Chalcogenide: From 3D to 2D and Beyond 9780081026878, 9780081027363, 2192202202, 2712712722, 0081026870

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Chalcogenide

Woodhead Publishing Series in Electronic and Optical Materials

Chalcogenide From 3D to 2D and Beyond

Edited by

Xinyu Liu Department of Physics, University of Notre Dame, Notre Dame, IN, United States

Sanghoon Lee Department of Physics, Korea University, Seoul, South Korea

Jacek K. Furdyna Department of Physics, University of Notre Dame, Notre Dame, IN, United States

Tengfei Luo Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, United States

Yong-Hang Zhang School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ, United States

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2020 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-102687-8 (print) ISBN: 978-0-08-102736-3 (online) For information on all Woodhead Publishing publications visit our website at https://www.elsevier.com/books-and-journals

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Contents

List of contributors

xi

1

1

2

The ubiquitous nature of chalcogenides in science and technology J.K. Furdyna, S.-N. Dong, S. Lee, X. Liu and M. Dobrowolska 1.1 Introduction 1.2 Chalcogenides in 3D form 1.2.1 Monocrystalline CdTe solar cells 1.2.2 II-VI magnetic semiconductors 1.2.3 Electronic and optical effects in II1-xMnxVI alloys 1.2.4 Miscellaneous II-VI-based diluted magnetic semiconductors 1.2.5 Chalcogenide lead salts 1.2.6 Chalcogenide spinels 1.3 Two-dimensional chalcogenide structures 1.3.1 Epitaxially-formed chalcogenides 1.3.2 2D “van der Waals” chalcogenides 1.3.3 Interface phenomena in chalcogenide structures 1.4 Chalcogenides beyond 2D 1.4.1 One-dimensional and quasi-one-dimensional chalcogenides 1.4.2 Zero-dimensional chalcogenide structures 1.5 Concluding remarks References Thermoelectric applications of chalcogenides Han Meng, Meng An, Tengfei Luo and Nuo Yang 2.1 Introduction 2.1.1 Thermoelectric effect 2.1.2 Thermoelectric efficiency 2.2 Nanostructure engineering 2.2.1 Bottom-up and top-down fabrication 2.2.2 Consolidation method 2.2.3 Introducing nanostructures 2.2.4 Introducing nanoprecipitates 2.3 Defect engineering 2.3.1 Normal doping 2.3.2 Introducing point defect 2.3.3 Introducing element deficiency 2.3.4 Other approaches

1 2 2 3 4 5 7 9 9 9 12 15 19 19 20 22 23 31 31 32 32 34 34 35 36 37 38 38 40 40 41

vi

3

4

5

Contents

2.4 2.5

Band structure engineering Crystal structure engineering 2.5.1 Original complex structure 2.5.2 Peierls distortion structure 2.5.3 Layered structure 2.5.4 Increase the degree of orientation 2.6 Outlook References

43 44 44 45 46 48 49 51

Lead salt photodetectors and their optoelectronic characterization D. Babic, L.W. Johnson, L.V. Snyder and J.J. San Roman 3.1 Introduction 3.2 Background 3.3 Lead salt detector fabrication 3.4 Lead salt detector characterization 3.5 Conclusions Acknowledgment References

57

Optical dispersion of ternary II VI semiconductor alloys Xinyu Liu and J.K. Furdyna 4.1 Introduction 4.1.1 The classical picture of dispersion 4.1.2 Electronic band structure and dispersion 4.1.3 The phenomenological dispersion model 4.2 Optical dispersion 4.2.1 Determination of the energy gap Eg(x) 4.2.2 Indices of refraction n(x) 4.3 Theoretical model 4.3.1 Semi-empirical model 4.3.2 Improvements of SEO model 4.3.3 Comparison between various semi-empirical fits for ZnTe 4.4 Data analysis and discussion 4.4.1 Experimental results for ternary II-VI alloys 4.4.2 Summary 4.5 Physical interpretation and discussion 4.5.1 Physical meaning of fitting parameters 4.5.2 Optical dispersion and ionicity References Appendix

67

Group-IV monochalcogenides GeS, GeSe, SnS, SnSe Lyubov V. Titova, Benjamin M. Fregoso and Ronald L. Grimm 5.1 Introduction 5.2 Crystal lattice and band structure calculations

57 57 59 60 64 64 64

67 67 69 71 72 73 76 83 83 85 88 91 91 95 99 99 105 107 109 119 119 120

Contents

5.3 5.4 5.5 5.6

Electronic band structure Electronic and optical properties Nonlinear optical properties Fabrication: single crystal growth and exfoliation; CVD, growth of 2D nanostructures Acknowledgment References Further reading 6

7

Epitaxial II-VI semiconductor quantum structures involving dilute magnetic semiconductors S. Lee, M. Dobrowolska and J.K. Furdyna 6.1 Introduction 6.2 Magneto-optical properties of ZnSe and ZnTe epilayers 6.2.1 Band structure and exciton 6.2.2 Exciton transitions in the absence of magnetic field 6.3 Landau level transitions and magneto-polaron effect 6.4 Composition modulated ZnSeTe sinusoidal superlattice 6.4.1 Band structure of superlattice with sinusoidal energy profile 6.4.2 Growth of ZnSeTe superlattices with sinusoidal composition modulation 6.4.3 Optical transitions in ZnSeTe sinusoidal superlattices 6.5 II-VI-based zero-dimensional structures 6.5.1 Spin polarization and relaxation of exciton in QDs 6.5.2 Spin-spin interaction between the coupled QDs 6.6 II-VI quantum structures involving DMSs 6.6.1 Zeeman splitting in II1-xMnxVI DMS epilayers 6.6.2 Mapping of exciton localization in QDs 6.7 Enhancement of spin polarization in non-DMS and DMS coupled QDs 6.8 Summary References 2D electron gas in chalcogenide multilayers A. Kazakov and T. Wojtowicz 7.1 Introduction 7.2 2DEG in magnetically doped QWs 7.2.1 2DEG in low-dimensional heterostructures 7.2.2 Spin interactions in chalcogenide DMS QWs 7.2.3 Magnetotransport in chalcogenide QWs 7.2.4 DMS QW in inhomogeneous magnetic fields 7.2.5 DMS QWs under terahertz and microwave radiation 7.3 Novel topological phases in chalcogenide multilayers 7.3.1 Domain walls and non-Abelian excitations 7.3.2 Wireless Majorana bound states 7.3.3 Quantum spin Hall effect in HgTe QWs

vii

123 126 131 137 140 140 150

153 153 155 155 157 158 161 162 163 165 166 167 171 173 173 174 178 181 182 189 189 190 190 193 195 206 209 211 212 217 217

viii

Contents

7.3.4 Quantum anomalous Hall effect in HgTe QWs 7.3.5 Topological phases in IV-VI materials 7.4 Summary and perspectives Acknowledgment References 8

9

Layered two-dimensional selenides and tellurides grown by molecular beam epitaxy Xinyu Liu, J.K. Furdyna, Sergei Rouvimov, Suresh Vishwanath, Debdeep Jena, Huili Grace Xing and David J. Smith 8.1 Introduction 8.1.1 Motivation 8.1.2 A survey of 2D chalcogenides 8.2 MBE growth of 2D materials 8.2.1 Advantages of MBE growth of 2D materials 8.2.2 Growth of layered selenide and telluride films and their heterostructures 8.2.3 Cross between 2D and 3D structures 8.2.4 Challenges 8.3 Physical characterization of 2D materials grown by MBE 8.3.1 Electronic structure of 2D materials 8.3.2 Phonon properties of 2D materials 8.3.3 Other optical properties of 2d materials 8.4 Concluding remarks Acknowledgment References Tailoring exchange interactions in magnetically doped II-VI nanocrystals Rachel Fainblat, Franziska Muckel and Gerd Bacher 9.1 Introduction 9.1.1 Theoretical background 9.1.2 Outline of the chapter 9.2 Two-dimensional (2D) colloidal nanocrystals 9.2.1 Giant magneto-optical response in Mn21-doped CdSe nanoribbons 9.2.2 Tuning magnetic exchange interactions by wavefunction engineering in core/shell nanoplatelets 9.3 Zero-dimensional nanocrystals 9.3.1 Valence-band mixing in doped nanocrystal quantum dots 9.3.2 Going to the limit: individual dopants in single nanocrystals quantum dots 9.4 At the border between quantum dots and molecules: magic sized nanoclusters 9.4.1 Smallest doped semiconductors

219 220 220 221 221

235

235 235 237 237 237 242 249 252 253 253 255 257 261 262 262

271 271 272 275 275 275 279 282 282 285 289 289

Contents

10

11

ix

9.4.2 Doped magic-sized alloy nanoclusters 9.4.3 “Digital” doping in nanoclusters 9.5 Conclusion and future trends Acknowledgments References

290 292 294 295 296

Chalcogenide topological insulators Joseph A. Hagmann 10.1 Introduction 10.1.1 The Z2 Topological insulator 10.1.2 Mercury telluride quantum wells 10.1.3 V2VI3-series 3D topological insulators 10.2 Synthesis 10.2.1 Mercury telluride quantum well growth 10.2.2 V2VI3-series 3D topological insulators 10.3 Experimental investigations 10.3.1 Spectroscopy 10.3.2 Electrical transport 10.3.3 Exotic topological states 10.4 Summary and outlook References Further reading

305

Thermal transport of chalcogenides Meng An, Han Meng, Tengfei Luo and Nuo Yang 11.1 Introduction 11.1.1 Basic theory of heat conduction 11.1.2 The structure characteristics of chalcogenides 11.2 Geometrical effect 11.2.1 Dimensional effect 11.2.2 Length dependence 11.2.3 Single-layer sheet 11.2.4 Discussion on the overall trend from single-layer to bulk 11.3 Extrinsic thermal conductivity of chalcogenide 11.3.1 Strain effect 11.3.2 Effect of atomic disorder and defect 11.3.3 Anisotropy 11.4 Fundamental insight into thermal transport 11.4.1 Resonant bonding 11.4.2 Lone pair electron 11.4.3 Rattling modes 11.5 Conclusion and outlook References

Index

305 306 307 309 312 312 314 318 318 321 323 327 329 336 339 339 339 342 343 343 346 348 351 352 352 355 358 360 360 361 363 364 364 371

List of contributors

Meng An College of Mechanical and Electrical Engineering, Shaanxi University of Science and Technology, Xi’an, China D. Babic Laser Components DG, Inc., Tempe, AZ, United States Gerd Bacher University of Duisburg-Essen, Duisburg, Germany M. Dobrowolska Department of Physics, University of Notre Dame, Notre Dame, IN, United States S.-N. Dong Department of Physics, University of Notre Dame, Notre Dame, IN, United States Rachel Fainblat University of Duisburg-Essen, Duisburg, Germany Benjamin M. Fregoso Department of Physics, Kent State University, Kent, OH, United States J.K. Furdyna Department of Physics, University of Notre Dame, Notre Dame, IN, United States Ronald L. Grimm Department of Chemistry and Biochemistry, Worcester Polytechnic Institute, Worcester, MA, United States Joseph A. Hagmann Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD, United States Debdeep Jena School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, United States L.W. Johnson Laser Components DG, Inc., Tempe, AZ, United States A. Kazakov International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Warsaw, Poland S. Lee Department of Physics, Korea University, Seoul, South Korea

xii

List of contributors

X. Liu Department of Physics, University of Notre Dame, Notre Dame, IN, United States Xinyu Liu Department of Physics, University of Notre Dame, Notre Dame, IN, United States Tengfei Luo Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, United States Han Meng State Key Laboratory of Coal Combustion and Nano Interface Center for Energy, School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan, P.R. China Franziska Muckel University of Duisburg-Essen, Duisburg, Germany Sergei Rouvimov Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, United States J.J. San Roman Laser Components DG, Inc., Tempe, AZ, United States David J. Smith Department of Physics, Arizona State University, Tempe, AZ, United States L.V. Snyder Laser Components DG, Inc., Tempe, AZ, United States Lyubov V. Titova Department of Physics, Worcester Polytechnic Institute, Worcester, MA, United States Suresh Vishwanath School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, United States T. Wojtowicz International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Warsaw, Poland Huili Grace Xing School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, United States Nuo Yang State Key Laboratory of Coal Combustion and Nano Interface Center for Energy, School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan, P.R. China

The ubiquitous nature of chalcogenides in science and technology

1

J.K. Furdyna1, S.-N. Dong1, S. Lee2, X. Liu1 and M. Dobrowolska1 1 Department of Physics, University of Notre Dame, Notre Dame, IN, United States, 2 Department of Physics, Korea University, Seoul, South Korea

1.1

Introduction

Chalcogenides (compounds and alloys of sulfur, selenium and tellurium) display an uncommon array of physical phenomena that range from interesting electronic [1,2], thermal [3], and optical properties [4,5] to novel forms of superconductivity [6,7] and magnetism [8]. This broad spectrum of intrinsic properties is further enhanced by contemporary technologies of creating lower-dimensional structures, that bring into play the ability to introduce enormous strains in epitaxial films, the possibility of forming quantum structures such as quantum wells, superlattices, and quantum dots, the formation of interfaces tailored for realizing a variety of proximity effects and related interfacial phenomena, and our ability to form atomic-scale two-dimensional chalcogenide systems [9,10] similar to graphene. The properties found in each of these classes can be further divided into diverse effects of interest to both fundamental science and to practical applications. In this chapter we will review novel phenomena that occur in the chalcogenide family. Because of the exceptional range of these diverse properties, we will organize this survey by focusing on specific examples, without dwelling on mathematical details, which the interested reader can easily find in the relevant literature that we will cite. Even by restricting ourselves to the specific topics, the range of topics turns out to be quite widespread. We will therefore further restrict this review primarily to crystalline metal chalcogenide compounds and alloys produced by equilibrium growth methods. This survey is organized as follows. We will first discuss novel optical, electronic and magnetic properties displayed by chalcogenides compounds and alloys in “three dimensional” (3D) form. We will then discuss the properties of 2D chalcogenide systems such as epitaxial films and more complex multilayers, whose formation is made possible by epitaxial methods. Next, we will provide examples of graphene-like chalcogenide 2D systems held together by van der Waals forces, that either form naturally or whose formation is coerced by layer-by-layer epitaxy, and that display unique properties of various forms. In this latter context we will pay special attention to topological insulators (such as Bi2Se3) and to transition metal Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00001-4 © 2020 Elsevier Ltd. All rights reserved.

2

Chalcogenide

di-chalcogenides (such as WSe2). We will then provide examples of novel properties occurring at interfaces between chalcogenides and other materials that are generated the energy band offset at the interface or by proximity effects. Finally, we will present a brief look into novel effects emerging as the dimensionality of the structures is reduced to 1D (nanowires and related structures), and to zero-D to form quantum dots by both self-assembly processes and colloidal chemical methods. This chapter is intended as a review of the field. While we will use examples obtained in the authors’ own laboratory, we will also use numerous examples from the published literature in order to present as complete picture as possible. In either case we will provide ample citations, so that the reader interested in a specific material can readily delve further into the topic of interest. While we cannot presume to cover a complete picture of crystalline chalcogenides, it is our hope that the examples presented in this review will at least serve as an illustration of the richness, diversity, and
a most of all
the enormous potential of physical effects that are made possible by chalcogenide materials.

1.2

Chalcogenides in 3D form

“Three-dimensional” (3D) metal chalcogenides are important materials group for modern information technology. Recent advanced applications of metal chalcogenides including: solar cell, microelectronic devices, catalysts, sensors, optical modulator, and laser sources [11,12]. In order to achieve high performance of above devices, good quality single crystals are needed. Growth of single crystals and their characterization towards device fabrication have been given great efforts due to their importance for both academic research as well as applied technology. Although with a given chalcogen (S, Se, or Te) many metals form various compounds, often with complex structures in a wide variety, for a large part the binary compounds, 3D metal chalcogenides commonly belong to the very basic structural types: the cubic NaCl rock salt and zinc blende, or the hexagonal NiAs and wurtzite types. The binary chalcogenide compounds often categorized as II-VI, VI-VI, and V-VI groups, which will be addressed in the following sections.

1.2.1 Monocrystalline CdTe solar cells To demonstrate the importance of 3D crystalline chalcogenides, we now refer to a recent breakthrough on the application of II-VI semiconductors for solar energy conversion. Specifically, a CdTe/MgCdTe double-heterostructure solar cell made of 1- to 1.5-μm-thick n-type monocrystalline CdTe absorbers is shown in Fig. 1.1 [13]. This design allows CdTe solar cells to be made thinner and more efficient. The best power conversion efficiency achieved in a device with this structure is 17.0% with open-circuit voltages of up to 1.096 V.

The ubiquitous nature of chalcogenides in science and technology

3

Figure 1.1 (Top) Device design and band diagram. (A) Layer structure of the CdTe/ MgxCd1-xTe double-heterostructure solar cell with an a-SiCy:H (y 5 0
6%) hole-contact layer. (B
D) Schematic band diagrams at equilibrium (B) and open circuit (C) and an equilibrium band diagram drawn to scale for the hero cell (D). (Bottom) Optimum device performance. (E) Measured J
V curve and associated device parameters. (F) Measured EQE and 1- reflectance (1 - R) with a calculated photo-current of 22.3 mAcm2. (G) Simulated absorptance spectrum for the highest-performing CdTe solar cell device with a calculated photo-current of 23 mAcm2. After Y. Zhao, M. Boccard, S. Liu, J. Becker, X.-H. Zhao, C.M. Campbell, et al., Monocrystalline CdTe solar cells with open-circuit voltage over 1 V and efficiency of 17%, Nat. Energy 1 (2016) 16067.

1.2.2 II-VI magnetic semiconductors Magnetic semiconductors have been pursued for more than 50 years because they combine two major components of modern information technology, semiconductor for logic and magnetism for memory [14,15]. In history, the list of candidate magnetic semiconductors can be grouped into two categories: undoped magnetic semiconductors, such as europium chalcogenides and semiconducting spinels that have a periodic array of magnetic elements on their own; [16,17] and doped magnetic semiconductors, such as (Cd,Mn)Se [18] and (Ga,Mn)As [19], which are achieved by doping magnetic elements into conventional nonmagnetic semiconductors to make them ferromagnetic. In the latter case, the Mn-VI sublattice contained implicitly in II1-xMnxVI alloys [18] such as Hg1-xMnxTe [20] differs from Mn chalcogenide compounds occurring naturally (e.g., MnTe and MnSe) in that in the alloy form they exist as tetrahedrally-bonded structures. Thus magnetic studies of II1-xMnxVI

4

Chalcogenide

Figure 1.2 Nuclear peak (0, 2, 0) and magnetic peaks (1, 1/2, 0) and (1, 3/2, 0) corresponding to antiferromagnetic type-III order for a Cd0.3Mn0.7Se epilayer. After T. Giebultowicz, P. Klosowski, N. Samarth, H. Luo, J. Rhyne, J. Furdyna, Antiferromagnetic phase transition in Cd12xMnxSe epilayers, Phys. Rev. B 42 (1990) 2582.

alloys allow us to investigate the properties of such “hypothetical” tetrahedrallycoordinated compounds such as MnSe or MnTe in either zinc-blende or wurtzite forms [18]. Using equilibrium crystal growth methods, the II1-xMnxVI alloys can be produced with Mn content which in some cases reaches 60% or more (i.e., x . 0.60) while retaining the structure of the parent II-VI compound. At low values of x these materials are paramagnetic, with increasingly dominant antiferromagnetic interactions between the Mn ions as x increases. This results in spin-glass-like behavior at concentrations of x . 0.17 (the magnetic percolation concentration), and gradually transform into short-range antiferromagnetic order with further increase of x [Fig. 1.2]. The limitations of Mn miscibility in the II-VI lattice in equilibrium crystal growth limit studies of these phenomena to at most x  0.60, but, as we will see later, non-equilibrium crystal growth processes such as molecular beam epitaxy allow us to extend these studies to higher values of x (including x 5 1.0!), which then allows one to study magnetic structures of MnTe and MnSe in zinc blende structure, that exhibit unique forms of long-range antiferromagnetic order [21].

1.2.3 Electronic and optical effects in II1-xMnxVI alloys The key property that makes the II1-xMnxVI alloys unique is the enormous Zeeman splitting of electronic levels in their band structure due to the so-called sp-d

The ubiquitous nature of chalcogenides in science and technology

2.48

5

d

Photon energy (eV)

2.46

2.44 c 2.42

b

2.40

2.38 a 2.36 0

5

10

15

20

Magnetic field (T)

Figure 1.3 Magnetic-field dependence of the energies of transitions a, b, c, and d of the 1s exciton in Zn1-xMnxTe (x 5 0.05), observed at 1.4 K in magnetoreflectance in the Faraday configuration. Solid and open circles correspond to transitions observed with opposite circular polarizations. After R. Aggarwal, S. Jasperson, P. Becla, J. Furdyna, Optical determination of the antiferromagnetic exchange constant between nearest-neighbor Mn21 ions in Zn0.95Mn0.05Te, Phys. Rev. B 34 (1986) 5894.

exchange interaction between the s and p states of the band structure determined by the II-VI “parent” compound and the d states of the incorporated Mn ions [18]. This in turn has led to a host of new optical and electronic phenomena, including enormous values of the g-factor, that are temperature dependent and result in spectacular thermo-magnetic effects; giant Zeeman shifts of excitonic transitions (that can achieve values as large as 100 meV) [22] [Fig. 1.3]; enormous Faraday rotations (of the order of thousands of degrees in a millimeter-thick sample) [23] [Fig. 1.4]; the presence of magnetic polarons; [24] and enormous values of magnetoresistance [25,26]. The wide range of the rather spectacular electronic and magneto-optical effects arising directly from the sp-d interactions in these alloys has been reviewed in Ref. [18].

1.2.4 Miscellaneous II-VI-based diluted magnetic semiconductors There has also been limited work carried out on other chalcogenide DMSs, such as II1-xCoxVI, II1-xCrxVI and II1-xFexVI, which exhibit interesting magnetic properties of their own. For completeness, we will briefly present examples of some of these materials and their properties in the present section.

6

Chalcogenide

10

Cd1–xMnxTe

Intensity (arbitrary units)

8

X = 0.23 T=5k H = 5 kG L = 0.351 cm

6

4

–585° 2

0 16,000

15,000

14,000

13,000

12,000

11,000

Wave number (cm–1)

Figure 1.4 Transmission as a function of wave number of light transmitted through a Cd1-xMnxTe (x 5 0.23) slab and sandwiched between two linear polarizers, displaying giant Faraday rotation. At 15,000 cm21 the rotation exceeds 28000 . The experiment is carried out at 5 K and 5 T. After D. Bartholomew, J. Furdyna, A. Ramdas, Interband Faraday rotation in diluted magnetic semiconductors: Zn12xMnxTe and Cd12xMnxTe, Phys. Rev. B 34 (1986) 6943.

1.2.4.1 Co-based DMS alloys Interesting results have also been obtained on dilute II1-xCoxVI alloys, which can contain significant concentrations of Co due to the tendency of that element toward divalent state. Alloys such as Zn1-xCoxS and Zn1-xCoxSe crystallize in hexagonal (wurtzite) structure, and have magnetic properties similar to their II1-xMnxVI counterparts [27]. An interesting aspect of these materials is that the nearest-neighbor AFM Co12-Co12 interaction is significantly stronger than in the Mn12 alloys, an effect that is not fully understood at the present time.

1.2.4.2 Cr-based DMS alloys II1-xCrxVI alloys represent yet another important group in this family of alloys, which also form with significant values of x because of the miscibility of Cr in the II-VI lattice arising from the preference of Cr to divalent state. Theoretical studies of these systems based on tight-binding approximation have been used to predict that magnetic properties of II1-xCrxVI DMS alloys are governed by ferromagnetic super-exchange, thus holding out the promise that these materials will exhibit ferromagnetic properties, thus making them different from their Mn-based counterparts [28]. Experimental studies did indeed reveal the presence of ferromagnetism in

The ubiquitous nature of chalcogenides in science and technology

7

Figure 1.5 Schematic diagram of Hg1-xFexSe band structure near the Γ point for several values of x. The Fe11 level is shown as a resonant donor. There are approximately 5 3 1018 cm23 conduction-band states available below the Fe11 level. For x . 3 3 1024, all these states are filled by ionization of Fe11 ions to Fe111 ions, which underlie the Van Vleck paramagnetism of this system. This also provides a natural limit for the Fermi level. After J. Furdyna, Diluted magnetic semiconductors: issues and opportunities, J. Vac. Science & Technol. A 4 (1986) 2002
2009.

various Cr doped II-VI systems, e.g. (Zn,Cr)Se [29] and (Zn,Cr)Te [30], and a large magnitude of observed magnetic circular dichroism (MCD) [30] observed in (Zn, Cr)Te has been used to suggest possible applications of this system in photonic devices, such as Faraday optical insulators. However, there still remains the unresolved issue that the observation of ferromagnetism in these systems was largely determined by aggregation of Cr cations (i.e. embedded Cr rich nanocrystals) [31].

1.2.4.3 Fe-based DMS alloys Finally, we will discuss the interesting case of II1-xFexVI alloys, which represent an altogether different magnetic system, in which magnetic properties arise from a competition between the populations of Fe13 and Fe12 ions in the system. This can be illustrated by the case of dilute Hg1-xFexSe, illustrated in Fig. 1.5 [32]. Briefly, it has been shown that the Fe12 level is a resonant state (in the case of Hg1-xFexSe this state is located at 230 meV above the bottom of the conduction band). Thus, at very low concentrations of x, all Fe states are self-ionized to Fe13, providing population of (surprisingly mobile) electrons, up to the point where all states up to the resonant Fe12 level are filled. This then leads to Van Vleck paramagnetism of this interesting system. In the interest of space, we refer the interested reader to several excellent discussions of Hg1-xFexSe and related alloys published in references [33
35].

1.2.5 Chalcogenide lead salts IV-VI compounds (PbTe, PbSe, PbS, SnTe, GeTe) and their alloys are narrow-gap semiconductors crystallizing in the rock-salt structure and known for very good

8

Chalcogenide

Figure 1.6 ARPES studies of the (001) surface of Pb0.77Sn0.23Se monocrystals. The temperature dependence of the ARPES spectra data along the k-space line (Г-Х-М). They clearly show the evolution of the gapped surface states (for T . 100 K) into the Dirac-like state upon lowering the temperature (T 5 9 K). After P. Dziawa, B. Kowalski, K. Dybko, R. Buczko, A. Szczerbakow, M. Szot, et al., Topological crystalline insulator states in Pb12xSnxSe, Nat. Mater. 11 (2012) 1023.

thermoelectric and infrared optoelectronic properties exploited, e.g. in mid-infrared p-n junction lasers and detectors [36]. Recently, these materials have been recognized as a new class of topological materials - topological crystalline insulators (TCI) [37,38]. The TCI surface states were discovered by angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling spectroscopy (STS) as well as observed in magneto-transport and magneto-optical studies [Fig. 1.6]. These states constitute a new type of two-dimensional (2D) electron system with unique properties brought about by strong relativistic effects (spin-orbit interaction). In particular, in Pb1-xSnxTe (x 5 0
1) [39] and Pb1-xSnxSe (x 5 0
0.4) [38] substitutional alloys the chemical composition, temperature and hydrostatic pressure induced band inversion is observed between conduction and valence bands. On the other hand, in parallel with the development of diluted magnetic II-VIbased semiconductors, very significant advances were also made in “making lead salts magnetic” by introducing Mn into the lead salt lattice. These studies involved material such as Pb1-xMnxTe [40], Pb1-xMnxSe [41] and, especially, Pb1-xySnxMnxTe [42]. Research on the latter quaternary proved to be particularly interesting because, unlike their II-VI-based counterparts discussed in the preceding sections, Pb1-x-ySnxMnxTe was shown to become ferromagnetic with increasing content of Sn, owing to increasing free carrier concentration which led to increasing carrier concentration, and thus to carrier-mediated RKKY-type ferromagnetism. At lower values of x this alloy has also exhibited spin-glass behavior, which has added to the interest of this material in the magnetism community. Undoubtedly, recent discovery of TCI makes the development of magnetic lead salts urgent need for realizing the quantum anomalous Hall effect in such materials.

The ubiquitous nature of chalcogenides in science and technology

9

1.2.6 Chalcogenide spinels More recently, chalcogenide compounds with spinel structure (such as FeCr2S4 [43], Fe0.5Cu0.5Cr2S4 [44], CdCr2S4 [45], ZnCr2Se4 [46], HgCr2S4 [47] and so on [48]) occupy a special place among magnetic chalcogenide family in that their complex crystal structure allows interesting colossal magnetoresistance (CMR) and multiferroic features. As an illustrative example we will use the case of CdCr2S4. CdCr2S4, in which ferroelectric and ferromagnetic transitions were found to occur at 57 and 85 K, respectively [49]. Interestingly, it has been found that a metalinsulator transition in CdCr2S4 can be triggered by an electrical field, making it a unique compound to possess the four important properties: colossal magnetocapacitive, electrocapacitive, magnetoresistive, and electroresistive properties [50]. These interesting results automatically open for speculation the possibility of finding similar effects in related crystalline structures of ACr2X4 (A 5 Cu, Cd, Zn, Hg, Fe, Co, etc.; X 5 S, Se, Te) [51].

1.3

Two-dimensional chalcogenide structures

1.3.1 Epitaxially-formed chalcogenides The emergence of epitaxial methods for materials deposition has had a profound effect on the development of crystalline metal chalcogenides. First, it has led the ability of forming new chalcogenide phases, that do not exist in nature; and, more importantly, it has resulted in a host of new physical phenomena, particularly those arising from quantum confinement, in which topological effects have played a special role. We review some of these striking effects below.

1.3.1.1 II
VI quantum cascade emitters II-VI semiconductors have direct bandgap with no intervalley scattering, making them good candidates for optical and optoelectronic devices. Since the conduction band offset (CBO) of ZnCdSe/ZnCdMgSe heterostructure is as high as 1.12 eV, IIVI system remains one of the most promising candidate for quantum cascade (QC) devices operating in a broad range of infrared wavelengths including wavelengths as low as 1.55 μm [52
55]. Recently works have shown a successful design of II
VI long-wave QC emitter [Fig. 1.7] [56]. The QC design was optimized and made robust by growing high quality II-VI heterostructures by MBE. Electroluminescence at 7.1 μm was observed up to 280 K. The results demonstrate the potential of long wavelength II
VI-based QC devices, suggesting that these structures are good candidates to pursue lasing from these materials [57].

1.3.1.2 Magnetic properties of epitaxial chalcogenides We recall that, although the miscibility of Mn12 in tetrahedrally-bonded II-VI lattices is impressively high, in equilibrium crystal growth methods the values of x are

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Chalcogenide

Figure 1.7 (A) Band diagram of active core design (ZnCdSe/ZnCdMgSe heterostructures: 42/10/34/10/26/10/21/12/15/12/14/12 in angstroms of one period of the active-injector region) used in device. (B) Current dependence on transverse magnetic polarized electroluminescence from device at 80 K. After T.A. Garcia, J. De Jesus, A.P. Ravikumar, C.F. Gmachl, M.C. Tamargo, II
VI quantum cascade emitters in the 6
8 μm range, Phys. Status Solidi (b) 253 (2016) 1494
1497.

still limited by the composition of DMS II1-xMnxVI alloys. Epitaxy (which is a non-equilibrium growth process) has changed that dramatically, resulting in our ability to form these alloys to much larger concentrations, in many cases allowing one to reach values of x 5 1.0! [21] This has opened the way to study magnetism of magnetically concentrated II1-xMnxVI alloys in the form of tetrahedrally-bonded zinc blende structure of the parent II-VI crystal, including the compounds MnTe and MnSe in zinc blende form [58,59]. This in turn has led to extended studies of magnetic order in these concentrated systems. Extensive neutron diffraction investigations carried out on these new systems have yielded a wealth of new information of antiferromagnetic properties of these systems. Specifically, II1-xMnxVI at very high values of x (including MnTe and MnSe) have revealed that magnetic ordering in these systems occurs in the form of so-called “Type-2” antiferromagnetic order, confirming neutron scattering results observed earlier at lower concentrations. Additionally, however, when films of these materials are grown by epitaxy, they internal experience strain, that can be regulated by the choice of substrates on which the II1-xMnxVI alloy (or Mn-VI compound) is deposited, which can also have a profound effect on magnetic order. It has been discovered that the choice of strain then results in new forms of ordering, including a helical incommensurate antiferromagnetic ordering that can be “tuned” by such strain [60]. Studies of superlattices of formed by layers of such magnetically-concentrated systems separated by non-magnetic “spacer” layers have further enabled studies of interlayer coupling. It has been shown, for example, that antiferromagnetic coupling between layers arises from appropriately doping of the spacer layers, thus leading to the conclusion that the inter-layer coupling is caused by the effects of free carrier.

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1.3.1.3 Epitaxial II1-xMnxVI magnetic semiconductor quantum structures The ability to form multilayers with atomic precision by modern epitaxial techniques has resulted in wealth of new research, particularly involving II1-xMnxVI DMSs, because of the giant Zeeman splittings which has already been discussed above. The Zeeman effect occurring in these materials has enabled the experimenter to tune band energies on the scale of tens of meV by an applied magnetic field, thus allowing on to vary and control depths of quantum wells, degree of confinement in quantum structures, band offsets, etc. Additionally, such tuning is different for different spins, thus making it possible to distinguish effects that occur for different spin polarization, especially in magneto-optical experiments [61]. Here we will briefly point to a few of the effects that have been made possible by this process in semiconductor quantum structures, referring the interested reader to the cited literature. Zeeman tuning and wave-function mapping: In quantum structures such as quantum wells and superlattices, comprised of adjacent DMS and non-DMS layers, the giant Zeeman effect in DMSs layers allows us to significantly change the band edge, and thus to magnetically control the band offset between adjacent layers of the quantum structure, automatically affecting the degree of confinement of the quantum well. This property has been very effectively used for mapping the localization of electrons and holes, providing us with valuable understanding of how wave functions are distributed in a specific quantum structure. We illustrate this by the results obtained by Lee et al. for a multiple quantum well structures [as seen in Fig. 1.8] [62,63]. Spin superlattices: Because the band offset can be tuned and is spin-selective, we can achieve a structure in which DMS layers can be so Zeeman-split that they act as confining quantum wells for carriers of one spin, while they act as barriers for carriers with the opposite spins. This has indeed been successfully achieved, forming in effect a superlattice with up- and down-spin states localized in alternating layers [Fig. 1.9] [64]. This approach also lends itself to other spin segregation phenomena, made possible by various strategies of DMS/non-DMS multilayer design. Determination of spin lifetimes: Of course the controllable localization of different spins in different regions automatically opens the possibility of determining the lifetimes of different spin states in a heterostructure, for example by time-resolved magneto-optical spectroscopy with circularly polarized light [65
68].

1.3.1.4 Tunable Dirac interface states in topological superlattices Quantum confinement in heterostructures of condensed matter also plays an interesting role regarding topological phases of matter, and thus the band engineering is one of promising approach for alternating band topology. Here, we use a topological superlattices made by TCI Pd0.75Sn0.25Se and Pd1-yEuySe as a model system to demonstrate tunable topological Dirac states at buried interfaces [69]. Topological interface states (TISs) are formed in epitaxial TCI superlattices consisting of multiple Pb12xSnxSe topological quantum wells (TQWs) separated by normal insulator (NI) Pb12yEuySe barriers [Fig. 1.10]. Using magnetoinfrared spectroscopy to probe the Landau-level (LL) structure of the quantum confined states formed at the buried interfaces, the behavior of the TIS is analyzed as a function of temperature.

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Chalcogenide

Figure 1.8 Wave functions for the three lowest states in a symmetric triple-quantum-well system. After S. Lee, M. Dobrowolska, J. Furdyna, L. Ram-Mohan, Wave-function mapping in multiple quantum wells using diluted magnetic semiconductors, Phys. Rev. B 59 (1999) 10302.

1.3.2 2D “van der Waals” chalcogenides 1.3.2.1 Topological insulators In recent years, the research on topological insulators (TIs) has attracted much attention due to their unique quantum phenomena [70
72]. The 3D TIs are characterized by insulating bulk and spin-momentum locked metallic surface states, often referred to as helical spin states. Such a unique electronic structure provides an ideal platform for fabrication of spintronic devices. The theoretical breakthrough in the previous decade led to the discovery of a large variety of topological materials which include 2D and 3D TIs characterized by Z2 topological numbers, as well as Chern insulators, Dirac semimetals, Weyl semimetals, and many others [73]. The TI chalcogenides [74], including Bi2Se3, Bi2Te3, and Sb2Te3, are currently the most widely researched TIs featuring a single Dirac cone on the surface. The spin-orbit coupled massless Dirac fermions give rise to numerous exotic phenomena with fruitful theoretical and experimental progresses accomplished in this field: such as weak antilocalization effect [75], Shubnikov-de Haas oscillations [76,77], the quantum

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Figure 1.9 A schematic diagram of the band structure of a magnetic-field-induced spin superlattice. In the Zn1-xMnxSe layers in the lower picture (B 6¼ 0), the dotted lines and the solid lines represent, respectively, spin-down and spin-up states of electrons and heavy holes. Zeeman splitting in ZnSe layers is negligible. The arrows show where particular spin states are localized. After N. Dai, H. Luo, F. Zhang, N. Samarth, M. Dobrowolska, J. Furdyna, Spin superlattice formation in ZnSe/Zn12xMnxSe multilayers, Phys. Rev. Lett. 67 (1991) 3824.

Figure 1.10 (A) Identical Dirac cones on the (111) surfaces of bulk Pb0.75Sn0.25Se. (B) Topological quantum-well Pb0.75Sn0.25Se/Pb12yEuySe superlattices of alternating TCI/NI layers studied. The band alignment and band profiles are shown in (C). L6 6 denote the conduction- and valence-band extrema at the L points in the rocksalt Brillouin zone of Pb0.75Sn0.25Se and Pb12yEuySe. εA ( , 0 at 4.2 K) is the L6 6 energy separation in the bandinverted Pb0.75Sn0.25Se quantum wells (A) and εB ( . 0) is that of the normal insulator Pb12yEuySe barriers (B). V is the conduction band offset and d is the well thickness. (D) Sketch of the evolution of the wave-function probability density and Dirac cones of the topological interface state as a function of temperature across the topological phase transition T , illustrating the tunability of the penetration depth and hybridization gap of the TIS. After G. Krizman, B. Assaf, T. Phuphachong, G. Bauer, G. Springholz, G. Bastard, et al., Tunable Dirac interface states in topological superlattices, Phys. Rev. B 98 (2018) 075303.

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anomalous Hall effect (QAHE) [78,79], spin
orbit torque [80], topological magnetoelectric effect [81], Majorana zero mode [82], magnetic monopole [83] and more.

1.3.2.2 Topological insulators and magnetism QAHE in topological insulators is especially interesting in that magnetic moments in the system combine with spin-orbit coupling to form a topologically nontrivial electronic structure that exhibits quantized Hall effect without an external magnetic field [84]. Because of its topological character, the QAHE can exist in topological insulators even when carrier mobilities are rather low (,1000 cm2/Vs). QAHE was recently discovered in MBE-grown topological insulator film Bi1-xSbxTe3 doped with Cr [79], as illustrated in Fig. 1.11. Numerous attempts have been made to introduce magnetic ions into the lattice of a topological insulator such as Bi2Se3 with an eye of controlling the role of inversion symmetry in these systems, which has resulted in several novel effects. For example, the time-reversal symmetry breaking was observed in the form of ferromagnetism arising from the interaction between magnetic impurities and the Dirac fermions in single crystals of Mn-doped Bi2Te3-ySey [85]. Ferromagnetism with well-defined ferromagnetic hysteresis in the magnetization and in the magnetoresistance has also been discovered in MBE-grown Cr-doped Bi2Se3, where films with good crystalline quality were obtained up to a Cr content of B5%, with a Curie temperature Tc  20 K in specimens with Cr content of 5.2% Cr [86]. Second-order ferromagnetic transition has also been reported in single crystals of Bi2-xMnxTe3, with Tc  9-12 K [87]. In our own work, we discovered that introducing a beam of elemental Mn during MBE growth of Bi2Se3 does not cause Mn to randomly enter the Bi2Se3 lattice, but results in the formation of layers of Bi2MnSe4 that intersperse between layers of pure Bi2Se3, forming a self-organized Bi2Se3/Bi2MnSe4 multilayer heterostructure, thus constituting an example of multicomponent quantum matter at the crossroads of magnetism, topological insulators, and two-dimensional materials [88].

Figure 1.11 The QAH effect of Cr0.15(Bi0.1Sb0.9)1.85Te3 film measured at 30 mK. Magnetic field dependence of ρyx and ρxx are shown for different gate voltage Vg. After C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, et al., Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator, Science 340 (2013) 167
170.

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1.3.2.3 2D dichalcogenide systems Until the day when graphene was exfoliated [89], thin film grown in a vacuum is the main route to fabricating optoelectronic devices ever since the pioneering discoveries of semiconductor heterostructures. Since then, researchers all over the world rapidly start to explore 2D layered compounds in order to create atomically thin optoelectronic device. Most of 2D layered compounds studied belong to the family of the transition metal dichalcogenides (TMDCs), forming TX2 is composed of the transition metals (groups 3
12 of the periodic table) “T” like titanium or molybdenum and chalcogenides (group 16) “X” like selenium, sulfur or tellurium [90]. Their electronic properties range from metals like VSe2 to insulators like HfS2. Current researches are often carried out at the 2D limit, in order to reduce the Coulomb screening and to achieve the unprecedented properties that are not possible in their 3D bulk forms.

1.3.2.4 Two-dimensional ferromagnetic materials Although the famous Mermin-Wagner theorem points out: Long-range magnetic order of isotropic Heisenberg model does not exist at nonzero temperature in dimensions d # 2 [91], in 2017, 2D ferromagnetic (FM) monolayer CrI3 and Cr2Ge2Te6 were successfully fabricated in experiments [92,93], which has attracted extensive attention both for fundamental research and for applied technology [94,95]. Since then, new 2D FM materials have been constantly predicted theoretically and synthesized experimentally, such as Fe3GeTe2 [96], VSe2 [97], MnSe2 [98], Fe3P [99], VI3 [100], and so on. The appearance of 2D FM materials provides a good platform for people to study physical properties of magnetic materials under 2D limit. For example, as shown in Fig. 1.12, the itinerant nature of ferromagnetism in Fe3GeTe2 allows more effective tuning of Curie temperatures and the hysteresis loop in the 2D magnet by controlling the carrier concentrations within via an ionic liquid gating [96]. Furthermore, it is well known that the spin-orbital toque (SOT) devices show much-improved energy efficiency as compared to spin transfer torque (STT) technique, and thus are of great importance for magnetic memory applications. A recent report shows that the vdW magnets can serve as a new material-base for the SOT multi-layer heterostructures [101]. Few-layered Fe3GeTe2 was chosen to be interfacially coupled to a metallic Pt thin film, thus giving rise to a non-magnetic/ vdW-magnetic interface. With the help of a small in-plane magnetic field, the outof-plane magnetization of Fe3GeTe2 can be switched at a critical current, as shown in Ref. [102].

1.3.3 Interface phenomena in chalcogenide structures 1.3.3.1 2D electron gas in chalcogenide multilayers Semiconductor interfaces are interesting because of their ability to form 2D electron gases (2DEGs), in which charge carriers behave differently than they do in the bulk. Of special interest for the topic of this review, in the presence of a magnetic

16

Chalcogenide

Figure 1.12 Ferromagnetism in an atomically thin Fe3GeTe2 flake modulated by an ionic gate. A, Conductance as a function of gate voltage Vg measured in a trilayer Fe3GeTe2 device. Data were obtained at T = 330 K. The inset shows a schematic of the Fe3GeTe2 device structure and measurement setup. S and D label the source and drain electrodes, respectively, and V1, V2, V3 and V4 label the voltage probes. The solid electrolyte (LiClO4 dissolved in polyethylene oxide matrix) covers both the Fe3GeTe2 flake and the side gate. B, C, Rxy as a function of external magnetic field recorded at representative gate voltages, obtained at T = 10 K (B) and T = 240 K (C). D, Phase diagram of the trilayer Fe3GeTe2 sample as the gate voltage and temperature are varied.E, Coercive field as a function of the gate voltage. Data were obtained at T = 10 K. After Y. Deng, Y. Yu, Y. Song, J. Zhang, N.Z. Wang, Z. Sun, et al., Gate-tunable roomtemperature ferromagnetism in two-dimensional Fe3GeTe2, Nature 563 (2018) 94
99.

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Figure 1.13 Left: Longitudinal (Rxx) and transversal (Rxy) resistance as a function of magnetic field at 50 mK of n-type modulation-doped CdMnTe quantum well structure. Right: Low magnetic field region, clearly showing the beating pattern and the nodes of the Rxx amplitude oscillations. Inset: zoom in the vicinity of Node2. After F. Teran, M. Potemski, D. Maude, Z. Wilamowski, A. Hassan, D. Plantier, et al., Coupling of Mn21 spins with a 2DEG in quantum Hall regime, Phys. E: Low-Dimens. Syst. Nanostruct. 17 (2003) 335
341.

field the Landau quantization of electronic levels in the 2DEG results in the quantum Hall effect (QHE), in which Hall conductance is quantized at values of e2/h. After its first discovery in a silicon metal-oxide-semiconductor field-effect transistor [103], QHE has been observed in a wide range of 2D electron systems, including semiconductor heterostructures [104], graphene [105], and topological insulators [106]. The physics of QHE has expanded into the fractional QHE (FQHE), in which the Hall conductance is quantized fractional values of e2/h [107]. An interesting question in this context is how the phenomenon changes when the Zeeman energy, which induces spin polarization of the carriers, becomes comparable to or larger than the Landau levels splitting. Group-II-based chalcogenides are very suitable for addressing this issue, since (1) they can form high-quality 2D electron structures [108
110], and (2) they provide the opportunity to incorporate magnetic elements which, as discussed earlier in this review, lead to giant Zeeman splittings. This is beautifully illustrated in Fig. 1.13 which shows the QHE observed in the CdMgTe/CdMnTe/CdMgTe QW system, in which CdMnTe is a well-known DMS characterized by giant Zeeman splittings of its energy levels. The data clearly shows the quantization of Hall resistance that is normally observed in 2D electron systems, but now modified by the enormous Zeeman splitting due to the sp-d exchange process. Note that a distinct feature of 2D systems involving DMSs is the beating in resistance shown in the inset of the Fig. 1.13. Such low field beating pattern is caused by the relative shift of spin-up and spin-down Landau levels arising from the large value of the Zeeman splitting. This wide range of tunability of spinsplitting in magnetic chalcogenides is unique, and can provide a means for manipulating and engineering Quantum Hall states in 2D electron gas systems.

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Chalcogenide

1.3.3.2 Magnetic proximity effects at interfaces Physics at interfaces also offers an important option by exploiting proximity effects, whereby materials properties are modified by the effect of one material on the material adjacent to it. In the case of magnetic properties, interfacial exchange coupling can lead to novel magnetic effects when two diverse materials are interfaced with one another. As an example, topologically enhanced interface magnetism was found by coupling a ferromagnetic insulator (EuS) to a topological insulator (Bi2Se3) in a bilayer system [111]. Importantly, this interfacial ferromagnetism persists up to room temperature, even though EuS orders ferromagnetically only below about 17 K. A possible explanation of this dramatic effect is that it is due to the large spin-orbit interaction and spin momentum locking at the topological insulator surface (Fig. 1.14). As another example, enhancement of magnetic ordering has also been reported in Cr-doped Bi2Se3 by the proximity of a 50-nm layer of a ferromagnetic YIG insulator (Y3Fe5O12), leading to a Curie temperature of 50 K in this combination of a magnetically-dope topological insulator and an adjacent ferromagnet [112]. Proximity-induced long-range ferromagnetism at ambient temperature has also been observed in Bi2-xMnxTe3 by effect of an adjacent Fe overlayer [113]. As a related phenomenon, we mention in passing the possibility of proximityinduced spin-splitting effects in monolayer transition metal dichalcogenides, which constitute the physical basis of field of “valleytronics”, that is rapidly gaining interest. By creating van der Waals heterostructures formed by an ultrathin ferromagnetic semiconductor (e.g., CrI3) and a monolayer of, e.g., WSe2, an unprecedented control of the spin and valley pseudospin has been observed, thus revealing a new platform for the study

Figure 1.14 Topologically-enhanced interface magnetism in Bi2Se3/EuS bilayers. After F. Katmis, V. Lauter, F.S. Nogueira, B.A. Assaf, M.E. Jamer, P. Wei, et al., A hightemperature ferromagnetic topological insulating phase by proximity coupling, Nature 533 (2016) 513
516.

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of the dynamics of layered magnets [114]. For further details in this and related areas we refer the interested reader to the comprehensive review by Schaibley et al. [115].

1.4

Chalcogenides beyond 2D

1.4.1 One-dimensional and quasi-one-dimensional chalcogenides In this section we briefly review dimensions beyond 2D, in which the physical properties of the parent chalcogenide alloys are further modified by the dimensionality of the structure. The clearest forms of lower dimensionality are group-II chalcogenide nanowires which can host magnetic ions (as in the case of II1-xMnxVI alloys already discussed), and thus can exhibit the interesting effects characterizing that materials family, but modified by their lower dimensionality. An excellent example of this is the core/shell Zn1-xMnxTe/Zn1-xMgxTe nanowires [116], exhibiting rather spectacular Zeeman splitting of their characteristic 1D band structure states (Fig. 1.15). A very interesting and novel structure has also been achieved by colloidal chemical methods in the form of “nanoribbons” [117], in effect lying between 2D and 1D geometries. This system, illustrated in Fig. 1.16, exhibits the well-known II-Mn-

Figure 1.15 Giant spin splitting in optically active ZnMnTe/ZnMgTe core/shell nanowires. After P. Wojnar, E. Janik, L.T. Baczewski, S. Kret, E. Dynowska, T. Wojciechowski, et al., Giant spin splitting in optically active ZnMnTe/ZnMgTe core/shell nanowires, Nano Lett. 12 (2012) 3404
3409.

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Chalcogenide

Figure 1.16 Giant Zeeman splitting in nucleation-controlled doped CdSe:Mn21 quantum nanoribbons. After J.H. Yu, X. Liu, K.E. Kweon, J. Joo, J. Park, K.-T. Ko, et al., Giant Zeeman splitting in nucleation-controlled doped CdSe:Mn21 quantum nanoribbons, Nat. Mater. 9 (2010) 47
53.

VI characteristics of giant Zeeman effects, but of course modified by the novel ribbon geometry. Finally, some layered chalcogenides (e.g., TaFe11xTe3) exhibit magnetic quasi1D chain-like configurations within their lattices [118]. In TaFe11xTe3 the Fe ions form two-stranded zigzag “ladders” along one crystal axis (the b axis), thus representing an intriguing quasi-1D magnetic system, in which the spins of adjacent “ladders” show an antiferromagnetic order with a Ne´el temperature of about 180 K (Fig. 1.17).

1.4.2 Zero-dimensional chalcogenide structures Solid state low dimensional structures have received a great deal of attention during the past decades owing to the possibility of improving storage capacity and increasing the speed of information processing in electronic devices. Among highly promising systems in this context are zero-dimensional spin structures, which have been proposed as building blocks for spintronic devices [119
121]. Highly promising candidates for this purpose are quantum dot structures consisting of II-VI-based alloys, which can easily host magnetic atoms such as Mn and Fe.

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Figure 1.17 Magnetic structure of quasi-one-dimensional antiferromagnetic TaFe11yTe3. (A) monoclinic crystal structure and (B) spin structure, and (C) detailed view of the zigzag ladders of TaFe11yTe3. After X. Ke, B. Qian, H. Cao, J. Hu, G.C. Wang, Z.Q. Mao, Magnetic structure of quasi-onedimensional antiferromagnetic TaFe11yTe3, Phys. Rev. B 85 (2012) 214404.

Magnetic QD systems, such as CdMnSe/ZnSe, CdSe/ZnMnSe, and CdMnTe/ ZnTe, were indeed successfully fabricated by both self-assembly processes and by colloidal chemical methods. Fig. 1.18 shows typical photoluminescence obtained from a self-assembled CdSe/ZnMnSe QD system, where three QDs are viewed by a 175 nm aperture, showing three distinct peaks photoluminescence (PL) peaks. Interestingly, the PL peaks shift to higher energies as the temperature increases, opposite to the behavior of the band gap. This phenomenon is caused by the formation of exciton magnetic polarons, in which spins of the Mn ions couple to the exciton spin in the QD, aligning themselves in one direction at low temperature in order to reduce the exciton energy [122]. Such polarons will, however, be dissociated as the temperature increases. This behavior is shown schematically in the inset of Fig. 1.18. Dynamic properties of such magnetic polarons in CdSe/ZnMnSe QDs have been further investigated by time resolve photoluminescence experiments [123] and the study has been even extended to a single QD in Cd12xMnxTe/ZnTe system [124]. Another noticeable fact is that the PL of the single QD is broadened relative to that of non-magnetic QD systems, such as CdSe/ZnSe and CdTe/ZnTe systems [122,125,126]. This is due to the presence of many magnetic ions even in a single

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Chalcogenide

Figure 1.18 PL spectra of CdSe/ZnMnSe quantum dots recorded with an aperture of 175 nm diameter are shown for different temperatures. After H. Scho¨mig, M.K. Welsch, G. Bacher, A. Forchel, S. Zaitsev, A.A. Maksimov, et al., Photoluminescence spectroscopy on single CdSe quantum dots in a semimagnetic ZnMnSe matrix, Phys. E: Low-Dimens. Syst. Nanostruct. 13 (2002) 512
515.

QD structure, which cause magnetic fluctuations in the QD [127,128]. In order to eliminate such magnetic fluctuation, it would be desirable to have only a single magnetic ion in a QD. This has been achieved in CdTe/ZnMnTe QD system via carefully controlled diffusion process of Mn ions during the QD growth [129,130]. This work demonstrated the ability to detect and manipulate individual spins in magnetic semiconductor QD structures, and thus paved the way for the future spintronic devices. Following these early DMS-based QD studies, there has been a great deal of activity in this area, including “bottom-up” colloidal DMS structures [130
133]. Unfortunately space limitations prevent us from doing justice to this work in the present review, but the interested reader will find relevant material in the references just listed.

1.5

Concluding remarks

As we illustrated by the examples discussed in this review, magnetic properties of chalcogenides span an unusually broad range behaviors. Because of space

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limitations, we have not been able to include in this paper the rich spectrum of novel phenomena displayed by rare-earth chalcogenides; [134] by Fe-based chalcogenide superconductors, that also exhibit interesting antiferromagnetic properties; [6,7] by 2d chalcogenide Fe1/4TaS2 which shows very good ferromagnetic properties below its TC 5 160 K; [135] or by other chalcogenide materials, such as NbFeTe2 [136], FePS3 [137], NiPS3 [138], CrSiTe3 [139], CrGeTe3 [140], Cr2Ge2Te6 [141]. Considering how this field has evolved in the last few years, it is a fair guess that research on chalcogenide materials will continue at its uncommonly rapid rate, due to the multi-faceted flexibility of chalcogenide bonds, and the variety of structures that emerge from this.

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Thermoelectric applications of chalcogenides

2

Han Meng1, Meng An2, Tengfei Luo3 and Nuo Yang1 1 State Key Laboratory of Coal Combustion and Nano Interface Center for Energy, School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan, P.R. China, 2College of Mechanical and Electrical Engineering, Shaanxi University of Science and Technology, Xi’an, China, 3Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, United States

2.1

Introduction

The metal chalcogenides, a series of materials composed of both metal and chalcogen elements (S, Se, Te), have been attracting significant attention in thermoelectric devices [1
4]. Compared with oxides, metal-chalcogenide-based thermoelectric materials possess a higher power factor due to the less covalent nature of their bonds, arising from low electronegativity. Their heavy atomic weight compared to some other thermoelectric materials is of benefit for reducing thermal conductivity. In addition, the chalcogenides are easy to form into different kinds of structures, which offer a good platform for investigating and improving thermoelectric performance. Moreover, the chalcogenides are easily doped into n-type (by halides) or p-type (by pnictides) materials, which is important for the fabrication of thermoelectric generators. It has been proved by many reports that metal-chalcogenide-based thermoelectric materials possess excellent performance. For example, hitherto, the highest ZT value (ZT 5 3.6 at 580 K) was obtained in PbSe0.98Te0.02/PbTe quantumdot superlattices grown by the molecular beam epitaxy approach [5], a ZT of 2.6 was obtained at 923 K for a single crystal of SnSe [6], and Na doped full-scale-structured PbTe exhibited a maximum ZT of 2.2 at 915 K [7]. The superiority of metal-chalcogenide-based thermoelectric materials also lies in their low cost, both for material fabrication and for operation. Yadav et al. calculated the efficiency ratios of different kinds of materials, using their ZT values divided by their fabrication cost [8]. The efficiency ratios of metal chalcogenides (typically Bi2Te3 and PbTe, which are marked by red circles) are slightly lower than for oxides (e.g., ZnO) and Zn4Sb3 (marked by green circle) despite the relatively high price of Te element. ZnO, NaCoO2, and Zn4Sb3 are unstable, however, at high temperature or in humid air [8]. LeBlanc et al. evaluated the operating costs for TERs made from different thermoelectric materials under a constant temperature difference [9]. They found that metal-chalcogenide-based materials possess both the highest ZT values and the lowest operating costs. Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00002-6 © 2020 Elsevier Ltd. All rights reserved.

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In this chapter, several fundamental concepts about thermoelectric effect are briefly introduced first. After that, all kinds of strategies to improve thermoelectric performance of chalcogenides are categorized and introduced according to the recent literatures, and the latest developments and mechanism behind their high performances are respectively discussed according to the classification. Finally, a brief outlook is given on the basis of the state-of-art chalcogenides-based thermoelectric materials.

