Designing Topological Phase of Bismuth Halides and Controlling Rashba Effect in Films Studied by ARPES (Springer Theses) 9811918732, 9789811918735

Table of contents :
Supervisor’s Foreword
Preface
Acknowledgments
Contents
1 Introduction
1.1 Outline of the Dissertation
2 Backgrounds
2.1 Electronic Structure at the Surface
2.1.1 Spin-Splitting and the Absence of Inversion Symmetry
2.1.2 Spin-Orbit Coupling in a Solid
2.2 Rashba Spin Splitting
2.2.1 Rashba Model
2.2.2 Spin Splitting at the Surface of Heavy Atoms: Beyond Conventional Rashba Model
2.3 Topological Phases of a Matter
2.3.1 2D and 3D Topological Insulators
2.3.2 Berry Phase and TKNN Invariant
2.3.3 Z2 Topological Invariant
2.3.4 Symmetry Indicator and Higher-Order Topology
2.3.5 Bulk-Boundary Correspondence in Nontrivial Insulators
References
3 Angle-Resolved Photoemission Spectroscopy
3.1 Principles of ARPES
3.1.1 Three-Step Model
3.1.2 Surface Sensitivity of ARPES
3.1.3 Orbital Selectivity in ARPES
3.1.4 Electron Analyzer
3.1.5 Efficient Measurements: Two-Dimensional Analyzer and Electro-Static Deflector
3.2 Light Sources
3.2.1 Laser Light Source for ARPES
3.2.2 Synchrotron Light Source for ARPES
3.3 Spin-Resolved Photoemission Spectroscopy
3.3.1 Detection of the Spin Polarization of Photoelectrons
3.3.2 Spin-Resolved ARPES Machine at ISSP, UTokyo
3.4 Spatial-Resolved ARPES
References
4 Revealing the Role of Wavefunctions in Rashba-Split States
4.1 Introduction
4.2 Giant Rashba Systems: Surface Alloys
4.2.1 Spin Textures Beyond the Rashba Model
4.2.2 Method
4.2.3 Orbital-Coupled Spin Textures in BiAg2 and BiCu2
4.2.4 Sample Preparation
4.2.5 Orbital-Selective ARPES of Surface Alloys
4.2.6 Orbital-Selective SARPES of Surface Alloys
4.2.7 Discussion
4.3 Quantum Well Films with Tunable Rashba Effects
4.3.1 Prototype Metallic Quantum Well Film Ag/Au(111)
4.3.2 Method
4.3.3 Sample Preparations of Ag/Au(111) Thin Films
4.3.4 SARPES of Ag/Au(111) Quantum Well Film
4.3.5 Thickness- and Quantum Number-Dependence of the Spin Splitting
4.3.6 Analysis of Charge Density Distributions Based on a 1D Pseudopotential Model
4.3.7 Discussion
4.4 Conclusion
References
5 Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides
5.1 Introduction
5.1.1 Quasi-1D Materials for the Search of New Topological Phases
5.1.2 Prediction of Rotation Symmetry-Protected TCIs with Topological Hinge States
5.1.3 A Versatile Platform to Realize Various Topological Phases: Bismuth Halides Bi4 X 4 (X = Br, I)
5.2 Basic Properties of Bi4 X 4 Crystals
5.2.1 Method
5.2.2 Crystal Structures and Transport Properties
5.2.3 Cleavage Surfaces of Bismuth Halides
5.3 A Weak Topological Insulator State in β-Bi4I4
5.3.1 Method
5.3.2 Possible Topological Phases of α- And β-Bi4I4
5.3.3 Bulk Band Structure Mapping by Synchrotron-ARPES
5.3.4 Observation of Topological Surface State by Laser-ARPES
5.3.5 Surface-Selective Measurements by Spatial-Resolved ARPES
5.3.6 Spin-Polarization of Topological Surface States in β-Bi4I4
5.3.7 Discussion
5.4 A Higher-Order Topological Insulator State in Bi4Br4
5.4.1 Method
5.4.2 Double Band Inversions in Double-Layer Bi4 X4
5.4.3 3D Band Structure Mapping by Synchrotron-ARPES
5.4.4 Observation of In-Gap States by Laser-ARPES
5.4.5 Spin-Polarization of In-Gap States Investigated by Laser-SARPES
5.4.6 Estimation of the Bulk Band Gap by K Deposition
5.4.7 Local Conductivity Mapping of Bi4Br4
5.4.8 Discussion
5.5 Conclusion
References
6 Summary and Future Outlook
6.1 Summary
6.2 Future Prospects
Appendix Curriculum Vitae

Citation preview

Springer Theses Recognizing Outstanding Ph.D. Research

Ryo Noguchi

Designing Topological Phase of Bismuth Halides and Controlling Rashba Effect in Films Studied by ARPES

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

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Ryo Noguchi

Designing Topological Phase of Bismuth Halides and Controlling Rashba Effect in Films Studied by ARPES Doctoral Thesis accepted by The University of Tokyo, Kashiwa, Japan

Author Dr. Ryo Noguchi Institute for Solid State Physics (ISSP) The University of Tokyo Kashiwa, Chiba, Japan

Supervisor Prof. Takeshi Kondo Institute for Solid State Physics (ISSP) The University of Tokyo Kashiwa, Chiba, Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-19-1873-5 ISBN 978-981-19-1874-2 (eBook) https://doi.org/10.1007/978-981-19-1874-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 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 spin-orbit coupling generates spin-split electronic states in solids even without applying an external magnetic field. As such states, topological surface states of topological insulators and the Rashba surface states of heavy-element solids have been intensively studied for many years. In particular, the control of these states has become a hot topic toward the development of future spintronics technology. In the present thesis, Dr. Noguchi utilized spin-resolved ARPES with a laser light source and spatial-resolved ARPES with synchrotron light sources and revealed in high precision the electronic states of quasi-one-dimensional (1D) topological insulators, bismuth halides, and metallic thin films exhibiting the Rashba effect. Through these researches, he has paved the way to control the spin-orbit interaction governing topological and spintronic properties of solids. Dr. Noguchi provided the first experimental realization of a weak topological insulator and a higher order topological insulator in solids by investigating quasi-1D bismuth halides. The topological edge states of these phases are expected to appear only at the side surfaces or hinges of crystals, which are hard to detect, thus had been eluding the experimental realization. In this thesis, the observation of these states was achieved by focusing on quasi-1D materials with two cleavage planes as well as by performing high-resolution and microscopic measurements on them. Thanks to these new findings, all four predicted topological insulators phases including the other two of the strong topological insulator and topological crystalline insulator which had been already reported have now become established by experiment. Interestingly, the topological phase of bismuth halides was revealed to change depending on the stacking sequence of chains that build the crystal. This novel property offers excellent functionality to control the spin current flowing along the edge of the crystal. Dr. Noguchi also investigated the spin splitting which occurs in quantum well films. Ag thin films on Au substrates have been known to form a typical quantum well film for more than 30 years. However, the spin splitting of such a system has been overlooked due to the lack of enough precision in experimental technology. Dr. Noguchi employed the state-of-the-art spin-resolved ARPES with high momentum and energy resolutions and successfully observed the spin-split band of Ag thin films. By tuning the thickness of the films, he demonstrated for the first v

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time that the Rashba effect can be controlled by the film thickness. The measurements of the systematic thickness dependence led him to find a scaling law in which the magnitude of the spin splitting is determined only by two parameters: the spinorbit coupling strengths of the atoms forming the films and the charge density at the interface between the films and substrate. In summary, this thesis written by Dr. Noguchi provides several key insights into the spin-splitting states induced by the spin-orbit coupling, based on new results for the novel quantum states obtained by applying advanced techniques of spin-resolved ARPES and spatial-resolved ARPES. His achievements not only highly contribute to the development of modern condensed matter physics but also provide valuable perception for engineering the next-generation spintronic devices with topological layered materials and quantum-well thin films. Kashiwa, Chiba, Japan March 2022

Takeshi Kondo

Preface

The manipulation of spin-split electronic states without magnetic field is one of the primary topics in condensed matter physics. In particular, copious studies have been given to understanding the Rashba effect and topological insulators, where spin-polarized electronic states appear at the surface via the interplay of Spin-Orbit Coupling (SOC) and asymmetry of the system. These electrons play an important role in realizing various quantum phenomena such as spin-charge conversion and emergent Majorana fermions, and hence the control of SOC effects will lead to the development of future spintronics applications. So far, the spin-polarized electrons have been detected in a variety of systems by Angle-Resolved Photoemission Spectroscopy (ARPES) and spin-resolved ARPES (SARPES). In particular, the Shockley surface states in noble metals and the Diraccone surface states in topological insulators have been intensively investigated as model systems with strong SOC effects. However, the complex spin-texture predicted in surface alloys beyond the simple Rashba model has not been revealed by conventional SARPES measurements without orbital sensitivity. Also, the demonstration of the systematic control of the Rashba effect has been limited due to the relatively low energy resolution in SARPES experiments. Moreover, some of the recently proposed topological insulator states have been elusive in experiments due to the lack of high spatial resolution in ARPES or appropriate surface preparation methods. In this thesis, we study the electronic structure of thin film systems with Rashba-split states and quasi-1D topological insulators bismuth halides. To overcome the above issues, we employ a high-resolution laser-SARPES machine recently constructed at the Institute for Solid State Physics, the University of Tokyo, in combination with synchrotron-based microscopic ARPES. First, we discuss the spin textures in surface alloys BiAg2 and BiCu2 with giant Rashba splittings. So far, complex spin textures beyond the Rashba model have been theoretically predicted in BiAg2 due to the hybridization between spin-split states, while it has been elusive in experiments. Here, we have utilized the linear polarization of laser light to perform orbital-selective SARPES measurements. As a result, orbital- and spin-resolved band structures have been clearly mapped in the momentum space, which allows us to confirms the complex spin texture in BiAg2 by vii

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simple analysis. The origin is attributed to the interband hybridization between the spz band and the pxy band, which modifies the orbital compositions of the spz band only in BiAg2 . In addition, we have revealed the Rashba-type spin splitting in prototype quantum well films Ag/Au(111), which can be systematically tuned through thickness engineering. We have utilized the high energy resolution of our SARPES machine to measure the Rashba splitting in quantum well states. The experimentally determined Rashba parameters in Ag/Au films show a systematic dependence on the film thickness and the quantum numbers, in good agreement with theoretical calculations. Moreover, by decomposing the splitting into the contribution from each layer, we find a scaling low for the Rashba effect in quantum well films; the magnitude of the effect is scaled by the charge density at the boundary and the SOC ratio between the film and the substrate regardless of the quantum numbers. This behavior originates in the envelope function of the well states, which governs the asymmetry of wavefunctions. Second, we present the investigation of the topological phases of quasi-1D bismuth halides: α-Bi4 I4 , β-Bi4 I4 and Bi4 Br4 . 3D bismuth halides can be viewed as van der Waals (vdW) stacked 2D topological insulators, offering a platform to study various topological phases depending on the different stacking sequences. Here, we present the first evidence for a Weak Topological Insulator (WTI) state and a Higher Order Topological Insulator (HOTI) state in real 3D crystals. Utilizing laser-SARPES, we have detected spin-polarized quasi-1D Dirac states in β-Bi4 I4 , while a gapped electronic structure is observed in a normal insulator α-Bi4 I4 . The quasi-1D states are attributed to the side surface states by spatial-resolved ARPES measurements, evidencing the WTI phase in β-Bi4 I4 . Furthermore, quasi-1D in-gap states have been detected in the cleaved surface of Bi4 Br4 by high-resolution ARPES measurements. This result is consistent with the expectation of topological hinge states for a HOTI state. The realization of the new topological phases is theoretically attributed to the slight difference in their stacking sequences. The A-stacking of β-Bi4 I4 protects the gappless surface states, while the A A -stacking in α-Bi4 I4 makes the system trivial. In contrast, the AB-stacking of Bi4 Br4 induces a unique situation where the higher order topology is stabilized. Our experimental demonstration of the stacking-dependent topology lays a foundation for the future stacking-engineering of spin-polarized states based on vdW topological insulators. Our work thus not only deepens the systematic understanding of the Rashba effect in thin films and newly confirmed topological insulator states, but also provides a new approach for the manipulation of spin-polarized electrons through thickness engineering of the Rashba effect and stacking-engineering of topological phases. California, USA

Ryo Noguchi

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

2.

3.

4.

R. Noguchi, K. Kuroda, K. Yaji, K. Kobayashi, M. Sakano, A. Harasawa, T. Kondo, F. Komori, and S. Shin, Direct mapping of spin and orbital entangled wave functions under interband spin-orbit coupling of giant Rashba spin-split surface states, Phys. Rev. B 95, 041111(R) (2017). R. Noguchi, T. Takahashi, K. Kuroda, M. Ochi, T. Shirasawa, M. Sakano, C. Bareille, M. Nakayama, M. D. Watson, K. Yaji, A. Harasawa, H. Iwasawa, P. Dudin, T. K. Kim, M. Hoesch, V. Kandyba, A. Giampietri, A. Barinov, S. Shin, R. Arita, T. Sasagawa, and T. Kondo, A weak topological insulator state in quasi-one-dimensional bismuth iodide, Nature 566, 518 (2019). R. Noguchi, M. Kobayashi, Z. Jiang, K. Kuroda, T. Takahashi, Z. Xu, D. Lee, M. Hirayama, M. Ochi, T. Shirasawa, P. Zhang, C. Lin, C. Bareille, S. Sakuragi, H. Tanaka, S. Kunisada, K. Kurokawa, K. Yaji, A. Harasawa, V. Kandyba, A. Giampietri, A. Barinov, T. K. Kim, C. Cacho, M. Hashimoto, D. Lu, S. Shin, R. Arita, K. Lai, T. Sasagawa, and T. Kondo, Evidence for a higher-order topological insulator in a three-dimensional material built from van der Waals stacking of bismuth-halide chains, Nat. Mater. 20, 473 (2021). R. Noguchi, K. Kuroda, M. Kawamura, K. Yaji, A. Harasawa, T. Iimori, S. Shin, F. Komori, T. Ozaki, and T. Kondo, Scaling law for the Rashba-type spin splitting in quantum well films, Phys. Rev. B 104, L180409 (2021).

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Acknowledgments

To begin with, I would like to express my sincere gratitude to my supervisor Prof. Takeshi Kondo for the unfailing support of my research and study. Five years of my graduate school work owe to his priceless guidance, patience, extensive knowledge, enthusiasm, and consideration. He also gave me various opportunities to participate in experiments worldwide and valuable chances to attend many conferences. Besides my advisor, I would like to thank my thesis committee: Profs. Shuji Hasegawa, Tohru Okamoto, Hirokazu Tsunetsugu, Iwao Matsuda, and Prof. Haruki Watanabe, for their encouragement, insightful comments, and challenging questions. My sincere thanks also go to our research associate, Dr. Kenta Kuroda. He constantly gave me helpful comments and generous support. From the basics of SARPES to the discussions on novel quantum materials, his wide range of knowledge provided appropriate advice when I got in trouble. It means a lot for me to have Prof. Shik Shin, whose mentoring and encouragement have always been valuable. It was also an immense delight to have the precious opportunity to manage state-of-the-art SARPES instruments in his favor. My appreciation also extends to Prof. Fumio Komori. Discussion with him launched a great part of this dissertation. He also gave me a chance to deal with SARPES and MBE systems. I also thank my collaborators for their kind help and wonderful interactions in research. I sincerely appreciate Prof. Takao Sasagawa, Mr. Takanari Takahashi, and Mr. Masaru Kobayashi for our well-designed collaborative research and highquality single crystals with fascinating physical properties. Dr. Koichiro Yaji and Ms. Ayumi Harasawa provided an ideal condition to perform SARPES experiments. Profs. Ryotaro Arita, Motoaki Hirayama, Dr. Masayuki Ochi, and Prof. Katsuyoshi Kobayashi discussed many theoretical parts with me. I also thank Dr. Masato Sakano, Dr. Peng Zhang, Dr. Cedric Bareiile, and Dr. Shunsuke Sakuragi for the collaborative measurements and valuable comments on the experimental results, our manuscripts and my research direction. A lot of experiments were performed at state-of-the-art ARPES instruments constructed at synchrotron facilities. I am grateful to Dr. Pavel Dudin, Dr. Timur Kim, Dr. Matthew Watson, Dr. Moritz Hoesch, Dr. Cephise Cacho, Dr. Hideaki xi

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Iwasawa, Dr. Donghui Lu, Dr. Makoto Hashimoto, Dr. Alexei Barinov, Dr. Viktor Kandyba, Dr. Alessio Giampietri, Dr. Shinichiro Ideta, and Dr. Kiyohisa Tanaka for all the supports in measurements. Throughout the studies, surface characterization experiments hugely support the conclusions. I am grateful to Dr. Tetsuroh Shirasawa for the GISAXS measurements. I also owe thanks to Dr. Takushi Iimori and Dr. Daisuke Hamane for many help during the film analysis by ARPES and SEM. Besides, I thank Profs. Yukio Hasegawa, Yoshinori Okada, Yasuo Yoshida, and Mikk Lippmaa for the supports in STM and AFM measurements. A debt of gratitude is owed to Prof. Keji Lai, Mr. Zhanzhi Jiang, Mr. Zifan Xu, and Mr. Daehun Lee for accepting me as a visiting student, for bits of help in MIM measurements, as well as beautiful daily life in Austin. I would like to represent my thanks to Prof. Kozo Okazaki, Dr. Takeshi Suzuki, and Dr. Yukiaki Ishida for useful discussions and advice in our group meetings and daily conversations. I also thank Dr. Suguru Ito for the valuable comments. The research for this thesis was financially supported by the graduate scholarship, “Advanced Leading Graduate Course for Photon Science (ALPS)”, and I appreciate Prof. Keisuke Goda for his various advice and valuable supports in my postgraduate study. I want to thank our group members for their kind help in daily life and enjoyable interactions in research: Dr. Mitsuhiro Nakayama, Dr. Shuntaro Akebi, Mr. So Kunisada, Mr. Chun Lin, Mr. Daiki Matsumaru, Mr. Kaishu Kawaguchi, Mr. Yuxuan Wan, Mr. Kifu Kurokawa, Mr. Yosuke Arai, Mr. Hiroaki Tanaka, Ms. Yuki Akimoto, Mr. Yuyang Dong, Mr. Kohei Aido, and Mr. Chi Feng; (in our adjacent rooms) Prof. Toshiyuki Taniuchi, Dr. Yuichi Ota, Dr. Yoshihito Motoyui, Dr. Takahiro Hashimoto, Mr. Yu Ogawa, Mr. Junpei Kawakita, Mr. Akihiro Tsuzuki, Mr. Shoya Michimae, Mr. Tsubaki Nagashima, Ms. Mari Watanabe, and Mr. Hao Wang. Each of our group members has impressed me with their unique individuality. The research life on the second floor, ISSP, with them was exciting and valuable in my life. Special thanks to Ms. Sachiko Shinnei, Ms. Akiko Fukushima for helpful advice and numerous supports during my Ph.D. life. Last but not least, I would like to express my deepest gratitude to my family, Kazuhiro Noguchi and Fuyumi Noguchi; Koki Nakamura and Mizuho Nakamura; ByoungHyun Lee and MyoungJa Choi; SeongGyun Lee, Jisun Lee, and Hakchan Lee. I express my sincere gratitude to my wife, Jiyeon N. Lee, for the endless encouragement, support, and tolerance as I pursue my dream. Kashiwa, Chiba, Japan December 2020

Ryo Noguchi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2

2 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Electronic Structure at the Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Spin-Splitting and the Absence of Inversion Symmetry . . . . 2.1.2 Spin-Orbit Coupling in a Solid . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Rashba Spin Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Rashba Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Spin Splitting at the Surface of Heavy Atoms: Beyond Conventional Rashba Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Topological Phases of a Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 2D and 3D Topological Insulators . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Berry Phase and TKNN Invariant . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Z 2 Topological Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Symmetry Indicator and Higher-Order Topology . . . . . . . . . 2.3.5 Bulk-Boundary Correspondence in Nontrivial Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 4 6 6

3 Angle-Resolved Photoemission Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 3.1 Principles of ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Three-Step Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Surface Sensitivity of ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Orbital Selectivity in ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Electron Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Efficient Measurements: Two-Dimensional Analyzer and Electro-Static Deflector . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Light Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Laser Light Source for ARPES . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Synchrotron Light Source for ARPES . . . . . . . . . . . . . . . . . . . 3.3 Spin-Resolved Photoemission Spectroscopy . . . . . . . . . . . . . . . . . . . .

7 8 9 11 12 13 16 17 21 21 23 24 26 27 28 29 30 32 32 xiii

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3.3.1 Detection of the Spin Polarization of Photoelectrons . . . . . . 3.3.2 Spin-Resolved ARPES Machine at ISSP, UTokyo . . . . . . . . . 3.4 Spatial-Resolved ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 34 36 37

4 Revealing the Role of Wavefunctions in Rashba-Split States . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Giant Rashba Systems: Surface Alloys . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Spin Textures Beyond the Rashba Model . . . . . . . . . . . . . . . . 4.2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Orbital-Coupled Spin Textures in BiAg2 and BiCu2 . . . . . . . 4.2.4 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Orbital-Selective ARPES of Surface Alloys . . . . . . . . . . . . . . 4.2.6 Orbital-Selective SARPES of Surface Alloys . . . . . . . . . . . . . 4.2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Quantum Well Films with Tunable Rashba Effects . . . . . . . . . . . . . . 4.3.1 Prototype Metallic Quantum Well Film Ag/Au(111) . . . . . . 4.3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Sample Preparations of Ag/Au(111) Thin Films . . . . . . . . . . 4.3.4 SARPES of Ag/Au(111) Quantum Well Film . . . . . . . . . . . . 4.3.5 Thickness- and Quantum Number-Dependence of the Spin Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Analysis of Charge Density Distributions Based on a 1D Pseudopotential Model . . . . . . . . . . . . . . . . . . . . . . . . 4.3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 39 42 43 43 45 46 50 52 54 56 58 59 59

5 Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Quasi-1D Materials for the Search of New Topological Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Prediction of Rotation Symmetry-Protected TCIs with Topological Hinge States . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 A Versatile Platform to Realize Various Topological Phases: Bismuth Halides Bi4 X 4 (X = Br, I) . . . . . . . . . . . . . . 5.2 Basic Properties of Bi4 X 4 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Crystal Structures and Transport Properties . . . . . . . . . . . . . . 5.2.3 Cleavage Surfaces of Bismuth Halides . . . . . . . . . . . . . . . . . . 5.3 A Weak Topological Insulator State in β-Bi4 I4 . . . . . . . . . . . . . . . . . . 5.3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Possible Topological Phases of α- And β-Bi4 I4 . . . . . . . . . . . 5.3.3 Bulk Band Structure Mapping by Synchrotron-ARPES . . . .

61 64 65 71 72 77 77 78 79 80 83 83 84 85 87 87 88 91

Contents

5.3.4 Observation of Topological Surface State by Laser-ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Surface-Selective Measurements by Spatial-Resolved ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Spin-Polarization of Topological Surface States in β-Bi4 I4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 A Higher-Order Topological Insulator State in Bi4 Br4 . . . . . . . . . . . 5.4.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Double Band Inversions in Double-Layer Bi4 X 4 . . . . . . . . . . 5.4.3 3D Band Structure Mapping by Synchrotron-ARPES . . . . . . 5.4.4 Observation of In-Gap States by Laser-ARPES . . . . . . . . . . . 5.4.5 Spin-Polarization of In-Gap States Investigated by Laser-SARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6 Estimation of the Bulk Band Gap by K Deposition . . . . . . . . 5.4.7 Local Conductivity Mapping of Bi4 Br4 . . . . . . . . . . . . . . . . . . 5.4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

93 95 98 98 101 101 102 105 107 108 110 111 112 115 116

6 Summary and Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.2 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Chapter 1

Introduction

The spin-orbit coupling (SOC) generates spin-polarized electrons in a system without space inversion symmetry. Since these electrons can play the central role in inducing quantum phenomena such as quantum spin Hall effect and Majorana fermionic states and achieving efficient spin to charge conversions, the microscopic understanding and the manipulation of spin-polarized states become one of the most important issues in the field of condensed matter physics. Providing an efficient method to control the SOC effects at the boundary of a solid will thus lay the foundation for future spintronics applications and Majorana-based quantum computation. So far, the spin-polarized electrons due to SOC have been found in a variety of systems including the so-called Shockley surface states and topological surface states. The importance of the asymmetric wavefunctions has been recognized through the early researches on the Rashba effect at the surface of noble metals, and the classification of topological insulators has enabled us to efficiently predict the existence or the absence of the spin-polarized surface states and edge states. Angle-resolved photoemission spectroscopy (ARPES) and spin-resolved ARPES (SARPES) have been intensively employed to investigate the SOC effects, due to their high momentum-resolution. In fact, the spin-resolved band dispersions of the Rashba systems and topological systems have been clearly visualized in various materials. However, spin polarized energy bands may show significant orbital dependence and complex spin textures beyond the simple Rashba model or small spin splittings, which are not detectable in conventional experiments. In addition, recently proposed topological boundary states may appear in particular side surfaces of small crystals that are difficult to measure. Therefore, orbital-sensitivity, high energy resolution, high spatial resolution, and appropriate surface conditions in ARPES and SARPES experiments are needed to deepen the understanding the role of electronic wavefunctions in such complicated Rashba systems or topological systems. To address these issues, we have employed recently developed laser-SARPES with orbital sensitivity and high energy resolution in combination with synchrotronbased microscopic ARPES to study the spin-polarized electronic structures in thin films and van der Waals (vdW) layered topological insulators. In this dissertation, the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 R. Noguchi, Designing Topological Phase of Bismuth Halides and Controlling Rashba Effect in Films Studied by ARPES, Springer Theses, https://doi.org/10.1007/978-981-19-1874-2_1

1

2

1 Introduction

observations of spin-polarized states in these systems are presented and the origins of the Rashba effect and nontrivial topological phases are discussed from the viewpoint of the role of the wavefunctions.

1.1 Outline of the Dissertation This dissertation is organized as follows. In Chap. 2, we review the basic properties of surface states, Rashba-type spin splitting, and topological phases of a matter. Chapter 3 summarizes the principles of ARPES and its application with spin detectors and cutting-edge light sources. ARPES is employed as the main experimental technique to investigate the electronic structures throughout this dissertation. In Chap. 4, we present the detailed studies on Rashba split states from the viewpoint of the modification of the wavefunctions. SARPES results of surface alloys and quantum well films are discussed. In Chap. 5, we study the topological electronic structures in quasi-one-dimensional (quasi-1D) bismuth halides. The formations of surface states in a weak topological insulator (WTI) and hinge states in a higher-order topological insulator (HOTI) are detected, which depends on the stacking sequences bonded by vdW forces. Chapter 6 summarizes the results of these investigations and give future prospects.

