottom) and its projection onto the spin space (top) [15]. The magnetization points in every direction, wrapping the unit sphere exactly once. b The emergent magnetic field accompanied by a single skyrmion corresponds to one flux quantum (φ0 = h/e) [1]. Adapted from Ref. [15] with permissions from American Chemical Society, copyright 2020 (a) and by permission from Springer Nature Customer Service Centre GmbH: Ref. [1] Copyright 2017, Nature Publishing Group (b)

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1 Introduction

Fig. 1.3 Schematics of experimentally identified topological spin textures with various dimensionality, topology, and internal structures [15, 34]. The direction of emergent magnetic fields (Beff ) with respect to the external magnetic field (Bext ) is related to the sign of the skyrmion number. Hedgehog and anti-hedgehog are the three-dimensional topological spin singularities, which serve as the source (monopole) and the sink (anti-monopole) of Beff , respectively. Adapted from Ref. [34] with permissions from AIP publishing (a,b,d) and Ref. [15] with permissions from American Chemical Society, copyright 2020 (c)

sion symmetry [35], where the antisymmetric exchange interaction due to spinorbit coupling (SOC), i.e. Dzyaloshinskii-Moriya (DM) interaction [36, 37] serves as the “twisting” interaction between spins. The internal spin structure depends on the underlying crystal symmetry: the Bloch-type skyrmion arises from properscrew helices in chiral magnets (e.g., B20-type alloys [23, 38–41], Cu2 OSe3 [42], EuPtSi [43, 44] and β-Mn-type CoZnMn alloys [45]) while the Néel-type skyrmion is formed by cycloidal helices in polar magnets (e.g., GaV4 S8 [46] and VOSe2 O5 [47]) or at heterostructure interfaces [48, 49]. Anti-skyrmion (w = +1), consisting of both the screw- and cycloid-type helices, can show up in crystals with D2d (e.g. Heusler compound [50]) or S4 symmetry. Recently it was theoretically and experimentally demonstrated that centrosymmetric magnets can also host skyrmions with ultrasmall size (∼a few nm) (e.g., Gd2 PdSi3 [51], Gd3 Ru4 Al12 [52], and GdRu2 Si2 [53]), which are stabilized via geometrical frustration and/or conduction-electron-mediated mechanisms such as Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction or higherorder exchange interactions [54–56] . Meron/anti-meron lattice (w = ±1/2) can be realized in chiral magnets with in-pane fvmagnetic anisotropy (Fig. 1.3b) [57] while

1.3 Berry Phase Engineering in Real Space

7

bi-skyrmion lattice can be realized by the competition between uniaxial magnetic anisotropy and the dipolar interaction (Fig. 1.3c) [58, 59]. Besides these two-dimensional spin textures (Fig. 1.3a–c), three-dimensional topological spin structures can also exist, such as the hedgehog (w = +1) and antihedgehog (w = −1) (Fig. 1.3d). These spin singularities act as the source and sink of quantized emergent magnetic field (∇ · Beff = 0), thus correspond to the emergent magnetic monopole and anti-monopole acting on conduction electrons [60–62].

1.3.3 Formation of Multiple-q State Besides the particle-like nature as discussed above, topological spin structures are often found as a crystalline lattice in magnetic materials. For the case of hexagonal skyrmion lattice (SkL), this can be viewed as the hybridization state of three helical structures (i.e. triple-q state), where n(r) is expressed as n(r) ≈ nuniform +

3 

nqi (r + ri ).

(1.9)

i=1

  Here, nqi (r) = A ni1 cos(qi · r) + ni2 sin(qi · r) represents each helical structure with their phase qi · ri , A is the magnetization of each helix with the modulationvector (q-vector) qi , and nuniform is the uniform magnetization induced by the external magnetic field. In bulk chiral magnets, the formation of the triple-q state is assisted by the thermal fluctuations, which has been explained by the phenomenological LandauGinzburg theory with Gaussian fluctuations [38]. A free energy can be expanded up to fourth order of magnetization m as  F[m] =

d 3r [r0 m2 + J (∇m)2 + 2Dm · (∇ × m) + U m4 − B · m],

(1.10)

where the third term represents DM interaction and the last term represents Zeeman term. The Fourier transform of the Eq. 1.10 on the order of m4 has the term 

(mf · mq1 )(mq2 · mq3 )δ(q1 + q2 + q3 ).

(1.11)

q1 ,q2 ,q3 =0

where mf is the uniform ferromagnetic component along the external magnetic field B. This leads to the formation of triple-q state in a plane perpendicular to B, with the three q-vectors satisfying the condition q1 + q2 + q3 = 0 (Fig. 1.5a). In SkL, the same magnetic texture is repeated along the B-direction, leading the formation of skyrmion tubes. It is worth noting that considering the spatial anisotropy of the coefficient U of the m4 in Eq. 1.11 can lead to versatile multiple-q topological spin crystals [63].

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1 Introduction

Fig. 1.4 a Various multiple-q states in two-dimensions and three-dimensions. b Ground state phase diagram for various multiple-q states as functions of magnetic field B and a parameter for magnetic anisotropy C to second order of gradient in magnetization. U and W are the parameters for magnetic anisotropy to second and fourth order of magnetization, respectively. Reprinted with permission from Ref. [64]. Copyright 2011 by American Physical Society

The possibility of diverse multiple-q states has been also demonstrated by a variational calculation, showing a ground-state phase diagram in the space of anisotropy, magnetic field, and interaction parameters (Fig. 1.4) [64]. Among them, a prominent example is the multiple-q state formed by three orthogonal q-vectors, which is the three-dimensional lattice of spin hedgehogs and anti-hedgehogs (i.e. cubic-3q hedgehog lattice (HL) state) as shown in Fig. 1.5b. Experimentally, the formation of multiple-q states can be identified through smallangle neutron scattering (SANS) experiments [38, 41] (Fig. 1.5c,d). For example, the hexagonal SkL was first reported in MnSi in 2009 [38], through the observation of the six-fold intensity pattern with the out-of-plane B; this result suggests the presence of three independent helices with mutual angles of 120◦ in a plane perpendicular to the B (Fig. 1.5c). On the other hand, the cubic-3q HL was first reported in polycrystalline MnGe in 2012 [41], showing the intensity perpendicular to the applied in-plane B (denoted by the black circle in Fig. 1.5d). In this case, one of the three q-vectors flips along the B-direction while the remaining two q-vectors produces the intensity perpendicular to the B. We note that the phase information of the helices (qi · ri ) is difficult to obtain with the current diffraction experiments, although it may have a significant impact on the consequent spin texture. For instance, the in-plane triple-q state can also give rise to a triangular lattice of meron/anti-meron instead of the SKL, if all the three helices have a phase shift of π/6 [51]. The averaged scalar-spin chirality of such meron/antimeron lattice should be zero, and hence we can distinguish them from the SkL in terms of the topological Hall effect. Furthermore, three-dimensional multiple-q states with four (or more) q-vectors may also have the phase degrees of freedom, which will require future theoretical investigations. From the viewpoint of such phase degrees

1.3 Berry Phase Engineering in Real Space

9

Fig. 1.5 Multiple-q states described by the superpositions of helical spin structures and their detection methods. a, b Hexagonal SkL formed by the in-pane triple-q state (a) and cubic-3q HL formed by three orthogonal q-vectors (b). c, d Characteristic SANS intensity patterns for SkL (c) and cubic-3q HL (d). The six-fold intensity appears under out-of-plane B for SkL [38], while the intensity perpendicular to the in-plane B emerges for cubic-3q HL [41]. e, f LTEM images for SkL with out-of-plane B [23] and cubic-3q HL seen from the direction along one of the q-vectors [65]. In panel (f), the projected positions of hedgehogs and anti-hedgehogs are shown with blue and red dots, respectively. Adapted from Ref. [38] with permissions from AAAS (c); Ref. [41] with permissions from American Physical Society, copyright 2012 (d); Ref. [23] with permissions from Springer Nature Customer Service Centre GmbH, Copyright 2018, Nature Publishing Group, a division of Macmillan Publishers Limited (e); and Ref. [65] with permissions from American Chemical Society, copyright 2021 (f)

of freedom which cannot be identified by any diffraction techniques, the real-space observation by Lorentz transmission electron microscopy (LTEM) is one potential probe to directly reveal the spin textures [23, 65] (Fig. 1.5e,f).

1.3.4 Emergent Electromagnetism and THE The density-wave picture of topological spin crystals as discussed above is essential for describing various collective excitations and emergent electromagnetism. In metallic compounds, for instance, the coupling between conduction electrons and topological spin textures gives rise to topological Hall effect (THE) [38, 66], topological Nernst effect [67], emergent electromagnetic induction [16], and nonreciprocal

10

1 Introduction

conductance [68, 69]. In insulators, on the other hand, SkL can give rise to multiferroics with magnetoelectric properties [42] and microwave directional dichroism due to electromagnon resonances [70, 71]. Here we focus on the THE, where the emergent magnetic Beff field induces the transverse motion of electrons (Fig. 1.6a), and explain its relation with the magnetic period and the dimensionality of topological spin textures. Experimentally, THE is detected as an additional Hall signal besides normal Hall effect (NHE) and anomalous Hall effect (AHE). The measured Hall resistivity is N A T N + ρ yx + ρ yx = R0 B + Rs M + P R0 Beff . Here, ρ yx , roughly expressed as ρ yx = ρ yx A T ρ yx , and ρ yx represent normal, anomalous, and topological Hall resistivity, while R0 , Rs , and P are the normal, anomalous Hall coefficients and spin polarization factor T depends not only on the bare Beff , but also on the of conduction electrons. Thus, ρ yx local spin polarization factor P and R0 , both of which are determined by the detail of band structures [72]. In particular, the polarization factors are usually small due to the coexistence of spin-up and spin-down conduction electrons in itinerant systems: e.g., P ∼ 0.025 (MnSi) [73, 74], P ∼ 0.1 (MnGe) [66], and P ∼ 0.07 (Gd2 PdSi3 ) [51]. T between different compounds would be Although a simple comparison of ρ yx difficult for the above reasons, it can still be a good measure of the magnitude of Beff . As described in the previous section, one skyrmion carries one flux quantum (φ0 = h/e) regardless of its size (ask ). The consequent Beff is in inverse proportion to ask (Beff = −φ0 /ask ) and hence can be extremely large for the small skyrmions (e.g., Beff ∼ 4, 000 T for ask = 1 nm2 .) In the case of topological spin textures, the Beff is enhanced in proportion to the sheet density of skrymion: Beff ∝ φ0 /λ2 where λ is the magnetic period of the helical structures. Thus, the large THE is generally expected in short-period topological spin textures. Indeed, topological Hall resistivity as large as 2.6 μ cm was identified in Gd2 PdSi3 (λ = 2.5 nm) at the lowest temperature [51], T ∼ 4 n cm, λ = 18 being one or two orders larger than that of the SkL in MnSi (ρ yx nm) near the transition temperature [73] (Fig. 1.6d). Note that a metastable SkL at the T ∼ 30 n cm), which is attributed low temperature in MnSi shows enhanced THE (ρ yx to the enhancement of P [25]. Since HLs also have short λ, large topological Hall resistivity, typically on the order of μ cm, has been observed in MnGe (λ = 2.8 nm) (Fig. 1.6e) [66] and in MnSi1−x Gex (λ = 1.9–2.3 nm for x = 0.4–0.8) [75] which will be discussed in Chap. 3. We finally comment on the B-dependence of THE in HLs, which is distinct from that of SkLs and is associated with the position change of monopoles and anti-monopoles. In HLs, emergent monopoles and anti-monopoles exist already at zero magnetic field, however, the entire Beff cancels out when averaged over the whole lattice. Upon application of external magnetic field, emergent monopoles (antimonopoles) move in the parallel (antiparallel) directions, creating longer skyrmion strings with a negative Beff (Fig. 1.6c) [62]. As a result, THE gradually emerges from the zero-field and remains up to the ferromagnetic (FM) transition (Fig. 1.6e). This is in stark contrast to SkLs (Fig. 1.6c), where THE appear as a step-like anomaly in T , only in a limited B-range where SkL is stabilized [73] (Fig. 1.6d). ρ yx