2.1.1 Thermoelectric effect The thermoelectric effect is the direct conversion of heat between electric energy via a thermocouple. A thermoelectric device creates voltage when there is temperature difference between each side. Conversely, when a voltage is applied to it, heat is transferred from one side to the other, creating a temperature difference. The term “thermoelectric effect” encompasses three separately identified effects: the Seebeck effect, Peltier effect, and Thomson effect. Seebeck effect, the conversion of heat directly into electricity at the junction of different types of wire due to the development of an electromotive force (EMF) across a material in response to the imposition of a temperature difference. As shown in Fig. 2.1, it was observed that a compass needle would be deflected by a closed loop formed by two different metals joined in two places, with a temperature difference between the joints. This was because the electron energy levels in each metal shifted differently and a potential difference between the junctions created an electrical current and therefore a magnetic field around the wires. Peltier effect is the presence of heating or cooling at an electrified junction of two different conductors. When a current is made to flow through a junction between two conductors, A and B, heat may be generated or removed at the junction. Thomson effect describes the heating or cooling of a current-carrying conductor with a temperature gradient. In different materials, the Seebeck coefficient is not constant in temperature, and so a spatial gradient in temperature can result in a gradient in the Seebeck coefficient. If a current is driven through this gradient, then a continuous version of the Peltier effect will occur.

2.1.2 Thermoelectric efficiency Thermoelectric materials can realize the direct conversion between thermal energy and electrical energy. As shown in Fig. 2.2, a thermoelectric circuit is comprised of

Figure 2.1 Schematic basic thermocouple.

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Figure 2.2 A thermoelectric circuit configured as a thermoelectric generator.

materials of different Seebeck coefficients (p-doped and n-doped semiconductors). The usefulness of a material in thermoelectric systems depends on its Seebeck coefficient, electrical conductivity and thermal conductivity under changing temperatures. Seebeck coefficient (also known as thermopower), a fundamental electronic transport property that generally varies as a function of temperature and depends strongly on the composition of material, is defined as S52

EEMF ; rT

where EEMF is electromotive force and rT is temperature gradient. In particular, Seebeck coefficient measures the entropy transported with a charge carrier as it moves, divided by the carrier’s charge. As such, the Seebeck coefficient is affected by charge carriers’ interactions with one another, with phonons and with the local magnetic moments of magnetic solids. Thermoelectric efficiency, the ability of a given material to efficiently produce thermoelectric power, is evaluated by the dimensionless figure of merit, which is given by ZT 5

σS2 T ; κ

where S is the Seebeck coefficient, κ is the thermal conductivity, σ is the electrical conductivity, and T is the temperature. There is a factor to determine the usefulness of a material in thermoelectric system. Advances in nanotechnology have recently boosted the ZT values of conventional thermoelectric materials to a new record, which not only demonstrates the nanoscale effects towards the improvement of thermoelectric performance of conventional materials, but also has led to a burst of research on thermoelectric nanomaterials. In the following section, all kinds of strategies to improve thermoelectric

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performance of chalcogenides are categorized based on the recent literatures, and the latest developments and mechanism behind their high performances are respectively discussed according to the classification.

2.2

Nanostructure engineering

Nanostructure engineering is an effective method to enhance thermoelectric performance due to the decreased thermal conductivity resulting from the presence of abundant grain boundaries. There are several effective routes in the nanostructure engineering of thermoelectric materials.

2.2.1 Bottom-up and top-down fabrication Bottom-up fabrication is widely used to synthesize the nanostructured thermoelectric materials with good thermoelectric property. Mehta et al. reported a microwave synthesis of nanostructured Bi(Sb)2Te(Se)3 with a ZT of 1.1, which is the highest reported value for an n-type BiSeTe material [10]. Furthermore, the product could be processed in air, which offers great convenience for further applications of these nanostructures. Liu et al. fabricated a net-like nanostructured Bi2S3 sample via a simple hydrothermal method, and it yielded a ZT of 0.5 at 623 K, which is the highest value for a pure Bi2S3 sample [11]. Wet chemical method, a large-scale fabrication method with both low cost and high controllability of the nanostructures, has been used to fabricate nanostructured thermoelectric materials. Flower-like n-type Bi2Te3 nanostructures were synthesized by a wet chemical method and sintered into a bulk, which exhibited a ZT of 1.16 at 420 K [12]. An alternative is the development of surfactant-free chemical synthesis routes. Han et al. developed a low-cost wet chemical method to synthesize different kinds of surfactant-free binary and ternary metal chalcogenide nanostructures [13,14]. The whole synthesis is conducted under ambient conditions, no surfactants are used, and it is easy to scale up. The thermoelectric performance of consolidated nanobulks from the obtained nanostructures is comparable to those of the corresponding bulk analogs. Another important method to enhance the thermoelectric performance of nanostructures obtained by wet chemical methods is the surface engineering of nanocrystals by using other inorganic materials to replace the capping ligands and absorbed organic solvents [15
18]. As shown in Fig. 2.3, by introducing precise amounts of doping in nanomaterials produced from the bottom-up assembly of colloidal nanoparticles, a significant enhancement of the thermoelectric figure of merit is achieved [16]. Talapin et al. simply replaced non-conductive organics with inorganic materials and proposed the concept of “nanocrystal glue” or “semiconductor solder” for mesoscale particles of thermoelectric materials [15,17,18]. Top-down process is another route to fabricate nanobulk materials with good thermoelectric performance [19
30]. Severe plastic deformation (SPD) is one of

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Figure 2.3 (A) Electrical conductivity, (B) Seebeck coefficient, (C) thermal conductivity and (D) thermoelectric figure of merit of nanocrystalline PbTexSeyS1-x-y obtained from the bottom-up assembly of PbTexSe1-x@PbS core-shell NPs with and without 2.3% HCl addition [16]. Copyright 2015, American Chemical Society.

the most important method, which can achieve materials with ultra-fine grains in the submicrometer or nanometer range. Several different mechanisms to enhance the thermoelectric properties involved in SPS, including (i) breaking of crystals down to the nanoscale [23,26]; (ii) enhanced formation of lattice defects, vacancies, and especially, dislocations [19,23]; (iii) and re-orientation of the crystals [23,24].

2.2.2 Consolidation method Consolidation method, e.g., the hot pressing or spark plasma sintering (SPS) techniques, which significantly influence the thermoelectric performance of the nanobulks. For example, although samples were prepared from same batch of Bi0.5Sb1.5Te3 nanopowder, the electrical conductivity of the hot pressed nanobulk decreased with increasing temperature, while in the case of the sample sintered by the SPS technique, an opposite trend was observed [31]. This difference is ascribed to the different features of the two techniques. During the SPS process, the local high temperature from the electric spark causes evaporation of elements and the formation of more defects, which significantly modifies the thermoelectric properties

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of the final product. On the contrary, the temperature during hot pressing is more controllable and uniform [31]. According to theoretical calculations, the ZT of CuGaTe2 could reach 1.69 at 950 K, with an optimized hole concentration of 3.7 3 1019 cm23 [32]. The maximum ZT of 1.4 was first achieved in polycrystalline CuGaTe2 by Plirdpring et al [33]. and later by Massaya et al [34]. via ball milling and hot pressing. The reason for such a high ZT value is mainly due to a sudden decrease in the thermal conductivity in the high temperature range.

2.2.3 Introducing nanostructures Introducing nanostructures into the matrix of bulk materials can influent the thermoelectric performance. As shown in Fig. 2.4, a new theory names energy filter effect (EFE) has been developed based on the introduction of nanostructures into the matrix. Through embedding nanostructured impurities with a slightly larger band gap than the matrix, a moderate energy barrier is introduced at the matriximpurity boundaries. Thus, only the charge carriers with higher energy and larger effective mass could pass through these barriers, leading to an increase in the Seebeck coefficient. In addition, the large surface area of nanograins also provides a high energy barrier caused by surface adsorbents, which could decrease electrical conductivity due to the EFE and scattering at grain boundaries. Therefore, there is a tradeoff between the benefits from EFE and the detrimental effects from the nanostructure. A typical example is a composite made of Cu3SbSe4 with p-type Bi0.5Sb1.5Te3 [35], in which Cu3SbSe4 particles not only played the role of phonon scattering centers, but also significantly increased the Seebeck coefficient due to EFE. Parallel to theory, several works that introduced different nanostructures and their effects will be reported in details. Liu et al. tried to enhance the thermoelectric performance of n- and p-type Bi2Te3 by introducing nanostructured SiC [36]. The ZT

Figure 2.4 Schematic diagram of energy filtering effect.

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of p-type Bi0.5Sb1.5Te3 was successfully enhanced from 0.88 to 0.97 at 323 K, but the ZT of n-type Bi2Se0.3Te2.7 was decreased after nano-SiC was introduced into it. This is because SiC is a p-type semiconductor, the mobility of electrons in the ntype matrix was strongly decreased by the SiC particles. Besides SiC nanoparticles, many kinds of other nanostructures have been investigated [37
40].

2.2.4 Introducing nanoprecipitates Endotaxial nanoprecipitates can be easily introduced by formation of solid solution to improve ZT value. All kinds of precipitates can significantly reduce the thermal conductivity by different mechanisms and different effects. The coherent precipitates act as point defects and scatter short wavelength phonons due to their slight mismatch with the matrix. They also have a slight effect on the electrical conductivity. The incoherent nanoinclusions possess a large mismatch with the matrix, they selectively scatter mid- to long-wavelength phonons. The maximum ZT of 2.3 was achieved in 3% Na doped (PbTe)0.8(PbS)0.2 due to the extremely low thermal conductivity arising from the various boundaries that ranged from coherent to incoherent, as well as the carrier concentration modulation [41,42]. The pseudo-binary PbSe-PbS [16,43,44] and PbTe-PbS [41,45,46] systems, in which nanostructures generated from nucleation and growth processes or spinodal decomposition did occur and could be used to account for the low expected thermal conductivity, have been studied and developed to a state of remarkably high performance. Besides the pseudo-binary system, the thermoelectric performance of ternary PbTeSeS was also investigated and proved to show high ZT over a broad temperature range [42,43,47]. Endotaxial nanoprecipitates can also be formed via self-formed inhomogeneities due to spinodal decomposition. Spinodal decomposition is an atomic level mechanism in which a metastable single phase generates a bi-phase nanoscale structure by phase segregation. Instead of resulting in a nucleation growth process, coherent nanodomains are uniformly created throughout the whole matrix via a diffusionsegregation procedure. Zhang et al. synthesized composites in situ with Ag2Te or Sb2Te3 embedded in the AgSbTe2 matrix, by utilizing the change in solid solubility near the single-phase region boundaries [48]. The nanodomain boundaries led to a further decrease in the thermal conductivity and a ZT of 1.53 was obtained in this material. Hsu et al. reported a high ZT value (2.20 at 800 K, n-type) for AgPb18SbTe20 alloys [49]. Li et al. synthesized AgPb21SbTe20 by combining mechanical alloying and the spark plasma sintering process [50]. With the advantages of the refined grains and nanostructures produced by repeating milling and SPS processes, simultaneous enhancement of the electrical conductivity and the Seebeck coefficient could be achieved, leading to a 50% increase in ZT value. Based on above two types of materials, several similar systems with spinodal decomposition have been developed, and they possess a similar crystal structure and exhibit spinodal decomposition at moderate temperatures and compositions, as described in a comprehensive review by the Kanatzidis group [51].

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In addition to nanoprecipitates, introducing other materials to form a composite is another important way to modify TE properties. For example, He et al. reported a new approach, displacement of the layered structure (PbSnS2) rather than nanostructures or a solid solution, to enhance phonon scattering in a PbTe based thermoelectric composite [52]. Extremely low lattice thermal conductivity (0.8 W/m  K) was obtained for the composite, which is almost 30% of that of bulk PbTe at room temperature. This is an excellent example of a bulk PbTe thermoelectric material without nanostructures that can still greatly enhance phonon scattering and provide very low thermal conductivity. Thermoelectric performance of CuGaTe2 was enhanced by introducing nanophase Cu2Se, which yielded a ZT of 1.2 at 834 K [53]. As we know, heat carrier phonons have a wide range of vibration frequencies, so carefully controlling the microstructure to form a full-scale scattering center is an effective way to minimize thermal conductivity to enhance ZT values [7,54,55]. As shown in Fig. 2.5, the ZT of 2.2 at 915 K was achieved via a combination of Na doping and nano-inclusions of SrTe in PbTe [7]. Pei et al. investigated the effective upper limit to the length-scale of microstructures where this effect becomes insignificant [56]. The highest ZT achieved in the Bi-S and Bi-Se systems was 0.8 at 773 K for the alloyed Bi2SeS2, which possesses extremely low thermal conductivity [57].

2.3

Defect engineering

Defect engineering is an effective way to modify the thermoelectric performance of. The ZT is reciprocal to the thermal conductivity, which is transferred via vibration of phonons. The heat-carrying phonons cover a broad spectrum of frequencies, and the lattice thermal conductivity can be expressed as a sum of the contributions from different frequencies. The point defects are highly efficient for scattering the high frequency phonons, and the grain boundaries, as well as nanoparticles, could be effective for scattering low frequency phonons, while dislocation arrays are effective for scattering phonons from low to high frequencies.

2.3.1 Normal doping In the most traditional way, doping with different elements could lead to enhancement of thermoelectric materials. The doping could result in defects in the matrix, change the carrier concentration, and decrease the thermal conductivity. Hitherto, the doping effects of many different elements, such as Fe, In, Cu, Ag, Cl, Ga, Sn, and Ag on Bi-Te based thermoelectric materials have been reported [58
66]. Br doped Bi2S3 yielded the maximum ZT of 0.72 at 773 K [63]. The improved thermoelectric properties can be attributed to the optimized carrier concentration and the lower thermal conductivity caused by Cu1 intercalation, Br substitution, and Cu nanoparticles. Cu2Ga4Te7 is a pseudo alloy of Cu2Te and Ga2Te3 with abundant

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Figure 2.5 (A) Maximum achievable ZT values for the respective length scales: the atomic scale (alloy scattering: red, Te; blue, Pb; green, dopant), the nanoscale (PbTe matrix, gray; SrTe nanocrystals, blue), and the mesoscale (grain-boundary scattering). (B) ZT as a function of temperature for an ingot of PbTe doped with 2 mol% Na (atomic scale), PbTe-SrTe(2 mol%) doped with 1 mol% Na (atomic plus nanoscale) and spark-plasmasintered PbTe-SrTe(4 mol%) doped with 2% Na (atomic plus nano plus mesoscale) [7]. Copyright 2012, Springer Nature.

ordered defects, the concentration and type of defects were optimized by doping with Zn to achieve a ZT of 0.47 at 770 K [67]. The maximum ZT values of 0.62 in Cu2Sn0.925In0.075Se2.1S0.9 [68], and 1.14 in doped Cu2Sn0.9In0.1Se3 [69] are achieved by via doping other elements to substitute Sn atoms, which contribute little to electrical conductivity. Doping is also an effective way to improve the thermoelectric properties of GeTe by reducing its hole concentration and/or introducing a secondary phase into the matrix to reduce its thermal conductivity, e.g., Mn, Yb doped GeTe [70
72]. Besides, normal doping effects were also observed in Sn doped p-type BiSbTe and n-type BiSeTe materials [61,62].

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2.3.2 Introducing point defect Although the effects of defects on thermoelectric performance have been investigated in terms of the doping effect, the important effect of point defects on thermoelectric performance has never been fully recognized. Hu et al. proposed defect engineering as an effective and important route to enhance the thermoelectric performance of the Bi-Te compounds [19]. They carefully controlled the concentrations and kinds of defects, and achieved the maximum ZT values of 1.3 and 1.2 for p-type Bi0.5Sb1.5Te3 and n-type Bi2Se0.3Te2.7, respectively. In 2015, Sang et al. reported a liquid-phase compaction method to create dislocation arrays between adjacent nanostructures to obtain obviously decreased thermal conductivity [54]. The maximum ZT of 1.86 at 323 K was reached for the liquid-phase compacted samples (Te-MS), and only 1.3 at 323 K was obtained for the solid-phase compacted sample (S-MS). The PbTe-PbSe system is also a high-performance thermoelectric system, its low lattice thermal conductivity has been explained on the basis of point defects created by the Te/Se mixed occupation in the rock-salt structure and by the presence of dopants [41,43,73]. As shown in Fig. 2.6, Korkosz et al. reported the pseudoternary 2% Na-doped (PbTe)0.86(PbSe)0.07(PbS)0.07 with low thermal conductivity deriving from alloy scattering and point defects, and a ZT values of 2.0 was achived at 800 K.

2.3.3 Introducing element deficiency The effects on thermoelectric performance of defects formed due to nonstoichiometric elemental ratio are also notable [74
76]. Typically, in the binary Cu2X system, it is easy to form Cu vacancies, and the composition changes from stoichiometric Cu2X to non-stoichiometric Cu22xX. The vacancies caused by Cu deficiency play several significant roles in the thermoelectric properties: (i) the vacancies as point defects directly tune the carrier concentration and conduction type of a PLEC [74,76]; (ii) the order-disorder phase transition temperature could also vary with x, because the vacancies may offer diffusion paths and accelerate the diffusion process [77]; (iii) the existence of these vacancies leads to a percolation effect on electrical conductivity, as charge carriers could also pass through them [75], which is very similar to the concept of the energy filtering effect. The vacancies can be regarded as a conductive medium with an energy barrier, and the conductivity of the vacancies could be described by the percolation law presented in the literature [75]. Thereby element deficiency is an efficient way to adjust thermoelectric performance of PLEC materials. As shown in Fig. 2.7, the thermoelectric performance of Sn2X and In2X can be further enhanced by the off-stoichiometric element ratio or doping effects, such as excess amount of Sn and deficiency of Se [78
86]. By reducing Ag content, thermoelectric properties of Ag12xGaTe2 was enhanced to be approximately two times than that of stoichiometric AgGaTe2 [87]. As an important route towards tuning electrical conductivity, deficiency of Bi is reported to be an effective approach for increasing the thermoelectric performance of the Bi-O-Se system [88]. The maximum ZT of 0.33 was achieved in

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Figure 2.6 TEM images (PbTe)0.86(PbSe)0.07(PbS)0.07. (A) Low magnification image shows homogeneous material. Inset shows the corresponding SAD. (B) High magnification image indicates the single-crystal lattice. (C) The existence of large precipitate in Z-contrast image and (D) the corresponding EDS spectra. The C, Cu, and O peaks in spectrum 2 are from the TEM grid underneath the sample [43]. Copyright 2014, American Chemical Society.

Cu0.97Fe1.03S2 at 700 K due to the increased lattice defects caused by element deficiency and the diminished carrier-magnetic-moment interactions at high temperature [89]. Qiu et al. reported the maximum ZT of 1.2 at 900 K for a composite of 0.8Cu8S4 1 0.2Cu5Fe2S4 [90].

2.3.4 Other approaches As mentioned above, defect engineering has been an important route to tuning the properties of thermoelectric materials [19,23]. It is a general method that can be

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Figure 2.7 Thermoelectric properties as a function of temperature for SnTe with different Sn content: (A) electrical conductivity; (B) Seebeck coefficient; (C) power factor; (D) total thermal conductivity; (E) lattice thermal conductivity; (F) ZT [80]. Copyright 2014, American Chemical Society.

used for other metal chalcogenides, as the defects can affect carrier concentrations and lower thermal conductivity. Besides, the defects engineering can be achieved in several different ways, such as by hydraulic pressure together with ion beams [91,92], bombardment by ion beams or high energy electromagnetic radiation including He21, Ar1, neutrons, γ-rays, X-rays, which are widely used for improving the thermoelectric performance of films, etc. [92,93]. Joonki et al. used He21 ion beam to induce native defects that dramatically enhanced the thermoelectric properties in Bi2Te3 film by simultaneously modifying all of them towards the desired direction [92]. This was enabled by the multiple functions of native defects, which act beneficially as electron donors, energy dependent charge scattering centers, and phonon blockers. Hydrogenation is another approach to improve the performance of thermoelectric materials. The introduction of hydrogen atoms onto the surface or into the lattice of the target materials can be performed in several ways, such as hydrogenation at high temperature in the presence of an H source; bombardment with a high energy H1 ion beam, direct hydrogenation using HCl, etc [91]. Besides the introduction of point defects, the injected electrons from the doped hydrogen atoms greatly influence the density of states (DOS), which leads to increased charge density across the Fermi surface to improve the electrical conductivity. Meanwhile, due to their very small ionic radius, the incorporation of protons can avoid large-scale structural distortion and maintain the lattice thermal conductivity. For example, in the case of VO2, after hydrogen incorporation, electrons from the hydrogen atoms were

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43

injected into the V-V chains, resulting in obviously enhanced electron-electron correlation effects. The hydrogenated VO2 with the highest hydrogen concentration had a ZT value of about 0.12 at 210 K, which was much higher than for the other two samples with less or no hydrogen incorporation [94].

2.4

Band structure engineering

The effect that could modify the band structure of the matrix, such as by causing band degeneracy or affecting the band gap, are summarized as a common route to enhance thermoelectric performance in band structure engineering. Resonant level doping, one of the most important strategies of band structure engineering, shows obvious advantages towards improving the thermoelectric performance, as it distorts the band structure and only increases the Seebeck coefficient without affecting the electrical conductivity. The formation of a resonant level depends critically on the amount of overlap between the impurity states and the band structure of the matrix, is very hard to realize and predict [45]. The distortion of the band must take place near the Fermi level (usually within 0.5 eV) after hybridizing with the impurity band. Too much overlap may distort the band too much and cause a shift in the Fermi level, while too little overlap may not distort the band. Only a few elements or specific materials have been identified as capable of resonant doping, such as Sn for Bi2Te3, Tl and Sb for PbTe, and In for SnTe [61,95
97]. Compared with cases where the same carrier concentration is obtained by doping with other impurities, Sn doping remarkably increased the Seebeck coefficient in Bi2Te3 single crystal due to resonant level doping, which is because Sn offers a resonant valence band state so that density of states (DOS) near the Fermi level was increased [61,99]. Thermoelectric performance of SnTe can be improved by adjusting the band gap via In doping and by modifying the band structure with Cd doping [80,95]. Reports from Kanatzidis’s group demonstrated that compared with single element doping, multi doping can lead to more efficient enhance on ZT of SnTe [80,100]. As shown in Fig. 2.8, Zhang et al. improved the thermoelectric performance of SnTe by rational doping of Gd and Ag together [98]. Doping Gd into SnTe effectively reduced the lattice thermal conductivity close to the theoretical minimum due to the introduction of nanoprecipitates. Further doping Ag into optimized Gd0.06Sn0.94Te significantly improved the Seebeck coefficient and power factor by tuning its composition to reduce the carrier density. It is very interesting that Ag doping led to the homogenous distribution of Gd in SnTe. The low thermal conductivity and large Seebeck coefficient in optimized sample (e.g., Ag0.11Gd0.06Sn0.94Te) resulted in the maximum ZT of B1.1 at 873 K. Resonant doping is also easy to achieve with different dopants in PbTe, such as Na, K, Cd, Sm, Sb, Ag, etc [46,56,97,101
105]. Tl was resonantly doped into p-type PbTe [96], then Sb was both theoretically and experimentally proved to be an effective resonant dopant for n-type PbTe [97,106]. Al was resonantly doped into PbSe, leading to a ZT of 1.3 at 850 K, which is close to the reported maximum value in the n-type Pb-X system

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Figure 2.8 Thermoelectric properties (electrical conductivity, Seebeck coefficient, total thermal conductivity, ZT) as a function of temperature for (A
D) GdxSn12xTe and (E
H) AgyGd0.06Sn0.94Te [98]. Copyright 2013, Royal Society of Chemistry.

[107,108]. It was found that Na is an effective dopant for the p-type Pb-X system. He et al. reported that Na has the highest solubility in PbS and the lowest solubility in PbTe and that doping with Na not only can lead to band degeneracy by tuning the carrier concentration, but also can introduce point defects and precipitates [109]. To enhance thermoelectric performance of CuInSe2, doping with Mn was used as an effective method, which modifies its band structure [110]. The thermoelectric performance of AgSbSe2 can be significantly improved by the convergence of the valence band valleys through proper carrier engineering by doping. It has been reported that a ZT of above 1 was achieved in AgSbSe2 when it was doped with Pb and Bi [111]. Besides tuning the carrier concentration to cause the band degeneracy, doping with elements such as K with a big size difference from the original Pb or Se atoms, may cause distortion of the original cubic lattice and lead to asymmetry of the Brillouin zone, which finally increases the Seebeck coefficient [101].

2.5

Crystal structure engineering

Chalcogens can react with metals to form different varieties of complex crystal structures, which offers a wealth of thermoelectric properties tuned through the crystal structure engineering.

2.5.1 Original complex structure The A-B-X systems, IB-IIIA-VIA, IB-IVA-VIA and IB-VA-VIA compounds, usually possess complex structures, which induce low thermal conductivity so that is good

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for thermoelectric performance. Cu2MSe3(M 5 Sn, Ge) with Cu-Se forms a complex charge carrying network, which has a great influence on the electrical conductivity, and the Ge or Sn atom acts like a filler atom, causing a reduction in lattice thermal conductivity. This structure leads to a low lattice thermal conductivity, moderate electrical conductivity, and thus a high ZT value. Cu3SbX4 (X 5 S, Se) compounds possess a tetragonal structure and have low thermal conductivity due to their complex structure. In addition to above, thallium-M-Te (M 5 Ag, Bi, Cr, Zr, etc.) has long been investigated as a family of good thermoelectric materials. It is found that the ZT of Tl9BiTe6 and Ag9TlTe5 reached 1.2 at 500 and 700 K, respectively [112,113]. Derived from A-B-X system, the quaternary Cu-C-B-X system also exists, in which “C” is another metal element from either the transition metal or the main groups, such as Ag, Zn, Fe, Hg, In, Ga, etc., while “B” is mainly Sn or Ge [114
120]. Due to their low thermal conductivity, originating from their extremely complex crystal structures, they have been widely investigated as thermoelectric materials. Although their wide band gap is detrimental for electrical conductivity, it can effectively suppress the bipolar effect at high temperature, which occurs very often in the narrow-bandgap materials. The bipolar effect is explained as follows. At high temperature, the main charge carriers, such as holes, in narrow-band-gap materials will hop from the valence band to the conduction band due to the strengthened random thermal motion and small band gap, which leads to compensation of carriers and changes in the DOS. Usually the bipolar effect leads to a decreased Seebeck coefficient and enhanced thermal conductivity. Due to the combination of these two advantages, high ZT values can be expected in the quaternary materials. Typically, ZT values of 0.95 and 0.90 are reported in Cu2ZnSn0.90In0.10Se4 and Cu2.1Zn0.9SnSe4, respectively [114,115].

2.5.2 Peierls distortion structure SnSe adopts a layered orthorhombic crystal structure at room temperature. Each layer is made up of two-atom-thick SnSe slabs. The two-atom-thick SnSe slabs are corrugated and create a zigzag accordion-like structure [6]. Similarly, the crystal structure of In4Se3 is formed by a layered structure, which are held together by van der Waals interactions [78]. In contrast to conventional Bi2Te3, the layered structure of SnSe and In4Se3 is a distorted quasi-one-dimensional chain-like structure. Along this chainlike structure, the charge transport is inherently low dimensional with strong electron-phonon coupling (a structure called the Peierls distortion), leading to a decrease in the thermal conductivity. The occurrence of this phenomenon distorts the chain to form a two-dimensional superlattice structure and charge density wave states [6,78]. Among the effects of the Peierls distortion, distortion of the lattice and electron states of the system also appearing due to the effects on the periodic arrangement of atoms of additional states and a small gap near the Fermi level. These are accompanied by decreased electrical conductivity and an increased Seebeck coefficient, so that the metallic material is transformed to a semiconductor or insulator.

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Figure 2.9 Band structure of (A) In4Se3 and (B) In4Se2.75 [121]. Copyright 2009, AIP Publishing.

Therefore, the zigzag arrangement of different atoms (Peierls distortion structure) leads to a tunable band structure. For example, in the off-stoichiometric In4Se32x, the periodical Peierls distortion of the In-Se chains is disturbed by Se vacancies, leading to a big difference in the band structure (as shown in Fig. 2.9) [121]. The semiconductor type band structure changes into a semi-metallic band with a deficiency of Se, and there is a flat band along the X-Γ and Y-S symmetry lines, indicating that the holes are highly localized. The Peierls distortion and charge density wave could also lead to decreased thermal conductivity [79]. In the SnX and In4Se3 systems, due to the off-stoichiometric valence of elements, an excess amount of charge carriers usually exists, for example, the hole concentration of SnTe is over 1021 cm23 [80]. Thus, the Se vacancies or element doping could simultaneously adjust the band structure and the carrier concentrations. Rhyee et al. reported that single crystal In4Se2.78 was a good n-type thermoelectric material with a ZT of 1.48 at 705 K [78]. A ZT of 2.6 at 923 K was obtained for p-type SnSe single crystal [6]. The high ZT of these materials is due to their unique zigzag bonding arrangement (i.e., Peierls distortion).

2.5.3 Layered structure Layered structure materials have long been recognized as good thermoelectric materials due to their special crystal structure and the possibility of manipulating charge carrier transport and phonon transport by modulation of different crystal planes. In addition to the aforementioned Bi-Te and Bi-Se systems, recent there is a great progress on the thermoelectric performance of layered ACrX2 (A 5 Ni, Cu, or Ag; X 5 S, Se), transition metal dichalcogenides (TMD), as well as the Bi-O-X system. A good comprehensive review on layered structure metal chalcogenides materials by Jood et al. was published in 2015 [122]. Here, we mainly focus on the following materials, which are not described in detail.

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As a group of magnetic materials, ACrX2 (A 5 Ni, Cu, or Ag; X 5 S, Se) compounds possess a trigonal layered structure, which can be described as CdI2-type layers of CrX-2 in which the Cr31 cations are in a distorted octahedral coordination to X2- anions, with the interlayer spaces being filled by the copper or silver ions occupying tetrahedral sites. The atoms inside the triple layers of CrX2 are bound to each other by strong ionic bonds, while the neighboring triple layers are bound to each other by weak van der Waals forces. Thus, CuCrX2 and AgCrX2, which show an order-disorder transition of Cu or Ag atoms around 400  C, are also PLEC materials. Tewari et al. and Gascoin et al. reported the maximum ZT of 2 at 300 K and 1 at 848 K for CuCrS2 and AgCrS2, respectively [123,124]. A ZT of 1.4 was achieved in sandwich-like (AgCrSe2)0.5(CuCrSe2)0.5 at 773 K [125]. The thermoelectric performance of this group of materials strongly depends on their synthesis, however. For example, Tewari et al. obtained highly textured CuCrS2 via a specific heat treatment process [123], which finally led to high electrical conductivity in the sample, but the CuCrS2 sample fabricated by ball-milling and SPS methods exhibited poor thermoelectric performance with a ZT of 0.11 [126] (Fig. 2.10). As one kind of TMD, TiS2 also has a layered structure with a trigonal space group. Atoms within the S-Ti-S sheets are bound by strong covalent interactions, whereas the bonding between the layers is only due to the weak van der Waals forces. Therefore, as a layer structured compound with a van der Waals gap, TiS2 is well known for its capability of intercalation by a wide range of both organic and

Figure 2.10 The thermoelectric transport properties of n-type SLMoS2 at 300 K, 400 K and 500 K. (A) The electrical conductivity and Seebeck coefficient; (B) the electrical thermal conductivity; (C) the power factor; (D) the figure of merit [127]. Copyright 2015, Springer Nature.

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inorganic materials to adjust its performance [128
134]. Detailed performances of TiS2 related materials have been reported [122]. Because of the wide van der Waals gap, excess Ti atoms can also be inserted into the layers, leading to the formation of Ti11xS2. The Seebeck coefficient and electrical resistivity of Ti11xS2 decrease with increasing titanium content because the carrier concentration increases with x. A high power factor was found for a near-stoichiometric composition [135]. Hence, despite the merits of its low cost, low toxicity, low density, and simple composition, the main difficulty for the wide application of this material is the synthesis of stoichiometric TiS2. MoS2 is a layered semiconductor and has a direct band gap, which enables the wide applications in electronic and optical devices. As shown in Fig. 2.10, Jin et al. studied both electron and phonon transport properties of single layer MoS2, and predicted a value of ZT as high as 0.11 at 500 K [127]. Besides, other transition metal disulfides (TMD) have also been investigated, such as WSe2, TiS3, etc. [136
138].

2.5.4 Increase the degree of orientation Bismuth chalcogenides such as Bi2Te3 have a layered structure, so they show anisotropic thermoelectric properties along different crystal directions and a high ZT along the in-plane direction. Commonly, Bi2Te3 based ingots with high ZT values are single crystalline or highly aligned polycrystals fabricated by the Bridgman, Zone melting, and Czochralsky techniques, with a preferred crystal orientation [139]. Therefore, one way to enhance the ZT and mechanical properties of the BiTe compounds is to increase the degree of orientation of crystals in their polycrystalline bulks, which has attracted considerable attention in recent years. The first approach is hot deformation, which has been proved to be an effective method to increase orientation degree in the polycrystalline bulk materials [19
22,139
141]. External strains may cause reorientation or induce preferential growth of crystals. For example, by combining the densification of nanostructured powders with two-step hot forging processes, the orientation degree of nanostructured p-type Bi0.5Sb1.5Te3 was significantly improved from less than 0.1 to 0.3, and the ZT value was increased from 1.0 to 1.5 [23,24]. The second approach to reorienting the crystals in bulk materials is the use of high magnetic field [139,142,143]. When a crystal with anisotropic magnetic susceptibility is placed in a strong magnetic field, it will be rotated to an angle that minimizes the magnetization energyassociated with the magnetic anisotropy of the crystal. Under a high magnetic field, the crystallographic c-axis of single crystal Bi0.6Sb1.4Te3 can be aligned perpendicularly to the direction of an applied magnetic field. Kim et al. fabricated polycrystalline Bi0.6Sb1.4Te3 under a magnetic field using the slip-casting method and then consolidated by SPS. The obtained bulk material indeed showed preferential orientation under a magnetic field of 10 T, and the maximum ZT reached 1.2 at 323 K, which is a 30% increase over the sample synthesized in the absence of magnetic field [143]. As shown in Fig. 2.11, Luo et al. imposed a variable intensity high static magnetic field during a traditional melt-solidification (MS) process to prepare p-type Bi0.5Sb1.5Te3 [139]. A c-axis

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49

Figure 2.11 ESEM images of a cross-section parallel to the magnetic field direction for p-type Bi0.5Sb1.5Te3 samples prepared under (A) 0 T, (B) 2 T, (C) 4 T, and (D) 6 T [139]. Copyright 2015, Elsevier.

alignment of the bismuth antimony telluride in the direction perpendicular to the magnetic field was observed in the sample fabricated under 2 T magnetic field, which is related to the enhanced convection of the melt under high magnetic field. Finally, this well aligned material yielded a ZT of 1.71 at 323 K, which is nearly two times higher than for the sample prepared without magnetic field. The third approach is a combination of chemical exfoliation and spark plasma sintering (CE-SPS). Puneet et al. used this approach to transform bulk n-type Bi2Te3 microparticles into well oriented layered nanostructures, which exhibited increased electrical conductivity and ZT [144]. Zhu et al. adopted a similar hydrothermal exfoliation and SPS process (HE-SPS) to obtain well oriented nanostructured p-type Bi0.48Sb1.52Te3 [145]. The maximum ZT of p-type Bi0.48Sb1.52Te3 increased from 1.2 to 1.4 at 300 K due to the increasing orientation degree.

2.6

Outlook

Although metal-chalcogenide-based thermoelectric materials shows relatively higher ZT values than other materials and significant progress has been achieved in recent years, several problems still need to be resolved. (i) Many high-ZT metal

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chalcogenides such as bulk SnSe single crystal [6], full scale structured PbTe bulk [7], mosaic CuSTe system [146], etc. are produced in the laboratory, they face many difficulties for making commercialization. For instance, it is more difficult and expensive to grow a single crystal or a Mosaic crystal than fabrication of a polycrystalline SnSe. (ii) Most metal chalcogenides (e.g., Pb
X based materials) with high thermoelectric performance show poor mechanical properties [147,148], which limits their real application. (iii) Due to the intrinsic volatility of chalcogen elements (S, Se, Te), some metal chalcogenides possess poor thermal stability, especially for those nanoscale metal chalcogenides fabricated from bottom up processes [14]. For PLEC materials, the fast movement of liquid like metal ions could also lead to their instability. The chalcogenide nanostructures originated from spinodal decomposition are also not stable during operation because spinodal decomposition is sensitive to temperature. (iv) Most high thermoelectric performance metal chalcogenides have the environmental issue or fabrication cost due to the use of toxic elements (e.g., Pb) or expensive precursors (e.g., Ag and Te) [149]. To fully develop the potential of thermoelectric materials or wide commercial applications in the near future, parallel development on both materials and thermoelectric devices are needed. From the viewpoint of materials, traditional Bi-Te and Pb-Te systems would continue dominate the commercial market because of their better performance than other metal chalcogenides and mature processing technologies. However, more and more efforts would be devoted to enhancing their mechanical strength, reducing fabricating cost and optimizing thermoelectric generators. In addition, low dimensional metal sulfides and selenides would attract considerable attention due to the possibility of simultaneous optimization of the thermal and electrical conductivity via separating the transfer of charge carriers from that of phonons, and their lower cost, environmental friendliness, as well as simple composition. Moreover, the stable PLEC materials with high performance would be a focus for future research. Meanwhile careful tailoring of nanostructures including shape, size, and interface would be still an important and efficient way to further enhance thermoelectric properties in near future. Partially introducing rationally designed nanostructures into bulk matrix would be a major development because of its distinct advantages. It can improve the thermal stability of metal chalcogenides, and the scattering of charge carriers can be suppressed. The hybrid of nanostrcutures and bulk may lead to full scale phonon scattering compared with single size of nanostructure. The introduced nanostructures could also lead to energy filtering effect. Chen et al. designed a coreshell structure to simultaneously achieve total non-scattering of charge carriers and enhance the Seebeck coefficient [150]. This effect is called “anti-resonant doping” or “invisible dopants” effect. Besides nanostructure engineering, defect engineering is another new highlight in the near future to further enhance thermoelectric performance of metal chalcogenides, because defects could effectively decrease their thermal conductivity and tune their carrier concentrations. Compared with traditional hot deformation method, irradiation could be a better way to introduce defects in thermoelectric materials because defects can be well controlled with different beams and their dosage. The effects of induced defects on thermoelectric properties of bulk materials

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51

have been rarely reported. For practical applications, more efforts should be devoted to fabricate efficient thermoelectric devices. An important development on thermoelectric devices is fabrication of flexible thin film thermoelectric generators (TEGs), which has several merits in comparison with rigid bulk modules [151
162]. It can be easily integrated into unusual topologies and maximize the heat absorbing area, leading to an enhanced practical efficiency compared to rigid TE devices. More importantly, the cost of thin film TEGs is low because the screen printing, inkjet printing or the roll to roll manufacturing have been widely used. The thin film TEGs can be used to supply power for small electronics due to their high power density. Their weight is also low. These supreme features of thin film TEGs have attracted considerable attentions and remarkable progress has been achieved [151
162]. However, most of thin film TEGs are fabricated from Bi-Te system and polymers, and used at room temperature [163
170]. A major problem of thin films TEGs is their much lower ZT compared with that of bulk materials, due to the less density of film and low sintering temperature. To further open up new market for metal chalcogenides based thermoelectric materials, one of the important developments of their application forms is combine the thermoelectric devices with other energy harvest devices such as solar cells. Many metalchalcogenide-based thermoelectric materials can also be used in solar cells, such as the Pb-X system, Cu-X system, Cu-In-Ga-Se system, etc [1,171,172]. Recently, Kraemer et al. reported a flat-panel solar thermoelectric generator (STEG) with a system efficiency of 4.6%
5.2% generated from a temperature difference of 180  C (Tc 5 20  C, Th 5 200  C) [173]. The thermoelectric legs are made from p-type and n-type ZnSb compound. The efficiency is 7
8 times higher than the best value previously reported [173]. With the development of nanotechnology and fabrication technology, with the discovery of new metal chalcogenide structures, as well as new theories leading to high thermoelectric performance, the ZT values of metal chalcogenides could be further enhanced to yield a comparable efficiency to conventional heat engines, and thus make greater contributions to energy saving and environmental protection.

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Lead salt photodetectors and their optoelectronic characterization

3

D. Babic, L.W. Johnson, L.V. Snyder and J.J. San Roman Laser Components DG, Inc., Tempe, AZ, United States

3.1

Introduction

Lead salt photodetectors are very popular detectors in the infrared spectral range due to their outstanding price to performance ratio and room temperature operation capability [1]. Commercially available lead salt photodetectors are lead sulfide PbS and lead selenide PbSe photoconductors that efficiently detect near infrared (NIR) radiation (PbS) and mid wavelength infrared (MWIR) radiation (PbSe). The main applications of lead salt photodetectors are medical and environmental gas analysis, process control to detect and analyze moisture and various other gases and materials, flame and fire detection and optical pyrometry. Laser Components DG, Inc. (LCDG) is a leading vendor of PbS and PbSe photodetectors for many listed applications. After reviewing the material physics of lead salt detectors, this chapter will review method of lead salt photodetector preparation and testing utilized by LCDG.

3.2

Background

Chalcogenide compounds generally show considerable photosensitivity [2]. Lead chalcogenides were among the first materials whose photosensitivity was discovered and studied. They show infrared (IR) sensitivity due to the size of their bandgap: 0.27 eV for PbSe and 0.42 eV for PbS, both at room temperature. PbTe is another IR sensitive chalcogenide that has also been extensively researched but has not found commercial applications as a photodetector and will be not be further discussed here. It is well known that Kutzscher discovered photoconductive properties of PbS in Germany in the 1930s making it the first infrared semiconductor material [1]. Production of PbS photodetectors started in Germany about 1943 for military and commercial applications. U.S. and British laboratories initiated lead salt research programs at the end of the World War II that continued until the 1970s. Cashman showed that PbSe has photoconductive properties similar to PbS in 1944. Work on lead salt photodetector development intensified after the World War II and lasted until approximately late 1960s. Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00003-8 © 2020 Elsevier Ltd. All rights reserved.

58

Chalcogenide

Practical lead salt photodetectors were established to be thin films of PbS or PbSe whose thickness was approximately 1 μm that were usually deposited on glass or quartz. Two major techniques for deposition of lead salt films have been established: vacuum deposition by evaporation and chemical bath deposition (CBD). The vacuum deposited lead salt films are n-type semiconductors while the CBD deposited films are p-type semiconductor materials. The films have to be sensitized by oxygen to become photosensitive and photosensitive lead salt films are p-type. In addition to oxygen sensitization, the PbSe films have been sensitized by a halogen, usually iodine, to increase their photosensitivity. The polycrystalline lead salt thin films were found to have superior performance to single-crystal films [1]. The CBD technique has offered significant cost advantages without compromising the performance so it has made feasible fabrication of large area lead salt detectors of the order of several mm squared. The lead salt photodetectors and PbS and PbSe materials were extensively characterized in order to improve their performance including developing several models of the photodetector operation. Lead sulfide and lead selenide are semiconductors that change their electrical conductivity when illuminated by photons of appropriate photon energy or wavelength, making them photoconductors. At room temperature, the PbS band gap is 0.42 eV making it IR light sensitive in the 1 3 μm range, while PbSe is sensitive to IR in the 1 4.5 μm range due to its band gap of 0.27 eV. Lead salt materials and detectors are very sensitive to environmental temperature variations and that is easily understandable due to their narrow band gap. However, the band gap decreases when the material is being cooled, contrary to what is generally observed in elemental and III V semiconductors. No clear explanation of the anomaly has been accepted although it is believed to be related to the exceptionally large dielectric constant, complexities of the band structure and anharmonicity of the phonon band structure of the material [2]. Several theoretical models of lead salt photoconductivity have been offered, mostly focused on PbS and tacitly assuming that they equally apply to PbSe [1,2]. The two most important models that will be briefly discussed here are the “numbers” model and the “barrier-modulation” model. In the “numbers” model photogenerated electrons get trapped at a site such as an oxygen atom while photogenerated holes are free for transport through the material. The electron and hole cannot recombine until the electron is freed from the trap giving the hole very long lifetime. The illumination induced increase of long lifetime hole concentration causes a sustained and measurable increase of the material conductivity making lead salt material an excellent photoconductor. No significant change of the hole mobility is assumed. The photogenerated carriers are generated in the bulk of the lead salt p-type crystallites and the current flowing through the detector is carried by the holes as the majority carriers. The “barrier-modulation” model focuses on the interface among different lead salt crystallites [3]. It assumes the existence of a barrier at the interfaces that still permits the flow of dark current. The photogenerated carriers are generated in the bulk of a crystallite flowing to the interface where they reduce the barrier height

Lead salt photodetectors and their optoelectronic characterization

59

enabling a larger number of carriers to cross the barrier and thus increasing the ‘effective’ carrier mobility and overall detector conductivity. Petritz [4] has proposed a model of PbS photoconductivity that combines the “numbers” and “barrier-modulation” models and suggested experiments that would elucidate relative contribution of each constitutive model to the photoconductivity mechanism. Bube and coworkers performed Hall, thermoelectric and photoresponse measurements on CBD PbS samples and concluded that the “barrier-modulation” model for PbS photoconductivity is inappropriate for CBD deposited and oxygen sensitized material [5]. Roberts and Baines [6] reached similar conclusions for CBD film of lead selenide. Recently, there has been renewed interest in lead salt photodetectors spearheaded in the USA by the University of Oklahoma group [7]. They have focused their attention on PbSe material preparing the detectors by a CBD technique. They found that iodine is the key to sensitize photoconductivity of PbSe and that oxygen is actually not required but supports more effective iodine sensitization. They also proposed a new model of the photoconductivity effect [8]. In this model the PbSe crystallites consist of p-type cores and n-type shells forming pn junctions due to the iodine sensitization. The spatial separation of photogenerated electrons and holes at the pn junctions prevents their recombination improving carrier lifetime and detector performance.

3.3

Lead salt detector fabrication

The lead salt detector fabrication process at LCDG starts by CBD deposition of the thin film (either PbS or PbSe) on quartz substrates. The PbSe film is subsequently sensitized by oxygen and by iodine. The Fig. 3.1 shows photographs of PbSe films at different stages of the processing: (A) as deposited, (B) after oxygen sensitization and (C) after iodine sensitization. The fully sensitized PbSe thin film is a p-type photosensitive semiconductor whose band gap is equal to the band gap of bulk PbSe. Standard photolithographic techniques are applied to pattern the sensitized PbSe film into desired detectors followed by gold contact deposition. Fig. 3.2 presents a photograph of a patterned and metallized PbSe film for illustration. The quartz substrates are diced to obtain self standing detectors. PbS detectors are fabricated by a very similar process. The main difference is that the PbS films are sensitized only once. The photolithography, contact deposition and dicing are performed in the same way as for the PbSe detectors. The difference in the sensitization between the PbS and PbSe films appears to be related to their different sensitivity to operation at elevated temperature. LCDG engaged in a project with the Arizona State University (ASU) to characterize the PbSe detector material in great detail using variety of techniques and equipment available at ASU. The characterization results verified that the manufacturing process used by LCDG achieves PbSe films having maximum photosensitivity for CBD prepared PbSe material.

60

Chalcogenide

Figure 3.1 Photographs of PbSe films at different stages of processing: (A) as deposited, (B) oxidized, (C) halogenated.

3.4

Lead salt detector characterization

The lead salt detectors are photoconductors that respond to IR illumination whose photon energy is larger than the band gap of the photoconductor material by lowering their resistivity. Measurement of the resistivity change requires application of an external bias. The most commonly applied biasing circuit consists of a voltage source and two resistors connected in series as a voltage divider. The divider consists of a

Lead salt photodetectors and their optoelectronic characterization

61

Figure 3.2 Photograph of patterned and metallized PbSe film.

resistor made from photoconductive material and another resistor (usually called the load) whose value is matched to the resistance of the photoconductor. The load resistor is typically 1 MΩ in applications involving lead salt photoconductors. Fig. 3.3 shows a schematic diagram of the test circuit consisting of the biasing circuit with an AC coupled amplifier in the non-inverting operational amplifier configuration. The reason behind the use of the amplifier is that the photoinduced resistance change of a lead salt photoconductive detector is small, generally below or around 0.1% of its resistance so the resulting voltage change signal is small and needs to be amplified for easy measurement. The measurement of the photoinduced signal by this circuit requires modulation of the IR illumination by chopping. The chopping shifts the signal of interest from DC to the chopping frequency enabling easier extraction of the AC signal from noise and background. The AC coupling of the amplifier permits amplification of AC components only, including the AC signal at the chopping frequency, while blocking the large DC component. A signal analyzer or a lockamplifier set at the chopping frequency are used to measure the AC signal induced in a lead salt photoconductor by the chopped IR illumination. Simple circuit analysis shows that the AC signal voltage is proportional to the photoinduced change of the photoconductor’s resistance and to the bias voltage. Another reason for applying the chopped IR illumination to lead salt detectors is a weak and slow drift of photodetector resistance. The drift tends to follow slow temperature variations of the detector environment and is probably due to very

62

Chalcogenide

Figure 3.3 Schematic diagram of lead salt testing circuit.

small band gap of the lead salts that makes them especially sensitive to minor temperature variations. The bias voltage applied to the voltage divider has to be carefully selected because its value affects not only the measured signal value but also the noise. The optimum bias voltage has been 50 V/mm of the detector electrode separation. The maximum bias is roughly double the optimum value; further bias increase would cause runaway thermal effects. The detector bias needs to be reduced at increased temperature to prevent detector heating and over currents. The heating and overcurrent are caused by reduction of the detector resistance at increased temperature. The minimum bias is set by the need to overcome system noise and, as quality of the amplifier electronics has been increasing and related noise decreasing, lowering of the bias voltage values has become acceptable. Lead salt detector noise has generation-recombination and 1/f noise as its most dominant components. Historically, detector specifications have been given at or close to 1 kHz to avoid the 1/f noise contribution. However, LCDG provides D detectivity data at much lower 90 Hz as well because the detectors are often used at this frequency. It appears that the noise increases roughly three times for the frequency change from 1 kHz to 90 Hz. Achievable D for PbS at room temperature is typically 1.1E11 while for PbSe it is 1.8E10. Prolonged exposure of the lead salt detectors to UV and visible light may change their performance or cause degradation and should be avoided to enable stable and repeatable application and characterization of detectors. LCDG catalog lists performance characteristics of PbS and PbSe photodetectors of different sizes. The excellent performance of PbS and PbSe photodetectors at room temperature is well known. In addition, temperature is ingeniously used to modify detector performance. The anomalous temperature dependence of the lead salt band gap and its high sensitivity to temperature variations has enabled a very sophisticated technique for controlling lead salt detector responsivity by adjusting its temperature. The peak

Lead salt photodetectors and their optoelectronic characterization

63

response wavelength is tuned and its responsivity increased by controlled cooling of the detector, usually by a thermoelectric cooler. Fig. 3.4 presents spectral response of a PbS detector measured by an FTIR instrument at different temperature illustrating the sensitivity especially of the cut off wavelength. Fig. 3.5 shows the spectral response of a PbSe detector in the same manner.

PbS FTIR relative spectral response (normalized) 1.00

Relative intensity

0.80 –50ºC –45ºC

0.60

–40ºC –35ºC –20ºC

0.40

0ºC 23ºC

0.20

73ºC

0.00 1.5000

2.0000

2.5000

3.0000

3.5000

Wavelength (µm)

Figure 3.4 Spectral response of PbS detector (normalized) measured at different temperatures.

PbSe FTIR relative spectral response (normalized) 1

Relative intensity

80ºC 0.8

23ºC 0ºC

0.6

–20ºC –35ºC

0.4

–40ºC –45ºC

0.2

–50ºC 0

1.5

2.5

3.5 4.5 Wavelength (µm)

5.5

–55ºC

Figure 3.5 Spectral response of PbSe detector (normalized) measured at different temperatures.

64

Chalcogenide

FTIR measurements require two measurements, one as a reference measurement using another detector type, usually a pyroelectric, to set the baseline and the other as a sample measurement using a lead salt detector in question. The FTIR spectrum of a sample is obtained by dividing the lead salt data with the pyroelectric detector baseline data and normalizing the peak to one. The spectra presented in Figs. 3.4 and 3.5 have been corrected for the imperfections of the pyroelectric detectors. To the best of our knowledge, this is the first time that the spectral response data for the lead salt detectors have been measured and collected in such wide temperature range with the proper corrections.

3.5

Conclusions

Lead salt photoconductors produced by chemical bath deposition are popular IR photodetectors due to their outstanding performance-to-cost ratio, high room temperature performance (D  . 1E10) and availability of large area detectors. History of lead salt detectors has been reviewed recognizing some more recent research developments. The chemical bath deposition technique for production of the lead salt detectors has been outlined, followed by a review of the basics for their characterization. Spectral responses of both PbS and PbSe detectors collected in a wide temperature range and properly corrected have been presented.

Acknowledgment The authors gratefully acknowledge Mr. Markus Leppelt for his help with the figures.

References [1] E.L. Dereniak, G.D. Boreman, Infrared Detectors and Systems, Wiley-Interscience, Hoboken, New Jersey, 1996. [2] T.S. Moss, G.J. Burrell, B. Ellis, Semiconductor Opto-Electronics, Butterworths, London, 1973. [3] J.C. Slater, Barrier theory of the photoconductivity of lead sulfide, Phys. Rev. 103 (1956) 1631. [4] R.L. Petritz, Theory of photoconductivity in semiconductor films, Phys. Rev. 104 (1956) 1508. [5] S. Espevik, Ch.-h. Wu, R.H. Bube, Mechanism of photoconductivity in chemically deposited lead sulfide layer, J. Appl. Phys. 42 (1971) 3513. [6] D.H. Roberts, J.E. Baines, Photoconductivity in chemically deposited films of lead selenide, J. Phys. Chem. Solids 6 (1958) 184.

Lead salt photodetectors and their optoelectronic characterization

65

[7] J. Qiu, B. Weng, Z. Yuan, Zh. Shi, Study of sensitization process on mid-infrared uncooled PbSe photoconductive detectors leads to high detectivity, J. Appl. Phys. 113 (2013) 103102. [8] L. Zhao, J. Qiu, B. Weng, C. Chang, Z. Yuan, Zh. Shi, Understanding sensitization behavior of lead selenide photoconductive detectors by charge separation model, J. Appl. Phys. 115 (2014) 084502.

Optical dispersion of ternary II
VI semiconductor alloys

4

Xinyu Liu and J.K. Furdyna Department of Physics, University of Notre Dame, Notre Dame, IN, United States

4.1

Introduction

4.1.1 The classical picture of dispersion Because the band-gap energy of most II-V semiconductors is between 1 eV and 3 eV, it makes them useful materials for fabrication of optoelectronic devices in the visible region of the spectrum. Since these materials are used as waveguides, laser media, distributed Bragg reflectors (DBRs) and cladding layers intended for optical confinement, it is essential to have a thorough understanding of their refractive index n and its dependence on the wavelength. In this chapter we will focus on the dispersion of n below or near their energy band gaps, because it is in that region of transparency where the knowledge of n is important in practice. All II-VI semiconductor compounds and their ternary alloys can be viewed as dielectric media, characterized by permittivity ε 5 εoε1, where εo is the permittivity of vacuum, and ε1 is the (relative) dielectric constant. Dielectric materials that absorb light at energies above the band-gap energy Eg are often represented phenomenologically by a complex electrical susceptibility [1], χ 5 χ0 1 iχ};

(4.1)

corresponding to a complex permittivity ε 5 εo (1 1 χ) 5 εo(ε1 1 ε2), where ε1 5 (1 1 χ’), ε2 5 χ’’ are real and imaginary parts of the dielectric function. The wave equation, r2E 1 k2E 5 0 remains applicable for this complex ε, with a complex wave vector k 5 ωðεμ0 Þ1=2 5 ð11χÞ1=2 k0 5 ð11χ0 1iχ}Þ1=2 k0 ;

(4.2)

where k0 5 ω/c0 is the wavenumber in free space. Since k is complex, it is useful to write k in terms of its real and imaginary parts, k=k0 5 ðεμ0 Þ1=2 c0 5 n 2 iκ 5 n 2 i

α 5 ð11χ0 1iχ00 Þ1=2 : 2k0

Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00006-3 © 2020 Elsevier Ltd. All rights reserved.

(4.3)

68

Chalcogenide

The medium therefore has an effective refractive index n and an absorption coefficient α (or an absorption index κ). Thus, Eq. (4.3) yields [2] "

ε1 1 n5 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#1=2 ε1 2 1ε2 2 ; 2

(4.4)

α 5 2 k0 ε2 5 2 k0 χ};

(4.5)

n2 2 κ2 5 ε1 5 ð1 1 χ0 Þ;

(4.6)

2nκ 5 2 ε2 5 2 χ}:

(4.7)

and

All II-VI semiconductor compounds and their ternary alloys are also dispersive media, characterized by a frequency-dependent susceptibility χ(ω) and refractive index n(ω), so that the speed of light c(ω) 5 c0/n(ω). Dispersion and absorption are intimately related by the Kramers-Kronig relations: [3,4] 2 ε1 ðωÞ 2 1 5 χ ðωÞ 5 π 0

ε2 ðωÞ 5 χ}ðωÞ 5

2 π

ðN 0

ðN 0

ω0 ε2 ðω0 Þ 0 2 dω 5 ω02 2 ω2 π

ðN

ω½ε1 ðω0 Þ 2 1 0 2 dω 5 ω2 2 ω02 π

0

ω0 χ}ðω0 Þ 0 dω ; ω02 2 ω2

ðN 0

ωχ0 ðω0 Þ dω0 : ω2 2 ω02

(4.8)

(4.9)

Thus ε1(ω) can be calculated at each photon energy if ε2(ω) is known explicitly over the entire photon-energy (or wavelength) range. As a consequence of Eqs. (4.4) and (4.5), the refractive index n(ω) is also related to the absorption coefficient α(ω), so that if one is known for all ω, the other may in principle be determined. For example, the dielectric medium can be completely characterized by its response to a harmonic (monochromatic) field (see Section 7.5 in Ref [1]). One can then obtain ð2 ω2 1 iσω 1 ω20 ÞP 5 ω20 εo χ0 E;

(4.10)

where χ0 5 e2N/mεoω02, ω0 is its resonance angular frequency, and σ is the damping coefficient. The form P 5 εoχ(ω)E will yield an expression for the frequencydependent susceptibility, χðωÞ 5 χ0 where Δω 5 σ.