Chapter 2

Backgrounds

2.1 Electronic Structure at the Surface At the surface of a solid, the electronic structure can be different from that of the bulk, since the translational symmetry of a three-dimensional (3D) crystal is broken at its surface. In the bulk, the wavefunction of an electron is expressed by Bloch functions ψk (r) as (2.1) ψk (r) ∝ exp (i k · r)u k (r), where u k (r) is a periodic function that satisfies the following equation u k (r + R) = u k (r) with R being the lattice vector of the system. In contrast, since the wavefunctions must vanish in the vacuum, the wavefunctions at the surface are classified into three types; surface states, and surface resonance states. The bulk state only decays into the vacuum while it becomes identical with the bulk electronic structure within the crystalline lattice. For the surface states, the wavefunction decays both in the vacuum and in the bulk, leading to a state localized at the surface. By projecting the bulk electronic structure E(k x , k y , k z ) to the k z = 0 plane, we can obtain the projected bulk band structure at the surface. A momentum region without any projected bulk band is called a projected band gap, where bulk states are not allowed to exist. However, two-dimensional (2D) electronic structures that have dispersions only within the k x -k y plane may be present, which is called the surface states. Conventional surface states (e.g. surface states in noble metals) are usually classified as either Shockley states or Tamm states, depending on the origin. The dispersion of the Schockley states resembles that of the bulk states whereas the energy positions are different and the wavefunction decays in the bulk, while the Tamm states are generated by the change of chemical bonding at the surface and hence the energy positions are usually significantly different from those of bulk states. Finally, the surface resonance states exist within the projected bulk band but have large density of states at the surface. The dispersions are 2D-like similarly to

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 R. Noguchi, Designing Topological Phase of Bismuth Halides and Controlling Rashba Effect in Films Studied by ARPES, Springer Theses, https://doi.org/10.1007/978-981-19-1874-2_2

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4

2 Backgrounds

the surface states, while the wavefunctions have much larger penetration depths into the bulk since the states are hybridized with bulk states [1]. Here we note that the importance of the band crossing in generating surface states was already hinted in the early papers by Shockley [2–4]. The band crossing was assumed between s and p bands, which means that the parity of the occupied states needs to be changed to support the original Shockley states. Therefore, the topological surface state, which we will discuss later, can be naturally linked to the original concept of the Shockley surface states.

2.1.1 Spin-Splitting and the Absence of Inversion Symmetry In a non-magnetic solid, the energy-momentum relation of electrons should satisfy E(k, ↑) = E(−k, ↓)

(2.2)

due to the time-reversal symmetry. In contrast, the space-inversion symmetry in the solid results in the relation E(k, ↑) = E(−k, ↑) (2.3) in the electronic structure. Thus, if the system possesses both of the time-reversal symmetry and the space-inversion symmetry, the system becomes spin-degenerate: E(k, ↑) = E(k, ↓).

(2.4)

Therefore, either of the symmetries must be broken to induce spin-polarized electrons in a solid. One example of a spin split system is the Zeeman splitting in a ferromagnetic material, where electrons become spin-polarized due to the breaking of time-reversal symmetry (the middle panel of Fig. 2.1). Another example is the surface or interface of a non-magnetic crystal, where the spin-orbit coupling (SOC) plays a crucial role in realizing momentum-dependent spin-splittings (the right panel of Fig. 2.1). The main target of this system is the role of SOC in spin-split systems.

2.1.2 Spin-Orbit Coupling in a Solid The Hamiltonian of SOC is obtained by calculating the non-relativistic limit of the Dirac equation as  σ · (∇V × p), (2.5) HS OC = 4m 20 c2 where the effective magnetic field (∇V × p) derived from the momentum p and the potential gradient ∇V is coupled to the electron spin σ. For a system with a spherically symmetric potential V = eφ(r ), the equation is simplified as

2.1 Electronic Structure at the Surface

5 Zeeman splitting

Spin-orbit splitting

0

0

0

E

Free electrons

k

k

k

Fig. 2.1 Schematics of the spin-splittings of electronic structures

HS OC =

e dφ S · L = ξ S · L, 2m 20 c2 r dr

(2.6)

where S = 2 σ is the spin angular momentum while L is the orbital angular momentum. Therefore, SOC is also called LS coupling. If SOC is large enough, a system is often interpreted using a total angular momentum J = L + S. Using Eq. 2.6, one can understand that the magnitude of SOC in a solid can change depending on the potential created by each atom in addition to the spin and orbital angular momentum of an electron. The coefficient ξ can be calculated using the radial distribution function Rnl (r ) as e dφ |Rnl  2m 20 c2 r dr    e dφ 2 2 = r dr |Rnl (r )| . 2m 20 c2 r dr

ξ = Rnl |

(2.7) (2.8)

By assuming a Hydrogen-like potential φ(r ) = −Z e/r where Z is the atomic number, we can obtain  Ze 1 |Rnl (r )|2 dr ξ= (2.9) r 2m 20 c2 =

Z 4 e2 , 2m 2 c2 a03 l(l + 1/2)(l + 1)

(2.10)

where a0 = 2 /m e e2 is the Bohr radius. Since ξ is proportional to Z 4 , the SOC effect becomes prominent in a solid with heavy elements like bismuth (Z = 83) [5]. Here we note that the above equation has ignored the effect of core electrons and anisotropy of the crystal lattice. Since the potential which valence electrons feel is

6

2 Backgrounds

shielded by core electrons, Z should be replaced by Z e f f to discuss the SOC effect on valence electrons. In addition, the anisotropy of the wavefunction or the potential should be taken into account to discuss the detailed band structures.

2.2 Rashba Spin Splitting 2.2.1 Rashba Model The Rashba model is a simple model to describe momentum-dependent spin-splitting induced by an electric field ∇V = −e E = −e(0, 0, E z ) [6–9]. Here the Hamiltonian for SOC becomes HS OC =

 σ · (∇V × p) 4m 20 c2

e2 (σ y k x − σx k y )E z 4m 20 c2 = αR (σ y k x − σx k y ), =

(2.11) (2.12) (2.13)

where αR is called the Rashba parameter which denotes the magnitude of the effect. Then the Hamiltonian for a free electron moving in the x y-plane is written as H = H0 + HS OC =

2 (k x2 + k 2y ) 2m

+ αR (k x σ y − k y σx ).

(2.14)

When the electron has a finite momentum only along x direction, the eigenstates and eigenvalues are calculated as (y)

ψk,± = exp (ik x x)χ± , E k,±

2 k x2 2 2 k02 ± αk x = (k x ± k0 )2 − , = 2m 2m 2m

(2.15) (2.16)

(y)

where χ± denotes spin-up or spin-down state along y-axis and k0 = m2 α is the magnitude of the splitting in the momentum space along k x direction. As shown in Fig. 2.2, the spin-degenerate parabolic dispersion of a free electron split into a pair of spin-polarized bands with opposite spin polarizations. The bottom of the parabola is shifted by E R = 2 k02 /2m, which is called the Rashba energy. Here the E R and k0 has a relation 2E R αR = , (2.17) k0 and this equation can be used to evaluate the Rashba coefficient. In addition, the eigenvalues at k indicate that the energy splitting at a momentum k is linearly pro-

2.2 Rashba Spin Splitting

7

Fig. 2.2 Spin splitting due to the Rashba model. The left panel illustrates the nearly parabolic E-k dispersion of Rashba split bands. In the right panel, a constant energy contour is shown with the direction of spin polarization

portional to k since E = 2αR k. Moreover, a constant energy contour is illustrated in the right panel of Fig. 2.2. Under the Rashba model, a circle splits into a pair of circles with opposite spin polarizations; one branch has a clockwise spin-texture, and the other has a anti-clockwise spin texture. Here one can notice that the Rashba band becomes spin degenerate at specific momentum points (k = 0 in Eq. 2.16). When the momentum k satisfies the relation k = −k + G where G is the reciprocal unit vector, the time-reversal symmetry is preserved for electrons since SOC itself does not break the time-reversal symmetry and therefore the spin polarizations become 0 at such momenta. Such momentum points are called time-reversal invariant momenta (TRIM). Importantly, the mass of atoms Z is not included in the above equations. Therefore, within the framework of the Rashba model, the magnitude of the effect should not depend on whether heavy elements are employed or not, while the electric field E z perpendicular to the surface determines the splitting.

2.2.2 Spin Splitting at the Surface of Heavy Atoms: Beyond Conventional Rashba Model In experiments, the Rashba effect was first demonstrated in a semiconductor heterostructure with applying gate voltage [10, 11]. In this case, the effect causes the spin splitting in the two-dimensional electron gas at the interface of the heterostructures, which can be detected by Shubnikov-de Haas oscillations as a function of the gate voltages. The Rashba parameter αR depends on the gate voltage, and typically takes a value of the order of 10 meVÅ. Later, spin split electronic structures were

8

2 Backgrounds

found at the surface of Au and W through the observation of the band structure by ARPES and the detection of spin-polarization by SARPES [12–15]. These observations have triggered a variety of researches in the field, and the Rashba effect has been confirmed in the surface of heavy metals, semiconductors, surface alloys, quantum well states and bulk electrons in a crystal without inversion symmetry [16–25]. Strikingly, the observed values of αR in these systems often exceed 1 eVÅ. Additionally, such “giant” spin splittings have been usually realized by using heavy atoms in the system. Thus, the conventional Rashba model fails to describe the microscopic mechanism of the observed splittings, whereas these large splittings are still usually classified as the Rashba spin splitting. To understand the mechanism of the large spin splitting, the contribution of atomic potentials have been reconsidered [26, 27]. Starting from Eq. 2.12, the spin splittings can be evaluated as ψ|HS OC |ψ 2k  3 ∂V (r) = d r |ψ(r)|2 , ∂z

αR ∝

(2.18) (2.19)

where ψ(r) is the wavefunction and the inversion symmetry breaking along z-axis is taken into account. The Coulomb term from the nucleus becomes suddenly large near each atom, and consequently the potential gradient ∂V /∂z shows an antisymmetric diverging behavior. Due to its antisymmetric nature, the integral becomes 0 if the wavefunction is symmetric, while it gives a finite contribution if the wavefunction is asymmetric. At the surface, the wavefunctions are modified due to the surface potential and can have large asymmetry around the nucleus. As a result, such asymmetric wavefunction coupled to the potential gradient from the Coulomb term determines the magnitude of the spin splittings. By density functional theory calculations, the integrand in Eq. 2.19 is mapped in the real space and the importance of the asymmetric wavefunction near the nucleus has been confirmed as schematically illustrated in Fig. 2.3.

2.3 Topological Phases of a Matter A topological insulator (TI) is a material that has a band gap in its bulk but hosts gapless surface/edge states due to the bulk-boundary correspondence stemming from the nontrivial bulk electronic structure [28–31]. Due to the broken inversion symmetry at the surface or the edge, the boundary states are spin-polarized similarly to the cases of the Rashba states. With the presence of the spin-polarized electrons, TIs have been intensively investigated searching for exotic quantum phenomena and spintronic applications.

Charge density/Potential gradient (arb. units)

2.3 Topological Phases of a Matter

9

∂V/∂z

2

∂V/∂z|ψ (z)|

2

∂V/∂z|ψ (z)|

2

|ψ (z)|

z

z

Fig. 2.3 Microscopic mechanism for the Rashba effect. The potential gradient from the Coulomb term is usually antisymmetric, and hence only the asymmetric part wavefunctions can give rise to spin splittings, as seen in the plot of ∂V /∂z|ψ(z)|2 in the right panel

2.3.1 2D and 3D Topological Insulators Historically, the quantum Hall insulator (QHI) is one of the first systems whose behavior is understood with the concept of topology [32]. When a strong magnetic field is applied to a 2D electron gas in a semiconductor at a very low temperature, the Hall conductivity σx y can be quantized as σx y = n

e2 , h

(2.20)

where n is an integer, e is the elementary charge and h is the Planck constant. This phenomenon is called the quantum Hall effect and has been attributed to the nontrivial topology characterized by TKNN invariants of electron wavefunctions. Furthermore, a gapless chiral edge state appears at the edges of the sample, as a result of the bulk-boundary correspondence. Later, the concept of topology in condensed matter physics was expanded and the existence of a quantum spin Hall insulator (QSHI) characterized by a Z 2 topological invariant was predicted [33, 34]. QSHI is a 2D system that hosts helical edge states at the edge of the sample and can be understood as a superposition of two quantum Hall systems with opposite chirality. Here the strong magnetic field in QHI corresponds to the SOC in QSHI. Therefore, the time-reversal symmetry is preserved in a QSHI while it is broken in QHI (Fig. 2.4). Since QSHI is theoretically expressed by the Z 2 invariant, it is also regarded as a 2D topological insulator (2D TI). In experiments, the quantization of electrical resistivity was observed in a semiconductor heterostructure composed of HgTe and CdTe, indicating the realization of the QSHI phase. Atomically thin films such as monolayer 1T -WTe2 are also considered to be in the

10

2 Backgrounds

Fig. 2.4 a Schematic of a quantum Hall insulator (QHI) and a quantum spin Hall insulator (QSHI). b Edge band structures for QHI and QSHI. Gapless chiral edge states connect the bulk conduction band and the bulk valence band in the QHI state, while gapless helical edge states are present in the QSHI state

QSHI phase [35], and the quantization of the resistivity was observed in transport measurements and the edge states were detected by scanning tunneling microscopy (STM) experiments and microwave impedance microscopy (MIM) experiments [36, 37]. The concept of Z 2 topology in two dimensions was again extended to three dimensions, and it was found that the Z 2 topology in 3D insulators can be characterized by four Z 2 invariants [38–40]. A 3D TI is classified as either a strong topological insulator (STI) or a weak topological insulator (WTI), and gapless surface states emerge in the bulk band gap on any surface for STI and on particular surfaces for a WTI [Fig. 2.5]. Such surface states are called topological surface states (TSSs), and the observation of the band dispersion of TSSs by angle-resolved photoemission spectroscopy (ARPES) has been the most powerful tool to determine the topology of 3D materials. So far, the STI phase has been confirmed in a number of materials such as Bi1−x Sbx and Bi2 Se3 [41–44]. Especially, Bi2 Se3 hosts a simple electronic structure with linearly dispersing TSS, offering an ideal platform to study topological phenomena in 3D STI. Importantly, TSSs in 3D TI are spin-polarized due to SOC and the broken inversion symmetry at the surface, similar to the Rashba-type spin splitting states.

2.3 Topological Phases of a Matter

11

Fig. 2.5 Schematic of 3D topological insulators. a A normal insulator without topological surface states (TSSs). b A strong topological insulator (STI) with TSSs on any surface. c A weak topological insulator (WTI) with TSSs on particular surfaces

2.3.2 Berry Phase and TKNN Invariant The concept of the Berry phase is an underlying mechanism in discussing topological invariants in a solid. The Berry phase term is obtained by considering the time evolution of wavefunctions as [31] T γn =

˙ dt R(t) · in, R(t)|∇R |n, R(t),

(2.21)

0

where R is a time-dependent parameter and |n is the nth eigenstate. The expression for the Berry phase can be simplified by considering a loop C as  γn =

dR · in, R|∇R |n, R C

(2.22)

 dS · n (R),

=−

(2.23)

S

where S is the surface enclosed by the loop C and  is the Berry curvature defined as n (R) = ∇R × An (R),

(2.24)

with the Berry connection An (R) = −in, R|∇R |n, R. The Berry curvature in the momentum space can be interpreted as an effective magnetic field B, which acts on the motion of electrons in the momentum space. For an 2D insulator with a band gap, the integral of the Berry curvature over the entire Brillouin zone is quantized due to the reasons similar to the quantization of the magnetic monopole:

12

2 Backgrounds

n=

1 2π

 d 2 k[∇k × A(k x , k y )]

(2.25)

BZ

with A(k) = −iu k |∇k |u k ,

(2.26)

where |u k  is the Bloch wavefunction. The quantized number n is called the TKNN invariant or Chern number, which is regarded as a topological invariant since it cannot be changed when the band dispersions are smoothly modified without closing the band gap. Here, the Hall conductivity can be calculated as Eq. 2.20 by using the Kubo formula.

2.3.3 Z2 Topological Invariant In discussing the QHI phase, the time-reversal symmetry is absent since a strong magnetic field is required to induce the phenomena. In a time-reversal invariant system, n must vanish due to the antisymmetric nature of the Berry curvature (k) and the QHI state cannot be realized. Instead, all the insulators can be characterized by the Z 2 invariant related to the time-reversal symmetry. Currently, it is known that there are several methods to calculate the Z 2 invariant [29, 31]. Following the Pfaffian approach, we define a unitary matrix ω as ωαβ = u α , −k||u β , k,

(2.27)

where  is the time reversal operator and α and β are the band indices. The matrix satisfies following equations ω† ω = 1 ωβα (−k) = −ωαβ (−k).

(2.28) (2.29)

At a TRIM point i in the Brillouin zone where k = 0 or k = π, ω(k = 0) and ω(k = π) becomes antisymmetric matrices. The determinants of antisymmetric matrices are the square of the Pfaffian P f [ω(i )] and therefore we can define P f [ω(i )] = ±1, δi = √ Det[ω(i )]

(2.30)

where P f [ω(i )] = ω12 (i )ω34 (i ) . . . ω2N −1,2N (i ). For a two-dimensional material, there are four TRIM points in the 2D Brillouin zone, and the Z 2 invariant is calculated as 4  P f [ω(i )] . (2.31) (−1)ν = √ Det[ω(i )] i=1

2.3 Topological Phases of a Matter

13

Here ν takes the value of either 0 or 1. The Z2 topological invariant calculated for all the occupied states classifies whether the insulator is in the trivial phase (ν = 0) or in the 2D topological phase (ν = 1). In three-dimensions, the Brillouin zone has 8 TRIM points and 6 planes where we can calculate the invariants, out of which only 4 invariants are independent. Then the Z 2 invariants a 3D insulator can be defined as  δ(n 1 ,n 2 ,n 3 ) (2.32) (−1)ν0 = n j =0,π νi

(−1) =



δ(n 1 ,n 2 ,n 3 ) (i = 1, 2, 3).

(2.33)

n j=i =0,π; n i =π

ν0 is the strong topological invariant which determines the overall topology of the occupied states, while ν1,2,3 are the weak topological indices which reflect the topology projected to the certain planes. When ν0 = 1, the system is in the strong topological insulator (STI) phase, where as it is in the weak topological insulator (WTI) phase when ν0 = 0 and any of νi (i = 1, 2, 3) is non-zero. If the system possesses the inversion symmetry, the calculation method can be simplified. Due to the inversion symmetry and the time-reversal symmetry, the Berry curvature vanishes for any k point. This results in the relation δ(i ) =

N 

ξ2n (i ),

(2.34)

n=1

where 2N is the number of occupied bands which are in general doubly degenerate, and ξ2n (i ) is the parity eigenvalue for the 2n-th band. With this formula, the Z 2 topological invariants for a 3D insulator with inversion can be predicted by calculations more easily than for an insulator without inversion symmetry. The relation between the parity eigenvalues and the topological invariants is illustrated in Fig. 2.6a–c. In (a), the parity eigenvalues of occupied states is same for all TRIM points, resulting in (ν0 ; ν1 ν2 ν3 ) = (0; 0 0 0). If one band at the point is inverted by another band with different parity, the system lies in the STI phase (ν0 ; ν1 ν2 ν3 ) = (1; 0 0 0) (Fig. 2.6b). In contrast, if the parity is switched at (0, 0, 0) and (0, 0, π), which is the 3D analogue of QSHI, the topological invariants become (ν0 ; ν1 ν2 ν3 ) = (0; 0 0 1), resulting in a WTI phase.

2.3.4 Symmetry Indicator and Higher-Order Topology Soon after the discovery of Z 2 topological insulators, other possible topological invariants in 3D insulators were sought for by theoretical calculations and experiments. In fact, the mirror Chern number can also be used to define the topological phases of a matter, which is given by the difference between two Chern numbers

14

2 Backgrounds

Fig. 2.6 Calculation of Z 2 and Z 4 topological invariants. a is an example of a normal insulator, while a strong topological insulator (STI) phase, a weak topological insulator (WTI) phase and a higher-order topological insulator (HOTI) phase are illustrated in b, c and d, respectively

obtained for the Hibert spaces with different mirror eigenvalues [45–48]. Besides, it was also proven that rotational symmetries can also generate nontrivial topology defined by the eigenvalues of rotation operators [49]. However, the discussion on these phases was specific to each symmetry, and more general formula was required to find unknown topological materials. In generalizing topological invariants, theorists have developed symmetry indicators with which we can efficiently predict the topological phases of a material [50–57]. In general, considering the wavefunction at high symmetry points ki ∈ K , a system is characterized by a vector b ∈ {BS}G as [58, 59] b = (n 1k1 , n 2k1 , . . . , n 1k2 , n 2k2 , . . .),

(2.35)

2.3 Topological Phases of a Matter

15

where n αki is the number of irreducible representations at ki with α labeling different irreducible representations. At the same time, similar vectors and the corresponding group {AI}G are defined for an atomic insulator, where the symmetry is kept but the hopping of electrons are perfectly suppressed. The atomic insulator cannot have nontrivial topology. Here, the criterion for topology is whether b is included in / {AI}G , the system has some kind of nontrivial topology. To use {AI}G or not. If b ∈ the criteria, quotient groups X GBS = {BS}G /{BS}G called symmetry indicators are obtained for all of the 230 space groups. It is known that the symmetry indicator X GBS is expressed as (2.36) X GBS = Z n1 × Z n2 × · · · × Z nN with ni being an integer larger than 2. Although the symmetry indicator does not directly reveal the related topological invariants since several different combinations of topological invariants are attributed to the same indicator, it is still useful to find topological materials beyond the framework of Z 2 topological insulators. Importantly, for three dimensional space groups with inversion symmetry, the symmetry indicator is (2.37) X GBS = (Z 2 )3 × Z 4 where Z 2 s are the weak topological indices while Z 4 corresponds to the strong topological index. The topological numbers can be written as k =0

k x =0 , νzxy (ν yz

k z =0 , ν yz , κ1 ),

(2.38)

where κ1 is an extension of ν0 mentioned in the 3D Z 2 topological insulators. With the inversion symmetry, κ1 is κ1 =

1 4



− (n + k − n k ) mod 4,

(2.39)

k∈T R I M

where n ± k is the number of occupied bands with ± party eigenvalues. The relation that k x =0 k x =π κ1 (−1) is equivalent to (−1)ν yz (−1)ν yz indicates that κ1 denotes the 3D nontrivial phase. While κ1 = 1, 3 means that the system is an STI as discussed before, the difference between κ1 = 0 and κ1 = 2 indicate that the systems previously regarded as normal insulators are divided into two classes; one is still normal insulators (κ1 = 0), and the other is higher-order topological insulators (HOTIs) with topological edge states (κ1 = 2). Figure 2.6d shows an example for the parity eigenvalues which realizes a HOTI phase. If there are two bands with odd parity eigenvalues at a TRIM point while two bands with even parities are occupied at the other TRIM points, κ1 becomes 2 resulting in the HOTI phase.

16

2 Backgrounds

2.3.5 Bulk-Boundary Correspondence in Nontrivial Insulators As a result of the nontrivial topology of the bulk electronic structures, topological boundary states should emerge in the bulk band gap of any topological insulators. This relation is called the bulk-boundary correspondence, and it is classified into three orders in a real crystal [60, 61]. The first order bulk-boundary correspondence is exactly what has been intensively investigated in recent years in the field of Z 2 topological materials, and the boundary states have (d − 1)-dimensions in d-dimensional crystals. As a consequence of the first-order topology, a nontrivial 1D system has zero-energy edge modes, while the gapless edge states or the gapless surfaces states emerge for 2D or 3D insulators. The second-order topology is defined in 2D or 3D systems, and zero-energy corner modes or gapless edge (hinge) states are present which are localized in a 0D or 1D space. Similarly, the third-order bulk-boundary correspondence guarantees the existence of 0D corner states for 3D systems. These consequences are schematically illustrated in Fig. 2.7. In this thesis, using angleresolved photoemission spectroscopy, the first-order topology and the second-order topology have been investigated in 3D crystals.