1.3 Berry Phase Engineering in Real Space

11

Fig. 1.6 a A schematic showing skyrmion motion and associated physical phenomena [11]. In particular, electrons are deflected by the Lorentz force due to the emergent magnetic field (i.e., THE). b, c Schematics illustration for the Beff in SkL (b) and HL (c). The Beff is generated along the skyrmion tubes in SkL (b), while it is generated by the elongated skyrmion strings connected by emergent magnetic monopoles and anti-monopoles in HL (c). d, e Topological Hall resistivity of SkL in MnSi [73]d and HL in MnGe [66]e. Adapted by permissions from Springer Nature Customer Service Centre GmbH: Ref. [11] Copyright 2013, Nature Publishing Group, a division of Macmillan Publishers Limited (a); adapted from Ref. [73] with permissions from American Physical Society, copyright 2009 (d); and adapted from Ref. [66] with permissions from American Physical Society, copyright 2011 (d)

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1 Introduction

To summarize, many topological spin textures with different dimensionality, topology, and internal structures have been identified so far. Moreover, these properties have direct consequences on the emergent properties. As can be seen from the THE in SkL and HL, the dimensionality of topological spin textures have a significant impact on the spatial distribution of Beff and hence the emergent electromagnetism. Such distinct differences between SkL and HL are one of the motivations of our study which will be discussed in Chaps. 3 and 4.

1.4 Berry Phase Engineering in Momentum Space 1.4.1 Momentum-Space Berry Phase and AHE As described in Sect. 1.2, the momentum space Berry phase Bn (k), associated with the twisting of the electron wavefunctions, gives the anomalous velocity to the particle, which leads to the dissipationless Hall currents. The Hall conductivity of this current is described by the Kubo formula as σxinty =

 e2  dk f (n (k))Bnz (k)  n (2π )d

(1.12)

The integral of the emergent magnetic field over the first Brillouin zone can be an integer number, leading to the quantized Hall conductivity as σxinty = νe2 / h. Here, ν is the so-called the Thouless, Khomoto, Nightingale, den Nijs (TKNN) number or Chern number, which was introduced in the context of quantum Hall effect in the two-dimensional electron system with broken time reversal symmetry [7, 8, 76, 77]. Importantly, this Chern number cannot be continuously transformed to the trivial state ν = 0 or a vacuum, and hence it provides a new scheme to classify the electronic states of matter, in contrast to the conventional orders associated with symmetries. Also, the gapless edge state is forced to appear at the sample edge, which is the phenomenon so-called the bulk-boundary correspondence. Before introducing topological electronic phases with finite Chern number or quantized Hall conductivity, we first summarize the basic backgrounds of the anomalous Hall effect (AHE) which are ubiquitously observed in magnetic materials. It is now well established that the AHE originates from three distinct mechanisms: (i) the intrinsic mechanism arising from momentum space Berry curvature, and extrinsinc mechanisms arising from electron scattering from impurity potential which include (ii) skew scattering and (iii) side jump mechanisms. These three contributions can be understood with an unifying picture by taking the Bloch state transport lifetime τ or conductivity (σx x ) as crossover parameter (Fig. 1.7). The intrinsic AHE, first proposed by Karplus and Luttinger [78], is solely determined by the Berry curvature of momentum space as discussed above, and hence is independent of τ (i.e., σxinty ∼const.). The order estimate of this intrinsic contribution

1.4 Berry Phase Engineering in Momentum Space

13

Fig. 1.7 Plot of Hall conductivity σx y and longitudinal σx x for various ferromagnets [82]. Three crossover regimes appear depending on the value of σx x or transport lifetime τ , which are dominated by the skew-scattering, intrinsic AHE, and side-jump, respectively. Reprinted with permission from Ref. [82]. Copyright 2008 by American Physical Society

is given by assuming the quantization limit in three dimensions σxinty = e2 / ha ∼ 1000 −1 cm−1 where a is the lattice constant. This gives the plateau-like behavior of σx y at 103 < σx x < 106 −1 cm−1 in the universal scaling curve of the AHE as shown in Fig. 1.7. On the other hand, electron scattering by the impurity potential can also lead to the AHE (i.e., extrinsic mechanisms). Although they are not related to the momentum space Berry curvature, we here briefly comment on their characteristics. The skewscattering mechanism was proposed by Smit [79], which is the asymmetric electron scattering due to the SOC at the impurity potential. The contribution of the skewscattering surpasses that of the intrinsic contribution in the high-conductivity regime ∝ σx x . However, the Hall (σx x > 106 −1 cm−1 ), with a linear scaling relation σxskew y angle (σxskew /σ ) remains small on the order of 0.1–1 % because the energy scale x x y of the SOC (E SO ) is much smaller than the width of the virtual bound state (i.e., ∼ σx x · (E SO vimp /W 2 )). Here, vimp is the impurity potential strength and W is σxskew y the electron bandwidth [10, 80–82]. Experimentally, the skew-scattering mechanism has been studied in super-clean systems with non-magnetic or single-spin impurities (e.g., Si or Co-doped Fe) [83, 84]. In contrast, the side-jump mechanism, proposed ∼const.) to that of intrinsic by Berger [85], yields an identical scaling relation (σxskew y AHE. However, the side-jump contributions become much smaller, again due to the ∼ (ha/e2 )/(E SO /E F )  small energy scale of the SOC, being on the order of σxskew y int σx y [10, 80–82].

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1 Introduction

1.4.2 Gapped Topological Phase and QAHE The quantum anomalous Hall effect (QAHE) is the quantized version of the AHE. The development of these two research fields was, however, rather independent in the early years until their close relationship was finally understood in a unified viewpoint from momentum space Berry curvature. Historically, the field of integer quantum Hall effect can be traced back to the discovery of QHE in the two-dimensional (2D) electron gas under strong magnetic field [7, 8, 76, 77]. A conceptual breakthrough was then provided by the Haldane’s model in 1988 [86], where he pointed out a possibility that a quantized Hall effect can exist in the absence of the external magnetic field. This proposal has led to the generalization of the concept of topological electronic phases, which is a big step forward the subsequent discovery of QAHE or Chern insulator states in many materials. Since then, theoretical works predicted QAHE in various synthetic 2D systems with broken time-reversal symmetry. Among them, the rapid research progress in the field of topological insulators (TIs) strongly stimulated the studies of the QAHE [87, 88]. The bulk band topology of time-reversal invariant 2D and 3D TIs are characterized by the Z2 topological invariant (0 or 1), which corresponds to the existence or absence of edge/surface modes [87, 88]. This is in contrast to the case of broken time-reversal symmetry (TRS), including the quantum Hall effect, which are characterized by an integer group Z. The Z2 topological invariant is related to the parity of band inversions between valence and conduction bands at time-reversal invariant momentum (TRIM) points. Due to the bulk-boundary correspondence, the surface of the 3D TI hosts massless Dirac fermions with spin-momentum locking [89, 90]. Magnetic doping in TI opens a mass gap in the Dirac surface state through the exchange interaction, where the Hamiltonian is written as [87, 91] H = vF (σx k y − σ y k x ) + mσz = R · σ, R = (vF k y , −vF k x , m).

(1.13) (1.14)

Here, σ is the pauli matrix of the electron spin and mσz is called a mass term which is proportional to the magnetization. The consequent anomalous Hall effect can be calculated by [76] σx y

e2 =− h

 BZ

dk ˆ R· 4π 2



ˆ ˆ ∂R ∂R × ∂x ∂y

ˆ = R. R R

,

(1.15) (1.16)

2

This gives a half quantized Hall conductance (± 21 eh ) for one surface of the 3D TI, where the summation of top and bottom surfaces leads to the quantized Hall 2 conductivity (± eh ) (Fig. 1.8a). Since the first experimental observation in 2013 [92],

1.4 Berry Phase Engineering in Momentum Space

15

Fig. 1.8 a The quantized Hall resistance in Cr-doped magnetic TI (Bi, Sb)2 Te3 . b A schematic drawing of the QAHE with dissipationless chiral edge mode. Reprinted from Ref. [92] with permission from AAAS

the QAHE has been realized in Cr or V doped (Bi, Sb)2 Te3 by appropriate Fermi energy tuning inside the gap through controlling the Bi, Sb composition ratio as well as by applying a gate voltage [92–96],. Also, the realization of the QAHE leads to various emergent phenomena including dissipationless chiral edge modes (Fig. 1.8b), spintronic functionalities, topological magnetoelectric and magneto-optical effects, as well as topological superconductivity [97].