ω20 ; ðω20 2 ω2 1 iω ΔωÞ

(4.11)

Optical dispersion of ternary II
VI semiconductor alloys

69

Figure 4.1 The real (upper) and imaginary (lower) parts of χ(ω) for a single damped harmonic oscillator.

As plotted in Fig. 4.1, the real and imaginary parts of χ(ω) are, χ0 ðωÞ 5 χ0

ω20 ðω20 2 ω2 Þ ; ðω20 2ω2 Þ2 1 ðω ΔωÞ2

χ}ðωÞ 5 2 χ0

ω20 ω Δω : ðω20 2ω2 Þ2 1 ðω ΔωÞ2

(4.12)

(4.13)

One can easily find that the real and imaginary parts of χ are inter-related by the Kramers-Kronig relations. Furthermore, the overall absorption coefficient and refractive index of the medium in question can be obtained by putting Eqs. (4.12) and (4.13) into Eqs. (4.4) to (4.7).

4.1.2 Electronic band structure and dispersion In the previous section, we described the well-known Kramers-Kronig (K-K) relations. These relations indicate that ε1(ω) can be calculated at each photon energy if ε2(ω) is known explicitly over the entire photon-energy range. Based on the electronic band structures of semiconductors, there are several absorption mechanisms that contribute to the imaginary part of the dielectric function. Hence, ε2(ω) can be written as [5] ε2 ðwÞ 5 ε2;g ðwÞ 1 ε2;ind ðwÞ 1 ε2;fc ðwÞ 1 ε2;ib ðwÞ 1 ε2;b ðwÞ 1 ε2;ph ðwÞ:

(4.14)

Here ε2,g(ω) is the contribution due to absorption by direct interband transitions near the fundamental absorption edge, ε2,ind(ω) is due to indirect interband absorption, ε2,fc(ω) is due to free-carrier absorption, ε2,ib(ω) is due to intraband absorption inside the valence or conduction bands, ε2,b(ω) is due to the interband absorption at

70

Chalcogenide

higher energies, and ε2,ph(ω) is due to the phonon absorption. The terms ε2,fc(ω) and ε2,ib(ω) represent carrier-related absorption mechanism that may affect the index of refraction at wavelength much longer than the fundament absorption edge; and phonon contribution ε2,ph(ω) to the dielectric function can be neglected for photon energies far above the phonon energy. Therefore, only the ε2,g(ω) and ε2,b(ω) terms, which are crucially important for calculation of refractive index below the energy gap, are considered below. G

G

ε2,g(ω) In direct-band-gap zinc blende semiconductors, absorption near the band gap includes the excitonic transitions and the valence-to-conduction band transitions at the Γ point of the Brillouin zone. Attraction between electrons and holes causes their motion to be correlated, and the resulting electron-hole pair is known as the exciton, which can be considered like a hydrogen atom, but with the particles having effective hole and electron masses. Because electrons and holes are only weakly bound, excitons become thermally ionized at room temperature. The contribution of the excitonic absorption is therefore very small, and one need only consider it at low temperatures. The continuum valence-toconduction band absorption at the Γ point includes transitions from the heavy-hole (hh), light-hole (lh), and spin split-off valence bands. Generally, the parabolic band model is a good approximation for most wide-gap zinc blende semiconductors. Here a proper description of band parabolicity is needed to calculated the joint density of states at the Γ point, which will yield ε2(ω) near the energy band gap Eg, as shown below. ε2,b(ω) Above the fundamental absorption edge, ε2 typically rises to an asymmetric peak E1 related to transitions occurring along the eight equivalent [111] directions of the Brillouin zone (direct optical transition at the L point). When the spin-orbit interaction in the valence bands is large, the E1 transition is split into the E1 and E1 1 Δ1 transitions. ε2(ω) reaches a strong absolute maximum known as the E2 peak for photo-energies greater than the bandgap energy and E1. This peak contains contributions from transitions occurring over a large region of the Brillouin zone close to the edges in the [100] and [110] directions (direct optical transitions at the Χ point). Band structure suggests that these transitions involve M1 or M2-type critical points (CP) correlated with their joint density of states. However, the nature of E2 transitions is more complicated, since it does not correspond to a single well-defined CP, and is characterized as a damped harmonic oscillator.

The energies [in eV] of the most basic parameter in the optical spectra of several common semiconductors are listed in Table 4.1 [6
9]. Table 4.1 The energies (eV) of several basic parameters in the optical spectra of some diamond and zinc-blende semiconductors. Transition

Si

Ge

GaAs

ZnSe

CdSe

ZnTe

CdTe

E0 E0 1 Δ0 E1 E1 1 Δ1 E2

4.185 4.229 3.45
4.330

3.006 3.206 2.22 2.42 4.49

1.5192 1.859 3.017 3.245 5.110

2.69 3.10 4.8 5.1 6.7

1.74 2.15 4.28 4.48 6.08

2.28 3.20 3.78 4.34 5.23

1.58 2.55 3.55 4.13 5.13

Optical dispersion of ternary II
VI semiconductor alloys

71

Figure 4.2 A comparison between the experimental and calculated dielectric function εr (A) and εi (B) of GaAs.

A comparison between experimental and calculated dielectric function of GaAs is shown in Fig. 4.2 [10]. Note that the value near the bandgap Eg is very much smaller than that of the E1 or the E2 peak. The principal direct transitions (E0, E1, E2) responsible for the structure in ε2 are marked in Fig. 4.2.

4.1.3 The phenomenological dispersion model The theoretical calculation of the refractive index (or the dielectric function) using Kramers-Kronig relations and electronic band structure usually leads to some complicated, lengthy and tedious equations [11
13]. Although such theoretical analysis can give reasonably good fits to the refractive index, a simple dispersion relation is

72

Chalcogenide

often desirable for more practical use. Several empirical models have been proposed to express the wavelength dependence of the refractive index n by various researchers: G

The Cauchy relation: n5A1

G

B C 1 4: λ2 λ

The Sellmeier-type relation: n2 5 A 1

G

(4.15)

Bλ2 : λ2 2 C

(4.16)

The Ketteler-Helmholz relation:

n2 5 a 1

bE2 : 1 2 ðE2 =c2 Þ

(4.17)

Note that all these relations are three-parameter empirical equations. Although these dispersion formulas can be used to successfully fit experimental data for most semiconductors, the parameters which they contain do not generally correspond to physically meaningful quantities.

4.2

Optical dispersion

In this section, ternary alloy Zn1-xMnxTe and Zn1-xMgxTe are chosen as examples to show the practical application of the above empirical dispersion formulas (in particular the Sellmeier-type relation) [14]. The maximum concentrations of the substitutional elements on the Zn-sublattice of the alloys investigated here were 75% for Mn and 67% for Mg. Alloy concentrations of Zn1-xMnxTe and Zn1-xMgxTe were determined from the lattice constant measurements by X-ray diffraction (XRD), by assuming a linear Vegard’s law dependence of the alloy lattice parameters on the concentration [15]. The uncertainty in the alloy concentration obtained from the XRD measurements is estimated to be about 1% of the parameter x for alloys of higher concentrations. The lattice constants of ZnTe, MnTe [16,17] and MgTe [18] used in the Vegard’s law interpolation are listed in Table 4.2. Here we first used the prism coupler [19,20] method to obtain the values of n(x) for the Zn1-xMnxTe and Zn1-xMgxTe alloys at four different wavelengths. Using these values together with the thicknesses of the epilayers (also obtained by the prism coupler measurement), the reflectivity spectra observed on these films were analyzed to determine the values of n at the numerous Fabry-Perot maxima and minima, thus providing a nearly continuous series of points as a function of wavelength. The resulting data were fitted to the Sellmeier-type equation, Eq. (4.16), to obtain the dispersion relations of n for both Zn1-xMnxTe and Zn1-xMgxTe alloy

Optical dispersion of ternary II
VI semiconductor alloys

73

Table 4.2 Energy gaps Eg and lattice constants a0 of binary compoundsa; and bowing parameters bMn and bMg for Zn1-xMnxTe and Zn1-xMgxTe alloys.b Material

T

Eg (eV)

˚) a0 (A

ZnTe

12 K RT 12 K RT 12 K RT 12 K RT 12 K RT

2.38 2.28 3.20 2.90 3.27 6 0.05 3.13 6 0.03 Eq. (4.18a)

6.103

MnTe MgTe Zn1-xMnxTe Zn1-xMgxTe a

bMn/Mg (eV)

6.344 6.42

Eq. (4.18b)

0.34 6 0.02 0.34 6 0.02 0.27 6 0.09 0.25 6 0.06

The energy gaps for ZnTe and MnTe are from Ref 14, and the value for MgTe is extrapolated from Fig. 4.4. These values are obtained from least-squares fits of Eq. (4.18a,b) to the data in Fig. 4.5.

b

systems. In addition to the above study of n, the energy gaps of these alloys have also been investigated by both photoluminescence (PL) and by optical reflectivity, in order to obtain the dependence of Eg on the alloy composition. It is interesting to note that
as will be shown
the indices of refraction n exhibit a surprisingly linear dependence on the band gap Eg of the alloys.

4.2.1 Determination of the energy gap Eg(x) All samples were examined by PL measurements to determine their energy gaps and their optical quality. The PL spectra for ZnMgTe and ZnMnTe epilayers are shown in Fig. 4.3. For most samples, there clearly exist PL emission peaks related to free exciton transitions, whose energies and line widths indicate the band gap of the material and its sharpness, respectively. In the case of ZnMgTe, the width of the PL peaks varies from 16 to 39 meV as the Mg content increases from 10% to 52%, indicating a rather good quality of the alloy layers even when the Mg concentration is relatively high. In the case of the 52% sample the PL peak is clearly split. It is likely that the lower-energy shoulder is caused by an impurity-bound exciton, and we therefore take the higher-energy peak as the position of the free exciton, from which we determine Eg. The optical sample quality of this alloy degrades rapidly at Mg concentrations above 52% (see PL spectrum of Zn0.33Mg0.67Te in Fig. 4.3). The PL line width for the ZnMnTe alloy is about 10
40 meV for Mn concentrations of 10%
56%, respectively, and the intensity decreases rapidly with increasing Mn content. There is no clear free exciton emission in this alloy series for Mn concentrations above 56%. It should be noted that the PL emission intensity in ZnMgTe is stronger than in ZnMnTe, and does not decrease as the Mg concentration increases. We ascribe the observed behavior of the PL intensity in ZnMnTe to

74

Chalcogenide

Figure 4.3 PL spectra of ZnMgTe (A) and ZnMnTe (B) at 12 K. The PL spectrum of ZnTe is shown in panel (B) for reference.

the effect of energy transfer between the band-gap exciton recombination and internal Mn11 transitions that typically occur in Mn-containing alloys near 2.1 eV [21]. Fig. 4.4 shows the band-gap energies of ZnMnTe and ZnMgTe alloys for various Mn and Mg concentrations at 12 K and at room temperature. The 12 K band gaps are measured by PL as discussed above, and the room temperature band gaps were determined from the band edge seen in the reflectivity spectra (see, e.g., Fig. 4.5B). We find that the energy gap as a function of concentration x for Zn1-xMnxTe and Zn1-xMgxTe can be expressed in the form Eg ðxÞ 5 ð1 2 xÞEZnTe 1 xEMnTe 2 bMn xð1 2 xÞ;

(4.18a)

Eg ðxÞ 5 ð1 2 xÞEZnTe 1 xEMgTe 2 bMg xð1 2 xÞ;

(4.18b)

where bMn and bMg are the so-called bowing parameters for ZnMnTe and ZnMgTe, respectively. The values of energy gap for ZnTe and MnTe are taken from Liu et al. [14]. In the case of zinc blende MgTe the band-gap was estimated by extrapolating the values for Zn1-xMgxTe obtained in the present work to x 5 1.00 (see Fig. 4.4), because we found the values cited in the literature for this binary alloy not to be consistent among themselves [22,23]. Here it is important to emphasize

Optical dispersion of ternary II
VI semiconductor alloys

75

Figure 4.4 Energy gaps of ZnMnTe (A) and of ZnMgTe (B) determined at 12 K and at room temperature (RT) as a function of Mn and Mg concentrations, respectively. The solid lines represent quadratic fits using Eq. (4.18a,b) and parameters given in Table 4.2.

the usefulness of optical reflectivity, which allowed us to extend the Eg(x) determinations to high values of x, where the PL signal either becomes difficult to interpret, or completely disappears. The energy gaps and lattice constants for the binary zinc blende compounds are collected in Table 4.2. The continuous curves shown in Fig. 4.4 are calculated using Eq. (4.18a,b) and the parameters given in Table 4.2. Because the thickness of the films in all cases exceeds 1 μm, we can assume that the films are fully relaxed, and that the values of Eg(x) plotted in Fig. 4.4 are therefore representative of bulk (i.e., unstrained) material. We note parenthetically an interesting feature seen in the dependence of Eg on concentration x for both alloys. While the curves representing Eg(x) at the two temperatures are essentially parallel to one another for Zn1-xMgxTe, the low temperature Eg(x) curve for Zn1-xMnxTe diverges toward higher values from the room temperature data as x increases. This is a consequence of magnetic exchange interaction between band electrons and magnetic moments of the Mn11 ions, which increases with increasing Mn content and decreasing temperature [24]. Discussion

76

Chalcogenide

Figure 4.5 (A) A typical spectrum obtained by the prism coupler method for a ZnTe film at λ 5 1300 nm. The spectrum yields n 5 2.748 and film thickness d 5 1.63 μm. (B) Room temperature reflectivity spectrum obtained for the same ZnTe film. The arrow marked Eg indicates the energy gap as determined from this spectrum. The dotted vertical line shows the position at which n and d were determined by the prism coupler, thus pinpointing the index m of the Fabry-Perot resonance at that wavelength.

of this effect is outside the intended scope of the present chapter, and we refer the interested reader to Ref [21] for further details.

4.2.2 Indices of refraction n(x) The indices of refraction in all of the alloy samples were measured by the prism coupler technique [19]. A typical spectrum obtained by this method at 1300 nm is shown in Fig. 4.5A for a ZnTe film. The index of refraction and the thickness of the sample are calculated by determining the angular positions at which the sharp dips occur in the reflected signal [19,20]. These sharp dips correspond to the

Optical dispersion of ternary II
VI semiconductor alloys

77

Table 4.3 Refractive indices of ZnTe obtained at four laser wavelengths by the prism coupler method for ZnTe film. Wavelength (nm)

N

Thickness d (nm)

1300 1152 632.8 594

2.748 6 0.0005 2.762 6 0.0006 2.989 6 0.0007 3.045 6 0.002

1632 6 5 1623 6 11 1636 6 4 1640 6 8

excitation of specific TE waveguide modes in the film medium enclosed between the air gap and the GaAs substrate. Table 4.3 shows the values of n obtained at four laser wavelengths for the ZnTe sample, together with the thickness of the film obtained with each prism coupler measurement. The internal consistency of the thickness values provides a measure of accuracy of this method. For the shorter wavelengths (λ 5 632.8 nm and λ 5 594 nm), the n values of ZnMgTe and ZnMnTe films are such that the first few waveguide modes (corresponding to the rightmost dips in Fig. 4.5) are either not observable (Fig. 4.6A), or difficult to observe (Fig. 4.6B, where we see that the modes become weak for low mode numbers). In that case the film parameters are calculated using the “mode offset” feature of the prism coupler technique. In this procedure one carries out the calculation by assigning consecutive trial integers to the observed mode sequence. The integer series giving the least standard deviation for the refractive index n and the film thickness d is then chosen as the correct series. The improvement in standard deviation is quite striking when the correct integer sequence is chosen. The reliability of this method is further reinforced by the fact that, when it is applied to a given film at different wavelengths, it results in the same value of d. Fig. 4.6 displays the prism coupler spectrum of Zn0.25Mn0.75Te and Zn0.33Mg0.67Te at 632.8 nm. The figure includes the mode number (note that in the case of Fig. 4.6A the first few modes are not observed, and the use of the “mode offset” is then essential). Since our software for analyzing the spectrum is limited to a maximum of eight waveguide modes, we have indicated by arrows the specific modes used in calculating the film parameters. However, we have confirmed by other trial calculations that any other sequence of eight modes yields identical results, provided that the integers are consistent with those shown in the figure. It should be noted that the λ 5 1152 nm and λ 5 594 nm measurements yield results with a somewhat higher uncertainty, as is evident from Table 4.3. This increased uncertainty is due to a higher absorption by the silicon prism (for λ 5 1152 nm) and by the ZnMgTe and ZnMnTe films themselves (for λ 5 594 nm). Reflectivity data were obtained at room temperature for all samples. A typical spectrum is shown in Fig. 4.5B for the same ZnTe epilayer as that used to illustrate the prism coupler spectrum in Fig. 4.5A. The oscillations seen in the spectrum are Fabry-Perot resonances arising from interference between rays that are reflected

78

Chalcogenide

Figure 4.6 Spectra obtained by the prism coupler method for ZnMnTe and ZnMgTe at λ 5 632.8 nm. The spectra yield n 5 2.71, d 5 3.23 μm for ZnMnTe; and n 5 2.52, d 5 3.53 μm for ZnMgTe. The ZnMgTe data were analyzed using the “mode offset” procedure described in the text. The arrows indicate the eight modes chosen to calculate the film parameters. The mode index numbers determined using the “mode offset” procedure are also given in the figure.

from air-film and film-substrate interfaces. It can be readily shown that the successive reflectivity extrema (i.e., maxima and minima) occur at the condition 4nd=λ 5 m

(4.19)

where m is an integer representing the optical thickness of the film in terms of quarter wavelengths. With the values of d and of n established for a specific wavelength by the prism coupler measurement, the value of m can be determined for a given Fabry-Perot resonance in the reflectivity spectrum. With the thickness d of the film and the values of m known, we can then use Eq. (4.19) to calculate the index of refraction n at each wavelength corresponding to an extremum of the reflectivity

Optical dispersion of ternary II
VI semiconductor alloys

79

Figure 4.7 Dispersion of room temperature indices of refraction for a series of Zn1-xMnxTe alloys. The dispersion of ZnTe is also plotted for reference (squares). The open symbols represent data obtained from reflectivity spectra, and the full symbols are from prism coupler measurements. The solid lines are fits obtained using Eq. (4.20).

spectrum (see Fig. 4.5B). Since the number of extrema is large, this provides a quasi-continuous wavelength dependence of n. As an illustration of this procedure, we obtained the dispersion of n for ZnTe using both the prism coupler and the reflectivity data, shown as the uppermost curve in Fig. 4.7. As can be seen from the figure, the reflectivity data are in excellent agreement with the four prism coupler points, providing independent verification for the reliability of the results. The results for ZnTe in Fig. 4.7 are also in excellent agreement with those of Marple [25], who investigated n for this material in the wavelength range from 570 to 2600 nm. This provides additional verification of the accuracy of the method, which we will now extend to ternary alloys. Applying the same procedure to the alloys, the reflectivity and the prism coupler data were analyzed for a series of concentrations of Mn in Zn1-xMnxTe (0 # x # 0.75) and of Mg in Zn1-xMgxTe (0 # x # 0.67). As in the case of ZnTe, n was calculated for each of these alloys at a series of wavelengths to generate the dispersion curve for that material. These data were then fitted using the Sellmeier relation, Eq. 4.20, for the index of refraction n [26], similar to that used in the analysis of n in other wide-gap II-VI alloys [23,27], n2 5 A 1 Bλ2 =ðλ2 2 CÞ;

(4.20)

80

Chalcogenide

Figure 4.8 Dispersion of room temperature indices of refraction for a series of Zn1-xMgxTe alloys. The dispersion of ZnTe is plotted (squares) for reference. Open symbols represent data obtained from reflectivity spectra, and full symbols are from prism coupler experiment. The solid lines are fits obtained using Eq. (4.20).

where A, B and C are constants, and the wavelength is expressed in micrometers. In Figs. 4.7 and 4.8, the refractive indices are shown as a function of wavelength for several selected compositions of Zn1-xMnxTe and Zn1-xMgxTe alloys. The full symbols in the figures represent data obtained from the prism coupler experiments, and the open symbols are from reflectivity spectra. The solid dispersion curves shown in the figure are obtained by least-squares fit of Eq. (4.20) to the data. The values of A, B and C obtained in this way are of course different for different alloy compositions, progressing monotonically with x. Plotting these values as a function of x, one can then generate fits which express these parameters as polynomial expressions in x for the two alloy families, in the manner described in Ref. [26]. The expressions for A, B and C obtained in the present work for Zn1-xMnxTe and Zn1-xMgxTe are collected in Table 4.4. Using these values in Eq. (4.20) enables one to determine n(x) for these ternary systems for any alloy composition and for any photon energy, and should thus prove useful in designing various optoelectronic devices based on these ternary systems. This is illustrated in the top panels of Figs. 4.9 and 4.10, where we have used Eq. (4.20) and the expressions in Table 4.4 to plot the indices of refraction n as a function of composition x for Zn1-xMnxTe and Zn1-xMgxTe at two selected

Optical dispersion of ternary II
VI semiconductor alloys

81

Table 4.4 Constants A, B, and C for Zn1-xMnxTe and Zn1-xMgxTe in terms of concentration x obtained from fits of Eq. (4.20).a Constant

Zn1-xMnxTe

Zn1-xMgxTe

A

(5.40 6 0.06) 2 (1.20 6 0.21)x 2(1.46 6 0.27)x2 (1.92 6 0.01) 1 (0.41 6 0.06)x 1 (0.51 6 0.06)x2 (0.184 6 0.002) 2 (0.071 6 0.004)x 2(0.011 6 0.005)x2

(5.39 6 0.06) 2 (1.43 6 0.06)x 2(0.60 6 0.03)x2 (1.92 6 0.01) 2 (0.56 6 0.02)x 2(0.24 6 0.01)x2 (0.184 6 0.002) 2 (0.096 6 0.004)x 1 (0.012 6 0.005)x2

B C a

A and B are dimensionless; C is in (μm)2.

Figure 4.9 Upper panel (A): plots of the index of refraction of Zn1-xMnxTe as a function of concentration x, using Eq. (4.20) and parameters A, B and C given in Table 4.4. Lower panel (B): plots of the index of refraction of Zn1-xMnxTe as a function of energy gap Eg, using Eq. (4.21) and parameters a, b and c given in Table 4.5. Note the nearly linear dependence of n on Eg.

82

Chalcogenide

Figure 4.10 Upper panel (A): plots of the index of refraction of Zn1-xMgxTe as a function of concentration x, using Eq. (4.20) and parameters A, B and C given in Table 4.4. Lower panel (B): plots of the index of refraction of Zn1-xMgxTe as a function of the energy gap Eg, using Eq. (4.21) and parameters a, b and c given in Table 4.5. Note the nearly linear dependence of n on Eg..

wavelengths. The actual data are also shown in the figures as solid points. The curvature of the plots in Figs. 4.9A and 4.10A is not surprising, reflecting the presence of quadratic terms in x in parameters A, B and C for ZnMnTe and ZnMgTe (see Table 4.4). Such a plot, although purely phenomenological, allows one to use the index of refraction as an alternative tool for determining concentrations x in the two alloy families. Finally — since many optical devices that involve cladding and/or waveguiding depend simultaneously on both n and Eg — it is useful to examine directly the relationship between these two parameters (i.e., what n to expect for a given Eg, and vice versa). This can be done using Eq. (4.20), and re-expressing listed x in Table 4.4 in terms of Eg via Eq. (4.18a,b). To accomplish this we divided the

Optical dispersion of ternary II
VI semiconductor alloys

83

Table 4.5 Constants a, b, and c (in eV) for Zn1-xMnxTe and Zn1-xMgxTe in terms of Eg obtained from fits of Eq. (4.21). Constant

Zn1-xMnxTe

Zn1-xMgxTe

a b c

(15.20 6 0.11) 2 (4.30 6 0.04)Eg (21.48 6 0.09) 1 (1.49 6 0.04)Eg (20.63 6 0.09) 1 (1.54 6 0.04)Eg

(10.84 6 0.22) 2 (2.39 6 0.09)Eg (4.06 6 0.06) 2 (0.94 6 0.02)Eg (0.20 6 0.09) 1 (1.18 6 0.04)Eg

second term in Eq. (4.20) by λ2 and express λ22 in terms of E2 (where E is photon energy corresponding to λ), obtaining     n2 5 a 1 b= 1 2 E2 =c2 :

(4.21)

Here E is in eV, and parameters a, b and c are listed in Table 4.5 as functions of Eg. We emphasize that parameters a, b and c are identical to A, B and C for any given value of x (except that the units of c are now in eV instead of μm2). In the lower panels of Figs. 4.9 and 4.10 we plot (as solid curves) the dependence of n on Eg calculated by Eq. (4.21) at two wavelengths for Zn1-xMnxTe and Zn1-xMgxTe, respectively. The corresponding experimental data are shown as solid points. We were surprised by the close-to-linear relationship between n on Eg. This near-linearity arises, of course, from the absence of quadratic terms in the parameters a, b and c, as shown in Table 4.5. While the physical significance of this linear dependence of the Sellmeier parameters on Eg is not clear, such Vegard’s-law-like relationship between n and Eg provides a convenient means for determining n for Zn1-xMnxTe and Zn1xMgxTe corresponding to any energy gap by a simple linear interpolation.

4.3

Theoretical model

4.3.1 Semi-empirical model The fitting parameters used so far to describe the index of refraction are entirely empirical. This is useful, but does not provide much physical insight. We will therefore attempt to approximate the general theoretical expressions for dielectric function in ways that explicitly display certain physically meaningful parameters. Some such semi-empirical methods for calculating the index of refraction of solids at energies below the direct band edge that involve meaningful physical concepts are summarized below. G

The Wemple-DiDomenico Single-Effective-Oscillator (SEO) method: [28]

In this approach the authors express n in the following form: χ 0 5 n2 2 1 5

½E02

Ed E0 ; 2 ðh ¯ ωÞ2 

(4.22)

84

Chalcogenide

where ¯hω 5 E is the photon energy, E0 is the single oscillator energy, and Ed is the so-called dispersion energy. This single effective oscillator model is successful in representing refractive index data for a wide variety of covalent, ionic, and amorphous materials at photon energies sufficiently below the direct band edge. A simple connection between the single-oscillator parameters E0 and Ed and the ε2(ω) spectrum can be obtained by equating Eq. (4.22) with Eq. (4.8) and comparing terms in an expansion in powers of ω. Wemple and DiDomenico defined the r-th moment of the optical spectrum by the relation [29] Mr 5 ð2=πÞ

ðN

Er ε2 ðωÞdE:

(4.23)

Eg

The parameters E0 and Ed are then given by E0 2 5 M21 =M23 ;

(4.24)

Ed 2 5 M21 3 =M23 :

(4.25)

and

The parameter Ed, which is a measure of the strength of inter-band optical transitions, is found to obey the simple empirical relation Ed 5 βNcZaNe, where Nc is the coordination number of cation nearest-neighbors to an anion, Za is the formal chemical valency of the anion, and Ne is the effective number of the valence electrons per anion (usually Ne 5 8). The parameter β is essentially two-valued, taking on the “ionic” value β i 5 0.26 6 0.04 eV for halides and most oxides, and the “covalent” value β c 5 0.37 6 0.05 eV for tetrahedrally-bonded ANB82N zinc-blende- and diamond-type structures, as well as for scheelite-structured oxides and some iodates and carbonates [29]. In the context of the present chapter, for zincblende II-VI semiconductor compounds and alloys Nc 5 4, Za 5 2, Ne 5 8, and the dispersion energy Ed is found to be around 25B27, giving β 5 0.40B0.42 eV. This suggests that II-VI semiconductor systems are “covalent” crystals. Furthermore, it is easy to show that this formula gives a straight line when the refractive index data are plotted as the reciprocal of polarizability, χ21 5 (n2-1)21 vs (E/E0)2. The data for most zinc blende semiconductors (i.e., GaAs, CdTe, etc.), however, depart substantially from the single-effective-oscillator curve when photon energy approaches the band edge, due to proximity of strong excitonic and interband absorption. G

M. A. Afromowitz’s Modified Single Effective Oscillator (MSEO) method: [30] χ0 5 n2 2 1 5

2E02 2 Eg2 2 E2 Ed Ed E 2 Ed E4 lnð 1 3 1 3 2 Þ; 2 E0 Eg2 2 E2 E0 2E0 ðE0 2 Eg Þ

(4.26)

where E 5 ¯hω is the photon energy, E0 is the single oscillator energy, Ed is the dispersion energy, and Eg is the direct band gap. A modified empirical form of the ε2 spectrum is

Optical dispersion of ternary II
VI semiconductor alloys

G

85

used in the MSEO method (ε2 5 ηE4 for Eg # E # Ef). Then η and Ef can be written in terms of E0 and Ed by comparing the power series expansions for ε1-1 or χ with that in the SEO model. The MSEO calculation could be fitted to the experimental refractive index data over a wide wavelength range for zinc blende III-V semiconductor compound and alloys (e.g., AlGaAs) [31]. However, so far there appears to be no report of this model being used for II-VI semiconductors. The Pikhtin-Yas’kov more detailed oscillator-model formula: [32] χ0 5 n2 -1 5

A E12 2 ðh ¯ ωÞ2 G1 G2 G3 1 2 1 2 ; ln 1 2 2 π Eg 2 ðh ¯ ωÞ2 E1 2 ðh ¯ ωÞ2 E2 2 ðh ¯ ωÞ2 E3 2 ðh ¯ ωÞ2

(4.27)

where E 5 ¯hω is the photon energy, E1 and E2 are the two characteristic maxima of the ε2 spectrum (see previous section), E3 5 ¯hΩTO is the transverse optical phonon energy, and Eg is the direct band gap. Parameters G1, G2, and G3 are proportional to the corresponding oscillator strengths, and are obtained by integrating over the band. In this model, ε2(ω) is represented by two undamped oscillators with natural frequencies ¯hω1 5 E1 and ¯hω2 5 E2, along with a contribution from the fundamental absorption edge,pexpressed by an uniform ffiffiffiffiffi absorption from energy Eg to E1 with an intensity of A 5 0.7/ Eg . Eq. (4.27) has been used to fit refractive index data for most semiconductors in a wide spectrum of frequencies (for example, for wavelength between 570 and 2500 nm for ZnTe). Because there are only two adjustable parameters G1 and G2 in the formula for E .. ¯hΩTO , the refractive index data still cannot be fitted well at photon energies approaching the band edge unless one varies parameter A.

4.3.2 Improvements of SEO model For pure dielectrics, the frequency dependence of optical constants may be described by the classical treatment proposed by Lorentz. That approach assumes the solid to be composed of a series of independent oscillators, which are set into forced vibrations by incident radiation. The Lorentz theory of absorption and dispersion for both insulating and semiconducting materials leads to the two familiar relations [see the Eqs. (4.12) and (4.13)], n2 2 κ 2 5 1 1

X i

Ai ðω2i 2 ω2 Þ ; ðω2i 2ω2 Þ2 1 ðω γ i Þ2

(4.28)

and 2nκ 5

X i

Ai ω γ i ; ðω2i 2ω2 Þ2 1 ðω γ i Þ2

(4.29)

where n is the refractive index, κ [κ 5 α=ð2k0 Þ] the absorption index, Ai (Ai 5 χ0ωi2 5 e2Ni/mεo) is a parameter associated with the oscillator strength of the i-th oscillator, ωi is the resonance frequency of the i-th oscillator, and γ i (γ i 5 σ) is

86

Chalcogenide

the damping constant of the i-th oscillator. In the region of high transparency, Eq. (4.28) can be simplified by neglecting the linewidth of the oscillators, thus reducing to n2 2 1 5

X fi E 2 i ; 2 2 E2 E i i

(4.30)

where Ei 5 ¯hωi , E 5 ¯hω, and fi 5 e2Ni/mεoωi2. Note that for most semiconductors the effect of free carriers and of phonon absorption are found to be negligibly small in the transparency region. The summation in Eq. (4.30) thus includes only contributions from interband transitions. We can combine these contributions into a single effective oscillator form with a resonance energy E0, which can be called an average band gap. Eq. (4.30) can then be rewritten as: n2 2 1 5

P X fi ðE2 2 E2 ÞE2 ð i fi ÞE02 0 i 1 ; 2 2 2 E 2 ÞðE2 2 E 2 Þ 2 E0 2 E ðE i 0 i

(4.31)

where the first term is the single-oscillator approximation, and the second term is the sum of the remaining terms. Since the remaining terms behave like a singularity at the energy Ei, the second term can be neglected when the photon energy E ,, Ei. In that range Eq. (4.31) can then be represented by the SEO relationship, n2 2 1 

F0 E02 E d E0 5 2 ; E02 2 E2 E0 2 E 2

(4.32)

P where F0 5 Ed/E0 5 i fi , and E0 and Ed are parameters from the SEO approximation. When the photon energy approches Ei, however, the refractive index n can deviate from the SEO form due to the sigularity at Ei. In that case, the SEO model does not describe the refractive index n near the band gap. Fig. 4.11 shows the behavior of the remaining terms in Eq. (4.31) corresponding to the oscillator energy Ei. Note that the effect of the sigularities at Ei is not trivial only when E approaches Ei, and ðE02 2 Ei2 Þ=ðE02 2 E2 Þ  1when E approaches Ei. Thus, we obtain the approximation fi ðE02 2 Ei2 ÞE2 fi E 2  ðE02 2 E2 ÞðEi2 2 E2 Þ ðEi2 2 E2 Þ

as

E ! Ei :

(4.33)

For illustration, in Fig. 4.11 the simplified expression on the right of Eq. (4.33) is compared with the full term on the left. Here the smallest Ei is the direct energy gap Eg. The energy range of most interest for semiconductor devices is often the transparent region below the energy gap. Since the effect of sigularities at larger Ei in this region is negligibly small compared with that at Eg, one can ignore the effect

Optical dispersion of ternary II
VI semiconductor alloys

87

Figure 4.11 The behavior of the second terms of Eq. (4.31) having the oscillator energy Ei. A simplified expression is also shown for comparison. Note that two of the curves are on top of each other.

of all sigularities except the singularity at Eg, which yields the improved SEO approximation, n2 2 1 

F0 E02 fg E2 ; 1 ðEg2 2 E2 Þ E02 2 E2

(4.34)

where fg 5 e2Ng/mεoωg2, which is a measure of the oscillator strength at the fundmental absorption edge. The first term of this formula is exactly the SEO formula, and the second term gives the correction when the photon energy approaches the band gap energy. The parameters F0, EdE0, and fg are then defined as F0 5 εN 2 1 5 nð0Þ2 2 1 5

X

fi 5

i

Ed E0 5 F0 E02 5 ð

X i

fg Eg2 ~Ng :

fi ÞE02 5

X e2 Ni ; mε0 ω2i i

X e2 Ni¯h2 i

mε0 Ei2

(4.35)

! E02 ~hNi i;

(4.36)

(4.37)

The dispersion of the refractive index below the interband absorption edge in IIVI semiconductor compounds and alloys are successfully analyzed using this improved SEO fitting formula. In addition to providing a measure of the parameters E0 and Ed, the fitting of the refractive index data by Eq. (4.34) also yields the value

88

Chalcogenide

of direct band-gap energy and oscillator strength associated with the fundamental absorption edge.

4.3.3 Comparison between various semi-empirical fits for ZnTe To illustrate the fitting procedures of the SEO and the improved SEO models, we will use the dispersion of the refractive index n for ZnTe as an example. We will then extend the improved SEO model to other II-VI semiconductor compounds and ternary alloys. Fig. 4.12 shows the dispersion of n obtained for ZnTe using both the prism coupler data (full circles) and the reflectivity data (open circles). It is seen from the figure that the reflectivity data are in excellent agreement with the three prism couple points, providing independent verification for the reliability of the results. In Fig. 4.12, we also show two empirical Sellmeier-type dispersion curves taken from the literature. Up to the 2.1 eV, our work is in rather good agreement with the dashed curve, which shows the results of Maple [23] (observed in the wavelength range 0.57
2.6 μm). However, there is a small deviation from Maple’s results at energies approaching the band edge. This may indicate a much stronger excitonic

Figure 4.12 Dispersion of the refractive index n obtained for ZnTe using both the prism coupler data (full circles) and the reflectivity data (open circles). Two empirical Sellmeiertype dispersion curves have been taken from the literature (dashed and dotted curve). The solid curve represents the Sellmeier-type fitting curve obtained by the authors.

Optical dispersion of ternary II
VI semiconductor alloys

89

Figure 4.13 Refractive index data of ZnTe in the form of a graph 1/(n2-1) vs E2.

transition for MBE-grown epilayer samples. The solid curve shows a fit to Sellmeier-type relation with the present fitting parameters. Although the Sellmeiertype curve-fitting procedure does describe the refractive index dispersion very well, no physical significance has been attached to its parameters, and the expression serves primarily as a formula for interpolation. By plotting 1/(n2-1) versus E2 in Fig. 4.13, we show that the refractive index data for ZnTe show a curve consisting of two parts. Far below the band gap energy Eg, the data show a typical straight-line behavior. At short wavelengths (approaching the energy band gap), a negative deviation from the straight line is observed due to proximity of the band edge and/or excitonic absorption. A solid straight line is plotted in Fig. 4.13, its linear pffiffiffi parameters being obtained by fitting the data with photon energies below Eg/ 2. One can easily obtain the values of parameters E0 and Ed for the SEO approximation from slope and intercept of the linear fit. A dotted curve presenting Afromowitz’s MSEO approximation with the same parameters E0 and Ed is also ploted in Fig. 4.13. Note, however, that there is significant deviation between the MSEO curve and experimental data. In addition, Fig. 4.13 also includes a dot-dashed curve obtained using a fit of Pikhtin-Yas’kov model. As mentioned before, the refractive index data still cannot be fitted well by these models at photon energies approaching the band edge. We will now attempt to illustrate the physical insights which can be extracted from analyzing the various semi-empirical models. By plotting 1/(n2-1) versus E2, the refractive index data for ZnTe are shown in Fig. 4.14A. A linear fitting pffiffiffi (solid straight line) is applied on the data with photon energies less than Eg/ 2, which correspond to the SEO approximation, i.e. to the first term of the improved SEO model. In Fig. 4.14B, the deviation between the experimental data and the SEO approximation is shown by plotting [(n2-1)-E0Ed/(E02-E2)] versus E2.

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Chalcogenide

Figure 4.14 A step-by-step fitting procedure using the improved single efficient oscillator (SEO) model.

It is clearly seen that large deviations appear at photon energies above 2 eV. One can fit this curvature deviation by using the second term of the improved SEO model, i.e., [fgE2/(Eg2-E2)]. An alternative method is to plot this deviation in the graph of 1/[(n2-1)-E0Ed/(E02-E2)] versus 1/E2 in the range of photon energies above 2 eV (Fig. 4.14C). When a linear fitting is applied here, parameters fg and Eg can be obtained by simple calculation using slope and intercept of the linear fit. If the direct band gap energy Eg is already known, the deviation can be plotted in the form of 1/[(n2-1)-E0Ed/(E02-E2)] versus 1/E2-1/Eg2. This will allow us to obtain fg by using the linear function y 5 ax. Thus, one can obtain all four parameters of the improved SEO model. In Fig. 4.14D, the solid curve presents the improved SEO formula with these parameters. Note that the experimental data are in rather good agreement with the present semi-empirical fitting. In general, the improved SEO model involves a 2 1 2 or 2 1 1 parameter linear fitting depending on our knowledge of Eg. Furthermore, in case of alloys, all the E0, Ed, fg and Eg parameters can be interpolated according to the alloy composition. This allows us to use the improved SEO model to determine the refractive index n directly for any II-VI ternary alloy and at any photon energy. In Table 4.6, we list the fitting parameters and formulas used in this section.

Optical dispersion of ternary II
VI semiconductor alloys

91

Table 4.6 Fitting formulas and parameters for ZnTe. Fitting formula (λ units: μm, E units: eV) Sellmeier-type formula: 2 n2 5 A 1 λ2Bλ2 C

Values of parameters Marple

A 5 4.27, B 5 3.01, C 5 0.142

Present work

A 5 5.43, B 5 1.92, C 5 0.182

PY formula: 1 0:7 E2 2 ðh ¯ ωÞ2 n2 2 1 5 ðpffiffiffiffiffiÞln 1 π Eg Eg2 2 ðh ¯ ωÞ2 1

G1 G2 G3 1 1 ½E12 2 ðh ¯ ωÞ2  ½E22 2 ðh ¯ ωÞ2  ½E32 2 ðh ¯ ωÞ2 

WD SEO model: n2 2 1 5

Ed E0 ½E02 2 ðh ¯ ωÞ2 

Improved SEO model: n 21  2

4.4

F0 E02 E02 2 E2

1

Eg 5 2.28, E1 5 3.7, E2 5 5.5, E3 5 0.0236, G1 5 36.634, G2 5 103.55, G3 5 0.000825

WD

E0 5 4.34, Ed 5 27.0

Present work

E0 5 4.225, Ed 5 26.20

Unknown Eg

E0 5 4.225, F0 5 6.199, fg 5 0.022, Eg 5 2.262

Known Eg 5 2.28 eV

E0 5 4.225, F0 5 6.199, fg 5 0.025

fg E2 ðEg2 2 E2 Þ

Data analysis and discussion

4.4.1 Experimental results for ternary II-VI alloys Refractive index data for II-VI semiconductor compounds and their ternary alloys (the available ternary alloys include ZnCdSe, ZnCdTe [33], ZnBeSe, ZnMgSe [13], ZnMnSe [13], ZnMgTe, ZnMnTe, and ZnSeTe [34]) are first plotted as a graph of 1/(n2-1) versus E2 from the near infrared region to the their direct band gap energy. All refractive index data sets are obtained as a combination of the prism coupler method and reflectivity. All measurements are made at room temperature (297
300 K). Then, for each alloy composition the improved SEO approximation is applied to fit the corresponding experimental data using the aforementioned procedures. For each ternary alloy composition, the fits yield one set of parameters. Finally, the result of the parameters E0, Ed, fg, Eg and εN for these materials are given in tables or plots (here, we only show those of ZnCdTe, ZnMgTe and ZnSeTe) as a function of alloy composition. The agreement between the fits and the experimental data is generally very good. Thus, a continuous variation of the refractive index n can be precisely obtained as a function of wavelength and alloy composition for the transparency region.

92

Chalcogenide

For each of the alloys, a table is given to demonstrate the change of the parameters as the alloy composition is varied. Importantly, from this listing the parameters for zinc-blende-type BeSe, MgSe, MnSe, MgTe, and MnTe can be obtained by extrapolation. This provides the opportunity to understand the physical properties of such “hypothetical” materials, which do not exist in nature, but have an implicit existence as part of the zinc blende alloy. The tables are listed in the sequence: ZnCdSe, ZnCdTe, ZnBeSe, ZnMgSe, ZnMnSe, ZnMgTe, ZnMnTe, and ZnSeTe (Tables 4.7
4.14).

Table 4.7 Fitting results for the dispersion of Zn1-xCdxSe films. Composition (x)

Eg (eV) (reflectivity)

E0 (eV)

Ed (eV)

ε0

fg

Eg (eV) (dispersion)

0 0.05 0.14 0.22 0.26 0.32 0.38 0.43 0.5 0.57 1

2.72 2.64 2.54 2.45 2.39 2.33 2.27 2.20 2.15 2.05 1.68

5.21 5.16 5.01 4.87 4.84 4.74 4.65 4.53 4.48 4.36 3.91

25.5 25.3 24.7 24.1 24.0 23.6 23.2 22.6 22.4 21.9 19.9

5.90 5.91 5.93 5.94 5.95 5.97 5.98 5.99 6.00 6.03 6.08

0.0470 0.0410 0.0409 0.0311 0.0349 0.0339 0.0316 0.0304 0.0367 0.0427 0.0417

2.72 2.66 2.55 2.46 2.40 2.34 2.26 2.20 2.14 2.05 1.68

Table 4.8 Fitting results for the dispersion of Zn1-xCdxTe films. Composition (x)

Eg (eV) (reflectivity)

E0 (eV)

Ed (eV)

ε0

fg

Eg (eV) (dispersion)

0 0.08 0.27 0.51 0.67 0.83 1

2.27 2.20 2.04 1.86 1.74 1.63 1.51

4.35 4.24 4.10 3.91 3.83 3.75 3.74

27.1 26.4 25.5 24.3 23.8 23.3 23.2

7.23 7.22 7.22 7.22 7.21 7.21 7.21

0.0342 0.0290 0.0314 0.0339 0.0465 0.0423 0.0240

2.28 2.21 2.05 1.86 1.75 1.64 1.52

Table 4.9 Fitting results for the dispersion of Zn1-xBexSe films. Composition (x)

Eg (eV)a [35], (calculated)

E0 (eV)

Ed (eV)

ε0

fgb

Eg (eV)b (dispersion)

0 0.076 0.114 0.164 0.24 0.368 0.529 0.661 0.824

2.72 2.88 2.95 3.10 3.27 3.57 3.94 4.20 4.80

5.21 5.41 5.49 5.59 5.71 5.92 6.20 6.39 6.56

25.5 26.4 26.7 27.2 27.6 28.2 29.0 29.3 29.6

5.90 5.88 5.87 5.86 5.83 5.76 5.67 5.59 5.51

0.0470 0.0557 0.0549 0.0466 0.0625 0.0555




2.72 2.92 2.98 3.13 3.25 3.53




a

Calculated from the formula provided by Chauvet et al. in Ref. [34]. Due to limitation of the reflectometry setup, reliable parameters Eg and fg can only be obtained for alloys with a direct band gap lower than 3.2 eV.

b

Table 4.10 Fitting results for the dispersion of Zn1-xMgxSe films. Composition (x)

Eg (eV) (reflectivity)

E0 (eV)

Ed (eV)

ε0

fg

Eg (eV) (dispersion)

0 0.078 0.133 0.166 0.240 0.270 0.374 0.492

2.71 2.79 2.84 2.88 2.95 2.99 3.07 3.15

5.20 5.25 5.29 5.31 5.42 5.47 5.54 5.68

25.5 25.2 25.0 24.8 24.4 24.3 23.9 23.4

5.90 5.81 5.73 5.67 5.50 5.44 5.31 5.11

0.0474 0.0641 0.0497 0.0429 0.0453 0.0638 0.0570 0.0590

2.72 2.79 2.82 2.87 2.95 3.01 3.08 3.17

Table 4.11 Fitting results for the dispersion of Zn1-xMnxSe films. Composition (x)

Eg (eV) (reflectivity)

E0 (eV)

Ed (eV)

ε0

fg

Eg (eV) (dispersion)

0 0.025 0.046 0.07 0.11 0.13 0.15 0.19 0.23 0.25 0.26 0.3 0.35 0.44 0.46 0.54

2.73 2.72 2.72 2.73 2.73 2.73 2.74 2.75 2.76 2.77 2.78 2.78 2.79 2.83 2.84 2.86

5.20 5.19 5.21 5.22 5.22 5.22 5.23 5.22 5.23 5.24 5.23 5.24 5.25 5.22 5.21 5.20

25.4 25.3 25.3 25.3 25.1 25.1 24.9 24.9 24.8 24.7 24.7 24.5 24.4 24.1 24.0 23.6

5.88 5.87 5.85 5.84 5.81 5.80 5.77 5.77 5.74 5.71 5.72 5.68 5.65 5.62 5.61 5.54

0.0495 0.0511 0.0551 0.0569 0.0579 0.0583 0.0603 0.0604 0.0553 0.0600 0.0610 0.0659 0.0598 0.0601 0.0570 0.0596

2.73 2.73 2.73 2.73 2.73 2.73 2.73 2.75 2.76 2.77 2.77 2.78 2.79 2.82 2.84 2.86

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Chalcogenide

Table 4.12 Fitting results for the dispersion of Zn1-xMgxTe films. Composition (x)

Eg (eV) (reflectivity)

E0 (eV)

Ed (eV)

ε0

fg

Eg (eV) (dispersion)

0 0.0988 0.118 0.169 0.221 0.411 0.516 0.667

2.28 2.33 2.35 2.40 2.44 2.57 2.64 2.80

4.22 4.29 4.32 4.41 4.45 4.66 4.71 4.98

26.2 25.9 25.7 25.3 25.0 24.2 23.8 22.5

7.20 7.03 6.96 6.72 6.62 6.19 6.05 5.52

0.0251 0.0269 0.0325 0.0334 0.0362 0.0262 0.0520 0.0311

2.28 2.33 2.37 2.41 2.47 2.54 2.70 2.82

Table 4.13 Fitting results for the dispersion of Zn1-xMnxTe films. Composition (x)

Eg (eV) (reflectivity)

E0 (eV)

Ed (eV)

ε0

fg

Eg (eV) (dispersion)

0 0.07 0.12 0.29 0.42 0.48 0.56 0.63 0.71 0.75 1

2.28 2.30 2.32 2.42 2.47 2.49 2.54 2.59 2.64 2.67 (2.90)

4.22 4.24 4.25 4.32 4.36 4.38 4.41 4.45 4.50 4.51

26.2 26.0 25.9 25.3 25.0 24.7 24.5 24.2 23.6 23.3

7.20 7.14 7.09 6.85 6.74 6.64 6.56 6.43 6.24 6.17

0.0251 0.0230 0.0233 0.0361 0.0344 0.0458 0.0438 0.0486 0.0449 0.0395

2.28 2.31 2.32 2.43 2.47 2.52 2.57 2.60 2.66 2.69

Table 4.14 Fitting results for the dispersion of ZnSexTe1-x films. Composition (x)

Eg (eV) (reflectivity)

E0 (eV)

Ed (eV)

ε0

fg

Eg (eV) (dispersion)

0 0.145 0.439 0.545 0.685 0.715 0.747 0.825 0.883 1

2.27 2.21 2.19 2.25 2.36 2.38 2.41 2.51 2.56 2.72

4.35 4.43 4.70 4.77 4.89 4.92 4.96 5.06 5.12 5.21

27.1 27.0 26.6 26.4 26.1 26.2 26.1 25.8 25.7 25.5

7.23 7.10 6.67 6.54 6.34 6.32 6.26 6.10 6.03 5.90

0.0342 0.0595 0.0422 0.0483 0.0575 0.0581 0.0664 0.0587 0.0413 0.0470

2.28 2.21 2.19 2.25 2.36 2.39 2.41 2.50 2.58 2.72

Optical dispersion of ternary II
VI semiconductor alloys

95

Figure 4.15 A plot of 1/(n2-1) versus E2 for Zn1-xCdxTe.

4.4.2 Summary Using both reflectivity and prism coupler techniques, we have obtained the wavelength (or frequency) dependence of the index of refraction n for a series of II-VI semiconductor alloys. As is evident from previous sections, the improved SEO formula used in producing dispersion relations for these zinc blende II-VI alloys gives a rather accurate fit, with very small chi-squared values. In Figs. 4.15
4.17 the refractive index data of ZnCdTe, ZnMgTe, and ZnSeTe are plotted in the form of graphs of 1/(n2-1) versus E2 for the range from near infrared region to their direct band gap energies. Those plots clearly show the accuracy of the improved SEO fitting. The data for other ternaries (e.g., ZnCdSe, ZnMgSe, ZnMnSe, ZnMnTe, and ZnBeSe) show similar behavior to ZnCdTe, ZnMgTe, and ZnSeTe. Having obtained the dispersion curves n(ω) for such a wide range of II-VI alloys, we also sought a phenomenological relation in which both the wavelength dependence and the alloy concentration could be incorporated for each alloy family [n(ω, x)]. This was achieved by performing a linear or polynomial fit to each of the parameters F0, E0, fg, and Eg for a given II-VI semiconductor alloy. The data for ZnCdTe are plotted here as an example. Figs. 4.18 to 4.23 show Eg, E0, F0, fg, Ed, and F0E02 for ZnCdTe as a function of Cd concentration. Clearly, except for fg, all other parameters give good fits to linear or polynomial functions.

96

Chalcogenide

Figure 4.16 A plot of 1/(n2-1) versus E2 for Zn1-xMgxTe.

The relations for these parameters are shown in Table 4.15 for all the II-VI alloy systems discussed in this chapter [36]. The dispersion results obtained from the present work thus enable one to determine n directly and very accurately for any alloy composition and at any photon energy, and should thus prove useful in designing various optoelectronic devices base involving these ternary systems that depend on the value of n. It should be noted that parameter fg depends strongly on the quality of the crystal, so that in determining the refractive index n we use average values of fg. Nevertheless, the formulas still give rather accurate fits for most dispersion data. In the Appendix, the dispersion data are shown in the graphs of refractive index n versus alloy concentration x at several different wavelengths. The refractive index data obtained by the prism coupler method are also plotted in the figures at several available laser wavelengths, giving strong evidence for the accuracy of the fits. In addition, the Appendix shows the lattice constants d as function of alloy concentration x for all alloys. The collection of such graphs should prove useful in designing various optoelectronic devices, such as distributed Bragg reflectors (DBRs). Although some of the binary II-VI semiconductor compounds (e.g., BeSe, MgSe, and MnSe etc.) have crystal structures other than zinc blende, their presence

Optical dispersion of ternary II
VI semiconductor alloys

Figure 4.17 A plot of 1/(n2-1) versus E2 for ZnSexTe1-x.

Figure 4.18 A plot of Eg versus Cd concentration x for Zn1-xCdxTe.

97

98

Chalcogenide

Figure 4.19 A plot of E0 versus Cd concentration x for Zn1-xCdxTe.

Figure 4.20 A plot of F0 versus Cd concentration x for Zn1-xCdxTe.

is implicit in the ternary alloys grown by MBE in the zinc blende structure. One can therefore obtain information on many of the physical properties of these “hypothetical” binary zinc-blende-type crystals by extrapolation. The parameters of several zinc blende type II-VI semiconductor compounds, obtained by using fitting equations shown in Table 4.15, are listed in Table 4.16. The calculated or measured lattice constants are also listed in the Table. A detailed discussion of these materials is included in the next section.

Optical dispersion of ternary II
VI semiconductor alloys

99

Figure 4.21 A plot of fg versus Cd concentration x for Zn1-xCdxTe.

Figure 4.22 A plot of Ed versus Cd concentration x for Zn1-xCdxTe.

4.5

Physical interpretation and discussion

4.5.1 Physical meaning of fitting parameters The refractive-index dispersion of II-VI semiconductor compounds and their ternary alloys summarized in Tables 4.15 and 4.16 shows several remarkable characteristics. First, parameter E0 can be seen as the average transition energy, which is defined in terms of moments Mr of ε2(ω). The average transition energy E0 can be

100

Chalcogenide

Figure 4.23 A plot of F0E02 versus Cd concentration x for Zn1-xCdxTe. Table 4.15 Summary of fitting results of parameters F0, E0, fg, and Eg in the improved Seo formula for arbitrary alloy concentrations. Alloy

F0

E0 (eV)

fg

Eg (eV)

Zn1-xCdxSe

4.903 1 0.191x

0.04

Zn1-xCdxTe

6.227
0.0214x

Zn1-xBexSe Zn1-xMgxSe Zn1-xMnxSe

4.907
0.299x 2 0.240x2 4.908
1.617x 4.878
0.611x

Zn1-xMgxTe

6.213
2.457x

5.228
1.686x 1 0.359x2 4.339
1.078x 1 0.471x2 5.232 1 2.193x 2 0.696x2 5.172 1 1.013x 5.194 1 0.277x 2 0.049x2 4.203 1 1.099x

Zn1-xMnxTe

6.192
0.792x 2 0.732x2 6.306
1.42x

4.226 1 0.244x 1 0.186x2 4.313 1 0.881x

0.035

2.728
1.338x 1 0.290x2 2.275
0.846x 1 0.0965x2 2.72 1 2.086x 1 0.505x2 2.718 1 0.949x 2.722 1 0.0995x 1 0.294x2 2.285 1 0.637x 1 0.248x2 2.28 1 0.359x 1 0.261x2 2.28
0.653x 1 1.108x2

ZnSexTe1-x

0.035 0.047 0.053 0.056 0.03

0.045

written as E02 5 M-1/M-3 directly from Kramers-Kronig relations. The oscillator energy E0 is therefore independent of the scale of ε2(ω), and is consequently an “average” energy gap of the material. Since the
1 and
3 moments are involved in the computation, the ε2(ω) spectrum is weighted most heavily near the interband absorption threshold. The average band gap is therefore intrinsically related to the lowest direct band gap Eg by E0  2.1 1 Eg. The graph of E0 versus Eg is plotted

Optical dispersion of ternary II
VI semiconductor alloys

101

Table 4.16 Extrapolated parameters for II-VI semiconductor compounds with zinc blende structure. ZB-type II-VI compound

Lattice constant a ˚) (A

F0

E0 (eV)

Ed (eV)

F0E02 (E0Ed)

Eg (eV)

ε0

BeSe MgSe MnSe ZnSe CdSe MgTe MnTe ZnTe CdTe

5.137 5.905 5.903 5.668 6.050 6.41 6.343 6.103 6.477

4.37 3.29 4.27 4.91 5.08 3.71 4.91 6.23 6.20

6.72 6.2 5.2 5.22 3.91 5.32 4.65 4.35 3.73

29.8 21.1 22.3 25.6 19.9 20.9 21.8 27.1 23.2

199 134 106 134 77.6 113 104 112 86

5.5 3.62 3.12 2.72 1.68 3.13 2.9 2.28 1.50

5.37 4.29 5.27 5.91 6.08 4.71 5.91 7.23 7.20

Figure 4.24 A plot of E0 versus Eg.

in Fig. 4.24, where the solid line represents the linear equation E052.1 1 Eg. Note that the data point for BeSe is slightly displaced from the line. Previous work suggests that the energy band structure of zinc blende BeSe may have an indirect band gap [33], which yields a smaller value of E0 than that of direct band gap Eg. It should be noted that the linear equation E052.1 1 Eg is only approximate for II-IV binary compounds. For ternary alloys, it is very difficult to give a simple equation relating E0 to Eg. For example, Eg for ZnSeTe shows a dramatic bowing as a function of Se content (Fig. 4.25), but the average transition energy E0 depends linearly on the Se content, as seen in Fig. 4.26.