Fig. 2.7 Bulk-boundary correspondence in higher-order topological insulators

References

17

References 1. Sakamoto K (2012) Physical properties at surfaces, vol 3. Kyoritsu Shuppan 2. Shockley W (1939) On the surface states associated with a periodic potential. Phys Rev 56(4):317–323 3. Foo EN, Wong HS (1974) Properties of interface Shockley states for a one-dimensional sp hybrid lattice. Phys Rev B 10(12):4819–4824 4. Yan B, Stadtmüller B, Haag N, Jakobs S, Seidel J, Jungkenn D, Mathias S, Cinchetti M, Aeschlimann M, Felser C (2015) Topological states on the gold surface. Nat Commun 6(1):10167 5. Weissbluth M (1978) Atoms and molecules. Elsevier, weissbluth 6. Bychkov YA, Rashba EI (1984) Properties of a 2D electron gas with lifted spectral degeneracy. JETP Lett 39(2):78 7. Bychkov YA, Rashba EI (1984) Oscillatory effects and the magnetic susceptibility of carriers in inversion layers. J Phys C Solid State Phys 17(33):6039–6045 8. Bihlmayer G, Rader O, Winkler R (2015) Focus on the Rashba effect. New J Phys 17(5):050202 9. Soumyanarayanan A, Reyren N, Fert A, Panagopoulos C (2016) Emergent phenomena induced by spin-orbit coupling at surfaces and interfaces. Nature 539(7630):509–517 10. Lommer G, Malcher F, Rossler U (1988) Spin splitting in semiconductor heterostructures for B0. Phys Rev Lett 60(8):728–731, 050202 11. Nitta J, Akazaki T, Takayanagi H, Enoki T (1997) Gate control of spin-orbit interaction in an inverted In0.53Ga0.47As/In0.52Al0.48As heterostructure. Phys Rev Lett 78(7):1335–1338 12. LaShell S, McDougall BA, Jensen E (1996) Spin splitting of an Au(111) surface state band observed with angle resolved photoelectron spectroscopy. Phys Rev Lett 77(16):3419–3422, 050202 13. Rotenberg E, Chung JW, Kevan SD (1999) Spin-orbit coupling induced surface band splitting in Li/W(110) and Li/Mo(110). Phys Rev Lett 82(20):4066–4069, 050202 14. Hochstrasser M, Tobin JG, Rotenberg E, Kevan SD (2002) Spin-resolved photoemission of surface states of W(110)-(1×1)H. Phys Rev Lett 89(21):216802 15. Hoesch M, Muntwiler M, Petrov VN, Hengsberger M, Patthey L, Shi M, Falub M, Greber T, Osterwalder J (2004) Spin structure of the Shockley surface state on Au(111). Phys Rev B 69(24):241401 16. Koroteev YM, Bihlmayer G, Gayone JE, Chulkov EV, Blügel S, Echenique PM, Hofmann P (2004) Strong spin-orbit splitting on Bi surfaces. Phys Rev Lett 93(4):046403 17. Ast CR, Henk J, Ernst A, Moreschini L, Falub MC, Pacilé D, Bruno P, Kern K, Grioni M (2007) Giant spin splitting through surface alloying. Phys Rev Lett 98(18):186807 18. Hirahara T, Miyamoto K, Matsuda I, Kadono T, Kimura A, Nagao T, Bihlmayer G, Chulkov EV, Qiao S, Shimada K, Namatame H, Taniguchi M, Hasegawa S (2007) Direct observation of spin splitting in bismuth surface states. Phys Rev B 76(15):153305 19. Rybkin AG, Shikin AM, Adamchuk VK, Marchenko D, Biswas C, Varykhalov A, Rader O (2010) Large spin-orbit splitting in light quantum films: Al/W(110). Phys Rev B 82(23):233403 20. Mathias S, Ruffing A, Deicke F, Wiesenmayer M, Sakar I, Bihlmayer G, Chulkov EV, Koroteev YM, Echenique PM, Bauer M, Aeschlimann M (2010) Quantum-well-induced giant spin-orbit splitting. Phys Rev Lett 104(6):1–4, 233403 21. Gierz I, Suzuki T, Frantzeskakis E, Pons S, Ostanin S, Ernst A, Henk J, Grioni M, Kern K, Ast CR (2009) Silicon surface with giant spin splitting. Phys Rev Lett 103(4):1–4, 233403 22. Dil JH, Meier F, Lobo-Checa J, Patthey L, Bihlmayer G, Osterwalder J (2008) Rashba-type spin-orbit splitting of quantum well states in ultrathin Pb films. Phys Rev Lett 101(26):266802 23. Yaji K, Ohtsubo Y, Hatta S, Okuyama H, Miyamoto K, Okuda T, Kimura A, Namatame H, Taniguchi M, Aruga T (2010) Large Rashba spin splitting of a metallic surface-state band on a semiconductor surface. Nat Commun 1(2):17, 266802 24. Ishizaka K, Bahramy MS, Murakawa H, Sakano M, Shimojima T, Sonobe T, Koizumi K, Shin S, Miyahara H, Kimura A, Miyamoto K, Okuda T, Namatame H, Taniguchi M, Arita R, Nagaosa N, Kobayashi K, Murakami Y, Kumai R, Kaneko Y, Onose Y, Tokura Y (2011) Giant Rashba-type spin splitting in bulk BiTeI. Nat Mater 10(7):521–526, 266802

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25. Sunko V, Rosner H, Kushwaha P, Khim S, Mazzola F, Bawden L, Clark OJ, Riley JM, Kasinathan D, Haverkort MW, Kim TK, Hoesch M, Fujii J, Vobornik I, Mackenzie AP, King PDC (2017) Maximal Rashba-like spin splitting via kinetic-energy-coupled inversion-symmetry breaking. Nature 549(7673):492–496 26. Bihlmayer G, Koroteev Y, Echenique P, Chulkov E, Blügel S (2006) The Rashba-effect at metallic surfaces. Surf Sci 600(18):3888–3891, 266802 27. Nagano M, Kodama A, Shishidou T, Oguchi T (2009) A first-principles study on the Rashba effect in surface systems. J Phys Condens Matter 21(6):064239 28. Moore JE (2010) The birth of topological insulators. Nature 464(7286):194–198 29. Hasan MZ, Kane CL (2010) Colloquium: topological insulators. Rev Mod Phys 82(4):3045– 3067, 064239 30. Qi X-L, Zhang S-C (2011) Topological insulators and superconductors. Rev Mod Phys 83(4):1057–1110, 064239 31. Ando Y (2013) Topological insulator materials. J Phys Soc Jpn 82(10):102001 32. Thouless DJ, Kohmoto M, Nightingale MP, den Nijs M (1982) Quantized Hall conductance in a two-dimensional periodic potential. Phys Rev Lett 49(6):405–408, 102001 33. Bernevig BA, Hughes TL, Zhang S-C (2006) Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314(5806):1757–1761, 102001 34. Konig M, Wiedmann S, Brune C, Roth A, Buhmann H, Molenkamp LW, Qi X-L, Zhang S-C (2007) Quantum spin hall insulator state in HgTe quantum wells. Science 318(5851):766–770, 102001 35. Qian X, Liu J, Fu L, Li J (2014) Quantum spin Hall effect in two-dimensional transition metal dichalcogenides. Science 346(6215):1344–1347 36. Tang S, Zhang C, Wong D, Pedramrazi Z, Tsai H-Z, Jia C, Moritz B, Claassen M, Ryu H, Kahn S, Jiang J, Yan H, Hashimoto M, Lu D, Moore RG, Hwang C-C, Hwang C, Hussain Z, Chen Y, Ugeda MM, Liu Z, Xie X, Devereaux TP, Crommie MF, Mo S-K, Shen Z-X (2017) Quantum spin Hall state in monolayer 1T’-WTe2 . Nat Phys 13(7):683–687, 102001 37. Shi Y, Kahn J, Niu B, Fei Z, Sun B, Cai X, Francisco BA, Wu D, Shen Z-X, Xu X, Cobden DH, Cui Y-T (2019) Imaging quantum spin Hall edges in monolayer WTe2 . Sci Adv 5(2):eaat8799 38. Moore JE, Balents L (2007) Topological invariants of time-reversal-invariant band structures. Phys Rev B 75(12):121306 39. Fu L, Kane CL, Mele EJ (2007) Topological insulators in three dimensions. Phys Rev Lett 98(10):106803 40. Fu L, Kane CL (2007) Topological insulators with inversion symmetry. Phys Rev B 76(4):045302 41. Hsieh D, Qian D, Wray L, Xia Y, Hor YS, Cava RJ, Hasan MZ (2008) A topological Dirac insulator in a quantum spin Hall phase. Nature 452(7190):970–974 42. Zhang H, Liu C-X, Qi X-L, Dai X, Fang Z, Zhang S-C (2009) Topological insulators in Bi2 Se3 , Bi2 Te3 and Sb2 Te3 with a single Dirac cone on the surface. Nat Phys 5(6):438–442, 045302 43. Chen YL, Analytis JG, Chu J-H, Liu ZK, Mo S-K, Qi XL, Zhang HJ, Lu DH, Dai X, Fang Z, Zhang SC, Fisher IR, Hussain Z, Shen Z-X (2009) Experimental realization of a threedimensional topological insulator, Bi2 Te3 . Science 325(5937):178–181, 045302 44. Xia Y, Qian D, Hsieh D, Wray L, Pal A, Lin H, Bansil A, Grauer D, Hor YS, Cava RJ, Hasan MZ (2009) Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat Phys 5(6):398–402, 045302 45. Fu L (2011) Topological crystalline insulators. Phys Rev Lett 106(10):106802 46. Hsieh TH, Lin H, Liu J, Duan W, Bansil A, Fu L (2012) Topological crystalline insulators in the SnTe material class. Nat Commun 3(1):982, 106802 47. Xu S-Y, Liu C, Alidoust N, Neupane M, Qian D, Belopolski I, Denlinger J, Wang Y, Lin H, Wray L, Landolt G, Slomski B, Dil J, Marcinkova A, Morosan E, Gibson Q, Sankar R, Chou F, Cava R, Bansil A, Hasan M (2012) Observation of a topological crystalline insulator phase and topological phase transition in Pb1−x Snx Te. Nat Commun 3:1192, 106802 48. Tanaka Y, Ren Z, Sato T, Nakayama K, Souma S, Takahashi T, Segawa K, Ando Y (2012) Experimental realization of a topological crystalline insulator in SnTe. Nat Phys 8(11):800– 803, 106802

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49. Fang C, Fu L (2019) New classes of topological crystalline insulators having surface rotation anomaly. Sci Adv 5(12):eaat2374 50. Po HC, Vishwanath A, Watanabe H (2017) Symmetry-based indicators of band topology in the 230 space groups. Nat Commun 8(1):50, 106802 51. Bradlyn B, Elcoro L, Cano J, Vergniory MG, Wang Z, Felser C, Aroyo MI, Bernevig BA (2017) Topological quantum chemistry. Nature 547(7663):298–305 52. Khalaf E, Po HC, Vishwanath A, Watanabe H (2018) Symmetry indicators and anomalous surface states of topological crystalline insulators. Phys Rev X 8(3):031070 53. Song Z, Zhang T, Fang Z, Fang C (2018) Quantitative mappings between symmetry and topology in solids. Nat Commun 9(1):3530, 031070 54. Zhang T, Jiang Y, Song Z, Huang H, He Y, Fang Z, Weng H, Fang C (2019) Catalogue of topological electronic materials. Nature 566(7745):475–479 55. Vergniory MG, Elcoro L, Felser C, Regnault N, Bernevig BA, Wang Z (2019) A complete catalogue of high-quality topological materials. Nature 566(7745):480–485 56. Tang F, Po HC, Vishwanath A, Wan X (2019) Comprehensive search for topological materials using symmetry indicators. Nature 566(7745):486–489 57. Tang F, Po HC, Vishwanath A, Wan X (2019) Efficient topological materials discovery using symmetry indicators. Nat Phys 15(5):470–476, 031070 58. Watanabe H (2020) Lecture note: from the basics of symmetry indicators to the applications to higehr-order topological insulators. Bussei Kenkyu 8(1) 59. Po HC (2020) Symmetry indicators of band topology. J Phys Condens Matter 32(26):263001 60. Benalcazar WA, Bernevig BA, Hughes TL (2017) Quantized electric multipole insulators. Science 357(6346):61–66 61. Schindler F, Cook AM, Vergniory MG, Wang Z, Parkin SSP, Bernevig BA, Neupert T (2018) Higher-order topological insulators. Sci Adv 4(6):eaat0346

Chapter 3

Angle-Resolved Photoemission Spectroscopy

After the theoretical description of the photoelectron effect was given by Albert Einstein in the early 20th century, many scientists have developed photoelectron spectroscopy (PES) to get an insight into the energy and the momentum of electrons in solids. PES is a photon-in/electron-out experiment utilizing the photoelectron effect. By measuring the energy of the emitted electrons from the surface with the knowledge about the wavelength (energy) of the incident light, we can extract the energy levels in a material. Moreover, if we measure the angle of the photoelectrons using an electron analyzer, we can discuss the energy-momentum relation of electrons in a solid. Such technique is called angle-resolved photoemission spectroscopy (ARPES), which is powerful in investigating the electronic structure of a solid and therefore has been intensively developed and employed to study the properties of many kinds of quantum materials such as high-temperature superconductors and topological insulators [1–3].

3.1 Principles of ARPES Here, we consider the transition process of electrons based on the simplest oneelectron approximation. When the incident photon source has an energy of h ν , by using the energy conservation law we obtain i + hν = φ + kin ,

(3.1)

for elastically scattered electrons, where i , kin , and ψ represent the initial state energy of the electron, the kinetic energy of the electron, and the work function of the sample, respectively. Thus, by measuring the kinetic energy of the emitted electron, we can determine the binding energy of the electron in the sample (Fig. 3.1). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 R. Noguchi, Designing Topological Phase of Bismuth Halides and Controlling Rashba Effect in Films Studied by ARPES, Springer Theses, https://doi.org/10.1007/978-981-19-1874-2_3

21

22

3 Angle-Resolved Photoemission Spectroscopy

Fig. 3.1 Transition process in photoelectron spectroscopy (PES). By illuminating the surface, electrons with positive binding energies are excited to proceed into the vacuum. The resultant photoemission spectra reflect the density of states (DOS) below the Fermi level (E F )

Moreover, by applying the law of conservation of momentum for the in-plane momentum of the electrons during the photoelectron process and by ignoring the momentum of photons since they are usually much smaller than those of electrons, the initial momentum k is expressed as k + G  = K  =

  2m e kin sin θ = 2m e (hν − φ −  B ) sin θ,

(3.2)

where K is the momentum of the electron in vacuum, m e is the mass of a electron,  B is the binding energy, θ is the angle between the sample surface and K , (Fig. 3.2), 2n π and G is the reciprocal lattice vector G = ( 2nax π , by , 2ncz π ). Using this equation and the knowledge about the energy and the angle of photoelectrons, we can determine the energy-momentum relation for one-dimensional or two-dimensional electrons, where the out-of-plane momentum can be ignored. For three-dimensional systems, the out-of-plane momentum should also be taken into account. Assuming that the final states are free electrons and that the energy loss of electrons at the surface is equivalent to the inner potential V0 , which is usually described by the width of the valence band, the out-of-plane momentum k⊥ is written as 1 2m e (E kin cos2 θ + V0 ). (3.3) k⊥ = 

3.1 Principles of ARPES

23

Since the inner potential cannot be directly measured by ARPES, one needs to determine it by comparing the hν-dependent ARPES spectra with the reciprocal lattice vectors calculated by the crystal structure. In these approximations, the measured energy-momentum relations correspond to that obtained by the Hartee-Fock equation (Koopmans’ theorem). As we have seen in this section, PES and ARPES can probe the dispersions of valence electrons as well as the energy levels of core electrons by applying high energy photons. Here, the energy levels denote the “absolute” levels where the Fermi level is set to be 0, while only relative energy levels can be investigated in most experiments such as optical absorption spectroscopy.

3.1.1 Three-Step Model Photoemission process is also described by a simple model called the three-step model [1]. This model consists of the following three steps; 1. Photo-excitation of electrons into final states in bulk 2. Transportation of photo-excited electrons to the sample surface 3. Transmission of electrons through the surface into vacuum. In the process 1, the transition probability ω is given by the Fermi’s golden rule as ω ∝ |ψ f ||ψi |2 δ(E f − E i − ω),

(3.4)

where ψi is the initial state wavefunction while ψ f is the final state wavefunction. Here, considering the second-order terms in the photo-excitation process induced by the electric field with vector potential A and scalar potential ψ E M ,  is expressed as

Fig. 3.2 Schematic of angle-resolved photoemission spectroscopy (ARPES). Photon source is incident to the surface to induce the photoelectron effect, and the emitted electrons are analyzed by a hemispherical analyzer [2]

24

3 Angle-Resolved Photoemission Spectroscopy

e e2 ( A · p + p · A) − eψ E M + A· A 2mc 2mc2 e e2 (2 A · p + i(∇ · A)) − eψ E M + = A · A. 2mc 2mc2

=

(3.5) (3.6)

By setting ψ E M to be 0, assuming ∇ A = 0 by the translation symmetry in bulk and ignoring the two-photon process given by A · A, we get ω ∝ |ψ f |

e A · p|ψi |2 . mc

(3.7)

In addition, considering the conservation of the in-plane momentum in the process 3 and the periodicity set by the reciprocal lattice vector G, the photoemission intensity I is expressed as I ∝ | M˜ i1f |2 δ(ki|| − k f || + G || )δ(ki − k f + G)δ(E f − E i − hν)δ(E kin − E f + φ), (3.8) 1 where M˜ i f is the matrix elements in the optical transition, which can be discussed based on initial state expressed by the Bloch states and free electron-like final states or final states with one hole due to the photoelectron effect. The process 2 can also be taken into account by considering a complex out-of-plane wave vector k⊥ = (1) (2) + ik⊥ , which induces the attenuation of photoelectrons during the process due k⊥ to the finite mean free path of electrons in a solid. Based on this simple model, the transition probability or the photoelectron intensity I is expected to be proportional to the density of states of initial states. In addition, inelastically scattered electrons during the process lose the information of the initial state energy and momentum, forming background signals. Thus, the photoelectron signals consist of peaks corresponding to the density of states and background signals, and the peak positions give the information about the band dispersions of electrons.

3.1.2 Surface Sensitivity of ARPES The probing depth of ARPES is mainly determined by the electron escaping depth, which is of the order of ∼1 nm since the probing light has much longer penetration depth. When the electrons are excited in a material, they may lose their energy by several processes such as plasmon excitation and electron-hole pair formation. These effects are summarized in the inelastic mean free path (IMFP) of electrons λ, which refers to the mean value of the path along which an electron moves between two inelastic scattering processes. Considering the situation that N electrons move along z-axis with IMFP of λ, the scattering probability for a path of dz is dz/λ and we get d N (z) = −

dz N (z). λ

(3.9)

3.1 Principles of ARPES

25

Therefore, the number of electrons that have not been inelastically scattered at a position z can be written as N (z) = N0 exp

z , λ

(3.10)

where N0 is the number of electrons excited at the initial position. Thus, the distribution function for the depth of photoelectrons φ(z) is  z φ(z) ∝ exp − , λ

(3.11)

and hence the photoelectron intensity I becomes ∞ I ∝

φ(z)dz.

(3.12)

0

Consequently, the probing depth of ARPES λ A R P E S , which is the mean value for the depth of the emitted photoelectrons, can be written as ∞ λ A R P E S = 0 ∞ 0

zφ(z)dz φ(z)dz

= λ.

(3.13)

Here the probing depth becomes the same as the IMFP. In this thesis, most of the ARPES experiments were performed with photons in the vacuum-ultra-violet (VUV) region, ranging from 7 eV obtained by laser to ∼100 eV generated by synchrotron facilities. For very low electron energies, the IMFP is mainly determined by the scattering probability that induces interband transition (electron-hole pair creation), and thus it is roughly given as λ ∝ E −2 ,

(3.14)

where the energy of electrons relative to the Fermi level. For higher energies about ∼150 eV, the main contribution to IMFP becomes the plasmon losses, and IMFP is approximated as E , (3.15) λ= a(log E + b) where a and b are material-dependent constants. This equation illustrates that the scattering cross-section for higher energy electrons is smaller. Combining these discussions, it has been known that the IMFPs of electrons in solids behave similarly for most materials, and the fitting to this behavior is called the universal curve of the IMFP of electrons. According to a previous study [4], the IMFP is well approximated by

26

3 Angle-Resolved Photoemission Spectroscopy 4

10

3

10

λ (nm)

2

10

1

10

0

10

-1

10

0.1

1

10

100

1000

10000

Energy (eV)

Fig. 3.3 Inelastic mean free path of electrons described by Eq. 3.16. The monolayer thickness of Au (∼2.4Å) is used to visualize the equation

λ=

2170 + 0.72(a E)1/2 monolayers, E2

(3.16)

where a is the monolayer thickness in nanometers. The universal curve has a minimum of ∼0.4 nm at about 40 eV, and it becomes longer as the energy becomes closer to 0 or as it goes to a larger value. For example, it is ∼4 nm at 7 eV, ∼0.6 nm at 100 eV, and ∼2 nm at 2000 eV (Fig. 3.3). As discussed above, the IMFP of electrons or the probing depth of ARPES is of the order of a few atomic layers, manifesting the surface sensitivity of ARPES. Therefore, while the bulk properties can be extracted by ARPES, the surface properties should also be considered in most materials, since the surface which breaks the translation symmetry of the bulk may yield different electronic structures than those of the bulk. However, the bulk electronic structures and the surface electronic structures can be separated in many results by carefully taking hν-dependence or comparing the ARPES signals with band structure calculations, since they are usually observed at different angles or different energy levels. Such a technique has been often used to discover topological insulators, which have an insulating bulk with gapless topological surface states.

3.1.3 Orbital Selectivity in ARPES The matrix elements in photoemission processes sensitively depends on the polarization of light, since photoemission occurs mainly due to the dipole transition as discussed before:

3.1 Principles of ARPES

27

Fig. 3.4 Schematic of polarization-dependent ARPES

I ∝ |ψ f | A · p|ψi |2 ,

(3.17)

where A is the vector potential which expresses the electric field of the light and p is the momentum operator. Let us consider the case with the reaction plane is parallel to the plane of incidence as shown in Fig. 3.4. In this geometry, both of the momentum operator p and the free-electron final states φ f | are even with respect to the mirror plane. Therefore, if A is even, the wavefunctions with even symmetry |ψeven  can give non-zero photoemission intensity, while photoelectrons are derived from the wavefunctions with odd symmetry |ψodd  if A is odd. For linearly polarized excitation light, therefore, p-polarized light (even symmetry) excites |ψeven  while s-polarized light excites |ψodd . When using circularly polarized lights or linearly polarized light with inclined polarizations, both of the wavefunctions |ψeven  and |ψodd  need to be taken into consideration. In summary, we can perform orbital-selective ARPES by employing linearly polarized light sources.

3.1.4 Electron Analyzer As an electron analyzer for ARPES, a hemispherical analyzer or a time-of-flight analyzer is often employed. In this thesis, all the measurements were performed with hemispherical analyzers. The photoelectrons emitted from a sample go through an electrostatic lens before entering the spherical part of the analyzer. The electrostatic lens allows electrons emitted from the focus point of the sample to be focused at the entrance slit of the analyzer, and decelerate or accelerate electrons to the so-called pass energy E Pass . In the hemispherical analyzer, the trajectory of electrons is curved by the voltages applied to the inner part E in and the outer part E out , so that the electrons with energy E Pass ± δ E can be transported to the micro-channel plate (MCP). The MCP amplifies the signals of photoelectrons so that they can be detected by a fluorescent plate and

28

3 Angle-Resolved Photoemission Spectroscopy

a CCD camera. Images taken by the CCD camera can be displayed and analyzed using computers. Here, the pass energy and the kinetic energy of an electron (E kin ) has a relation expressed as E Pass = E kin + W − Wana − eVR ,

(3.18)

where W is the work function of the sample, Wana is the work function of the analyzer, and VR is the retarding voltage at the electrostatic lens. Hence, by scanning VR , we can take an energy distribution curve of photoelectrons in a wide energy range. The theoretical energy resolution of the analyzer E ana is given as  E ana = E Pass

Sent α2 + 2R 4

 ≈

E Pass Sent , 2R

(3.19)

where R is the radius of the analyzer, Sent is the width of a entrance slit and α is the incident angle of photoelectrons to the slit (Fig. 3.5). One can understand that the smaller pass energy and the larger analyzer lead to higher energy-resolution.

3.1.5 Efficient Measurements: Two-Dimensional Analyzer and Electro-Static Deflector When ARPES was first invented in the late 20th century, only photoemission signals at a certain emission angle were collected at a certain geometry, and the analyzer or the sample had to be rotated to obtain the angle-dependent signals; that is, the signals taken by the analyzer was one-dimensional spectra. Later, due to the development of two-dimensional hemispherical analyzers which have the capability of taking an energy-angle image (E − k x image) at the same time, ARPES has become a more efficient and powerful technique with high energy and momentum resolution. Currently, two-dimensional analyzers are widely employed at ARPES end stations of many research institutes and synchrotron facilities. Recently, electron analyzers have been further developed to detect the constant energy contour of photoelectrons without rotating samples. For example, a hemispherical analyzer with an electron deflector has such capability, enabling us to measure a tiny sample without measuring different domains after sample rotations. The analyzer mainly used in this thesis is ScientaOmicron DA30L, which can detect the photoelectrons within the range of ±15◦ along x-axis and y-axis. Thus, a constant energy contour (E − k x − k y image) can be obtained in a short time. This advantage is also evident in spin-resolved measurements, where one-dimensional measurements are still common and the sample needs to be rotated before each measurement. However, when analyzing data taken with an electron deflector, the transformation of angles to momentum should be more carefully performed than 1D or 2D ARPES experiments, since the tilt angle or the polar angle of the sample stage does not

3.1 Principles of ARPES

29 Deceleration/ acceleration to EPass

E E-δE

E~EPass

E+δE VR

Vin

Vout

EPass-δE EPass EPass+δE Fig. 3.5 Schematic of an electron trajectory in a hemispherical analyzer. In the strong paths before the entrance of the spherical part, electrons are decelerated or accelerated by the electric field E R . In the spherical part, the trajectory of the electrons is curved by the voltage Vin and Vout so that the electrons Electrons with energies around E Pass can reach the micro-channel plate at the end of the analyzer. In this method, the energy range detected at the same time is set by |E − E Pass | < E δ , and the Vin and Vout change according to the pass energy. The electron energies detected by the analyzer can be scanned by changing the deceleration/acceleration voltage V R

exactly correspond to the deflector angle αx and α y . With the experimental geometry schematically shown in Fig. 3.6, the in-plane momentum k x and k y are obtained as    k x = 0.512 E k (cos θ · αx − sin θ sin φ · α y )sinc α2x + α2y  

− sin θ cos φ cos α2x + α2y , and 

  k y = 0.512 E k cos φ · α y sinc(α2x + α2y ) − sin φ cos α2x + α2y , (3.20) where θ is the polar angle and φ is the tilt angle of the sample [5]. The deflector angle shown in a software is usually related to the analyzer normal angle by a polar coordinate with angles α and β as illustrated in Fig. 3.6. Here αx = α cos β and α y = α sin β. An rotation transformation should also be included if the sample has a certain azimuthal angle.

3.2 Light Sources To detect photoelectrons, electrons in a solid need to be excited by light so that they can escape from the sample surface into the vacuum. The potential barrier at the surface is described by the work function of a sample. Since it usually takes a value

30

3 Angle-Resolved Photoemission Spectroscopy

Fig. 3.6 ARPES with an electron-deflector system. The sample is rotated by angles θ and φ with respect to the sample manipulator, while the deflector angle α is defined to the analyzer coordinate system

of 4 ∼ 5 eV, the energy of the incident light source should be higher than 5 eV to perform ARPES experiments. Moreover, since the momentum measured in ARPES √ is expressed as 0.512 × E kin sin θ Å−1 , the measurable momentum space depends on the energy of the probing light. If the photon energy is higher, ARPES signals can be taken over several Brillouin zones, while the angular resolution may become worse. Therefore, to discuss the band structure within a Brillouin zone, the photon energy used for ARPES typically ranges from 6 eV to 1000 eV. Such light can be generated by a lamp, a laser, or a synchrotron. A Helium discharge lamp is one of the most common light sources in ARPES, since it can provide 21.2 eV photons using He-I resonance line, with which moderate energy and angular resolutions with high-stability have been achieved. In this thesis, however, lasers and synchrotrons are employed to achieve high energy resolution or high spatial resolution with high flux light sources.

3.2.1 Laser Light Source for ARPES Laser light has several characteristics differently from ordinary light. First, it has a high degree of coherence, a high directionality, and a high monochromaticity. In addition, the output intensity of a laser can be very high. Moreover, the light obtained by laser is highly polarized depending on the mechanism of the generation. Consequently, lasers have become a powerful tool to perform various kinds of measurements in condensed matter physics. For example, the high intensity enables us to perform measurements with high resolution in a short time. The high spatial coherence is also powerful compared with conventional lamps since small samples (typically smaller than 1 mm) can be selectively illuminated using the focused laser light. However, the wavelength of a laser light is often longer than 500 nm, with

3.2 Light Sources

31

Fig. 3.7 Laser system for 7 eV generation with a KBBF crystal [6]

which photoelectric effects cannot be induced. Therefore, in ARPES experiments, a pulsed laser is usually employed and the frequency is multiplied by using non-linear optical crystals. In addition, if many electrons are ejected from the surface at the same time by pulsed light, the information on the energy and momentum of electrons will be lost due to the Coulomb interaction between photoelectrons, which is called the space charge effect. Therefore, a good laser light source for ARPES needs • High repetition rate with a large number of photons (quasi-continuous laser) • Photon energy higher enough than the work function of a solid (∼5 eV which corresponds to ∼200 nm). Due to These requirements, the use of laser in ARPES is strictly limited. In this thesis, a frequency tripled Nd:YVO4 laser (Paladin compact, Coherent) is used since it generates 355 nm laser light (3.497 eV) with a repetition rate of 120 MHz. As shown in Fig. 3.7, the intensity of the laser is controlled using a half-wave plate and a polarizing beamsplitter. In addition, by putting three half mirrors between the Nd:YVO4 laser and the nonlinear crystal, the repetition rate is effectively multiplied up to 960 MHz. The light is now focused by a lens and then becomes incident on a KBe2 BO3 F2 (KBBF) crystal to double the frequency. Here, we can obtain high intense light of 6.994 eV (147 nm) when the incident angle between KBBF and the 355 nm light is a phase-matching angle of 66.2◦ [6, 7]. Since the 6.994 eV light is easily absorbed by water in the atmosphere, the path between the KBBF and the ARPES chamber is filled with nitrogen gas. The polarization is horizontal after the KBBF crystal, and the direction of the linear polarization can be manipulated by a half-wave plate while it can be circularly polarized using a quarter-wave plate. By using a lens before the ARPES chamber, the light is focused into a spot smaller than 50 µm, which enables us to measure a small crystal.