1.4.3 Gapless Topological Phase–Dirac and Weyl Semimetals The field of topological electronic phases has recently expanded to include gapless metals or semimetals, which feature three-dimensional (3D) Dirac or Weyl node protected by symmetry [98–101]. Here we briefly summarize the definition of Weyl and Dirac fermions as well as the consequent emergent phenomena. Also, their close relationship to the Chern insulating state of the QAHE will be discussed. The canonical Hamiltonian for the Weyl fermion reads [101, 102]  H=

dkψ † (k)h(k)ψ(k),

(1.17)

where ψ(k) is the two-component field operator and h(k) is a 2 × 2 matrix defined as h(k) = ηvσ · k. (1.18) Here, the sign η = ±1 represents the chirality of the Weyl fermion. The Weyl point is assumed to be located at the origin for simplicity. The eigenvalue problem for the two-component wave function at each k-point h(k)|φnk  = E n (k)|φnk  leads to the two energy eigenvalues E n (k) = ηv|k|, resulting in a linear dispersion with right- or left-moving particles. Such chirality of the Weyl fermions manifest itself in transport phenomena under magnetic field, such as the negative magnetoresistance

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1 Introduction

due to chiral anomaly (i.e., charge transfer proportional to E · B between two Weyl fermions) [103]. The calculation of the Berry curvature with the Hamiltonian shown in Eq. 1.17 leads to the momentum space Berry curvature as [2] Bn (k) = nη

k , 2|k|3

(1.19)

for the upper (n = +) and lower (n = −) energy eigenstates. This suggests that the divergence of Bn (k) is non-zero (i.e., ∇k · Bn (k) = 0), and the total flux of the gauge field is quantized to be ±2π for any surface enclosing the Weyl point. Hence, the Weyl points act as the (anti-)monopoles of Berry flux (Fig. 1.9). This was first studied in terms of accidental band crossing in ferromagnetic metals [3, 10, 104]. Also, a promising platform of Weyl fermions includes the phase transition between trivial insulator to topological insulator [105], as well as the multilayered topological insulators [106]. Owing to the “no-go theorem” [107], the Weyl nodes with opposite chirality need to appear in pairs, where the separation length in momentum space determines the consequent anomalous Hall conductivity as σx y =

e2 2π h

    ki  . 

(1.20)

Here, ki is the distance between the ith pair of Weyl nodes. Owing to the nontrivial bulk topology, the Weyl semimetals host characteristic surface states socalled the Fermi arc states, which terminate on the surface projections of the band touching points (Fig. 1.9). This exotic surface states have been directly identified through surface-sensitive ARPES measurements [99] and transport measurements (e.g. Shubnikov-de Haas oscillation arising from the Weyl orbits) [108]. Note that the realization of Weyl points requires breaking of either inversion symmetry (IS) or time-reversal symmetry (TRS), since the Berry curvature vanishes (Bn (k) = 0) if Bn (k) = −Bn (−k)(IS) and Bn (k) = Bn (−k)(TRS) are simultaneously satisfied. The minimum number of Weyl points required for IS- and TRSbroken systems are four and two, respectively (Fig. 1.10a, b). In systems with both IS and TRS, the Kramers degeneracy exists for every momentum k and the effective Hamiltonian is described by the 4 × 4 matrix [110]. This is so-called the 3D Dirac fermion which can be regarded as the superposition of two Weyl nodes with opposite chiralities. While the Weyl fermion is robust against perturbations since it uses all three of the Pauli matrices (i.e., perturbation only results in the position shift of Weyl points), a perturbative mass term can be introduced to the 3D Dirac nodes to open a gap. Such massive Dirac states have been reported at the surface of magnetically doped TI [111] or in magnetic Dirac metals with layered kagome lattices [112, 113]. On the other hand, an additional crystalline symmetry can again protect the 3D Dirac state from such gap opening, leading to the 3D Dirac metal or semimetal state [114]. Among the experimentally identified Weyl semimetals, the magnetic Weyl semimetals with broken TRS offer a fertile playground for the interplay between magnetism and band topology, which may lead to rich quantum states and func-

1.4 Berry Phase Engineering in Momentum Space

17

Fig. 1.9 Schematics of a pair of Weyl fermions with linear dispersions in the 3D Brillouin zone and the connection of the surface states to the bulk Weyl points (Fermi-arc state) [99, 109]. Reprinted with permission from Ref. [99]. Copyright 2018 by American Physical Society

Fig. 1.10 a, b Schematics showing the simplest Weyl semimetals with broken TRS (a) and IS (b), which contain one pair and two pairs of Weyl points, respectively. The arrows illustrate the Berry curvature while k and s represent the momentum and spin, respectively. c Various quantum states neighbouring the magnetic Weyl semimetal state. The transitions can be achieved by tuning the parameters such as magnetism, thickness, or electron correlation. Reprinted from Ref. [117] with permission from AAAS

tionalities by tuning parameters (Fig. 1.10c). One of the hallmarks of magnetic Weyl semimetal is the emergence of large anomalous Hall effect at zero magnetic field [10]. In particular, the materials with large anomalous Hall angle may be a promising platform to study their topological phenomena (e.g. Co3 Sn2 S2 (∼20 % in) [115] and B-induced Weyl semimetal state in GdPtBi (∼ 15 %) [116]), in a sense that the contribution from trivial bands are small. Finally, it is worth noting that the magnetic Weyl semimetals are indeed closely related to the Chern insulators and provides an alternative way to realize the

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1 Introduction

Fig. 1.11 The Chern number evaluated for each 2D plane with fixed k z showing a jump when the plane moves across the Weyl points [118]. Reprinted by permission of the Taylor & Francis Ltd

QAHE [118, 119]. Here we consider a single pair of Weyl points along the k z directions in TRS-broken systems. We assume a 2D plane for a fixed k z with fully gapped band structure, where the integral of Berry curvature is quantized with a well-defined integer Chern number within the plane. Because the gap must be closed and reopened when moving across the Weyl point (i.e., topological phase transitions), the Chern number should show a jump as shown in Fig. 1.11. The total Hall conductivity of the π 1 dk z σx y (k z ). Hence, if we are system is given by the integral of k z as σxtotal y = 2π −π

able to make a 2D system of magnetic Weyl semimetal where k z is quantized and only particular values of k z are allowed, a quantized Hall conductivity may appear, leading the emergence of the QAHE.

1.4.4 Towards High-Temperature QAHE Realization of the quantum anomalous Hall effect (QAHE) [91, 120] at high temperature is one of the most important goals of condensed matter physics. As mentioned earlier, the QAHE has been mainly studied in magnetically-doped topological insulators (TIs) (e.g., Cr or V doped (Bi,Sb)2 Te3 ) [92–96]. However, their quantization temperature (TQAHE ∼ 2K at maximum [121]) has been much smaller than the Curie temperature (a few tens of Kelvin) or the size of exchange gap (hundreds of Kelvin).

1.4 Berry Phase Engineering in Momentum Space

19

This is partly due to the inhomogeneity of the magnetization gap with randomly distributed magnetic dopants [122]. On the other hand, QAHE was recently achieved at relatively high temperatures in intrinsic magnetic TIs such as MnBi2 Te4 -family (TQAHE ∼1.4 K at zero magnetic field (B) and TQAHE ∼ 6.5 K under B) [123] and twisted bilayer graphene (TQAHE ∼ 3 K) [124]. In these cases, however, the relatively low magnetic ordering temperatures hinder the realization of high-temperature QAHE. Hence, it would be of great importance to search for topological materials with intrinsic magnetism and high magnetic ordering temperature.

QAHE Due to Quantum Confinement Effect As mentioned earlier, one promising direction for the realization of high-temperature QAHE is the quantum confinement of magnetic Weyl semimetal by controlling the dimensionality. This strategy has some advantages compared to the case of magnetically doped TIs thin films. Firstly, the stoichiometric material avoids the difficulty of doping, which is in general good for raising the quality of the sample. Second, the quantization temperature may be raised up to the magnetic ordering temperature of the material. Moreover, it offers the platform to explore QAHE with a high Chern numbers. Since Dirac and Weyl Fermions are only defined in the 3D momentum space, the breaking of the translational symmetry along one direction leads to the gap-opening through the formation of subbands with discretized energy levels (Fig. 1.12) [119, 125]. The QAHE shows up when the band inversion is maintained and the Fermi level (E F ) is tuned inside the gap. Importantly, the quantum confinement effect shows up not only in the 2D limit but also survives up to a critical thickness determined by the Fermi wavevector of the material t ∼ 2π/kF (e.g., t ∼ 23 nm in Dirac semimetal Cd3 As2 thin films) [126]. To summarize, various gapped and gapless topological electronic phases have been identified so far. The bulk-boundary correspondence results in characteristic surface modes/states for each topological phase, which may be exploited for dissipationless electronics. In that sense, the realization of high-temperature QAHE is crucial, where the dimensionality control of magnetic Weyl semimetal offers an alternative solution. This is the motivation of our study which will be discussed in Chap. 5.

1.5 Purpose of This Thesis As explained in the above sections, versatile topological magnetic and electronic phases have been identified. Furthermore, each topological phase exhibits unique emergent phenomena associated with their topological invariants. Nevertheless, topological phase transition and their critical phenomena have yet to be fully investigated.

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1 Introduction

Fig. 1.12 Theoretical calculation on the QAHE in ferromagnetic HgCr2 Se4 thin film with a single pair of Weyl fermions [119]. a Formation of subbands with discretized energy levels as a function of film thickness. b The calculated Hall conductance as a function of film thickness. Reprinted with permission from Ref. [119]. Copyright 2011 by American Physical Society

In this thesis, we focus on the emergent magnetic phases and phenomena through engineering the topology of spin texture and electronic bands. For this aim, we have developed appropriate material platforms and explored their characteristics mainly through transport measurements and neutron scattering experiments. In Chap. 3, we study the transition between two prominent topological spin textures, i.e. skyrmion- and hedgehog-lattice states, in new chiral magnets MnSi1−x Gex prepared by high-pressure synthesis. By controlling the lattice constants through substitution between Si and Ge, we systematically reveal the evolution of spin textures and topological Hall effects by means of neutron scattering, Lorentz transmission electron microscopy, magnetization and transport measurements. This study not only led to the discovery of new topological spin texture at the intermediate composition range, but also offered new insights into the formation mechanism of hedgehog lattice states with unusually short magnetic period which cannot be explained by the conventional DMI-based models. In Chap. 4, we investigate transport properties at the topological phase transition between hedgehog lattice state and the ferromagnetic (FM) state in MnGe. This is partly motivated by the assumption that the annihilation process of topological spin singularities (i.e., hedgehogs and anti-hedgehogs) can be highly non-trivial: in a sense that dramatic change in the spin texture may lead to critical magnetic fluctuations. Indeed, we discover unusual magnetic-field-induced Seebeck effect in bulk samples as well as the large anomalous Hall effect in single-crystalline thin films, both of which cannot be explained by the conventional mechanisms. By performing highmagnetic-field measurements, we demonstrate that these phenomena are strongly related to magnetic fluctuations which seem to survive even in the nominally ferromagnetic region, leading to unconventional electron scattering. In Chap. 5, we explore transport properties in thin flakes of magnetic Weyl semimetal Co3 Sn2 S2 synthesized by the CVT method for the first time. The final goal

1.5 Purpose of This Thesis

21

of this project is the realization of high-temperature quantum anomalous Hall effect through the quantum confinement effect, i.e. topological phase transition induced by controlling the dimensionality of electronic structure. Our transport measurements suggest not only the high sample quality of the thin flakes but also some unusual features absent in bulk samples, providing a viable platform for studying the non-trivial crossover from the three-dimensional topological semimetal to the two-dimensional quantum anomalous Hall insulator. Finally, we summarize the results of the thesis in Chap. 6.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

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24 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111.