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Chalcogenide

Figure 4.25 Graph of Eg versus Se concentration x for ZnSexTe1-x.

Figure 4.26 Graph of E0 versus Se concentration x for ZnSexTe1-x.

Second, parameter Ed can be written as Ed2 5 M-13/M-3 from Kramers-Kronig (K-K) relations. Thus, E0 depends on the scale of ε2(ω), and serves as an interband transition strength parameter. From Wemple and DiDomenico’s work, Ed equals

Optical dispersion of ternary II
VI semiconductor alloys

103

βNcZaNe, and β is essentially two-valued, taking on the “ionic” value β i 5 0.26 6 0.04 eV or the “covalent” value β c 5 0.37 6 0.05 eV. For most our samples, β has the vale between 0.32 and 0.47, which indicates that all the semiconductor alloys studied here are essentially covalent tetrahedrally bonded ANB82N zinc-blende systems. It is interesting to note that BeSe has the largest value of β, suggesting that the nature of the band structure of BeSe is fundamentally different from other II-VI compounds. Third, the total effective-oscillator strength F0E02 (E0Ed) may be related to the “average” electron density of valence electrons, while fgEg2 is only related to the optical transition at the direct energy band gap. From the fitting results, the values of fgEg2 are two orders of magnitude smaller than those of F0E02. This indicates that optical absorption at the fundamental gap is indeed also a small fraction of all the possible interband transitions. However, this weak transition at the direct band gap clearly affects the dispersion of the refractive index near the energy gap. Considering the optical absorption at the fundamental gap, we know that the absorption intensity is directly proportional to the optical joint density of states corresponding to direct energy gap. The optical joint density of states can be written as ρðνÞ 5

ð2mr Þ3=2 ðhν2Eg Þ1=2 πh ¯2

hν $ Eg ;

(4.38)

where 1/mr 5 1/me 1 1/mh, me and mh being the effective conduction and valence band masses. If both heavy hole and light hole transitions are accounted for, one can see that by plotting fgEg2 versus 3mr,hh3/2 1 mr,hh3/2 in Fig. 4.27, the values of fgEg2 are closely related to the absorption coefficient α ~ (mr,hh3/2|dcv,hh|2 1 mr,hh3/2| dcv,lh|2) [37], where dcv,hh and dcv,lh are the dipole matrix elements of heavy hole

Figure 4.27 Graph of fg Eg2 versus (3mr,hh3/2 1 mr,lh3/2).

104

Chalcogenide

Table 4.17 Near-band-gap oscillator strength and values of conduction and valence band effective Masses (in units of electron Mass m0) in zinc-blende II-VI semiconductor compounds. II-VI compound

fg

Eg

fgEg2

mc

mlh

mhh

mr,lh

mr,hh

ZnSe CdSe ZnTe CdTe

0.047 0.04 0.035 0.02

2.72 1.68 2.28 1.51

0.348 0.113 0.182 0.0456

0.14 0.09 0.11 0.08

0.24 0.1 0.16 0.1

0.79 0.9 0.6 0.9

0.088 0.047 0.065 0.044

0.119 0.082 0.093 0.073

and light hole transitions, respectively. Note that |dcv,hh|2 5 3|dcv,lh|2 for zinc blende II-VI compounds. Therefore, fgEg2 does indeed provide some energy band information relevant to the fundamental gap (e.g., the density-of-states or effective conduction band mass). The relevant data are listed in Table 4.17. In addition, for II-VI ternary alloys the values of fgEg2 also depend on the quality of the crystal, which suggests that the electronic density of states and the effective masses of semiconductor alloys are strongly affected by growth conditions. The last remarkable property of the dispersion data is that there are three different types of behavior in the graphs of 1/(n2-1) versus E2. For ZnCddSe and ZnCdTe alloys, the F0 [n(0)2-1] is not significantly changed as the compositions of the alloys vary, but F0E02 is changed. In contrast, the values of F0E02 (E0Ed) does not depend significantly on the alloy composition for ZnMgSe, ZnMgTe, ZnMnSe, ZnMnTe and ZnSeTe, but they all have different n(0)2-1. For the alloy ZnBeSe, both n(0)2-1 and E0Ed depend on the concentration of Be. Considering the position of Be, Mg, Mn, Zn, and Cd in the periodic table, we can immediately see that the parameter F0E02 (or E0Ed) depends on the row number of the cation in the periodic table. This indicates that for higher row numbers, the valence electrons are much farther away from the atomic core, and the “average” electron density of the valence electrons is thus much smaller, resulting in weaker interactions. Therefore, no matter what kind of anion we have (Se or Te), Zn and Mn behave very similarly to each other: they have almost the same effective valence electron density, and thus the same value of E0Ed. Furthermore, using the f-sum-rule integral and the K-K relation, it is easy to show that ε1(0) 5 1 1 M-1, which yields the total number of effective valence electrons. Note that both Zn and Cd are in the same column of the periodic table, having a full d-shell and 2 outer valence electrons. Their valence electrons experience the same effect from their full d-shell electrons, which yield similar values of ε1(0). On the other hand, Mn has a half-filled d-shell, and Mg has no d-shell electrons. In Table 4.16 one can thus see the effect of this progression of d-shell occupation on ε1(0). Finally, zinc-blende BeSe shows some special characteristic (not following any of the trends previously identified), suggesting again that BeSe has energy band characteristics that distinguish it from other II-VI semiconductor compounds. In the

Optical dispersion of ternary II
VI semiconductor alloys

105

next section we will show that the “ionicity” of II-VI binary compounds also show trends that “follow” the periodic table.

4.5.2 Optical dispersion and ionicity Using time-dependent perturbation theory, the frequency dependence of the real part of electronic dielectric constant can be approximated in the form ε1 ðωÞ 5 1 1 Ep2

X n

ðEn2

fn : 2 E2 Þ

(4.39)

Here e and m are, respectively, the electronic charge and mass; and Ep2 5 nve2h2/4π2mε0 is the plasma frequency of the valence electrons, nv being the effective density of valence electrons. Phillips and Van Vechten have shown that ε1(0) 5 1 1 Ep2DA/Eav2, where A 5 1 - Eg/4Ef 1 (Eg/4Ef)2/3, Ef is the Fermi energy, and D is a d-state correction factor. Thus the average energy gap Eav can be sensibly decomposed into homopolar and heteropolar parts Eh and C that obey the quadrature relation Eav2 5 Eh2 1 C2. Furthermore, these authors defined a useful scale of ionicity fi 5 C2/(Eh2 1 C2) within the framework of the single-gap dielectric model [38
42]. Since we have obtained static dielectric constants for several zinc-blende II-VI semiconductor compounds, we now apply the Philips theory to achieve the scale of ionicity for these materials. We have calculated Ep2 5 nve2h2/ 4π2m using nv 5 8 N/a3, where N 5 4 and a is the lattice constant, and thus determine Eav2 5 Ep2DA/[ε1(0) - 1]. By assuming that C 5 0 for C, Si, and Ge, and that Eh only depends on the lattice constant [in the form of a relation log(Eh) 5 2.502 2.479 3 log(a)], we obtain the values of Eh and C. The results are listed in Table 4.18. This procedure depends sensitively on the value of the static dielectric constant ε(0), but it clearly yields reasonable values. Note that some of our results are different from the results of Phillips and Van Vechten. This is because the different crystal structures considered in the two cases have different Ep and ε1(0). Our results show that BeSe is more covalent than ZnSe and MgSe, but is much more ionic than that was predicted by Phillips and Van Vechten [43]. Shaw has given another operational definition of ionicity, based on experimentally determined moments of ε2(ω) [43]. This author defined a normalized disper2 2 sion energy Ed in the form Ed 5 β m,n21ε2d , where ε2d 5 Ep2[ε1(0) - 1], and n and m designate the rows in the periodic table for the binary constituents. The normalizing 2 coefficients β m,n2 are determined by requiring (arbitrarily) that Ed 5 1 for the nonionc single-constituent diamond-structure materials C, Si, Ge, and Sn. This allows 2 one to calculate β m,n2 5 2β n,n2β m,m2/(β n,n2 1 β m,m2) and hence Ed . Therefore, 2 Shaw defined ionicity as fi 5 1 - Ed . Results calculated using dispersion data taken in this chapter are also listed in Table 4.18. In addition, the last column shows results calculated by a qualitative fit of the formula Ed  60[ξ(1 - fi)]1/2 given by Shaw, where Ed is the dispersion energy defined by Wemple and DiDomenico, and

Table 4.18 Ionic character fi for zinc-blende II-VI semiconductor compounds.a ZB-type II-VI compound

Lattice ˚) constant a (A

Ep (eV)

Eh (eV)

C (eV)

Ed (eV)

Eav (eV)

m

fi (1)

fi (2) C2/ (Eh2 1 C2)

fi (3)

fi (4)

BeSeb MgSeb,c MnSe ZnSeb CdSeb,c MgTeb,c MnTe ZnTeb CdTeb

5.137 5.905 5.903 5.668 6.050 6.410 6.343 6.103 6.477

18.04 14.64 14.65 15.57 14.12 12.94 13.15 13.93 12.74

5.50 (5.7) 3.89 (3.3) 3.89 4.31 (4.3) 3.66 (3.6) 3.17 (3.2) 3.26 3.59 (3.6) 3.09 (3.1)

6.23 (3.4) 6.57 (6.4) 5.85 5.56 (5.6) 5.26 (5.5) 5.68 (3.6) 5.11 4.50 (4.5) 4.42 (4.4)

29.8 21.1 22.3 25.6 19.9 20.9 21.8 27.1 23.2

8.31 7.64 7.03 7.04 6.41 6.50 6.06 5.76 5.40

1.63 1.64 1.98 1.81 2.56 1.51 1.71 1.65 1.88

0.299 0.790 — 0.626 0.699 0.554 — 0.608 0.675

0.562 0.741 0.693 0.625 0.674 0.762 0.711 0.612 0.672

0.649 0.787 0.747 0.674 0.698 0.797 0.747 0.639 0.675

0.507 0.753 0.741 0.636 0.780 0.757 0.736 0.592 0.701

a For purposes of comparison, the last four columns contain fi values predicted by theories or definitions of Phillips and Shaw. fi (1) is cited from Van Vechten’s work (see Refs. [37] and [38]) using the theory of Phillips. fi (2) is calculated from the present work using the same model of Phillips. fi (3) is calculated from this work using the ionicity definition of Shaw. fi (4) is calculated from the parameter Ed using a qualitative fit formula from Shaw’s work. b The values calculated by Van Vechten are provided in parentheses. c In Van Vechten’s work, the crystal structure of these materials is not zinc-blende.

Optical dispersion of ternary II
VI semiconductor alloys

107

ξ  1/2. These two sets of results serve as additional evidence for the scale of ionicity of the II-VI semiconductor compounds. Finally, by comparing Eqs. (4.39) and (4.22), the model of Phillips and Van Vechten suggests that the value of Ep2 should be equal to EdE0 (F0E02). The point is that model does not give a good description of dispersion; so that Ep2 is actually not equal to EdE0. We then calculate the effective mass m 5 Ep2/EdE05M1/(M-12/ M-3) defined by Shaw, which describes the distribution of valence electrons. From the calculated results in Table 4.18, one sees shows that m has roughly the value of 2, decreases as the row number of the anions increases in the periodic table (i.e., it is larger for Se than for Te), and increases as the row number of cations increases in the periodic table (i.e., it is smaller for Zn than for Cd).

References [1] J.D. Jackson, Classical Electrodynamics, 2nd ed., Wiley, New York, 1962. [2] B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, Wiley, New York, 1991. [3] H.A. Kramers, Nature 117 (1926) 775 [abstract]; H.A. Kramers, Atti Congr. Intern. Fisici, Como-Pavia-Roma (N. Zanichelli, Bologna) 2 (1928) 545
557; H.A. Kramers, Phys. Z. 30 (1929) 522.. [4] R. de, L. Kronig, J. Opt. Soc. Am. 12 (1926) 547. [5] P.P. Paskov, Refractive indices of InSb, InAs, GaSb, InAsxSb1-x, and In1-xGaxSb: effects of free carriers, J. Appl. Phys. 81 (1997) 1890. [6] K. Sato, S. Adachi, Optical properties of ZnTe, J. Appl. Phys. 73 (1993) 926. [7] S. Adachi, T. Taguchi, Optical properties of ZnSe, Phys. Rev. B 43 (1991) 9569. [8] S. Adachi, T. Kimura, N. Suzuki, Optical properties of CdTe: experiment and modeling, J. Appl. Phys. 74 (1993) 3435. [9] L. Vin˜a, S. Logothetidis, M. Cardona, Temperature dependence of the dielectric function of germanium, Phys. Rev. B 30 (1984) 1979. [10] L.X. Benedict, E.L. Shirley, R.B. Bohn, Theory of optical absorption in diamond, Si, Ge, and GaAs, Phys. Rev. B 57 (1998) R9385. [11] S. Adachi, Model dielectric function of hexagonal CdSe, J. Appl. Phys. 68 (1990) 1192. [12] S. Adachi, Model dielectric constants of Si and Ge, Phys. Rev. B 38 (1988) 12966. [13] S. Adachi, Model dielectric constants of GaP, GaAs, GaSb, InP, InAs, and InSb, Phys. Rev. B 35 (1987) 7454. [14] X. Liu, U. Bindley, Y. Sasaki, J.K. Furdyna, Optical properties of epitaxial ZnMnTe and ZnMgTe films for a wide range of alloy compositions, J. Appl. Phys. 91 (2002) 2859. [15] J.K. Furdyna, M. Dobrowolska, H. Luo, Semiconductors, diluted magnetic, Encycl. Appl. Phys. 17 (1996) 373. [16] J.K. Furdyna, J. Kossut, A simple lattice-matching guide for superlattices and heterostructures of tetrahedrally-boned semiconductors, Superlattices Microstructures 2 (1986) 89. [17] J.K. Furdyna, Diluted magnetic semiconductors, J. Appl. Phys. 64 (1988) R29. [18] J.M. Hartmann, J. Eymery, L. Carbonell, Y. Wang, Large and small angle x-ray scattering studies of CdTe/MgTe superlattices, J. Appl. Phys. 86 (1999) 1951.

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[19] F.C. Peiris, S. Lee, U. Bindley, J.K. Furdyna, A prism coupler technique for characterizing thin film II
VI semiconductor systems, J. Appl. Phys. 84 (1998) 5194. [20] H.J. Lee, C.H. Henry, K.J. Orlowsky, R.F. Kazarinov, T.Y. Kometani, Refractive-index dispersion of phosphosilicate glass, thermal oxide, and silicon nitride films on silicon, Appl. Opt. 27 (1988) 4104. [21] C.S. Kim, M. Kim, S. Lee, J. Kossut, J.K. Furdyna, M. Dobrowolska, CdSe quantum dots in a Zn1-xMnxSe matrix: new effects due to the presence of Mn, J. Crystal Growth 214/215 (2000) 395. [22] K. Watanabe, M. Th. Litz, M. Korn, W. Ossau, A. Waag, G. Landwehr, et al., Optical properties of ZnTe/Zn1
xMgxSeyTe1
y quantum wells and epilayers grown by molecular beam epitaxy, J. Appl. Phys. 81 (1997) 451. [23] J.M. Hartmann, J. Cibert, F. Kany, H. Mariette, M. Charleux, P. Alleysson, et al., CdTe/MgTe heterostructures: growth by atomic layer epitaxy and determination of MgTe parameters, J. Appl. Phys. 80 (1996) 6257. [24] T. Dietl, (Diluted) Magnetic semiconductors, in: T.S. Moss (Ed.), Handbook of Semiconductors, Vol. 3b, North Holland, Amsterdam, 1994, p. 1251. [25] D.T.F. Marple, Refractive index of ZnSe, ZnTe, and CdTe, J. Appl. Phys. 35 (1964) 539. [26] M. Born, E. Wolf, Principles of Optics, 3rd ed., Pergamon, New York, 1965, p. 97. [27] F.C. Peiris, S. Lee, U. Bindley, J.K. Furdyna, Precise and efficient ex situ technique for determining compositions and growth rates in molecular-beam epitaxy grown semiconductor alloys, J. Vac. Sci. Technol. B 18 (2000) 1443. [28] S.H. Wemple, M. DiDomenico Jr., Optical dispersion and the structure of solids, Phys. Rev. Lett. 23 (1156) (1969). 24, 193(E) (1970). [29] S.H. Wemple, M. DiDomenico Jr., Behavior of the electronic dielectric constant in covalent and ionic materials, Phys. Rev. B 3 (1971) 1338. [30] M.A. Afromowitz, Refractive index of Ga1-xAlxAs, Solid State Commun. 15 (1974) 59. [31] Y. Kaneko, K. Kishino, Refractive indices measurement of (GaInP)m/(AlInP)n quasiquaternaries and GaInP/AlInP multiple quantum wells, J. Appl. Phys. 76 (1994) 1809. [32] A.N. Pikhtin, A.D. Yas’kov, Dispersion of refractive index of semiconductors with diamond and zinc-blende structures, Sov. Phys. Semicond 12 (1978) 622. [33] F.C. Peiris, S. Lee, U. Bindley, J.K. Furdyna, Precise measurements of the dispersion of the index of refraction for CdZnTe alloys, J. Elect. Mater. 29 (2000) 798. [34] Hosun Lee, S.M. Kim, B.Y. Seo, E.Z. Seong, S.H. Choi, S. Lee, et al., Optical study of ZnSexTe1
x alloys using spectroscopic ellipsometry, Appl. Phys. Lett. 77 (2000) 2997. [35] C. Chauvet, E. Tournie´, J.P. Faurie, Nature of the band gap in Zn1
xBexSe alloys, Phys. Rev. B 61 (2000) 5332. [36] X. Liu, J.K. Furdyna, Optical dispersion of ternary II
VI semiconductor alloys, J. Appl. Phys. 95 (2004) 7754. [37] W. Sch¨afer, M. Wegener, Semiconductor Optics and Transport Phenomena, SpringerVerlag, Berlin, 2002. [38] J.C. Phillips, Dielectric definition of electronegativity, Phys. Rev. Lett. 20 (1968) 550. [39] J.C. Phillips, Dielectric theory of cohesive energies of tetrahedrally coordinated crystals, Phys. Rev. Lett. 22 (1969) 645. [40] J.A. Van Vechten, Quantum dielectric theory of electronegativity in covalent systems. I. Electronic dielectric constant, Phys. Rev. 182 (1969) 891. [41] J.A. Van Vechten, Quantum dielectric theory of electronegativity in covalent systems. II. Ionization potentials and interband transition energies, Phys. Rev. 187 (1969) 1007. [42] J.C. Phillips, Ionicity of the chemical bond in crystals, Rev. Mod. Phys. 42 (1970) 317. [43] R.W. Shaw Jr., Optical dispersion and ionicity, Phys. Rev. Lett., 25, 1970, p. 818.

Optical dispersion of ternary II
VI semiconductor alloys

Appendix

Figure A.4.1 Refractive index n versus Cd concentration for Zn1-xCdxSe.

109

110

Figure A.4.2 Refractive index n versus Cd concentration for Zn1-xCdxTe.

Chalcogenide

Optical dispersion of ternary II
VI semiconductor alloys

Figure A.4.3 Refractive index n versus Be concentration for Zn1-xBexSe.

111

112

Figure A.4.4 Refractive index n versus Mg concentration for Zn1-xMgxSe.

Chalcogenide

Optical dispersion of ternary II
VI semiconductor alloys

Figure A.4.5 Refractive index n versus Mn concentration for Zn1-xMnxSe.

113

114

Figure A.4.6 Refractive index n versus Mg concentration for Zn1-xMgxTe.

Chalcogenide

Optical dispersion of ternary II
VI semiconductor alloys

Figure A.4.7 Refractive index n versus Mn concentration for Zn1-xMnxTe.

115

116

Figure A.4.8 Refractive index n versus Se concentration for ZnSexTe1-x.

Chalcogenide

Optical dispersion of ternary II
VI semiconductor alloys

117

Figure A.4.9 Lattice constant a versus alloy composition x for II-VI semiconductor ternary alloys.

Group-IV monochalcogenides GeS, GeSe, SnS, SnSe

5

Lyubov V. Titova1, Benjamin M. Fregoso2 and Ronald L. Grimm3 1 Department of Physics, Worcester Polytechnic Institute, Worcester, MA, United States, 2 Department of Physics, Kent State University, Kent, OH, United States, 3Department of Chemistry and Biochemistry, Worcester Polytechnic Institute, Worcester, MA, United States

5.1

Introduction

Group-IV monochalcogenides, a class of layered van der Waals materials, includes GeS, GeSe, SnS and SnSe. Since decades ago, bulk group-IV monochalcogenides have been studied extensively for their exceptional thermoelectric and electronic properties [1 4]. Interest in two-dimensional (2D) materials and new approaches to growth or isolation of high-quality monolayer and few-layer structures have sparked renewed interest in re-examining this class of materials. Recent theoretical investigations suggest that monolayer- and few-layer GeS, GeSe, SnS and SnSe exhibit a combination of extraordinary properties unlike any other 2D materials studies thus far: a giant in-plane switchable spontaneous ferroelectric polarization [5 8], multivalley band structure [9], large spin-orbit splitting of 19 86 meV [5], exciton binding energy up to 0.6 eV [7,10] and carrier mobility that rivals that of crystalline Si [11]. Possibility of tuning the band gap and inducing semiconductor to metal transition with external electric field has also been predicted for GeS [12]. Broken inversions symmetry associated with a spontaneous ferroelectric polarization is expected to result in anisotropic electronic and optical properties, a pronounced zero-bias inplane shift current upon above the band gap photoexcitation as well as large second harmonic generation (SHG) [6,13 17]. Combining ferroic properties with semiconducting nature makes these materials truly unique and opens opportunities for applications in nonvolatile memory, where reading can be achieved by using the anisotropy of electronic conduction and optical properties [18]. High carrier mobility and band gaps in the visible and near-infrared range holds promise for applications in photovoltaics [19 25]. Finally, recent measurements of unprecedented thermoelectric figure of merit ZT as high as 2.6 6 0.3 at 923 K in SnSe single crystals [26] are brining renewed focus on thermoelectric properties of these and other group-IV monochalcogenides [27 29]. Calculations also predict low lattice thermal conductivities and high ZT, comparable to the value measured in SnSe single crystals, in all monolayer group-IV monochalcogenides [30,31]. Thermoelectric properties will not be the emphasis of this Chapter as we instead focus on overviewing the current state of the art in our understanding of the structure of group-IV Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00007-5 © 2020 Elsevier Ltd. All rights reserved.

120

Chalcogenide

monochalcogenide monolayer and few-layer structures, their electronic, linear and non-linear optical properties, as well as of the approaches to their fabrication.

5.2

Crystal lattice and band structure calculations

While group-IV monochalcogenides under different conditions can adopt a number of crystal structures including litharge, inverted litharge and the buckled hexagonal structures, a highly distorted rock salt is the thermodynamically stable structure for GeS, GeSe, SnS, SnSe at room temperature and up to 600  C [32]. It belongs to an orthorhombic Pnma 62 space group and consists of puckered layers separated by a ˚ [5,33,34]. In the bulk, the unit cell covers two adjavan der Waals gap of about 4 A cent puckered layers (Fig. 5.1). In a monolayer form, group-IV monochalcogenides have a symmetry of the point group mm2. Each atomic species is covalently bonded to three neighbors of the other species, leaving one lone pair per atom. As shown in Fig. 5.1, distortion of the orthorhombic

Figure 5.1 (A) Top and (B and C) side views of crystal structure of group-IV chalcogenides with a distorted orthorhombic lattice structure. Corresponding short (d1) and long (d2) bond lengths and bond angles (θ1 and θ2) are given in Table 5.1. Distortion of orthorhombic structure results in a spontaneous ferroelectric polarization ( 6 P0) in armchair direction in each monolayer (B). In a multilayer structure, layer stacking is antiferroelectric. (D) Top view of a distorted orthorhombic unit cell. (E) Brillouin zone corresponding to (D).

Group-IV monochalcogenides GeS, GeSe, SnS, SnSe

121

structure manifests in the deviation from 90 of the cation anion bond angles θ1 (defined in the top view) and θ2 (defined in the side view in armchair direction), resulting in the three short (d1) and three long (d2) bonds about the cation. Table 5.1 lists structural parameters including lattice constants, long and short bond lengths, and angles θ1 and θ2 for the bulk materials. The difference between the lattice parameters in the bulk structure, in the bilayer and even in the monolayer limit is predicted to be very small, less than 2%, as the lattice parameters are dominated by the ionic radii of the constituents [5]. While similar in structure to black phosphorus, the symmetry of group-IV monochalgogenides’ lattice is lower and lacks inversion center, giving rise to the predicted ferroelectricity and ferroelasticity of the monolayers, as well as in-plane anisotropy of electronic and optical properties. In the monolayer limit, structure of group-IV monochalcogenides is also characterized by the spontaneous tensile strain along the armchair direction and compressive strain along the zigzag direction relative to the centrosymmetric paraelastic structure [7]. In an intuitive picture, the intrinsic polarization of the monolayers results from a dipole moment in armchair direction, as shown in Fig. 5.1B. It is coupled to the spontaneous ferroelastic lattice strain that manifests in the distortion of the orthorhombic structure [7,8,13,14,17,39,40]. As in the case of the traditional ferroelectrics, the spontaneous polarization at temperatures below the Curie temperature is driven by the soft optical phonons associated with the displacive instability [39,41,42]. As temperature approaches Tc from above, the frequency of the soft modes becomes imaginary, resulting in a transition from a high-symmetry structure with zero polarization to the symmetry-broken phase. Fei et al. have investigated the ferroelectric phase transition in group-IV monochalcogenide monolayers via the Monte Carlo simulations using an effective Landau-Ginsburg Hamiltonian [39]. The characteristic double-well schematic diagram of the total energy as a function of deviation of the angle θ2 from 90 or, equivalently, as a function of the polarization, is shown in Fig. 5.2, along with the theoretically predicted configurational potential barrier height EG and spontaneous polarization in group-IV monolayers at zero temperature. TC is above the room temperature for all group-IV monochalcogenide monolayers and increases with the EG, going from 326 K in SnSe to 1200 K in SnS, to 2300 K in GeSe, and 6200 K in GeS. Other theoretical studies of ferroelectricity in group-IV monochalcogenide monolayers arrived at the same qualitative conclusions and comparable quantitative results [18,43]. A finite energy barrier separating the states with opposite polarization (Fig. 5.2) and the direct coupling between the ferroelectic and ferroelastic orders implies that the polarization can be switched by the application of the external electric field or strain. Indeed, while no experimental confirmation has yet been reported, several DFT studies have investigated multistability and reversible phase transitions in group-IV monochalcogenide monolayers [7,18,43]. They have demonstrated that application of electric field along the armchair direction can reverse the direction of electric polarization, and external tensile strain in the armchair direction increases electrical polarization. Furthermore, in-plane tensile strain in zigzag direction can reorient the lattice, switching armchair and zigzag directions [7,43].

Table 5.1 Crystal structure parameters for group-IV monochalcogenides. Parameters calculated or measured for the bulk structure as opposed to the monolayer structure are marked by an asterisk ( ). Experimentally obtained parameters are identified by a label “Exp.”. ˚ ) ( ) c (A

˚) b (A

˚) b (A

˚) d1 (A

˚) d2 (A

θ1

θ2

GeS

10.47 (Exp. [34]) 10.81 [5]

2.45 [36]

96.8 [36]

91.7 [36]

10.82 (Exp. [34])

4.30 (Exp. [34]) 4.40 [5] 4.40 [5] 4.43 [32] 4.26 [5] 4.38 [32] 4.39 (Exp. [37]) 4.40 [34] 4.45 [5]

2.44 (Exp. [35]) 2.47 [5]

GeSe

3.64 (Exp. [34]) 3.67 [32] 3.68 [5] 3.68 [5] 3.834 (Exp. [37])

2.57 (Exp. [35]) 2.62 [5] 2.9 (Exp. [37,38])

2.56 [36]

97 [37,38] 96.3 [36]

91.3 [36]

4.24 [5] 4.26 [32] 4.34 (Exp. [34]) 4.35 [5]

2.66 (Exp. [35]) 2.69 [5]

2.63 [36]

96.8 [36]

89.0 [36]

4.36 [5] 4.41 [32] 4.44 [26] 4.45 (Exp. [34])

2.79 (Exp. [35]) 2.84 [5]

2.74 [36]

96.0 [36]

89.0 [36]

10.84 (Exp. [37]) 11.31 [5] SnS

11.14 [34] 11.37 [5]

SnSe

11.49 (Exp. [26]) 11.50 (Exp. [34]) 11.81 [5]

3.85 [34] 3.91 [5] 3.95 [32] 3.99 [5] 3.97 (Exp. [34]) 4.02 [5] 4.06 [32] 4.07 [5] 4.135 (Exp. [26]) 4.15 (Exp. [34]) 4.27 [32] 4.22 [5] 4.30 [5]

4.47 [5]

Group-IV monochalcogenides GeS, GeSe, SnS, SnSe

123

(A)

Total energy

Polarization

0

PS EG 0 θ2 – 90°

(B)

Material

EG (mev)

GeS GeSe SnS SnSe

–580.77 –111.99 –38.30 –3.758

PS (10–10 C m–1) 5.06 3.67 2.62 1.51

Figure 5.2 (A) Schematic of a double-well potential of group-IV monochalcogenide monolayers as a function of deviation from the orthorhombic structure and the corresponding polarization. (B) Theoretically predicted zero-temperature potential barrier and the spontaneous polarization calculated by Fei et al. [39].

Predicted room-temperature ferroelectricity in the monolayer limit makes groupIV monochalcogenides unique in the ferroelectric materials family as the depolarization field destroys ferroelectric polarization in the conventional ferroelectrics such as perovskite oxides when their thickness is less than several monolayers [44 46]. Group-IV monocalcogenide ferroelectrics hold promise for realizing ultrathin, flexible nonvolatile ferroelelastic memory devices. Furthermore, structural anisotropy in group-IV monochalcogenides naturally results in anisotropic electronic and optical properties, as discussed below. Possibility of controlling optical and electronic anisotropy the application of external strain and electric fields opens yet new possibilities for optoelectronic devices.

5.3

Electronic band structure

Electronic band structure of group-IV monochalcogenides in the bulk form, as well as in the bilayer and monolayer limit, has been investigated using firstprinciples calculations based on density functional theory by several authors [2,5,9,32,33,47 51]. Calculations show that the top of the valence band is formed mainly by the p orbitals of the chalcogen, partly hybridized with the cation’s s orbitals, while the bottom of the conduction band is comprised from the empty p orbitals of cation. The hybrid functional band structure calculations using the Vienna ab initio simulation package (VASP) with the Heyd-Scuseria-Ernzerhof (HSE06) [52] exchange-correction term yields gap energies that are in a good agreement with

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Figure 5.3 Electronic band structures for monolayer, bilayer, and bulk group-IV monochalcogenides calculated using the HSE hybrid functional. The valence band maxima (VBM) and conduction band minima (CBM) are highlighted by full circles. Dashed black arrows indicate possible direct transitions (T1 and T2) to points very close in energy to the VBM and CBM. Triangles indicate the position of the CBM when spin-orbit coupling effects are considered. Reproduced with permission from L.C. Gomes, A. Carvalho, Phosphorene analogues: isoelectronic two-dimensional group-IV monochalcogenides with orthorhombic structure, Phys. Rev. B 92 (8) (2015) 085406.

available experimental data [5]. Fig. 5.3 shows the band structures for the bulk, bilayer and monolayer GeS, GeSe, SnS and SnSe calculated along the high symmetry paths of the Brillouin zone using the HSE functional by Gomes and Carvalho [5]. While the crystal structure is strikingly different along the zigzag and the

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armchair direction, the dispersion of the bands nearest to the gap is nearly the same along the Γ-X and the Γ-Y directions. In addition, multiple valence- and conduction band valleys are present, and many of them are very close in energy. These results are in reasonable agreement with DFT calculations of the energy band structure of group-IV monochalcogenides by the other authors [32,53]. Calculations show that while almost all of the compounds have an indirect band gap, except for the monolayer SnSe and GeSe, and bilayer GeSe, possible direct transitions that are close in energy to the indirect gaps occur in all compounds. In fact, the difference between direct and indirect energy gaps that is comparable to the room-temperature thermal energy of 26 meV can explain the observed room-temperature photoluminescence in the bulk GeS, as will be discussed later [51]. In the case of GeSe, calculations by Song et al. have also showed that the monolayer and bilayer stuctures possess the direct band gap, while the bulk GeSe is an indirect gap material; furthermore, they found that GeSe remains direct band gap when its thickness is increased up to six monolayers [49]. Table 5.2 compiles the calculated gap energies, and compares them to the available experimental measurements of bulk gap energies determined from optical experiments. While comparing calculated and experimental parameters, it is important to keep in mind that theoretical calculations assume temperature 0 K and experiments are typically carried out at temperatures varying from 4 K to 300 K. However, available temperature-dependent measurements suggest that temperature-dependent band gap renormalization in group-IV monochalcogenides is small, such as 0.03 eV for SnS [56]. Reduced dielectric screening in 2D materials, especially in their monolayer form, results in pronounced excitonic effects [58,59]. While the mean-field, independent-particle DFT is notoriously inadequate in predicting quasiparticle Table 5.2 Band gap energies (Eb) for monolayer, bilayer, and bulk group-IV monochalcogenides. Unless otherwise specified, gap energier were calculated by Gomes and Carvalho [5] using DFT with the HSE hybrid functional. LDA stands for the Local Density Approximation to DFT [53], and PBE is the Perdew, Burke and Ernzerhof functional used in DFT [49]. The star ( ) indicates direct band gaps. Results from several experimental studies of optical absorption are also shown. Material

Theory: monolayer Eb (eV)

Theory: bilayer Eb (eV)

Theory: bulk Eb (eV)

Experiment: bulk Eb (eV)

GeS

2.32

2.20

1.58 [54]

GeSe

SnS

1.54 1.15 (PBE [49]) 1.96

1.45 1.15 (PBE [49]) 1.60

1.81 1.74 (LDA [53]) 1.07 0.81 (PBE [49]) 1.24 1.26 (LDA [53])

SnSe

1.33

1.20

1.08 1.18 [55] 1.47 [56] 1.049 1.076 [57] 0.989 0.903 [57]

1.00

1.14 [54]

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Table 5.3 Calculated exciton binding energies obtained from the effective-mass model for the lowest excitations along the armchair and zigzag directions for various group-IV monochalcogenide monolayers [61]. Material

Eex (eV)

GeS GeSe SnS SnSe

0.98 0.38 0.50 0.28

armchair (a) direction

Eex (eV)

zigzag (b) direction

0.77 0.36 0.50 0.28

energies, approaches that directly account for many-body electron-electron interactions such as the Bethe-Salpeter Equation (BSE) [60] combined with GW approach have been successfully applied to predict exciton binding energies in group-IV monochalcogenide monolayers [10,61]. There calculations predict large exciton binding energies, ranging from 0.9 eV in SnS to 1.2 eV in GeS [10]. An alternative approach to calculating excitonic effects in group-IV monochalocogenide monolayers and taking into account in-plane anisotropy has been developed by Xu et al. based on the effective mass approach [61]. Table 5.3 lists the resulting exciton binding energies for the lowest energy excitations in the armchair and zigzag directions. In the case of multilayer and bulk structures, relatively strong, compared to other 2D materials, interlayer interactions in group-IV monochalcogenides result in significantly weaker exciton binding energies. Quantitative predictions of the excitonic effects in the bulk group-IV monochalcogenides have been obtained by including interlayer interactions via the PBE 1 vdW [10,62]. These calculations yield bulk excitons binding energies that are almost at order of magnitude lower that monolayer binding energies, 60 100 meV [10,62].

5.4

Electronic and optical properties

Based on predicted band structure, onset of optical absorption for monolayer groupIV monochalcogenides occurs in the near-infrared (for SnSe) to the green (GeS) part of the visible spectrum. Highly anisotropic crystal structure with significant differences between the armchair and zigzag directions in monolayers as well as the in the bulk form of groupIV monochalcogenides suggests that electronic and optical properties may also be anisotropic. Linear optical response of the materials is well characterized by the imaginary part of the dielectric function, ε2 , which is a measure of the density of available states and hence characterizes the materials ability to absorb radiation. Larger values of ε2 indicates shorter penetration depths and a large absorption constant which are highly desirable in the visible regime (1.5 3 eV) to avoid energy waste in solar energy applications. Fig. 5.4 shows ε2 for monolayers of the group-IV monochalcogenide calculated within the independent particle

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Figure 5.4 Imaginary part of the dielectric function of monolayer group-IV monochalcogenides, calculated using DFT-BBE by Rangel et al. [13]. Here, the armchair direction is designated as z, zigzag direction as y, and x corresponds to the direction perpendicular to the base plane of the monolayers. Reproduced with permission from T. Rangel, B.M. Fregoso, B.S. Mendoza, T. Morimoto, J.E. Moore, J.B. Neaton, Large bulk photovoltaic effect and spontaneous polarization of single-layer monochalcogenides, Phys. Rev. Lett. 119 (6) (2017) 067402, DOI: 10.1103/ PhysRevLett.119.067402.

approximation from the band structures obtained from the DFT-PBE [13]. It shows that ε2 can reach large values, B50 2 200, over a wide range of visible frequencies. It is interesting that the response is roughly isotropic for Sn compounds [5,13,61]. This is due to the similar lattice constants in the armchair and zigzag directions of the materials and hence similar band structures in the Brillouin zone. While the experimental studies of linear optical properties of group-IV monochalcogenide monolayers are yet to be carried out, limited by the availability of monolayer samples, measurements on high quality bulk crystals are now available and confirm the in-plane anisotropy of optical properties. In one study, Hsueh, Li and Ho carried out detailed angle-resolved polarized optical absorption measurements on B 80 nm thick GeS crystal [62] and demonstrated that the absorption edge varies between the armchair and zigzag direction. Polarized transmittance spectra of a multilayer GeS crystal as a function of angle from the armchair direction (a) are shown in Fig. 5.5A, and derivative transmittance spectra in Fig. 5.5B. Derivative transmittance spectra clearly demonstrate existence of two absorption edges, Eg|| 5 1.60 eV and Eg\ 5 1.65 eV, with Eg|| most prominent near E || a (0 ) polarization at 1.6 eV while the Eg\ feature is dramatically enhanced close to the E || b (90 ). The first-principles BSE calculations with GW self-energy correction included to account for the many-electron screening effects yields a direct band gap of 1.69 eV near the Γ point, increased from the DFT value of 1.20 eV that reproduces experimental observations within the bounds of the experimental error. BSE-GW calculations also show the energetically lowest excitonic state at 1.62 eV that corresponds to the excitonic transition from the maximum of valence bands (VBM) to the minimum of conduction bands (CBM) near Γ point. It is weakly bound with a binding energy of 60 meV due to a relatively strong, compared to other 2D materials, interlayer interactions in bulk GeS. The predicted oscillator strength of this excitonic transition varies for polarizations parallel and perpendicular to the armchair direction, and can account for the observed absorption edge anisotropy. The emission due to the bright exciton at 1.62 eV is polarized along the

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Figure 5.5 Angular-dependent spectra of (A) transmittance and (B) derivative transmittance for multilayer GeS. Reproduced with permission from H.-C. Hsueh, J.-X. Li, C.-H. Ho, Polarization photoelectric conversion in layered GeS. Adv. Opt. Mater. 6 (4) (2018) 1701194, DOI: 10.1002/ adom.201701194.

armchair direction, while the calculated oscillator strength of an exciton transition at 1.69 eV with the E || b polarization is negligible [62]. Polarized photoluminescence and thermoreflectance measurements confirm these predictions, as shown in Fig. 5.6. Photoluminescence of both a 40 nm and a 270 nm GeS nanoflakes is fully polarized along the armchair direction. Optical emission peak matches well the optical transition near the band edge as identified in E || a polarized thermoreflectance spectrum (Fig. 5.6C) [63]. Similar observations were reported by Tan et al. who studied anisotropic optical absorption and emission properties of GeS flakes at room temperature and observed band edge emission at energy B1.6 eV that does not shift in energy for different thickness of GeS nanoflakes, as well as anisotropy of band-edge absorption between the armchair and zigzag directions [51]. Photoluminescence in has been observed in a few studies of bulk-like samples of group-IV monochalcogenides including GeS [51,63] and SnS [64], and has been ascribed to pronounced many-body effects. In GeSe, transition from the indirect to direct band gap as the thickness is reduced to three layers and less has been predicted by the first principles calculations and confirmed by the photoluminescence measurements by Zhao et al. [65]. Quantum confinement in all three dimensions

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Figure 5.6 (A) Polarized micro-photoluminescence spectra of multilayer GeS with thickness around of 270 nm (A) and with thickness of 40 nm (B). The measurements were done with the linearly polarized light along and perpendicular to the armchair (A) axis. (B) Optical microscope image and AFM results of the GeS multilayer nanoflakes A and B. B is about 40 nm and A is a three-step stack to about 270 nm. (C) The polarized thermoreflectance and micro-photoluminescence spectra of the multilayer GeS near band edge at 300 K. Reproduced with permission from C.-H. Ho, J.-X. Li, Polarized band-edge emission and dichroic optical behavior in thin multilayer GeS. Adv. Opt. Mater. 5 (3) (2017) 1600814, DOI: 10.1002/adom.201600814.

can dramatically blue-shift the band gap and may result in a bright photoluminescence in all group-IV monochalcogenides, as already been demonstrated for SnSe colloidal quantum dots [66] and SnS nanoparticles [67]. Along with the anisotropic optical absorption in group-VI monochalcogenides across the visible range, calculations predict an ultra-high carrier mobility, especially in the monolayer limit [11,30]. Table 5.4 lists effective carrier masses and intrinsic room temperature charge carrier mobility, limited only by phonon scattering, for the monolayer group-IV monochalcogenides, calculated by different DFTbased approaches [11,30,61]. While the lack of convergence between the different studies underscore the need for more detailed studies incorporating the feedback from the experimental measurements of carrier transport, all studies show exceptionally high carrier mobility. Predicted phonon-limited carrier mobilities, shown in Table 5.4 along with the calculated anisotropic carrier effective mass values, surpass the high mobility in other 2D materials such as MoS2 (B200 cm2 V21 s21) [68], and rival the mobility in single crystalline silicon (B cm2 V21 s21) [11]. They also exhibit considerable in-plane anisotropy, with lower effective masses and correspondingly higher mobility along the armchair direction. Similarly high and anisotropic carrier mobility is predicted for the bulk SnSe: at room temperature, the mobilities of electrons along the armchair and zigzag direction, and normal to the basal plane, are 801, 623, and 325 cm2 V21 s21, respectively, whereas those of holes are 299, 291, and 100 cm2 V21 s21, respectively [26]. Combination of strong optical absorption and high carrier mobility naturally lends itself to applications of group-IV monochalcogenides in photodetectors and field effect transistors. Indeed, Tan et al. have demonstrated a highly polarizationsensitive, broadband multilayer GeS photodetector with a photoresponsivity of 6.8 3 103 A W21 and an ultrahigh specific detectivity of 5.6 3 1014 Jones [69].

Table 5.4 Calculated carrier effective masses and mobility values. Material

me in armchair direction (m0)

me in zigzag direction (m0)

mh in armchair direction (m0)

mh in zigzag direction (m0)

μe in armchair direction (cm2 V21 s21)

μe in zigzag direction (cm2 V21 s21)

μh in armchair direction (cm2 V21 s21)

μh in zigzag direction (cm2 V21 s21)

GeS

0.20 [11] 0.19 [30] 0.22 [61] 0.27 [30] 0.13 [61] 0.24 [30] 0.19 [61] 0.15 [30] 0.13 [61]

0.41 [11] 0.37 [30] 0.50 [61] 0.30 [30] 0.40 [61] 0.28 [30] 0.20 [61] 0.16 [30] 0.12 [61]

0.23 [11] 0.23 [61]

0.61 [11] 0.92 [61]

50 [11] 100 [64]

0.44 [61]

580 [61]

100 [64]

0.22 [61]

0.27 [61]

530 [61]

420 [61]

0.12 [61]

0.14 [61]

2950 [11] 529 [30] 930 [61] 465 [30] 240 [61] 419 [30] 1930 [61] 965 [30] 32,080 [61]

160 [11] 60 [61]

0.14 [61]

3680 [11] 1045 [30] 8080 [61] 541 [30] 6220 [61] 623 [30] 1580 [61] 1036 [30] 2390 [61]

3620 [61]

840 [61]

GeSe SnS SnSe

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131

Multilayer GeS field effect transistor with high photoresponsivity of 206 A W21 under 1.5 μW cm22 illumination at 633 nm, with gate voltage of 0 V and drainsource voltage of 10 V [70]. High on-off ratio, photoresponsivity comparable to the best existing 2D photodetectors (3.5 A W21) at 4 V forward bias, and quantum efficiency of 553% has been also demonstrated in several studies of GeSe multilayer nanosheet photodetectors [37,38,71]. These results demonstrate that the optical and electronic properties of group-IV monochalcogenides, such as strong anisotropy, high carrier mobility and optical absorption across the entire visible range, make these 2D materials highly promising candidates for a myriad of applications in optoelectronic and solar energy conversion devices.

5.5

Nonlinear optical properties

Pronounced nonlinear optical effects in various layered 2D materials have inspired increasing research efforts aimed at uncovering the fundamental mechanism behind them as well as on developing new photonics applications [72]. Most of these efforts have been focused on graphene, transition metal dichalcogenides (MoS2, WS2, WSe2 and others), and more recently on black phosphorus. With the same orthorhombic structure as black phosphorus, structural anisotropy and predicted robust multiferroic nature, group-IV monochalcogenides hold significant promise for nonlinear optics. Moreover, group-IV monochalcogenides are distinctly oxidation-resistant and thus less vulnerable to degradation under ambient conditions in the presence of high power optical beams than other 2D materials [73]. We now describe features of the nonlinear optical response common to all group-IV monochalcogenides based on analytical and numerical density functional theory calculations. We concentrate on experimentally relevant quantities that are likely to remain true even if particle interactions are included. Second harmonic generation (SHG): As the intensity of radiation increases, nonlinear processes occur, such as the second harmonic generation (SHG). In the SHG, two photons are absorbed producing a single photon of twice the incident frequency. Single-layer group-IV monochalcogenide monolayers have point group mm2 and since the SHG is a third rank tensor only seven components are nonzero, namely zxx, zyy, zzz, yyz, xzx, xxz and yzy. In Fig. 5.7 we show the magnitude and imaginary part of the SHG tensor components relevant for incident linearly polarized. Note that the magnitude of the SHG is very large and it is highly anisotropic. The largest SHG is obtained for light polarized along the basal plane but perpendicular to the spontaneous polarization, or armchair, direction (Fig. 5.1). The magnitude of the SHG can reach values up to 10 nm V21 which is one order of magnitude larger than prototypical nonlinear semiconductor GaAs (1 nm V21) [74] and among the largest reported in 2D materials [14,17]. What is the origin of such large response? Is it correlated to the material spontaneous polarization?The short answer is that it is not known. Recent studies suggest many factors are involved including reduced dimensionality and strong in-plane

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Figure 5.7 Second harmonic generation (SGH) tensor for group-IV monochalcogenides. Armchair directions (and direction of the spontaneous polarization) is labeled z, zigzag direction y, and x is perpendicular to the basal plane. Reproduced with permission from S.R. Panday, B.M. Fregoso, Strong second harmonic generation in two-dimensional ferroelectric IV-monochalcogenides, J. Phys.-Condens. Matter 29 (43) (2017), DOI: 10.1088/1361-648X/aa8bfc.

polarization (see for example, [75]). Historically, the SHG is notoriously hard to predict from first principles alone. The SHG of GaAs, for example, does not agree with experiments after almost 60 years of research. For single-layer materials, however, the SHG may be easier to understand because they have less degrees of freedom and all macroscopic responses tend to be strongly correlated. This seems to be the case for single-layer group-IV monochalgogenides as the SHG is correlated to spontaneous polarization [15,17] and ferroelastic order [14]. Experimental studies will shed the light on the SHG in group-IV monochalcogenides in the near future. Injection and shift current: The so-called shift and injection dc-current are second order photovoltaic effects that differ from the conventional photovoltaic effect by the fact that interfaces or barriers, e.g., semiconductor pn-junctions, are not present. Because of this unique characteristic, they are usually called bulk photovoltaic effects (BPVEs). The practical interest in the BPVE is in the novel design of photovoltaic cells with efficiency is not constrained by the Shockley Queisser limit [76,77]. Injection current: The microscopic origin of the injection current has been clear for long time. It is due to unequal rate of carrier pumping into states with timereversed momenta in the Brillouin zone [78]. The injection current for the nonzero component of group-VI monochalcogenides is shown in Fig. 5.8. Just like the case of SHG, only seven components of the injection current tensor are non-zero, as prescribed by the point group mm2. However, the injection current is antisymmetric in the last two indices, which reduces the number of independent component two yyz and xxz. Since we choose the x-axis as the vacuum direction the component xxz is very small (strictly zero in the simulations), and hence we focus on the yyz component, where, as described before, y is along the zigzag, and z is along the armchair direction. First, note that the magnitude of the injection current in all these materials has a broad maximum of magnitude B1011 A V21 s22 in the visible regime

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Figure 5.8 Injection current of group-IV monochalcogenides. Large values of the injection current are obtained in the visible regime 1 3 eV. Comparison with density of states shows no obvious relation. This means injection current is controlled by the geometry of the Bloch wavefunction. Reproduced from S. R. Panday, S. Barraza-Lopez, T. Rangel, and B. M. Fregoso, Injection current in ferroelectric group-iv monochalcogenide monolayers, ArXiv:1811.06474 [condmat.mes-hall].

1 3 eV followed by shallow maxima with the opposite sign. The maximum of injection current is of the same magnitude across the group-IV monochalcogenides, and, surprisingly, is very large despite the 2D nature of these materials. For example, it is two orders of magnitude larger than that of CdS [79]. Experimentally, injection photocurrent can be generated either in a two-color excitation due to the interference of light with carrier frequencies at ω and 2ω when 2h ¯ ω crosses the band gap, or in a single color scheme, as a result of quantum interference between absorption pathways for orthogonal polarization components of a single-frequency beam [79 82]. Thus far, injection photocurrents in group-IV monochalcogenides have not yet been explored experimentally. The density of states, discussed earlier (Fig. 5.4), is also show in Fig. 5.8 in the expanded energy range for comparison. While there are some peaks common to both responses, there is no obvious relation, indicating that the form of the wave function plays a significant role in determining the magnitude of the injection current. Shift current: The shift current is a second order effect in the optical field and hence, similar to the injection current and SHG, requires breaking of the inversion symmetry, e.g., it occurs in ferroelectric materials. In the shift-current mechanism, the photoexcited electrons are thought to jump from one atom to the next one nearby. This occurs because the centers of charge of the conduction and valence bands are spatially separated [81,83 85]. The scale the electron jumps is given by the so-called shift vector, which can roughly be interpreted as a k-dependent band polarization difference [15]. In Fig. 5.9 we show the nontrivial tensor components of the shift current for the group-VI monochalcogenides. We find that the magnitude of the effective (bulk) shift current is very large for all materials reaching values of 150 μA V22 in GeSe over the visible range 1 2 eV. This value is the largest reported in any material [76].

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Chalcogenide

Figure 5.9 (A D) Shift current, integrated shift vector and imaginary part of the dielectric function of group-IV monochalcogenides. Very large values of shift current are obtained which are related to the ferroelectric order parameter via the shift vector but not directly related to the density of states. Reproduced with permission from T. Rangel, B.M. Fregoso, B.S. Mendoza, T. Morimoto, J.E. Moore, J.B. Neaton, Large bulk photovoltaic effect and spontaneous polarization of single-layer monochalcogenides, Phys. Rev. Lett. 119 (6) (2017) 067402, DOI: 10.1103/ PhysRevLett.119.067402.

While the peaks in the shift current match some peaks in the density of states, there is no straightforward relation between the two. What is the origin of the large shift current in group-VI monochalcogenides? As can be seen from the second row in Fig. 5.10, the shift vector integrated over the entire Brillouin zone follows the shift current closely. Since the integrated shift vector is proportional to band polarization differences (Rangel, 2017), ferroelectrics, which have finite spontaneous polarization, would potentially yield large shift current. Indeed this is what we find in the numerical experiments of single-layer monochalcogenides in Fig. 5.9. In the recent experiments on two of the group-VI monochalcogenides, GeS and GeSe, we have provided experimental evidence of shift photocurrent in these materials using THz emission measurements [86,90]. THz emission spectroscopy allows electrical contact-free, all-optical monitoring of the photoexcited carrier dynamics and transient photocurrents with sub-picosecond time resolution by detecting, in the far field, THz radiation emitted by those currents [81,85 90]. We have studied ultrafast photoinduced carrier dynamics in two systems: an array of GeS nanosheets

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Figure 5.10 (A) Schematic diagram of THz emission spectroscopy experiment. THz waveforms generated by an ultrafast shift current following 100 fs, 400 nm excitation in (B) GeS (15 μJ cm22 excitation fluence) and (C) GeSe(190 μJ cm22 excitation fluence). Waveforms recorded at different sample orientation show that rotating sample by 180 reverses polarity of the emitted pulse. (D) Normalized amplitude spectra of the THz waveforms generated by GeS and GeSe. The amplitude spectrum corresponding to the THz emission from ZnTe optical rectification source in response to B 1000 μJ cm22, 800 nm pulses, is also shown, offset for clarity.

of 10 30 μm length and ,1 μm width synthesized on sapphire substrates using the vapor-liquid-solid method [91] and a GeSe crystal of B2 mm in lateral dimensions grown using vapor transport and thinned to sub- μm thickness by mechanical exfoliation. In both cases, 2D structures contained many layers, with thickness from ,1 μm (GeS) to few μm (GeSe). Samples were excited at normal incidence with 400 nm, 100 fs laser pulses from a 1 kHz amplified Ti:Sapphire source. A pair of off-axis parabolic mirrors focused the THz pulses emitted by the photoexcited transient currents onto a [110] ZnTe crystal where they were coherently detected by free-space electro-optic sampling [92]. With excitation and detection at normal incidence, this experimental arrangement allows detecting exclusively the radiation from real or polarization photocurrents flowing in the basal planes of GeS or GeSe layers. A polarizer placed in the path of the THz beam ensured detection of only one linearly polarized component of the generated THz pulses. Sample orientation was varied by rotation of a sample stage through an angle θsample as illustrated schematically in Fig. 5.10A and the direction of the linear polarization of the optical pump pulse relative to the THz detection was varied by using a half-wave plate (not shown).

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Chalcogenide

Excitation with 400 nm (3.1 eV) pulses ensures direct interband excitation in both GeS and GeSe. Both systems emit nearly single-cycle electromagnetic field transients (Fig. 5.10B and C). Normal incidence geometry rules out currents flowing normal to the surface, such as Dember effect currents, as a possible origin. Linearly polarized single color excitation rules out injection currents, and above band gap excitation suggests that a real photocurrent rather than a transient polarization as in the case of the optical rectification, is responsible for the observed emission [83,88]. These observations, along with the linear excitation fluence dependence of the emitted THz pulse amplitude and tell-tale excitation polarization and sample orientation dependence confirm that the observed effect indeed originates in the shift current upon above the band gap photoexcitation. In both GeS and GeSe, the pump polarization has negligible effect on the emitted THz pulses. However, rotating the sample by 180 reverses the polarity of emission for both GeS and GeSe while the temporal shape of the waveform show only minimal change. This suggests the polarity of the emitted pulses, which carry the information about the direction of the shift current, is governed by the symmetry breaking in the orthorhombic lattice (Fig. 5.1), which is in turn gives its intrinsic ferroelectric polarization [15]. Unlike the monolayers, bulk group-IV monochalcogenides do possess inversion symmetry, ruling out a bulk shift current response. However, inversion symmetry is broken at the surface, indicating that the observed THz emission is due to a surface rather than bulk shift current. Shift current in response to photoexcitation suggests applications of these layered materials in BPVE photovoltaics and THz sources. Surface selectivity of the THz emission in GeS and GeSe may also lead to new applications of these 2D materials in chemical sensing. The temporal behavior and the corresponding bandwidth of the emitted pulses provide additional insight into the propertied of GeS and GeSe. We find that the bandwidth of the THz pulses emitted by GeSe is narrower, with the spectral weight limited to the low frequencies, as shown in the Fourier transform amplitude spectra (Fig. 5.10D). The shift current can be phenomenologically modeled as a convolution of the temporal derivative of the charge displacement with the pump intensity envelope [88]. For the pump pulse of B 100 fs duration it is expected to vary on sub-picoseconds time scale with the bandwidth of the emitted THz radiation extending to B10 THz [86,88]. In the case of GeS nanosheets, we find the bandwidth of the ZnTe detector crystal limits the observed bandwidth of THz emission, as comparison of the emitted amplitude spectrum with the spectrum of the THz pulse emitted by the ZnTe crystal identical to the detector crystal, which is a representation of the detector response function [86]. However, in the case of GeSe, we find that strong THz absorption by the sample itself is a limiting factor in uncovering the true ultrafast transient behavior of the surface shift currents from the THz pulses detected in the transmission geometry [87]. GeSe crystal exhibits strong absorption in the THz range, with absorbance increasing nearly fivefold between 0.2 1.8 THz, which can be attributed to low frequency B3u and B1u infrared active phonons centered in the 2.5 2.6 THz range [93]. B3u phonons in particular are associated with opposite motion of Ge and Se along the armchair direction and couple strongly to the THz radiation polarized along this direction. As a result, GeSe crystal itself acts

Group-IV monochalcogenides GeS, GeSe, SnS, SnSe

137

as a low pass filter, attenuating THz pulses emitted by the surface layer and broadening them to 1 2 ps in duration (Fig. 5.10C). Same phonon modes in GeS occur at higher frequencies and do not affect the observed bandwidth. Despite strong attenuation, efficiency of a B100 200 μm thick GeSe crystal, as well as that of the GeS nanosheets, as a THz source is comparable to the efficiency of a standard 1 mm thick [110] ZnTe source. Reducing the thickness of GeS or GeSe to minimize re-absorption of the emitted THz radiation would result in a broadband source with efficiency surpassing that of conventional sources. Experimental demonstration of THz emission by the surface shift current in group-IV monochalcogenides suggests these 2D materials as candidate for the next generation shift current photovoltaics, nonlinear photonic devices and THz sources.