32

3 Angle-Resolved Photoemission Spectroscopy

3.2.2 Synchrotron Light Source for ARPES The light source used in this thesis is not limited to laser, but a variety of ARPES experiments have been performed using Synchrotron light sources. In a synchrotron facility, an undulator is inserted into a storage ring where a bunch of electrons is running to generate pulsed light with high intensity. The emitted light has a very wide range in energy before monochromatized by a diffraction grating. Thus, it is easier to change the photon energy of synchrotron light than laser light, providing an ideal platform to study the three-dimensional electronic structure of a material. In addition, the energy of synchrotron light can range from vacuum-ultra-violet light to a hard X-ray. Since the energy resolution of the light is expressed by E/E, lower energy light usually achieves a higher energy resolution. Therefore, a synchrotronbased ARPES branch with low energy photons (300 eV) are suitable for bulk-sensitive ARPES which can minimize the surface effect as well as k z broadening. In this thesis, high-energy resolution ARPES experiments were performed at Diamond light source in the United Kingdom [8] and at Stanford Synchrotron Radiation Lightsource at SLAC national accelerator laboratory in the United States. In addition, spatial-resolved microscopic ARPES experiments were performed at Diamond Light Source and Elettra SynchrotronTrieste in Italy.

3.3 Spin-Resolved Photoemission Spectroscopy An electron in a solid does not only possess information about its energy and momentum but also has spin or orbital angular momentum. Photoelectrons proceeding in the vacuum can also be spin-polarized due to the initial state properties or the final state properties, while the orbital angular momentum is not defined in the vacuum. Since electron spins play an important role in magnetism as well as spintronics, the direct measurement of the spin polarization of electrons in a solid is essential to understand their mechanism of such quantum effects. Spin-resolved ARPES (SARPES) is a powerful tool with the capability of detecting electron spins in the momentum space [9], while most spin-sensitive techniques usually measure the total spin effect integrated over the whole Brillouin zone.

3.3 Spin-Resolved Photoemission Spectroscopy

33

3.3.1 Detection of the Spin Polarization of Photoelectrons In typical SARPES measurements, the spins of electrons are measured after resolving the emission angle and the kinetic energy of electrons. A Mott-type spin detector and a VLEED-type (Very low energy electron diffraction type) spin detector are the representative systems used for the spin detection [9, 10]. A Mott-type spin detector utilizes the Mott scattering which is a relativistic effect based on the spinorbit coupling and has been developed earlier and widely used since the 1980 s due to its high stability. To enhance its efficiency, photoelectrons are accelerated to ∼10 keV and made incident on a target composed of heavy elements such as Au with large spin-orbit coupling coefficients. By counting the number of backscattered electrons, we can calculate the spin polarization of photoelectrons based on the knowledge about the spin-dependent scattering probability. However, since electrons must be accelerated to a very high energy to make use of the relativistic effect, the efficiency of the detection is about 10000 times lower than spin-integrated ARPES and thus the development of high energy resolution SARPES with a Mott spin detector has been limited. In contrast, a VLEED-type spin detector employs a ferromagnet such as Fe as the target, and the intensity of diffracted electrons at the surface which depends on the magnetization direction of the target is measured to obtain information on the spin polarization. In this technique, the (00) diffraction pattern in low energy electron diffraction (LEED) is measured, and hence the efficiency has been predicted to be higher than the Mott system. Generally, the figure of merit, the efficiency of a spin detector, is defined as I (3.21) FOM = Se2f f , I0 where I0 is the intensity of incident electrons, I is the intensity of diffracted electrons at the target, and Se f f is the so-called effective Sherman function. Se f f represents the detection power of an electron spin and has a relation P=

Asym , Se f f

(3.22)

where P is the spin polarization while Asym is the asymmetry of spin-dependent signals. For a VLEED detector, Se f f is around 0.3 and I /I0 ∼ 0.1. Therefore, the FOM is of the order of 10−2 . For a typical Mott detector, FOM is of the order of 10−4 , meaning that a VLEED detector is about 100 times more efficient than a Mott detector. Nevertheless, the use of VLEED detectors was very limited due to the instability of Fe surfaces. To increase the stability, an oxygen-terminated Fe(001)p(1 × 1)-O surface is employed as the target instead of a pure Fe(001) surface. This oxygenterminated surface is stable under the ultra-high vacuum for over several weeks while the pure Fe surface can degrade in a few hours. Moreover, the quality of the target can be recovered by flashing and annealing, making it suitable for the SARPES target. The procedure for preparing the target is as follows:

34

1. 2. 3. 4.

3 Angle-Resolved Photoemission Spectroscopy

Cleaning of MgO substrate Deposition of thick Fe films on the MgO substrate Annealing of the Fe film under an oxygen atmosphere Confirmation of the (1 × 1) pattern of the Fe(001)p(1 × 1)-O surface by LEED.

Since even a small amount of impurities can prevent the formation of the target, it must be very carefully prepared. The contamination of carbon usually leads to the formation of a (2 × 2) pattern in LEED experiments. The Fe(001)p(1 × 1)-O shows a large asymmetric spin-dependent diffraction at 6 eV. The asymmetry Asym in Eq. 3.22 is defined as Asym =

I+ − I− , I+ + I−

(3.23)

where I+ (I− ) is the photoelectron intensity reflected from the target with parallel (antiparallel) magnetization to the spin quantization axis. With the spin polarizations P calculated by Asym and the effective Sherman function Se f f , the photoelectron intensities with up- or down-spin (I↑,↓ ) are obtained as I↑,↓ =

1 (1 ± P)(I+ + I− ). 2

(3.24)

Using a VLEED target, spin-resolved measurements are possible along two different axes that are parallel to the surface of the target. Consequently, a combination of two VLEED targets enables us to obtain the full information of 3D spin polarizations of photoelectrons.

3.3.2 Spin-Resolved ARPES Machine at ISSP, UTokyo Due to the development of the VLEED-type spin detectors in combination with the third generation synchrotron light sources or laser light sources, the energy resolution of SARPES machines has been rapidly improved. Meanwhile, researchers have found spin-orbit split electronic structures at the surface of heavy elements and the surface of topological insulators, enriching the field of spin-dependent phenomena which should be investigated by SARPES. In these circumstances, Yaji et al. constructed a high-resolution laser-SARPES machine at the Institute for Solid State Physics, the University of Tokyo (Fig. 3.8) [11, 12]. The machine employs KBBF-based 7 eV laser system as the main light source and two VLEED-type spin detectors with ScientaOmicron DA30L analyzer, with which 3D spin polarizations of photoelectrons at θx = ±15◦ in the horizontal direction and θ y = ±10◦ in the vertical direction can be measured without sample rotations. The best energy resolution is 1.7 meV, and highly efficient measurements can be performed with the energy resolution of 10 meV to 30 meV by using different pass energy, analyzer slit and a larger aperture between the hemispherical analyzer and the VLEED detectors.

3.3 Spin-Resolved Photoemission Spectroscopy

35

Fig. 3.8 Schematic of SARPES machine constructed at the Institute for Solid State Physics at the University of Tokyo. Reprinted from Ref. [11], with the permission of AIP Publishing

In Fig. 3.9, ARPES and SARPES results for the Shockley surface state of Au(111) taken with the laser-SARPES machine is summarized. Here, p-polarized light was used. A normal ARPES image is shown in Fig. 3.9a. The observed band structure of the surface state is consistent with previous studies. By using spin-resolving mode, ¯ − spin-dependent energy distribution curves (EDCs) were taken at ±11◦ along M ¯ ¯  − M direction (Fig. 3.9b and c). The quantization axis of spin polarization was set to y-axis, which is perpendicular to the analyzer slit. The spin-dependent EDCs show that the peak positions of the up-spin spectra and the down-spin spectra are separated by E ∼ 0.1 eV, and the relation of two peaks get inverted between +11◦ and −11◦ . The spin polarization at the peaks has been found to close to 100%. By changing the deflector angle θx , we can map the spin-dependent photoelectron intensities in a 2D image. While Fig. 3.9d shows the spin-integrated intensity mapping which is basically the same as Fig. 3.9a except the resolution, Fig. 3.9e shows a spin-polarization mapping which can only be obtained by SARPES. To enhance the visibility, a 2D color code is employed in Fig. 3.9f to illustrate both the total intensity and the spin-polarization in the same image. A pair of spin-split surface states with nearly parabolic dispersions is confirmed in the image, which is also consistent with previous reports.

36

3 Angle-Resolved Photoemission Spectroscopy (b)

0

-0.2 -0.3 -0.4 -0.5 -0.6 -15 -10 -5 0 5 10 15 Angle (deg)

Spin-integrated ARPES

(c)

(e)

-0.2 -0.1 E-EF (eV)

0

Spin-polarization map

SARPES EDC at θ=+11 1.0 0.5 0 -0.5 -1.0

-0.3

(f)

0

0

0

-0.1

-0.1

-0.2

-0.2

-0.2

-0.3 High

-0.4

-0.3 Sy

-0.4

1.0

-0.2 -0.1 E-EF (eV)

0

SARPES image

-0.1

E-EF (eV)

E-EF (eV)

(d)

-0.3

E-EF (eV)

E-EF (eV)

-0.1

SARPES EDC at θ=-11 1.0 0.5 0 -0.5 -1.0

Intensity (arb. units) Ppolarization

Normal ARPES image

Intensity (arb. units) Polarization

(a)

-0.3 Sy

-0.4

1

0.5

-0.5

0

-0.5

0

-0.5

-0.5

-0.6 -10

-5 0 5 Angle (deg)

10

Low

-1.0

-0.6 -10

-5 0 5 Angle (deg)

10

-1 Low High

-0.6 -10

-5 0 5 Angle (deg)

10

Fig. 3.9 SARPES of Au(111). a Conventional ARPES image taken for a clean Au(111) surface. b, c Spin-resolved intensity (I↑ and I↓ ) taken at θ = ±11◦ . d Spin-integrated ARPES image (I↑ + I↓ ) taken with spin detection mode. e Spin polarization map taken for d. f SARPES image generated from d and e with a 2D color code

3.4 Spatial-Resolved ARPES Due to the development of light sources with high spatial coherence, we can now focus the probe light into a spot smaller than 1 µm to perform scanning photoemission microscopy experiments. As a focusing system, a Fresnel zone plate [13] or Schwarzschild objectives [14] can be employed before the sample at synchrotron facilities, as schematically illustrated in Fig. 3.10. The technique which employs a nano-focused beam and ARPES is usually called nano-ARPES, and it has been widely applied to the study of the electronic structures of samples with small domains or thin film devices. Currently, several synchrotron beamlines have nano-ARPES setups in the world. In this thesis, the electronic structure on the side surface of quasi-one-dimensional crystals have been measured at the nano-ARPES branch at the beamline i05 at Diamond light source and Spectromicroscopy beamline at Elettra Synchrotron Trieste. At the beamline i05, a Fresnel zone plate is employed, which makes a wide range of photons energies applicable to ARPES experiments. Besides, a hemispherical analyzer with an electron-deflector system (ScientaOmicron DA30L) enables Fermi surface mapping without sample rotations. However, the intensity of photons

3.4 Spatial-Resolved ARPES

37

Fig. 3.10 Schematics of focusing systems for nano-ARPES. a A Fresnel zone plate with a pinhole and an order sorting aperture. b Schwarzschild objectives

is reduced to about 1/700 compared with the normal ARPES branch. Therefore, a single experiment needs a much longer time than conventional ARPES experiments. At Spectromicroscopy beamline, Schwarzschild objectives are used to focus the beam, and consequently, the available photon energies are limited. Instead, the reduction of the intensity of the light is only about 10 times, which is important in performing relatively quick measurements. The electron analyzer at Elettra is facility-made and has a capability of rotation inside the chamber, which also enables Fermi surface mapping without sample rotation. In nano-ARPES measurements, the alignment of samples is more important since the effective spot size becomes larger if the sample is out of focus. To locate the sample at the focus point, a combination of the sample movement within the sample holder plane and the sample movement normal to the light is usually repeated, which is the key in measuring small samples or small domains by nano-ARPES.

References 1. Hüfner S (1996) Photoelectron spectroscopy, vol 82. Springer series in solid-state sciences. Springer, Berlin 2. Damascelli A, Hussain Z, Shen Z-X (2003) Angle-resolved photoemission studies of the cuprate superconductors. Rev Mod Phys 75(2):473–541 3. Lv B, Qian T, Ding H (2019) Angle-resolved photoemission spectroscopy and its application to topological materials. Nat Rev Phys 1(10):609–626 4. Seah MP, Dench WA (1979) Quantitative electron spectroscopy of surfaces: a standard data base for electron inelastic mean free paths in solids. Surf Interface Anal 1(1):2–11 5. Ishida Y, Shin S (2018) Functions to map photoelectron distributions in a variety of setups in angle-resolved photoemission spectroscopy. Rev Sci Instrum 89(4):043903 6. Kiss T, Shimojima T, Ishizaka K, Chainani A, Togashi T, Kanai T, Wang XY, Chen CT, Watanabe S, Shin S (2008) A versatile system for ultrahigh resolution, low temperature, and polarization dependent Laser-angle-resolved photoemission spectroscopy. Rev Sci Instrum 79(2) 7. Shimojima T, Okazaki K, Shin S (2015) Low-temperature and high-energy-resolution laser photoemission spectroscopy. J Phys Soc Jpn 84(7):072001

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8. Hoesch M, Kim TK, Dudin P, Wang H, Scott S, Harris P, Patel S, Matthews M, Hawkins D, Alcock SG, Richter T, Mudd JJ, Basham M, Pratt L, Leicester P, Longhi EC, Tamai A, Baumberger F (2017) A facility for the analysis of the electronic structures of solids and their surfaces by synchrotron radiation photoelectron spectroscopy. Rev Sci Instrum 88(1):013106 9. Okuda T (2017) Recent trends in spin-resolved photoelectron spectroscopy. J Phys Condens Matter 29(48):483001 10. Okuda T, Takeichi Y, Maeda Y, Harasawa A, Matsuda I, Kinoshita T, Kakizaki A (2008)A new spin-polarized photoemission spectrometer with very high efficiency and energy resolution. Rev Sci Instrum 79(12) 11. Yaji K, Harasawa A, Kuroda K, Toyohisa S, Nakayama M, Ishida Y, Fukushima A, Watanabe S, Chen C, Komori F, Shin S (2016) High-resolution three-dimensional spin- and angle-resolved photoelectron spectrometer using vacuum ultraviolet laser light. Rev Sci Instrum 87(5):053111 12. Kuroda K, Yaji K, Harasawa A, Noguchi R, Kondo T, Komori F, Shin S (2018) Experimental methods for spin- and angle-resolved photoemission spectroscopy combined with polarizationvariable laser. J Vis Exp 136:e57090 13. Avila J, Razado-Colambo I, Lorcy S, Lagarde B, Giorgetta J-L, Polack F, Asensio MC (2013) ANTARES, a scanning photoemission microscopy beamline at SOLEIL. J Phys Conf Ser 425(19):192023 14. Dudin P, Lacovig P, Fava C, Nicolini E, Bianco A, Cautero G, Barinov A (2010) Angleresolved photoemission spectroscopy and imaging with a submicrometre probe at the SPECTROMICROSCOPY-3.2L beamline of Elettra. J Synchrotron Radiat 17(4):445–450

Chapter 4

Revealing the Role of Wavefunctions in Rashba-Split States

4.1 Introduction The observation of spin-split band structures by ARPES and SARPES allows us to directly observe the Rashba effect and determine its magnitude. In addition to the surface states of heavy metals such as Au, W, and Bi, large spin-splittings have been reported in surface alloys, surfaces of semiconductors with heavy elements, and bulk crystals without inversion symmetry [1–10]. As we have seen in Sect. 2.2.2, the origin of spin splittings induced by heavy elements is mainly attributed to the asymmetry of the wavefunction around the atoms [11, 12]. Therefore, the essential key to understand the detailed mechanism which governs the magnitude of splitting and spin texture lies in the behaviors of wavefunctions. Thus, we will discuss the characteristic behavior of spin-orbit coupled wavefunctions in Sect. 4.2 by utilizing laser-SARPES with a polarization-variable 7 eV laser [13]. In Sect. 4.3, we demonstrate the control of the Rashba splittings in quantum well states by modifying the charge density distributions around the film. We have further found the scaling law for the Rashba effect in quantum well films, by considering the microscopic Rashba effect originating in the envelope function of the well state [14]. Through these studies, the role of wavefunctions in determining the spin texture and the magnitude of the splitting is discussed.

4.2 Giant Rashba Systems: Surface Alloys Surface alloys are one of the most heavily investigated systems among the materials with the Rashba effect. Especially, √a 1/3 monolayer of Bi deposited on top of a bulk √ Ag(111) crystal, Ag(111)-( 3 × 3)R30◦ -Bi surface alloys has gained significant attention due to the large spin splitting and its spin polarization (Fig. 4.1) [6, 15–32] as well as its potential applications in spintronics [33, 34]. One of the interesting © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 R. Noguchi, Designing Topological Phase of Bismuth Halides and Controlling Rashba Effect in Films Studied by ARPES, Springer Theses, https://doi.org/10.1007/978-981-19-1874-2_4

39

40

4 Revealing the Role of Wavefunctions in Rashba-Split States

BiAg2 E

pxy

EF k

BiCu2

pxy

EF spz

spz

Fig. 4.1 Schematics for the band structures of Bi-based surface alloys BiAg2 (left) and BiCu2 (right) around the Fermi level (E F ). The lower bands mainly consist of spz orbitals, while the upper bands mainly consist of px y orbitals

aspects of this system is that the Rashba effect in the alloy is larger than that found at the surface of Bi(111) or other planes, although the SOC effect of Ag is expected to be much smaller than that of Bi due to the lighter mass of an Ag atom. The observation of the band structures on Bi/Ag(111) has triggered the field, and the detailed electronic structures of various surface alloys have also been investigated both by experiments and by theoretical calculations [35–41]. In Fig. 4.2, the top view and the side view of the topmost surface atoms of Ag(111) and Bi/Ag(111) are shown. The topmost atoms of Ag(111) are placed in a flat layer, while the bismuth atoms in Bi/Ag(111) are located higher than the Ag toms by d = 0.65 Å [20]. Since the band dispersions of the surface states can be well captured by only considering the topmost BiAg2 layer, this surface alloy is also written as BiAg2 /Ag(111). Here we review the formation of the surface states in BiAg2 /Ag(111) [15]. The unit cell of BiAg2 consists of two Ag atoms and one Bi atom. As a result, two Ag 5s orbitals and Bi 6 p orbitals contribute most to the band structures around the Fermi level (E F ). Theoretical calculations suggest that before the inclusion of SOC, Ag 5s orbitals and Bi 6 pz orbital are mixed to form a hole-like spz -derived state, which is almost fully occupied. In contrast, the px orbital and the p y orbital of bismuth form two bands which are degenerate at the ¯ point. For the px state and p y state, most of the dispersions are located above E F . After the inclusion of SOC, the px state and p y state are hybridized to form a band with m j = 1/2 and a band with m j = 3/2. The energy level of the m j = 1/2 band is lower than the m j = 3/2 band and thus the m j = 1/2 band becomes partly occupied, while the m j = 3/2 band is fully unoccupied. All of the Bi-based states show large spin splittings, among which the splitting of the spz -derived state is the largest. The band dispersions of these states and their spin-polarized behaviors have been confirmed by ARPES and inversephotoemission spectroscopy [18, 29]. Due to the mechanism mentioned above, two pairs of Rashba-split states exist around E F as shown in Fig. 4.1. In this thesis, the fully occupied band is called the spz band while the partially occupied band is called the px y band due to the orbital compositions.

4.2 Giant Rashba Systems: Surface Alloys

41

Fig. 4.2 Atomic structures of the topmost layer of a Ag(111) and b BiAg2 /Ag(111) visualized by VESTA [42]. The black lines denote the lattice vectors. In b, d shows the height difference between Bi atoms and Ag atoms

Similar but smaller Rashba-split bands have also been observed in other systems such as BiCu2 /Cu(111) and PbAg2 /Ag(111) [37, 40], while the surface states in SbAg2 /Ag(111) show almost spin degenerate dispersion within the experimental resolution of ARPES [38, 39]. Since the atomic numbers of Bi and Pb are similar, the difference of the magnitude of the Rashba effect in BiAg2 and PbAg2 indicates that the origin of the Rashba effect cannot be solely explained by the large atomic SOC coefficient of Bi. According to reports based on density functional theory (DFT) calculations [27], the most plausible explanation for the large Rashba effect is the combination of the large SOC effect of Bi and the displacement of the Bi layer from the topmost Ag layer. It has been reported that the magnitude of the effect has a systematic dependence on the displacement by calculating the band structures for hypothetical BiAg2 with various corrugation parameters and that the effect becomes negligible if the SOC of Bi is turned off in theoretical calculations [27]. Particularly, the spin splitting becomes smaller when the corrugation parameter is close to 0 or larger than 1Å. This result can be interpreted as follows. First, when the Rashba effect is mainly derived from the atomic SOC, we have  αR =

d 3r

∂ V (r) |ψ(r)|2 . ∂z

(4.1)

Since the dominant factor for creating the asymmetry in the integrand of Eq. 4.1 is the wavefunction of the surface states, the surface wavefunctions should have a large asymmetry around the bismuth atoms in the topmost layer. Considering the fact that the splitting is well reproduced by theoretical calculations for the BiAg2 monolayer without the Ag(111) substrate, the asymmetry can be attributed to the mixture of Bi 6 p orbitals and Ag 5s orbitals, which are present near the Fermi level. Therefore, if

42

4 Revealing the Role of Wavefunctions in Rashba-Split States

the corrugation parameter is close to 0, the asymmetry wavefunctions become small since the overlap integral of p-orbitals and an s-orbital decreases. In addition, if the bismuth layer is separated by the Ag layer with a large corrugation parameter, the hybridization also decreases due to the large distance between Ag atoms and Bi atoms, resulting in a smaller Rashba effect.

4.2.1 Spin Textures Beyond the Rashba Model Apart from the large Rashba parameters, another interesting aspect of this system is the complex spin texture predicted by DFT calculations. Based on the conventional Rashba model, a pair of spin-split states should have opposite spin polarization, and the direction of the spin polarizations remain constant regardless of the momentum. Thus, one can draw two pairs of spin-split states with the conventional Rashba effect as in the left panel of Fig. 4.3. However, the spin-texture predicted by DFT calculations is more complicated [15]: first, the band structures are modified due to the hybridization between the spz state and the px y state, since they are located very 1 closely at a large momentum point k x ∼ 0.15Å . Although the spin texture is similar to what is based on the Rashba model around the ¯ point, the spin-polarization of the spz states are reversed around the momentum point where it intersects with the px y state. Furthermore, two branches of the px y state at E F have parallel spin polarizations, which is also against the Rashba model. So far, the spin textures have been directly investigated by conventional SARPES and spin- and angle-resolved inverse photoemission experiment [18, 29]. In previous SARPES studies, the spin polarization of the spz band near the ¯ point have been detected, while the reversal of the spin polarization has not been discussed. Also, the spin polarization of the outer px y band could not be evaluated due to the weak photoemission intensity. In the inverse photoemission study, the measured

Fig. 4.3 Schematic of the complex spin texture in BiAg2 /Ag(111). Simple spin texture based on the Rashba model is presented in the left panel while the spin texture predicted by DFT calculations is summarized in the right panel

4.2 Giant Rashba Systems: Surface Alloys

43

spin polarizations for the px y bands (the m j = 1/2 band and the m j = 3/2 band) did not correspond well with the predicted spin texture, and the comparison with orbital-resolved spin texture indicates the importance of the light polarization which couples to the orbital symmetry of wavefunctions through the inverse photoemission process. In addition, the complex spin texture was indirectly discussed by quantum interference through scanning tunneling spectroscopy [28, 30], while the results are still controversial. To summarize, the complex spin texture of BiAg2 /Ag(111) has been theoretically attributed to the unconventional orbital texture in the surface band structures. However, the lack of the orbital sensitivity has prevented us from directly revealing the full spin information, which is still under discussion, and no conclusive understanding of the SOC-induced interband hybridization between the spz band the px y band has been obtained.

4.2.2 Method DFT calculations were performed using the VASP code [43]. The projector augmented wave method [44] is used in the plane-wave calculation, and the generalized gradient approximation by Perdew, Burke, and Ernzerhof [45] is used for the exchange-correlation potential. The spin-orbit interaction is turned on to reproduce the Rashba effect. 28- and 19-layer symmetrical slabs were used to simulate the surface electronic structures of BiAg2 and BiCu2 , respectively. The atom positions of BiAg2 are optimized, while those of BiCu2 are taken from a previous experimental report Ref. [46]. The laser-ARPES and laser-SARPES measurements were performed at ISSP, the University of Tokyo. The main light source is a high-flux 6.994-eV laser, while He discharge lamp was also used. The main chamber is equipped with a ScientaOmicron DA30L photoelectron analyzer [47]. The experimental setup is shown in Fig. 4.7. For laser-based measurements, the p- and s-polarizations (horizontal polarization and vertical polarization, respectively) were used in the experiment. The photoelectrons ¯ ¯ lines of the surface Brillouin zones of BiAg2 /BiCu2 [¯ M ¯ K were detected along lines of Ag(111) and Cu(111)]. The spin polarizations were resolved along y-axis, which is perpendicular to mirror plane of the measurements. The sample temperature was kept at ∼15 K. The instrumental energy (angular) resolutions of the setup were set to be 2 meV (0.3◦ ) and 20 meV (0.7◦ ) for ARPES and SARPES, respectively. In this section, all the DFT calculations were performed by Prof. Kobayashi.

4.2.3 Orbital-Coupled Spin Textures in BiAg2 and BiCu2 As mentioned in the previous section, it has been pointed out that the complex spin texture is related to the orbital texture of the surface band dispersions. In order to discuss the experimental results, the orbital composition and the orbital-coupled

44

4 Revealing the Role of Wavefunctions in Rashba-Split States BiAg₂/Ag(111) pxy 0 spz

-0.3 -0.6 -0.9

0

Even orbitals

(b) E-EF (eV)

E-EF (eV)

(a)

0 -0.3 -0.6 -0.9

0.1 0.2 0.3 -1

0

0 -0.3 -0.6 -0.9

0

0.1 0.2 0.3 -1

kx (Å )

0

(d)

0.1 0.2 0.3 -1

kx (Å )

E-EF (eV)

E-EF (eV)

Spin polarizations

0.1 0.2 0.3 -1

kx (Å )

(c)

Odd orbitals

kx (Å )

Orbital-coupled spin plarizations

0 -0.3 -0.6 -0.9

0

0.1 0.2 0.3 -1

kx (Å )

0

0.1 0.2 0.3 -1

kx (Å )

Fig. 4.4 Orbital decomposed band structures of BiAg2 . a Calculated band structure and b calculated orbital weight of even-orbitals (left) and odd-orbitals. The weights of orbitals with each symmetry are expressed by the size of the circles. c Calculated total spin texture and d the orbital-coupled spin texture decomposed into even-orbitals (left) and odd-orbitals (right). We note that electron-like dispersions are quantum well states of the substrate. Reprinted with permission from Ref. [13] ©2017 by the American Physical Society

spin polarizations are obtained by DFT calculations for BiAg2 /Ag(111) and a related surface alloy BiCu2 /Cu(111). The calculated band structures of BiAg2 are shown in Fig. 4.4a. The intersection of the outer spz band and the inner px y band is reproduced. Here, odd-symmetry orbitals and even-symmetry orbitals with respect to the x-z plane are separately calculated, in accordance with the orbital selectivity in ARPES. Figure 4.4b shows that the inner part of the spz band, the px y band, and the quantum well states of Ag(111) mainly consist of even-orbitals, while odd-orbitals contribute much to the outer part of the spz band and the outer branch of the px y band. In addition, the reversal of the spin polarization of the spz band is found at k x ∼ 0.2 Å−1 in Fig. 4.4c. As expected by the symmetry rule, even-orbitals and odd-orbitals are coupled to opposite spin polarization as shown in Fig. 4.4d. In contrast to BiAg2 , the spz bands of BiCu2 do not cross other surface bands except the quantum well states of the substrate (Fig. 4.5a). The spin polarization of the spz band does not reverse in the calculated region, while the px y bands show the reversal above E F , corresponding to the variation of the orbital compositions. This simple spin texture of the spz band in BiCu2 allows us to discuss the complex spin texture of BiAg2 through the comparison of these systems.