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1 Introduction Tokura Y, Yasuda K, Tsukazaki A (2019) Nat Rev Phys 1:126 Vafek O, Vishwanath A (2014) Ann Rev Condensed Matter Phys 5:83 Armitage NP, Mele EJ, Vishwanath A (2018) Rev Mod Phys 90:015001 Yan B, Felser C (2017) Ann Rev Condensed Matter Phys 8:337 Nagaosa N, Morimoto T, Tokura Y (2020) Nat Rev Mater 5:621 Weyl H (1929) Zeitschrift für Physik 56:330 Nielsen H, Ninomiya M (1983) Phys Lett B 130:389 Fang Z, Nagaosa N, Takahashi KS, Asamitsu A, Mathieu R, Ogasawara T, Yamada H, Kawasaki M, Tokura Y, Terakura K (2003) Science 302:92 Murakami S (2007) New J Phys 9:356 Burkov AA, Balents L (2011) Phys Rev Lett 107:127205 Nielsen H, Ninomiya M (1981) Nucl Phys B 185:20 Zhang C, Zhang Y, Yuan X, Lu S, Zhang J, Narayan A, Liu Y, Zhang H, Ni Z, Liu R, Choi ES, Suslov A, Sanvito S, Pi L, Lu H-Z, Potter AC, Xiu F (2019) Nature 565:331 Witczak-Krempa W, Chen G, Kim YB, Balents L (2014) Ann Rev Condensed Matter Phys 5:57 Dirac PAM, Fowler RH (1928) Proceedings of the royal society of London. Series A, Containing Papers of a Math Phys Charac 117:610 Chen YL, Chu J-H, Analytis JG, Liu ZK, Igarashi K, Kuo H-H, Qi XL, Mo SK, Moore RG, Lu DH, Hashimoto M, Sasagawa T, Zhang SC, Fisher IR, Hussain Z, Shen ZX (2010) Science 329:659 Ye L, Kang M, Liu J, von Cube F, Wicker CR, Suzuki T, Jozwiak C, Bostwick A, Rotenberg E, Bell DC, Fu L, Comin R, Checkelsky JG (2018) Nature 555:638 Yin J-X, Ma W, Cochran TA, Xu X, Zhang SS, Tien H-J, Shumiya N, Cheng G, Jiang K, Lian B, Song Z, Chang G, Belopolski I, Multer D, Litskevich M, Cheng Z-J, Yang XP, Swidler B, Zhou H, Lin H, Neupert T, Wang Z, Yao N, Chang T-R, Jia Zahid Hasan SM (2020) Nature 583:533 Yang B-J, Nagaosa N (2014) Nat Commun 5:4898 Liu E, Sun Y, Kumar N, Muechler L, Sun A, Jiao L, Yang S-Y, Liu D, Liang A, Xu Q, Kroder J, Süß V, Borrmann H, Shekhar C, Wang Z, Xi C, Wang W, Schnelle W, Wirth S, Chen Y, Goennenwein STB, Felser C (2018) Nat Phys 14:1125 Suzuki T, Chisnell R, Devarakonda A, Liu Y-T, Feng W, Xiao D, Lynn JW, Checkelsky JG (2016) Nat Phys 12:1119 Liu DF, Liang AJ, Liu EK, Xu QN, Li YW, Chen C, Pei D, Shi WJ, Mo SK, Dudin P, Kim T, Cacho C, Li G, Sun Y, Yang LX, Liu ZK, Parkin SSP, Felser C, Chen YL (2019) Science 365:1282 Weng H, Yu R, Hu X, Dai X, Fang Z (2015) Adv Phys 64:227 Xu G, Weng H, Wang Z, Dai X, Fang Z (2011) Phys Rev Lett 107:186806 Qi X-L, Hughes TL, Zhang S-C (2008) Phys Rev B 78:195424 Mogi M, Yoshimi R, Tsukazaki A, Yasuda K, Kozuka Y, Takahashi KS, Kawasaki M, Tokura Y (2015) Appl Phys Lett 107:182401 Lee I, Kim CK, Lee J, Billinge SJL, Zhong R, Schneeloch JA, Liu T, Valla T, Tranquada JM, Gu G, Davis JCS (2015) Proc Nat Acad Sci 112:1316 Deng Y, Yu Y, Shi MZ, Guo Z, Xu Z, Wang J, Chen XH, Zhang Y (2020) Science 367:895 Serlin M, Tschirhart CL, Polshyn H, Zhang Y, Zhu J, Watanabe K, Taniguchi T, Balents L, Young AF (2020) Science 367:900 Wang Z, Weng H, Wu Q, Dai X, Fang Z (2013) Phys Rev B 88:125427 Uchida M, Nakazawa Y, Nishihaya S, Akiba K, Kriener M, Kozuka Y, Miyake A, Taguchi Y, Tokunaga M, Nagaosa N, Tokura Y, Kawasaki M (2017) Nat Commun 8:2274

Chapter 2

Experimental Methods

2.1 Bulk Crystal Growth 2.1.1 MnSi1−x Ge x Poly Crystals Polycrystalline samples of MnSi1−x Gex were prepared by the high-pressure synthesis technique. Mn, Si, Ge chips were first mixed with a stoichiometric ratio and melted in an arc furnace under an argon atmosphere. Afterwards, it was heated at 1073 K for 1h under a high pressure of 5.5–6.0 GPa (x = 0–0.9) and 4 GPa (x = 1.0(MnGe)), using cubic-anvil-type high-pressure apparatus (Fig. 2.1a). The assembly of the highpressure cell is shown in Fig. 2.1b. The powder x-ray analysis confirmed B20-type crystal structure.

2.1.2 Co3 Sn2 S2 Single Crystal Single-crystalline samples of Co3 Sn2 S2 were prepared by Bridgman method. Co, Sn, and S powders were mixed in a stoichiometric ratio and sealed in a quartz tube under vacuum. Afterwards, it was heated to 1323 K and cooled down to 973 K with a rate of 4mm per hour. The single phase with a shandite-type structure was confirmed by using the powder X-ray analysis.

2.2 Magnetization and Electrical Transport Measurement Magnetization measurement and electrical transport measurement were performed with Quantum Design Magnetic Property Measurement System (MPMS) and Physical Property Measurement System (PPMS), respectively. The resistivity measure© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 Y. Fujishiro, Exploration of Quantum Transport Phenomena via Engineering Emergent Magnetic Fields in Topological Magnets, Springer Theses, https://doi.org/10.1007/978-981-16-7293-4_2

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2 Experimental Methods

Fig. 2.1 a Cubic-anvil-type high-pressure apparatus installed at RIKEN, CEMS. b A schematic illustration for the high-pressure cell assembly. A pressure is applied from six directions of the cubic gasket, while the sample is heated by the carbon heater through applying the electric current from the Mo electrodes

ment was performed with a four-probe method and the electrical contacts were made with indium solder or silver paste. The AC-transport option was used for low-resistivity metallic samples (MnSi1−x Gex ) while the DC-transport option was used for the measurement of Co3 Sn2 S2 bulk samples and thin flakes.

2.3 Thermoelectric Measurement Seebeck coefficient of MnGe was measured in vacuum (< 1 × 10−4 Torr) with a set up shown in Fig. 2.2. The sample was fixed on a copper block which served as the heat bath by silver paste (H20E), and varnish was put below the copper block to make the system electrically insulated. Magnetic field was applied along the longest side of the sample, which is parallel to the temperature gradient generated by 1 k chip resistor. The temperature difference (0.5–2 K) was read by two Cernox thermometers attached to the sample with varnish, which was calibrated before the measurement. Voltage was measured through Manganin wires attached to the sample by indium solder since the Seebeck coefficient of Manganin is negligibly small (less than 1 µV/K below 300 K) [1]. The whole setup was covered by a copper tube to filter radiation heat from outside. The measurement was performed at settled temperature and magnetic field, which were controlled by PPMS.

2.4 High-magnetic-field Measurement

27

Fig. 2.2 Experimental setup for the measurement of Seebeck effect in MnGe polycrystalline sample

2.4 High-magnetic-field Measurement 2.4.1 Magnetization and Resistivity Measurements High-magnetic-field measurements of magnetization (MnSi1−x Gex ) and transport properties (MnGe bulk samples and thin films) were performed with non-destructive mid-pulse magnets with pulse duration of 30 ms energized by capacitor banks installed at International MegaGauss Science Laboratory of Institute for Solid State Physics (ISSP), University of Tokyo, Japan (Fig. 2.3). Magnetization was measured up to 56 T by the conventional induction method using coaxial pickup coils (Fig. 2.4a), while resistivity was measured by the four probe method (Fig. 2.4b, c). Transport properties of MnSi1−x Gex (x = 0.7–0.9) were measured with nondestructive long-pulse magnets (pulse duration of 1s) energized by a flywheel DC generator installed at ISSP. The thickness of the sample was reduced to ∼ 100 µm for increasing the accuracy of the measurement. Also the whole setup shown in Fig. 2.4b, c was thoroughly covered with Apiezon (N) grease to reduce the noise due to the vibration of Au wires during the pulsed shot.

2.4.2 Thermoelectric Measurement The high-magnetic-field measurement of Seebeck effect in MnGe was performed utilizing 25 T Cryogen-free Superconducting Magnet (CSM) installed at High Field Laboratory for Superconducting Materials of Institute for Materials Research (IMR), Tohoku University, Japan [3]. The 25T-CSM consists of high-temperature and lowtemperature superconducting coil as shown in Fig. 2.5a, which can generate a stable magnetic field up to 24 T. The setup for thermoelectric measurement is basically the

28

2 Experimental Methods

Fig. 2.3 Non-destructive pulse magnet energized by capacitor banks installed at ISSP, University of Tokyo [2]

Fig. 2.4 a Powder sample of MnSi1−x Gex packed in a quartz tube for the high-magnetic-field magnetization measurement. b, c Typical experimental setups for high-magnetic-field resistivity measurement of MnSi1−x Gex . Gold wires are attached to the sample by indium solder and magnetic field (B) is applied perpendicular the electric current

same as the one used in PPMS and a sapphire substrate was put below the copper block to make the system electrically insulated (Fig. 2.5b). Magnetic field was applied parallel to the thermal current.