5.6

Fabrication: single crystal growth and exfoliation; CVD, growth of 2D nanostructures

Among the primary literature that established the physiochemical properties of these IV-VI monochalcogenides, syntheses primarily included high temperature melt or transport methods. Melt methods included a zone melt [1], Bridgman growth [42] [94 96], and simply melting the elements together in evacuated ampoules [3,97]. Transport methods included both chemical vapor transport [35] that followed earlier work on other II-valent metal chalcogenides such as CdS and ZnS [98], and physical vapor transport such as evaporation/sublimation [4,35,99 104]. While recently applied to yield single-layer graphene from graphite [105], a quarter of a century earlier researchers employed adhesive tape to cleave thin samples from single crystal germanium(II) sulfide for fundamental studies [106]. Exfoliation remains an active method for producing GeS [51,107]. Recent efforts targeting monochalcogenide group-IV materials for broader applications to energy conversion and storage, as well as to emergent phenomena enabled by nanoscale structures have similarly expanded the synthetic methodologies for yielding interesting shapes, structures, and phases. High-temperature methods remain actively employed. Recent studies employed high-temperature Bridgman growth (GeS [108] and SnSe [29]), or stationary melts for SnS [109,110] and SnSe [27,28,111]. Van der Waals epitaxy produced SnS on graphene [112], while laser photolysis yielded GeSe and SnSe [113]. Chemical vapor deposition (CVD) often utilizing a vapor-liquid-solid (VLS) mechanism has yielded GeS nanowires [114,115], nanosheets [70], and amorphous structures [116]. Physical vapor transport methods have included evaporation from the IV-VI monochalcogenide [21,117 123], co-evaporation of atomic precursors [22,124], and RF sputtering [125,126]. Researchers have correlated deposition substrate temperatures with orientation control [119] and amorphous [127] vs orthorhombic vs cubic phase growth [123]. Many recent applications utilize lower-temperature methods to access metastable phases and nanostructures. Popular methods include hydrothermal

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(aqueous) and solvothermal (nonaqueous) synthesis at elevated temperatures in high-pressure reactors. Synthesis of SnS often includes tin(II) chloride as the tin precursor and alternatively thioacetamide [128], thiourea [67,129], or sodium sulfide [130] as the sulfur precursor. SnSe synthesis has employed elemental selenium as a precursor [131]. Nontraditional solvents recently included ionic liquids [67]. Shapes have included quantum dots [132], dendritic nanoparticles, nano-flakes, and nanobelts [54] and other high-surface-area structures that are implicated as battery electrodes [133 135]. Importantly, researchers note changes in growth mode with the concentration of the tin chelator, triethanolamine [128]. As with hydrothermal/solvothermal methods, chemical bath deposition [136] has broadened the scope of “low” temperature and ambient-pressure deposition methodologies that also accesses non-traditional shapes and phases. Many chemical bath recipes include similar precursors [136 138] to hydrothermal/solvothermal methods. Recent efforts include bath deposition from a single-source precursor compound containing both germanium xanthates for GeS [139], and a tetrakis-thioureatin(II)-chloride complex for SnS [140]. Interesting morphologies have included quantum dots via the inclusion of capping agents [141], as well as oriented, vertically stacked platelets of SnS [142] and of SnS/SnSe [143]. Formation of the metastable cubic phase has alternatively included SnCl2 and sodium thiosulfate with EDTA as a complexing reagent [144,145], and SnCl2 and thioacetamide [23,146,147]. High-temperature bath methods have enabled single-crystal nanosheets of GeS and GeSe [54]. Other near-ambient methods have included successive ionic layer adsorption and reaction (SILAR) to deposit SnS; [24,148] as well as electrodeposition of GeS in ionic liquids [149], SnS [150 152], and Sn(S, Se) [153]. Annealing has further played a critical role in the production of desired IV-VI monochalcogenide materials. Annealing conditions have included sulfurization of tin [20,154,155] and germanium [156] for synthesis as well as post-synthesis treatment. Post-synthesis annealing environments include sulfur [125,126], helium [157], argon/ethanol for carbon doping [158], vacuum annealing [121], and conversion of SnS2 to SnS under vacuum [25] or forming gas [159]. Air annealing [160] has effected a conversion of p-type SnS to n-type material [122]. These recent advances in fabrication of high quality, single crystalline group-IV nanostructures are finally enabling experiments aimed at characterizing their unique properties predicted by theory and development of a new generation of group-IV monochalcogenide-based optoelectronic and solar energy conversion devices. An example of single crystal micrometer-size GeSe nanosheets fabricated by a facile solution-phase synthesis is shown in Fig. 5.11 [38]. The observed rectangular, truncated triangular or elongated hexagonal shapes of nanosheets with average lateral dimensions of 9 10 μm by 4 5 μm area shapes reflect the distorted orthorhombic structure, as can be seen in the scanning electron (Fig. 5.11A) and transmission electron (Fig. 5.11B). The selected area electron diffraction (SAED) shows that the surface normal is along the [100] direction, in agreement with the most energetically stable plane (100) being the top surface of the crystal (Fig. 5.11C). Lattice ˚ and the in-plane bond angle θ1 is plane spacing d1 (defined in Fig. 5.1) is 2.9 A  97 , as high-resolution TEM (HRTEM, Fig. 5.11D) shows. Photocurrent

Group-IV monochalcogenides GeS, GeSe, SnS, SnSe

139

Figure 5.11 (A) SEM image of micrometer-sized GeSe nanosheets. (B, E) TEM and SEM image of a single GeSe nanosheet. (C) SAED pattern of a single GeSe nanosheet. (D) HRTEM image of a single GeSe nanosheet. Reproduced with permission from D.J. Xue, J. Tan, J.S. Hu, W. Hu, Y.G. Guo, L.J. Wan, Anisotropic photoresponse properties of single micrometer-sized GeSe nanosheet, Adv. Mater. 24 (33) 2012 4528 4533. https://doi.org/10.1002/adma.201201855.

measurements on GeSe nanosheet devices in this study revealed large anisotropy of photoresponse in directions parallel and perpendicular to the layers, with the on/off switching ratio for photocurrent perpendicular to the layers was 3.5 times higher than that parallel to the layers. In summary, we have overviewed the current state of knowledge of the properties of group-IV monochalcogenides GeS, GeSe, SnS and SnSe, that are an emergent class of 2D van der Waals materials. Optical and electronics properties of group-IV monochalcogenides are strongly anisotropic, and can potentially be engineered by strain and controlled by external fields. Group-IV monochalcogenide monolayers are predicted to combine robust room temperature ferroelectricity and ferroelasticity with giant spontaneous electric polarization and lattice strain.

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Consequently, they are expected to exhibit pronounced nonlinear optical effects. THz emission experiments suggest that surfaces of multilayered and bulk structures retain those unique properties. Band gaps ranging from the visible to the near-IR, in concert with ultra-high carrier mobility, puts these materials forth as candidates for applications in solar energy conversion, from photoelectrochemistry to the thirdgeneration solar cells based on bulk photovoltaic effects. In the near future, aided by the recent progress in fabrication of high quality, defect free bulk crystals as well as monolayer and few layer nanosheets, this promising new class of 2D materials is poised to become a platform for fundamental physics and transformative applications in communications, photonics, and solar energy conversion.

Acknowledgment LVT acknowledges the support from National Science Foundation under Grant No. DMR1750944.

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Further reading M. Freitag, M. Steiner, A. Naumov, J.P. Small, A.A. Bol, V. Perebeinos, et al., Carbon nanotube photo- and electroluminescence in longitudinal electric fields, ACS Nano 3 (11) (2009) 3744 3748. Available from: https://doi.org/10.1021/nn900962f.

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P. Ramasamy, D. Kwak, D.-H. Lim, H.-S. Ra, J.-S. Lee, Solution synthesis of GeS and GeSe nanosheets for high-sensitivity photodetectors, J. Mater. Chem. C 4 (3) (2016) 479 485. Available from: https://doi.org/10.1039/c5tc03667d. D. Vaughn, D. Sun, S.M. Levin, A.J. Biacchi, T.S. Mayer, R.E. Schaak, Colloidal synthesis and electrical properties of GeSe nanobelts, Chem. Mater. 24 (18) (2012) 3643 3649. Available from: https://doi.org/10.1021/cm3023192. S.M. Yoon, H.J. Song, H.C. Choi, p-Type Semiconducting GeSe Combs by a vaporization condensation recrystallization (vcr) process, Adv. Mater. 22 (19) (2010) 2164 2167. Available from: https://doi.org/10.1002/adma.200903719.

Epitaxial II-VI semiconductor quantum structures involving dilute magnetic semiconductors

6

S. Lee1, M. Dobrowolska2 and J.K. Furdyna2 1 Department of Physics, Korea University, Seoul, South Korea, 2Department of Physics, University of Notre Dame, Notre Dame, IN, United States

6.1

Introduction

The elements of group six of the periodic table, such as sulfides (S), selenides (Se), and tellurides (Te), are known as the chalcogens. Chalcogenide materials are chemical compounds involving at least one chalcogen ions. The best example is II-VI compound semiconductors consisting of elements of the second and sixth columns in the periodic table [1
4]. The crystal structure of the II-VI compound is such that each atom is surrounded tetrahedrally by four other atoms. This is called the zincblende structure. In this type of crystal, the lattice is formed with tetrahedral s-p3 bonding [5] involving the two valence s electrons of the group II elements and the six valence p electrons of the group VI elements. The wavefunctions of the core electrons are tightly bound so that only the eight valence electrons per each pair of atoms contribute to the chemical bonding and thus it is those eight electrons that determine the electrical and the optical properties of the semiconductor. The orbital of every atom (s- or p-like) hybridize with an orbital of a neighboring atom thus producing two levels which are called bonding and anti-bonding levels. As there is an effectively infinite number of atoms in the crystal the bonding and anti-bonding levels broaden into bands. Those states at the center of the Brillouin zone that constitute the highest lying valence band correspond to bonding combinations of p- like functions whereas the lowest lying conduction band is formed from antibonding combinations of s-like functions The energy difference between the conduction and the valence band edges is called the “band gap”. Due to the large energy gap in II-VI compounds such as ZnSe, CdSe, CdTe, ZnTe, and ZnS, these materials provide the possibility of optical device applications in the visible range (blue light emitters at panel displays, etc.). Specially, significant improvements in the growth of II-VI thin films have been achieved by epitaxial growth technique (e.g., molecular beam epitaxy (MBE)) [6,7], where single atomic layers of a semiconducting material can be grown on an appropriate substrate. A series of monolayers of the same material is called an epilayer and the quality of such epilayers has been shown to be superior to that of bulk crystals. This breakthrough in the growth technology has produced II-VI materials on which highly Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00009-9 © 2020 Elsevier Ltd. All rights reserved.

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informative optical and electrical measurements can be performed. Strong emphasis is given in the first part of this chapter to clearly understand the physical concepts of the band structure and optical properties of II-VI semiconducting epilayers, since a complete understanding of these aspects is also required to achieve the full device potential offered by II-VI materials. In contrast to the epilayers, the structure where two epilayers of different materials are grown on top of one another is called a heterojunction. When heterojunctions are repeated, the whole system becomes a heterostructure. This situation can be conveniently visualized by plotting the energy gap of the successive layers as a function of distance in the growth direction. The simplest such structure is the situation where two heterojunctions are grown ‘back to back’: e.g., an epilayer of a smaller band gap material is ‘sandwiched’ between two epilayers with larger band gap. If the dimensions in the growth direction of the central layer are reduced until its thickness is comparable to the Compton wavelength of charge carriers in the semiconductor, quantum mechanical effects dominate, and hence the central layer is called a quantum well (QW). These quantum mechanical effects include the confinement of wavefunctions of the carriers in the quantum well and the quantization of energy levels of the charge carriers in the QW. Using compound semiconductors, such quantum mechanical effects have been studied in various heterostructures, such as the single quantum well (SQW), multiple quantum wells (MQW), superlattices (SLs), and quantum dots (QDs) [8
11]. Many studies are done by investigating interband optical transitions which take place between eigenstates in the valence band and those in the conduction band. In these spectroscopic methods, transition energies in the spectrum depend on the band alignments of the quantum structures. Thus, the ability of externally controlling the band alignment of the structures provides tremendous advantages in studying such quantum effects. If II-VI compounds are used in the heterostructures, there exists the possibility of combining the structures with II-VI-based diluted magnetic semiconductors (DMSs), which shows a giant Zeeman splitting in the presence of external magnetic field [12
14]. One can then take advantage of such band gap tuning ability to investigate quantum effects in heterostructures using external magnetic field. For example, the dependence of inter-well coupling between the wells on barrier height can be investigated in a double quantum well (DQW) coupled by a DMS barrier, in which we have the ability to continuously vary the barrier height by changing magnetic field [15,16]. Furthermore, using DMS layers for specific wells in multiple quantum wells (triple or quintuple quantum wells), the wavefunction distributions of the eigenstates can be mapped out in experiments by observing the Zeeman splitting of transitions involving each of the eigenstate [17]. The achievement of such continuous variation of inter-well coupling in DQWs, and Zeeman mapping of wavefunction distributions of the eigenstates in MQWs was simply not possible without using DMS layers. In this chapter, we further review the Zeeman tuning ability of the DMSs in studying localization and spin polarization of carriers in IIVI-based quantum dot (QD) structures.

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6.2

155

Magneto-optical properties of ZnSe and ZnTe epilayers

6.2.1 Band structure and exciton II-VI semiconducting compounds crystallizing in the zinc-blende structure, such as CdTe, ZnTe, and ZnSe, consist of two interpenetrating face-center-cubic (f.c.c.) lattices. The band structures of such crystals has been formulated using various theoretical methods (tight binding, pseudo-potential, OPW methods, etc.) [18,19]. These methods are quite general, applying throughout the whole Brillouin zone in kspace. They are therefore quite complicated, and considerable computer time is therefore needed for their calculation. In many situations, however, we are only interested in a very small region of k-space -- for example that near the band edge, where all the free carriers are located. One can then achieve highly accurate results by simpler calculation procedures. One of the most accurate and convenient methods for calculating band structure in the vicinity of a specific point in k-space is the so called k  p method. This method was used by Seitz [20] to derive the expression for the effective mass, which accounts for the effect of the periodic field in crystals. This approach was then extended to the more complicated case of degenerate bands [21], including spin-orbital interaction [22], and was eventually established as a reliable basis for determining of band structure near special points in the Brillouin zone. In most II-VI compounds, only four bands are closely positioned in energy at the center of the Brillouin zone (the Γ point), where the bands have extrema: the s-like conduction band, and the three p-like valence bands. In the zinc blende crystals, the s-like band has the total symmetry of the group Td at k 5 0 and is related to the sstate of a free atom. The three p-like bands transform under the Td group as the coordinates x, y, and z, and are degenerate at the Γ point if the spin-orbit interaction is ignored, in analogy with atomic states. If, in addition, we consider the spin of the states (up and down for each band), we have 8 bands which we must treat exactly. It is well known that the spin-orbit interaction is non-zero in semiconductors, and that it strongly affects the band structure. The spin-orbit interaction partially removes the valence bands degeneracy, while the conduction band (symmetry Γ6 ) remains unaffected. Thus, the valence bands split into a quadruplet (symmetry Γ8 ) corresponding to J 5 3/2, and a doublet (symmetry Γ7 ) corresponding to J 5 1/2, where J is the total angular momentum. These bands are, accordingly, labeled Γ6 , Γ7 , and Γ8 , reflecting their transformation properties, as shown in Fig. 6.1. The energy separation between the minimum of Γ6 band and the maximum of Γ8 band is defined as the energy gap, Eg 5 Γ6 2 Γ8 . The Γ7 band is the spin-orbit split-off valence band, which originates as one of the three p-like bands, but has been shifted below the other two Γ8 bands by the spin-orbit splitting Δ. The ground state of the electronic system of a perfect semiconductor consists of a completely filled valence band and a completely empty conduction band. We can optically excite electrons from the valence band to the conduction band by photons with energies in the band gap region. In this process, we bring the system of N

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Figure 6.1 Band structure of a direct-gap II-VI compound semiconductor in the vicinity of the center of the Brillouin zone.

electrons from the ground states to excited states. What we need for understanding of the optical properties of a semiconductor is therefore a description of the excited states of the N-particle problem. The quanta of these excitations are called “excitons”. In this sense, excitons are the lowest-lying excited states of semiconducting crystals. The original suggestion of the existence of such excited electronic states in crystals was made by Frenkel [23,24]. In his model an electron and a hole as a pair of opposite charge experience a Coulomb attraction and are bound together in an appropriate orbital, the pair traveling through the crystal. The exciton with the large radius is known as a Wannier exciton and behaves like the hydrogen atom showing a hydrogenic spectrum. It becomes evident that the Wannier model is appropriate for II-VI semiconductors in which the binding energies are smaller than the energy of longitudinal optical phonons and the characteristic exciton radius is much larger than the lattice constant. The motion can be described by the effective masses me and mh and the Coulomb interaction between the electron and the hole reduced the by the static dielectric constant E of the medium. The energy extrema of the conduction and valence bands in direct-gap semiconductors occur at the same k values, while they are positioned at different k values in the indirect-gap semiconductors. The formation of excitons usually leads to the appearance of narrow peaks in the absorption edge of direct-gap semiconductors (so called direct-exciton), or of steps in the absorption edge in the case of indirectsemiconductors [25]. The most general description of exciton states in semiconductors was given by Knox [26], who used a single conduction and a single valence band, both isotropic, to obtain the hydrogenic levels of this electron-hole system. Exciton states can be represented by using a reduced mass μ0 in a4 medium with a dielectric constant E. If we use the effective Rydberg, R0 5 2hμ¯02eE2 , and effective

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Bohr radius a0 5 μEh¯e2 as units of energy and length, respectively, the corresponding 0 exciton states are En;i 5 2 n12 . When an electron is placed in a uniform static magnetic field, the electron motion parallel to the field direction remains unaltered, but the components of motion perpendicular to the field exhibit circular motion due to the Lorentz force. In quantum mechanical description, the energy of the electron is independent of magnetic field in the longitudinal direction, while the energy associated with the transverse motion can be described in terms of a simple harmonic oscillator, as shown by Landau [27]. In semiconductors, since the carriers in bands affected by the magnetic field as above, each band splits into n Landau levels, with equal energy space ¯hωc between successive levels, while the energy associated with motion along the magnetic field direction retains its parabolic nature. This results in n parabolic bands -- the so called Landau subbands. Such a simple band model gives us a qualitative understanding of the band structure of semiconductors in a magnetic field. However, these bands interact with each other, and cannot be treated separately. The most accurate description of Landau levels at the Γ point in semiconductors with a zinc-blende lattice, taking account of the interaction between bands, is provided by the model of Pidgeon and Brown [28]. The Pidgeon-Brown model treats the closely spaced Γ6 , Γ7 , and Γ8 , bands together using the effective mass approximation [29], and applying a procedure of nearly degenerate states. The interactions with higher bands are included to order k2, as in the Luttinger model [30]. Following Kane’s analysis of the zero-field case [31], the k  p interactions between spaced Γ6 , Γ7 , and Γ8 bands are treated exactly, so that the nonparabolicity effects of the energy levels are also included. The problem was then solved with harmonic oscillator functions. Interband transitions between the Landau levels can be described by the Pidgeon Brown model, which leads to the selection rules for the optical transitions in the presence of a magnetic field [28]. 2

6.2.2 Exciton transitions in the absence of magnetic field Since most optical transitions take place by forming excitons the observation of exciton states is very important for the understanding of optical properties of semiconductors. The most reliable band parameters of semiconductors such as the precise value of the fundamental band gap, masses of carriers, and valence band parameters are obtained by studying the exciton fine structure. Crystals of very high quality are required for meaningful measurements so that the linewidths of exciton peaks are sufficiently narrow to be resolved excitonic fine structures. A molecular beam epitaxy (MBE) growth techniques make it possible to grow high quality II-VI semiconductors on III-V compound substrates (e. g., GaAs). A typical transmission spectrum for MBE grown ZnTe and ZnSe epilayer observed at 1.5 K is shown in Fig. 6.2 [32]. The ZnTe spectrum shown in Fig. 6.2(a) is characterized by two distinct peaks around 2.38 eV and 2.39 eV. The strongest doublet peaks (at 2.3789BeV and at 2.3804BeV) are the 1s excitonic states, and the next doublet transitions (at 2.3883BeV and at 2.3900BeV) are the 2s exciton states

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Figure 6.2 Transmission spectra of (A) ZnTe and (B) ZnSe epilayers in the band gap region at 1.5 K. Excitonic fine structures involving the heavy and the light hole bands are observed up to 2s states. The intensity difference between the ground and the excited states of the excitons are clearly seen. After S. Lee, F. Michl, U. Ro¨ssler, M. Dobrowolska and J.K. Furdyna, Magnetoexcitons and Landau levels in strained ZnTe and ZnSe layers, J. Cryst. Growth 184-185, 1998, 1105
1109. https://doi.org/10.1016/S0022-0248(98)80231-5.

from the two top valence bands, i.e., the heavy-hole and the light-hole band. Another peak at 2.4048 eV is the LO-phonon-assisted transition: it lies 25.9 meV above the 1 s free exciton. This value is in good agreement with the energy of LOphonons (26.1BmeV), obtained by other measurements [33]. Optical transitions observed in the bandedge region of a ZnSe epilayer at 1.5 K in the absence of a magnetic field show similar exciton peaks to those of the ZnTe epilayer except for the appearance of the impurity-bound exciton at 2.7973 eV, and the absence of a clear energy splitting between the heavy and the light hole excitons in the 1s state. Since ZnSe has a larger exciton binding energy than the ZnTe, it is easier to resolve higher exciton states (2s and 3s) from the onset of the exciton continuum. The peaks at 2.8173 eV and 2.8182 eV are the 2s states for the heavy and the light hole excitons, respectively. The peak at 2.8334 eV is the LO-phononassisted transition, exactly 30.6 meV above the 1s free exciton line [34].

6.3

Landau level transitions and magneto-polaron effect

The application of magnetic field results in especially remarkable enrichment of spectroscopic detail in the region above the band gap. The new features begin to appear at moderate magnetic field of 2 T and show an oscillatory behavior with greater and greater amplitude as the field increases [35]. Fig. 6.3 shows transmission spectra observed at 3 T for ZnTe for σ1 and σ2 polarizations. In the figure, the low-energy part of the spectrum is dominated by exciton absorption (diamagnetic shift of hydrogenic levels), and the high energy part clearly shows periodic structure arising from the formation of Landau levels.

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Figure 6.3 ZnTe spectrum showing Landau level transitions observed in the σ1 (A) and σ2 (B) polarizations at 3 T, together with calculated positions of the allowed transitions. The calculated transition energies are indicated by different symbols defined in the figure, the length of the vertical bars indicating the oscillator strength of the transitions. After S. Lee, F. Michl, U. Rossler, M. Dobrowolska and J.K. Furdyna, Interband magnetoabsorption in strained epitaxially grown ZnTe and ZnSe, Phys. Rev. B 57, 1998, 9695
9704. https://doi. org/10.1103/PhysRevB.57.9695.

A quantitative theoretical description of these transitions in a magnetic field is given by the method developed by in Ref [36]. in which Landau level energy and dipole transition probabilities for σ1 and σ2 polarizations were calculated. Dipole transitions from valence to conduction band Landau levels elicited by circularly polarized photons ðσ 6 Þ obey the selection rule ΔN 5 6 1, where N 5 n 1 M 1 23 is a good quantum number under axial symmetry [37]. Here n and M are the quantum numbers of the Landau oscillator and of the z component of angular momentum, respectively. Among transitions corre
the transitions
allowed for ΔN
5 6 1, those
sponding to n; 2 23 v ! n; 2 12 c for σ1 and n; 23 v ! n; 12 c for σ2 polarizations have oscillator strengths almost a factor of two larger than other transitions [35]. The peaks in the spectra can be assigned to the corresponding Landau level transition as shown in Fig. 6.3. The transitions of ZnSe spectra can also be identified by calculating oscillator strengths as shown in Fig. 6.4. The transition energies for the strong oscillator strength can be fitted to the prominent features of the spectra for large n ( . 5, where the Coulomb corrections are negligible), using the heavy hole band edge energy as a fitting parameter [35]. The calculated fan chart together with experimental points, is shown in Fig. 6.5. The best fit was obtained for the heavy hole band edge energy of 2.3928 6 0.0005 eV for the ZnTe epilayer. This give hh 1s exciton binding energy and energy gap for unstrained ZnTe as Eex 5 12:4 6 0:5meV and Eg 5 2:3934 6 0:0005eV, respectively. Both Eg and Eex are in close agreement with values reported in the literature [34,38]. The best fit to experimental data for ZnSe epilayer yields the 1s exciton energy of Eex 5 22:2 6 0:5meV and the band gap energy for unstrained ZnSe of Eg 5 2:8246 6 0:0005eV. Exciton energy is somewhat bigger than the values

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Figure 6.4 ZnSe spectrum showing Landau level transitions observed in the σ1 (A) and σ2 (B) polarizations at 4 T, together with calculated positions of the allowed transitions. The calculated transition energies are indicated by different symbols defined in the figure, the length of the vertical bars indicating the oscillator strength of the transitions. After S. Lee, F. Michl, U. Rossler, M. Dobrowolska and J.K. Furdyna, Interband magnetoabsorption in strained epitaxially grown ZnTe and ZnSe, Phys. Rev. B 57, 1998, 9695
9704. https://doi. org/10.1103/PhysRevB.57.9695.

Figure 6.5 Magnetic field dependence of Landau level transition energies in ZnTe (A) and ZnSe (B) for the σ1 polarization. Experimental data are shown as points connected with dotted lines, and calculated results are given by the solid curves. The Landau quantum numbers involved in the transitions are indicated for each line. The polaron anti-crossing is depicted as a thick solid line at the top of the fan chart. After S. Lee, F. Michl, U. Rossler, M. Dobrowolska and J.K. Furdyna, Interband magnetoabsorption in strained epitaxially grown ZnTe and ZnSe, Phys. Rev. B 57, 1998, 9695
9704. https://doi.org/10.1103/ PhysRevB.57.9695.

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reported in the literature [34,38], but the exciton binding energy derived from the difference between the band edge and the 1 s exciton energy is more accurate than that determined from the 1s
2s exciton separation by assuming a Rydberg series. Systematic deviations of the calculated fan charts from the spectral positions make it obvious that the electron-hole correlation plays an important role in magneto-optical properties observed near the band gap. Careful examination of Figs. 6.5A,B reveals that, for higher Landau quantum numbers and increasing magnetic field, the experimental points start to bend off again from the calculated lines. This can be attributed to the electron-LO phonon interaction. This interaction becomes relevant whenever the optical phonon is equal to an integral number of Landau level spacing, that is, nh ¯ ωec 5 ¯hωLO . Thus interband transition energies allowed by the electron-LO phonon interaction (“phonon-assisted” transitions) are [35] ΔE 5 Ecn 2 Ehn 5 Eg 1

 
1 ¯hωh ¯hωec 1 ¯hωhc 1 ¯hωLO 1 1 ce 2 ¯hωc

(6.1)

The thick solid line near the top of Figs. 6.5A,B indicates a resonance with
the 1-LO phonon level for single-particle transitions originating from the n; 6 23 hole states. The anticrossing of the observed transition energies with the 1-LO phonon level line due to resonant LO-phonon coupling is clearly seen in the case of ZnTe data. This so called “magneto-polaron effect” is less pronounced for ZnSe than for ZnTe, because the LO-phonon energy is larger in ZnSe (30.6 meV) than in ZnTe (25.9 meV). Such anticrossing, or magneto-polaron effect, has been also observed in GaAs and CdTe materials [39,40].

6.4

Composition modulated ZnSeTe sinusoidal superlattice

Semiconductor heterostructures (quantum wells and superlattices) have been received great attention during last several decades in the field of semiconductor due to their superior electro-optical properties over bulk semiconductors. Such structure was first introduced by Esaki and Tsu in 1970 [41], and there has been tremendous effort in the field of crystal growth to achieve diverse heterostructures. Epitaxial crystal growth techniques, such as a molecular beam epitaxy (MBE) [6] in which one can control precisely the thickness of layers consisting the structures, made it possible to fabricate semiconductor heterostructures. Most studies, however, focus on the quantum structures in the form of a succession of square-wells, resulting from abrupt interfaces between constituent layers. A very limited number of investigations address quantum phenomena arising from structures with non-abrupt interface [42,43]. However, non-abrupt modulation of the potential profile in the superlattice structures, specifically, with sinusoidal profile, is expected to lead to

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new and unique physical properties. In this section, we describe a novel quantum effect arising for the sinusoidal band profile.

6.4.1 Band structure of superlattice with sinusoidal energy profile SLs with a sinusoidal potential profile are especially amenable for the analytical description in terms of the nearly-free-electron (NFE) approximation. Consider a single band of a sinusoidally modulated SLs schematically shown in Fig. 6.6. Such sinusoidal potential is given by,   1 2πz V ðzÞ 5 Vcos 2 L

(6.2)

where L is the SL period, z is the SL growth direction, and V is the peak-to-peak value of the band offset of the band under consideration. In order to solve Schro¨dinger equation, which describes the behavior of an electron in the band profile, the potential V ðzÞ can be expanded as a Fourier series, with the Fourier coefficients Vn of the series given by Vn 5

1 Vδn;1 4

(6.3)

i.e., all Fourier components of V ðzÞ vanish except V1 . This results in a major simplification of the band structure. Since in sinusoidal SLs only V1 6¼ 0, it is readily shown that, to first order, only one minigap opens: the first minigap at k 5 6 πL with the minigap width given by [44,45]

Figure 6.6 Energy band profile of the ZnSe1-xTex sinusoidal superlattice for the conduction and for the valence band with type-II alignment. The valence band offset is assumed larger, to qualitatively simulate the case of ZnSe1-xTex superlattices.

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Figure 6.7 Band structure of superlattices with sinusoidal energy profile. Only a single band gap appears at the zone edge in first order perturbation analysis. Note the shift of the band edge below the compositional average by the amount mc L2 Vc2 =4h2 . After G. Yang, S. Lee and J.K. Furdyna, Band structure and optical properties of sinusoidal superlattices: ZnSe12xTex, Phys. Rev. B 61, 2000, 10978
10984. https://doi.org/10.1103/ PhysRevB.61.10978.

Eg;1 5 2jV1 j 5

1 V: 2

(6.4)

It is interesting to note that this minigap is determined only by the value of the band offset. Such characteristic band structure is shown schematically in Fig. 6.7 in the extended zone form. The wave functions associated with the two energy extrema of this minigap are cosðnπz=LÞ and sinðnπz=LÞ, their order (which is upper, which is lower) depending on the sign of V1 [44].

6.4.2 Growth of ZnSeTe superlattices with sinusoidal composition modulation Non-uniform flux distribution of elemental source over the substrate exist in the MBE system and results in the different growth rate depending on the position of substrate (i.e., obtained film has different thickness at the center and at the edge of substrate). Such non-uniform flux distribution will also affect even seriously in the

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growth of heterostructures where several elemental sources are necessary involved. Even though the system was designed such way that each elemental source is in the same distance from the center of substrate, flux distribution of each source over the substrate depends on its relative position to the substrate. This then results in the different coverage of the layer and more importantly chemical composition in the case of multicomponent alloy layer [46
48]. The presence of an uneven distribution of constituent fluxes (Te and Se in the present case) during MBE growth can be strategically used to grow a novel ZnSeTe “sinusoidal superlattice (SSL)”, i.e., periodic structure in which the composition of a semiconductor alloy varies sinusoidally (or very nearly so) in one direction. Three different ZnSeTe SSL were grown by MBE under identical growth condition by varying only the rate at which the substrate was rotated in the chamber. The rotation speed will be reflected as period of modulation superlattices in the set of samples. The radial ðθ 2 2θÞ X-ray scans of the samples are shown in Fig. 6.8, in which the data typically contain four peaks. The strongest, sharpest peak is the GaAs substrate (004) Bragg reflection. The next most intense peak, displaced to the left from the substrate peak, is the (004) fundamental reflection from the ZnSeTe overlayer. The position of the fundamental ZnSeTe peak indicates the average composition of the overlayer [49]. The two weak peaks on either side of (and equally displaced from) the fundamental ZnSeTe peak are the first-order superlattice peaks, due to the compositional modulation within the overlayer. The presence of only a single superlattice Fourier component in the X-ray indicates sinusoidal compositional profile in these superlattice -- in sharp contrast with diffraction spectra of superlattices with abrupt interfaces, which are characterized by multiple satellite peaks [50].

Figure 6.8 X-ray scans for the three different superlattices grown with different rotation speed. Superlattice periods are plotted as a function of rotation rate in the inset. After S. Lee, U. Bindley, J.K. Furdyna, P.M. Reimer and J.R. Buschert, Controlled growth of ZnSeTe superlattices with sinusoidal compositional modulation, J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 18, 2000, 1518
1521. https://doi. org/10.1116/1.591417.

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Based on the existence of such sinusoidal profile, we know that regions richer in Se than the average composition are equal in thickness to those richer in Te. The separation in reciprocal space ΔQ between the superlattice peaks is inversely proportional to the superlattice modulation distance Λ, i.e., the superlattice period. Inset of Fig. 6.8 show the superlattice period Λ as a function of the period of rotation used in fabricating the respective specimens. It is very clear that the superlattice period Λ is directly proportional to the period of rotation T: the faster the sample is rotated, the shorter is the repeat distance of the superlattice modulation. The superlattice periods obtained from X-ray measurement are L 5 13.5, ˚ for three samples, respectively. The same technique has been used 26.4, and 58.1 A to fabricate GaAsSb sinusoidal SL [51].

6.4.3 Optical transitions in ZnSeTe sinusoidal superlattices Optical properties of sinusoidal SLs can be conveniently investigated by using photoluminescence measurements. The photoluminescence spectrum is expected to dominate by transitions associated with the band edge (i.e., the Γ point). The ZnSe1-xTex sinusoidal SLs with two different periods, but with the same modulation amplitude of x, as determined by X-ray studies (x varies from 0.35 to 0.65) [47] are measured for comparison. In this specific ZnSe1-xTex system, the largest band gap difference between the two layers (i.e., the band gap difference between ZnSe0.65Te0.35 and ZnSe0.35Te0.65) is about 130 meV [52]. The photoluminescence (PL) spectra for the two ZnSeTe sinusoidal SLs are shown in Fig. 6.9, where the left and right panels represent the results obtained on ˚ and 26.4 A ˚ , respectively. In each the two superlattices with periods L 5 13.5 A panel, there are two spectra, one taken on a sinusoidal SL, and one on the alloy ZnSeTe whose composition is the same as the average composition of the SL. It is clear that the red shift of the superlattice PL position relative to the corresponding alloy is much larger in the case of the SL with the longer period. This behavior of the PL energy is as anticipated by the band edge shift obtained in the NFE model (see Fig. 6.7). The shift of transition energies at the Γ point can be expressed explicitly in terms of band offsets (Vc and Vv), superlattice period (L), and effective mass (mc and mv ); [53] hh10 ! e10 :E0 2


L2 Vc2 mc 1 Vv2 mv 2 4h

(6.5)

Note that, as a good approximation, one can ignore the contribution from the conduction band in Eq. (6.5), since the electron mass is five times smaller, and the band offset is at least two times smaller, than the corresponding quantities in the heavy-hole band. Fig. 6.10 shows the plot of the energy shift as a function ˚ periods obtained of the valence band offset for sinusoidal SLs with 13.5 and 26.4 A using the value of 0.7 m0 for the heavy-hole mass based on NFE model [53]. This energy shift between the average alloy (V 5 0) and the SL can then be used to find

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(A)

(B)

Figure 6.9 PL spectra for ZnSeTe alloy and for ZnSeTe SLs with sinusoidal modulation. ˚ and for 26.4 A ˚ periods, respectively. Panels (A) and (B) are for the SLs with the 13.5 A After G. Yang, S. Lee and J.K. Furdyna, Band structure and optical properties of sinusoidal superlattices: ZnSe12xTex, Phys. Rev. B 61, 2000, 10978
10984. https://doi.org/10.1103/ PhysRevB.61.10978.

the valence band offset in each sinusoidal SL. The energy shifts observed experi˚ BSLs (4meV and 17 meV, respectively are mentally for the 13.5 and 26.4 A marked as horizontal dotted line in Fig. 6.10). The crossing point of these lines with the corresponding calculate curve then gives a very close estimate of the valence band offset. The values obtained in this way are V 5 202meV for the SL ˚ period, and V 5 205meV for the 26.4 A ˚ SL. Recalling that the two with the 13.5 A SLs have identical compositional modulation amplitudes, it is very gratifying that the value of the valence band offsets obtained from the two specimens, 202 and 205, are very close to one another. This value for the valence band offset automatically gives the conduction band offset to be about 70meV, since the maximum band gap difference for this modulation (0.35 , x , 0.65) is 130 meV.

6.5

II-VI-based zero-dimensional structures

Advances in nano-technology permit the fabrication of even zero-dimensional geometries -- the so called “quantum dots” (QDs) -- in which the motion of carriers and/or excitons is confined in all three directions. One method for the fabrication of QDs with high optical quality is using the process of spontaneous self-assembled island formation driven by the lattice mismatch between the QD material and the substrate [54,55]. This QD growth requires the deposition of several atomic layers of one material on the top of another of a different lattice

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Figure 6.10 Energy shifts of band edge transition as a function of valence band offset, ˚ and 26.4 A ˚ periods, as discussed in the text. Experimental calculated for SLs with the 13.5 A PL energy shifts of two SLs relative to the alloy are shown as horizontal lines. The intersection of the line and the calculated curve gives an estimate of the valence band offset for the two SLs (both very close to 200 meV). After G. Yang, S. Lee and J.K. Furdyna, Band structure and optical properties of sinusoidal superlattices: ZnSe12xTex, Phys. Rev. B 61, 2000, 10978
10984. https://doi.org/10.1103/PhysRevB.61.10978.

constant (since the process depends on strain) under precisely controlled conditions. Recent developments in epitaxial techniques, such as metal-organic chemical vapor deposition (MOCVD) and molecular beam epitaxy (MBE), have finally made it possible to control layer deposition to the degree required for the formation of QDs. This spontaneous growth process has already been shown to produce QDs in the form of coherent nanoscale islands in various combinations of lattice-mismatched IV-IV, III-V, and II-VI materials. The II-VI QDs have been receive special attention because of their device potential in the short-wavelength visible range of the electromagnetic spectrum. Here, we review optical properties of self-assembled QDs in II-VI materials by focusing on the spin related phenomena of CdSe/ZnSe QDs systems.

6.5.1 Spin polarization and relaxation of exciton in QDs Spin states of carriers and their dynamics in QDs are of central interest, because of the promise that they hold for spin-polarized microelectronics (so called spintronics) [56]. Specially, spin states of semiconductor QD structures offer the possibility of using them as quantum bits for quantum computation [57,58]. In this context, a key challenge for realizing such a spintronic device is then the ability to manipulate spins in these quantum structures, which in turn requires a clear understanding of spin phenomena in the QDs. Polarization-selective magneto-photoluminescence study is a suitable experimental technique for studying spin phenomena of QD system, in which spin relaxation time τ s is significantly longer than the exciton recombination time τ r [59].

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Chalcogenide

40,000 514 nm Excitation

PL intensity (arb. unit)

514 nm 458 nm 30,000

3 LOResonance

20,000 2 LOResonance

10,000

0 2.20

2.25

2.30

2.35

2.40

2.45

Energy (eV)

Figure 6.11 PL spectra observed at 1.5 K on a CdSe QD ensemble using with two different excitation energies. A typical broad PL peak is observed for the 458 nm excitation. In the case of resonant excitation (i.e., 514 nm excitation, below the energy gap of the ZnSe barrier), LO-phonon resonance lines are seen to appear at the higher energy side of the broad PL line. After S. Lee, M. Dobrowolska and J.K. Furdyna, Polarization-selective magnetophotoluminescence study on CdSe self-assembled quantum dots by resonant excitation, J. Cryst. Growth 275, 2005, E2301
E2306. https://doi.org/10.1016/j.jcrysgro.2004.11.368.

Fig. 6.11 show photoluminescence (PL) spectra CdSe/ZnSe DQs obtained from the resonant (514 nm line) and the non-resonant (458 nm line) excitations. The PL line emitted by the QDs is very strong and is centered around 2.3 eV, which is typical for CdSe islands embedded in a ZnSe matrix [60]. The broad PL line width of 50 meV is due to inhomogeneous broadening arising from the size and composition distributions of the QD ensemble [60]. Note that the line obtained with resonant excitation appearing to be shifted to a higher energy due to the more effective excitation of the QDs on the high-energy side of the PL under resonant excitation, as manifested by the appearance of LO-phonon resonance lines marked by arrows in the figure. It is well known that interband optical transitions are governed by spindependent selection rules, i.e., transitions between the spin-up states of the conduction and the valence bands correspond to the σ2 polarization, and transitions between the spin-down states to the σ1 polarization [61]. When the semiconductor is placed in an external magnetic field, the conduction as well as the valence bands edges undergo a Zeeman splitting [62]. The removal of the degeneracy by the applied magnetic field will then automatically lead to a difference in the populations of the spin-up and the spin-down carriers. Since the PL intensity directly indicates the number of carriers participating in the transition, polarization selective PL will then reflect the relative numbers of spin carriers with the two spin states in QD system (i.e., the net spin polarization of the QD) at the instant of exciton recombination.

Epitaxial II-VI semiconductor quantum structures involving dilute magnetic semiconductors

169

The carrier spin relaxation between the Zeeman sublevels in the QDs can be investigated by polarization-selective magneto-PL experiments using resonant excitation. For example, the PL can be excited using unpolarized light, and the PL emission can be measured using polarization-sensitive (either σ2 or σ1 ) detection. When a magnetic field is applied, the PL intensities of the two polarizations varies due to the Zeeman splitting, increasing in the σ2 polarization, and decreasing in σ1 [59]. This behavior is due to the relatively low energy position of the spin-up state in the Zeeman sublevels, so that carriers excited by unpolarized light relax more effectively to that state. Moreover, spin-down carriers can flip their spin and transfer to spin-up states, thus also contributing to the σ2 PL intensity [63]. The difference of the PL intensities between the two polarizations at a given energy expressed by the standard formula P 5 ðI 1 2 I 2 Þ=ðI 1 1 I 2 Þ, where I 1 and I 2 are the PL intensities of the σ1 and σ2 polarizations, respectively. The magnetic field dependence of the degree of polarization for the CdSe QDs emitting at different energies is shown in Fig. 6.12. The PL spectrum is also shown in the figure to identify the LO-phonon resonances. The degree of polarization of the PL is zero in the absence of an applied field, but it monotonically increases as the field increases. This polarization behavior of the CdSe QDs in a magnetic field can be understood by considering two levelsystem as shown in the inset of Fig. 6.13. The degree of spin polarization in a twolevel system is given by [62]

0T 1T 2T 3T 4T 5T 6T PL

Polarization

0.00

–0.08

–0.16

–0.24 2.26

LOresonance 2.28

2.30

2.32

2.34

Energy (eV)

Figure 6.12 The degree of polarization P obtained for the PL of the CdSe QD ensemble at different magnetic fields. It is clear that the degree of polarization is different within the band, depending on the energy separation from the LO-phonon resonance line. After S. Lee, M. Dobrowolska and J.K. Furdyna, Polarization-selective magneto-photoluminescence study on CdSe self-assembled quantum dots by resonant excitation, J. Cryst. Growth 275, 2005, E2301
E2306. https://doi.org/10.1016/j.jcrysgro.2004.11.368.

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Chalcogenide

Figure 6.13 The values of τ s =τ r obtained from the fitting process using Eq. (6.6) for 5 selected energy positions. The ΔE in the x-axis is the separation of the PL energy from the LO-phonon resonance line. It is clear that the spin relaxation time strongly depends on the separation of the PL energy from the LO-phonon resonance line. After S. Lee, M. Dobrowolska and J.K. Furdyna, Polarization-selective magneto-photoluminescence study on CdSe self-assembled quantum dots by resonant excitation, J. Cryst. Growth 275, 2005, E2301
E2306. https://doi.org/10.1016/j.jcrysgro.2004.11.368.

P52

1 2 e2ΔE=kT ; τ s =τ r 1 1 1 e2ΔE=kT

(6.6)

where τ s corresponds to an effective spin relaxation time between the two Zeeman levels; and τ r is the exciton recombination time. Here ΔE 5 gμB B is the Zeeman energy splitting, where g is the Lande g-factor, and μB is the Bohr magneton. As we can be seen from Eq. (6.6), the degree of polarization depends strongly not only on the g-factor determining the Zeeman splitting, but also on the ratio of τ s =τ r . Since it is known that the spin-flip time in CdSe (and actually in most other nonmagnetic semiconductors) is about one order of magnitude longer than the exciton recombination time [64], the factor τ s =τ r is expected to significantly affect the net degree of spin polarization of the carriers within the CdSe QDs. The dependence of PL polarization for 5 different energy positions, shown by dotted vertical lines in the Fig. 6.12, has been fitted using Eq. (6.6) by treating g-fac tor and τ s =τ r as a fitting parameter. The fitting gives that the parameter g 5 0:42 6 0:08 and τ s =τ r ranges from 2 to 8 as shown in Fig. 6.13, depending on the separation of the PL energy from the LO-phonon resonance line. This indicates that the spin relaxation time τ s is much longer than the exciton recombination time τ r for the QDs. If the exciton recombination time of CdSe QDs is taken as 450 ps [65], the spin relaxation time for this QD system is in the range of a few nanoseconds, in good agreement with studies done by others on similar QD systems [66,67]. It is interesting that the g-factor (0:42 6 0:08) obtained from the fitting is almost identical to

Epitaxial II-VI semiconductor quantum structures involving dilute magnetic semiconductors

171

those reported for the conduction electron g-factor (g 5 0.42) in CdSe [68]. It is well known that spin polarization of the holes in the valence band undergoes rapid dephasing [69]. It is therefore likely that the selection rules determining the recombination process are primarily dominated by the spin polarization of conduction electrons.

6.5.2 Spin-spin interaction between the coupled QDs As the interest of the scientific research turns to spin Q-bit in QDs, coupling between the QDs automatically become important since spin states in coupled double QDs can be used as two-quantum-bit gates for quantum computation, as proposed by Loss and DiVincenzo [57]. Here we discuss the phenomenon of spin polarization of carriers that arises from inter-dot coupling between QD pairs which form when two adjacent QD layers are deposited. A strategically designed double layer QD structures, in which the PL signals of two QD layers are separated, enables one to study the polarization of the PL from each QD layer separately. Since the polarization of the PL provides information on the spin polarization of the carriers in the dots, its behavior in the presence of a magnetic field reflects variations of how the different spin states of the QDs are populated. In order to investigate spin coupling between the QDs, two different double layer QD (DLQD) systems, in which CdSe and of CdZnSe QD layers are separated by either 20 or 60 ML of ZnSe barrier, are prepared by using MBE technique. The magneto-PL spectra obtained at 1.5 K from the two DLQD samples are plotted in Fig. 6.14, in which the upper and lower panels correspond to samples with the 60 and 20 ML ZnSe barrier samples, respectively. The red and blue spectra in the figure represent σ1 and σ2 polarizations. The lower-energy peak (around 2.3 eV) originates from the CdSe QDs, a typical signal for 2.5 ML CdSe deposition; [60] and the higher energy peak (at 2.49 eV) corresponds to the CdZnSe layer. The energy difference of about 200 meV between the CdSe QD emission and the CdZnSe QDs gives an estimate of the Zn content in the CdZnSe layer to be around 20% [70]. The difference of the PL linewidth of each QD layers are due to the inhomogeneous broadening caused by maturity of QDs and strain condition [71]. The PL spectra of both CdSe and CdZnSe QDs exhibit noticeable changes in the peak intensities when an external magnetic field is applied, as can be seen from Fig. 6.14. In both QD systems, the peak intensity of the PL taken at 6 T with σ 1 polarization is weaker than that of the σ 2 polarization, giving a σ2 polarization of the QD systems in the magnetic field. This is a common behavior of CdSe and CdZnSe QDs, because the spin-up states correspond to slightly lower energies in both conduction and valence bands and thus the carriers excited by unpolarized light would relax more effectively to spin-up states; and, furthermore, spin- down carriers can also flip their spin and transfer to spin- up states [72,73]. The degree of polarization can be obtained from the difference of the PL intensities between the two polarizations as described in the previous section. The magnetic field dependence of polarization for the two PL peaks of DLQD systems are summarized in Fig. 6.15. The CdSe and CdZnSe QD layers in 60 ML DLQD show

172

Chalcogenide

PL intensity (arb. unit)

(A)

σ+ σ+

6T 60 ML X 20

(B) PL intensity (arb. unit)

σ– σ–

6T 20 ML

2.2

σ–

σ–

σ+

σ+

2.3 2.4 Energy (eV)

2.5

Figure 6.14 Magneto-PL spectra detected with σ2 andσ1 polarization at a magnetic field of 6 T. The upper and lower panels represent spectra taken on DLQD systems with 60 ML and 20 ML barrier, respectively. While the difference of PL intensities between σ2 andσ1 polarizations for the CdSe and CdZnSe layers is similar in the DLQD with 60 ML, it is significantly different in DLQD with 20 ML barrier, indicating that the QD layers separated the 20 ML ZnSe barrier are coupled to each other. After S. Lee, M. Dobrowolska and J.K. Furdyna, Coupling-dependent spin polarization of quantum dots in double layer geometry, J. Cryst. Growth 275, 2005, e2295
e2300. https://doi.org/10.1016/j.jcrysgro.2004.11.367.

very similar field dependence of polarization. The degree of polarization of PL peaks increases monotonically with magnetic field for both CdSe and CdZnSe QDs and it reaches about 11% at 6 T. The degrees of polarizations are very close to that for independent CdSe and CdZnSe layers [62]. This indicates that the CdSe and CdZnSe QD layers in 60 ML DLQD are not coupled. The values of P are, however, significantly different for CdZnSe and for CdSe layers in the DLQD with the 20 ML separating layer. While the degree of polarization for CdSe QD increases up to 17% at 6 T it decreases to 4% for CdZnSe QDs. The differences in P between the CdSe and CdZnSe PL peaks monotonically increases with magnetic field, reaching a value of about 13% at 6 T. This indicates that in the case of the DLQD with 20 ML barrier there exists a spin interaction between carriers localized in CdSe and CdZnSe QD layers; i.e., that in this case the spin state in one layer appears to influence the net spin polarization of the adjacent layer. In such coupled system, spin-spin interaction affects the rate of spin redistribution between the carriers in the two QD families. That is, the population of spin-up carriers in the CdSe QDs increases faster than in the CdZnSe QDs. Furthermore, the polarization dependence of the two peaks in the DLQD structure shifts opposite

Epitaxial II-VI semiconductor quantum structures involving dilute magnetic semiconductors

173

0.00

Polarization

–0.04 –0.08 –0.12

60 MLCdSe 60 MLCdZnSe 20 MLCdSe 20 MLCdZnSe

–0.16 –0.20 0

1

2

3

4

5

6

Magnetic field (T)

Figure 6.15 Polarizations of PL peaks for two DLQD systems in the magnetic field. Solid and open circles represent CdSe and CdZnSe QDs in DLQD with 60 ML spacer. Solid and open squares represent CdSe and CdZnSe QDs in DLQD with 20 ML spacer. While the polarizations of CdSe and CdZnSe QDs is similar for the 60 ML DLQD, they are significantly different for 20 ML DLQD. After S. Lee, S.R.E. Yang, M. Dobrowolska and J. K. Furdyna, Influence of inter-dot coupling on spin polarization of carriers in double quantum dots, Semicond. Sci. Technol. 19, 2004, 1125
1130. https://doi.org/10.1088/02681242/19/9/010, Pii S0268-1242(04)78041-4.

direction with respect to uncoupled QD layer. While the polarization degree for low energy CdSe peak (solid square) increases faster than that for the uncoupled CdSe QDs, the polarization of high energy CdZnSe peak (open square) increases slower than that of uncoupled CdZnSe QD layers with increasing magnetic field. The fact that the PL peaks of CdSe and CdZnSe QDs in DLQD show opposite behavior of polarization relative to the uncoupled layer QD structures indicates the existence of an antiparallel spin interaction between coupled QDs in this double layer QD structure [62].

6.6

II-VI quantum structures involving DMSs

6.6.1 Zeeman splitting in II1-xMnxVI DMS epilayers Diluted magnetic semiconductors (DMSs) are semiconducting alloys in which a part of the semiconductor crystal lattice is substitutionally replaced by magnetic transition metal ions. II-VI semiconductors in which a fraction of the group-II atoms is replaced by Mn21 are the best-known examples of such alloys (e.g., Zn1xMnxSe, Zn1-xMnxTe, Cd1-xMnxSe, etc.). The random distribution of magnetic ions over the cation sublattice in DMSs leads to novel important magnetic effects, e.g., the formation of spin-glass-like phase at low temperature, extremely large Zeeman splittings of electronic levels, giant Faraday rotation, magnetic-field-induced

174

Chalcogenide

metal-insulator transition, and formation of bound magnetic polarons. These enhanced spin-dependent properties occurring in DMS systems were widely studied in the literature and are already summarized in many excellent review articles [12
14]. [74
76], The presence of localized magnetic ions in DMSs leads to an exchange interaction between the sp band electrons and the d electrons associated with Mn11, resulting in extremely large Zeeman splitting of electronic levels. The Zeeman splitting of band edges of a DMS layer, ΔE, can be described by a modified Brillouin function, as follows: [12]  ΔE 5 CB5=2

5μB gMn B 2kB ðT 1 T0 Þ

 (6.7)

where C 5 ðα 2 βÞN0 xS0 , and B5/2(x) is the Brillouin function for spin S 5 5/2. Here C is a temperature-independent parameter determined by the well-known exchange integrals for electrons ðαN0 Þ and holes ðβN0 Þ, the Mn concentration x, the effective spin of the localized Mn21 ions S0. The most dramatic effect is the very large energy difference between the spin states, which can be selectively observed with the σ1 and σ2 polarizations in optical transition. The splitting is of the order of 100 meV
a magnitude which in a nonmagnetic semiconductor would require fields of the order of a megagauss. Owing to such large energy splitting, this feature provides an exceptionally convenient experimental tool for “tuning” the energy gap of a semiconductor system by using an external magnetic field. This ability to tune the bandgap has, in turn, important implications for studying quantum effects in DMS heterostructures, providing the experimenter with a powerful tool for varying the relative band alignment in a controlled and continuous manner simply by varying the applied field.

6.6.2 Mapping of exciton localization in QDs One of the prominent properties of QDs is their ability to strongly localize excitons by three-dimensional confinement. Such strong exciton confinement significantly increases temperature stability of the photoluminescence (PL) emitted by the QDs [77,78] as compared to that observed for higher dimensional structures, such as quantum wells (QWs) or bulk systems. Even though the temperature stability of excitons in QDs is now well established by many optical experiments, experimental determination of the degree of carrier localization in QDs is rather difficult due to the uncertainty in their shape and size. Since the degree of carrier localization is an major factor not only for the temperature stability of QD excitons, but also for the interaction between coupled QDs [62,79], it is important to establish how much of the carrier wavefunction is actually localized within the dots. The issue of wavefunction localization in zero-dimensional structures can be directly investigated by using the system involving the diluted magnetic semiconductor (DMS). Since the DMS alloy exhibits giant Zeeman splitting of the band

Epitaxial II-VI semiconductor quantum structures involving dilute magnetic semiconductors

175

edges when a magnetic field is applied [12] the Zeeman shift observed on the optical transitions emitted by the quantum structures reflects the degree to which the wavefunctions of the excitons in the quantum structures overlap with the DMS barrier. Such wavefunction mapping using DMSs has been extensively applied to study charge carrier localization in a wide range of other semiconductor heterostructures, primarily those involving multiple quantum wells [17]. Here this unique feature of DMSs is used to determine the degree of localization of exciton wavefunctions in QD structures [80]. Fig. 6.16 shows PL spectra obtained at 1.5 K from the three CdSe/ZnMnSe QD structures in which the CdSe coverage varied from 1.5 to 2.5 ML. The three PL peaks clearly resolved in the 1.5 ML CdSe QD system as one can see from the spectrum in the bottom. The highest energy peak at 2.8 eV corresponds to the ZnMnSe barrier; the middle energy peak around 2.35 eV originates from the CdSe QDs; the lowest energy peak around 2.1 eV is identified as Mn internal transition. Note that the energy position of 1.5 ML CdSe QD appears as relatively low energy compared to typical 1.5 ML CdSe QDs in pure ZnSe matrix [81]. Such low energy shift of PL in CdSe QDs grown on ZnMnSe is due to the nucleation effect of Mn atom presented on the surface [82,83] where CdSe is deposited. The relatively weak PL intensity of PL observed in the 1.5 ML QD system is due to the well known exciton energy transfer to intra-Mn transitions near 2.1 eV [82,84]. Even though we are not able to clearly resolve QD signal in the high CdSe coverage (i.e.,

Figure 6.16 PL spectra taken at 1.5 K on a series of CdSe/ZnMnSe QD systems. PL peaks from the Mn internal transition, CdSe QDs, and ZnMnSe barrier are clearly resolved. A systematic red shift of the PL peak from CdSe QDs with increasing CdSe coverage is observed. After S. Lee, M. Dobrowolska and J.K. Furdyna, Growth and magneto-optical properties of CdSe/ZnMnSe self-assembled quantum dots, J. Cryst. Growth 301
302, 2007, 781
784. https://doi.org/10.1016/j.jcrysgro.2006.11.276.