4.2 Giant Rashba Systems: Surface Alloys BiCu₂/Cu(111) pxy

0.4 0 spz

Even orbitals

(b) 0.8 E-EF (eV)

E-EF (eV)

(a)0.8

45

0.4 0

-0.4

-0.4

-0.8

-0.8 0

0.2

0.4

0

0.4 0

Spin polarizations

0.8 0.4

(d)

0

-0.4

0.2

0.4 -1

kx (Å )

E-EF (eV)

E-EF (eV)

0.2 -1

-1

kx (Å )

(c)

Odd orbitals

kx (Å )

Orbital-coupled spin plarizations

0.8 0.4 0

-0.4

-0.8

-0.8 0

0.2

0.4 -1

kx (Å )

0

0.2

0.4 0 -1

kx (Å )

0.2

0.4 -1

kx (Å )

Fig. 4.5 Orbital decomposed band structures of BiCu2 . a Calculated band structure and b calculated orbital weight of even-orbitals (left) and odd-orbitals. The weights of orbitals with each symmetry are expressed by the size of the circles. c Calculated total spin texture and d the orbital-coupled spin texture decomposed into even-orbitals (left) and odd-orbitals (right). We note that electron-like dispersions are quantum well states of the substrate. Reprinted with permission from Ref. [13] ©2017 by the American Physical Society

4.2.4 Sample Preparation In this section, clean surface alloys were prepared by using the molecular beam epitaxy apparatus connected to the laser-SARPES machine. First, the clean Ag(111) substrate and the clean Cu(111) substrate were obtained by cycles of Ar sputtering, and annealing at ∼500 K. The quality of the substrates were checked by low energy electron diffraction (LEED) (Fig. 4.6) and ARPES of the Shockley surface states. Second, 1/3 ML of Bi was evaporated using a resistively heated Knudsen cell on to the

Fig. 4.6 LEED patterns of Ag(111) and BiAg2 /Ag(111). a The three fold pattern (quasi-six fold pattern) from Ag(111) surface taken at 59.5 eV. b √ √ ( 3 × 3)R30◦ pattern from BiAg2 taken at 47.5 eV

46

4 Revealing the Role of Wavefunctions in Rashba-Split States

clean Ag/Cu surfaces to form the surface alloys following √ procedures presented √ the in the literatures [6, 19]. The formation of the the ( 3 × 3)R30◦ surface alloys were confirmed by LEED (Fig. 4.6) and the Fermi surface mapping by ARPES.

4.2.5 Orbital-Selective ARPES of Surface Alloys Let us start with the ARPES results of Ag(111) taken with p-polarized laser light (Fig. 4.7). The sharp feature with a strong photoemission intensity at ∼100 meV below E F at the ¯ point is the Shockley surface state in the projected bulk band gap (Fig. 4.8a), which is also called the L-gap due to the absence of the occupied bands at E F along the -L line. The spin splitting of the Shockley surface state is not resolved in ARPES spectra due to the small Rashba effect at Ag(111) surface [48]. Apart from the ¯ point, another large parabolic feature is detected with a large dispersion along k x . This feature has been detected in previous studies and attributed to the direct transition between occupied bulk sp states and unoccupied bulk sp states (the sp-sp band in Fig. 4.8a) [49]. In addition, the edge of the projected bulk band gap is also detected at ∼400 meV below E F , which becomes visible with an enhanced color contrast (Fig. 4.8b). Below the edge of the projected gap, weak photoemission signals have been detected, which originate from a bulk occupied state with large k z dispersion. Having confirmed the clean √ of Ag(111) substrate, 1/3 ML of bismuth √ surface was deposited to form the ( 3 × 3) surface alloy. The ARPES intensity maps for BiAg2 taken with the p- and s-polarized laser light and with the He lamp are shown in Fig. 4.9. The spz -derived bands and px y -derived bands have been observed,

Fig. 4.7 Experimental geometry of ARPES/SARPES measurements of the surface alloys. The black hexagons denote the BiAg2 (BiCu2 ) surface Brillouin zones, while the blue dotted hexagon shows the surface Brillouin zone of Ag(111) (Cu(111)). The linearly polarized 6.994 eV light is used to excite photoelectrons. The red and blue arrows below S y are the spin polarization vectors used in this experiment

4.2 Giant Rashba Systems: Surface Alloys

(a)

47

(b)

0.0

0.0

-0.3

E-EF (eV)

E-EF (eV)

SS

Max

Max

-0.3

-0.6 sp-sp

-0.9

-0.2

0.0

0.2

Min 0.6

0.4

-0.6

Min -0.2

0.0

0.2 -1

-1

kx (Å )

kx (Å )

Fig. 4.8 Laser-ARPES of Ag(111) taken with p-polarized light. a Wide ARPES intensity map ¯ M¯ line. b Magnified ARPES image with an enhanced color contrast to emphasize the edge along of the projected bulk band gap, which is denoted by the red dotted line. The surface state (SS) is observed in the projected bulk band gap, whereas the photoemission signals from the transition between bulk sp states (sp-sp) are overlapping with the bulk states BiAg2/Ag(111), p-pol

(a)

pxy

spz

BiAg2/Ag(111), s-pol

(b)

pxy

pxy

0

0

-0.3

-0.3

-0.6

-0.6

spz

pxy

(c)

BiAg2/Ag(111), He-I pxy

Umklapp

spz

pxy

0.0

-1.2

Max

-0.9

E-EF (eV)

Max

-0.9

E-EF (eV)

E-EF (eV)

-0.5

Max -1.0

-1.2 sp-sp

-1.5

-1.5

-1.5

Min -0.2

0

0.2 -1

kx (Å )

0.4

Min -0.2

0

0.2 -1

kx (Å )

0.4

Min -0.3

0.0

0.3

0.6

-1

kx (Å )

Fig. 4.9 ARPES of BiAg2 . Each image is taken with p-polarized laser (a), s-polarized laser (b) or using a He lamp (c)

which disperse downwards in energy showing the Rashba-type spin splittings. In addition, photoemission signals from the sp-sp direct transition is again observed at ∼1.5 eV below E F only for p-polarized laser light. This reflects the nature of the direct transition since it is allowed for lower photon energies with even-symmetry. Interestingly, the photoemission signals show a strong dependence on the linear polarization, indicating the different orbital compositions for each branch of the spz bands and px y bands. In addition to the surface bands which are present in the

48

4 Revealing the Role of Wavefunctions in Rashba-Split States BiCu2/Cu(111), p-pol

spz

spz

BiCu2/Cu(111), s-pol

(b)

pxy

spz

0

0

-0.5

-0.5

Max -1.0

-1.5

sp-sp

E-EF (eV)

E-EF (eV)

(a)

pxy

Max -1.0

-1.5 Min

-0.2

0

0.2 -1

kx (Å )

0.4

0.1

0.2

0.3

0.4

Min 0.5

-1

kx (Å )

Fig. 4.10 ARPES of BiCu2 . The images are taken with p-polarized light (a) and s-polarized light (b)

calculated band structures, another hole-like dispersion is observed at ∼0.3 Å−1 in the ARPES image taken with s−polarized light. This feature is attributed to a Umklapp band, which stems from the Umklapp scattering of the bulk electrons due to the surface superstructure. In the ARPES experiments of BiCu2 , the surface states have also been detected, while the energy levels of the surface states are higher than BiAg2 and the spz bands become partly unoccupied (Fig. 4.10). Here, strong linear polarization dependence and the Umklapp band are also observed. Considering the orbital selection rule in photoemission experiments, p-polarized light and s-polarized light excite different orbitals: since the surface states composed of Bi6 p orbitals and Ag 5s or Cu 4s orbitals, p-polarized light is sensitive to the s, px and pz orbitals which have even symmetry with respect to the mirror plane, while s-polarized light mainly excites the p y orbital. The linear polarization dependence of BiAg2 is summarized in Fig. 4.11a. In the ARPES results taken with p-polarized light, we observe strong intensities from the pair of the spz bands and the inner branch of px y band. The data clearly shows the band crossing of the outer spz band and the inner px y band around k x ∼ 0.15 Å−1 . The spectral intensity of the outer spz band is strongly suppressed around the crossing point. In contrast, by using s-polarized light, the spz band is clearly observed at large k x (> 0.15 Å−1 ) together with the outer px y band. With these results, the overall dispersion of the spz bands and px y bands are determined, which was absent in previous studies [6, 19, 50]. The right panel of Fig. 4.11a shows a differential intensity map. The magenta color corresponds to even orbitals, while the green color corresponds to odd orbitals. These results are in good agreement with the orbitals compositions obtained by theoretical calculations (Fig. 4.4), and the change of the orbital character of the spz band at the band crossing point is clearly visible.

4.2 Giant Rashba Systems: Surface Alloys

(a)

49

High Low

Low

spz

pxy

High

spz

Even

Odd

pxy

spz

pxy

E-EF (eV)

0 -0.2 -0.4 -0.6 -0.8 BiAg2 0

(b)

BiAg2 0.1

0.2

spz

0.3

0

BiAg2 0.1

0.2

spz

pxy

0.3

0

0.1

0.2

spz

pxy

0.3

pxy

E-EF (eV)

0 -0.2 -0.4

BiCu2 -0.6

0.1

BiCu2 0.2

-1

kx (Å )

0.3

0.1

BiCu2 0.2

-1

kx (Å )

0.3

0.1

0.2

-1

0.3

kx (Å )

Fig. 4.11 Polarization dependent ARPES of the surface alloys. a and b The magnified ARPES images taken with (left) p- and (middle) s-polarized light, and (right) the differential intensity maps for BiAg2 and BiCu2 , respectively. The differential intensity is calculated as I p − Is where I p and Is are the photoelectron intensity taken with p- and s-polarized light. The dashed lines indicate the edges of the projected bulk bands of Ag(111) and Cu(111). Reprinted with permission from Ref. [13] ©2017 by the American Physical Society

By contrast, the linear polarization dependence in BiCu2 is rather simple (Fig. 4.11b). The spz bands the px y bands are mainly composed of even orbital and odd orbitals, respectively, which is indicated by their sensitivity to the p-polarized light and s-polarized light. These bands are well separated in the momentum space, avoiding the band crossing and the hybridization. In addition, we see the reduction of the spectral intensity at the edge of the projected bulk band gap indicated by the black dotted lines, especially for the even orbitals of the spz band. This feature suggests the strong interaction between the pz orbital at the surface and the s orbital in the substrate, which can also explain the suppression of the weights of even orbitals of the spz band of BiAg2 (Fig. 4.11a). The significant k-dependence of the linear polarization dependence is found to be unique for the spz bands of BiAg2 . This result suggests that the band hybridization between the outer spz band and the inner px y band due to SOC modifies the orbital texture of the spz band, although the gap opening between these two bands, which is expected by the hybridization [15, 25, 27, 28], is not clearly visible in our ARPES experiments.

50

4 Revealing the Role of Wavefunctions in Rashba-Split States

4.2.6 Orbital-Selective SARPES of Surface Alloys To investigate the complex spin texture and the effect of the interband hybridization. We performed laser-SARPES with linearly polarized light. The SARPES spectra obtained for BiAg2 and BiCu2 are shown in Figs. 4.12 and 4.13, respectively. The corresponding SARPES images are also shown with a 2D color code which illustrates both of the spin polarization and the ARPES intensity [51]. For p-polarized light, the spin polarizations of the inner spz band and the outer spz band are negative and positive around the ¯ point, consistent with the conventional Rashba-splitting [21, 50]. The observed spin polarization is found to be large up to nearly 80 and 60 % for BiAg2 and BiCu2 , respectively. Remarkably, the sign of the spin polarizations flips depending on the linear polarization. This is clearly seen in the outer spz band of BiAg2 at around k x = 0.15 Å−1 , where the positive and negative spin polarizations are detected with p-polarized light and s-polarized light, respectively. Most remarkably, we find that the sign of the spin polarization sensitively depends on the linear polarization. This can be seen in the SARPES spectra particularly for k x = 0.12 Å−1 in BiAg2 . The spectral weight of the spin-up is considerably larger than the spin-down for p-polarized light and achieves nearly +80 % spin-polarization. For s-polarized light, the intensity relation turns to the opposite and the resulting spin-polarization is

Fig. 4.12 SARPES results of BiAg2 /Ag(111). a and b SARPES spectra with spin polarizations resolved along y-axis taken by p- and s-polarized light, respectively. The solid (open) triangles indicate the peak positions of the spz bands (the px y bands). c and d Spin polarization and intensity maps with the two dimensional color code [51]. Reprinted with permission from Ref. [13] ©2017 by the American Physical Society

(a)

p-polarization

(b)

s-polarization -1

Spin-resolved intensity (arb. units)

kx (Å ) -0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24

-0.8 -0.6 -0.4 -0.2 0 E-EF (eV)

(c)

p-polarization

-0.8 -0.6 -0.4 -0.2 0 E-EF (eV)

(d)

s-polarization Sy

0

E-EF (eV)

1

-0.2

0

-0.4 -0.6

-1

-0.8

0

0 .1 -1

kx (Å )

0 .2

0.3

0

0 .1 -1

kx (Å )

0 .2

Low High 0.3 Intensity

4.2 Giant Rashba Systems: Surface Alloys Fig. 4.13 SARPES results of BiCu2 /Cu(111). a and b SARPES spectra with spin polarizations resolved along y-axis taken by p- and s-polarized light, respectively. The solid (open) triangles indicate the peak positions of the spz bands (the px y bands). c and d Spin polarization and intensity maps with the two dimensional color code [51]. Reprinted with permission from Ref. [13] ©2017 by the American Physical Society

(a)

51 p-polarization

s-polarization

(b)

-1

Spin-resolved intensity (arb. units)

kx (Å ) 0.07 0.10 0.13 0.16 0.19 0.22 0.25 0.28 0.31 0.34

-0.6

(c)

-0.4 -0.2 E-EF (eV)

0

-0.6

(d)

p-polarization

-0.4 -0.2 0 E-EF (eV)

s-polarization Sy

0

E-EF (eV)

1

-0.2

0

-0.4

-1 -0.6

0.1

0.2

kx (Å )

0.3

0.1

0.2

kx (Å )

0.3

Low High Intensity

found to be −70 %. Since the linear polarization is sensitive to different orbital symmetry, our laser-SARPES umambiguously reveals that the spin direction strongly depends on the orbital character. Considering the dipole selection rule in SARPES, the strong linear polarization dependence unambiguously shows the realization of orbital-dependent spin polarizations in the surface alloys. Assuming the presence of the mirror plane, the wavefunctions of spin-orbit coupled states can be generally written as |±i  = |even, ↑ (↓) + |odd, ↓ (↑),

(4.2)

where the quantization axis of spinors | ↑ and | ↓ are taken along y-axis, which is perpendicular to the mirror plane. ± denotes the mirror eigenvalues. This equation shows that even orbitals and odd orbitals are generally coupled to opposite spin polarizations under the presence of SOC. In fact, the orbital-coupled spin polarizations have been found in topological insulators [52–57] and Rashba systems [58, 59]. In BiAg2 , the positive spin polarization coupled to even orbitals predominates the outer spz band around the ¯ point, while negative spin polarization coupled to odd orbitals is dominant at larger k x as shown in Fig. 4.12c and d. Apparently, the results deviate from the complex spin texture predicted by DFT calculations

52

4 Revealing the Role of Wavefunctions in Rashba-Split States

(Fig. 4.4c), while they show good agreement with orbital-coupled spin polarizations (Fig. 4.4d). The results therefore indicate that the momentum dependence of the orbital weights play the significant role in determining the total spin polarizations, which can be probed by SARPES with linearly polarized light sources. For BiCu2 , the SARPES results shown in Fig. 4.13c and d show qualitative agreement with the calculated orbital-coupled spin polarizations (Fig. 4.5d), further demonstrating that the orbital-dependence of the spin polarizations play essential roles in understanding the SARPES results with polarized light.

4.2.7 Discussion So far, the orbital-coupled spin texture in the surface alloys has been revealed by orbital-selective SARPES. However, the question remains whether or not the total spin polarization can be experimentally extracted. Here we will show that the total spin texture can be traced back by using a combination of SARPES results taken with p-polarized light and s-polarized light. Owing to the orbital symmetry detected in our laser-SARPES setup, the orbital-dependence is eliminated by the integrations of SARPES mapping as follows: Ptotal =

(I↑, p + I↑,s ) − (I↓, p + I↓,s ) , (I↑, p + I↑,s ) + (I↓, p + I↓,s )

(4.3)

where Ptotal is the total spin polarization and I↑, p (I↑,s ) and I↓, p (I↓,s ) are the spinresolved photoemission intensity obtained by p-polarized light and s-polarized light. The results of the above analysis are shown in Fig. 4.14. The reversal of the spin polarization is found in the outer spz band of BiAg2 at k x ∼ 0.15Å−1 , while the

(a)

p-pol.+s-pol.

p-pol.+s-pol. BiCu2

BiAg2 0

Sy

1

E-EF (eV)

0

(b)

-0.2 -0.2

-0.4

0

-0.6

-0.4

-1

-0.8 0

0 .1

-1

kx (Å )

0 .2

0 . 3 -0.6 0.1

0.2

-1

kx (Å )

0.3

Low High Intensity

Fig. 4.14 Experimentally determined orbital-integrated spin polarizations for a BiAg2 and b BiCu2 . We have assumed that the photoemission cross sections for p- and s-polarized light are identical. Reprinted with permission from Ref. [13] ©2017 by the American Physical Society

4.2 Giant Rashba Systems: Surface Alloys

53

Fig. 4.15 Spin texture in the BiAg2 surface alloy. a Schematic of the theoretically obtained spin textures of the spz bands and the px y bands. The red (blue) color denotes the positive (negative) spin polarizations. b–d Spin textures for the constant energy contours at E = E F , E 1 , and E 2 shown in a. The black arrows indicate the directions of the spin polarizations. The black rectangles mark the spin polarizations that are confirmed by SARPES experiments

signs of the spin polarizations of other bands remain constant in the region of interest. Moreover, the analysis shows that signs of the spin polarizations of the inner and the outer px y bands in BiAg2 are the same at positive k x in clear contrast to the simple Rashba model. These results are consistent with the predicted total spin textures of the surface alloys, revealing the complex spin texture realized only in BiAg2 . The comparison between the theoretically and the experimentally obtained spin textures are summarized in Fig. 4.15. Notably, one can expect that the directions of the spin polarizations for the pair of Fermi surfaces are the same as shown in Fig. 4.15b. These complex spin textures discussed above are likely due to the interband spin-orbit hybridization, which leads to the momentum-dependent modulation of the orbital components in the surface bands. Since the orbital mixing is widely expected in a material with strong SOC, this polarization-dependent analysis offers a general advantage in understanding the complex spin texture due to strong SOC effects. Here we note that in the above analysis, we assumed that the photoemission cross sections for p-polarized light and s-polarized light are the same. However, since the matrix elements sensitively depends the angle of the incident light and the final state effects, photon energy or geometrical dependence and photoemission calculations are required to do more qualitative analysis [60, 61]. Another question is about the absence of the hybridization gap between surface states, although the orbital-selective SARPES strongly suggests the interband hybridization. Here we discuss the possible origin of the closing of the hybridization gap due to the presence of the substrate. It was shown that the hybridization gap could be clearly observed by ARPES if the thickness of Ag(111) substrate is reduced to less than 4 ML [25]. Since the continuum electronic state in bulk is quantized in thin films, it is expected that the closing of the band gap is due to the hybridization between the surface states and the bulk continuum state. In fact, the effective mass of the outer spz band is suddenly changed at the edge of the projected band gap of Ag(111) as shown in Fig. 4.16, the bulk bands clearly affect the band structure of the surface states. In addition, the interaction between the bulk continuum states and the surface states can be modeled by effectively adding a non-Hermitian Hamiltonian to the surface Hamiltonian. This gives rise to the finite lifetime of surface electrons,

54

4 Revealing the Role of Wavefunctions in Rashba-Split States

Fig. 4.16 Interaction between the surface bands and the bulk bands. a and b ARPES intensity maps for BiAg2 taken with p- and s-polarized light overlaid with the peak positions of each band (red circles). c First derivative of the peak positions of the outer spz band. A kink-like feature is found, indicating the sudden change of the effective mass. d The energy positions of the surface states of BiAg2 overlaid with an ARPES image of Ag(111). The kink-like feature is present around the edge of the bulk band

which allows the band hybridization without gap opening, as is reported in image potential resonances [62, 63]. This result suggests that the interband hybridization can generally happen without the hybridization gap, which can be investigated by looking at the orbital compositions or spin polarizations. In conclusion, we have mapped the spin and orbital wavefunctions of the Rashbasplit states in Bi-based surface alloys by laser-SARPES with linearly polarized light. Although the SARPES results only reflect the orbital-coupled spin texture, the full spin information can be discussed by the integration of SAPRE results taken with different polarizations. Moreover, we have revealed the pronounced interband hybridization effect between the surface states only in BiAg2 , which leads to the complex orbital and spin texture predicted by DFT calculations.

4.3 Quantum Well Films with Tunable Rashba Effects When electrons are confined in a potential well in a certain direction, the energy levels of electrons are quantized with the formation of discrete states. Such a situation is common in the textbooks on quantum mechanics, which deepens our understanding of the energy levels of electrons. In the simplest infinite quantum well, the quantized wavefunctions ψn (z) are written as ψ(z) ∝ sin

 nπ z  d

,

(4.4)

where d is the thickness of the width of the well and n is an integer number, and forms the standing wave patterns. The energy levels are given by

4.3 Quantum Well Films with Tunable Rashba Effects

55

Fig. 4.17 Schematics of quantum well states. (left) An textbook infinite quantum well. (middle) A finite quantum well where electrons can penetrate outside the well. (right) A quantum well film which beaks the space inversion symmetry (IVS)

En =

2  nπ 2 2m d

(4.5)

with the wave vector k quantized as k = nπ/d.

(4.6)

Here, the charge density ρ(z) = |ψ(z)|2 is perfectly localized within the well. However, if the height of the well is finite, the wavefunctions are modulated so that they can penetrate into the well (Fig. 4.17). In general, the decay of the wavefunctions outside the well depends on the quantum number n, and states with larger n show larger penetrations. In these simplest models, the space inversion symmetry is preserved for electrons; thus, no spin splitting is expected. The quantized states also appear in solid state physics, especially in the field of thin films [64]. If a metallic film is sandwiched by two insulators or vacuum, the electrons in the film are confined to form the standing wave patterns. Let us consider a quantum film consisting of a substrate, a film, and vacuum as shown in the right panel of Fig. 4.17. The potential barrier between the vacuum and the film is relatively high for an electron, although that between the substrate and the film can be much smaller and sensitively depends on the choice of the compounds. Thus, one can expect the spin splitting in quantum well states (QWSs), since the shape of the potential become asymmetric along the z-direction, which breaks the space inversion symmetry. Notably, the tunability of film thicknesses offers an ideal platform to gradually modify the inversion asymmetry of electrons to control the magnitude of the Rashba effect, which is difficult to achieve in surface states without applying an external field or inducing band bending effects by surface adsorption. Here we note that Eq. 4.6 is only valid for infinite wells, and phase shifts at the boundary must be included to evaluate E-k relation along the z-axis. So far, quantum well effects in solids have been found in a variety of systems, including semiconductor heterostructures and metallic heterostructures. In semiconductors, the potential barrier which forms the well is the fundamental gap of semi-

56

4 Revealing the Role of Wavefunctions in Rashba-Split States

conductors. Therefore, it is quite common to realize the quantum well confinement by varying the compositions of the films. In contrast, it has been shown that metallic heterostructures can form quantum wells, although there is no fundamental gap in metal. This is due to the presence of the projected band gap in the substrate. For example, the L-gap of Au(111) supports the existence of QWSs in Ag films and Cu films which cannot propagate into the substrate, similar to the behavior of the Shockley surface states of Au(111) [65, 66]. In addition, the so-called symmetry gap can accomplish the confinement in a metallic system. This happens when the symmetries of the bands in the film and the substrate are different. For instance, if sp-orbitals of Ag cannot propagate into Fe substrate with d-orbitals near E F , which also offers a platform to study quantum well phenomena [67]. In this thesis, as the simplest case of QWSs with tunable Rashba effects, Ag films grown on Au(111) has been chosen where the projected bandgap of Au(111) mainly confines the sp electrons of Ag. Although the Rashba effect is generally expected in quantum well films without space inversion symmetry, the reports on the Rashba-split states in QWSs have been limited mainly due to the small splitting in the real systems. An example with the Rashba effect is Pb films grown on Si(111) substrate, where the large SOC effect of Pb atoms induce the energy splitting [68]. While the splitting is still small to be detected by spin-integrated ARPES measurements, SARPES results clearly demonstrate the spin splittings of QWSs and their dependence on the interface engineering. However, the reported values of the Rashba parameters do not show systematic dependence on the film thickness while the energy positions shift as the thickness changes. Another example is Al/W(110) with exceptionally large spin splittings, which is due to the hybridization with the interface states [8]. In spite of the large Rashba effects, the tunability utilizing the film thickness has not been reported since the complex band structures have prevented us from systematically modifying the wavefunctions and the band structures. Therefore, quantum well films with simple band structures are needed to demonstrate the control of the effect and to reveal the mechanism.