2.5 Neutron Scattering Experiment

29

Fig. 2.5 a The world highest cryogen-free superconducting magnet which can generate high precision and stable magnetic fields up to 24 T, installed at IMR, University of Tohoku [3]. b Experimental setups for the high-magnetic-field thermoelectric measurement. c A picture of measuring probe with the sample mounted at the bottom

2.5 Neutron Scattering Experiment 2.5.1 Small- and Wide-angle Neutron Scattering Experiment Small- and wide-angle neutron scattering experiments for MnSi1−x Gex were performed with an instrument (TAIKAN) [4] built at BL15 of Materials and Life Science Experimental Facility (MLF) in Japan Proton Accelerator Research Complex (J-PARC). A powder sample of MnSi1−x Gex was packed in an aluminum container filled by He gas, and installed in a cryomagnet. Magnetic field up to 4 T was applied perpendicular to the incident neutron beam. The diffracted neutron beam with the wavelength of 0.5 < λ < 7.8 Åwas collected by four detector banks of small-, middle-, and high-angle and backward detector banks (Fig. 2.6), and analysed by using time-of-flight (TOF) method. The direction of q-vectors with respect to crystalline axes was determined by analyzing the distribution of the so-called “satellite peaks” which appear around the nuclear peaks in the wide-angle detector. The reciprocal vector for the satellite peaks qs is given by  qs = qn ± qmag

2π 2π 2π h ± qx , k ± qy , l ± qz a a a

 (2.1)

where qn is the reciprocal vector for (hkl) nuclear reflection in cubic crystal with lattice constant a, and qmag is the normalized reciprocal vector for magnetic component. The schematic illustration for the case for (q00) satellite peaks around (110) nuclear peak is shown as an example in Fig. 2.7. The complete analysis of the satellite peaks is summarized in Appendix.

30

2 Experimental Methods

Fig. 2.6 Schematic illustration of the TAIKAN installed at BL15 of MLF, J-PARC with four c detector banks covering a wide range of neutron wave length of 0.5 < λ < 7.8 Å[4]. Copyright  2014 The Physical Society of Japan

Fig. 2.7 A schematic illustration for (q00) satellite peaks emerging around (110) nuclear peak

References 1. Rathnayaka KDD (1985) Scientific instruments. J Phys E 18:380 2. Akiba K (2015) InlineImage master’s thesis. University of Tokyo 3. Awaji S, Watanabe K, Oguro H, Miyazaki H, Hanai S, Tosaka T, Ioka S (2017) Superconductor Sci Technol 30:065001 4. Ichi Takata S, Ichi Suzuki J, Ohishi K, Iwase H, Shinohara T, Oku T, Tominaga T, Inamura Y, Ito T, Nakatani T, Suzuya K, Aizawa K, Otomo T, Sugiyama M, Arai M (2014) Hamon 24:281

Chapter 3

Topological Transitions Between Skyrmion- and Hedgehog-Lattice States in MnSi1−x Ge x

3.1 Introduction Here we briefly summarize basic properties of B20-type compounds which will be discussed in Chaps. 3 and 4. While MnSi is one representative material which shares generic features with other B20-type compounds, MnGe occupies a unique position with its ditstinct magnetic properties.

3.1.1 Crystal Structure B20-type compounds consist of transition metal and group-14 elements with a composition ratio of 1:1 (Fig. 3.1). The crystal structure belongs to the chiral space group P21 3, where the lattice chirality is characterized by the stacking direction of the atoms (either transition metals or group-14 elements) as viewed from [111] crystal axis (Fig. 3.1). The lack of inversion symmetry allows Dzyaloshinskii-Moriya (DM) interaction [1, 2] which is essential for the formation of twisted spin structures such as helical and non-coplanar structures in B20-type compounds [3, 4].

3.1.2 Generic Aspects of B20-Type Compounds The typical magnetic ground state of B20-type compound is a long-period (λ = 10– 100 nm) helical structure which can be understood by three magnetic interactions with well-separated energy scales; ferromagnetic exchange interaction, DM interaction, and magnetic anisotropy [3]. While the strongest ferromagnetic exchange interaction (J Si · S j ) favors parallel  spin alignment, the competition with much weaker DM interaction D · (Si × S j ) favoring mutually orthogonal spin alignment leads to © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 Y. Fujishiro, Exploration of Quantum Transport Phenomena via Engineering Emergent Magnetic Fields in Topological Magnets, Springer Theses, https://doi.org/10.1007/978-981-16-7293-4_3

31

32

3 Topological Transitions Between Skyrmion …

Fig. 3.1 The unit cell of B20-type chiral crystal structure (space group P21 3). The chirality is characterized by the stacking direction of atoms as viewed from [111] direction

the moderately twisted helical structure (λ ∼ a · J/D (a is lattice constant)). Here, Si and S j are the spin vectors on the neighboring atomic sites of i and j. The helicity of the helical structure is fixed by the sign of DM interaction [5]. The weakest cubic anisotropies, i.e., fourth-order perturbation terms of spin-orbit interaction, fixes the direction of the helical modulation vector (q-vector) to one of specific crystalline axes [6], leading to the multidomain helical state at zero magnetic field. The well-separated energy hierarchy of the magnetic interactions leads to a universal magnetic phase diagram in B20-type compounds [3] (e.g., MnSi [7], Mn1−x Fex Si [8], Mn1−x Cox Si [8], Fe1−x Cox Si [9], FeGe [10, 11],Cu2 OSeO3 [12]). as shown in Fig. 3.2. The multi-domain helical phase at zero field shows a spin-flop transition to the so-called conical phase by the application of small magnetic field (B). This is a single-domain state with q-vector along the B-direction and also the spins tilting towards the B-direction (Fig. 3.2c). With further increasing B, the cone angle gradually closes and the field-polarized ferromagnetic (FM) state is realized at a critical field Hc . The value of Hc (typically less than 0.5 T) represents the energy costs to unwind the spin spiral structures as well as the strength of DM interaction; Hc ∼ D 2 Ms /J (Ms is the saturation magnetization) [13]. The two-dimensional (2D) hexagonal skyrmion lattice (SkL) appears in a small T –B region just below the transition temperature TN with the assistance of thermal fluctuations, which has been historically referred to as A-phase. Despite intensive research efforts since the 1970s, the spin texture of A-phase had remained elusive until the small-angle neutron scattering (SANS) experiment [14] revealed the presence of SkL in MnSi for the first time in 2009 [7]. The observed six-fold SANS intensity pattern suggests the presence of three independent helices with mutual angles of 120◦ in a plane perpendicular to the applied B (i.e., triple-q state), leading the 2D hexagonal SkL with a magnetic period of λ ∼ 18 nm in MnSi.

3.1 Introduction

33

Fig. 3.2 Generic magnetic phase diagram of typical B20-type magnets hosting SkL, which is a consequence of the well-separated energy hierarchy of the underlying magnetic interactions [3]. The ferromagnetic transition field (Bc ) is relatively small (Bc bz , bz  (Fig. 4.2e) is also consistent with the experimental observation of MR (Fig. 4.2c). Also, mixing of the magnon modes and acoustic phonon in the monopole crystal results in a elastic softening as shown in Fig. 4.2f. As summarized in contour mappings shown in Fig. 4.3, anomalies of MR and elastic constant are critically enhanced around the FM transition, while the characteristic peak or kink structures appear slightly below the FM phase boundary (denoted by triangles in Fig. 4.3). We note that this may be related to the fact that the pair annihilation and the FM transition are not strictly equal in this system. Indeed, a recent theoretical work [1] has demonstrated that the annihilation process is highly sensitive to the B-direction and also there may be topologically-trivial multiple-q states before the FM transition. Detailed study including such anisotropic responses in single crystals of MnGe [2, 3] would be of great interest. Lastly, we would like to comment that the anomaly of MR (Fig. 4.3a) only gradually disappears and seems to remain even in the nominally FM region. This will be discussed in more detail through our high-magnetic-field transport measurements in this chapter.

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Fig. 4.3 Contour mappings of anomalous positive contribution of longitudinal MR a and elastic constant with propagation direction parallel to the magnetic field b in the temperature-magneticfield plane. The FM transition field is defined as the inflection points in magnetization curves at respective temperatures. The characteristic peak or kink structure are denoted by triangle markers. Reprinted from Ref. [4] by the author licensed under CC BY 4.0

4.2 Large Magneto-Seebeck Effect in Bulk MnGe Here we report an usually enhanced magneto-Seebeck effect in polycrystalline MnGe, which we attribute to the strong energy dependence of charge-transport lifetime caused by the unconventional carrier scattering via the dynamics of emergent magnetic field. High-performance thermoelectric materials provide a viable solution towards environmental issues since they realize efficient electricity generation from waste heat without greenhouse gas emissions [11]. In long history of the research, extensive efforts have been made to enhance Seebeck coefficient S with minimal increase in electrical resistivity (ρ) to improve thermoelectric figure of merit Z T = S 2 T /ρκ, where T and κ represent temperature and thermal conductivity, respectively. However, non-trivial topology of quantum states has been scarcely exploited for improving the efficiency of heat-electricity conversion. Seebeck coefficient, i.e., electromotive force per unit temperature gradient, can be interpreted as averaged entropy flow per charge carrier on the basis of the Onsager relations [12]. In the framework of the band structure picture, semi-classical treatment using the Boltzmann transport equation provides a guiding principle for obtaining the efficient entropy flow, which is described by Mott formula [11]; by neglecting the T -dependence of chemical potential μ, i.e., setting μ = E F , it reads π 2 kB2 T S=− 3e



   ∂ ln D(E)  ∂ ln τ (E)  +  ∂E ∂ E  E=EF E=E F

(4.2)

where kB and e are Boltzmann constant and elementary charge, respectively. The first and second terms in the brackets, respectively, represent energy derivatives of density of states D(E) and relaxation time τ (E) at Fermi energy E F . It commonly occurs in metals that entropy flows of electrons with their potential energy above and below E F cancel out with each other, resulting in small S on the order of a few µVK−1 . This is because the Fermi distribution function permits only electrons within

4.2 Large Magneto-Seebeck Effect in Bulk MnGe

57

their energy range approximately between E F ± kB T to be involved in heat transport phenomena, and electrons above and below E F carry heat (product of entropy and temperature) with opposite signs. The Mott formula suggests that such cancellation can be avoided in the presence of difference in number, velocity, and scattering rate are measured by the between electrons above and below E F , the first two of which  ∂ ln D(E)  ∂ ln τ (E)  and the last of which by ∂ E  . Indeed, asymmetric derivative ∂ E  E=E F

E=E F

band structures around E F generate large S (e.g., pseudogap structures in Heusler compounds [13]), whereas Kondo scattering creates strong energy dependence in τ and the consequent exotic Seebeck effect in rare-earth compounds [14]. Also, application of magnetic field (B) generally leads to the suppression of S, basically by quenching the internal degrees of freedom of carriers, such as spin and orbital entropy, spin-dependent scatterings, and so on, as exemplified by the observed reduction of S in Nax CoO2 , where spin and orbital degeneracy is lifted by B [15, 16].