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2.0 and 2.5 ML) samples due to the overlap with signal from Mn internal transition, whose energy position is marked as dotted line in the figure, one can tell that the QD signal of the 2.0 ML sample is still in higher energy side of Mn line while the PL energy of 2.5 ML QD sample is in lower energy side of Mn line. Therefore, it appears that the PL energy from CdSe QDs in a ZnMnSe matrix systematically shift to lower energy with increasing CdSe coverage, as observed in pure binary system of CdSe/ZnSe [81], indicating systematic development of CdSe QDs with increasing CdSe coverage within the range investigated in this study. The PL spectra of these QD systems exhibit significant changes in the peak intensities and energy position when a magnetic field is applied. Fig. 6.17 shows magneto-PL spectra for the CdSe QDs with 2.0 ML coverage. The PL intensity of CdSe QDs, which was comparable to Mn transition at zero field, becomes much stronger than the intensity of Mn transition in the magnetic field due to the suppression of energy transfer [82,84]. The PL signals from the CdSe QDs and ZnMnSe barrier exhibits large energy shift with increasing magnetic field. In quantum system composed of non-DMS and DMS material, the Zeeman shift of the non-DMS structure is determined by the empirical parameter γ that represents the overlap between the carrier wave function of the structure and the DMS layer. In the first order approximation, the Zeeman shift of the CdSe QD in our system can be written as

Figure 6.17 Magneto-PL spectra taken for CdSe/ZnMnSe QDs obtained with 2.0 ML CdSe coverage. It is clear that the PL peaks from CdSe QDs and from ZnMnSe both show large red shifts as a function of magnetic field, as expected for DMS-based systems. Note also that the intensities of the PL peaks from CdSe QDs and from the ZnMnSe barrier increase with magnetic field, while the intensity of Mn internal transition decreases, as shown by the vertical arrows in the figure.After ] S. Lee, M. Dobrowolska, J.K. Furdyna. Zeeman mapping of exciton localization in self-assembled CdSe quantum dots using diluted magnetic semiconductors. Sol. Stat. Comm. 141, 2007, 311.

Epitaxial II-VI semiconductor quantum structures involving dilute magnetic semiconductors CdSeQD ZnMnSe ΔEZeeman 5 γΔEZeeman

177

(6.8)

ZnMnSe is the Zeeman shift of ZnMnSe barrier, which can be described by where ΔEZeeman Eq. (6.7). Since we have observed the Zeeman shifts of both CdSe QDs and ZnMnSe barrier, we can experimentally determine the empirical parameter γ, representing wave function overlap of CdSe QDs with ZnMnSe barrier, by taking ratio of the two values. The values of γ for three QD samples are shown in Fig. 6.18, where the values of γ are systematically larger with decreasing CdSe coverage for QD structures in all magnetic field region investigated. Since the portion of carrier wave function penetrating ZnMnSe barrier in the QD structures expected to be larger with smaller CdSe QDs, the dependence of γ on CdSe coverage is consistent with the results that inferred from the evolution of PL energy peak position with CdSe coverage as shown in the inset of Fig. 6.18, where the values of γ obtained at 6.0 T are plotted as a function of the PL energy for the QDs. Note that the values of γ for all three QD samples show slight dependence on the magnetic field, which is not expected from Eq. (6.8). The observation of magnetic field dependence of γ then implies that there exist yet other contributions to the Zeeman shift of CdSe QDs. One such contribution may arise from changes in the band offset between CdSe QDs and the ZnMnSe barrier when the field is applied. Since the Zeeman splitting of the band edges of ZnMnSe reduces the

Figure 6.18 Magnetic field dependence of the Zeeman shift ratio g of CdSe QDs peaks with respect to the PL peaks from the ZnMnSe barrier. The values of g show a systematic dependence on the CdSe coverage. Inset shows the values of g obtained at 6.0 T as a function of the PL energy of QD peaks. The monotonic dependence of g on the PL energy indicates that the penetration of the exciton wave functions into the barrier is stronger in the less developed QDs. After S. Lee, M. Dobrowolska and J.K. Furdyna, Growth and magnetooptical properties of CdSe/ZnMnSe self-assembled quantum dots, J. Cryst. Growth 301
302, 2007, 781
784. https://doi.org/10.1016/j.jcrysgro.2006.11.276.

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barrier height, the confinement potential of the QD becomes gradually smaller as the field increases. Such reduction in confinement would allow a larger fraction of the exciton wave function to penetrate into the ZnMnSe barrier, resulting in an increase of g with increasing magnetic field. Another possible—and perhaps stronger—source of such additional contribution is the diffusion of Mn ions into the CdSe QDs. The existence of Mn ions inside CdSe dot would automatically lead to direct sp
d exchange interaction between electronic states within the QDs and Mn ions, and would cause an additional Zeeman shift of the PL emitted by the QD when a magnetic field is applied. One can include this contribution by considering the diffusion profile of Mn ions at the QD/barrier interface, as was done for DMS quantum wells by other workers [16,85,86]. However, incorporation of Mn diffusion profile is rather difficult in the case of self-assembled QDs due to the uncertainty of shape and size of the QDs.

6.7

Enhancement of spin polarization in non-DMS and DMS coupled QDs

QDs involving DMS are interesting because of the strong spin-based effects which DMS materials bring into the picture, thus allowing highly promising options for spin manipulation in the QD geometry [57]. The giant Zeeman splittings of the band edges of DMSs in turn leads to a strong spin polarization of carriers in DMS quantum structures [87,88], thus providing a powerful tool for the study and manipulation of spin phenomena. The spin-spin interactions between QDs can be effectively studied by using such spin properties of DMS QDs. An asymmetric QD double-layer structures involving DMS material are very suitable to address interQD spin interactions. Fig. 6.19 shows magneto-PL spectra observed at 1.5 K on the CdZnSe and CdMnSe double-layer QD (DLQD) system. The sample shows strong luminescence at 2.49 eV and at 2.33 eV, corresponding to emissions from CdZnSe and CdMnSe QDs, respectively. The emission from the CdZnSe QDs occurs at a higher energy because of incorporation of about 25% Zn in those dots. The relatively weak PL intensity observed on CdMnSe QDs is due to the well known exciton energy transfer to intra-Mn transitions near 2.1 eV [82,84]. The PL spectra of both CdZnSe and CdMnSe QDs exhibit conspicuous changes in the peak intensities depending on detected polarization when a magnetic field is applied. It is a striking feature of the data that the PL intensity from the CdZnSe QDs clearly increases in theσ2 polarization and decreases in the σ1 polarization, while the intensity emitted by the CdMnSe QDs shows the opposite magnetic field dependence, decreasing in the σ2 polarization and increasing in σ1 (see inset of Fig. 6.19). The magnetic field dependence of PL intensities for both QD families is summarized in Fig. 6.20. In the non-magnetic system (CdZnSe in our case) the spin-up states are shifted down in energy in the conduction band and up in the valence band, thus becoming the ground states in both bands [72]. In DMS systems, on the

Epitaxial II-VI semiconductor quantum structures involving dilute magnetic semiconductors

6T

PL intensity (arb. units)

6T

σ+

179

1.5 K

σ– 0T

σ+ 6T

σ– 6T 2.30

2.35

2.40

CdZnSe QD

Energy (eV)

CdMnSe QD 2.30

2.35

2.40

2.45

2.50

Energy (eV)

Figure 6.19 Magneto-PL spectra taken on DLQD system, where the PL peaks from CdZnSe QD and CdMnSe QD are clearly resolved. The peak intensities of the two PL peaks taken using σ1 and σ2 polarization at 6 Tesla show opposite behavior indicating opposite net spin polarization of excitons in the dots. The magnification of CdMnSe QDs peaks are shown in the inset. After S. Lee, M. Dobrowolska, J.K. Furdyna. Spin relaxation of excitons in nonmagnetic quantum dots: Effect of spin coupling to magnetic semiconductor quantum dots. J. Appl., 99, 08F702 (2006) [89].

other hand (CdMnSe QDs in our case) the sequence of spin states is reversed relative to the CdZnSe QDs [89,91] due to the sp-d interaction between spins of the extended band states and the Mn ions, so that the spin-down states in both bands become the ground states of the system [12]. Although both spin orientations are excited with approximately equal probabilities, in each QD system a fraction of the higher-energy spin states will thermalize to the ground state via spin-flip transitions before recombining. Thus, the emission from CdZnSe QDs is dominated by the σ2 polarization, and the CdMnSe QD emission by the σ1 polarization, as is observed in Fig. 6.19. Since the intensities of the emitted σ1 and σ2 polarizations, I 1 and I 2 , respectively, are directly determined by the spin-down and spin-up populations of the carriers at the instant of recombination, one can obtain the net spin polarization of a given QD family from the measured degree of circular polarization of the PL peak emitted by that QD family. The magnetic field dependence of polarization P observed for the CdZnSe and CdMnSe QDs from the double QD layer is shown in the inset of Fig. 6.20. The observed values of P indicate that the PL from CdZnSe is mostly σ2 -polarized (i.e., dominated by spin-up transitions), while the emission from CdMnSe QDs is mostly σ1 -polarized, indicating that spin orientation of the carriers in DMS QDs is predominantly spin-down. The polarization behavior observed on this QD-double layer structure thus appears at first glance to agree with the respective ground states of the two QD families.

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Chalcogenide

Figure 6.20 The magnetic field dependence of the PL peak intensities for the CdZnSe QDs (top panel) and the CdMnSe QDs (bottom panel) of DLQD structures. The peak intensities of the two PL peaks vary in opposite directions for two different polarizations. The degree of polarization is shown in the inset for each QD layer. Note that the direction of polarization of CdMnSe QDs (i.e., positive P value) is opposite to the CdZnSe QDs (i.e., negative P). The lines are a guide for the eye. After S. Lee, M. Dobrowolska, J.K. Furdyna. Spin relaxation of excitons in nonmagnetic quantum dots: Effect of spin coupling to magnetic semiconductor quantum dots. J. Appl., 99, 08F702 (2006).

However, the degree of polarization of the PL emission from CdZnSe QDs in our QD double-layer system, shown in the inset of Fig. 6.20, is unexpectedly large for non-magnetic QDs. The degree of σ2 polarization is about 25% at 6 T, which is more than twice the value typically observed in a single layer of non-DMS QDs [64,92]. To confirm such enhancement of polarization of non-DMS QDs in DLQD, the polarization obtained from the CdZnSe QD single layer system, which was grown as a reference sample, is plotted in Fig. 6.21 together with P for CdZnSe QDs observed in the DLQD structure. The polarization of CdZnSe QDs is significantly enhanced in the coupled QD structure indicating that the presence of spinspin interaction between two types of QDs. This enhancement of spin polarization of CdZnSe QDs in the DLQD structure can be discussed in terms of anti-parallel spin interaction between the neighboring dot layers [62]. It is well known that the carriers in the DMS quantum structures

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181

Figure 6.21 The magnetic field dependence of the degree of polarization P. The open triangles and solid squares represent the results obtained from the CdZnSe SLQD and CdZnSe QDs in DLQD structure. The significant enhancement of P observed for the CdZnSe QDs of the DLQD structure indicates antiferromagnetic interaction between the carrier spins localized in coupled pairs of QDs from the two layers. The lines are a guide for the eye. After S. Lee, M. Dobrowolska, J.K. Furdyna. Spin relaxation of excitons in nonmagnetic quantum dots: Effect of spin coupling to magnetic semiconductor quantum dots. J. Appl., 99, 08F702 (2006).

are strongly spin polarized even in the moderate magnetic field less than 1 T (see also inset of lower panel in Fig. 6.20) [87]. Once the carriers in the DMS dots relax to their lowest-lying states (which corresponds to the spin-down orientation), the preference for anti-ferromagnetic spin alignment of carriers in coupled dots accelerates the spin-flip process of spin-down carriers in the non-DMS dots into the lowerlying spin-up states, resulting observed enhancement of spin polarization in the CdZnSe QDs. This phenomenon was also observed in the CdSe and CdZnMnSe double layer QD system, a reciprocal structure of current QDDL structure [88], where the value of P and its dependence on the magnetic field are very similar to that shown in Fig. 6.21. Therefore, the results indicate the existence anti-parallel spin interaction between the neighboring dot layers and provides a handle for spin manipulation of non-magnetic QDs by controlling the spin states in an adjacent magnetic QDs.

6.8

Summary

We have described magneto-optical properties of II-VI semiconductor epilayers and their heterostructures involving DMSs. Owing to the high quality of MBE grown ZnTe and ZnSe epilayers, we are able to observe strong exciton peaks including excited states. When the magnetic field is applied, all the exciton states in the epilayers show a diamagnetic shift and optical spectrum develops to Landau level

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transitions, from which information on the band structure are obtained. In addition, polaron effects are observed at large magnetic fields and large Landau quantum numbers, due to the strong electron-phonon interaction. This polaron effect leads to increased carrier masses, thus reducing the experimental transition energies as the Landau quantum number or magnetic field are increased. This also generates an anti-crossing effect in the observed transitions, and no transitions are observed at energies higher that the resonant-polaron energy. The observed the polaron effect and the Landau level transitions contribute to understand detail feature of II-VI semiconductor band structure. The band structure of sinusoidal superlattices is unique, characterized by a single minigap separating two wide free-electron-like subbands for both electrons and holes. Such composition modulated ZnSeTe sinusoidal superlattice has been grown by the process of rotating the substrate in the presence of an inhomogeneous flux distribution of Se and Te in MBE system, in which the period of the modulation is directly controlled by the rate of the substrate rotation. Interband selection rules are derived for optical transitions involving conduction and valence-band states at the superlattice Brillouin-zone center, and at the zone edge. The position of the PL showed the predicted dependence on the period of the SL. The investigation will stimulate exploring the effects of other than sinusoidal band-edge profiles on the superlattice band structure. As the semiconductor heterostructure approaches to zero-dimensional structure (i.e, quantum dot), its spin state becomes very interest because of the promise that it holds for application as q-bits for quantum computation. As investigated in this study, the spin relaxation time of excitons in CdSe/ZnSe QDs is to be significantly longer than exciton recombination time in the presence of magnetic field, so that the carrier spin can be manipulated before its recombination. Coupled QD pairs are especially interesting since spins localized in one QD tend to align antiparallel with those in the other. Such antiferromagnetic interaction between the spins in coupled QDs can provide an effective tool for control of localized spins. Strong enhancement of spin polarization of nonmagnetic CdSe QDs was indeed observed when they are neighbored by CdZnMnSe diluted magnetic semiconductor (DMS) QDs in double-layer geometry.

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62.

2D electron gas in chalcogenide multilayers

7

A. Kazakov and T. Wojtowicz International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Warsaw, Poland

7.1

Introduction

In condensed matter physics, a two-dimensional electron gas (2DEG) is a model system to study electron motion in reduced dimensions. Electrons are free to move in two dimensions but are strictly confined in the third. Such confinement leads to quantized energy levels in the third dimension. These conditions are usually achieved in a quantum well (QW) or transistor-like semiconductor heterostructures and recently in atomically-thin 2D materials. At low temperatures, the mobility of a 2DEG can reach extremely high values, up to several 107 cm2 =Vs for GaAs based QWs [1,2]. Such high-quality QWs opened access to new physics, such as integer [3] and fractional [4] quantum Hall effects, fractional statistics [5], and many other phenomena. In QWs made of chalcogenides, carrier mobility is generally lower than found in III-V semiconductor QWs. However, the advantage of chalcogenide QWs is that they can be doped with Mn without affecting the carrier concentration and without strongly reducing carrier mobility. Manganese is a neutral impurity in II-VI semiconductors but is an acceptor in III-V semiconductors. The exchange interaction between conduction electrons and magnetic ions give rise to a giant spin splitting. Thus, chalcogenide QWs made of diluted magnetic semiconductor (DMS) materials offer the possibility to study highly spin-polarized low-density 2DEGs [6
8]. In addition, QWs based on narrow-gap semiconductors, such as HgTe, offer 2DEG with a giant spin-orbit splitting [9] and the possibility to study a 2D system, where both electrons and holes coexist [10]. 2DEGs in DMS-based QWs have been the subject of intensive research in terms of possible applications in spintronic devices [11
13]. Different devices implementing giant spin splitting have been proposed, e.g., spin aligners and injectors. Among the proposed spin injectors is a device based on a resonant tunneling diode (RTD) [11,12], consisting of a (Zn,Mn)Se DMS QW sandwiched between thin, non-magnetic Zn12x Bex Se barriers, which were separating QW from ZnSe epilayers. In such a device one can selectively bring the spin-up or spin-down states into resonance thus increasing the transmission probability of electrons with selected spin alignment. In recent years there is renewed interest in 2DEG in chalcogenides due to the prediction of various topological phases in such compounds. The quantum spin Hall Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00011-7 © 2020 Elsevier Ltd. All rights reserved.

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effect (QSHE), which is a hallmark of topological insulator (TI) phase, was realized in HgTe QWs [14]. Ferromagnetism introduced into a suitable TI can break timereversal symmetry and lead to the quantum anomalous Hall effect (QAHE). It also was experimentally realized in 2D systems based on chalcogenides, namely in Cr0:15 ðBi0:1 Sb0:9 Þ1:85 Te3 [15] and in Hg12x Mnx Te QWs [16]. One of the main focus in the research of topological materials is the realization of so-called topological superconductor (TSC) phase, with boundary states obeying non-Abelian statistics. Though at the moment there is no readily available material which is in the TSC phase (except the debated occurrence of the TSC phase in Sr2 RuO4 [17,18]), it can be realized in hybrid semiconductor-superconductor systems. Many different proposals have been developed in recent years, several among them are based on DMS QWs [19,20]. Most of the results obtained on 2D and bulk DMS systems in the 1990s and 2000s can be found in extensive reviews [21,22], while transport studies in HgTe and PbTe QWs have been reported in Refs. [23,24] and [25], respectively. In this review, we will focus mainly on experimental results and proposals published in recent years. In the first part, we will concentrate on the consequence of giant spin splitting for magnetotransport. The second part of the current review is devoted to topological phases found in chalcogenide QWs. In particular, we will review several concepts concerning the creation of Majorana states in CdTe QWs and briefly mention QSHE and QAHE in HgTe QWs. We will not discuss here 2D TI phases found in ðBi; SbÞ2 Te3 and ðBi; SbÞ2 Se3 thin films, where the top and bottom surface states hybridize.

7.2

2DEG in magnetically doped QWs

7.2.1 2DEG in low-dimensional heterostructures Most of the chalcogenide heterostructures hosting 2DEG are based on the II-VI semiconductors, such as ZnSe(ZnTe), CdTe, and HgTe. All of them have a zincblende crystal structure with direct band gap at the Γ point. ZnSe and ZnTe have the low temperature highest band gap values of 2.82 and 2.40 eV respectively. The band gap of CdTe is equal to 1.6 eV, while HgTe has the so-called inverted band structure with the band gap of 20.3 eV. It is worth to mention that by varying the composition of Hg12x Cdx Te, the band gap also varies from a positive to a negative value through a zero gap for a particular composition. It was found in Ref. [26] that such zero gap energy spectrum in Hg0:9 Cd0:1 Te has linear dispersion E~k, i.e., Dirac cone. Refs. [27
30] were the first, which recognized the formal analogy between the dispersion relation EðkÞ of relativistic electrons in vacuum and electrons in semiconductors, where the parameter corresponding to the light velocity is proportional to the band gap. After years of development [31], this analogy lead the discovery and understanding of graphene and topological insulators (TIs). The concept of TIs is discussed at the end of the current chapter (Section 3.3).

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Other material systems which we will only briefly mention are IV-VI semiconductors, such as (Pb,Sn)Te and (Pb,Sn)Se. (Pb,Sn)Te and (Pb,Sn)Se crystallize in the cubic rock-salt structure, provided that Sn content is low enough. Their multivalley band structure consists of four elongated ellipsoidal valleys at the L-points of the first Brillouin zone, one for each equivalent [111] direction. The growth of QWs is usually done on (111) BaF2 along the [111] crystallographic axis, which results in lifting the fourfold L-valley degeneracy. Therefore, the ground-state of the 2D subband originates from a single valley with the long axis oriented along the [111] growth direction (longitudinal valley), and the lowest states formed by another three valleys obliquely oriented to the growth direction at an angle of 70:533 (oblique valleys) has higher energy than the longitudinal valley first excited state [32,33]. It is worth to mention that low temperature band gap value in PbTe is 0.18 eV and SnTe has an inverted band structure with 20.3 eV band gap value [34]. Variation of Tin content in Pb12x Snx Te (Pb12x Snx Se) alloy result in similar physics as for Hg12x Cdx Te alloys, i.e., linear dispersion law. 2DEG in chalcogenide materials is usually achieved in heterostructures containing a QW. These heterostructures are made of semiconductors with unequal band gaps, so the resulting band structure can be adjusted to one’s need by an engineered layering of different materials [35]. Such layered structures are usually fabricated with the use of the molecular beam epitaxy (MBE) technique, which allows growing thin films with atomic precision. To restrict the motion of carriers to a 2D plane a confining potential is needed. Confinement is achieved either in type-I heterojunctions together with doping of the wider band gap semiconductor, or in QW structures. In a QW structure, a lower band gap semiconductor layer is sandwiched between wider band gap semiconductor layers. A standard way to separate free charge carriers and charged impurities is the modulation-doping scheme [36]. In such a scheme, dopants are spatially separated from the confining potential well. Thus, carrier scattering at ionized impurities (donors and acceptors) is greatly reduced, which results in higher low-temperature carrier mobility — a conventional measure of 2DEG quality. In most of the works devoted to QWs based on wide band-gap II-VI semiconductors, the QW was grown along the [001] direction on a GaAs substrate. Though there have been studies of QWs grown along other crystallographic directions, they are rather scarce. The big progress in realizing 2D systems based on II-VI materials was achieved after the introduction of halogen dopants. The main advantage of halogen doping over previously more traditional indium doping is its smaller tendency to produce DX-like centers with deep in-gap states [37] and higher doping efficiency [38]. Implementation of halogen doping allowed to create 2DEGs in nonmagnetic QWs CdTe/CdMgTe [39,40], ZnTe/CdSe [41] as well as in their DMS analogues: CdTe/CdMnTe [38], ZnSe/(Zn,Cd,Mn)Se [6,42], CdMnTe/CdMgTe [43], ZnTe/CdMnSe [44], etc. In the case of DMS QWs, the magnetic ions (Mn ions are usually used for this purpose) can be placed either into the barrier materials or into the QWs themselves. In both cases, the electron wavefunction in the QW overlaps with that of the magnetic ion giving rise to s-d exchange interaction, which modifies QW energy levels, especially in the presence of an external magnetic

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field. There are two approaches to place magnetic dopants into QW. The first is the so-called “disordered alloy”, in which the magnetic impurities are randomly distributed over the whole QW width. In the other approach [45,46], called “digital alloy”, nonmagnetic layers (e.g., CdTe or ZnCdSe) are separated by atomically thin magnetic layers (e.g., CdMnTe or MnSe). Such a technique allows achieving higher local concentrations of magnetic ions without the transition to antiferromagnetic or spin glass phases [6]. It was also observed that in various versions of “digital alloy” and “disordered alloy” doping schemes, for the same number of incorporated paramagnetic spins (resulting in the same average concentration of Mn in the QW) the spin splitting of excitonic states is the same while the magnetization dynamics is different [47]. The mobilities of the 2DEGs in digital alloy QWs were high enough to allow observation of Shubnikov-de Haas (SdH) oscillations in magnetoresistance (MR) and the quantum Hall effect (QHE). Here it is worth to note some peculiarities of the band alignment in HgTe/ HgCdTe QWs. Calculations performed with the eight-band kUp method [48,49] have shown that the positions of E1 and H1 bands — E and H label the electron and heavy hole subbands — strongly depend on the QW thickness, d. Above a critical thickness, dc C6:3 nm H1 lays above E1 so the band structure becomes inverted. In the case of dCdc linear dispersion relation, i.e., Dirac cone is observed, thus making HgTe QW with critical thickness a single valley Dirac semimetal [50]. HgTe QWs are usually grown in [001] direction with iodine or indium doping, however, there are several works devoted to (013) QWs [10,23]. Due to the difference between HgTe and CdTe lattice constants, HgTe QWs are strained. It was found that in wide (013) HgTe QWs with dB20 nm [51] this strain leads to overlap between H1 and H2 bands with extrema points at different k, thus constituting a 2D electron-hole system. The most investigated IV-VI semiconductor QWs are PbTe/PbEuTe heterostructures [52,53]. They are usually grown on (111) BaF2 substrate and Bi is used as an n-type dopant [54,55]. Without intentional doping IV-VI semiconductors are typically p-type due to non-stoichiometry; cation vacancies are a natural source of holes [56]. PbTe has been chosen as a QW material since it possesses the highest mobility among lead-tin selenides and tellurides [57]. In addition, PbTe has a very large dielectric constant (EPbTe 5 1300 2 3000 at helium temperatures [58]), hence the Coulomb potential from defects and impurities is efficiently screened. For this reason doping within QW hardly affects the mobility, thus making remote doping unnecessary. The high mobility makes PbTe QWs perfect to study quasiballistic transport phenomena [25,59,60]. Already the first PbTe/PbEuTe QWs were of good enough quality to allow for the observation of SdH oscillations and signatures of QHE [52]. Nevertheless, studies of QWs based on these materials remain rather scarce. It is known that doping of QWs induces a certain disorder which affects the transport properties of 2DEG. This disorder is partially responsible for the persistent photoconductivity (PPC) effect which is observed after illumination of a semiconductor at low temperature. QWs based on II-VI semiconductors exhibit similar effects [61,62]. In the work of Ray et al. [61] PPC in magnetically doped ZnSe QW

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was interpreted as due to the formation of DX centers at Mn sites. However, it is known that n-type dopants also create DX centers in CdTe, CdMnTe, and CdMgTe [37,63
67]. Temperature dependent PPC was also observed in PbTe/PbEuTe QWs with contributions from both barriers and QW [68]. At high temperatures transport via barriers was dominating, while in the low-temperature range conduction proceeded within the QW.

7.2.2 Spin interactions in chalcogenide DMS QWs As already mentioned, the advantage of II-VI chalcogenides over III-V semiconductors is the fact that Manganese (Mn) form a neutral impurity in II-VI compounds, though providing local magnetic moments. Mn has the highest magnetic moment S 5 5=2 among transition metals because of its half-filled d-shell. The direct exchange interaction between neighboring Mn ions is antiferromagnetic, but here we discuss only systems with low Mn content and no magnetic ordering. The giant spin splitting of a 2DEG in the magnetic field stems from the exchange interaction between d-electrons, which are localized on Mn ions, and conduction s- and p-electrons. Exchange contribution to the spin-splitting energy depends on magnetization M of the Mn-subsystem in the applied magnetic field B: [69]
M 5 xeff N0 gμB SBs

 gμB SB ; kb ðT 1 TAF Þ

(7.1)

Here g 5 2 is the free electron g-factor; μB is the Bohr magneton; N0 is the total number of cations per unit volume; Bs ðB; T Þ stands for the Brillouin function; xeff is the effective Mn concentration which is lower than the actual concentration xMn due to antiferromagnetic correlations between Mn magnetic moments at higher doping levels; [70] TAF . 0 accounts for the effect of long-range antiferromagnetic interactions between Mn pairs [71]. In reduced dimensions, under energy confinement conditions, the exchange interaction constant may differ from its bulk value. Indeed, modification of exchange parameters with QW thickness was observed in Cd12x Mnx Te=Cd12x2y Mnx Mgy Te heterostructures with xMn  0:04 2 0:11 [72]. These results agree with those of Ref. [73] where an explanation of such an effect was suggested. The exchange interaction between conduction s-electrons from the center of the Brillouin zone and localized d-electrons has potential character and is positive. For electrons away from the Γ point symmetry is no longer purely s-wave, because p-like states in the valence band admix to Bloch functions of the conduction band. This admixture introduces kinetic exchange correction to the exchange constant, and the magnitude of this correction depends on the degree of admixture. In the case of narrow QWs, kinetic exchange correction can even lead to a sign reversal of the exchange parameter. However, in recently measured Cd12x Mnx Te=Cd0:7 Mg0:3 Te QWs with much lower Mn content (xB0:00146 and B0:00027) no change in exchange constants was observed [74]. Magnetooptical investigations revealed also a pronounced anisotropy of the in-plane

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Zeeman effect [75,76] due to the valence band mixing. The anisotropy of the transverse hole effective g-factor g\ in CdTe/CdMnTe QWs could be so large that it can even change the sign for different directions of the in-plane magnetic field [77]. Besides the exchange interactions, the carrier spins in the QWs experience also spin-orbit (SO) coupling. SO splitting of band states occurs in solids lacking a center of inversion, such as crystals with a zinc-blende structure (Dresselhaus term [78]), as well as systems with structural inversion asymmetry (Rashba term [79]), e.g., asymmetric QWs. SO coupling acts on a carrier spin as an effective magnetic field, which strength and direction depend on electron motion. Dynamics of carrier spins and localized magnetic moments are usually considered together since they are coupled to each other either directly, through exchange scattering [45,80,81], or indirectly, through the lattice (phonons) [82]. The first process is quite fast with characteristic times of the order of 10212 2 10211 s for electrons [45,80] and one order of magnitude slower for holes [80,83], as established by time-resolved Faraday rotation measurements. Spin-lattice relaxation times were found to depend strongly on Mn content, ranging from 1023 s (for xMn  0:4%) to 1028 s (for xMn  10%) [82]. Spin excitations in CdMnTe QWs were probed by the Raman scattering technique and their collective character was established [84]. A very large Knight shift was observed in the 10-nm CdMnTe QWs (xMn B0:2% and B0:3%) for a specific magnetic field strength at which the energies of free carrier and Mn spin excitations were almost equal. This ferromagnetic coupling between Mn spins and conduction electrons was theoretically described in Ref. [85]. The mean-field description of spin excitations developed in this work agreed quantitatively with the experimental data of Ref. [84]. It was noted, however, that the long-range magnetic order in (Cd, Mn)Te QWs can be stabilized only by the spin-orbit coupling. Resonant Raman scattering investigations also proved to be useful in a study of both collective and single-particle spin-flip excitations [86,87]. Careful measurements of angle-resolved magneto-Raman scattering on spin-flip waves in CdMnTe QWs [88] allowed clarification of the dispersion law of the damping rate η, which was found to be η 5 η0 1 η2 q2 [89], where q is the spin-wave in-plane momentum. The quadratic in q enhancement of the damping rate is described by a single-particle dynamics and occurs because of disorder and spin-Coulomb drag. Moreover, it was found that SO coupling affects spin waves in DMS QWs reconstructing them to chiral spin waves, which are invariant under inversion of both magnetic field and wavevector [90]. Observation of chiral spin waves in DMS QWs is possible when Rashba SO coupling is of the order of spin-splitting energy. This condition may be fulfilled by adjusting the QW width, carrier density, and xeff . Indeed, under this condition, chiral spin waves were observed in CdMnTe QWs [91]. It was found that the dispersions of energy and damping rate are both shifted by a wave vector q , which depends on the relative strength of the SO field. Thus, a change of a SO coupling strength (e.g., through gating or PPC effect) can alter the group velocity of spin waves in a system. Control over the propagation direction of spin waves can find its application in magnonic devices. Spin waves observed in CdMnTe QWs were also found to be a perspective source of spin-based THz radiation [92].

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In Refs. [93,94], it was found that interplay between localized moments, phonons, and carriers can even lead to long-range, ferromagnet-induced proximity effect in a ferromagnet-semiconductor hybrid system. In the studied system, a cobalt layer was deposited on the cap layer of CdTe/CdMgTe QW which did not contain any magnetic doping and spin polarization of the acceptor bound hole was detected through photoluminescence spectra. The thickness of the cap layer was varied up to 30 nm which should rule out any proximity effect due to the overlap of the carrier wavefunction with the magnetic layer. It was suggested that elliptically polarized phonons are responsible for angular momentum transfer between the Co layer and the semiconductor leading to polarization of hole spins due to strong SO interaction in the valence band. Polarized phonons couple to transitions between split heavy-hole and light-hole states. This effect is absent in the case of electron gas because the SO interaction is much weaker in the conduction band. The longrange p-d exchange constant was directly measured by spin-flip Raman scattering [94] and its value was found to be 50 2 100 μeV. Application of electric field across such a ferromagnet-semiconductor hybrid structure changes the splitting of the heavy-hole and light-hole states thus affecting phonon coupling. It was shown [95] that electric field strengths of the order of 104 V=cm are sufficient to bring heavy-to-light hole transition out of resonance with the magnon-phonon resonance of the ferromagnet. Therefore, low voltage electric control of the conceptually new long-range exchange coupling mediated by elliptically polarized phonons is possible.

7.2.3 Magnetotransport in chalcogenide QWs 7.2.3.1 Low-field magnetotransport in DMS QWs The first grown DMS QWs had usually low mobilities [6,39], which was caused mainly by poorly developed growth technology and in part by disorder introduced by magnetic ions. All of these QWs had similar features in magnetoresistance (MR) [6,38,42,96], i.e.: (a) significant low-field positive MR and (b) high-field negative MR. Qualitatively the same MR behavior was observed in both perpendicular and parallel fields [6,96], which points to the importance of the giant spin splitting in scattering processes. In 3D DMS such positive MR behavior was explained [97] by the destructive effect of giant exchange-induced spin splitting on the quantum correction to conductivity in the weakly localized regime. The observed increase of the resistivity with lowering temperature indicated the formation of magnetic polarons (MPs) [98
100], ferromagnetic clouds of Mn ion spins that are polarized by a quasilocalized electron spin within its localization orbit. These MPs constitute centers of effective spin-disorder scattering that increases resistivity. Application of high magnetic field destroys MPs, and hence suppress scattering, which leads to the positive magnetoresistance. However, in modulation-doped QWs charged donors are removed from the conductive channel, thus the formation of MPs is less probable. Indeed in magneto-optical studies [101
103] no signatures of bound MP formation in DMS QWs have been found. It was debated that the existence of so-called free

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magnetic polarons — spin-dressed carrier states — may explain the experimental data [6,102]. Strong, high-field, negative MR in DMS QWs, observed close to the metal-insulator transition [104], was explained by the formation of ferromagnetic bubbles and corresponding electronic phase separation. At low temperatures, there is a competition between the insulating DMS phase characterized by antiferromagnetic interactions and the metallic phase which is governed by carrier-mediated ferromagnetic correlations [105]. Thus, ferromagnetic metallic bubbles are formed within a carrier-poor, magnetically disordered matrix. Such behavior resembles that of the colossal MR phenomenon observed in manganites [106]. Low-field positive MR in DMS QWs is usually explained [6,41,42] by the quantum corrections to the conductivity originating from electron-electron interactions [107,108], modified by a giant spin splitting. Magnetic impurities also affect transverse magnetotransport in DMS QWs, leading to the anomalous Hall effect (AHE) which occurs in all magnetic systems. Edwin Hall measured anomalously high “rotational coefficients” in nickel and cobalt [109] almost a year after he discovered the Hall effect [110]. Nowadays this phenomenon is known as AHE and its contribution to total Hall resistance is proportional to the magnetization of the studied material. Thus total Hall resistivity ρxy is described by the empirical formula: ρxy 5 RH B 1 RAHE M, where RH is ordinary Hall coefficient related to carrier density and RAHE is anomalous Hall coefficient. However, the mechanism of AHE is rather complex and consists of three contributions: intrinsic, side jump, and skew-scattering; [111] the latter two mechanisms strongly depend on the strength of SO coupling. Since AHE occurs in magnetic materials, DMS-based QWs are a natural place for studying this effect in 2D. AHE contribution to the Hall effect was extracted in the case of paramagnetic Zn12x2y Cdy Mnx Se QW in the Ref. [112]. A temperature-independent carrier density (and thus the ordinary Hall contribution) was calculated from the period of SdH oscillations, and magnetization was measured by magneto-luminescence. The temperature dependence of the extracted AHE contribution was found to follow the paramagnetic Brillouin-like magnetization of Mn ions (Eq. (7.1)). It was found that the AHE coefficient RAHE Bρxx , which was interpreted as the dominant skewscattering mechanism despite the fact that the SO parameter in the ZnSe system is small [113]. Closely related to AHE is the spin Hall effect (SHE) [114]. While AHE is a phenomenon observed in systems with spin-imbalance, where spin-dependent scattering creates spatial charge separation, SHE occurs in systems with equally populated spin subbands, where spin-dependent scattering creates a spatial spin separation. Therefore, AHE and SHE are governed by the same mechanisms [114] and materials with a high SO constants are best suited for studying SHE. One of such systems is HgTe QW, in which a large value of the SO parameter was found [9]. In Ref. [115] a high mobility HgTe/HgCdTe QW was used in a device geometry proposed in [116]. In the applied experimental scheme the inverse SHE was used for the detection of a spin accumulation induced by SHE. HgTe QW was etched to form an H-shaped device, with two legs connected by a short channel. The electrical current applied to one of the legs generated a spin current through

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the connecting channel. The generated spin current, in turn, generated transverse voltage due to inverse SHE. This non-local voltage induced in the other leg of the device can dominate over local contributions if the separation between the legs is large enough. The device size was chosen to ensure operation in a quasiballistic regime. Gate voltage was used to control the strength of the Rashba SO coupling by variation of the electric field across the QW and the Fermi energy EF in the QW. Inverse SHE was found to be an order of magnitude higher in p-type regime with strong SO coupling than in weakly SO coupled n-type regime. Here it is worth noting that SHE was also detected using Kerr rotation measurements in 1.5 μm thick epilayer of ZnSe [113] despite the small value of the SO parameter in this material.

7.2.3.2 Quantum Hall effect in 2D systems Integer QHE (IQHE) was observed by von Klitzing in 1980 [3] as an appearance of plateaus in a Hall resistance Rxy while varying the gate voltage in a Si-MOSFET. It was found that these plateaus in Hall resistance correspond to quantized values of Rxy 5 νeh2 , where ν is an integer number, h is the Planck constant, and e is the electron charge. The precision of the QHE plateau quantization is 1 part per billion, which makes it a perfect resistance standard. Owing to the rapid progress in growing clean structures, soon after the discovery of IQHE the fractional QHE (FQHE) was discovered [4]. Each of the discoveries — integer and fractional QHE — was awarded the Nobel prize. QHEs were extensively studied mostly in GaAs/(Ga,Al) As QWs and recently in 2D materials, such as graphene. QHE is observed in 2-dimensional systems, where carrier movement in z-direction (perpendicular to the 2D plane) is quantized. From quantum mechanics it is known that confinement of electron motion in a potential well leads to quantization of energy eigenvalues Ei (called subbands) in the growth direction, while the motion in the 2D plane remains unaffected. Further, we will consider only the lowest subband (i 5 0) to be occupied. Two phenomena lay in the basis of IQHE, namely Landau quantization of energy levels in a strong magnetic fields, and the disorder-induced localization. Application of a magnetic field splits  the 2DEG’s  energy levels into Landau levels (LLs) with energies: EcN 5 ¯hωc N 1 12 . Here ωc 5 eB=m is the cyclotron frequency, ¯h — reduced Planck constant, m — effective mass and N is the LL index. Each LL is further split into two spin-resolved energy levels due to Zeeman effect. In wide-gap semiconductors the intrinsic Lande´ g-factor g is relatively small (e.g., in ZnSe gZnSe 51 1:14; [117] in CdTe gCdTe 5 2 1:6 [118]) so the Zeeman splitting Ez of LL is generally much smaller than the cyclotron gap. In a magnetic field B 2DEG will split into the following spin-resolved LLs:
 1 1 EN;m;k 5 EcN 1 Ez 5 ¯hωc N 1 6 g μB B; 2 2

(7.2)

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where EcN is the cyclotron energy, Ez is the Zeeman energy and 6 12 stands for spin orientation m and k. In real samples, these discrete LLs are smeared into bands due to both thermal broadening and scattering. Thus, the resulting density of states in a high magnetic field is split into a sequence of broadened δ-peaks at energy values corresponding to spin-split LLs. The states close to the center of the LL band are extended, while the states in the tails of LLs are localized and do not contribute to transport. Only the extended states can carry current at zero temperature. When these extended states overlap the Hall resistance Rxy is not quantized and the longitudinal resistance Rxx exhibits SdH oscillations due to the oscillating density of states. With increasing of a magnetic field this overlap diminishes. If EF lays between LL bands, Rxy exhibits plateaus at the quantized values νeh2 , where ν (called filling factor) corresponds to number of LL bands below EF , and Rxx 5 0. Whenever EF passes the center of a LL band, Rxy makes a transition between plateaus and Rxx 6¼ 0 during this transition. A widely used description of electrical transport in the QHE regime is the edge channel picture [119] and the Landauer-Bu¨ttiker formalism [120]. Within this picture, 1D edge channels are formed at the intersection of EF with LLs bending at the physical edge of the sample. This can be intuitively understood by treating the exterior of the sample as even higher energy barriers than those which form the QW. As a result, energy levels in the exterior of the sample lie much higher than those inside QW and LLs inside the QW have to smoothly level out to their outside values. The number of edge channels is equal to the number of occupied bulk LLs, 2 and the conductance of each edge channel is G 5 eh , independent of its length. They are also chiral, which means that their direction is determined by the direction of the applied magnetic field. QHE in semiconductor heterostructures is usually observed at helium temperatures and high magnetic fields when the thermal broadening of LLS is lower than the cyclotron energy. Hence, lower m (and consequently the higher EcN ) reduce requirements for the temperature and magnetic field. This is important for the metrology applications since it makes QHE achievable at lower magnetic fields and higher temperatures. QHE at room temperature and 30
45 T magnetic field was observed in graphene, because of the huge LL splitting for massless Dirac fermions [121]. The Dirac semimetal phase can be also achieved in chalcogenide HgTe/ HgCdTe QWs for a particular critical QW thickness dc [50]. QHE states with large energy gaps were indeed observed in QWs with thicknesses close to dc [122,123]. It was argued that further optimization can be obtained with the use of a strained HgTe QW, which would result in a better QHE performance. Thus, achieving precisely quantized plateaus would be possible at magnetic fields accessible with permanent magnets [124]. In the DMS-based QW the LL energies are modified because of the presence of the Mn-subsystem. The spin-splitting term in Eq. (7.2) in this case contains an additional exchange term Eexch , which is proportional to the magnetization of the Mnsubsystem (Eq. (7.1)):

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EN;m;k 5 EcN 1 Ez 1 Eexch 
 1 1 5 ¯hωc N 1 6 g μB B 1 xeff Esd SBs ðB; T Þ ; 2 2

(7.3)

where Esd stands for s-d exchange energy [69] and the Brillouin function Bs ðB; T Þ has the same argument as in Eq. (7.1). Introducing an effective g-factor geff : geff 5 g 1

xeff Esd SBs ðB; T Þ ; μB B

(7.4)

allows to rewrite Eq. (7.3) in the same form as Eq. (7.2). The exchange term Eexch in DMS QWs leads to a strong modification of the LL chart, see Fig. 7.1. First of all, it can be seen that for xeff . 1% the spin-splitting energy Ez 1 Eexch is comparable to the cyclotron energy ¯hωc . In conventional, nonmagnetic QWs such condition is usually achieved by tilting 2DEG by large angles in high magnetic fields [125,126], since Ez depends on the total B and EcN depends only on the perpendicular component B\ of B. In Mn-doped chalcogenide QWs it can be achieved solely by controlling Mn doping [6
8,127] or by gate voltage in asymmetrically doped QWs [62,128]. In both cases, B is perpendicular to the 2DEG plane, which allows to avoid using very high magnetic fields, that are usually applied to keep B\ large enough to maintain QHE quantization. Also, it should be noted that LL spectrum can be further distorted in CdTe QWs by the correction to the spin gap between fully occupied LLs [129]. (A) 15

3

2

Ez

0

4

6

B (T)

8

10

1

0

5

Ec Ez + Eexch

0

2

2

1

E (meV)

0

0

2

10

0

-5

3

1

Ec

5

Filling factor 12 8 6 5 4

15

1

10

E (meV)

(B)

Filling factor 12 8 6 5 4

-5

0

0

2

4

6

8

10

B (T)

Figure 7.1 Calculated LL fan diagram (blue and red lines denote different spin orientations) and chemical potential (Fermi energy, green line) for (A) QW without paramagnetic impurities and (B) QW with paramagnetic impurities. Calculations were performed for carrier density 3U1011 cm22 ; LLs were calculated according Eq. (7.3) using Esd 5 220 meV for (Cd,Mn)Te [69] and xeff 5 1:85%. Here, for the simplicity, we neglected corrections which arise from electron-electron interactions [129].

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7.2.3.3 Modification of Shubnikov-de Haas oscillations The first studies of quantum transport in chalcogenide DMS 2DGEs were actually performed not on QW structures but on the inversion layers created on the surface of (Hg,Mn)Te bulk crystals [130] or naturally occurred in the grain boundaries of (Hg,Cd,Mn)Te [131
134]. Ref. [130] reported probably the first evidence of the modification of SdH oscillations in a 2DEG due to the presence of magnetic impurities, i.e., strong temperature dependence of the positions SdH oscillations. Also, a QHE in DMS was observed in grain boundaries in bulk (Hg,Cd,Mn)Te crystals [132,133]. Well developed QHE plateus were reported in a single isolated grain boundary. Analysis of activation energies, which corresponded to different filling factors, also confirmed the modification of LLs in the presence of magnetic impurities. As mentioned above, SdH oscillations in MR have been already observed in the first obtained chalcogenide QWs. Soon after that, with increasing quality of the 2D systems grown, QHE with well developed plateaus in Rxy was observed in both nonmagnetic and magnetic QWs [6,40,135]. Already the first measurements revealed a modification of the QHE states sequence in magnetic QWs compared to non-magnetic ones [6]. Only even number plateaus (ν 5 2, 4, 6), with the only exception of ν 5 1, were observed up to 10 T in nonmagnetic Zn0:8 Cd0:2 Se=ZnSe QW, indicating that spin splitting is rather small in the studied system. On the other hand, both even and odd filling factors were clearly observed in DMS-based QWs starting from 2 T. This is a clear manifestation of spin splitting enhancement by the s-d exchange term (Eexch in Eq. (7.3)). Moreover, other anomalies in the SdH pattern were observed in DMS-based QWs, such as additional resistance peaks and deviations from the periodicity of SdH versus 1/B [135,136]. The anomalies observed in magnetotransport data agreed well with magnetization measurements, i.e., with the de Haas-van Alphen oscillation pattern obtained using cantilever magnetometry [137]. The above mentioned anomalies in SdH oscillations were thoroughly studied in later works [7,8] which have shown that they also stem from giant spin splitting, which modifies the energy spectrum of LLs (see Fig. 7.1). At low Mn concentrations, when Eexch (Eq. (7.3)) is rather small and the carrier mobility is relatively high, SdH oscillations are well resolved already at low magnetic fields (B # 2 T). A beating pattern in SdH oscillations is present in such samples [7,138,139]. At low magnetic fields the spin-splitting energy is larger than the cyclotron energy and the actual arrangement of LLs depends on the applied magnetic field (see Fig. 7.2A
D). In the case when the spin-splitting energy Ez 1 Eexch is an integer multiplier of the cyclotron energy ¯hωc , LLs from both spin subbands coincide (Fig. 7.2A
C) and a maximum in the SdH amplitude is observed. When Ez 1 Eexch is a half-integer multiplier of ¯hωc , the centers of LLs from one spin subband are aligned between LLs from the opposite spin subband (Fig. 7.2D). Therefore, there are no oscillations in the density of states and a node in the beating pattern is observed. Usually, such beating is visible at low fields B , 1 2 2 T, when QHE plateaus are not well resolved. Furthermore, dips in SdH oscillations between nodes correspond to an alternating sequence of even and odd

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Figure 7.2 (A-C) Possible arrangements of spin-up and spin-down LLs when Zeeman energy is a multiple of the cyclotron energy. In these cases, the Fermi energy EF lies in the gap of the density of states which corresponds to a minimum in Rxx . (D) Arrangements of spin-up and spin-down LLs for a magnetic field corresponding to the case when the Zeeman energy is half-integer of the cyclotron energy (e.g. B5=2 ). In this case, there is no gap in the density of states for any filling factor and a node in the beating pattern of Rxx is observed. (E) Beating pattern in SdH oscillations in the low magnetic field region. Adapted with permission from F. Teran, M. Potemski, D. Maude, T. Andrearczyk, J. Jaroszynski, G. Karczewski, Pauli paramagnetism and Landau level crossing in a modulation doped CdMnTe/CdMgTe quantum well, Phys. Rev. Lett. 88 (18) (2002) 186803.

filling factors, which depends on the difference between the number of filled LLs for opposite spins, see Fig. 7.2A
C. Due to the fact that at such low fields and elevated temperatures the Brillouin function contained in the exchange term (Eq. 7.3) is not yet saturated, TAF can be obtained from the temperature dependence of the node positions. Analyses of SdH beating patterns provide slightly different values for TAF : 180 mK [7] and 40 mK [139] in (Cd,Mn)Te QWs with xMn 5 0:3%, and 2.6 6 0.5 K [138] in a HgMnTe QW with xMn 5 2%. Similar behavior, namely a low-field SdH beating pattern, can also arise in the systems where spin splitting is induced solely by SO coupling [140]. In a QW structure SO effects are mainly governed by structure asymmetry, e.g., produced by gating. In the case of the (Hg,Mn)Te QW, SdH beating was used to separate contributions to the spin splitting arising from the gate-dependent Rashba effect and temperature-dependent spin splitting caused by exchange interactions [138]. In Ref. [139] the exchange term from Eq. (7.3) was incorporated into the Lifshitz-Kosevich formalism in a mean-field approach in order to describe quantitatively the low-field SdH oscillations. Beside the s-d exchange and Zeeman contributions to the spin-splitting energy, additional terms were considered, such as the contribution of antiferromagnetic interactions within pair clusters of Mn atoms [141] and contribution of electron-electron interactions [129]. Considering these terms allowed a better fit of the detailed temperature and field dependencies of node positions as well as determination of cluster formation probability, which was found to be 20% (for xMn B0:3 %). The obtained temperature and field

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dependencies of LL broadening in (Cd,Mn)Te QWs suggested that there is an additional source of LL broadening, caused by inhomogeneity of Mn distribution on the local scale. However, comparison of SdH oscillations in CdTe and (Cd,Mn) Te QWs gives approximately the same values for the total LL broadening in the two systems, thus LL broadening originating from other sources should be smaller in (Cd,Mn)Te QW than in CdTe QW. This can be explained by the suppression of spin-flip scattering between LLs in (Cd,Mn)Te. Indeed, in CdTe QW LLs with opposite spins and the same N are close to each other. On the other hand, in (Cd, Mn)Te the giant spin splitting brings into coincidence LLs with different indexes N and spin orientations. Thus, spin-flip scattering would occur between LLs with different orbital quantum numbers, which may affect the scattering rate between them. Altogether, the proposed modifications allowed achieving a quite good fit to the exchange-induced SdH beating pattern at low B.

7.2.3.4 Quantum Hall ferromagnetic transition At high magnetic fields, as shown in Fig. 7.1B, LLs with different spin orientations cross each other. Thus, it is possible to have 2DEG being partially or fully spinpolarized. Transitions between QHE states with different spin polarization but the same ν are called quantum Hall ferromagnetic (QHF) transitions [8,126,142]. Such transitions were observed both in the integer [8,126] and in the fractional [125,143] QHE regime. The QHF transition is accompanied by an additional peak in the longitudinal resistance Rxx . The properties of QHF transitions in (Cd,Mn)Te QWs were extensively studied in the 2000s [8,144], for a popular description see Ref. [142] and for a short review see Ref. [145]. When two LLs are brought into coincidence at a sufficiently low temperature they split into one fully occupied and one empty LL, provided that the energy gain is larger than the LL band width. This happens below a certain critical temperature Tc , since LL widths decrease with temperature. Below Tc the Fermi energy lies in the energy gap between two LLs and at the lowest temperature, when the thermal broadening of LLs is lower than the energy gap, the QHE state with Rxx 5 0 and Rxy 5 1n eh2 recovers. In magnetic fields on either side of the crossing (in Fig. 7.1B, e.g., the j0mi state crosses the j1ki state at B 5 8 T) 2DEG has different spin polarizations of the top filled LL. The behavior of the peak in Rxx which accompanies the QHF transition at finite non-zero temperature was studied in details in Ref. [8]. The following features of the resistance peak were reported: 1. Dependence of the peak amplitude on the filling factor ν with a maximum at ν 5 2 6 0:02 (see Fig. 7.3A); 2. Hysteretic behavior of the peak position, which manifests itself as a change of the crossing field B under an upward and downward magnetic field sweep (see Fig. 7.3C); 3. Strong temperature dependence: the peak amplitude increases with the temperature reaching its maximum value at Tc and then decreases with a further increase of temperature, while the hysteresis in the peak position gradually decreases with temperature and disappears at Tc (see Fig. 7.3B).

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Figure 7.3 (A) Longitudinal resistance ρxx measured at 0.33 K for carrier density in the range of 2:31 2 3:44 1011 cm-2 . Traces are shifted vertically for clarity. Dashed lines mark additional resistance spikes associated with LL crossings. Numbers in brackets indicate indexes of crossing LLs. (B) Temperature dependence of the spike height for two different carrier densities (n 5 2:97 (squares) and 2:54 1011 cm22 (open circles)). Full circles show the temperature variation of the difference between peak positions ΔB obtained for opposite field sweep directions (presented in (C) for different temperatures). Adapted with permission from J. Jaroszy´nski, T. Andrearczyk, G. Karczewski, J. Wro´bel, T. Wojtowicz, E. Papis, et al., Ising quantum Hall ferromagnet in magnetically doped quantum wells, Phys. Rev. Lett. 89 (26) (2002) 266802.

While Ref. [8] points out that one-electron anti-crossing effects driven by SO interactions are of minor importance, in Ref. [128] the temperature dependence of the peak maximum was interpreted as SO-induced anti-crossing of LLs. Numerical calculations of the SO-induced energy gap between j1ki and j0mi LLs yielded value in quantitative agreement with experimental data. A more thorough experimental study of the anti-crossing gaps is presented in Ref. [144] where energy gaps of QHF transitions for different filling factors were reported. However, to achieve LL crossings at different filling factors the sample had to be tilted. It was found that the energy gap depends on the filling factor — e.g., an anti-crossing energy gap ΔS was closed for filling factors ν 5 5 and 7=2 and opened for ν 5 3; 5=2 and 3=2. These results were explained assuming that ΔS is modified by disorder mediated electron-electron interactions. The effect of disorder was also noticed in Ref. [62] for the filling factor ν 5 2. It is worth to note that a peak in Rxx , which would correspond to a crossing between states j0ki and j0mi, was not observed in transport measurements. A possible reason for that is the high value of the energy gap for this anti-crossing — approximately 10 K, according to the calculations of SO-induced gap [128].

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QHF transitions have also consequences for the configuration of edge channels in the vicinity of the sample edge. The bulk filling factor in the QHE regime reduces in a step-like manner when approaching the sample edge [146]. In a DMS-based QW, during the QHF transition, the arrangement of LLs changes with the magnetic field (Fig. 7.1B). A corresponding change of the edge channel arrangement is hence expected. A theoretical study [147] has shown that edge channel crossings are soliton-like, one-dimensional, Bloch magnetic domain walls, which can be induced on-demand under certain conditions. Recently, such phenomena were studied experimentally through observing magnetoconductance resonances [148]. Experimental data, interpreted in terms of crossings of edge channels originating from different LLs, agreed well with the theoretical picture, though not completely.

7.2.3.5 Fractional quantum Hall effect in DMS QWs As it was said above, if the samples are higher quality, i.e. higher mobility, plateaus with fractional filling factor ν emerge between plateaus in Rxy with an integer ν. In the experiment [4] on a GaAs/AlGaAs QW a QHE plateau with ν 5 13 was observed, which was quite unexpected since there is no energy gap below ν 5 1 in IQHE picture. Explanation of this effect required accounting electron-electron interactions, which were excluded before. A widely accepted theory, which describes FQHE, is a composite fermion (CF) picture [149,150]. Within the CF description, each electron attaches an even number (2p) of quantized vortices, and electronic states are transformed into composite fermion states. CFs are weakly interacting quasi-particles which are moving in reduced magnetic field: BCF 5 B 2 2pnφ0 , where n is carrier density, and φ0 5 he is flux quantum. CFs are forming their own LLs in the effective BCF , which are called Λ levels or ΛLs. Filling factor for CFs CF ν CF 5 jBnφCF0 j is related to electron filling factor: ν 5 2pννCF 6 1 , where minus sign corresponds to a case when BCF is opposite to B. Recent progress in the growth of CdTe/CdMgTe heterostructures made it possible to observe FQHE not only in non-magnetic CdTe QWs [151] but also in DMS (Cd,Mn)Te QWs [143] (see Fig. 7.4A). Fractional filling factors were observed in CdTe-based 2DEG samples with slightly lower mobilities than those needed in GaAs QWs to observe FQHE. The quantum scattering time τ q , which is considered a predictor for the strength of FQHE states [152], turns out to be higher in both non-magnetic CdTe and magnetic (Cd,Mn)Te QWs (τ qCdTe  3 6 0:3 ps [143,151]) than in GaAs QWs of same mobility. Another important difference between FQHE states in GaAs and CdTe QWs is the different Zeeman energy scale. Due to the fact that the g-factor for CFs, gCF , in CdTe is B 2 1:99 [143], which is larger than in GaAs (gCF GaAs 5 2 0:61 [153]), FQHE states were found to be completely spin polarized [151]. The incorporation of paramagnetic Mn ions in (Cd,Mn)Te QWs leads to crossing of ΛLs, which can be described by Eq. (7.3), modified for the CF case:
EN;m;k 5 ¯hωCF c

  1 1 N1 6 gCF μB B 1 xeff Esd SBs ðB; T Þ : 2 2

(7.5)

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Figure 7.4 (A) Longitudinal and Hall resistances in a high mobility (Cd,Mn)Te QW in perpendicular magnetic field. Fractional filling factors are indicated by dashed lines. (B) Angular evolution of FQHE states in the vicinity of ν 5 3=2. Remarkably, 5=3, 4=3, and 8=5 FQHE states first disappear at intermediate tilt angles and then reappear at Larger angles. In contrast, the ρxx minimum corresponding to the 7=5 state can be only observed in the intermediate range at tilt angles around 383 . Adapted with permission from C. Betthausen, P. Giudici, A. Iankilevitch, C. Preis, V. Kolkovsky, M. Wiater, et al., Fractional quantum Hall effect in a dilute magnetic semiconductor, Phys. Rev. B 90 (11) (2014) 115302.