4.3.1 Prototype Metallic Quantum Well Film Ag/Au(111) Among many kinds of QWSs reported so far, we focus on Ag/Au(111) for three reasons [64–66, 73–77]. First, the band structures of QWSs are quite simple and resemble those of free electrons. The absence of complicated hybridization between the substrate and the film helps us to trace the thickness evolution of the band structures. Second, since the Shockley state of Au(111) is known for the typical Rashba effect, we can also expect the spin splitting in QWSs induced by the large SOC of Au atoms. Third, it has been reported that the penetration of QWSs into the substrate systematically depends on the film thickness [77]. This is the key feature in unveiling the scaling law for the Rashba effect in quantum well films, which will be discussed later. In Au and Ag, the valence bands consist of 1 sp electron and 10 d electrons. As shown in Fig. 4.18a, the d electrons form fully occupied states ∼4 eV below E F ,

4.3 Quantum Well Films with Tunable Rashba Effects

(a)

Bulk band of Au

10

5

Energy (eV)

Energy (eV)

0

Γ

U

X Γ

Au(111)

0

-10

L

Γ

U

X Γ

L

(c) Au(111) Ag film Vacuum E

Ag(111) QWS

-1

Energy (eV)

0

-5

-5

(b)

Bulk band of Ag

10

5

-10

57

QWS Au sp

Resonance -2

Ag sp

z V(z)

(d) |ψenvelope|²

-3

|ψ(z)|² -4

Γ

L Γ

L

Fig. 4.18 Formation of quantum well states in Ag/Au(111). a Theoretical band disperions of bulk Au and bulk Ag. b Magnified images of the theoretical band dispersions along the -L line, which is projected to the ¯ point of the Au(111) surface and the Ag(111) surface. c Schematic of the quantization of Ag electrons close to the valence band maximum. d Schematic of the charge density distribution |ψ(z)|2 and the well potential V (z). ψenvelope is the envelope function of ψ(z). In a and b, DFT calculations were performed using OpenMX including the spin-orbit coupling [69–72]

while the sharp dispersion of the sp electron cross the Fermi level, making the system metallic. By projecting the bulk bands onto the (111) surface, we get the projected band gap around the ¯ point. Since the -L line is projected to the ¯ point, we focus on the band structures along the -L line (Fig. 4.18b). The top of the valence band in Au is located at the L point at around 1.2 eV below E F , forming the L-gap above this energy. A similar gap is also formed in Ag but with a different energy scale; the top of the valence band of Ag is around 0.3 eV below E F . Accordingly, if the Ag layer is set adjacent to Au(111), only the electrons with energy lower than −1.2 eV

58

4 Revealing the Role of Wavefunctions in Rashba-Split States

can propagate into the substrate, while the electrons with energy between −0.3 eV and −1.2 eV are reflected at the interface. This leads to the partial confinement of valence electrons in the Ag film-forming QWSs (Fig. 4.18c). A schematic of the wavefunction of QWS in Ag/Au(111) is illustrated in Fig. 4.18d [64]. Since the electronic states near the L point are quantized, the wave vector k z of QWS is close to π/t with t being the thickness of a monolayer film. Therefore, the wavefunction of QWS oscillates rapidly around each atom, differently from the textbook quantum well states. However, the envelope function of this state shown by the black curve in Fig. 4.18d resembles that of a simple model. When the envelope function of QWS has ν(= 1, 2, 3 . . .) maxima, the state is often referred to as ν = 1, 2, 3 . . . well states. We will also use this notation since ν allows us to trace the thickness evolution of QWSs easily, and in fact, ν plays a significant role in evaluating the Rashba effect. Here we note that in a N layer film, the ν = n QWSs have N − n and hence can be understood as the (N − n )th eigenstate in this system. After the first observation of QWSs in Ag/Au(111) [65], many groups have reproduced the results and reported the details of the band structure. For films thinner than 6 ML, only the surface state is present in the projected bulk gap, while quantum well resonance states may have been formed. Above 7 ML, QWSs emerge from the bulk state, showing nearly parabolic dispersion. With increasing the film thickness, QWSs gradually shift upward in energy, and finally, the number of QWSs increases [76]. Theoretical calculations have suggested that the penetration of QWSs, which breaks the space inversion symmetry, shows systematic dependence on the film thickness [77]. Thus, Ag/Au(111) is an ideal system to investigate the thickness-dependent Rashba effect and demonstrate the modulation of wavefunctions for the engineering of the Rashba effect.

4.3.2 Method The laser-ARPES and SARPES measurements were performed with a laser-SARPES machine constructed at the Institute for Solid State Physics, the University of Tokyo [47, 78]. The light source used in this study is the p-polarized light emitted from a high-flux 6.994 eV laser. The photoelectrons were detected by a hemispher¯ line of the projected ¯ M ical analyzer (Scienta Omicron, DA30L) along with the surface Brillouin zone of Ag(111) or Au(111). The samples were transferred to the analysis chamber in situ and measured at ∼30 K. The instrumental energy (angular) resolutions were set to about 5 meV (0.3◦ ) and 15 meV (0.7◦ ) for ARPES and SARPES, respectively. In DFT calculations, the slab models of Ag/Au(111) consist of 60 ML of 1 × 1 Au units to describe the substrate and 6–34 ML of 1 × 1 Ag units. The relaxation of the interlayer distance at the surface is included by structural optimization. We employ OpenMX [69] with norm-conserving pseudopotentials [70] and pseudo-atomic localized basis functions [71, 72]. The exchange-correlation energy proposed by Perdew-

4.3 Quantum Well Films with Tunable Rashba Effects

59

Burke-Ernzerhof [45] is used through the generalized gradient approximation. These DFT calculations were performed by Dr. Kawamura.

4.3.3 Sample Preparations of Ag/Au(111) Thin Films The Au(111) substrate was cleaned by cycles of Ar sputtering and annealing at ∼500 ◦ C. The quality of the surface was verified by low-energy electron diffraction and ARPES observation of the Shockley surface state. Ag films were deposited on the Au(111) substrate using a resistively heated Knudsen cell, and the substrate temperature was kept at ∼100 K during the deposition using liquid nitrogen. The thickness of the films was estimated by the deposition rate obtained by a quartz crystal microbalance and determined through the comparison of the photoemission spectra of quantum well states with previous reports [76, 77]. After the deposition, the samples were annealed at room temperature to obtain high-quality crystalline Ag films [75]. The laser-ARPES images of Au(111) and Ag/Au thin films are displayed in Fig. 4.19a. For the pristine substrate, the Shockley surface state (SS) is observed at E − EF ∼ 0.5 eV at the ¯ point, and the edge of the bulk band is found at around E − EF = 1.1 eV. After the deposition of Ag films on the substrate, the energy position of the surface states shift upward in energy, and the position of SS in 12 ML film is close to that of the Ag(111) substrate. This indicates that most of the wavefunction of SS stays in the film, and thus the effect of the substrate on the surface state is small compared with thinner films. However, for films thicker than 12 ML, the energy positions of SS are higher than that of SS in Ag(111), with the broadening of the spectra from SS, implying the relatively rough surface conditions. Nevertheless, the edge of the projected bulk band of Au(111) and the sharp spectra from QWSs have been detected, indicating the high quality of the film and the interface. The QWSs which are present in the projected band gap have isotropic dispersions as evident in the constant energy contours shown in Fig. 4.19b. The energy positions of SSs and QWSs observed in this study is summarized in Fig. 4.19c. The thickness evolution of the band structures reported previously has been reproduced in our experiments [76, 77]. Importantly, the dispersions of QWSs are similar to those of free electrons, reflecting the simple band structures showing systematic dependence on the film thickness, which offers an ideal platform to investigate the Rashba effect in quantum well films.

4.3.4 SARPES of Ag/Au(111) Quantum Well Film The ARPES intensity maps presented in Fig. 4.19 failed to visualize the Rashbatype spin splitting in QWSs.. Therefore, to investigate the spin splitting in QWSs, we performed laser-SARPES experiments with high energy resolution. If the spin

60

4 Revealing the Role of Wavefunctions in Rashba-Split States

(a)

Au(111)

2 ML Ag

4.5 ML Ag

7 ML Ag

0

E-EF (eV)

-0.2

SS

SS

-0.4

SS

-0.6

SS

-0.8

ν=1

-1.0 -1.2 -0.2

-0.1

0

0.2 -0.2

0.1

-0.1

-1

0

0.1

0.2 -0.2

-0.1

-1

0

0.1

0.2 -0.2

-0.1

-1

0

0.1

0.2

-1

kx (Å )

kx (Å )

kx (Å )

kx (Å )

12 ML Ag

19 ML Ag

24 ML Ag

30 ML Ag

SS

SS

SS

SS

0

E-EF (eV)

-0.2 -0.4

ν=1

ν=1 -0.6

ν=1

-0.8 -1.0

-0.1

0

0.2 -0.2

0.1

-0.1

-1

19 ML Ag

0

0.1

0.2 -0.2

-0.1

-1

E=EF

0

0.1

30 ML Ag

(c)

E=EF

High

0

-0.1 -0.2 -0.2

-0.1

ν=2 0

0.1 -1

kx (Å )

0.2 -0.2

-0.1

Low

0

0.1 -1

kx (Å )

0.2

0

0.1

0.2

-1

kx (Å )

0.0 Surface state QWS ν=1

-0.4 -0.6

ν=2

-0.8

ν=1 ν=1

E-EF (eV)

SS

-1

ky (Å )

SS

-0.1

kx (Å )

-0.2 0.1

0.2 -0.2

-1

kx (Å )

kx (Å ) 0.2

ν=3

ν=2

-1.2 -0.2

(b)

ν=2 ν=2

ν=1

ν=3

ν=4

-1.0 0

5

10

15

20

25

30

35

Ag film thickness (ML)

Fig. 4.19 Thickness evolution of ARPES spectra of Ag/Au(111). a Representative ARPES intensity maps were taken for Au(111) and Ag/Au(111) thin films with various thicknesses. SS denotes the surface state, and v = 1, 2, 3 shows the QWSs with corresponding quantum numbers. b The constant energy contour at the Fermi level (E F ) for 19 ML film and 30 ML film. The surface state is located at the center of the images, while the ring-like features outside the surface state correspond to the quantum well states. c The energy positions of the surface state and the quantum well states at the ¯ point as a function of film thickness

polarization of the states is close to 100%, which is expected for the Rashba-split state composed of spz -derived states along the mirror plane, the Lorentzian fitting to SARPES spectra measured for opposite spin polarization enables us to determine the peak positions of each branch with the energy resolution smaller than the bandwidth. The resolution can also be better than the experimental resolution referring to the full width at half maxima (FWHM) of the measured spectrum. Hence, SARPES has the capability of detecting spin splittings, which are smeared out by the natural broadening in conventional ARPES measurements.

4.3 Quantum Well Films with Tunable Rashba Effects

61

The experimental geometry of SARPES is shown in Fig. 4.20a. The spin quantization axis is set to be the y-axis. In this section, we focus on the 12 ML film as a representative example. Although the splitting of QWS is not found in the ARPES image (Fig. 4.20b), by taking the second derivative of the ASRPES image of 12 ML, a pair of peaks in a single QWS is found as shown in Fig. 4.20c, which can be the signature of the Rashba splitting. Remarkably, this feature is clearly resolved with spin polarizations by SARPES, as shown in SARPES intensity and polarization map in the top panel of Fig. 4.20d. Here, the single parabolic feature observed in spin-integrated ARPES is divided into two parabolic features with opposite spin polarization. As evident in the middle panel of Fig. 4.20d, the peak positions of spinup spectra are located at higher energy than those of spin-down spectra at negative momenta, while the spin-up states and spin-down state become almost degenerate around the ¯ point and the relation is reversed at positive momenta. This corresponds to the typical Rashba-type spin-texture, and the orientation of the spin polarization vectors is found to be the same as that of the Shockley surface state of Au(111). Since the Rashba effect induces the splitting E = 2αR k at the in-plane momentum k, the Rashba parameter αR is obtained as αR = 135 meVÅ. This splitting is larger than that of the surface state of Ag(111), which is reported to be 31 meVÅ [48], indicating that only the surface effect cannot explain the origin of the spin splitting. In fact, the spin-splitting of the surface state in 12 ML film is also resolved by our laser-SARPES, and the Rashba parameter is obtained as αR = 37 meVÅ (Fig. 4.20e), which is about three times smaller than that of QWS and slightly larger than that of the surface state of bulk Ag(111). These results suggest that the consideration of the whole wavefunctions over the film and the substrate is necessary to understand the mechanism of the spin splitting observed by SARPES.

4.3.5 Thickness- and Quantum Number-Dependence of the Spin Splitting Having confirmed the spin splitting of QWS in 12 ML Ag/Au(111), we have investigated the thickness evolution of the Rashba effect. The SARPES results of representative films are summarized in Fig. 4.21. As we have seen in Fig. 4.19c, the ν = 1 QWSs shift upward in energy, and QWSs with larger ν emerge with increasing the film thickness. Our SARPES results reveal that the magnitude of the spin splitting of ν = 1 QWSs monotonically decreases as the film thickness increases: the peak separations of spin-up and spin-down spectra at the emission angle θ = −6.7◦ is ∼25 meV at 9 ML while it decreases to ∼2 meV at 30 ML, leading to the almost degenerate spectra within the experimental resolution (Fig. 4.21d). Next, we turn to the momentum-dependence of the spin splitting. In Fig. 4.21e–f, SARPES spectra for each QWS observed in the bulk gap of Au(111) are presented for three different films. By tracing the peak positions of the spectra, the energy positions of the spin-split bands have been determined. Thus, the energy splittings for all the bands at

62

4 Revealing the Role of Wavefunctions in Rashba-Split States y

(a)

x

e

(b)

12 ML Ag/Au(111)

Sy -0.6

1 0

-0.8

0

E-EF (eV)

-0.1 -0.3 -0.6

ν=1

Intensity (arb. units.)

Low -0.2 -0.1 0 -1 0.1 0.2 kx (Å ) 2nd derivative 0

E-EF (eV)

0.1

Sy 1

-0.04

0 -0.08

-1 Low High Intensity

-0.05

High

-0.9

(c)

0 -1 kx (Å )

E-EF (eV)

spin-up spin-down

-0.3

θ(°)

θ(°)

-6.5

-6.5

-4.3

-4.3

-2.0 0.2 2.5

-0.6

-1 Low High 0 0.05 Intensity -1 kx (Å )

Intensity (arb. units.)

-

Surface state

0.00

50° z

(e)

ν=1 QWS -0.4

E-EF (eV)

hν

(d)

p-pol.

-2.0 0.2 2.5

4.7

4.7

6.9

6.9 9.2

9.2 -0.9 -0.2 -0.1 0 -1 0.1 0.2 kx (Å ) -0.4

-0.6 -0.7 0.05

0.10-1 kx (Å )

0.15

25

0.04

20

αR=135 meV Å

0 -25 -50

ΔE (meV)

ΔE (meV)

E-EF (eV)

50 -0.5

-0.08 -0.04 0 E-EF (eV)

-0.8 -0.7 -0.6 -0.5 E-EF (eV)

10 0 -10 -20

-0.1 0 -1 0.1 kx (Å )

αR=37 meV Å -0.05

0

0.05 -1

kx (Å )

Fig. 4.20 SARPES results of 12 ML Ag/Au(111). a Experimental geometry of SARPES. b ARPES intensity map for 12 ML Ag/Au(111). c Second derivative of b. d and e (top) SARPES images, (middle) corresponding SARPES spectra and (bottom) momentum-dependent energy splitting for ν = 1 QWS and the surface state. The red and blue curves in the middle panels correspond to spin-up and spin-down spectra, respectively. The red lines in the bottom panels are the linear fit to obtain αR . Reprinted with permission from Ref. [14] ©2021 by the American Physical Society

several momentum points versus the momentum have been obtained, and the Rashba parameters are calculated by the linear fitting (Fig. 4.21h–j). The signs of the spin splittings do not show thickness dependence, indicating the same spin texture as the surface state of Au(111) is constantly realized. Interestingly, the magnitudes of the spin splitting in a film shows dependence on ν if multiple QWSs are present. The thickness- and the quantum number-dependence of the Rashba parameters is displayed in Fig. 4.22. The experimentally determined Rashba parameters for dif-

4.3 Quantum Well Films with Tunable Rashba Effects 9 ML (b)

0

(c)

21 ML

0

30 ML

0

-0.6 ν=1

-0.3

E-EF (eV)

E-EF (eV)

E-EF (eV)

SS -0.3

ν=1

-0.6 ν=2

ν=1

-0.3

ν=2

-0.6

ν=3

-0.9

-0.9

-0.9

-1.2

-1.2

-1.2

(d)

Intensity (a.u.)

(a)

63 30 ML

ν=1 θ ~ -6.7°

24 ML 19 ML 12 ML 9 ML

-0.1

0 0.1 -1 kx (Å )

21 ML

(f)

θ(°) -6.7

ν=1

0 0.1 -1 kx (Å )

ν=2

2.2

6.7

(i)

αR=174 meV Å 0 0.1 -1 kx (Å )

θ(°) -6.7

ν=1

ν=2

6.7 -1.0 -0.9 -0.8 -0.7

ΔE (meV)

αR=101 meV Å -0.1

2.2

αR=36 meV Å

0 0.1 -0.1 0 0.1 -1 -1 kx (Å ) kx (Å )

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 E-EF (eV)

30 ML 40

0

-40

0.0

4.5

ν=1

20

-20

-2.2

(j)

21 ML 40

ΔE (meV)

ΔE (meV)

0 -20

ν=2

-4.5

-0.9 -0.8 -0.7 -0.5 -0.4 -0.3 E-EF (eV)

ν=1

-0.1

2.2 4.5

20

-40

0.0

6.7

9 ML 40

-2.2

4.5

-0.9 -0.8 -0.7 E-EF (eV) (h)

ν=3

-4.5 Intensity (a.u.)

Intensity (a.u.)

0.0

0 0.1 E-E1 (eV)

30 ML

(g)

θ(°) -6.7

ν=1

-4.5 -2.2

-0.1

Intensity (a.u.)

9 ML

(e)

-0.1

-0.1 0 0.1 -1 kx (Å )

ν=3

ν=2

ν=1

αR=74 meV Å

αR=44 meV Å

αR=15 meV Å

20 0 -20 -40

-0.1

0 0.1 -0.1 0 0.1 -0.1 0 0.1 -1 -1 -1 kx (Å ) kx (Å ) kx (Å )

Fig. 4.21 Thickness-dependence of the Rashba parameters. a–c ARPES intensity maps for 9 ML, 21 ML and 30 ML films. d Thickness-dependent SARPES spectra for ν = 1 QWSs at the emission angle θ = 6.7◦ . e–g Angle-dependent SARPES spectra for QWSs in 9 ML, 21 ML and 30 ML films. h–j Corresponding k-dependent spin splittings. Red lines are the linear fits to obtain αR . Reprinted with permission from Ref. [14] ©2021 by the American Physical Society

ν=1 (exp.) ν=2 (exp.) ν=3 (exp.) ν=4 (exp.)

200

150

αR (meV•Å)

Fig. 4.22 Experimental Rashba parameters αR s of QWSs in Ag/Au(111) films with systematic dependence on the film thickness and the quantum number ν. For the same ν, αR decreases as the film thickness increases, while the QWSs with larger ν shows large αR in the same film. Reprinted with permission from Ref. [14] ©2021 by the American Physical Society

100

50

0 10

15

20 25 Film thickness (ML)

30

35

64

4 Revealing the Role of Wavefunctions in Rashba-Split States

ferent film thickness and different quantum numbers νs show a clear tendency that αR monotonically decreases as the film thickness increases, although αR increases for QWSs with larger ν in the same film. This simple dependence of the Rashba parameters on the film thickness and the quantum numbers allows us to discuss the origin of spin splitting via the comparison with theoretical calculations.

4.3.6 Analysis of Charge Density Distributions Based on a 1D Pseudopotential Model Since the asymmetry of wavefunctions is the key to enhance the Rashba effect, the analysis of the charge density distributions should deepen the understanding of the origin of the thickness dependence. As the charge density distribution along the out-of-plane direction is not easily accessed by experiments, we have performed theoretical calculations to evaluate the wavefunctions of QWSs. In this section, a one-dimensional pseudopotential (1DPP) model is employed. Although SOC is not included, the 1DPP model can describe the shape of the overall wavefunction without large discrepancy since SOC can be regarded as a perturbation term to the Coulomb potential, which mainly forms the potential well. The 1DPP model was proposed to calculate the energy levels of the image potential states at the surface of a metal by constructing a one-dimensional model potential, including the surface term. It has been employed to evaluate the wavefunctions of the surface states of metallic heterostructures [73], and the energy positions of QWSs have also been reproduced with small modification [76]. For the region near the surface, we employed the surface potential proposed Chulkov et al. [79, 80], and the free electron mass was replaced by the effective mass to reproduce the multiband effects [76]. In this model, the potential around each atom is expressed as 

 2π z , V1 (z) = A10 + A1 cos as

(4.7)

where as is the interlayer spacing, and A10 and A1 are independent values that depend on the atomic species. The parameters are adjusted to reproduce the energy levels of QWSs. For the Ag/Au interface, the linear combination of Eq. 4.7 is adopted, ignoring the effect of the interface inside the film and the substrate. The constructed model potentials for 12 ML Ag/50 ML Au and 21 ML Ag/50 ML Au are displayed in Fig. 4.23a. By solving the Schrödinger equation numerically using Numerov’s method, we obtain the eigenenergies and wavefunctions of the bound states. The calculated charge density distributions corresponding to the surface states and QWSs are shown in Fig. 4.23b. Notably, the penetration of QWSs depends on the film thickness and the quantum number. By tracing the thickness dependence of the energy levels of QWSs shown in Fig. 4.24a, one can understand that the 1DPP model captures the key features

4.3 Quantum Well Films with Tunable Rashba Effects Au

Ag

Vacuum

Au

4

4

0

0

E-EF (eV)

E-EF (eV)

(a)

-4 -8 -12 -30

-20

-10

0

65

10

Ag

Vacuum

-4 -8 -12 -30

Distance to surface (atomic layer)

-20

-10

0

10

Distance to surface (atomic layer)

-30

Normalized charge density

Normalized charge density

(b)

Surface state

QWS =1 -20 -10 0 Distance to surface (atomic layer)

10

Surface state QWS =1

-30

=2 -20

-10

0

10

Distance to surface (atomic layer)

Fig. 4.23 1D pseudo potential (1DPP) model for Ag/Au(111). a The potentials constructed for 12 ML Ag/50 ML Au and 21 ML Ag/50 ML Au heterostructures. b The square values of the wavefunctions for the potentials shown in a, which correspond to the charge density distributions

of the quantum well films. By utilizing the simple calculation method available with the 1DPP model, we have searched for the values which can be correlated to the Rashba parameters. Thus, we found that the charge density at the interface shows dependence on the film thickness and the quantum number as shown in Fig. 4.24b, which is quite similar to that of the experimentally obtained Rashba parameters (Fig. 4.22). Here, we can imagine that the charge density at the interface determines the Rashba parameters because the presence of the interface breaks the inversion symmetry. However, this may be a bit surprising since the value of the charge density distribution itself does not appear in the equation describing the microscopic origin of spin splittings (Eq. 4.1), while its gradient is involved in the integrand.

4.3.7 Discussion To discuss the microscopic origin of the relationship between the Rashba parameters and the charge density at the interface, we have performed DFT calculations. The first step is to reproduce the band structures by DFT calculations. In Fig. 4.25a, the band ¯ line is displayed, which corresponds to ¯ M dispersions for 24 ML Ag film along the -

66

4 Revealing the Role of Wavefunctions in Rashba-Split States

(a)

Calculation

0.0

Surface state

E-EF (eV)

QWS =1

-0.6

=2 =3

-0.8

=4

-1.0 10

15

20

25

Film thickness (ML)

30

35

Interface charge density

0.10

-0.2 -0.4

(b)

Experiment

QWS =1

0.08

=2

0.06

=3 =4

0.04 0.02 0.00

Surface state 10

15

20

25

30

35

Film thickness (ML)

Fig. 4.24 Interface charge density obtained by the 1DPP model. a The theoretical energy levels of the bound states were obtained for various films (red lines) together with the experimentally obtained values (black circles). b Charge density at the interface for QWSs with various νs and film thicknesses

the ARPES result shown in Fig. 4.19a. The dispersions of the surface state of Ag and QWSs show good agreement with the experiments. The charge density calculations show the localization of the surface state and QWSs within the film, as expected for the bound states. Thus, overall electronic structures of the quantum well films are consistent with the experimental results and previous suggestions [64]. Based on the slab structures, the Rashba parameters of QWSs are obtained as a function of film thicknesses and quantum numbers, showing quantitative consistency between calculations and experiments (Fig. 4.25c): The tendency that αR increases as the thickness reduces or ν increases is reproduced. Considering the finite potential well, the portion of wavefunctions outside the well becomes larger if when the width of the well is narrower or when the wavefunctions are expressed with larger quantum numbers. Therefore, QWSs in thinner Ag films or with larger νs are expected to mix more with Au electrons, which leads to the enhancement of spin splittings with the help of large SOC effects in the Au substrates. To investigate the Rashba effect microscopically, the total Rashba effect is decomposed into the contribution of each Ag/Au layer αR . As shown in the top panels of Fig. 4.26, αR has its maximum value at the interface, while it shows oscillating behavior in the film and decaying behavior in the substrate. The layer dependence can be related to the standing wave patterns of QWSs. Through the comparison with the standing wave patters and the layer-resolved Rashba parameters, we have found that αR becomes almost 0 at the layer where the envelope function of the charge density distribution takes its maximum or minimum values. The above results can be understood within the framework of Eq. 4.1 with the envelope function of the well states. Since ∂ V (r)/∂z shows divergent behavior near the nuclei and QWSs mainly consist of the pz orbitals, the integrand of Eq. (4.1) can be schematically illustrated as a product of two components as Fig. 4.27a. Around i-th atom, the wavefunction can be approximated as ψ(r) ≈ ψ pz ,i (r − ri )ψenv (z), where ψ pz,i is the wavefunction of pz orbitals at i-th atom, and ψenv is the envelope

4.3 Quantum Well Films with Tunable Rashba Effects (a)

67

(b) 0

Charge density (arb. units.)

Ag SS

E-EF (eV)

-0.2 Au SS -0.4 QWS ν=1 -0.6 QWS ν=2

-0.8

Ag SS QWS ν=1 QWS ν=2 QWS ν=3

-1.0 -0.2

0.1

0 -1 kx (Å )

-0.1

0.2

(c)

-60

-40 -20 0 20 40 Distance to topmost Ag layer, z (Å)

ν=1 (exp.) ν=2 (exp.) ν=3 (exp.) ν=4 (exp.)

200

ν=1 (theory) ν=2 (theory) ν=3 (theory) ν=4 (theory)

αR (meV•Å)

150

60

100

50

0 10

15

20

25

30

35

Film thickness (ML)

Fig. 4.25 Band structures and the Rashba parameters obtained by DFT calculations. a Theoretical band structures of 24 ML Ag/60 ML Ag. The sizes of the red solid circles show the weight of wavefunctions in Ag films. b The charge density distributions of the surface states and ν = 1, 2, 3 QWSs. The solid circles denote positions of Ag and Au layers. c Rashba parameters for QWSs in Ag films. The energy splitting at k x = 0.043 Å−1 is used to calculate the Rashba parameters using the relation αR = E/2k. Reprinted with permission from Ref. [14] ©2021 by the American Physical Society

function of QWSs. Now the layer decomposition of the Rashba parameters is be written as αR ≈

∞   i=−∞

d 3r

∞  ∂ Vi (r − ri ) |ψ pz ,i (r − ri )|2 |ψenv (z)|2 ≡

αR,i , ∂z i=−∞

(4.8)

where Vi and ri are defined around ith atom. Due to the gentle slope of the envelope function, we can expand |ψenv |2 into the form of |ψenv (z)|2 = |ψenv (z i )|2 + d|ψenv |2 /dz|z=zi (z − z i ). Thus, αR is simplified as

68

4 Revealing the Role of Wavefunctions in Rashba-Split States Au

 R (meV•Å)

80

Ag

Vacuum

=1

40

Ag

Vacuum

Au

=2

12 ML 18 ML 26 ML 34 ML

60

Au

Ag

Vacuum

=3

26 ML 34 ML

18 ML 26 ML 34 ML

20

2

| env(z)| (arb. units.)