4.2.1 Observation of Field-Induced Large Seebeck Effect In contrast to the above general features of S, we have discovered that magnetoSeebeck effect in the polycrystalline MnGe exhibits strong enhancement with increasing B below the magnetic transition temperature TN (Fig. 4.4a). Here the temperature gradient is applied parallel to the magnetic field. Its increment ratio becomes prominent at low temperatures. While S shows a typical value for ordinary metals at zero field (e.g., −5.5 µVK−1 at 15 K), it develops an order of magnitude larger at 14 T (e.g., 26 µVK−1 at 15 K). Phase transition from hedgehog lattice to ferromagnetic (FM) state at the critical magnetic field Bc is recognized as a kink in the S-B curve (indicated by the black triangle in Fig. 4.4a), followed by a saturating behavior. This clearly indicates the strong correlation between thermoelectric property and spin texture in this compound. By contrast, the thermopower in MnSi decreases with B (Fig. 4.4b), which is a behavior generically expected for magnetic materials; this highlights the unconventional magneto-Seebeck effect in MnGe. We also noticed some anomalous behaviors of S at low temperatures in MnGe; it keeps on increasing even above 14 T, although the spin texture should have almost turned into the FM state. The suppression of S is, however, eventually realized in the low-T and high-B region, along with the reduction of positive MR, which we attribute to the suppression of magnetic fluctuations caused by annihilation of HL. This will be discussed in a later section. Another point is that non-monotonous structure appears around 6 − 8 T (shown as gray triangles in Fig. 4.4b), however, these anomalies gradually disappear with the elevation of T . We speculate that this anomalous structure may be related to the B-dependence of emergent magnetic field in MnGe, whose magnitude becomes the maximum around the corresponding magnetic field [17].

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Fig. 4.4 a Field-induced large Seebeck coefficient S observed in MnGe. Topological phase transition from hedgehog lattice to ferromagnetic state manifests itself as a kink in the S-B curve (shown as black triangles), suggestive of its link to the observed Seebeck effect. There are also observed anomalies in the S-B curves at low temperatures, indicated by gray triangles. The temperature gradient is applied parallel to the magnetic field. b S as a function of magnetic field in MnSi as the contrasting example to MnGe, showing a monotonic decrease generically found in magnetic materials. c Profiles of S in MnGe in the temperature-magnetic-field space. The contour mappings are displayed in the bottom plane with the white line representing the phase boundary between ferromagnetic (FM) state and hedgehog lattice (HL). d Profiles of S in MnSi in the temperaturemagnetic-field space. Reprinted from Ref. [19] by the author licensed under CC BY 4.0

Comparison between T -B variations of S in MnGe and MnSi gives a clear summary of the features listed above, as shown in Fig. 4.4c, d. In MnGe, we can confirm the increasing behavior of S towards the phase boundary between HL and FM (a white line in Fig. 4.4c) at every T and the saturating behavior in the FM state. The profile of S forms a broad peak structure around the phase boundary; this suggests a widespread effect of the large fluctuations around the HL-FM phase boundary, where the topological transition occurs as accompanied by the annihilation of hedgehog and anti-hedgehog magnetic textures [17]. A former study showed that hedgehog (anti-

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Fig. 4.5 Temperature dependence of (S) for several samples with different residual resistivity ratio [RRR= ρ(300 K)/ρ(2 K)]. a S of the sample which is mainly presented in the thesis. b S of other samples with various RRR values for comparison. Reprinted from Ref. [19] by the author licensed under CC BY 4.0

Fig. 4.6 Temperature dependence of power factor (PF) for MnGe and its sample dependence. a Temperature dependence of PF(= S 2 /ρ) at 0 T and 14 T for sample which exhibits the largest Seebeck effect at low temperatures. b Temperature dependence of PF at 14 T for various samples with different sample quality. Reprinted from Ref. [19] by the author licensed under CC BY 4.0

hedgehog) is viewed as emergent magnetic monopole (anti-monopole) and that the HL-FM topological transition corresponds to the pair annihilation of monopole and anti-monopole [4]. In contrast, MnSi shows nearly featureless profile of S (Fig. 4.4d) and no discernible structure in its narrow skyrmion phase region [18, 19]. We note that S and thermoelectric conversion efficiency [power factor S 2 / ρ(µWK−2 cm−1 )] largely vary in different samples. The maximum power factor obtained in our study reaches as large as S 2 /ρ = 65 µWK−2 cm−1 at B = 14 T, T = 19 K, although it shows the large sample dependence and is apparently related to the value of residual resistivity or to the background (B = 0) transport lifetime (Figs. 4.5 and 4.6).

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4.2.2 Photoemission Spectroscopy and Band Structure Calculation To discuss what physical parameters mainly contribute to the large S in terms of the Mott formula, electronic structures and other magneto-transport properties have been investigated for MnGe. First, we examine the band structure of MnGe by performing photoemission spectroscopy (PES) and band calculation. Figure 4.7 shows T -dependence of photoemission spectra of MnGe near the Fermi level E F . When a metallic system has a large Fermi surface with tiny energy dependence of D(E), the spectrum obeys Fermi distribution function with respect to E F . In the case of MnGe, deviation from the typical Fermi distribution function profile (represented by the spectrum in Au, indicated by a black line in Fig. 4.7a) becomes discernible with decreasing temperature. To further evaluate this T -dependence, we divided the spectra by resolution-convoluted Fermi distribution function to obtain the effective D(E) (Fig. 4.7b). Here the formation of narrow pseudogap (∼ 40 meV) is clearly observed especially below the transition temperature of MnGe (TN ∼ 170 K), suggesting its relationship with magnetic ordering in this system. We estimated the upper limit of its contribution to S by assigning the steepest downward slope of D(E) curve at 11 K to the first term of π 2k2 T Mott formula − 3eB ∂ ln∂D(E) (see the dashed line in Fig. 4.7b for the corresponding E slope). It turns out that the narrow pseudogap generates thermopower of S ∼ 0.13T (µVK−1 ). Under the assumption that application of B effectively causes a shift in E F while keeping the pseudogap structure robust, our estimation of the upper limit of S should be also valid for the FM state; the estimated value of S is far short of the experimental one. We also calculated the band structure in the FM state using the density functional theory (DFT) to theoretically derive S (Fig. 4.8a). We found that D(E) does not present any distinct structures like the pseudogap detected by PES [19]. Therefore the corresponding S calculated with the approximation of constant relaxation time represents only a small value of −0.9 µVK−1 at 50 K ( Fig. 4.8b). Thus, electronic structure of MnGe alone cannot be the dominant source for the unconventional thermoelectric response as observed.

4.2.3 Specific Heat and Magneto-Resistivity The field-induced behavior of S is suggestive of enhancement of entropy, which can be best illustrated by the field-dependent specific heat (C). Temperature dependence of C under various B in MnSi and MnGe are shown as the curves of C/T versus T 2 in Figs. 4.9a, b, respectively. We also convert these data to the change in ratio of specific heat [C(B)/C(0) = (C(B) − C(0))/C(0)] as functions of normalized magnetic field (B/Bc ) at various temperatures (Fig. 4.9c for MnSi and Fig. 4.9c for MnGe). There is again stark contrast between these two systems. As to MnSi, we

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Fig. 4.7 a Temperature dependence of photoemission spectra for MnGe with the reference spectrum of Au. b Effective density of states obtained by dividing the photoemission spectra by resolutionconvoluted Fermi distribution function. Here the narrow pseudogap (∼ 40 meV) is discernible especially below the transition temperature. The dashed line is a fitted linear function used for the estimation of the Seebeck coefficient. Reprinted from Ref. [19] by the author licensed under CC BY 4.0

Fig. 4.8 aThe calculated electronic band structure of MnGe assuming FM state. b Seebeck coefficient calculated on the basis of the corresponding band structure in ferromagnetic state shown in (a), which is far short of the experimental value. Vertical thin lines indicate the position of Fermi energy. Reprinted from Ref. [19] by the author licensed under CC BY 4.0

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Fig. 4.9 Specific heat C in MnSi a, c and MnGe (b, d). a, b C/T plotted against T 2 under various magnetic fields. c, d Change in the specific heat C(B) as a function of external magnetic field normalized by the critical field Bc at respective temperatures. MnGe shows a sharp enhancement at the topological phase transition (B/Bc = 1.0). Reprinted from Ref. [19] by the author licensed under CC BY 4.0

observe a monotonous decrease of specific heat with B at every temperature. Here we note that the significant decrease in C(B)/C(0) is due to release of latent heat associated with the first-order transition from the conical to collinear (ferromagnetic) spin structure (see Fig. 4.9c, 28 K and Ref. [20]). In contrast, specific heat of MnGe shows a clear increase around Bc , where the spin hedgehogs and antihedgehogs undergo the pair annihilation and the topological transition into the FM state occurs [4]. There obviously exist strong fluctuations unique to the topological phase transition in MnGe. Now we can estimate the maximum of thermoelectric contribution of the entropy enhancement in thermal equilibrium. If we can take full advantage of the increase in specific heat C(B), S can change by S(B) = C(B)/ne, whatever the mechanism is (e.g., phonon or magnon drag [7]). With the largest C (0.181 JK−1 mol−1 at 25 K) and carrier density n (∼ 1.3 × 1023 cm−3 ) [17], we obtain S(B) ≤ 0.52 µVK−1 , which is again quantitatively insufficient to be a dominant origin of the large S in MnGe.

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Fig. 4.10 Magnetoresistivity (MR) normalized by the value at zero field at respective temperature for MnGe (a) and MnSi (b). Large positive anomaly observed in MnGe a is attributed to the fluctuations of emergent magnetic field. Reprinted from Ref. [19] by the author licensed under CC BY 4.0

As all the above possible origins have failed to explain the large B-induced S in MnGe, some scattering processes are likely to make a major contribution, producing π 2k2 T a large value of the second term in Mott formula − 3eB ∂ ln∂τE(E) . In other words, the carriers should acquire the large entropy by getting scattered. With increasing B, such B-enhanced scattering of carriers manifests itself in a large positive MR around the B-induced HL-FM topological transition, as observed in a previous study [4] and reproduced for the present sample in Fig. 4.10a. Note that enhanced spin fluctuations accompanied by those of the emergent magnetic field around the topological phase transition, are evidenced also by the result of specific heat (Fig. 4.9d). The characteristic B-dependence of MR in MnGe (Fig. 4.10a) shows a broad peak structure around Bc at every temperature below TN . This positive MR is also unconventional since external magnetic fields basically suppress spin-dependent scattering, which leads to the monotonously decreasing negative MR as observed in MnSi (Fig. 4.10b). The anticipated strong energy dependence of τ may also be rooted in such a B-dependent enhancement of the fluctuating emergent magnetic field.