Here almost all notations are the same as in Eq. (7.3), except g for electrons is e2 replaced by the gCF for CFs, N being now the index of CF ΛL, and ωCF c B εlB the pffiffiffiffiffiffiffiffiffiffiffiffiffi CF cyclotron energy with ε — the dielectric constant and lB 5 ¯hc=eB — the magnetic length. Addition of the exchange term for CFs made it possible to reproduce the observed angular dependence of the energy gap for ν 5 53, i.e., closing and reopening of the gap (see Fig. 7.4B), which corresponds to a change of spin polarization. An important result of the observation of FQHE in DMS QWs is that incorporation of a relatively large number of magnetic impurities (xeff 5 0:3%) does not inhibit the formation of FQHE states. However, only fractional states with relatively large energy gaps (ν 5 23 ; 43 ; 53) were observed in experiments [143,151]. (Cd,Mn)Te QWs with even higher quality are needed to study how modification of the CF spin splitting energy affects the versatile FQHE phenomena [150].

7.2.3.6 Magnetotransport in wide HgTe QWs In a wide HgTe (013) QW with d 5 20:5 nm a non-linear Hall resistance was observed at low magnetic fields [10]. It was interpreted as a coexistence of two types of carriers — electrons and holes. It was argued that in such a wide QW the conduction and valence bands are formed by the H1 and H2 bands, respectively. The latter has a local maximum away from the center of the Brillouin zone and there is an overlap with the conduction band, which minimum is at k 5 0. The overlap is of the order of 2
5 meV and depends on a QW thickness. When EF

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coincides with this overlap mixed-type conductivity is observed. Thus, a wide HgTe QW constitutes a 2D electron-hole system. Calculations have shown that the strain (caused by the lattice mismatch between HgTe and CdTe) plays a crucial role in the formation of the overlap between conduction and valence bands [51]. Extensive transport measurements [154] later confirmed this picture qualitatively, however, quantitatively they gave different results for electron and hole effective masses as well as the position of the local maximum for the H2 band in k-space. QHE was measured in the region of mixed-type conductivity [155]. It was found that though there was no plateau in Rxy at high fields, there was a plateau for ν 5 0 in σxy , accompanied with non-zero σxx . However, this state did not exhibit the usual activation behavior with ρðTÞBeΔ=2kT . Moreover, the longitudinal MR was strong, while the Hall resistance was 0. It was claimed, that the origin of this QHE state is the formation of electron-hole “snake states” at the boundary between electron and hole puddles, caused by smooth fluctuations of the local filling factor. These “snake states” are similar to those which arise in the systems with inhomogeneous magnetic fields [156].

7.2.3.7 Magnetotransport in IV-VI QWs In the past, there were several studies of magnetotransport in PbTe/PbEuTe QWs. In Ref. [157] SO coupling was studied by means of the weak antilocalization effect. Weak antilocalization is a result of quantum interference, which manifests itself as a positive correction to the low-field conductivity [107,158,159]. Values of the SO gap ΔSO were found to be in the range from 0:17 to 0:60 meV, depending on QW thickness. These values are comparable to those found in III-V semiconductor QWs. The Rashba term was dominant in the SO splitting. QHE was also observed in IV-VI QWs. The main problem in the studies of QHE in these QWs is the significant parallel conductance [160] resulting from growth on BaF2 substrates, which is responsible for non-local signals and prevents observation of precise QHE quantization. In early studies [52] neither quantized plateaus in Rxy , nor Rxx 5 0 were observed in the field range up to 10 T. The problem of parallel conductance can be solved in structures with higher europium content in the barriers. QHE quantization was achieved in later experiments [53] performed on such structures, though at much higher magnetic fields (B . 10 T). In the cited study an unusual sequence of QHE states was observed (ν 5 15; 11; 10; 6; 5; 4), which was explained by assuming that there are three occupied electrical subbands in the QW, two originating from the longitudinal valley and one from the oblique valleys.

7.2.4 DMS QW in inhomogeneous magnetic fields Now we turn to the behavior of 2DEG in spatially non-uniform magnetic fields. The dynamics of two-dimensional electrons in microscopically inhomogeneous magnetic fields exhibits some peculiarities and differs from electron motion in a spatially uniform magnetic field [156]. There are several ways how an

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inhomogeneous magnetic field can be created. First of all, a microstructure of magnetic elements can create an inhomogeneous stray field in the plane of electron motion. Another way is to place a superconductor on top of the QW. Superconducting elements made of type-II superconductors are penetrated by the magnetic fields in the form of flux tubes if the applied magnetic field is higher than the first critical field Hc1 . These tubes modulate the magnetic field in the QW plane in which electrons are moving [161,162]. Transport properties of such hybrid structures with periodic inhomogeneities [163,164] were extensively studied for nonmagnetic 2DEGs in the 90s. Weak 1D electrostatic periodic modulation of the QW potential has led to the discovery of a new type of MR oscillations, called commensurability oscillations [163]. The influence of 1D periodic modulation on 2D electron motion was treated with perturbation theory. Within this approach [165], LLs broaden into bands whose widths depend on the applied uniform magnetic field. Due to the finite bandwidth a non-zero contribution to transverse conductivity appears which leads to an increase of longitudinal resistance. Every time the Landau bands at the Fermi energy becomes flat, the transverse conductivity vanishes and Rxx exhibits minimum. In the case of DMS QWs, magnetic field inhomogeneity can lead to new topological phenomena [166,167] and to new applications in spintronics [168]. Experimental studies of hybrid structures involving DMS QWs are rather scarce and limited mainly to optical studies [169
172]. It was shown in Ref. [169] that photoluminescence spectra changes after deposition of an iron film on the cap layer of the ZnðMnÞSe=Zn0:80 Cd0:20 Se QW. However, it was claimed that these changes are caused by the strain induced by the iron layer rather than by stray fields. The absence of the observable effects due to stray fields was explained by the difference between the size of the laser spot in the experiment (B10 μm) and the size of the edge region of the deposited magnetic layer in which the magnetic field drops off (B100 nm). Results of local photoluminescence measurements [171,172] were interpreted in terms of excitons trapped locally in the potential minimum produced by the fringe magnetic field of the micrometer size iron island. Non-uniform magnetic fields are also at the root of a new type of spin transistor made of DMS QW, which has been demonstrated in Ref. [168]. The first spin transistor was proposed by Datta and Das [173]. In this proposal, a spin polarized current is injected from the ferromagnetic source into a narrow channel of 2DEG where a gate-controlled SO field induces precession of the carrier’s spins. If the spins of carriers approaching ferromagnetic drain-analyzer after precession are aligned parallel to its magnetization, then the transistor is in the “on” state, if they are aligned antiparallel, the transistor is “off”. However, low efficiency of spininjection and short spin lifetimes result in a quite small signal level, which limits practical realizations of such a transistor. In Ref. [168] a new type of spin transistor was proposed. The operation of such a spin transistor relies on diabatic LandauZener transitions that are controlled by a combination of the homogeneous, externally applied magnetic field and stray fields from a magnetized array of Dysprosium (Dy) stripes (see Fig. 7.5A).

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Figure 7.5 (A) Sample fabricated in the form of a Hall-bar with Dy stripes grating on its surface. (B) MR in studied devices exhibits pronounced maximum at a particular field for different spacing between stripes. (C) Electron spin rotation in the adiabatic regime; the spin direction follows that of the slowly varying total magnetic field, which is a superposition of the externally applied field and stray fields from magnetized Dy stripes. Since the external field does not compensate completely the stray fields anywhere on the path of an electron, the adiabatic theorem applies and the electron remains in the ground state during the motion (“on” state). (D) Electron spin dynamics in the diabatic regime. In the middle between Dy stripes the external field completely compensates the stray fields, thus creating a discontinuity in the direction of the total field. Electrons moving from the source to drain cannot adjust to such a sudden change of its spin direction in order to remain at the lowest energy level, and this leads to backscattering and increase of the device resistance (“off” state). Adapted with permission from C. Betthausen, T. Dollinger, H. Saarikoski, V. Kolkovsky, G. Karczewski, T. Wojtowicz, et al., Spin-transistor action via tunable Landau-Zener transitions, Science 337 (2012) 324
327.

In order to create stray fields, Dy stripes were magnetized at a high magnetic field. The stray fields form a modulated pattern, which slowly rotates the spin of the propagating electron. If the strength of the applied external magnetic field is lower that of the stray fields between the Dy stripes, then the electron spin slowly, adiabatically rotates in the superposition of external and stray fields and remains in the ground state throughout its motion along the channel, Fig. 7.5C. The spin transistor is in the “on” state. The situation is different if the external and stray fields cancel each other between Dy stripes. In this case, a degeneracy point occurs between the Dy stripes and the electron feels a discontinuity in the direction of the total field, Fig. 7.5D. Thus, in order to maintain the same spin direction, the electron would have to transfer to the upper Zeeman band. Since the potential energy

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rises above EF , this leads to back-scattering and spin transport is blocked. The transistor is in the “off” state. Creation of a Dy stripes grating increases the probability of such a process giving rise to a noticeable MR effect. Indeed, for a period of 1 μm between magnetized Dy stripes, a pronounced maximum of  10% in MR was observed at  0:07 T (see Fig. 7.5B). Worth mentioning are also theoretical papers proposing deposition of a type-II superconductor layer on top of the DMS QW containing 2DEG [167,174]. In a clean (i.e., without pin centers) superconductor placed in the magnetic field $ Hc1 vortices form a triangular lattice which is a source of a periodic modulation of the magnetic field in the QW plane. This small modulation of the magnetic field creates spin and charge texture due to the giant spin-splitting energy. Such a periodic charge texture, being an image of the vortex lattice, forms a superlattice in the 2D plane, which leads to the phenomenon of Hofstadter butterfly [175] in a triangular lattice [176]. In a 1D system (e.g., a narrow superconductor stripe of the width wBλ, where λ is the penetration depth), such periodic modulation would lead to Bloch oscillations, whose period will depend on the applied magnetic field. Though similar phenomena were studied in different systems [177
180], the proposed platform may offer an interesting alternative for studying these effects since vortex lattices can be manipulated both with applied magnetic and electric fields. In Ref. [166] a 2-dimensional triangular lattice of ferromagnetic nanocylinders was proposed to induce the so called topological field in the DMS QW. An anomalous Hall contribution, called topological Hall effect, would arise due to electron motion in the topological stray field from magnetized nanocylinders. This topological field is zero on average but has some spatial variation which is solely determined by geometrical parameters of the nanocylinder array. Topological Hall effect, which arises in such a system, is expected to have a very distinct field dependence with several jumps of the Hall conductivity around field values which correspond to the change of topology of the stray field. We also note that in the FQHE regime CFs are moving in an effective magnetic field BCF 5 ð1 2 2pν CF ÞB which depends on the local carrier density. Thus, any spatial variation of carrier density (e.g., created by non-uniformity of remote doping) will result in the inhomogeneous BCF and hence modification of magnetotransport behavior, as observed in GaAs QWs [181,182]. However, in the case of DMS QWs, there are no experimental or theoretical studies of the effect of giant spin splitting on CF magnetotransport in inhomogeneous BCF .

7.2.5 DMS QWs under terahertz and microwave radiation 7.2.5.1 Radiation induced spin currents Many different methods have been proposed to generate spin currents, which are at the heart of spintronics, including the spin Hall effect (SHE). In the context of DMS QWs very interesting is the method of generating spin currents by terahertz (THz) [183] or microwave [184,185] radiation. This approach is based on the spin dependent scattering of carriers by static defects or phonons. In gyrotropic media

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(such as e.g., GaAs or CdTe QWs) an asymmetric, spin-dependent term due to SO interaction exists in the scattering probability. This additional term is linear in respect to the wavevector k and the Pauli matrices σ. Thus, both at the stage of Drude absorption of radiation by carriers followed by scattering (polarization dependent excitation mechanism), and at the stage of energy relaxation process of heated carriers (relaxation mechanism) an imbalance in the distribution of photoexcited spin-up and spin-down carriers between positive and negative k is created. Each of the equally occupied spin subbands creates an equal electric current, but with the direction that depends on the spin orientation. Therefore, electric currents from spin-up and spin-down subbands cancel each other leaving a pure spin current. These spin currents can be converted into spin-polarized electrical currents by applying an external magnetic field, which polarizes the carriers and reduces the compensation of electrical currents from opposite spin subbands. Magnetic-fieldinduced electrical currents are determined by spin imbalance in the system and, consequently, by the spin-splitting energy: [183,186] jx 5 2 e Ez 1kBETexch Js , where Js is the spin current and jx is the converted electrical current. Indeed, using QWs with higher g (e.g., narrow-gap InAs QW), and especially DMS-based QWs (e.g. (Cd, Mn)Te chalcogenide QW) with giant effective g-factor geff , results in very strong amplification of the spin to electrical current conversion [183
186]. In (Cd,Mn)Te QWs this amplification occurs for two reasons [185]. The first is the giant spin splitting, which determines redistribution between spin subbands, and the second reason is the spin-dependent scattering of carriers on Mn21 spins which are polarized in an applied magnetic field. Exchange contribution to the spin splitting is also responsible for the particular temperature dependence of the converted electrical current, which reverses its sign upon cooling to helium temperatures. That is because exchange spin-splitting Eexch depends on the Brillouin function Bs (Eq. (7.3)), and at elevated temperatures, when Bs is negligible, the spin current is determined solely by the intrinsic Zeeman term Ez . At low temperatures, however, the exchange term takes over reversing the sign of the spin current and strongly increases its amplitude.

7.2.5.2 Magnetic quantum ratchet effects Now we will turn to THz radiation induced photocurrents (ratchet currents) in the 2DEG system subjected to spatially periodic, non-centrosymmetric, lateral potentials [187]. Such potential can be achieved, e.g., by a metallic, asymmetric lateral dual-grating gate superlattice produced at the surface of the 2DEG structure. This grating screens the 2DEG underneath the metallic fingers from the THz radiation, while the unprotected 2DEG is heated. Thus, a metal grating leads to inhomogeneous heating of 2DEG inducing modulation of the local electron temperature. Because of the temperature gradient, electrons diffuse from the hot to cold regions forming a non-equilibrium density profile. The ratchet current at zero magnetic field can be represented as a drift current of the electrons in the electric field of the spatially modulated electrostatic potential. This mechanism of creating polarization

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independent ratchet current, called thermoratchet (or Seebeck ratchet) [188], dominates at zero magnetic field and is also most relevant to the magnetic quantum ratchet current observed in DMS QWs [189]. In the absence of magnetic field the position dependent relative correction to the concentration, which defines ratchet current, is much smaller than the temperature correction, which results in a rather weak ratchet current. However, the situation changes when the magnetic field is applied [189,190]. The magnetic field induces ratchet currents in directions both perpendicular and parallel to the stripes of the metal grating. In the presence of quantizing magnetic fields ratchet current exhibit 1=B-periodic oscillations with the same period as SdH oscillations and has strongly enhanced magnitude as compared to ratchet current at zero field. This follows from the fact that now the photocurrent arises from heating induced correction to conductivity rather than from correction to electron density [189]. Near the Dingle temperature variations of the electron temperature result in an exponential increase of conductivity, thus making ratchet current much stronger. Oscillations of the ratchet currents in magnetic and non-magnetic QWs are different, because of the alteration of the LL diagram caused by the magnetic impurities (Fig. 7.1). Strong temperature-dependent modulation of 1=B-periodic oscillations of ratchet currents in (Cd,Mn)Te QWs were reported in Refs. [189,191]. Inclusion of the exchange term Eexch in the spin-splitting energy for magnetic QWs allowed also to explain such experimental findings as the beating pattern of oscillations and weakening of the high field ratchet current at low temperatures. None of these three characteristics of magnetic quantum ratchet have been observed in non-magnetic CdTe QWs.

7.3

Novel topological phases in chalcogenide multilayers

In recent years there has been a burst of interest in new phases of quantum matter, which are characterized by a topological invariant of their ground state [192,193]. These phases are topologically protected, which means that they are protected against any perturbation by a certain symmetry. One of the examples of such materials is the topological insulator (TI), which has non-trivial boundary states that are protected against back-scattering by time-reversal symmetry [193]. Surface and edge states of topological insulators can find application in spintronics because of the spin-momentum locking [192,194]. Another gapped topological state, namely the topological superconductor, can be used in the field of quantum computation [195,196]. Boundary states of topological superconductors obey nonAbelian statistics and are known in the literature as Majorana fermions [197
200]. More information about topological insulators (TIs) can be found in other chapters of this book. In the current chapter we will focus on different proposals [19,20] on creating Majorana fermions in DMS QWs and briefly mention 2D topological phases which were proposed to exist [201
203] in II-VI and IVVI semiconductor heterostructures and the ones, which were experimentally found [14,16] in HgTe QWs.

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7.3.1 Domain walls and non-Abelian excitations As we discussed in the previous section, QHF transition is accompanied with an additional maximum in Rxx ðBÞ. According to the QHE picture, it means that 2DEG goes through a conductive state. It was argued that during such transition a network of conductive domain walls (DWs) is formed and conduction occurs through these conductive channels [204
207]. These DWs separate domains of opposite spin polarization of top filled LL (Fig. 7.6A), and they are not chiral anymore since they are separating QHE states with the same filling factors — in the simplified picture, they consist of two counter-propagating chiral channels, which correspond to LL with opposite spins (Fig. 7.6C). In such a viewpoint, these DWs are helical [20,62], similar to edge states of TIs. There was speculation about possible transport properties of a single DW in the theoretical works at the beginning of 2000s [204
208], but no experimental study of a single DW was done at that time. The study of an individual DW is a challenge since the conventional way to induce the QHF transition is either by tilting B [126,144] or by changing the magnetic dopant concentration [127]. Both approaches drive the QHF transition in the entire 2DEG, controlling thus the global spin polarization of the QHE system. Local control of spin polarization becomes possible in DMS QWs with magnetic ions introduced asymmetrically to the QW region along the growth direction. It was shown [128] that in such a structure the electric field created by an electrostatic gate changes the Eexch contribution to the spin-splitting energy (Eq. (7.3)). This becomes visible when considering Eexch in the hÐ mean-fieldi approximation of the P exchange Hamiltonian: Jsd Ri δðr 2 Ri ÞSiUσ~ ½Mn jϕðzÞj2 dz hSi, where interaction of an electron at a position r with a large number of Mn ions at positions Ri is approximated as an overlap of the electron probability density jϕðzÞj2 with a uniform Mn background within zA½Mn and an average magnetization hSi 5 hSz i 5 SBs ðB; TÞ. Within this approach it is easy to see that in QWs with homogeneous Mn distribution the overlap of the electron wave function with Mn21 ions does not change, even when the wave function is shifted due to the electric

Figure 7.6 (A) Domain structure during QHF transition. Arrows and color denote spin polarization of the top filled LL. (B) Chiral domain boundary between QHE states with different ν. (C) Helical DW between QHE states with different spin polarization but the same ν. It consists of two counter-propagating edge channels with opposite spins.

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field created by voltage applied to the gate, provided that the potential barrier is high enough, so that the wave function does not penetrate the barrier material. Thus exchange splitting remains unchanged [8]. On the contrary in QWs with Mn introduced only into some specific region of QWs an application of gate voltage shift the electrons with respect to this Mn containing region, changes the overlap integral and hence changes exchange contribution to the effective g-factor geff of 2DEG (either it increasing or decreasing). Indeed, it was demonstrated [128] that applying a gate voltage to asymmetrically Mn doped QWs changes geff of a 2DEG. This voltage control of geff was revealed using two manifestations of magnetic interaction in the QWs discussed in the previous section, i.e., low-field SdH beating pattern (for low Mn content, Fig. 7.7B) and high field additional QHF resistance cusp (for larger Mn content, Fig. 7.7A). Thus, electrostatic gating can be used to define an isolated DW at the gate edge, which separates LL of opposite spin polarization. The first study of DW transport properties was reported in Ref. [62] where a (Cd,Mn)Te QW with Mn introduced asymmetrically was used to form an isolated DW. In this experiment, samples were patterned into alternating gated and ungated Hall bar sections. These sections were separated by narrow constrictions along which gate boundaries were aligned. The lithographical length of the DW formed along the gate boundary in the constriction varied with constriction size from B8 μm to B0:8 μm. This experiment employed QHF transition at the ν 5 2 QHE state, i.e., LL crossing between j0mi and j1ki states (Fig. 7.1B). Transport properties of the single DW were probed indirectly by measuring a longitudinal resistance RDW across the constriction in the presence of a phase boundary. The wide Hall bar sections enable independent measurements of resistance spikes associated with QHF transitions in ungated and gated regions (upper panels in Fig. 7.8A). At low temperatures, QHF resistance spikes vanish in these regions. For large size constrictions (thus long DWs) the voltage drop also vanishes at low temperatures (the lowest panel in Fig. 7.8A). In contrast, for narrow constrictions (thus short DWs with lengths of , 4 μm), the voltage drop across the constriction saturates at low temperatures (middle panel in Fig. 7.8A). This indicates that a conductive channel is formed along the gate boundary. However, the low-temperature saturation value of Rxx does not depend on the constriction size (Fig. 7.8B,C). It was also shown that these channels exhibit the necessary for helical channels symmetry with respect to magnetic field reversal. Under reversed magnetic fields the resistance of these DWs remains the same [62], while chiral channels formed at the boundary of QHE states show quantized resistance in one field direction and zero resistance in the opposite direction [209]. Simple model calculations following Landauer-Bu¨ttiker formalism [120] showed, that these channels have relatively high resistance at low temperatures — . h=e2 . Nevertheless, electronic transport through short DWs shows mesoscopic fluctuations (Fig. 7.9A,B), which are clearly seen at low temperatures. Analysis of these fluctuations allowed extracting phase coherence length, which value was comparable to the length of the DW. It was found that the fluctuation pattern changes drastically if the magnetic field is ramped outside the ν 5 2 QHE state (Fig. 7.9B).

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Figure 7.7 (A) The QHF transition is clearly seen in the longitudinal resistance (Rxx ) at ν 5 2 QHE state (seen from Hall curve (Rxy )), measured in the (Cd,Mn)Te QW with xeff B1:7% for zero gate voltages. The QHF transition separates j1mi and j0ki states. (B) SdH beating pattern clearly changes its shape with variation of the back gate voltage from 2200 V to 1200 V in the (Cd,Mn)Te QW with xeff B0:2%. Nodes are shifting to lower fields (blue arrow) and SdH extrema are shifting to higher fields (red arrow) with increasing the back gate voltage. (C,D) Evolution of the QHF peak shown in (A) under application of back (front) gate voltage with fixed front (back) gate voltage, respectively. The position of the QHF transition is highlighted by the white line. In (C) the green line marks the value of B for which polarization of the top LL can be switched between m and k by gate voltage. Reprinted with permission from A. Kazakov, G. Simion, Y. Lyanda-Geller, V. Kolkovsky, Z. Adamus, G. Karczewski, et al., Electrostatic control of quantum Hall ferromagnetic transition: a step toward reconfigurable network of helical channels, Phys. Rev. B 94 (2016) 075309.

It was concluded that dynamic fluctuations rather than static impurities define an actual conduction path in the DW. Activation type behavior of electronic transport through the DWs suggests that its energy spectrum is also gapped, as that of 2DEG LLs. Together with the mesoscopic fluctuations and the suppression of the conduction in long DWs at low temperatures, it suggests the following mechanism of conduction in the DW.

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Figure 7.8 (A) In the devices designed for studying DW transport in the QHE regime Rxx was measured both in the large ungated and gated areas of the Hall bar (upper panel), at the same time RDW was measured across a narrow constriction in the presence of a DW. In the middle panel, the saturating behavior of RDW in the 4 μm constriction and in the lowest panel the vanishing of RDW in the 6 μm constriction upon lowering temperature are presented. (B,C) Arrhenius plots for the temperature dependencies RDW ðTÞ in two different devices for various size constrictions. Solid lines are fits to R 5 R0 1 AUe2Δ=2kT , dashed lines are fits to thermally activated conduction in large areas. Reprinted with permission from A. Kazakov, G. Simion, Y. Lyanda-Geller, V. Kolkovsky, Z. Adamus, G. Karczewski, et al., Mesoscopic transport in electrostatically defined spin-full channels in quantum Hall ferromagnets, Phys. Rev. Lett. 119 (2017) 046803.

At low temperatures electron states inside the energy gap become localized, thus suppressing transport through long DWs ( . 6 μm). However, if the length of DW is less than the localization length, transport through localized in-gap states becomes possible even at low temperatures. This scenario is visualized in Fig. 7.9C, where a single DW is formed by the localized states in the SO-induced anticrossing gap between broadened LLs. These localized states provide a conduction path which defines observed mesoscopic fluctuations (Fig. 7.9D). An isolated DW offers the potential opportunity to create and manipulate nonAbelian excitations, namely Majorana quasi-particles [20] which are important for topological quantum computation [195,196]. In Ref. [20] a hybrid structure was considered, which consisted of s-superconductor and QHF DW at the filling factor ν 5 2. The magnetic field, as well as the global gate voltage, were set to bring the system

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Figure 7.9 (A,B) RDW measured for a 2 μm constriction exhibits mesoscopic fluctuations. The resistance pattern remains similar if the magnetic field is repeatably swept in a narrow range within ν 5 2 (A) and changes considerably if B was changed is swept in a wider range, 5 2 10 T (B). (C) Energy diagram of a DW formed at the gate boundary. Wiggling lines indicate schematically the role of disorder and shaded areas are localized states in the tails of LLs. At low temperatures, conduction occurs via localized states in the gap. (D) Schematic representation of the conducting channel formed by coupled ν 5 2 edge states. Electron tunneling via in-gap states (magenta) provides several interfering trajectories resulting in observed mesoscopic fluctuations of RDW in (A) and (B). Dark shaded area represents a gate. Adapted with permission from A. Kazakov, G. Simion, Y. Lyanda-Geller, V. Kolkovsky, Z. Adamus, G. Karczewski, et al., Mesoscopic transport in electrostatically defined spin-full channels in quantum Hall ferromagnets, Phys. Rev. Lett. 119 (2017) 046803.

close to the QHF transition. EF was in the middle of the SO induced gap so the 2DEG conductivity, except DWs, was exponentially suppressed at low temperatures. As it was discussed previously, in-gap impurity states are responsible for conductivity along short enough DW. Each impurity generates two energy levels for every LL, so in the vicinity of QHF transition at ν 5 2, there are four energy levels bound to the impurity, which are generally non-degenerate unless cyclotron splitting is compensated by the sum Ez 1 Eexch . In the case of such compensation, the impurityinduced states form pairs, each doubly degenerate. Along a DW these impurities form a spatial chain of in-gap states, which is responsible for the conductivity at low temperatures. Together with superconducting pairing, it results in topological superconductivity along the DW. The resulting system resembles generalized Kitaev chain [210] that supports two Majorana localized modes at its ends if jμk 2 Lk j , maxðtk;k11 ; Δk;k11 Þ;

(7.6)

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where k denotes the spatial index of the impurity along DW; μk is the chemical potential at the impurity; Lk is the angular momentum dependent on the gateinduced potential gradient (the latter parameter is also responsible for removing fermion doubling in the DW); tk;k11 is the effective tunneling amplitude between adjacent impurity-induced states; Δk;k11 is the induced superconducting pairing. Thus, topological superconductivity can be controlled exclusively by the gate voltage gradient, provided that DW is proximatized by superconducting contacts. With an appropriate multiple gate structure, Majorana modes can be manipulated, fused and braided [20]. In the end, it should be noted that similar physics is expected in case of FQHE. The difference is that in the FQHE regime and with induced superconductivity DWs in the vicinity of a spin transition would lead to the formation of higher order non-Abelian excitations — parafermions [20,211,212].

7.3.2 Wireless Majorana bound states In Ref. [19] another interesting proposal for creating and braiding non-Abelian excitations was presented. The proposal was based on using a semiconductor with a large g , such as e.g., (Cd,Mn)Te QW, and taking advantage of its giant spin splitting. In the modeled system a superconducting gap in the DMS QW was induced through an s-wave superconductor located underneath the 2DEG. A two-dimensional array of magnetic tunnel junctions (MTJs) was placed on the top of the QW. MTJs can be in two states: either ON (parallel configuration) or OFF (antiparallel configuration). Topologically nontrivial regions are created by inducing a spatial gradient of the perpendicular magnetic field. Thus, magnetic texture produced by the switchable magnetization of MTJ provides a way to control topological phase transitions for confining and braiding Majorana bound states (MBSs). A chain of MTJs in the ON state with the alternating direction of vertical magnetization creates an effective 1D wire which supports MBSs on its ends. By switching the magnetization of a single MTJ from OFF to the proper ON state and vice versa one can move, braid, and fuse MBSs. By measuring the change of a charge (probed either by single-electron transistor spectroscopy or by STM) in a single quantum dot under an initialized MTJ non-Abelian statistics can be probed. An interesting feature in this proposal is the absence of rigid 1D channels: all manipulations with MBSs are performed purely in the 2D plane.

7.3.3 Quantum spin Hall effect in HgTe QWs HgTe QWs have recently attracted a lot of attention, since they were first proposed [213] and then confirmed experimentally [14] to be 2D topological insulators. A topological insulator (TI) is a novel state of matter which is insulating in the bulk but has gapless states at its boundaries, i.e., edge states in 2D and surface states in 3D TIs, respectively.

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Figure 7.10 (A) Dependence of E1 and H1 energy bands on HgTe QW thickness d. (B) Energy dispersion of E1 and H1 bands at thicknesses corresponding to the trivial regime (left), band crossing at critical QW thickness dc (center), and to the inverted regime (right). The color scheme indicates the type of symmetry and is the same as applied in (A). (C) Experimental evidence for QHSE in HgTe/CdTe QW. For Hall bars of tens of microns size (devices I and II) an insulating behavior is observed (two terminal conductance G , 1 e2 =h) while for Hall bars of one micron and less (devices III and IV) conductance quantization G 5 2 e2 =h is observed. (A, B) Reprinted with permission from B.A. Bernevig, T.L. Hughes, and S.-C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314 (2006) 1757
1761. (C) Reprinted with permission from M. Ko¨nig, S. Wiedmann, C. Bru¨ne, A. Roth, H. Buhmann, L.W. Molenkamp, et al., Quantum spin Hall insulator state in HgTe quantum wells, Science 318 (2007) 766
770.

The central role in the theoretical proposal for realizing 2D TI and its hallmark, quantum spin Hall effect (QSHE) [213] is the band inversion. It was already known [214,215] that the barrier material CdTe has normal band alignment with the s-type Γ6 band lying above the p-type Γ8 band, while in HgTe the band alignment is inverted. The Γ6 band lies below the Γ8 band. In HgTe/CdTe QWs these bands form E1, H1, and L1 subbands (the latter is separated from the first two and can be disregarded). In a HgTe/(Hg,Cd)Te QW with thickness d lower than the critical thickness dc C6:3 nm “normal” band arrangement occurs (E1 lies above H1), as in CdTe, but when d . dc , the band structure is inverted (E1 lies below H1) [213], see Fig. 7.10A
B. In the ‘inverted’ regime (QSHE state) “helical” edge states appear at the sample edges. Their main property is that they are “spin filtered”, i.e., the up spin propagates in one direction, while the down spin propagates in another. A quantized value of two-terminal conductance G 5 2e2 =h was noted as an experimental consequence of QSHE [216]. Indeed, a value of two-terminal conductance close to the predicted quantized value was observed experimentally [14] in the 7:3 nm thick HgTe=Hg0:3 Cd0:7 Te QW at zero magnetic field (Fig. 7.10C), however, only over distances shorter than 1 μm. Longer distances lead to a vanishing conductivity of edge channels [14,217]. Meanwhile, theoretical predictions [218] state that helical

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edge states are topologically protected against back-scattering unless the timereversal symmetry is broken. Different mechanisms [219
227] of inelastic backscattering were proposed, which limit the length over which quantized transport can be observed. However, the mechanism responsible for inelastic backscattering in helical channels has not been yet determined experimentally. One of the exciting proposal to implement TI helical edge states in future devices is to use them as a platform for the realization of topological superconductivity [228]. That’s why the investigation of the interplay between 2D TI phases and superconductivity attract a lot of attention in recent years [229
231]. HgTe QWs are one of the main focuses of current research in this field. It was shown by studying the Fraunhofer interference that Josephson supercurrent indeed can be induced to QSHE edge states [229]. Later study revealed details of the electron pairing, which occurs in the materials with non-trivial spin structure proximatized by a superconductor [230]. Recently it was experimentally shown that induced superconductivity has indeed a different nature in trivial and topological regimes, which was controlled by applying in-plane magnetic field [231]. Thus, HgTe QWs are a promising platform in building scalable networks of Majorana devices for fault-tolerant quantum computation.

7.3.4 Quantum anomalous Hall effect in HgTe QWs It was predicted [232] that magnetic doping may break time-reversal symmetry and leave only one spin-polarized edge state out of the two forming QSHE helical edge state. This can be understood as follows. Magnetic doping introduces a spin-splitting term to the QSHE Hamiltonian and if one of the spin blocks moves to the trivial regime while the other one remains in the ‘inverted’ regime, then the system will be in so-called QAHE state. This imposes certain constriction on the spinsplitting gaps 2GE and 2GH of E1 and H1 states of HgTe QW respectively, i.e. GE GH , 0. This condition simply means that g of E1 and H1 subbands must have opposite signs, which fortunately is the case for HgTe QWs [233]. Analysis of the QAHE parameter space for Mn-doped HgTe QW [232] showed that it can be in the QAHE state as long as the magnetization is large enough and perpendicular to the QW plane. The main problem for QAHE realization in (Hg,Mn)Te/CdTe QWs is the fact that the exchange field does not drive the magnetic system into the ferromagnetic state since QWs are paramagnetic up to Mn content of  15% [234]. Thus, a small external field ( 70 mT) is still required to drive the system to the ν 5 2 1 QAHE state [16]. However, any small magnetic field has also an orbital effect and such observation alone cannot unambiguously prove the existence of the QAHE state. Nevertheless, experiments in tilted magnetic fields showed different evolution of the ν 5 2 1 state under an in-plane field in nonmagnetic HgTe and magnetically doped (Hg,Mn)Te QWs. This observation indicates that these states have a different origin.

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7.3.5 Topological phases in IV-VI materials In recent years there is a renewed interest in IV-VI semiconductors containing tin (e.g., (Pb,Sn)Se). It was predicted [235] and then confirmed by ARPES measurements [236,237] that they belong to a new class of topological insulators — topological crystalline insulators (TCIs), whose surface states are topologically protected by crystalline symmetries [235,238]. Calculations show that while Pb12x Snx Te and Pb12x Snx Se are topologically trivial for xSn 5 0 [235], band inversion at low temperatures occurs with increasing Sn content at critical values of xcSn 5 0:35 [34] for tellurides and xcSn 5 0:15 for selenides [239]. It was shown theoretically that a QW structure made of TCI material will also host helical edge states [202,203], and in the case of a ferromagnetic order — the QAHE phase with a higher Chern number [240]. Topologically trivial nature of PbTe/PbEuTe QWs was established in Ref. [160] by careful analysis of non-local measurements interpreted in terms of parallel conductance through PbEuTe barriers. However, studies of Pb12x Snx Te or Pb12x Snx Se QWs with the composition in the topological regime still await realization.

7.4

Summary and perspectives

Despite being studied for almost 30 years, 2D systems based on chalcogenide multilayers still attract a lot of attention. Almost all recent developments concerning 2DEG reviewed here are based on the fact that magnetic ions can be easily incorporated into the chalcogenide QW structure, thus giving rise to giant spin splitting due to exchange interaction between carriers and magnetic ions. Moreover, with the use of MBE growth technique, spin splitting (or in other words geff ) can be engineered not only by changing concentration of magnetic component in disorder alloys, but also by placing magnetic ions in strictly predefined positions, so as to produce digital alloys, graded potential QWs [127] or magnetically asymmetric QWs [128]. In the latter case, spin splitting can be externally controlled not only by magnetic field and temperature but also by the electric field from the voltage applied to electric gates in nanodevices [62]. Very important is also the fact that incorporation of magnetic ions into chalcogenides, at already useful level of concentration of localized spins, does not lead to any strong reduction of 2DEG mobility [139,143]. Finally, the concentration of 2DEG can be varied by donor doping independently of magnetic doping. All these facts lay at the basis of many proposals to implement DMS QWs in future devices and realization of novel topological phases. Some of these proposals are still awaiting realization. The very important direction is studying superconducting proximity effect in 2DEG in chalcogenide QWs. Several platforms for realization of a topological qubit, based on materials reviewed in the current chapter, have been proposed in recent years. The very interesting direction in this area is utilizing DWs which emerge during QHF transitions to create and manipulate non-Abelian excitations [20]. This particular goal would also require inducing superconductivity in 2DEG

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through the proximity effect. Superconductivity induced in a semiconductor has attracted a lot of attention nowadays because of potential application for topological quantum computation [195,196]. Also, there are only a few studies of the interplay between a superconductor and QHE states [241
243]. Investigation of such interplay would be particularly interesting in the case of DMS QWs with a giant spin splitting. Quite interesting would be a practical realization of topological phases in a non-uniform magnetic field created by a periodic lattice of vortices in a superconductor proposed already many years ago [167]. The increasing quality of grown structures would allow observing more fragile FQHE states, which versatile physics can also bring unexpected results in the case of 2DEGs with giant spin splitting. Moreover, the realization of 2D topological phases in QWs of IV-VI semiconductors will introduce another member to the small group of 2D TI materials [14,244,245]. There is also hope, that further improvement in the quality of heterostructures, which is to be achieved by progress in the MBE growth technology, will help to develop new ideas for interesting physical studies and for novel devices based on 2DEGs in chalcogenide multilayers.

Acknowledgment This work was partially supported by the Foundation for Polish Science through the IRA Programme co-financed by EU within SG OP and by the National Science Centre (Poland) through Grant No. DEC- 2012/06/A/ST3/00247.

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79.

Layered two-dimensional selenides and tellurides grown by molecular beam epitaxy

8

Xinyu Liu1, J.K. Furdyna1, Sergei Rouvimov2, Suresh Vishwanath3, Debdeep Jena3, Huili Grace Xing3 and David J. Smith4 1 Department of Physics, University of Notre Dame, Notre Dame, IN, United States, 2 Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, United States, 3School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, United States, 4Department of Physics, Arizona State University, Tempe, AZ, United States

8.1

Introduction

8.1.1 Motivation To develop new system that can integrate, improve, and even replace complementarymetal-oxide-semiconductor (CMOS) technology, monolayers (MLs) of graphene have been extensively studied [1], and it were demonstrated to perform as twodimensional (2D) systems with ultrahigh mobility [2,3], unique optical properties [4
6], and high mechanical in-plane strength [7
9]. However, the absence of a band gap in single-layer graphene has been a major challenge for integration into digital logic circuits [10]. Alternatively, 2D layered chalcogenides have recently attracted widespread attention as candidates for novel electronic device applications because of the wide scope of their electrical properties, that can be superconducting, magnetic, half-metallic, and semiconducting with band gaps ranging from far infrared to near ultraviolet [9,11
15]. 2D layered chalcogenides also exhibit distinct dependence of their electronic and optical properties on the number of layers in the system. For example, monolayers of MoS2 and MoSe2 have larger direct band gaps than their multilayers [11
14,16
20]. Furthermore, the 2D topology of these layered materials enables heterojuction stacking without inducing strain between the layers. This feature makes possible the fabrication of transistors and diodes scaled to atomic thicknesses with tunable band gaps and excitonic effects [21
23]. The family of 2D chalcogenides currently consists of over 40 different species [24
27]. Various 2D materials growth methods, characterization of physical properties, and device applications have been intensely pursued by multiple research groups. Initial investigations on 2D chalcogenides focused on exfoliated bulk crystals that were naturally formed or were grown by chemical vapor Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00012-9 © 2020 Elsevier Ltd. All rights reserved.

236

Chalcogenide

transport (CVT) or chemical vapor deposition (CVD) [28
30]. More recently molecular beam epitaxy (MBE) has also been employed for growth of highpurity, electronic grade crystals of 2D materials [9,31
36], with very promising results. In a 3D semiconductor system, the growth of heterostructures is limited primarily by the lattice mismatch between various layers. Hence, a figure like Fig. 8.1A plotting band gap versus lattice constant sets the constraints on the choice of layers in a heterostructure, including the substrate [36]. However, in the case of a layered material grown on another layered material, as reported in the case of MBE MoSe2 grown on graphene and on highly-ordered pyrolytic graphite (HOPG) substrate [34], the growth of 2D materials is by Van der Waals (VdW) epitaxy process. This is supported by the fact that we do not see any discernible changes in lattice constants in the layers as one moves away from the interface in transmission electron microscopy (TEM) images. This gives added freedom to design electronic or optical devices by using band alignments, as shown in Fig. 8.1B [36]. Promise of few-layer heterostructures has been demonstrated in form of WSe2/MoS2 p
n junctions [37], black phosphorus/SnSe2 tunnel diodes [23]. The near broken band alignment of WSe2/SnSe2, MoTe2/SnSe2, GaSe/SnSe2 seen in Fig. 8.1B is especially interesting for electronic applications such as highly scaled tunnel field effect transistors (Thin-TFET) [19]. With the recent observation of direct band gap in few-layer structures of MoTe2 [38] and in bulk ReS2 [39], few layered materials and heterostructures would likely be strong candidates for optoelectronic devices as well.

Figure 8.1 (A) Plot showing band gap as a function of lattice constant of various bulk layered materials. Materials in red are layered 2D materials of interest in this chapter, in orange are presented quasi-2D layered materials, blue dashed lines represent lattice constants of common bulk 3D substrates with hexagonal symmetry, and green lines show those with cubic symmetry. (B) Cumulative plot of experimentally determined band gaps (BGs) and electron affinity (χ) of various layered materials in bulk form, compared to common compound semiconductor substrates. Adapted from S. Vishwanath, X.Y. Liu, S. Rouvimov, L. Basile, N. Lu, A. Azcatl, et al., Controllable growth of layered selenide and telluride heterostructures and superlattices using molecular beam epitaxy, J. Mater. Res. 31 (2016) 900
910.

Layered two-dimensional selenides and tellurides grown by molecular beam epitaxy

237

8.1.2 A survey of 2D chalcogenides 2D and quasi-2D materials don’t yet have uniformly accepted definitions. In this chapter, In Table 8.1 we group most common 2D materials by the number of atomic layers comprising the monolayer [36]. In the Table 8.1 we have also listed ratios of the lattice constant perpendicular to the cleavage plane of the layered material (c) to the sum of the ionic radii of the constituent atoms of the monolayer. We see that for layered materials having 3 atomic layers in a monolayer, all materials are clustered together irrespective of the group in the periodic table to which the cation belongs. On the other hand, in materials comprised of 2 atomic layers or 5 atomic layers the ratios change dramatically with change in the group to which the cation belongs. The definition of the ratio holds well for layered materials having greater than one atomic layer per monolayer. It gives us a rough estimate of the fraction ascribed to the Van der Waals gap per monolayer height. Hence, in spite of the 2D-like layer structure of quasi-2D materials, the cation in one monolayer likely has more in this case significant interaction with the cation in the next monolayer as compared to true 2D materials. In addition, Table 8.1 also lists MBE-grown 2D materials reported in the literature to date. High-quality growth of 2D layered chalcogenides with well-controlled properties is clearly at the heart of research on these materials. This, however, is still a significant challenge owing to difficulty of growing these atomically-thin layers, to the inevitable substrate effect, and to the material stability itself. The objectives of this chapter is therefore to provide an overview of recent experimental approaches developed to address these issues, with particular focus on MBE-growth of 2D layered chalcogenides and their physical characterization. This chapter will also point out the challenges in this field, and provide our viewpoint on what the future holds for the 2D layered chalcogenides. The rest of this chapter is organized as follows: In Section II we will discuss the fundamentals of MBE growth of 2D materials. Section III covers the physical properties of 2D materials grown by MBE. And finally in Section IV we lay out the opportunities which this exciting research area holds in store, as well as the challenges that still lie ahead.

8.2

MBE growth of 2D materials

8.2.1 Advantages of MBE growth of 2D materials Molecular beam epitaxy (MBE) growth of 2D materials, although pioneered in early 1990s [59,78
81], has been blooming recently [31,34,64,68,82
84], owing to some extent to the idea that 2D materials can be stacked as Lego blocks, so as to realize various structures desirable to probe novel physics and device applications [85], as illustrated by Geim in his paper on Van der Waals heterostructures [86], Historically, one advantage of MBE is to achieve abrupt semiconductor heterostructures with monolayer precision. In-situ monitoring of thickness and composition of

Table 8.1 A survey of 2D layered materials. Substrate & reference

Ionic radius of cation ˚) (A

Ionic radius of anion ˚) (A

Sum of atomic radii in each monolayer ˚) (A

(c/2)/ Sum of atomic radius in each monolayer

3.3

0.41

1.46

1.46

2.28082

1

3.4

0.7

0.7

0.7

4.78571

GeSe

2

5.395

0.73

1.84

2.57

2.099

SnSe

2

5.7

0.69

1.84

2.53

2.27075

TiTe2

3

6.5

0.67

2.07

4.81

1.35135

ZrTe2

3

6.6

0.72

2.07

4.86

1.35802

InAs(111) [41]

HfSe2

3

6.2

0.85

1.84

4.53

1.3596

AlN(0001)/Si(111) [42,43]

Materials

Total atomic layers per monolayer

h-BN

1

Graphene

c/2 lattice constant/ ˚) 2 (A

GaAs(111) [40]

MoS2 or HOPG [33,44] HfS2

3

5.8

0.85

1.7

4.25

1.36471

TiSe2

3

6.0

0.67

1.84

4.35

1.37931

graphene/SiC [45] bilayer graphene/ SiC [46
48]

ZrSe2

3

6.2

0.72

1.84

4.4

1.39773

AlN(0001)/Si(111) [49]

TiS2

3

5.7

0.67

1.7

4.07

1.40049

SnSe2

3

6.1

0.69

1.84

4.37

1.40275

MoS2, NbSe2, graphite [50] GaAs(111) [51]

ZrS2

3

5.8

0.72

1.7

4.12

1.40777

MoTe2

3

7.0

0.79

2.07

4.93

1.41582

sapphire; [52] MoS2 [53] bilayer graphene/SiC [54] CaF2(111) or GaAs(111) [55]

SnS2

3

5.9

0.69

1.7

4.09

1.43863

mica [56]

WSe2

3

6.5

0.8

1.84

4.48

1.443

HOPG [9]

NbSe2

3

6.3

0.68

1.84

4.36

1.44495

GaAs(111)B [57] Au(111) [58]

MoSe2

3

6.5

0.79

1.84

4.47

1.44642

SnS2; [59] MoS2 [60,61] S-terminated GaAs(111) [62] GaAs(111); [34] CaF2(111) [34] sapphire;[52] HOPG [31,34,63
65] graphene/SiC [34,65,66] bilayer graphene/SiC [31,67] AlN(0001)/Si(111) [42] HfSe2/AlN(0001)/Si(111) [43] ZrSe2/AlN(0001)/Si(111) [49] AlN(0001)/Si(111) [68]

MoS2

3

6.1

0.79

1.7

4.19

1.46539

WS2

3

6.2

0.8

1.7

4.2

1.47619

SiTe2

3

6.7

0.4

2.07

4.54

1.47797

Au(111) [69]

(Continued)

Table 8.1 (Continued) Materials

Total atomic layers per monolayer

GaSe

4

c/2 lattice constant/ ˚) 2 (A 8

Ionic radius of cation ˚) (A

Ionic radius of anion ˚) (A

Sum of atomic radii in each monolayer ˚) (A

(c/2)/ Sum of atomic radius in each monolayer

Substrate & reference

0.76

1.84

5.2

1.53846

GaAs(111)B; [70] GaAs(111) [71,72] H-terminated Si(111) [73,74] Si(111), Si(110), Si(100) [75,76] InSe(buffer)/InSe(0001) [77]

InSe

4

12.473

0.94

1.84

5.56

2.24335

Bi2Te3

5

15.2

1.17

2.07

8.55

1.78304

Bi2Se3

5

14.4

1.17

1.84

7.86

1.8257

Sb2Te3

5

15.2

0.9

2.07

8.01

1.894

In2Se3

5

14.38

0.94

1.84

7.4

1.9432

InSe(0001) or GaSe(0001) [77]

Adapted from S. Vishwanath, X.Y. Liu, S. Rouvimov, L. Basile, N. Lu, A. Azcatl, et al., Controllable growth of layered selenide and telluride heterostructures and superlattices using molecular beam epitaxy, J. Mater. Res. 31 (2016) 900
910.

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241

epitaxial layers by various techniques is an important additional advantage made possible by MBE methodology. While 3D heterostructures with atomic scale thickness have been successfully fabricated by MBE and other epitaxial growth techniques, high quality 3D heteroepitaxial growth was typically restricted to only limited combinations of materials due to severe lattice matching conditions that need to be satisfied. In his pioneering work [87], Koma proposed a new kind of epitaxial technique, Van der Waals epitaxy (VdWE) [88], to overcome such restriction. As shown in Fig. 8.2A, usually there appear dangling bonds on a clean surface of a substrate, and this makes it difficult to grow good heteroepitaxial films without good lattice match in the constituent materials. There are, however, materials having no dangling bonds on their clean surfaces, as is seen in Fig. 8.2B, on which epitaxial growth proceeds via Van der Waals forces [73,82,89,90]. In this case, the requirement for lattice mismatch is less strict, given by the weak VdW or Coulombic interactions along direction normal to the 2D plane. Koma also highlighted two potential benefits from Van der Waals epitaxy: 1. good heterostructures can be grown by such epitaxy even between materials having significant lattice mismatch; 2. using Van der Waals epitaxy, a

Figure 8.2 Interfaces connected by (A) dangling bonds, (B) Van der Waals gap, and (C) quasi Van der Waals gap. Adapted from A. Koma, Van der Waals epitaxy for highly lattice-mismatched systems, J. Cryst. Growth 201 (1999) 236
241.

242

Chalcogenide

very abrupt interfaces with small amounts of defects can be obtained because of the nonexistence of dangling bonds. Thus, Van der Waals epitaxy provides a powerful strategy for fabricating high-quality heterostructures, even layers with thicknesses of one atomic monolayer. The primary advantages of VdWE over ex-situ approaches, such as micromechanical exfoliation, include minimal ambient contamination, atomically sharp interfaces and less constraint in lattice mismatch. In this section, we will discuss the step by step process for achieving high quality Van der Waals materials and their heterostructures using a standard MBE system.

8.2.2 Growth of layered selenide and telluride films and their heterostructures All layered selenide and telluride films in this chapter were grown using a dualchamber Riber 32 MBE system. In our case, one of the chambers of the machine is dedicated to the growth of III-V-based compound semiconductor, and is equipped with Al, Ga, In, As, Sb, and P elemental sources. The second chamber originally intended exclusively for the growth of 2D chalcogenides, is currently operating with Bi, Sn, Se and Te elemental effusion cells, and contains an Oxford Applied Research e-beam evaporator, which serves as a source of high-melting-point metals, such as Mo, W, and Nb. The two chambers are inter-connected by an ultrahigh vacuum channel, allowing transfer of wafers between the two chambers without exposure to the atmosphere. The growth processes in each chamber are monitored in-situ using reflection-high-energy electron diffraction (RHEED). We will demonstrate MBE growth of 2D chalcogenides using the following selected materials as examples.

8.2.2.1 Tin selenide Tin selenide thin films were grown on GaAs(111)B substrates. The growth sequence was as follows: First an epi-ready substrate was deoxidized by heating to 600  C in the III
V MBE chamber, and a 60-nm-thick GaAs buffer layer was then deposited. The sample was subsequently transferred via an ultra-high-vacuum loadlock system to the second chamber for tin selenide deposition. The substrate was immediately selenated by exposure to Se flux for 15 min at 600  C, and the RHEED pattern was observed to become streaky. The substrate was then cooled in the presence of Se flux to 150  C, and growth of the SnSe(x) films was initiated. The tin selenide growth was carried out at a substrate temperature of B150  C, using selected Se:Sn flux ratios, as stated below. Finally, the samples were transferred back to the III-V chamber for deposition of a protective GaAs capping layer. Four growths were carried out using Se:Sn flux ratios of 3:1, 4.6:1, 10:1, and 40:1. The materials were characterized ex situ by high-resolution x-ray diffraction (HRXRD) and transmission electron microscopy (TEM) [51]. Fig. 8.3A shows HR-XRD patterns obtained from tin selenide films grown with two different Se:Sn flux ratios. For the film grown with a Se:Sn ratio of 3:1, only reflections from (200)-type lattice planes of SnSe are visible; and for the film

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Figure 8.3 (A) High-resolution X-ray diffraction patterns for tin selenide films grown on GaAs (111)B substrates. SnSe forms when the Se flux is weak (Se:Sn 5 3), while conditions for formation of SnSe2 are favorable in the presence of strong Se flux (Se:Sn 5 40). (B) High-resolution electron micrograph (HR-TEM) of tin selenide thin film grown with flux ratio of Se:Sn 5 3:1, showing a region where both SnSe and SnSe2 phases are present. (C) HR-TEM of SnSe2 thin film grown with flux ratio of Se:Sn 5 10:1. (D) HR-TEM of SnSe2 thin film grown with flux ratio of Se:Sn 5 40:1. The substrate projection is GaAs [1
10] for all TEM images. Adapted from B.D. Tracy, X. Li, X.Y. Liu, J. Furdyna, M. Dobrowolska, D.J. Smith, Characterization of structural defects in SnSe2 thin films grown by molecular beam epitaxy on GaAs (111)B substrates, J. Cryst. Growth 453 (2016) 58
64.

grown with Se:Sn ratio of 40:1, only reflections from (0001)-type lattice planes of SnSe2 are observed. Fig. 8.3B
D shows cross-section electron micrographs of the tin selenide films grown with three different Se:Sn flux ratios. Overall, the tin selenide films generally showed flat selenide/GaAs interfaces and relatively flat top surfaces, while improved crystallinity was apparent for samples grown with higher Se: Sn flux ratios. As an example, Fig. 8.3B shows two different areas of tin selenide film for the sample grown with the flux ratio Se:Sn 5 3:1. The film shows reasonable crystallinity, and the structure is consistent with layer-by-layer growth. However, a high density of horizontal stacking faults and dislocations is visible,

244

Chalcogenide

and the film appears to have several distinctive areas of different atomic layer thickness, which can be identified respectively as SnSe oriented along the [011] direction, and as SnSe2 oriented in the [0
110] direction. Fig 8.3C and D show HR-TEM images of tin selenide films grown with flux ratios of Se:Sn 5 10:1 and 40:1, respectively, as viewed in direction identified as [0
110] for hexagonal SnSe2. Overall the SnSe2 crystal lattice shows high quality in both cases, and fewer defects are seen in both cases compared with samples grown with lower flux ratios. However, the density of structural defects is seen to vary from place to place across the films. Note, however, that the observed growth defects are mainly confined to a region within a few nanometers from the GaAs substrate surface, most likely due to the lattice mismatch between GaAs and SnSe2. The threading defect visible within the bulk of the film (arrows in Fig. 8.3D) can be traced back down to the termination of an extra GaAs monolayer step at the interface, suggesting the possibility of columnar growth. In summary, this study shows that high quality SnSe2 films can be grown on GaAs(111)B substrates when high Se:Sn ratios ( . 10:1) are used during MBE growth. Importantly, as the flux ratio is increased, the presence of SnSe rapidly diminishes; and the crystal quality of the 2D SnSe2 continues to improve with increasing Se flux intensity.