0

-10

0

20

10

30 30 20 10 0 -10 Distance to topmost Au layer, z (ML)

-10

0

10

20

30

Fig. 4.26 Layer-decomposed Rashba parameters αR for ν = 1, 2, 3 QWSs are shown in the top panels. In the bottom panels, the envelope function of the charge density distributions |ψenv |2 are displayed. The kink-like features are due to the surface effect. The dashed lines show the positions where αR becomes 0 and the envelope function takes its maximum values. Reprinted with permission from Ref. [14] ©2021 by the American Physical Society



αR,i ≈ |ψenv (z i )|2  d|ψenv |2  =   dz

∂ Vi (r) |ψ d 3r ∂z

 



d|ψenv |2  2  pz ,i (r)| + dz



d 3r z=z i

δV i ,

∂ Vi (r) |ψ pz ,i (r)|2 z ∂z

(4.9)

z=z i

where 

δV i ≡

d 3r

∂ Vi (r) |ψ pz ,i (r)|2 z. ∂z

(4.10)

After the integration, only the asymmetric part of the wavefunctions remain in the equation because the potential gradient is nearly antisymmetric around each atom. With Eq. (4.9), one can understand that αR becomes 0 when the envelope function takes its maximum or minimum values. In contrast, αR becomes infinite when the gradient of the envelope function is finite, and the sign of αR depends on the sign of the gradient. This leads to the oscillating behavior of αR inside the Ag film for ν = 2, 3 QWSs. With Eq. (4.9), the total Rashba effect can be written as

4.3 Quantum Well Films with Tunable Rashba Effects (b)

(c)

αAu

ΔαR 0

αAg

ν=2

0.08

|ψ (0)| 2 |ψ (1)| αR

200 150

ν=3

100

0.04

αR (meV•Å)

z

|ψ (z)| (arb. units.)

(a)

69

50

0

0 10 15 20 25 30 35 Film thickness (ML)

(d)

100 50

200

200

150

150

100

ν=1 ν=2 ν=3 ν=4

50

0

αR (meV•Å)

αR (meV•Å)

150

αR (meV•Å)

ν=1 ν=2 ν=3 ν=4

200

0 0

0.02 0.04 0.06 0.08 0.10 2

|ψ (0)|

100

ν=1 ν=2 ν=3 ν=4

50 0

0

0.02

0.04

0.08

0.06

2

0

2

0.06

0.04

0.02 2

|ψ (0)| - 0.30|ψ (1)|

2

|ψ (0)| - 0.50|ψ (1)|

Fig. 4.27 Scaling law for the Rashba effect in quantum well films. a Relation between the envelope function of QWSs and the layer-resolved Rashba effect. Blue curves indicate the charge density distributions, while the red curves denote the potential gradient ∂ V /∂z. b Decomposition of αR into the contributions from the Au substrate and the Ag film. d Thickness- and quantum numberdependent charge density distribution at the boundary Au layer (|ψ(z 0 )|2 ) and Ag layer (|ψ(z 1 )|2 ). The magnitude of αR is added by the black dashed lines for the comparison. (d) Plot of the Rashba parameters of QWSs versus the charge density at the interface layer with a varying coefficient c in Eq. (4.13). Reprinted with permission from Ref. [14] ©2021 by the American Physical Society

 ∞  d|ψenv |2  αR =

δV i dz z=zi i=−∞

(4.11)

Here, δV i takes different values for Ag and Au but does not change on other parameters. Therefore, we can divide the total splitting into the contribution from the Au substrate and the Ag film as depicted in Fig. 4.27b, and we get αR ≈ δV Au

  0 ∞   d|ψenv (z)|2  d|ψenv (z)|2  +

δV  . Ag   dz dz z=z i z=z i i=−∞ i=1

(4.12)

Assuming that the the envelope function |ψenv (z)|2 changes slowly throughout the system, the summation can be approximated by the integration and we obtain 0 αR ∝ δV Au −∞

d|ψenv (z)|2 + δV Ag dz dz

∞ dz

d|ψenv (z)|2 dz

1

= δV Au |ψenv (0)|2 − δV Ag |ψenv (1)|2 ∝ |ψenv (0)|2 − c|ψenv (1)|2 , (4.13)

70

4 Revealing the Role of Wavefunctions in Rashba-Split States

where c is a constant, and 0(1) is the position of the boundary Au (Ag) atomic layer. Here we assume that the envelope function vanishes at z ± ∞. This equation indicates that αR is determined by the charge density at the interface and atomic species of the film and the substrate, regardless of the detailed charge density distribution inside the well as long as the envelope function varies slowly. Therefore, the similarity between the thickness dependence of the Rashba parameters and that of the charge density is the consequence of the integration of the microscopic Rashba effect. In addition, one can understand that the difference of δV  in the substrate and the film is the key to enhance the Rashba effect, since |ψ(0)|2 and |ψ(1)|2 usually takes similar values. Here, we note that δV  is proportional to the SOC coefficient defined for each atomic orbital. To confirm our discussion, the local charge densities at the boundary Au and Ag layers of QWSs are extracted (Fig. 4.27c), showing thickness dependence similar to that of the Rashba parameters. The thickness-dependence of the local charge densities at the interface shows a similarity to that of αR . Furthermore, αR is plotted against the charge density at the interface layers using Eq. (4.13) (Fig. 4.27d). The coefficient is varied between 0 and 0.5, and we have found that the Rashba parameters are linearly displayed when c = 0.30 regardless of quantum numbers, suggesting the scaling law of the Rashba parameters on the interface charge density. Here we note that the value of c = 0.30 corresponds to the ration of the SOC coefficients of Ag 5 p orbital to that of Au 6 p orbital [12], validating our analysis. Finally, we review the formation of Rashba-split quantum well states through the thickness evolution. When the film is too thin to confine the electrons in the potential well, quantum well resonance states emerge outside the projected band gap, which is the precursor of QWSs (left panels of Fig. 4.28). The wavefunctions of the resonance states expand over the film and the substrate, and hence the charge density at the interface remains small. Therefore, although the inversion symmetry of the electronic state is broken, the states are expected to be almost spin degenerate. When the film becomes thick enough to confine the electrons, the electronic states are quantized. Just after the formation of QWSs, the width of the well is still narrow, and the charge density at the interface can be relatively large, leading to the Rashba effect observed in this study (center panels of Fig. 4.28). However, if the film becomes thick enough, the system behaves like an infinite potential well, which confines almost all the charge density in the film. Then the effect of the inversion symmetry breaking on the electrons is small, and the Rashba effect also becomes small (right panels of Fig. 4.28). This evolution can also be interpreted in terms of the change of the energy levels of the state. If the projected band dispersions of the film and the substrate overlap a lot in the momentum space, only resonances states are formed with small Rashba parameters. In addition, if the states are separated by far in the momentum space, only small hybridization is allowed, which again results in small Rashba parameters. Therefore, to enhance the Rashba effect, the energy dispersions of the film and the substrate should have some deviation, but the deviation should not be too large to prohibit the hybridization between the two states. Considering the mechanism stated above, there are three conditions required to enhance the Rashba effect using the scaling law (Eq. (4.13)):

4.3 Quantum Well Films with Tunable Rashba Effects

Quantum well resonance Substrate

71

Quantum well state I

Film

Substrate

ρ(z)

Quantum well state II

Film

Substrate

ρ(z)

Projected band gap

Film

ρ(z)

Projected band gap

Projected band gap

Occupied states

Momentum

Large splitting

Energy

Energy

Energy

Small splitting Almost degenerate

Occupied states

Momentum

Occupied states

Momentum

Fig. 4.28 Formation of QWSs with Rashba splitting. The Rashba parameter depends on the degree of the hybridization between the film and the substrate

1. The gap which supports QWS is not a symmetry gap but a relative gap, which allows the hybridization of the electronic states between the film and the substrate. 2. The band dispersions of the film and the substrate have an overlapping region in the 3D limit (bulk limit), as the valence bands of Ag and Au do. Otherwise, the potential well becomes too high for electrons to penetrate into the substrate. 3. The SOC coefficients of two atomic species are substantially different. When the ratio of SOC coefficients is close to 1, the spin splitting remains small even though atoms with large masses are selected. If the above conditions are satisfied, we can design quantum well states with large Rashba splittings not only in metallic quantum well films but also in any heterostructures.

4.4 Conclusion In conclusion, we have studied the role of wavefunctions in the Rashba-split states by means of laser-SARPES utilizing the orbital sensitivity and high energy resolution. For the surface alloys, spin-orbit coupled wavefunctions have been detected in both BiAg2 and BiCu2 by using linearly polarized light. The changes of the orbital components and corresponding spin polarizations are observed in the spz band of BiAg2 , indicating that the hybridization between the spz band and the px y band modifies the wavefunctions of the spz band significantly. By a simple analysis, the complex

72

4 Revealing the Role of Wavefunctions in Rashba-Split States

spin texture in BiAg2 beyond the Rashba model has been reproduced, although the hybridization gap is not detected in our study. The result presents the important role of orbital hybridizations in realizing a novel spin texture. In the study of quantum well films, we have detected the Rashba splitting in Ag/Au(111) QWSs, and further shown the systematic thickness- and quantum number-dependence of the Rashba effect. These results show qualitative agreement with DFT calculations. Also, we have found the scaling law for the Rashba effect: the spin splitting in a quantum well film is simply expressed by the charge density at the boundary layers and the ratio of SOC strengths of the film and the substrate. Here the envelope function of the wavefunctions plays an essential role in the formation of the spin-split states and the scaling law. Hence, it can in principle be applied to any quantum well films where the envelop function has a moderate slope. Thus our results can be extended to design and control the spin splitting of QWSs for future spintronics measurements.

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61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74.

75.

76. 77.

78.

75

C, Xu Z, Zhou XJ (2014) Orbital-selective spin texture and its manipulation in a topological insulator. Nat Commun 5:3382 Kuroda K, Yaji K, Nakayama M, Harasawa A, Ishida Y, Watanabe S, Chen C-T, Kondo T, Komori F, Shin S (2016) Coherent control over three-dimensional spin polarization for the spin-orbit coupled surface state of Bi2 Se3 . Phys Rev B 94(16):165162 Bawden L, Riley JM, Kim CH, Sankar R, Monkman EJ, Shai DE, Wei HI, Lochocki EB, Wells JW, Meevasana W, Kim TK, Hoesch M, Ohtsubo Y, Le Fèvre P, Fennie CJ, Shen KM, Chou F, King PDC (2015) Hierarchical spin-orbital polarization of a giant Rashba system. Sci Adv 1(8):e1500495 Maaß H, Bentmann H, Seibel C, Tusche C, Eremeev SV, Peixoto TRF, Tereshchenko OE, Kokh KA, Chulkov EV, Kirschner J, Reinert F (2016) Spin-texture inversion in the giant Rashba semiconductor BiTeI. Nat Commun 7(1):11621 Heinzmann U, Dil JH (2012) Spin-orbit-induced photoelectron spin polarization in angleresolved photoemission from both atomic and condensed matter targets. J Phys Condens Matter 24(17):173001 Krasovskii EE (2015) Spin-orbit coupling at surfaces and 2D materials. J Phys C 27(49):493001 Winter M, Chulkov EV, Höfer U (2011) Trapping of image-potential resonances on a freeelectron-like surface. Phys Rev Lett 107(23):236801 Höfer U, Echenique PM (2016) Resolubility of image-potential resonances. Surf Sci 643:203– 209 Chiang T-C (2000) Photoemission studies of quantum well states in thin films. Surf Sci Rep 39(7–8):181–235 Miller T, Samsavar A, Franklin GE, Chiang TC (1988) Quantum-well states in a metallic system: Ag on Au(111). Phys Rev Lett 61(12):1404–1407 McMahon WE, Miller T, Chiang TC (1993) Avoided crossings of Au(111)/Ag/Au/Ag doublequantum-well states. Phys Rev Lett 71(6):907–910 Smith NV, Brookes NB, Chang Y, Johnson PD (1994) Quantum-well and tight-binding analyses of spin-polarized photoemission from Ag/Fe(001) overlayers. Phys Rev B 49(1):332–338 Dil JH, Meier F, Lobo-Checa J, Patthey L, Bihlmayer G, Osterwalder J (2008) Rashba-type spin-orbit splitting of quantum well states in ultrathin Pb films. Phys Rev Lett 101(26):266802 http://www.openmx-square.org/ Morrison I, Bylander DM, Kleinman L (1993) Nonlocal Hermitian norm-conserving Vanderbilt pseudopotential. Phys Rev B 47(11):6728–6731 Ozaki T (2003) Variationally optimized atomic orbitals for large-scale electronic structures. Phys Rev B 67(15):155108 Ozaki T, Kino H (2004) Numerical atomic basis orbitals from H to Kr. Phys Rev B 69(19):195113 Cercellier H, Fagot-Revurat Y, Kierren B, Reinert F, Popovi´c D, Malterre D (2004) Spin-orbit splitting of the Shockley state in the Ag/Au(111) interface. Phys. Rev. B 70(19):193412 Popovi´c D, Reinert F, Hüfner S, Grigoryan VG, Springborg M, Cercellier H, Fagot-Revurat Y, Kierren B, Malterre D (2005) High-resolution photoemission on Ag/Au(111): spin-orbit splitting and electronic localization of the surface state. Phys Rev B 72(4):045419 Cercellier H, Didiot C, Fagot-Revurat Y, Kierren B, Moreau L, Malterre D, Reinert F (2006) Interplay between structural, chemical, and spectroscopic properties of Ag/Au(111) epitaxial ultrathin films: a way to tune the Rashba coupling. Phys Rev B 73(19):195413 Luh D-A, Cheng C-M, Tsuei K-D, Tang J-M (2008) Atomic-layer-resolved bound states in quantum wells analyzed using a pseudopotential approach. Phys Rev B 78(23):233406 Forster F, Gergert E, Nuber A, Bentmann H, Huang L, Gong XG, Zhang Z, Reinert F (2011) Electronic localization of quantum-well states in Ag/Au(111) metallic heterostructures. Phys Rev B 84(7):075412 Kuroda K, Yaji K, Harasawa A, Noguchi R, Kondo T, Komori F, Shin S (2018) Experimental methods for spin- and angle-resolved photoemission spectroscopy combined with polarizationvariable laser. J Vis Exp 136:e57090

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Chapter 5

Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides

5.1 Introduction A band insulator is generally classified by its topological phase, which predicts the boundary mode due to the bulk-boundary correspondence. Among topological insulators (TI) in 3D, a first-order TI and a second-order TI host spin-polarized surface and edge states. Thus, the discovery of new topological insulator states have broadened our understanding of spin-polarized electrons, which may lead to the advancement in spintronics applications or the observation of spin-related quantum phenomena. After the theoretical prediction on the 3D Z 2 topological insulators [1, 2], which manifest the first order non-trivial band topology, a number of materials such as Bix Sb1−x and Bi2 Se3 have been confirmed to be strong topological insulators (STIs) through the observation of the topological surface states on any surface of the crystal [3–5]. Topological surface states connect the conduction band and the valence band in the momentum space, and hence the system becomes metallic even if the bulk is insulating. In STIs, the presence of topological surface states (TSS) is guaranteed by the so-called topological protection of the system, and hence TSSs are expected to emerge regardless of the surface structure or the disorder as long as the time-reversal symmetry is protected. In contrast, a weak topological insulator (WTI), which was predicted at the same times as an STI, has attracted less attention since the topological protection in a WTI was considered as “weak”, which implies that the surface states are easily destroyed by perturbations, and thus the system becomes insulating [1, 2]. Later, however, theoretical studies have shown that TSSs in a WTI is robust against nonmagnetic disorders [6–8], and several compounds have been predicted to be WTIs with gapless surface states on particular surfaces [9–13]. In a WTI, certain surfaces are still insulating depending on the topological invariants, and the dispersions of the TSSs on side surfaces can be highly one-dimensional, differing from 2D TSSs in STIs [14]. Nevertheless, due to experimental difficulties, TSS on particular surfaces, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 R. Noguchi, Designing Topological Phase of Bismuth Halides and Controlling Rashba Effect in Films Studied by ARPES, Springer Theses, https://doi.org/10.1007/978-981-19-1874-2_5

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which is the hallmark of a WTI, has been elusive despite many experimental efforts [10, 13, 15–19]. During the advancement of the understanding of Z 2 topological insulators, it has also been recognized that chiral hinge states, which are localized around the “hinge” of a crystal, can be stabilized in a magnetic topological insulator [20]. This view has been further developed into the prediction of a higher-order topological insulator (HOTI), which is a class of a topological crystalline insulator (TCI) [21, 22]. In artificial 2D systems, HOTI phases with corner states have been established [23– 28] In 3D crystals, semimetal bismuth has been claimed to be in the C3 -protected HOTI phase through the observation of the topological hinge states [29], although the topological phase of bismuth is still controversial [30, 31]. At the same time, a HOTI with its insulating bulk has been sought for the observation of transport signatures of topological hinge states. However, such a bulk-insulating HOTI has not been found in 3D crystals so far due to the difficulty in the prediction of new HOTIs by calculating eigenvalues for specific symmetry operations. In this thesis, quasi-1D compounds bismuth halides Bi4 X 4 (X = Br, I) [32–40] have been chosen as a platform to investigate new topological phases. In the following sections, we review our investigation of the electronic structures of bismuth halides and present the evidence for the WTI state in β-Bi4 I4 [41] and the HOTI state in Bi4 Br4 [42]. The advantage of employing quasi-1D compounds lies in the possibility of observing the surface electronic structures on the side surface and the hinge electronic structures.

5.1.1 Quasi-1D Materials for the Search of New Topological Phases The simplest scheme to design a WTI is to build a stacking of 2D TI layers without breaking the symmetry [9]. In this way, the edge states of a 2D TI are piled up to form quasi-1D surface states on the side surface with the top surface kept gapped, as schematically illustrated in Fig. 5.1. Assuming the weak interlayer coupling, one can also understand the Z 2 invariant of a 2D TI naturally evolves into those of s WTI as we have seen in Sect. 2.3.3. However, the two-dimensionality of a 2D TI prevents us from detecting the surface states: since the clean side surfaces are hard to obtain, surface-sensitive probes such as ARPES or STM cannot be easily performed, although the surface electronic structure of the topologically dark top surface can be well investigated. Bi1 4Rh3 I9 is one of the candidate WTIs, which consists of 2D TI layers of (RhBi4 )3 I separated by insulating layers of Bi2 I8 [10, 13]. To date, the gapped electronic structure of the top surface and the existence of edge states on the monolayer edges have been demonstrated, which supports the WTI phase in this compound. However, the observation of topological surface states stemming from the weak topology has not been reported due to the lack of cleavable side surfaces.

5.1 Introduction

79

Side plane

Stacked 2D TI

Weak TI Top plane

(weak interlayer coupling)

1D chains

Fig. 5.1 Layer construction of a weak topological insulator (WTI). (left and middle) A WTI can be regarded as a stacked 2D TIs with weak interlayer coupling. (right) Schematic of a quasi-1D compound with two cleavable surfaces

One solution to this difficulty is to-employ a quasi-1D material, which consists of 1D chains standing parallel in two-dimensions (right panel of Fig. 5.1). If the single layer of these 1D chains is a 2D TI, we can expect a WTI phase in the 3D crystal. Notably, both the top surface and the side surface are expected to be cleavable in this case, and then TSSs in the side surfaces which originate in the WTI phase can be experimentally detected depending on the topological indices.

5.1.2 Prediction of Rotation Symmetry-Protected TCIs with Topological Hinge States In conventional approaches, the non-trivial topology of a TCI was found by the analysis of the band structures based on a particular symmetry operation [43]. Correspondingly, the absence of the non-trivial topology in the calculation results only implies that the target system is trivial in the particular TCI phase, which may lead to the overlooking of other non-trivial TCI phases. However, the recent development on the theory of symmetry indicators has enabled us to perform a comprehensive analysis of the band topology [44–49]. With symmetry indicators, one can classify a band insulator as either a trivial insulator (normal insulator) or a topological insulator, while further calculations may be needed to determine which TCI phase is realized in the target material. The symmetry indicators are defined for all the 230 space groups, and thus a variety of topological materials have been predicted by combining crystallographic database and symmetry indicators. Especially for systems with space inversion symmetry, the band symmetry is given by Z 23 × Z 4 , which can be regarded as an extension of the Fu-Kane parity criterion for Z 2 topological insulators. The strong index in the conventional topological index is promoted to a Z 4 factor κ1 . If κ1 = 2, it implies that the material is an inversion-symmetric TCI, which is a higher-order TI (HOTI) with

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5 Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides

(b)

(a) E

Trivial

HOTI

+

-

+

SOC

C2-protected TCI

Hinge states

-

EF -

+

-

+ Surface states

Fig. 5.2 A C2 -protected TCI. a A general scheme to realized HOTI phases with κ1 = 2. b Schematic of a C2 -protected TCI with two hinge modes and two surface states in the cross-section

helical hinge states. The simplest scheme to realize phases with κ1 = 2 is illustrated in Fig. 5.2a. If the valence bands consist of a pair of bands with negative parity eigenvalues and the conduction bands consist of a pair of bands with positive parity eigenvalues, the inversion of the two valence bands and the two conduction bands leads to a phase with κ1 = 2. This corresponds to two copies of an STI with gapped surface states and gapless hinge states. One of the new topological phases which can be predicted by symmetry indicators is Cn -rotation (n = 2, 4, 6) symmetry protected TCIs [50, 51]. In a Cn -protected TCI, n pairs of Dirac-like gapless surface states are expected in the surface perpendicular to the rotation axis, while n helical hinge modes connect two surfaces with gapless surface states, as illustrated in Fig. 5.2b. As the hinge states are separated by the gapless surface states and other hinge states in the real space, the Cn -protected TCIs offer an interesting platform to study the behavior of 1D hinge states, for which protection from the backscattering is expected. Remarkably, a HOTI can be constructed by stacking 2D TI layers similarly to the case of a WTI [52, 53], while the doubling of the unit cell is one of the strategies for the edge state of 2D TI to evolve into topological hinge states. In the detection of hinge modes, quasi-1D materials are advantageous again since the natural cleavage can yield long hinges where the topological hinge modes are stabilized.

5.1.3 A Versatile Platform to Realize Various Topological Phases: Bismuth Halides Bi4 X 4 (X = Br, I) The quasi-1D compounds α-Bi4 I4 , β-Bi4 I4 and Bi4 Br4 share the similar crystal structures formed by a build-up of bismuth halide Bi4 X 4 chains with same monoclinic space group C2/m (No. 12), differing in the stacking sequences [32–34]. While the atoms are strongly bonded within each quasi-1D chain, the interchain bonding is expected to be quite small, leading to highly anisotropic electronic structures. In

5.1 Introduction

(a)

81

(b)

Single layer Bi4X4

z y

Biin-antibonding

Biex-antibonding x Bi X (=Br, I) Primitive cell of β-Bi4I4

Bi px

Biin Biin (odd)

Biex

EF

Biex-bonding (even)

Biex2+

Biex (even)

Biin0

Biin-bonding (odd) Biin0 Biex2+

Atomic+ crystal field

Intrachain (x/y)

SOC

Fig. 5.3 Single layer Bi4 X 4 as a 2D TI candidate. a (top) The crystal structure of a single layer Bi4 X 4 and (bottom) the primitive cell of β-Bi4 I4 which is important for the consideration on the band structures visualized by VESTA [54]. b Schematic of the band evolution of single layer Bi4 X 4 . The energy bands around E F mainly consist of Bi px orbitals

understanding the topological properties of bismuth halides, we can theoretically consider a single layer Bi4 X 4 , of which the unit is shown in Fig. 5.3a. Importantly, these single layer compounds have been predicted to be 2D TIs which support spinpolarized helical edge states [35, 36]. The band structure can be understood as following: first, we consider the atomic energy levels and the symmetry of the system. The 6 p electrons of bismuth are located around E F , and here the px orbital and the pz orbitals are allowed to hybridize while p y orbitals are independent at the  and L points due to the mirror symmetry of the material. Also, due to the strong electronegativity of iodine atoms, the outside bismuth atoms (Biex ) are hall doped compared ti the inside bismuth atoms (Biin ) (Fig. 5.3a). Thus the electrons around different bismuth atoms are energetically different. As a next step, we consider the intrachain hopping along x and y directions. The energy levels split into the bonding states and the anti-bonding states derived from Biin atoms and Biex atoms, respectively, since one primitive unit cell consists of two Biin atoms and two Biex atoms (the primitive cell for a 3D case is drawn in the bottom panel of Fig. 5.3a). Here, the valence band maximum is mainly made from the px orbitals of Biin atoms while the conduction band minimum from the px orbitals of Biex atoms. After the inclusion of SOC, the conduction band and valence band shift in energy oppositely as illustrated in Fig. 5.3(b), realizing the band inversion at the X point in the 2D Brillouin zone (BZ), which corresponds to the M and L points in the 3D BZ. Since the conduction band and the valence band in the singlelayer Bi4 X 4 have different parity eigenvalues, the band inversion induces non-trivial topology in 2D, offering a variety of 3D topological phases once it is stacked threedimensionally. Previous theoretical researches have predicted that the single layer Bi4 Br4 is stabilized in the 2D TI phase, while the single-layer Bi4 I4 is located in the proximity of the topological phase transition [35].

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5 Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides

Fig. 5.4 Crystal structure of 3D bismuth halides. a–c Crystal structure with the conventional unit cell for α-Bi4 I4 , β-BI4 I4 and Bi4 Br4 , respectively, visualized by VESTA [54]. The green and magenta arrows are the schematics of the AA’-, A- and AB-stackings. d 3D Brillouin zone of β-Bi4 I4 . Adopted from Refs. [41, 42]

As shown in Fig. 5.4a–c, the stacking sequence changes the unit cell of Bi4 X 4 in 3D. The simplest case is realized in β-Bi4 I4 , where the conventional unit cell contains a single Bi4 I4 block (A-stacking), while the c axes of α-Bi4 I4 and Bi4 Br4 are doubled due to the AA’-stacking and AB-stacking, respectively. These layers are only weakly connected by the van der Waals forces (vdW forces). In the AA’-stacking, each layer is shifted by a single 1D chain, while in AB-stacking, each layer is alternately rotated by 180◦ . Interestingly, different topological phases have been predicted in these compounds. Previous studies indicated that β-Bi4 I4 is in the proximity of topological phase transition between an STI (1;110) and a WTI (0;001), while α-Bi4 I4 and Bi4 Br4 are in their normal insulator phases [37, 38]. Here we note that the previous ARPES study on β-Bi4 I4 did not separate signals from the top surface and the side surface [38], and hence further investigations are needed to unambiguously determine the band topology. In these studies, however, the concept of HOTI was overlooked, and recent theoretical works have indicated that a C2 -protected TCI phase, which is one of HOTI phases, can be realized in Bi4 Br4 [39, 40]. Importantly, the hallmarks of the WTI phase and the HOTI phase, which are the topological surface states on the side surface and topological hinge states, are expected to be accessible by experiments due to the quasi-1D crystal structures. Here we also note that TSSs in the C2 -TCI phase are not experimentally accessible since the cross-section (010) is not

5.1 Introduction

83

a cleavable plane. Therefore, quasi-1D bismuth halides, which can be regarded as stacked 2D TI layers, provide a platform to investigate the topological phases which have been hypothetical so far by making use of the two cleavable planes. Furthermore, the understanding of stacking-dependent topology may lead to the development of vdW-engineering of topological phases, where topological phases can be selected by the manipulation of the stacking sequence of vdW layered materials.

5.2 Basic Properties of Bi4 X 4 Crystals Let us move to the basic properties of bismuth halides crystals obtained in our experiments.