4.2.4 High-magnetic-field Measurement The positive MR due to the fluctuations of emergent field only gradually falls down and still remains well above Bc as shown in (Fig. 4.10a). This implies that there may exist robust or pinned excitations of spin states with non-coplanar spin arrangements like hedgehogs even in the FM phase, causing large magnetic fluctuations as a source of the non-diminishing behavior of S. To corroborate this interpretation, we performed high-magnetic-field measurements on MR (up to 33 T) and S (up to 24 T) at low temperatures (below 10 K), where such magnetic fluctuations is anticipated to be sufficiently suppressed. Here, the electric current and temperature gradient are applied parallel to the magnetic field. Figures 4.11a–d show the results for MR

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Fig. 4.11 Magneto-resistivity (MR) and Seebeck coefficient (S) in MnGe at high-magnetic fields. The electric current and temperature gradient are applied parallel to the magnetic field. a–d Longitudinal MR measured by pulsed magnet at low temperatures (T = 2, 5, 10, 30 K). Thick-line curves are the results on the same sample by steady-field measurements (PPMS) up to 14 T. Bold black curves are the estimated conventional MR associated with the variations of magnetization shown in the black thin lines. Large noises, in particular for the low-field and low-temperature region, in the pulse-field results are due to the low resistivity of the sample (< 10 µcm). e–h The estimated contributions of the emergent-field fluctuations to positive MR, which correspond to the color-shaded regions in (a–d). i–l Magnetic-field dependence of S at low temperatures measured with a 25 T superconducting magnet (T = 2, 5, 10 K) and PPMS (T = 30 K). Reprinted from Ref. [19] by the author licensed under CC BY 4.0

and magnetization (M). As the resistivity is generally expressed as [W · τ ]−1 with W being Drude weight, the MR can be decomposed to the field-induced respective changes of W and τ . The MR due to the field-change of W is well known for the double-exchange system (e.g., colossal magnetoresistance manganites [21]) and can be well scaled with M, as confirmed also for the present case of MnGe [4]. Here, with use of the corresponding magnetization data, we estimate the conventional negative MR due to the field-increase of W , as shown with black lines in Figs. 4.11a–d. Then, we can deduce the effect of magnetic fluctuations on MR, i.e., the field-induced change of τ −1 , as the deviation from the conventional negative MR [4]. The estimated deviations, which correspond to the color-shaded regions of Fig. 4.11a–d, are displayed in Fig. 4.11e–h. They clearly show the residual magnetic fluctuations to scatter the conduction electrons in the FM state, which appear to be steeply enhanced with increasing temperature across T = 10 K. As for S in the high-field regime, we found a strong correlation with the observed MR. A decreasing behavior of S with

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the field is clearly identified across the HL to FM transition (shown in dashed lines) for T = 2 K and 5 K (Fig. 4.11i,j) along with the strong suppression of magnetic fluctuations as evidenced by the MR measurement (Fig. 4.11e,f). When the temperature is elevated, by contrast, the non-decreasing behavior of S (Fig. 4.11j–l) takes over due to the finite magnetic fluctuations surviving in the FM phase (Fig. 4.11f–h). Even a magnetic field higher than 24 T seems to be required to fully suppress the enlarged magnetic fluctuations above 5 K. Here we note that the enhancement of S does not measure the variation of scattering rate τ −1 itself but its energy dependence ∂ ln τ/∂ E as described by Mott’s formula. Hence, the enhancement of S can happen in principle as long as there exist any finite magnetic fluctuations (Fig. 4.11f–h) affecting τ , although the quantitative connection between τ and ∂τ/∂ E is difficult to verify at the moment. We speculate that the most important integrant for the observed thermoelectric phenomena in MnGe should be the dense lattice of magnetic singularities like spin hedgehogs and anti-hedgehogs, where their large emergent fields and fluctuations critically affect the motion of electrons. The paradigm presented in this work, that is the efficient heat-electricity conversion of topological origin, may lead to new guiding principles of achieving high thermoelectric performance in topological magnets.

4.3 Large Anomalous Hall Effect from Spin-Chirality Skew Scattering In this section, we report the observation of giant anomalous Hall effect (AHE) in MnGe single-crystalline thin film [22], which we attribute to a new type of skew scattering via thermally-excited spin-clusters with scalar-spin chirality (SSC). The AHE is an important phenomenon which potentially benefits applications, since a large Hall response provides a pathway to energy efficient electronic or spintronic devices through the suppression of the longitudinal current which entails the Joule heating. Its mechanism is roughly classified into two categories: intrinsic and extrinsic mechanisms [23]. The intrinsic AHE arises from the gauge field of electronic bands and magnetic orders, however, the Hall conductivity has the upper threshold set by the Berry curvature. In the case of momentum-space Berry curvature, the consequent Hall conductivity (σx y ) should be less than e2 / ha (h and a being Planck’s constant and a typical lattice constant values), and hence of the order of σx y = 102 –103 −1 cm−1 [24]. On the other hand, the contribution of electron scattering (conventionally termed “extrinsic” mechanism) is not restricted by the Berry curvature. Nevertheless, extrinsic mechanisms have rarely been studied in the context of the large AHE responses due to its small Hall conductivity. The only exception occurs in the extremely high conductivity regimes (empirically above σx x > 5 × 105 −1 cm−1 ), where the skew scattering (asymmetric electron scattering due to the spin-orbit coupling (SOC) at impurities) dominates σx y with a characteristic scaling relation (σx y ∝σx x ) [24, 25].

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Even in that case, the Hall angle (θH = tan−1 σx y /σx x ) remains a constant value, i.e., σx y ∝σx x , which is as small as 0.1–1% because the SOC is usually smaller than the width of the virtual bound state and also skew scattering requires the phase shifts of different orbital angular momenta [24–29].

4.3.1 Observation of Giant Hall Response in MnGe Thin Film In contrast to the above well established features of AHE, we observed giant Hall conductivity and Hall angle of ∼ 40, 000 −1 cm−1 and 18% in the ferromagnetic region of MnGe thin film, exceeding the conventional limits of AHE of intrinsic and extrinsic origins, respectively. As shown in Fig. 4.12a, σx y shows a striking enhancement reaching ∼ 40, 000 −1 cm−1 at the temperature T = 2 K, with a sharp peak structure at around the magnetic field B = 14 T for 160 nm-thick sample. Whereas the longitudinal conductivity (σx x = 2.02 × 105 −1 cm−1 at 2 K) belongs to the empirical intrinsic regime where the Berry curvature mechanism is dominant, the observed σx y far exceeds the threshold value of the intrinsic AHE from the momentum space, which is roughly estimated from the quantization limit (∼ e2 / ha = 800 −1 cm−1 for MnGe) in three dimensions (denoted by the dashed line in Fig. 4.12a). Here, a = 4.795 Å is the lattice constant of MnGe. Upon increasing the temperature, σx y is rapidly suppressed while showing a broader peak structure. Above T ∼ 70 K, σx y follows the conventional behavior of the intrinsic AHE, which scales with the magnetization. To overview this Hall response in the B-T plane, the contour map of σx y is shown with the magnetic phase diagram in Fig. 4.12b. The striking enhancement of the Hall response appears at low temperatures below 50 K, in proximity to the ferromagnetic (FM) phase boundary. Note that the negative component in σx y below 50 K in the low-B region is ascribed to the topological Hall effect of the hedgehog lattice (Fig. 4.12c).

4.3.2 Scaling Relation and Temperature-magnetic-field Profile To understand the characteristics of this large Hall response observed at low temperatures, we first plotted the data of σx y against σx x to investigate their scaling relation, which has been typically used to identify the mechanism of the AHE [23– 25]. Figure 4.13a shows the plot for various film samples with different thicknesses (t = 80, 160, and 300 nm) for T = 2–30 K. The displayed data points (σx x , σx y ) are those exhibiting the maximum Hall angle at each temperature. In every sample, the scaling relation between σx y and σx x is linear (σx y ∝σx x ), which is a typical feature expected for skew scattering mechanism, while showing a maximum Hall angle of 18–22% (shown by the dashed gray line in Fig. 4.13a).

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Fig. 4.12 Hall conductivity (σx y ) in MnGe thin film for the thickness t = 160 nm. a High-magneticfield data of σx y at various temperatures, showing a large enhancement. The value of the quantization limit (∼ e2 / ha) in three-dimensions is denoted by a dashed line. The observed giant Hall response far exceeds the value allowed by the intrinsic anomalous Hall effect. b Contour plot of σx y in the B-T space with the magnetic phase diagram consisting of the hedgehog lattice (HL), helical, and the ferromagnetic (FM) phases. c σx y at various temperatures for the low-B region. The negative dip structures are attributed to the topological Hall effect arising from the formation of spin hedgehogs and anti-hedgehogs bridged by skyrmion-strings (shown in the inset). Reprinted from Ref. [30] by the author licensed under CC BY 4.0

Fig. 4.13 Characteristics of the anomalous Hall effect observed in MnGe thin film, suggesting a spin-chirality skew scattering. a Plot of Hall conductivity (σx y ) versus conductivity (σx x ) for the thickness of t = 80, 160, and 300 nm. The data are taken from the peak positions of the Hall angle below 30 K. The linear relation (σx y ∝σx x ) appears in a certain B–T region with a large Hall angle of ∼ 18% shown by the dashed gray line. b Constant-B cut of σx y showing a peak structure at a finite temperature. The triangles denote the peak positions, which shifts to a higher temperature at a higher B. These results are suggestive of the relation to the thermal excitation of the scalar-spin chirality as illustrated in the inset. Reprinted from Ref. [30] by the author licensed under CC BY 4.0

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The above features of the skew scattering in MnGe are distinct from the conventional skew scattering induced by the non-magnetic chemical defects or single-spin impurity [23–29], while showing the similar linear (σx y ∝σx x ) scaling relation. In the conventional skew scattering, the perturbatively small SOC compared to the electron bandwidth results in a small Hall angle of 0.1–1%. Also, σx y should be monotonously increased to lower temperature with higher σx x . Hence, not only the giant Hall angle of 18–22% (Fig. 4.13a) but also the non-monotonous B − T profile (Fig. 4.13b) are both beyond the conventional understanding of skew scattering. One possibility for this unconventional skew scattering is the recently proposed “spin-chirality skew scattering” mechanism, where the thermally fluctuating spins in the ferromagnetic state act as the spin-clusters with the SSC (inset of Fig. 4.13b) [31]. Hence, the magnetic scattering contributes to the AHE. This is a quantum interference effect of electrons scattered by multiple-spins and is completely different from the conventional skew scattering mechanism. One outcome of this new mechanism is the giant Hall angle (θH ). In the conventional skew scattering, θH is typically in the order   2  E SOC , of 0.1% because of the small energy scale of SOC (E SOC ) as θH ∝ EJF EF where J is the coupling strength between impurity and conduction electron while E F is the Fermi energy. In contrast, the scattering process of the spin-chirality skew  3 scattering is independent of SOC: θH ∝ EJF [43]. Hence, θH can be larger by J = 10–100, compared to the conventional skew scattering. Note the factor of ESOC that these assumptions are clear only for the weak-coupling case where impurity potentials are treated perturbatively [43], while the recent theoretical calculation in the strong-coupling regime also predicted the giant Hall angle for the spin-cluster skew scattering [31]. In particular, the Hall angle in the order of 10% is anticipated to show up for this spin-cluster skew scattering when the size of the spin-cluster is comparable to the inverse of the Fermi-wave-vector (∼ 1/kF ), which resembles the resonance condition of the interference effect [31]. We found that the observed B − T profile of the AHE is also consistent with the spin-cluster scenario. Under fixed B, the spin-cluster AHE is expected to be maximized at a finite temperature as shown in Fig. 4.13b. Two types of spin excitations with opposite SSC are thermally excited, being responsible for skew scatterings with opposite Hall angles. Because those SSC excitations have different activation energies due to the presence of DMI, the cancellation of AHE signal only occurs above the temperature where the SSC excitations with the higher activation energy start to proliferate. To confirm the above assumption and see how the SSC excitations are generated in a high magnetic field in terms of a tractable mode, we performed a Monte Carlo simulation with a two-dimensional model for chiral magnet [32, 43] with the following Halmiltonian;