5.2.1 Method Single crystals of Bi4 I4 and Bi4 Br4 were grown by the chemical vapor transport (CVT) method, similar to the Ref. [34]. The transport agents used were HgI2 , and HgBr, and the optimized temperatures were 285◦ C and 188◦ C for the source and growth zones, respectively. For Bi4 I4 , we found that the α-phase (β-phase) is stable below (above) around 300 K. We also found that the structural transition from the β-phase to the α-phase is suppressed through rapid quenching using liquid nitrogen from around 400 K. This quench technique allowed us to characterize the β-phase at low temperature by ARPES and transport measurements. The electrical resistivity along the b-axis of α-Bi4 I4 , β-Bi4 I4 and Bi4 Br4 were measured by a standard fourprobe technique. Grazing-incidence small-angle X-ray scattering (GISAXS) experiments were performed at the beamline 3A and the beamline 18B of the Photon Factory at KEK. The X-ray energy was 11 keV and 10.8 keV, and the slit-cut beam size was 0.1 × 0.1 mm. The glancing angle of the X-rays was 0.1◦ with respect to the top plane (001). The scattered X-rays were detected with a PILATUS-100K located 1 m downstream of the sample at the beamline 3A and with a PILATUS 1M located 1.2 m downstream of the sample at the beamline 18B. For the alignment of sample orientation, the 001 and 200 Bragg reflections were used. All the measurements were performed at room temperature. In this section, Mr. Takahashi and Mr. Kobayashi in Prof. Sasagawa’s group grew the single crystals and conducted X-ray diffraction and transport measurements. Dr. Shirasawa performed GISAXS measurements.

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5 Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides

5.2.2 Crystal Structures and Transport Properties

006

008

003

004

005

004

002

α-Bi 4I 4

β-Bi 4I 4

10

002

x (-1)

001

Intensity ( a.u. )

First of all, the previously reported crystal structures have been confirmed in our samples, as indicated by X-ray diffraction (XRD) results (Figs. 5.5 and 5.6). The lattice parameters of α-Bi4 I4 , β-BI4 I4 and Bi4 Br4 are obtained as follows: a = 14.345, b = 4.443, c = 20.112 Å and β = 92.974◦ for the α-BI4 I4 , a = 14.406, b = 4.432, c = 10.475 Å and β = 107.463◦ for the β-BI4 I4 , and a = 13.090Å, b = 4.340Å, c = 20.110Å, and β = 107.13◦ for Bi4 Br4 . The A-stacking phase of Bi4 Br4 , which was theoretically proposed [37], has not been found in our experiments. For Bi4 I4 , the structural transition is found around the room temperature by XRD. Despite the small difference in α-Bi4 I4 and β-Bi4 I4 , they show distinct transport properties: the α-phase exhibits a typical semiconductor-like resistivity while the βphase contrary presents conductive channels (Fig. 5.7a). Moreover, the crystal phase transition is observed as the hysteresis in the resistivity around 300 K if the crystal temperature is changed mildly with 0.3 K/min (Fig. 5.7b). We have also found that the high-temperature β-phase can be pinned even at low temperature if the crystal is rapidly cooled from ∼400 K to liquid nitrogen temperature, which enables us to perform ARPES of these two phases equally at low temperature with high energy resolutions. The resistivity curve for Bi4 Br4 is also indicative of metallic conductance channels in a semiconductor. In a typical insulator, ρ(T ) exponentially increases with decreasing temperature. In contrast, the result shows a deaccelerating behavior toward the lowest temperature (Fig. 5.7a). As a result, the variation of resistivity from 300 K to 2 K is only a factor of four, which is not expected in ordinary insulators and semiconductors. This implies that ρ(T ) around the room temperature is dominated by thermally excited bulk carriers, whereas topologically-originated edge carriers play a dominant role at below 50 K.

20 30 2θ ( deg., CuKα )

Fig. 5.5 X-ray diffraction results of α- and β-Bi4 I4 . These two results are differentiated by the 005 peak only observed in the α-phase. Adopted from Ref. [41]

5.2 Basic Properties of Bi4 X 4 Crystals

85

Fig. 5.6 X-ray diffraction (XRD) results of Bi4 Br4 . a XRD patterns from the ab-planes of single crystals. b XRD patterns from powders. The experimental results and theoretical results are indicated by the red and the blue curves, respectively. Adopted from Ref. [42]

(a)

(b)

1

-Bi4I4

b (cm)

0.6

0.008

0.1

-Bi4I4 0.4

0.01 0.001

0.0 0

0.006

-Bi4I4

Bi4Br4 0.2

α -p hase

Bi4Br4

0

100

200

0.004 300

0.002

-Bi4I4

0 200

100

300

T(K)

β-p hase 290

300 T (K)

310

Fig. 5.7 Transport properties of bismuth halides. a ρ(T ) curves for α-Bi4 I4 , β-BI4 I4 and Bi4 Br4 shown by the green, red and blue curves, respectively. Inset is the same data shown with log scale. b Magnified image of a around 300 K. A hysteresis behavior has been found due to the crystal phase transition. Adopted from Refs. [41, 42]

5.2.3 Cleavage Surfaces of Bismuth Halides As a next step, we have investigated the cleavage properties of bismuth halides since the presence of cleavable side surfaces is essential in detecting the WTI phase or HOTI phase by experiments. Both for Bi4 I4 and Bi4 Br4 , we have confirmed that the quasi-1D crystals naturally exposed two surfaces by cleaving the largest a-b plane (Fig. 5.8a). In a cleaved β-Bi4 I4 crystal, the top surface (001) is mainly observed

86

5 Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides (b) Aer cleaved b

(a) ~500 μm

c

a b

~2.0 mm

(d)

(001) 50 μm

(100)

Scattered X-rays

1.0

a

[001]

0.8

(c) 16 [μm] 8 0

a b 60

0

qz/nm-1

Incident X-ray 0.6

[100]

20

40 0

[100] 72˚

0.2

40

20

5 72˚ [001]

72˚

0.4

0.0 -0.8

6

4

3 2

Log(intensity)/a.u.

~30 μm

Beam stop -0.4

0.0 qy/nm-1

0.4

0.8

Fig. 5.8 Cleavage surfaces of β-Bi4 I4 . a Typical dimensions of β-Bi4 I4 samples. b Scanning electron microscopy (SEM) and c laser-microscope image of a cleaved surface, respectively. Black circle indicates the typical spot size of laser light used in our ARPES experiment. d Grazing-incidence small angle X-ray scattering (GISAXS) result of a cleaved surface after ARPES experiments. The chain direction of the sample (b axis) was aligned parallel with the incident X-ray. Adopted from Ref. [41]

as terraces in scanning electron microscope (SEM) and laser-microscope images (Fig. 5.8b and c). A number of the facets oriented along the 1D chain direction can be assigned to the side-surface (100), which is further confirmed by grazingincidence small-angle X-ray scattering (GISAXS) measurement of the cleaved surface, as shown in Fig. 5.8d. The data shows two strong reflection peaks separated by ∼ 72◦ , which corresponds to the edge angle between the (001) and the (100) planes of β-Bi4 I4 . This observation, therefore, suggests the presence of the experimentally accessible side-surface in a cleaved surface, which is clearly different from other WTI candidates [10, 13, 15, 17–19, 55]. The same experiments have been conducted for Bi4 Br4 crystals (Fig. 5.9a). Our SEM image and laser-microscope images exhibit many 1D terraces and facets, which are attributed to the top surface (001) and the side surface (100) by GISAXS experiments. Therefore, in the cleaved surface of Bi4 Br4 , numerous hinges are formed between these two surfaces, with which we can expect the emergence of highdense topological hinge states (Fig. 5.9e). Thus, the macroscopic experiment such as ARPES can be applied to the detection of hinge states, whereas microscopic experiments such as STM is usually employed to investigate the edge states of 2D topological insulators. In fact, the spot size of our laser setup is about 50 µm, which cannot resolve the single edge or the single side surface in these materials. However, by accumulating photoelectrons from all the side surfaces or the edges in the illuminated region, we can obtain the knowledge on the electronic structure of interest: the side surface states in a WTI and the hinge states in a HOTI.

5.3 A Weak Topological Insulator State in β-Bi4 I4 (a)

(b)

Pristine crystals

87 (c)

After cleaved

10 [μm] 5 0 20

20 μm

0 10

10 20

1 mm

30

(d)

qz (nm-1)

0.2 0.0

6 5 4 3 2

[001]

[100]

73˚

Scattering from slits

-0.2 -0.8

-0.4

-0.0 qy (nm-1)

0.4

Log(intensity) (a.u.)

z [001] [100] 107˚

0.4

0

(e)

0.8

Fig. 5.9 Cleavage surfaces of Bi4 Br4 . a As-grown Bi4 Br4 crystals. b SEM and c laser-microscope image of a cleaved surface, respectively. d GISAXS result of a cleaved surface after ARPES experiments. e Schematic of the cleaved surface with topological hinge states (marked by the black lines). Adopted from Ref. [42]

5.3 A Weak Topological Insulator State in β-Bi4 I4 In this section, we present a detailed study on the electronic structures of Bi4 I4 and provide evidence for the weak topological insulator state in the β-phase.

5.3.1 Method Laser-based ARPES and SARPES measurements were performed at the Institute for Solid State Physics, the University of Tokyo [56]. The laser system for 6.994-eV photons [57] was employed, and photoelectrons were detected using a ScientaOmicron DA30L analyzer. The samples were kept at about 20 K. The angle resolution was 0.3◦ (0.7◦ ), and the overall energy resolution was set to less than 5 meV (20 meV) in ARPES (SARPES). Synchrotron-based high-resolution ARPES measurement was performed at the HR-ARPES branch of the beamline I05 of the Diamond Light Source. The photoelectrons were detected by a Scienta R4000 analyzer. The sample temperature was about 10 K. The angular resolution was 0.2◦ , and the overall energy resolution was better than 20 meV. Microscopic nano-ARPES measurements were performed at a nano-ARPES branch of the beamline I05 of Diamond Light Source and at the Spectromicroscopy beamline of Elettra light source [58]. A zone plate was used to focus the photon beam to a spot with less than 1 µm size in Diamond Light Source, while Schwarzschild objectives were used in Elettra, respectively. The photoelectrons were detected with a ScientaOmicron DA30L in Diamond Light Source, which gives a capability to per-

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5 Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides

form Fermi surface mapping without sample rotation. The detector in the Elettra light source is facility-made and rotatable inside the measurement chamber, with which the Fermi surface mapping was performed without sample rotation. The photon energy was set at 85 eV and 74 eV in Diamond Light Source and Elettra, respectively. The samples were kept at ∼30 K in Diamond Light Source and at ∼100 K in Elettra light source. The overall energy resolutions were set to be better than 60 meV. To investigate the possible topological phases of Bi4 I4 m, we have performed DFT calculations using the full-potential linearized augmented plane-wave method as implemented in the WIEN2k code [59]. Based on the generalized gradient approximation of Perdew, Burke and Ernzerhof (GGA-PBE) of exchange-correlation functional [60], the modified Becke-Johnson exchange potential (mBJ) [61] with manually varying the two adjustable parameters were used to evaluate the effect of the gap size correction on the topological phases. Spin-orbit coupling (SOC) was included, and the experimentally determined lattice constants were used for all the calculations. The plane-wave cutoff energy was set to R M T K max = 7, where the muffin tin radii were R M T = 2.5 a.u. for both Bi and I. The Brillouin zone was sampled with the Monkhorst-Pack scheme [62] with momentum grids finer than k = 0.012 Å−1 (i.e. 20 × 8 x 20 k-point mesh for β-Bi4 I4 ). For the calculations of surface spectra, we extracted the Wannier functions of the Bi- and I- p orbitals using the Wien2wannier [63] and Wannier90 [64] codes and constructed semi-infinite-slab tight-binding models from these Wannier functions [65]. In this section, the DFT calculations were performed by Prof. Sasagawa and Dr. Ochi.

5.3.2 Possible Topological Phases of α- And β-Bi4 I4 To investigate the possible topological phases of Bi4 I4 m, we have performed DFT calculations. The band topology calculations are shown in Fig. 5.10. We have found that the topological phase of the β-phase varies sensitively depending on the computational parameters. The bulk band structure without spin-orbit coupling (SOC) (the left panel of Fig. 5.10b) shows that the conduction and valence band edges with small band gap (E G ) is located at L and M points, and these two states exhibit opposite parities derived from Bi 5 p orbitals as discussed before. The inclusion of SOC induces parity inversions realizing different topological phases. Taking account for the band inversions at time-reversal invariant momenta (TRIMs) in 3D Brillouin zone (BZ), the Z 2 topological invariants (ν0 :ν1 ν2 ν3 ) are obtained using the Fu-Kane parity criterion [2]. Thus, the topological phase of β-Bi4 I4 is estimated to be an STI (1;110), a WTI (0;001), or even a trivia phase (0;000) reflecting the band gap E G corrections (the right panel of Fig. 5.10b), which is consistent with the previous studies reporting the electronic structure of β-Bi4 I4 is in proximity of three different phase [37, 38]. In contrast to the complications in β-Bi4 I4 , α-Bi4 I4 is found to be in the normal insulator phase, since its double block crystal structure brings the even-numbered parity inversions at every TRIM (Fig. 5.10a).

5.3 A Weak Topological Insulator State in β-Bi4 I4

89

Fig. 5.10 Topological phases of α- and β-Bi4 I4 . a (left) DFT calculation for the α-phase without the spin-orbit coupling (SOC) obtained by (gray) GGA approximation and (green) GGA+mBJ approximation. (right) The band dispersions after the inclusion of SOC. b (left) DFT calculation for the β-phase without the spin-orbit coupling (SOC) obtained by (gray) GGA approximation and (green) GGA+mBJ approximation. (right) Band inversions of the β-phase after the inclusion of SOC. The red and blue circles label the parities for the bulk bands at M and L point. The gap size was adjusted by varying parameters in mBJ exchange potential. Adopted from Ref. [41]

The STI state manifests gapless surface states in any surface which have 2D Dirac-like dispersions, as in a number of Bi-based chalcogenides (Fig. 5.11b) [4, 66]. On the contrary, it should emerge only at particular surfaces in the WTI state [1, 2], which is topologically protected [6–8]. According to DFT calculations on the WTI in β-Bi4 I4 state, quasi-1D Dirac-like TSS emerges at the side-surface (100) (Fig. 5.11a), whereas the top-surface (001) is topologically dark with absence of TSS (Fig. 5.12c), [67]. This quasi-1D TSS can be regarded as the signature of the 3D stacking of 2D TI state with weak interlayer couplings. Notably, in contrast to the helical spin texture of the Dirac-cone in the STI phase, the quasi-1D Dirac-state exhibits the characteristic spin texture in k-space: the spin direction is locked almost along the segment of quasi-1D Fermi surface (FS), and they align almost antiparallel at ±k z . Therefore, the spin-texture most likely prohibits the electron-backscattering similarly to the robust spin-current at the edge of a 2D TIs [68, 69]. At the side-surface (100), the quasi-1D Dirac SS features gapless at ¯ and Z¯ points (Fig. 5.11c), while the 2D Dirac SS of the STI state shows the large gap opened at Z¯ point (Fig. 5.11d). This difference becomes further implication to clarify either the STI or WTI. Therefore, to unambiguously distinguish the two non-trivial phases, the systematic surface investigations not only at the top surface (001) [38] but also at the side-surface (100) is the only way. In contrast, no gapless surface state is expected in the α-phase. Here we note that the α-phase and the β-phase can also be distinguished by carefully observing the bulk band dispersions. Considering the bulk band dispersion projected onto the top surface (001) in the β-phase, it shows a single Dirac-like dispersion with small ¯ point due to the weak interlayer bonding, regardless of k z dependence at the M the topological phases (Fig. 5.12a and b). In the α-phase, the k z dependence can be significantly suppressed due to the bilayer structure, which instead leads to the

90

5 Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides

(a)

(b)

(001) c

b

kz

(100)

Ζ

c

(d)

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kz/ky

(100) b

ky/kx

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E

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ky

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Ζ

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E-EF (eV)

High 0 -0.2 -0.4

0 -0.2 -0.4 Low

-0.6

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0 -1 ky (Å )

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0 -1 ky (Å )

-0.6

0.1

-0.1

0 -1 ky (Å )

0.1 -0.1

0 -1 ky (Å )

0.1

Fig. 5.11 Calculated surface spectra at the side surface of β-Bi4 I4 . a and b Schematics of the topological surface states (TSS) of the β-Bi4 I4 in the real space and their band dispersions in reciprocal space for the WTI phase and the STI phase, respectively. c and d Calculated surface spectra at the side-surface (100) for the WTI phase and the STI phase, respectively, along distinct high-symmetry lines (red and yellow lines in Fig. 5.4d). Adopted from Ref. [41]

Γ

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Su rf a c e i n t e n s i t y

E-EF (eV)

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kz 0 0.2π/c 0.4π/c 0.6π/c 0.8π/c π/c

-0.2

0 0.2 -1 ky (Å )

Μ

0.2

STI (001) plane 0.2

E-EF (eV)

(a)

0.1

Fig. 5.12 Calculated surface spectra at the top surface of β-Bi4 I4 . a and b The (001) surface projected bulk band for the WTI phase and for the STI phase, respectively. c and d Calculated surface spectra at the top-surface (001) for the WTI phase and for the STI phase, respectively. Adopted from Ref. [41]

5.3 A Weak Topological Insulator State in β-Bi4 I4

91

Fig. 5.13 Bilayer splitting in α-Bi4 I4 . The (001) surface projected bulk bands around ¯ point and the ¯ point. the M Adopted from Ref. [41]

bilayer splitting of the bulk bands as presented in Fig. 5.13. Hence, the observation of the bilayer splitting becomes a definitive signature of the α-phase.

5.3.3 Bulk Band Structure Mapping by Synchrotron-ARPES To determine the band structures in wide momentum space, we conducted synchrotron-based ARPES experiments at hν = 85 eV at the high-resolution branch of i05 beamline at Diamond Light Source. Here, the spot size of the light is about 50 µm, and hence the photoelectrons from both the top surface (001) and the sidesurface (100) are expected (Fig. 5.14a). The two crystals are selectively obtained by the quench. During the measurements, the top surface (001) is aligned almost normal to the analyzer. In this setup, we have observed strong photoemission intensities with the periodicity corresponding to the top surface (001), as indicated in the Fermi surface mapping with the (001) surface Brillouin zone shown by the black dashed lines in Fig. 5.14b and c for both of the α- and the β-phases. Thus, the observed photoemission patterns are attributed to the band structure projected on the top surface (001). The overall band structures are very similar for the α-phase and the β-phase: a hole-like dispersion is found around the ¯ point at E − E F ∼ 0.4 eV, while the hole band shifts in energy with sample rotation along θ (Fig. 5.14a) and forms a ¯ point (Fig. 5.14d and e). At the M ¯ point, island-like V-shape feature around the M intensities are detected in the Fermi-surface mapping, which likely comes from the bottom of the conduction band, implying that the samples are slight n-doped. The good agreement with the observed band structures and the bulk band dispersions obtained by DFT calculation indicates that photoelectrons mainly originate from the bulk bands, although ARPES is a surface-sensitive technique. In this experimental geometry, the possible topological surface state of the STI phase in β-Bi4 I4 , which ¯ point, has not been clearly detected. Therefore, is expected to emerge around the M we cannot conclude the topological phase of β-Bi4 I4 at this stage. However, despite the similarity of the bulk bands, a weak feature is observed in the bulk band gap only

5 Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides

α-phase

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0.5 1.0 -1 ky (Å )

-6 0.5 1.0 -1 ky (Å )

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k x (Å )

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-1

(a) analyzer slit (b)

k x (Å )

92

EDC at Γ -0.10

0

β α -0.6 -0.4 -0.2 E-EF (eV)

0

Fig. 5.14 Synchrotron-based high-resolution ARPES results of α- and β-Bi4 I4 measured at hν = 85 eV. a Schematic of the experimental geometry. b and c Constant energy contour maps at E F of alpha and β-Bi4 I4 , respectively, with the 1D chains aligned along k y . The ARPES intensities are integrated within 30 meV. The solid lines denote the (001) surface Brillouin zone. d and e ¯ for the α-phase and the β-phase, ¯ M’ ARPES intensity maps along the high-symmetry direction respectively. f Magnified intensity map around ¯ point near E F . g Momentum distribution curves (MDCs) at E F around ¯ point. h Energy distribution curves (EDCs) at ¯ point. Adopted from Ref. [41]

in the β-phase, as shown in Fig. 5.14f. This feature is weak but clearly observed in the momentum distribution curves (MDCs) at E F and the energy distribution curves (EDCs) around the ¯ point (Fig. 5.14g and h). However, the weak feature is absent even in the calculated surface band dispersions regardless of the topological phase, pointing out the important role of the side-surface (100) in discussing the observed photoemission signals from these compounds.

5.3 A Weak Topological Insulator State in β-Bi4 I4

93

5.3.4 Observation of Topological Surface State by Laser-ARPES To investigate the electronic structures in both of α-Bi4 I4 and β-Bi4 I4 in more detail, we performed laser-ARPES measurement with high energy and momentum resolutions. In this experiment, we observe the superposition of photoelectrons from the top surface (001) and the side-surface (100), since the incident laser light with the spot size of ∼50 µm illuminates the two planes simultaneously (Fig. 5.15). In contrast to the synchrotron-based ARPES results, the constant energy contours of observed photoelectron intensity clarify the different topologies of Fermi surfaces (FSs) for the two materials (Fig. 5.15b and c). Clearly, the data for the β-phase presents the quasi-1D FS, while for the α-phase we see the intensity distribution only around ¯ of the (001) surface BZ. In accordance with the theoretical conclusion about the M band topology in α-Bi4 I4 , we experimentally demonstrate its trivial band structures. Figure 5.15d displays the ARPES band maps in the α-phase at different emission angles (θ ) with respect to the normal to the top surface (001). The valence band shows the energy dispersion along k x , which is observed as its θ evolution with the ¯ point of the (001) surface valence band top found at θ = 36◦ which corresponds to M BZ. The conduction band is also seen in the data, which results in the photoelectron ¯ distribution in the FS (Fig. 5.15b). No in-gap Dirac-state is observed at ¯ and M points (the left panel and the right panel of Fig. 5.15d), which is consistent with the trivial band topology. In addition, we find the signature for the bilayer splitting as shown in EDC taken at θ = 28◦ (Fig. 5.15f): two peaks are detected at a momentum space, reflecting the bilayer structure of the crystal. In contrast, we see the Dirac-cone like energy dispersions near the Fermi energy E F for each θ in the β-phase (Fig. 5.15e). The weak θ evolution of the Dirac-state displays its quasi-1D character as already seen in Fig. 5.15c. It is surprising that the Dirac-state is clearly observed even at θ = 0◦ , which corresponds to ¯ in the BZ for the (001) surface since there should be no state according to the DFT calculation in any case of the band topology (Fig. 5.12). This can be explained only if the observed Dirac-state is considered to derive from the side-surface (100). In particular, much different from the trivial band topology of α-phase, the quasi-1D Dirac-state is observed only in β-phase despite their similar crystal structures. The quasi-1D dispersion also corresponds to the weak intensity around ¯ detected by the synchrotron-based measurements (Fig. 5.14). These results signify that the quasi-1D state is derived from the side surface of β-Bi4 I4 displaying its non-trivial band topology. The non-trivial band topology, either STI or WTI, can be determined by the observed surface dispersions since they should reflect the bulk band topology as theoretically demonstrated in Fig. 5.11. We here show, in Fig. 5.15e, the ARPES band maps obtained at θ = 49◦ and 72◦ , which cut Z¯ and ¯ points of the (100) surface BZ along the k y line (see red and yellow lines in Fig. 5.15c. The data clearly shows

94

5 Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides (b) α-phase

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θ =36°

θ =28°

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θ =72°

E-EF (eV)

-0.1 -0.2 -0.3 -0.4 -0.5 -0.1

0

0.1-0.1

0

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0 0.1-0.1 -1 ky (Å )

0

0.1 -0.1

0

0.1-0.1

0

0.1

Fig. 5.15 Experimental band structures of α- and β-Bi4 I4 by laser-ARPES. a Experimental geometry of laser-based ARPES. We collect the photoelectrons at different emission angle θ with the respect to the normal of the top-surface (001). b and c Photoelectrons intensity distributions at E F for the two phases, respectively. The intensities are integrated within 40 meV. The black and red dashed lines display the surface Brillouin zones of (001) and (100) surfaces, respectively. d and e, APRES band maps taken at different θ-s in the α-phase and the β-phase, respectively. f Energy distribution curves extracted from images for θ = 28◦ indicated by the red dashed lines in d and e. Adopted from Ref. [41]

that the observed Dirac-state dispersions feature gapless at both high-symmetry k points, which appears to be consistent with the DFT calculation of the WTI state (Fig. 5.11c). Therefore, these results clarify the band topology of the β-phase as the WTI phase.

5.3 A Weak Topological Insulator State in β-Bi4 I4

95

5.3.5 Surface-Selective Measurements by Spatial-Resolved ARPES However, since the side-surface (100) shows up only a small area (Fig. 5.8), it is difficult to unambiguously discriminate the pure signals from the side-surface (100) in the ARPES with the spot size of ∼50 µm, which is relatively larger than the height of the facet with typically ∼2 µm. Therefore, to further confirm our conclusion about the band topology in β-Bi4 I4 , we used surface-selective ARPES measurement with zone plate achieving the spot size less than 1 µm, the so-called nano-ARPES. To obtain a large side surface for nano-ARPES measurements, the crystal is mounted so that the top plane is perpendicular to the sample holder and cleaved by the scotchtape method (Fig. 5.16). Figure 5.17a and c show the experimental geometry for the surface selective investigation on the side-surface (100) and the top surface (001), respectively. For the surface-selective nano-ARPES for the (100) surface, the sidesurface (a-c plane) of the crystal with a small thickness < 50 µm was cleaved. As shown in Fig. 5.17b, the microscopic intensity map of nano-ARPES captures the very thin topography of the side-surface (100). We shine a light on the part of the bright area with relatively flat intensity distribution within ∼10×100 µm2 (white circle in Fig. 5.17b). This surface-selective experiment unambiguously reveals the quasi-1D FS topology even at hν = 85 eV (Fig 5.17d), where no quasi-1D signals have been detected in the conventional ARPES measurement. As shown in Fig. 5.17f, the nano-ARPES band maps cut along the high-symmetry momentum lines at ¯ and Z¯ (red and yellow lines in Fig. 5.17d) display the linear dispersions crossing the Fermi level. In MDCs at E F , two peaks are clearly resolved both at the ¯ point and the Z¯ point, which is consistent with the predicted quasi-1D TSS for the WTI phase. In contrast, no quasi-1D TSS is observed in the top-surface (Fig. 5.17e and g). Apparently, all of these results show good agreement with our conclusion about the WTI state determined by laser-ARPES results. So far, the nano-ARPES experiments were performed with a zone plate to focus the light, which significantly reduced the intensity of the excitation light. Thus, one

(a)

cleavage

(b)

side plane

tape

sample ~2.0 mm ~30 μm

cleavage

~500 μm

holder

top plane

tape sample ~2.0 mm ~500 μm

holder

~30 μm

Fig. 5.16 Schematics of the cleavage of the sample by the scotch-tape method. a For the measurement of the side plane, the crystal is mounted vertically. b The ordinary cleaving for the top plane. Adopted from Ref. [41]

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5 Stacking-Dependent Topological Phases in Quasi-1D Bismuth Halides

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