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Fig. 4.14 Monte Carlo simulation for chiral magnets at high magnetic field above the ferromagnetic transition. The snapshots of the spins at T = 0.2 (a, b), T = 0.5 (c, d), and T = 1.0 e, f at the magnetic field of B = 3.5. The magnetic ordering temperature (Tc ) and the ferromagnetic transition temperature (Bc ) for this model are Tc ∼ 0.9 and Bc ∼ 3.0, respectively (see Refs. [30, 32, 43] for detail).The color bars correspond to the z-component of spins a, c, e and the scalar-spin chirality b, d, f, respectively. Reprinted from Ref. [30] by the author licensed under CC BY 4.0

H = −J



 r ) · S(  r + x) + S(  r ) · S(  r + y) S(

r

−D

 r

 r ) × S(  r + x) + yˆ · S(  r ) × S(  r + y) − B xˆ · S(



S z ( r ). (4.3)

r

Note that the effective spin model of MnGe has not been established. Figure 4.14 √ shows the result for D/J = 6 and B/J = 3.5. This model shows three phases in magnetic fields: helical phase in the low field, skyrmion crystal phase in the intermediate field, and the field-forced FM phase in the high field. The transition from the skyrmion crystal phase to the FM phase occurs at B/J ∼ 3 for the parameters mentioned above. While the SSC excitation is suppressed at a low temperature (T /J = 0.2), it emerges in the snapshots for T /J = 0.5. However, the skyrmionic excitations with the opposite chirality (or anti-skyrmionic excitation) also appear in

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the higher temperature (T /J = 1.0), reducing the total SSC density. These results are consistent with our experimental observation, where the anomalous Hall effect shows a maximum at a finite temperature in a fixed magnetic field. Although the temperature dependence of the Hall signal in a fixed magnetic field is consistent with our analytical calculations and Monte Carlo simulations (Fig. 4.14), the reason why the Hall signal is enhanced at high magnetic fields under fixed temperature remains elusive and needs further theoretical investigations. This may be attributed to the divergent spin correlation and the criticality around the FM transition. The former invalidates the short-correlation-length assumption used in the skew scattering theory, while the latter may affect the transport phenomena in a nontrivial way. Study on such effects is interesting but requires intensive theoretical investigations that go beyond the scope of this work.

4.3.3 Film-Thickness Dependence To elaborate on the possibility of the spin-cluster mechanism, we have controlled the magneto-crystalline anisotropy by changing the film-thickness. The previous neutron scattering experiment has revealed that MnGe thin film has triple-q state where the three q-vectors form the rhombohedral deformation (Fig. 4.15a) and that the tilting angle from the film-normal direction ([111]) can be controlled with film-thickness (Fig. 4.15b) [22]. In specific, in-plane magnetic anisotropy is expected to enhance with decreasing film-thickness (Fig. 4.15b). Therefore, the spins can “tilt” easier from the field-polarized direction in the FM state in thinner films and we expect that the SSC excitation in the collinear spin background has a corresponding lower energy. In other words, larger SSC can be produced due to the enhanced in-plane anisotropy in the thinner films as schematically illustrated in the insets of Figs. 4.15c–e. The B-T profile of the AHE shows a clear variation with film-thickness of t = 80, 160, and 300 nm, while the maximum value of the Hall angle is almost independent of the film-thickness (Fig. 4.15c–e). In thinner films with enhanced in-plane anisotropy, a large Hall angle shows up at lower T and higher B, where the SSC excitation cost more energy.

4.3.4 Discussion on Other Mechanisms Although we have focused on the three-spin correlation with SSC, the two-spin correlation with vector spin chirality can also produce the AHE. However, we note that the two-spin correlation is not the dominant origin of the observed Hall signal. This is because both the intrinsic [33, 34] and the extrinsic AHE [35] induced by the two-spin correlation require SOC, as in the case of the conventional skew-scattering induced by single-spin or non-magnetic impurities. If these SOC-related AHE are the dominant contributions, the Hall angle would be small and also the Hall response

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Fig. 4.15 a The SANS intensity pattern of MnGe thin films with the film-thickness t = 160 nm [22]. The three q-vectors are tilted to the film-normal direction [111], due the uniaxial anisotropy from the substrate. b The effect of the film-thickness on the direction of the q-vectors, where θ is the angle between the q-vectors and the film-normal direction [22]. It suggests that the inplane anisotropy of spin is enhanced in thinner films. c–e Contour plot of tan θH = σx y /σx x for the thickness of t = 80 (c), 160 (d), and 300 nm (e). The solid diamond markers connected by a dashed line represent the maximum point of tan θH at each temperature. The insets are the intuitive schematics to explain the fact that the larger scalar spin chirality excitation is more favourable in thinner films with enhanced in-plane magnetic anisotropy. Reprinted from Ref. [30] by the author licensed under CC BY 4.0

would not be suppressed in the high field limit. Hence, the giant Hall angle as well as the rapid quench of σx y as shown in Fig. 4.12a suggest the dominant contribution of three-spin correlation which does not require SOC. One another possibility for the large Hall response observed in MnGe films is the emergence of high-mobility carriers in the FM region, which can result in σx y with a sharp peak structure, as typically observed in Dirac or Weyl semimetals [36]. We found that the observed Hall conductivity could be roughly reproduced by two-carrier Drude model, by assuming the presence of high mobility (∼ 690 cm2 V−1 cm−1 ) and low-carrier-density (∼ 9.4 × 1020 cm−3 ) hole pocket. However, this assumption that the observed Hall response is dominated by the normal Hall effect is less plausible,

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Fig. 4.16 Hall conductivity at various temperatures for thin film, single crytstal, and poly crystal of MnGe. The enhancement of Hall conductivity in the ferromagnetic region is observed in singlecrystalline thin films and bulk samples [2], while it is not observed in polycrystalline bulk samples [17]. Reprinted from Ref. [30] by the author licensed under CC BY 4.0

partly because the giant Hall response is not observed in bulk polycrystalline MnGe which has almost the same σx x (= 1.6 × 105 −1 cm−1 at 2 K) [17], reflecting the similar carrier density or mobility, with that of the thin film (Fig. 4.16). In contrast, the enhancement of the Hall respose is observed in bulk single crystals [2]. This may suggest the possibility that the SSC excitation effect may be canceled out due to the randomly oriented crystalline domains (Fig. 4.17). Further experiments such as the direct observation of electronic structure, especially in the FM region, would be necessary to discuss the possibility of large normal Hall effect or the presence of magnetic Weyl points in MnGe. Moreover, the characteristic features discussed in this work, such as the scaling relations, non-monotonous B-T profile, and magnetocrystalline anisotropy dependence of the Hall response, strongly suggest the spinchirality skew scattering mechanism argued above. We also investigated the formation of enantiomorphic twins in MnGe thin films by the dark-field TEM measurement, since these lattice-chirality domain boundaries may serve as the pinning potential for the thermal excitation of the SSC. As shown in Fig. 4.17, we observed a typical domain size of ∼ 2–3 µm, regardless of the thin film thickness. This result suggests that they may have small effects, if any, on the observed Hall effects since the Hall response dramatically changes with the filmthickness while the domain sizes remain nearly constant.

4.3.5 Comparison with Other Materials The discovery of this new type of the AHE, showing giant Hall conductivity and Hall angle simultaneously, provides a distinct exception in the universal scaling curve of σx y versus σx x established for various ferromagnets [23, 24] (Fig. 4.18). The Hall angle of 18% in MnGe gives one- or two-orders of magnitude larger σx y for a given σx x , compared to the conventional skew scattering, resulting in an upward shift of

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Fig. 4.17 Dark-field TEM measurement on MnGe thin films. a Electron diffraction pattern with [111] incidence. The MnGe domains of interest are found by rotating around the [112] axis by approximately 22◦ . b The diffraction pattern observed at such a tilt, with the peaks corresponding to the two chiral domains, labeled A and B. c–e Dark-field TEM images of film-thickness t = 80 nm c, 160 nm d, and 300 nm e, using diffraction peaks A (upper panels) and B (lower panels), respectively. Bright parts correspond to the domains A and B, respectively. Reprinted from Ref. [30] by the author licensed under CC BY 4.0

the scaling plot with a linear relation (σx y ∝σx x ). We also emphasize that σx y from the intrinsic AHE cannot reach this regime, due to the limitations set by the Berry curvature in momentum space. Recently, the spin-cluster AHE was also reported in a frustrated magnet KV3 Sb5 , with large Hall conductivity of 15,507 −1 cm−1 and Hall angle of 1.8% [37]. We assume that the difference in the Hall angle between MnGe and KV3 Sb5 may be related to the resonance condition for the spin-chirality skew scattering. For MnGe, the unusually short magnetic period (λ ∼ 2.8 nm) has been discussed in terms of the conduction-electron mediated mechanisms [38] such as the Rudermann-KittelKasuya-Yosida (RKKY) interaction; this can automatically tune the size of the spin cluster to the typical size of ∼ 1/kF , satisfying the resonance condition. Hence the RKKY magnets or the metallic spin-glass systems would be the promising candidates

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Fig. 4.18 The shaded region with a red dashed line indicates a linear relation (σx y ∝σx x ) in MnGe thin films with a giant Hall angle reaching 18–22% for the thickness of t = 80, 160, and 300 nm. The data are taken from the peak positions of the Hall angle below 30 K. The large Hall conductivity and the Hall angle realized in MnGe thin films result in a clear deviation from the conventional scaling plots established for ferromagnets [23–25]. The reported data for other materials are cited from literature [17, 24, 25, 28, 37, 39–42]. Reprinted from Ref. [30] by the author licensed under CC BY 4.0

for achieving a large Hall angle. However, the true nature of the excitation modes for the SSC in MnGe remains to be a challenge for future study. For instance, the origin of the SSC can be either the slightly tilted spin clusters or the pairwise excitations of spin hedgehogs and anti-hedgehogs connected by the skyrmion strings. We speculate that the latter may be the case, given the fact that the magnetic ground state of MnGe at low temperature (T < 50 K) is an ordered state of these spin singularities. Furthermore, spin-chirality skew scattering would provide an opportunity to explore giant AHE responses in various materials, partly because most of the experimentally available materials belong to the intrinsic regime (103