8.2.2.2 Molybdenum telluride Systems comprised of a few layers of 2H-MoTe2 can be successfully grown on GaAs (111)B and CaF2 (111) substrates at a low substrate temperature (340  C) at a growth rate of B6 min/ML, and with Te:Mo flux greater than 100 [55]. Before the growth, GaAs substrate is treated as described in the previous section, with presence of Te flux. The CaF2 substrate is first heated to 800  C in the growth chamber for 30 min in vacuum (B5 3 10210 Torr) to allow desorption of weakly bound surface contaminants, and then cooled to the growth temperature of 340
440  C. Once the growth temperature is stabilized, Mo and Te are deposited on the substrate simultaneously from the MBE e-beam evaporator and effusion cell. During the growth, clear RHEED streaks of MoTe2 (Fig. 8.4(a)) are observed along [11
20] direction for growth on either of the substrates. The in-plane lattice spacing ˚ , which of MoTe2, as determined from the ratio of RHEED streak spacing, is B3.5 A ˚ is very close to the value of 3.52 A [91] that corresponds to bulk 2H-MoTe2. For samples grown on GaAs (111)B substrates, a pair of faint RHEED streaks with a spacing less than 2H-MoTe2 were observed, which could indicate the presence of ordered defects, such as Te interstitials. Fig. 8.4(b) shows RHEED intensity oscillations of the RHEED spectral point during the growth of 2H-MoTe2, indicating nearly layer-bylayer growth, the period between crests corresponding to approximately one monolayer. The likely cause for the decay of RHEED oscillation intensity in figure probably arises from increasing roughness or waviness of the film surface. The HR-TEM image (Fig. 8.4(c)) also illustrates the high quality of the MoTe2 film, and a c-axis lattice ˚ , closely consistent with 2H-MoTe2. spacing of 13.9 A In order to get a better estimate of the in-plane lattice constant of MBE grown 2H-MoTe2 on CaF2 as compared to the estimate obtained from RHEED streak

Layered two-dimensional selenides and tellurides grown by molecular beam epitaxy

245

Figure 8.4 (A) RHEED pattern from post growth/anneal of 2H-MoTe2 on GaAs (111)B substrate. The dashed lines are a guide for the eye, yellow for MoTe2 and red for the additional set of lines observed. (B) RHEED oscillations of the spectral point during MBE growth of MoTe2, showing that the growth is nearly layer-by-layer with a period of B218 s per monolayer, which we use for determining layer thickness. (C) Cross-sectional TEM, showing an abrupt interface between GaAs and MoTe2, and pure tellurium crystallite with a significantly different lattice constant and contrast. The pair of white lines and the pair of red lines are a guide for marking lattice constant of Te and 2H-MoTe2. Adapted from S. Vishwanath, A. Sundar, X. Liu, A. Azcatl, E. Lochocki, A.R. Woll, et al., MBE growth of few-layer 2H-MoTe2 on 3D substrates, J. Cryst. Growth 482 (2018) 61
69.

spacing, and to understand the preference of in-plane rotational orientation, grazing incidence X-ray diffraction (GI-XRD) measurements were carried out. The GIXRD from our sample (see Fig. 8.5A) shows an extended line corresponding to the overlap of (10-10) and (10
11) planes of MoTe2, while the sharp high intensity (red) peak is from the CaF2 substrate. The in-plane lattice constant of 2H-MoTe2 ˚ . The in-plane grain size of calculated from the (10-10) peak corresponds to 3.638 A ˚ is estimated from the full width at half maximum (FWHM) of the 2HB92 A MoTe2 (10-10) peak [92]. From Fig. 8.5B, showing an in-plane phi (φ) scan, we see that MBE-grown MoTe2 undergoes significant twinning, resulting in 2 sets of 6-fold symmetry diffraction patterns. The excess Te ( . 2 Te:Mo film stoichiometry) and the presence of stacking faults, which results in a mixture of 2Hb and 3R phases, could be the contributing factor for the observed larger ‘a’ and ‘c’ lattice constants in MBE-grown 2H-MoTe2 compared to bulk 2H-MoTe2, that may result from excess Te incorporation into few-layer 2H-MoTe2 during MBE growth with elemental Mo and uncracked Te sources.

8.2.2.3 Molybdenum selenide MoSe2 are grown in a similar manner as MoTe2 on HOPG [34]. The cross-sectional TEM image of a 9 ML MoSe2 specimen is shown in Fig. 8.6A. A sharp interface between the MoSe2 film and the HOPG substrate is observed. The interlayer spacing is calculated to be B0.65 nm, very close to the reported value of 0.647 nm for bulk MoSe2 [93]. Fast Fourier transform (FFT) confirms that the MoSe2 crystal structure is indeed 2H, and that the crystal plane perpendicular to the view direction

246

Chalcogenide

Figure 8.5 (A) In-plane ω-2θ scan showing a peak corresponding to CaF2 and to (10-10) of MoTe2. The extended streak in the perpendicular direction is due to overlap of the extended columns from (10-10) and (10
11) in reciprocal space due to the B5 monolayer thin film. ˚ 21 corresponding to Below it is the integrated intensity in the q\ range of 0.02
0.15 A (10-10) peak. (B) The in-plane phi (φ) scan of the (10-10) peak of MoTe2 carried out to understand the rotational alignment and the extent of twinning in the MBE-grown film. ˚ 21. The figure below shows integrated intensity in the q\ range 0.02
0.15 A Adapted from S. Vishwanath, A. Sundar, X. Liu, A. Azcatl, E. Lochocki, A.R. Woll, et al., MBE growth of few-layer 2H-MoTe2 on 3D substrates, J. Cryst. Growth 482 (2018) 61
69.

is close to (1120). In-plane TEM was performed by exfoliating MoSe2 grown on HOPG onto a TEM grid (Fig. 8.6B); small triangular domains of B5 nm stitched together are observed, resulting locally in a nearly single crystal diffraction pattern (electron beam diameter is in this case B150 nm). Formation of triangular grains during CVD growth of layered materials consisting of two different elements, such as h-BN, MoS2, MoSe2 etc [94
96], has been previously observed. What is surprising here is the high degree of local rotational alignment because of VdW epitaxy taking place in MBE growth. Although such triangular features have recently been reported by many groups, and have been studied by scanning tunneling microscopy

Layered two-dimensional selenides and tellurides grown by molecular beam epitaxy

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Figure 8.6 9 ML MoSe2 grown on HOPG: (A) cross-sectional TEM and (B) in-plane MoSe2 TEM images. Adapted from S. Vishwanath, X. Liu, S. Rouvimov, P.C. Mende, A. Azcatl, S. McDonnell, et al., Comprehensive structural and optical characterization of MBE grown MoSe2 on graphite, CaF2 and graphene, 2d Mater. 2 (2015).

(STM) [63,97], the fact that these triangles are inherent in the as-grown material, and not are Moire´ patterns arising from interactions with underlying layers or substrates, is evident from diffraction patterns corresponding to HR-TEM of MoSe2 alone. The similarities and differences in MBE growth of MoSe2 on HOPG and epitaxial graphene (VdW substrates) and on CaF2 (quasi-VdW substrate due to inert surface-fluorine termination) were also investigated [34]. The growth of films occurs by VdW epitaxy in both cases, and results in close to stoichiometric 2H oriented films. But grains in the two cases are very different. Whether the underlying cause of this discrepancy is the quality of the substrate or something more fundamental is, however, yet unclear and needs further investigation.

8.2.2.4 Layered heterostructures and superlattices We will now comment on the advantages of MBE as a growth method for layer materials and their heterostructures and superlattices (SLs) involving MoSe2, MoTe2, Bi2Se3, Bi2Te3, MoSe2, SnSe, and SnSe2 [36]. As an example, a TEM image of Bi2Se3/MoSe2 SL samples is presented in Fig. 8.7A. The growth temperature is selected as 340  C to avoid any dissociation of Bi2Se3. To explore the effects of substrate temperature on the Bi2Se3/MoSe2 heterostructure, growths of MoSe2 at 340  C and 400  C were carried out on Bi2Se3, which was grown on sapphire at 340  C. Atomic force microscopy (AFM) imaging of the as grown Bi2Se3 at 340  C shows triangular domains consistent with other reports [98]. Significant differences

248

Chalcogenide

Figure 8.7 (A) HAADF-STEM image of the Bi2Se3/MoSe2 SL. (B) HAADF-STEM image of the Bi2Se3/MoSe2/SnSe2 heterostructure. Adapted from S. Vishwanath, X.Y. Liu, S. Rouvimov, L. Basile, N. Lu, A. Azcatl, et al., Controllable growth of layered selenide and telluride heterostructures and superlattices using molecular beam epitaxy, J. Mater. Res. 31 (2016) 900
910.

in surface morphology are, however, observed in MoSe2 grown at 400  C. MoSe2 grown at 340  C follows the contours of the underlying Bi2Se3, but when the growth is done at 400  C, we observe features similar to those seen in thick MoSe2 grown on HOPG [34], i.e., tall protrusions enclosing large smooth regions. TEM imaging of the two samples reveals that MoSe2 grown at 400  C has a greater degree of waviness, but the interface between Bi2Se3 and MoSe2 is sharp in both cases. Energy dispersive x-ray spectrum (EDX) line scan is consistent with the expected 40% Bi in Bi2Se3 and 33.3% Mo expected in MoSe2.

Layered two-dimensional selenides and tellurides grown by molecular beam epitaxy

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As another example, we have grown a heterostructure involving three materials: Bi2Se3, MoSe2, and SnSe2. The cross-sectional TEM [Fig. 8.7B] image reveals that majority of the deposited Sn
Se was in the form of SnSe2, but there were also some regions of SnSe. We speculate that even though the growth temperature was lowered from 340  C to 240  C for the growth of SnSe2, that lower temperature of was still higher than optimal for SnSe2 growth. These results clearly illustrate the importance of carefully controlling growth temperature during the growth of 2D layered heterostructures.

8.2.3 Cross between 2D and 3D structures As mentioned earlier, HOPG, CaF2 and epitaxial graphene on SiC are the three representative substrates frequently used for growing 2D materials. HOPG is a polycrystalline non-polar layered crystal with no out-of-plane dangling bonds. CaF2 is a polar 3D crystal with an inert surface termination. Electronic grade graphene (a 2D material) prepared on a single crystal substrate (SiC) is a natural Van-der-Waals (VdW) substrate [34]. All these substrates are not of the standard epi-ready character, and therefore their crystallinities are not as good as those of sapphire, Si, and GaAs substrates. Thus, one is often motivated to use epi-ready substrates to further improve the crystalline quality of the 2D materials. In that case, however, careful preparation of surface of 3D substrates is critical for the growth of high quality 2D materials. For example, it is found that correct Se/Te passivation of GaAs (111)B surface (i.e., exposing it to Se/Te flux prior to the growth, or overpressure of Se/Te during the initial stages of growth) leads to replacement of surface As with Se/Te. Such replacement then removes dangling bonds, forming a surface termination of Ga-Se or Ga-Te, a pseudo-VdW surface, and thus providing an excellent platform for growth of 2D materials. It is interesting that the Van der Waals growth mode promotes epitaxy even when there is considerable lattice mismatch, or even a symmetry changes between the film and substrate [90,99]. For example, (001) GaAs substrates have served successfully as templates for the growth of Bi2Se3 and Bi2Te3 [98,100,101]. In an earlier report [102], by combining atomic resolution imaging and spectroscopy, Dycus et al. determined that a bilayer of Ga
Te exists at the interface between Bi2Te3 and GaAs (100), terminating GaAs dangling bonds. In Fig. 8.8 we show a more interesting electron microscopy from a Bi2Te3 and GaAs (100). Clearly, in addition to the formation of a Ga-Te bilayer of Ga2Te3 to provide the template for Van der Waals epitaxy, we observe Te intercalates at the interface, as shown in Fig. 8.8. Our results indicate that the interfacial chemistry and structure between the 6-foldsymmetry Bi2Te3 film and the 4-fold-symmetry GaAs substrates, while quite complicated, are critical to integration of the two district symmetries. This interfacial phase clearly requires further exploration, including computational studies, but its understanding is crucial for optimization of the quality of crystals grown with such mixed symmetry combination. Growth of ternary and more complex alloys in 2D-layered form is also important for device applications of interest for fields of spintronics, thermoelectrics,

250

Chalcogenide

Figure 8.8 Aberration-corrected electron micrographs showing Bi2Te3 layer grown on GaAs (001) substrate: (A) high-angle annular-dark-field image
atomic columns with bright contrast; (B) bright-field image
atomic columns with dark contrast. (C) The in-plane lattices of Bi2Te3 c-plane and GaAs (100) surface, blue dots showing the intercalation sites for Te atoms.

interconnects, high-speed transistors, as well as quantum computing [103]. Here we use the ternary (Sn,Mn)Se alloy [104] to illustrate how 2D material “accepts” significant concentrations of “foreign” ions in the form of a uniform ternary alloy, without forming precipitates. An important feature of the ternary (Sn,Mn)Se alloy is that the “parent” 2D CdI2-type structure of SnSe2 progressively and seamlessly transforms to the 3D NaCl structure of MnSe as the Mn concentration increases. This transformation is made possible by two factors. First, the hexagonal positions of Se on the basal plane of SnSe2 are nearly identical to those of Se or Mn atoms of the rock-salt MnSe in their respective (111) planes. And second, the 2D planes of

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SnSe2 are bound by weak VdW inter-planar bonding, which allows intercalation of Mn ions between the planes as their concentration increases, thus also allowing the small degree of inter-planar adjustment needed to meet the requirements of the rock-salt MnSe. The process is schematically summarized in Fig. 8.9 as follows. Fig. 8.9A represents pure SnSe2. Then, at low concentrations, Mn ions simply substitute for Sn (Fig. 8.9B). As Mn flux during MBE growth is increased, some Mn ions also enter spaces between the 2D layers, as indicated in Fig. 8.9C. It appears that such intercalation process in the VdW spaces is quite stable, and continues to take place even for relatively high Mn content (as high as 22 at.%). As the Mn concentration is further increased, however, the system as a whole gradually acquires

Figure 8.9 Suggested structural evolution of the (Sn,Mn)Se system from 2D to 3D with increasing Mn concentration: (A) SnSe2, (B
E) (Sn,Mn)Se with increasing Mn at.%, ending (F) as rock-salt MnSe. (G
I) Corresponding cross-sectional HR-TEM images of the (Sn,Mn) Se film with 0, 28, and 100 at.% of Mn, respectively, viewed along the [112] zone-axis projection of the GaAs (111) substrate. Adapted from V. Kanzyuba, S.N. Dong, X.Y. Liu, X. Li, S. Rouvimov, H. Okuno, et al., Structural evolution of dilute magnetic (Sn,Mn)Se films grown by molecular beam epitaxy, J. Appl. Phys. 121 (2017) 075301.

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Chalcogenide

the rock-salt structure (see Fig. 8.9D and E), leading to a pure MnSe rock-salt structure shown in Fig. 8.9F. Fig. 8.9G
I show the corresponding cross-sectional HRTEM images of the (Sn,Mn)Se film (0, 28, and 100 at.% of Mn, respectively), providing clear evidence for the structural transformation of 2D SnSe2-like structure to the 3D rock-salt structure as the Mn concentration increases.

8.2.4 Challenges MBE is a powerful technique for growing a wide range of materials systems with electronic grade quality. As described in this chapter, we have demonstrated successful growth of 2D materials on both 2D and 3D substrates. These efforts are aimed to optimize growth conditions for achieving high quality 2D VdW films, precise doping, formation of heterostructures with abrupt interfaces, as well as integration with other groups of materials for device applications such as Thin-TFETs. Here the precise growth control provided by MBE becomes crucial due to the complicated growth conditions required for 2D films of a few atoms thick. Relative to other more common methods, such as physical and chemical exfoliation and chemical vapor deposition, MBE provides a unique combination of (i) wafer-size scalability, (ii) atomic control over crystallinity and thickness, and (iii) UHV compatibility. That last advantage has become essential for realizing true intrinsic properties of a wide range of 2D materials. As shown earlier, MBE growth of 2D layered chalcogenides involves coevaporation of high-purity metal and chalcogen (Se or Te) from e-beam evaporators and standard Knudsen cells. In most cases an elevated substrate temperature is used in order to promote efficient crystallization of the deposited film. Importantly, the stoichiometry of the 2D layered chalcogenides is believed to be self-regulating, in that the extra chalcogen atoms cannot be incorporated into the films after their material sites are occupied when the substrate temperature is higher than the sublimation temperature of the chalcogen. However, In order to achieve electronic grade 2D materials, the following issues must still be considered and overcome. 1. Although monolayer-by-monolayer growth mode appears to occur, as indicated by the observation of RHEED oscillation in most growths, the step-flow growth mode is not yet confirmed in MBE growth of most 2D layered materials due to the lower transition metal adatom mobility. 2. Precise control of monolayer deposition is still a great challenge for realizing interlayer thin film transistors. This is because growth of 2D chalcogenides to date is not selflimiting, and hence needs to be timed using RHEED intensity oscillations, which does not give perfect monolayer control. 3. The small grain size and twinning are observed in most MBE-grown 2D layer materials, which could present a problem for achieving higher electron mobility. 4. In addition, precise control of stoichiometry of 2D layer materials in monolayer form is still an issue for some compounds, making it difficult to optimize crystalline quality and to control doping. 5. The substrate/interface-induced effects, which naturally expected to be present, are still poorly understood.

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Finally, compared to their bulk counterparts, 2D materials are generally characterized by their lower density of states, which leads to two major consequences: (i) reduced screening effects of the deposited 2D materials, and (ii) significant interface charge redistribution. In the case of heterostructures, additional factors, such as polarity and polarizability between different materials, have to be taken into account as well. Therefore, understanding the role of each interaction and their interplay is clearly important, and will be especially critical for tailoring the monolayer growth.

8.3

Physical characterization of 2D materials grown by MBE

8.3.1 Electronic structure of 2D materials 8.3.1.1 Scanning tunneling microscopy To realize the potential applications of layered chalcogenides, such as optoelectronic or logic devices, electronic structures of layered chalcogenides are intensely investigated using ultrahigh-vacuum (UHV) scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS). Here we use such studies of MBEgrown WSe2 and SnSe2 as examples. Among the layered chalcogenides, WSe2 is interesting because of its electronic properties such as a large spin 2 orbit coupling of .400 meV [105], valley coherence [106], and small direct-indirect gap crossover energy [107]. In this work [9], the electronic structures of MBE-grown WSe2 are characterized on atomic scale by STM and STS at 100 K. After Se capping layer is sublimated by annealing the sample at 773 K in the UHV STM, STM imaging is carried out at different conditions, showing a hexagonal structure with little noticeable point defects or dislocations. In Fig. 8.10a, STM image shows the moire´ pattern of ML WSe2 due to electron orbital overlap with underlying HOPG [86,108]. The atomic resolution STM image in Fig. 8.10b reveals hexagonal arrays of the top Se layer of the three atomic planes in WSe2. The lattice parameter of WSe2 was determined from STM line traces to be 0.32 6 0.01 nm, in good agreement with crystal structure data. The electronic band gap (Eg) was then determined for ML and bilayer (BL) WSe2 using STS [31,35,109
111]. Fig. 8.10c shows the averaged (dI/dV)/(I/V) versus V data on the basal plane of WSe2 [112], far away from step edges of the second WSe2 layer. A standard fitting method [112] was employed to extract the band gap, shown as dashed lines in figure. The quasi-particle band gap for ML WSe2 is determined to be 2.18 6 0.03 eV, close to the reported theoretical value [110]. Employing the same method, the band gap of bilayer WSe2 is determined to be 1.56 6 0.02 eV [107,111,113]. Note that in all measurements of ML and BL the spectra reveal conductance extending from band edges into the band gap, suggesting some sort of “band tail” states.

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Figure 8.10 UHV atomic resolution STM image and STS of decapped MBE WSe2 on HOPG. (A) STM image of hexagonal moire´ pattern and corresponding Fourier transform (Vsample: 2 V, IT: 60 pA). (B) Atomically resolved STM image showing hexagonal atomic arrays of Se atoms in WSe2 and corresponding Fourier transform (Vsample: 1.5 V, IT: 80 pA). As the sample bias is changed, the moire´ pattern disappears in (B). (C) (dI/dV)/(I/V) of ML (black) and bilayer (red) WSe2, showing electronic band gaps of 2.18 6 0.03 eV for ML WSe2 and 1.56 6 0.02 eV for BL WSe2. Adapted from J.H. Park, S. Vishwanath, X. Liu, H. Zhou, S.M. Eichfeld, S.K. FullertonShirey, et al., Scanning tunneling microscopy and spectroscopy of air exposure effects on molecular beam epitaxy grown WSe2 monolayers and bilayers, Acs Nano 10 (2016) 4258
4267.

During device fabrication layered chalcogenide materials are typically exposed to ambient air; therefore, it is critical to understand the effect of air exposure on the structural and electronic properties of layered chalcogenides. However, studies on the air stability of these materials, and the effect of air on their surface properties, such as morphology or band structure, has been initiated at the atomic scale. Here we use SnSe2 as an example to illustrate the effect of air exposure. SnSe2 has been selected from among other 2D materials because of its large electron affinity [79] to be able to form a broken band gap alignment with WSe2, which could make it a candidate for high efficiency two-dimensional heterojunction interlayer tunneling field effect transistors. SnSe2 grown on HOPG via MBE was investigated with STM and STS as an example of air-exposed metal dichalcogenides (MDs) with weak internal bonding [114]. After thermally removing the Se capping layer in the UHV chamber at 523 K for 15 min, an atomically flat SnSe2 layer can be observed, as shown in the STM image of Fig. 8.11A for ML, bilayer (BL), and trilayer growth. In Fig. 8.11B, a line trace following the white dashed line in Fig. 8.11A shows B0.7 nm height for each layer, consistent with three atomic layers (Se 2 Sn 2 Se) shown at the bottom of Fig. 8.11A. In an atomically resolved STM image of Fig. 8.11C, a nearly hexagonal structure is observed with about 0.34 nm spacing. Fig. 8.11D shows the band gap of the ML SnSe2 to be about 1.51 eV, while BL SnSe2 has a band gap of about 1.19 eV. Note that both ML and BL have the Fermi level pinned near the CB [115
117].

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After exposure in ambient air for 1 day, STM imaging shows that the entire SnSe2 layer is decomposed. As shown in the STM image in Fig. 8.11E, the twodimensionally layered SnSe2 disappears with air exposure and only particles are observed across the surface, consistent with the decomposition of SnSe2. As a result of this decomposition, broad band-edge states appear in the STS band gap, as shown in Fig. 8.11F: the band edge states extend to the Fermi level (0 V), consistent with a near-zero band gap. Briefly, then, SnSe2 is extremely unstable in ambient air: after exposure to air for an equivalent duration, all SnSe2 layers were decomposed and oxidized to SnOx and SeOx. On the other hand, similar studies on WSe2 and MoS2 show a selective reactivity at step edges to ambient air with partial oxidation (WSe2) and adsorption of hydrocarbons (MoS2), while the terraces on both layered materials are nearly inert [9,114]. These comprehensive surface studies across the 2D layered chalcogenides can provide fundamental understanding of the effects of air exposure on these materials.

8.3.1.2 Angle-resolved photoemission spectroscopy Electronic band structure features, such as dimensionality of the band structure and locations of their conduction band minima (CBM) and valence band maxima (VBM), provide important insights into the understanding 2D layered chalcogenide materials. As an example, we will use MBE-grown SnSe2 to present its electronic structure obtained by using angle-resolved photoemission spectroscopy (ARPES) [118]. Specifically, we will show the agreement between the ARPES results (Fig. 8.12A) and density function theory (DFT) calculations (Fig. 8.12B), highlighting the importance of spin-orbit coupling (SOC) for SnSe2. Fig. 8.12A shows the image plot of energy distribution curves (EDCs) and momentum distribution curves (MDCs) measured along the K ! Γ ! M path of the surface Brillouin zone (SBZ). To better understand the ARPES data, ab initio band structure calculations are performed using the WIEN2k DFT software package [118,119] and shown in Fig. 8.12B, with spin-orbit coupling (SOC) included. A detailed description of these results can be found in Ref. [118]. Clearly, the SnSe2 electronic structure is well described by the DFT calculations; and we conclude that the CBM (VBM) lies along the Γ 2 K (M 2 L) direction in momentum space. Overall, these results demonstrate the validity of studying the SnSe2 band structure with DFT, and resolve several inconsistencies found in the literature.

8.3.2 Phonon properties of 2D materials 8.3.2.1 Molybdenum selenide Raman spectroscopy is the most common method for studying atomic dynamics (i.e., phonon modes) of crystal films. For bulk 2H
MoSe2, theoretical analysis predicts three Raman-active in-plane modes, E1g, E12g and E22g, one active out-ofplane mode A1g, and two inactive B1u and B2g modes [120]. The phonon properties

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Figure 8.11 (A) Large area STM image of clean SnSe2 on HOPG (VS 5 2 V, IT 5 20 pA). Cross-sectional atomic structure of SnSe2 is shown below. (B) Line trace following the white dashed line in (A). (C) Atomically resolved STM image of clean ML SnSe2 (VS 5 1.2 V, (Continued)

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of the MBE-grown MoSe2 of varying thicknesses along with bulk MoSe2 are shown in Fig. 8.13 [34]. As shown in the figure, strong E12g and A1g Raman peaks and weak E1g and B2g peaks are observed for all MBE-grown films, but no B2g for bulk MoSe2, in agreement with the literature [94,121]. The E1g peak might arise due to a slight deviation from the laser beam normally-incident on the basal plane, a two photon process [122], or an appreciable crystallographic disorder. The inactive mode B2g has been reported to become Raman-active in few-layer 2H
MoSe2 due to the breakdown of translation symmetry [123], which is perhaps the reason for observing it in the MBE-grown materials. The shift and broadening of both A1g and E12g peaks could be attributed to the small grain size of MBE-grown MoSe2 films, causing localization of phonons [124]. A detailed analysis of the experimental results of this type can be found in Ref. [34].

8.3.2.2 Molybdenum telluride Raman spectra of a series of MoTe2 films grown on CaF2 are shown in Fig. 8.14 (a), which confirm an evolution from the 2H phase to a new phase as we progress from samples grown with high flux ratio of Te:Mo to samples obtained with low Te:Mo flux ratio [55]. The 2H phase is confirmed by comparing Raman from sample A (grown with flux ratio Te:Mo 5 297) with Raman from CVT grown bulk 2HMoTe2, although the FWHM of the peaks from MBE-grown MoTe2 is several times wider than that of the CVT-grown MoTe2, pointing to a significant disorder in the MBE-grown material. The new phase found in sample C (grown with Te:Mo 5 71) is labeled as unidentified phase MoTex marked as “UP” in the figure, because its peak positions are not consistent with the reported values of 2H-MoTe2 or 1T’ MoTe2 [125,126]. Sample B shows a mixed phase, containing Raman signatures from both the 2H and the unidentified phases.

8.3.3 Other optical properties of 2d materials 8.3.3.1 Photoluminescence and absorption spectroscopy

L

Room temperature (RT) photoluminescence (PL) and absorption spectroscopy from ML MoSe2 grown by MBE on CaF2 and epitaxial graphene on SiC are shown in Fig. 8.15(a) [34]. A PL peak at B1.563 eV is observed on the sample grown on graphene, and B1.565 eV is measured on the sample grown on CaF2, which is close to the reported value of B1.57 eV obtained at RT for exfoliated ML MoSe2 on SiO2 [127] and 1.55 eV at RT for MBE-grown MoSe2 on bilayer epitaxial IT 5 140 pA). (D) STS of clean ML and BL SnSe2 showing 1.51 and 1.19 eV band gaps, respectively. (E) SnSe2 layer grown on HOPG air-exposed for 1 day. (F) STS of air-exposed SnSe2 surface. Adapted from J.H. Park, S. Vishwanath, S. Wolf, K. Zhang, I. Kwak, M. Edmonds, et al., Selective chemical response of transition metal dichalcogenides and metal dichalcogenides in ambient conditions, Acs Appl. Mater. Interfaces 9 (2017) 29255
29264.

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Figure 8.12 (A) ARPES image plot measured along K ! Γ ! M. Filled circles indicate band positions extracted by Lorentzian fit to EDCs and MDCs, where the errors due to fitting are comparable to the symbol size. All experimental data measured at room temperature from a 69 ML SnSe2 film using He-I light. (B) Comparison between ARPES (open circles; from (A)) and DFT (continuous lines) for different kz, calculated with SOC. The DFT band structure curves are shifted downwards in energy by 1.35 eV to match the ARPES data. Adapted from E.B. Lochocki, S. Vishwanath, X.Y. Liu, M. Dobrowolska, J. Furdyna, H.L.G. Xing, et al., Electronic structure of SnSe2 films grown by molecular beam epitaxy, Appl. Phys. Lett. 114 (2019) 091602.

graphene [31]. This suggests that MoSe2 does not chemically interact with the underlying substrate. Since the 1 ML MoSe2 grown by MBE is not suitable for large-area absorption spectroscopy measurements, the absorption coefficient (alpha) was measured on a 9 ML MoSe2 on CaF2 and plotted in Fig. 8.15(b). On the semilog scale, a sharp band-edge with a 1000 3 increase in alpha over B60 meV increase in the photon energy is observed, corresponding to a slope of about 20 meV/decade. We note here that a sharp density of states distribution near the band edge is critical for achieving sub-60 mV/decade steep slope transistor applications [19].

8.3.3.2 X-ray photoemission spectroscopy X-ray photoemission spectroscopy (XPS) spectra corresponding to Mo, Te, O, Ca, F and C are detected from three MoTe2 samples MBE-grown on CaF2 with different Te:Mo flux ratio [55]. The results in Fig. 8.16 indicate that Te:Mo stoichiometry in MBE-grown MoTe2 is much greater than 2, a sign of excess Te incorporation into MoTe2 films during MBE growth. Careful analysis of peaks in XPS spectra

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Figure 8.13 (A) Raman spectra of 9 ML MoSe2 grown on CaF2 and HOPG compared to bulk MoSe2. (B) Table of Raman peak positions obtained by Lorentzian fitting. (C) Evolution of Raman spectra of MoSe2 grown on epitaxial graphene/SiC, and (D) Raman shift in 2D peak of epitaxial graphene after deposition of MoSe2. Adapted from S. Vishwanath, X. Liu, S. Rouvimov, P.C. Mende, A. Azcatl, S. McDonnell, et al., Comprehensive structural and optical characterization of MBE grown MoSe2 on graphite, CaF2 and graphene, 2d Mater. 2 (2015).

shown in Fig. 8.16 indicates that 2H-MoTe2 phase is observed for sample (A) grown with a high flux ratio of Te:Mo (297:1); an unidentified phase MoTex is dominated for sample (C) grown with high flux ratio of Te:Mo (71:1); and a mixture of these two phases occurring for sample (B) grown with a flux ratio of Te:Mo (98:1). The Te:Mo ratios are B2.57, 2.54 (2.66) and 2.61 (2.72) after correction for attenuation from the oxide overlayer for the 2H-MoTe2 (MoTex) phase in samples A, B, and C, respectively. Note that unidentified phase MoTex cannot be identified as 1T’-MoTe2 phase, since the peaks assigned to unidentified phase MoTex are not fully consistent with the reported peak positions of 1T’-MoTe2 [125,128,129]. Note that the chemical bonding state of MoTe2 is homogeneous through the analyzed depth, as the Mo-Te peak width remains constant with changing angle. We also note that the extent of oxidation in the telluride system is much greater than previously reported for MBE-grown MoSe2 [34]. In particular, in sample C the Mo 3d peak intensity associated with Mo oxide is much higher than that for MoTe2, as

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Figure 8.14 Raman measurements on samples grown on CaF2. Samples A, B, and C are grown by MBE with Te:Mo flux ratios 297, 98, and 71, respectively. Adapted from S. Vishwanath, A. Sundar, X. Liu, A. Azcatl, E. Lochocki, A.R. Woll, et al., MBE growth of few-layer 2H-MoTe2 on 3D substrates, J. Cryst. Growth 482 (2018) 61
69.

Figure 8.15 (A) RT PL from B1 monolayer MBE-grown MoSe2 on epitaxial graphene and CaF2. (B) Semi-log plot of absorption coefficient measured on 9 ML MoSe2 grown on CaF2 (inset: linear plot of the same data). Adapted from S. Vishwanath, X. Liu, S. Rouvimov, P.C. Mende, A. Azcatl, S. McDonnell, et al., Comprehensive structural and optical characterization of MBE grown MoSe2 on graphite, CaF2 and graphene, 2d Mater. 2 (2015).

well as the oxide intensity observed on other samples. This suggests that, in spite of employing a large overpressure of uncracked Te (dimers) during growth, not only does Mo form predominantly the MoTex phase, but that the majority of Mo has a strong tendency for oxidation. The presence of the resulting molybdenum oxide in sample C then exhibits two different Mo oxidation states of 15 and 16.

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Figure 8.16 XPS spectra measured on molybdenum telluride samples, showing various phases and different extent of oxidation for samples grown with different Te:Mo flux ratios. The pink line corresponds to Mo16 oxide, maroon line to 2H-MoTe2, blue line to unidentified-phase MoTex, red line to TeO2, and green line to Mo15 oxide. Samples A, B and C were grown by MBE with Te:Mo flux ratios 297, 98, and 71, respectively. Adapted from S. Vishwanath, A. Sundar, X. Liu, A. Azcatl, E. Lochocki, A.R. Woll, et al., MBE growth of few-layer 2H-MoTe2 on 3D substrates, J. Cryst. Growth 482 (2018) 61
69.

8.4

Concluding remarks

2D layered chalcogenides recently saw rapid advances in understanding, quality, and fabrication; they now begin to meet the stringent requirements of a wide range of new device applications. As we have pointed out in previous sections, in this context MBE offers a unique advantage by enabling the combination of wafer-size scalability, high quality crystallinity and atomically-precise thickness control. These features are expected to make high-quality 2D layered chalcogenides and their heterostructures to be accessible in practice, thus opening important new opportunities both for fundamental research and for applications. As described in this chapter, the electronic, structural, optical and chemical properties of MBE-grown films such as MoSe2, WSe2, MoTe2, SnSe2 have been investigated using STM, STS, HRXRD, TEM, PL, XPS, Raman, ARPES etc. Interestingly, some of these films show entirely new characteristics that could not be foreseen from their bulk counterparts. It should be pointed out that it is difficult to establish fully complete properties of these atomically-thick films before we fully understand the role of interlayer interactions and their interplay with both ambient air and with underlying substrates on which the films are grown. In this regard, further and comprehensive research on these MBE-grown 2D materials is necessary for achieving the current vision of “Lego”-type atomically-thin electronic devices. Here, one major challenge which must still be overcome is to achieve controlled monolayer growth, which is hampered by our limited understanding of mechanisms governing this type of precision in epitaxy. Furthermore, the interfacial interactions

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between 2D chalcogenides and foreign molecules provide another process whereby the properties of these 2D materials are modified. This concept is not new, and has already been well demonstrated in the case of graphene [130,131]. Finally, it should be mentioned that growth of 3D semiconductors with zinc blende and wurtzite structures on 2D chalcogenides to form 3D/2D hybrid heterostructures [132,133], also holds promise of a variety of interesting novel phenomena. Developing such growth methodologies is therefore expected to lead to development of entirely new functionalities, and thus to entirely new applications.

Acknowledgment This work was supported by NSF Grant DMR 1400432 and NSF-EFRI 2DARE Grant DMR 1433490.

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127.

Tailoring exchange interactions in magnetically doped II-VI nanocrystals

9

Rachel Fainblat, Franziska Muckel and Gerd Bacher University of Duisburg-Essen, Duisburg, Germany

9.1

Introduction

Since the preparation of highly monodisperse II-VI nanocrystals with controllable size and outstanding optical properties more than two decades ago [1], chemical synthesis routes have been refined to a high degree of perfection and extended to different materials, geometries and architectures, opening a wide field of applications [2
4]. Among others, one of the most interesting challenges was (and still is) controlled doping of the nanocrystals, i.e. replacing individual atoms in the crystal matrix by impurities. This endows the host material with a certain functionality, such as a magnetic, an optical or an electrical one [5
7]. The proof of a giant magnetooptical response in colloidal ZnSe quantum dots doped with manganese by Norris et al. in 2001 [8] was an important breakthrough in the field. Doping was initially assumed to be challenging for small nanocrystals (, 2 nm for Mn21-doping of ZnSe, see Ref. [9]) due to surface segregation and self-purification processes. However, successful incorporation of impurities into very small clusters with a well-defined number of atoms has been achieved recently via different synthetic approaches [10
16]. Successful incorporation of, e.g., Mn21 on lattice sites of II-VI compounds leads to the so-called sp-d exchange interaction between the dopant spins and the spins of the charge carriers in the conduction band (CB) and the valence band (VB) of the host. This results in giant magneto-optical effects, like huge effective g-factors exceeding 2900 at low temperatures [17] or the formation of magnetic polarons, i.e. a collective alignment of the transition metal ion spins by the exchange field of optically [18
20] or electrically [21] injected charge carriers. Only recently, Fainblat et al. succeeded in proving giant excitonic exchange splittings at zero field in single CdSe quantum dots doped with individual magnetic impurities [22]. Besides Mn21-doped II-VI nanocrystals, analogue II-VI nanostructures doped either with Co21 or with Cu21 are the most popular materials, for which spin-spin interactions between dopants and charge carriers of the host material have been reported [23
26]. Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00014-2 © 2020 Elsevier Ltd. All rights reserved.

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Doping with transition metals is not only attractive for achieving giant magnetooptical responses in semiconductor nanocrystals but in addition opens other fields of potential applications. In so-called “dual emitters”, simultaneous emission of the bandgap and the luminescent dopants has been realized via a careful adjustment of the energies of the lowest excitonic state and the d-d emission of Mn21 [27]. These dual emitters might be attractive for nanoscale temperature sensors [28], because the energy transfer between excitonic states of the host material and Mn21 luminescent states is strongly temperature dependent, thus impacting the overall emission spectrum. The beauty of colloidal synthesis lies in the ability to engineer shape, size and composition of transition metal doped II-VI nanomaterials. This offers the chance to tune both, the electronic structure as well as magnetic exchange interactions in a much wider range than achievable for their epitaxially grown counterparts, which initially dominated the field of magnetically doped semiconductors (also known as dilute magnetic semiconductors, DMS) for decades [29
35]. It is the purpose of this contribution to demonstrate the unprecedented control of magneto-optical functionality in transition metal doped nanocrystals - down to the level of single-atom doped nanocluster - by making use of the flexible protocols in colloidal synthesis.

9.1.1 Theoretical background We start with a brief description of the interactions causing the giant magnetooptical responses of DMSs, namely exchange interactions between the d-type electrons of the transition metal impurities and the charge carriers in the CB and the VB, called s-d and p-d exchange interactions, respectively. At zero external magnetic field, these interactions can lead to excitonic magnetic polaron formation [18
20,29,33,36
39] and giant zero-field excitonic exchange splittings [34,40
44], the most reported zero-field magnetic phenomena, vastly investigated by different groups. Applying an external magnetic field B aligns the magnetic moments of the Mn21 dopants, leading to a splitting of spin-degenerated states, called giant Zeeman effect. Eq. (9.1) describes the total exciton Zeeman splitting ΔEtot observed in DMS [35]. ΔEtot

5 ΔEint 1 ΔEsp2d
  5 gint μB B 1 xeff N0 γ e2Mn ∙a∙α 1 γ h2Mn ∙b∙β Sz

(9.1)

The first term ΔEint represents the temperature independent intrinsic Zeeman splitting, where gint is the exciton’s intrinsic g-factor (e.g., B 1 1.0 for the first excited state of CdSe nanocrystals) [45] and μB is the Bohr magneton. Note that in this book chapter we will refer to excited states as optically generated electron-hole pairs
i.e. excitons - with the first excited state being composed of an electron in the lowest CB and a hole in the highest VB electronic state. The ground state represents the system prior to the photoexcitation, i.e. no electronhole pair is present.

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273

ΔEsp2d is related to the sp-d exchange interaction, thus being proportional to the amount of magneto-optically active dopants xeff , and the dopant-electron (-hole) wavefunction overlap γ e2Mn (γ h2Mn ). The exchange constants N0 α and N0 β are material-dependent and describe the s-d and p-d mean-field dopant-carrier exchange coupling, where N0 is the lattice cation density. The weighting factors a and b depend on the exact nature of the CB and the VB electronic states involved in the excited state. For bulk DMSs with zinc blende crystal structure, a 5 1 and b 5 2 1 for the first excited state involving an electron in the CB and a hole in the heavy hole VB subband. For the excited state involving a charge carrier of the light hole (split-off) band in addition to the CB electron a 5 2 1 and b 5 2 1=3 (a 5 2 1 and b 5 1=3) [30]. Sz describes the dopant’s spin expectation value quantized along the magnetic field axis, which is a Brillouin-type function (see Eq. (9.2)) for dopants with spin-only ground states (such as Mn21) [46].        1 gTM μB B gTM μB B Sz 5 2 ð2S 1 1Þcoth ð2S 1 1Þ 2 coth 2 2kT 2kT

(9.2)

Herein gTM represents the dopant’s g-factor, k is the Boltzmann constant, and T 5 Tbath 1 TAF the effective temperature. The term “effective temperature” has been first introduced by Gaj et al. [47], showing that, depending on the dopant concentration, a small correction of the bath temperature Tbath is necessary to account for the antiferromagnetic interaction among the impurities,   which is described by the antiferromagnetic temperature TAF . By convention Sz is a negative number [48] saturating at
5/2 for fully magnetized Mn21. As usual N0 α (N0 β) is positive (negative), ΔEsp2d opposes its relatively weak intrinsic counterpart ΔEint , e.g., in CdSe, in case of the first excited state, i.e. the heavy hole exciton. ΔEsp2d follows the paramagnetic behavior of Mn21, being therefore temperature dependent and can be (depending on the different parameters explained above) orders of magnitude larger than its intrinsic analogue. Thus, the temperature dependent behavior of ΔEtot is quite complex and given by the competition between two components ΔEsp2d and ΔEint [49,50]. At low temperatures and sufficiently large values of xeff , negative total Zeeman splittings ΔEtot can be observed for the first excited state in Mn21-based II-VI DMSs. Details about the spectroscopic quantification of the Zeeman splitting by magneto-optical measurements, including the sign conventions used in this book chapter can be found in Ref. [48]. In the limit of small magnetic fields, ΔEtot is linearly proportional to B. In this case, an effective g-factor geff can be extracted using the following equation: ΔEtot 5 geff μB B. As one decreases at least one dimension of a semiconductor below the excitonic Bohr radius, quantum effects begin to play an essential role, strongly impacting the structure of electronic states in both CB and VB. In II-VI-based two-dimensional (2D) materials, the VB is composed of three subbands with well-defined hole character, i.e. heavy-hole (hh), light-hole (lh) or split-off (so) hole states, similar to bulk analogues. The one-dimensional confinement and the step-like density of states creates a multitude of well-defined excited states, which can be denoted by VBnhene.

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Depending on the subband involved, VB equals to hh, lh or so, while nh and ne are the principal quantum numbers of VB and CB electronic states, respectively. In this model hh1e1 denotes the first excited state in the most general case, i.e. in absence of strong anisotropic effects or strain. Each excited state is four-fold degenerated, and the degeneracy can be lifted e.g. by exchange interactions and/or an external magnetic field. Ground and excited states of 2D nanostructures are depicted schematically in the left panel of Fig. 9.1. Compared to bulk or 2D materials, the structure of both CB and VB becomes more complex in zero-dimensional (0D) nanostructures. Efros and co-workers first developed a theoretical treatment of this issue [51,52], which has since then been used to describe and assign excited states observed in absorption spectra in II-VI nanocrystals [2,53]. Recently, long-range exchange interaction has been taken into account to provide a more precise quantitative description of the excited states in undoped nanocrystals [54]. According to this theory, each CB state can be described using spherical harmonics and Bessel functions, thus resulting in the notation neLe (ne represents the radial quantum number and Le the angular momentum of the envelope wavefunction of the electron). For the total angular momentum of the envelope wavefunction L 5 0 (L 5 1, L 5 2), the electronic state will be S-like (P-like, D-like). Due to the coupling between the angular momenta of atomic and envelope wavefunctions, the description of true wavefunctions for VB states in spherical nanocrystals requires the introduction of a “new” quantum number called total angular momentum (F), which is a sum of Lh (the angular momentum of the hole envelope wavefunction) and J (total unit cell angular momentum) [51]. The reason why the theoretical treatment of VB states

Figure 9.1 Scheme of ground (GS) and excited states in two- (quantum wells
left panel) and zero-dimensional (quantum dots
middle and right panel) nanostructures. In case of the quantum wells, each excited state is four-fold degenerated and most excited states exhibit defined hole character. For spherical confinement (colloidal quantum dots) in contrast, the theoretical description of the excited states requires the consideration of valence band mixing effects. Each excited state might in addition be affected by fine structure splitting due to exchange interaction (see right panel), resulting in a multitude of states with different optical and magneto-optical activities. Notations are introduced in the text.

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(described by nhLF) becomes very complex is related to the fact that these states are affected by mixing between hh, lh and so subbands. This effect is called valence band mixing, whereby the word “mixing” implies that the hole character in colloidal quantum dots is not as clearly defined as in bulk and 2D materials. The excited states are composed of a combination of one electronic state in the VB and one in the CB, thus having the nomenclature nhLFneLe. In this model the first excited state is assigned as 1S3/21Se (see Fig. 9.1, middle panel). In contrast to the four-fold degeneracy of the first excited state in 2D nanostructures, the first excited state in 0D nanocrystals is eight-fold degenerated due to the spherical confinement. Shape and crystal structure anisotropies as well as electronhole exchange interactions lead to the lift of the degeneracy of 1S3/21Se in 0D nanostructures, finally resulting in five sublevels in the absence of an external magnetic field [52]. The five levels (see Fig. 9.1, right panel) are named by their exciton total angular momentum projection N 5 F 1 sz (F and sz are the hole and electron total angular momentum projection, respectively), complemented by a superscript U/L for “upper”/“lower” according to their energy: 6 2, 6 1L, 6 1U, 0L and 0U. While the states 6 2, 6 1L, 6 1U are expected to split in presence of an external magnetic field according to their two-fold degeneracy, 0L and 0U are non-degenerate, being magneto-optically inactive. The exact order of the five sublevels with increasing energy depends on the shape anisotropy, whereby the order depicted in Fig. 9.1 is valid for spherical wurtzite-structured II-VI nanocrystals.

9.1.2 Outline of the chapter The chapter is organized as follows: Section 9.2 focuses on two-dimensional nanostructures, where the valence band states exhibit a well-defined hole character, and the thickness of the individual layers can be defined with monolayer precision. Section 9.3 summarizes recent results on magnetic exchange interactions in quantum dots, where the electronic structure is strongly modified by the spherical carrier confinement and valence band mixing occurs. Dopant concentrations down to the level of a single Mn21 atom per nanocrystal can be reached, leading to a characteristic fingerprint in the optical emission spectra. In Section 9.4 transition metal doped magic-size nanoclusters are discussed
the smallest doped semiconductors so far. Due to the extreme quantum confinement, electron-hole exchange interaction leads to an additional splitting of the energy states. Giant magneto-optical responses induced by single dopants can be traced up to room temperature. The chapter closes with a brief summary and an outlook.

9.2

Two-dimensional (2D) colloidal nanocrystals

9.2.1 Giant magneto-optical response in Mn21-doped CdSe nanoribbons Due to the generation of strongly coupled electron-hole pairs (so-called excitons), the absorption spectrum of two-dimensional nanostructures is known to exhibit

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spectrally well resolved features, whose energies reflect the bandgap of the material, the quantization energy of the involved states, as well as the exciton binding energy. In II-VI-based 2D materials, those features can be assigned to excited states involving VB states with well-defined hole character, i.e. hh, lh or so hole states (see Fig. 9.1, left panel). Hence, the model of a one-dimensional potential well corrected by the respective exciton binding energy - can be used to calculate the energies for resonances observed in absorption measurements. The absorption feature of lowest energy is related to an electron from the heavy hole subband being photoexcited to the lowest CB state, i.e. a heavy-hole exciton (hh1e1) is created. With increasing energy, heavy hole excitons involving higher subbands as well as light-hole exciton (lh1e1) or split-off exciton (so1e1) resonances can be observed, whose exact order depends on several parameters such as layer thickness, presence of strain and crystal structure. Mn21 doped CdSe nanoribbons synthesized at low temperatures (ca. 70  C) via colloidal chemistry from a Lewis acid 2 base reaction of cadmium chloride and manganese-(II) chloride with octylammonium selenocarbamate represent the first realization of transition metal doped two-dimensional chalcogenide nanocrystals [55]. Since the first report on successful incorporation of magnetic impurities into CdSe nanoribbons, the synthesis has been adapted enabling the fabrication of alloyed two-dimensional Mn21: Zn1-xCdxSe structures of different compositions [11]. Self-organization of bundles containing up to 20
30 individual layers arises from strong van der Waals forces among the 1.4 nm thick nanoribbons. These bundles exhibit a total thickness of 40 2 80 nm, a width of 10 2 50 nm, and a length in the micrometer range (see Fig. 9.2A). An external magnetic field results in the lift of degeneracy of the excited states (see Eq. 9.1). The energy difference between these excited states, the Zeeman splitting, can be accessed using different techniques. A particularly versatile tool to investigate the magneto-optical response (and thus the Zeeman splitting) of excited states is magnetic circular dichroism (MCD) spectroscopy, an optical technique, which probes the difference in absorption of right and left circularly polarized light in the presence of an external magnetic field [57,58]. Taking advantage of MCD spectroscopy, Yu et al. investigated the giant magneto-optical response of Mn21 doped CdSe nanoribbons at T 5 1.8 K, reporting on effective g-factors ranging from 230 to 2600 depending on the Mn21 concentration [55]. A great advantage of MCD spectroscopy compared to magnetic field-dependent photoluminescence experiments is the ability to quantify the Zeeman splitting of the first excited state and of excited states of higher energy. Fig. 9.2B schematically depicts the origin of the MCD signal in the specific case of the studied nanoribbons. At zero magnetic field, no spin splitting of the hh1e1 and the lh1e1 states is obtained. This situation changes in the presence of an external magnetic field. Due to the Zeeman splitting, spin-polarized hh1e1 and lh1e1 excitons are photogenerated at different energies via the absorption of right and left circularly polarized photons. As the MCD signal is proportional to the differential absorbance ΔA 5 AL 2 AR , a derivate-shaped MCD signal is expected in the most common case, i.e. nondegenerate ground state and degenerate excited state. According to the sign

Figure 9.2 (A) Schematic sketch of a nanoribbon bundle including the miller indices. (B) Magnetic field induced lift of degeneracy of excited states (hh1e1 and lh1e1) and the different absorption of left (green arrows) and right (red arrows) circularly polarized light. (C) Absorption (top) and magnetic-field dependent MCD signal (bottom) of Cd0.92Mn0.08Se nanoribbons measured at T 5 4.5 K. (D) Temperature dependent giant Zeeman splitting of Cd0.96Mn0.04Se (black) and Cd0.92Mn0.08Se (red) nanoribbons for both, hh1e1 and lh1e1 excited states. Inset: magnetic-field dependent giant Zeeman splitting of the hh1e1 exciton in Cd0.96Mn0.04Se (black) and Cd0.92Mn0.08Se (red) nanoribbons measured at room temperature (A, C, and D). Adapted with permission from R. Fainblat, J. Frohleiks, F. Muckel, J.H. Yu, J. Yang, T. Hyeon, et al., Quantum confinement-controlled exchange coupling in Manganese(II)-doped CdSe two-dimensional quantum well nanoribbons. Nano Lett. 12 (2012) 5311
5317. Copyright 2012 American Chemical Society [56].

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convention of Piepho and Schatz [48], a MCD signal exhibiting a maximum followed by a minimum (with increasing energy) is related to a negative g-factor. Combining absorption and MCD data, the exciton Zeeman splitting ΔEtot can be extracted based on the following relationship [48]: ΔEtot 5

pffiffiffiffiffi 2e ΔA σ 2 A0

(9.3)

Herein 2σ represents the Gaussian linewidth and A0 the amplitude of the absorption feature. The electronic absorption of those nanostructures exhibits spectrally well resolved features assigned to hh1e1, lh1e1 and so1e1 excited states (see Fig. 9.2C, top). A pronounced magneto-optical response of nanoribbons containing an average Mn21 doping concentration of 8% has been observed (see Fig. 9.2C, bottom). The MCD amplitude is expected to increase with applied magnetic field up to the limit, where all magnetic moments of the transition metal dopants are aligned along the applied external field. In the case of Mn21-doped CdSe nanoribbons, the saturation limit has only been observed in experiments conducted at significantly larger magnetic fields and lower temperatures [55]. Fitting the experimental data (red dotted line Fig. 9.2C) allows the conclusion, that the g-factors of hh1e1 and lh1e1 excitons have opposite sign, an unusual signature compared to earlier works on bulk DMS materials [30,59,60]. In CdMnSe N0 α 5 0:23eV and N0 β 5 2 1:27eV [35], therefore for the ideal case of zinc blende structured bulk CdMnSe one would expect the Zeeman splitting of an excited state involving a heavy hole to be proportional to N0 ðα
2 β Þ 5 1:50eV, while its analogue involving a light  hole should be proportional to N0 2α 2 β=3 5 0:19eV (see Eq. 9.1). The fact that the Zeeman splittings of hh1e1 and lh1e1 excited states have opposite sign can be explained by anisotropy effects related to the wurtzite crystal lattice and the extreme quantization in one direction [61,62]. As theoretically expected, the extracted absolute value of ΔEtot for the hh1e1 excited state is larger than for the lh1e1. However, for a quantitative analysis of these values one must consider the orientation of the nanoribbons on the substrate, which finally will impact both optical selection rules and Zeeman splitting of the different electronic states. For the sake of brevity, we shall not further discuss these aspects here, which have been addressed in detail in Ref. [56]. Temperature-dependent experiments enable analyzing the influence of thermal effects on the magneto-optical response in Mn21-doped CdSe nanoribbons. The decrease of ΔEtot with increasing temperature following a temperature-dependent Brillouin function is a clear evidence of an efficient sp-d exchange interaction [56]. No sign flip has been observed for ΔEtot of the hh1e1 excited state over the whole temperature regime from 4.5 to 300 K. Due to the fact that ΔEsp2d strongly decreases as the temperature increases while ΔEint is temperature independent, this observation is a result of a predominant giant Zeeman splitting compared to its intrinsic counterpart up to room temperature. In the low-field limit ΔEtot 5 geff μB B, therefore one can extract an effective g-factor geff for different temperatures via a linear fit of the magnetic field-dependent Zeeman

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splitting. As discussed previously, the largest geff observed for Mn21-doped CdSe nanoribbons so far has been measured at 1.8 K and is in the order of 2600 [55]. Even at room temperature, geff 5 213 for a Mn21 doping concentration of 8% and geff 5 23 for a smaller Mn21 doping concentration of 4% are found for the hh1e1 excited state, significantly larger (and of opposite sign) than the intrinsic g-factor reported for undoped CdSe nanocrystals (gint 5 1:0) [45]. The pronounced magneto-optical response observed all the way to room temperature is remarkable and particularly interesting for potential device applications.

9.2.2 Tuning magnetic exchange interactions by wavefunction engineering in core/shell nanoplatelets While nanoribbons are formed by bundles of layers with strictly predefined thickness due to their template-mediated growth from coalescence of metastable clusters, 2D nanoplatelets offer precise synthetic control over thickness and composition [63
68], including core/shell [69,70] and core/crown [71,72] architectures. This synthetic flexibility in combination with transition metal doping allows tailoring the sp-d exchange interactions of specific states via atomic layer precise variation of the thickness. Magnetically doped nanoplatelets can be obtained by coating an undoped CdSe core (grown from cadmium acetate dehydrate dissolved in a mixture of oleic acid and octadecene by addition of trioctylphosphine-selenium) with Mn21: CdS using colloidal atomic layer deposition [73]. Here, the anionic and cationic layers of the shell are added subsequently by exchanging the precursor solution between ammonium sulfide in N-methylformamide (NMF) and cadmium acetate combined with manganese(II) acetate in NMF. With this technique, the location of the doped layers within the platelet architecture can be chosen with monolayer precision. Initial studies by Delikanli et al. on CdSe/Mn21: CdS/CdS multishell nanoplatelets (undoped CdSe cores surrounded by a doped Mn21: CdS shell covered with an additional undoped CdS shell) have proven magnetic exchange interactions via photoluminescence (PL) experiments conducted in the presence of an external magnetic field [73]. A Zeeman splitting exhibiting a Brillouin dependence with temperature and magnetic field (see Eq. 9.2) was found
an indication of sp-d exchange interaction in the emissive state. By changing the thickness of the undoped core and the outer shell, the electron and hole wave function overlap with the doped regions could be precisely controlled. This results in a modification of the strength of the magnetic exchange interactions, which has been experimentally addressed via the degree of polarization of the PL signal in a magnetic field. However, the origin of this emissive state
weather related to a band-to-band transition or stemming from localized states - could not be finally clarified [74]. As shown in the previous chapter for Mn21: CdSe nanoribbons, MCD spectroscopy enables direct access to the magneto-optical activity of both, the first excited state as well as higher excited states. This technique has been applied to investigate sp-d exchange interactions between the dopants and band charge carriers for a

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multitude of excited states in CdSe/Mn21: CdS core/shell nanoplatelets. The MCD spectrum of a sample consisting of a 2 monolayer (ML) thick CdSe core surrounded by 8 ML (4 ML on each side) of doped Mn21: CdS (see Fig. 9.3A) exhibits a variety of distinct features, which are virtually absent in the spectrum of the undoped counterparts (compare black and gray MCD signal in Fig. 9.3A). The MCD signal in addition reveals several excited states above 2.5 eV, which cannot be resolved in the absorption data (see top panel in Fig. 9.3A). By calculating the energies of the confined electronic states in the CB and VB using a twostep, infinite potential barrier model, the first three excites states can be assigned to hh1e1, lh1e1 and so1e1 (at 2.16 eV, 2.30 eV and 2.58 eV, respectively) excitons, similar to the case of the nanoribbons discussed in the previous section. According to Eq. (9.1), a negative (positive) sign for the giant Zeeman splitting can be expected for excited states
involving a hh (a so) hole, since ΔEsp2d ~ðN0 α 2 N0 β Þ for hh1e1 and ΔEsp2d ~ 2N0 α 1 13 N0 β for so1e1 - remind the different signs for N0 α and N0 β). For excitonic states including a lh, the sign of the giant Zeeman splitting is expected to depend on the degree of quantum confinement [56,76] and on strain [77], and can either be positive or negative. Thus, the sign of a spectral feature in the MCD signal offers additional information about the nature of an excited state. Taking into account the sign of the Zeeman splitting indicated by the shape of the MCD feature and considering the calculations, the excited state at 2.67 eV, exhibiting a negative splitting, is attributed to the transition from the third hh state in the VB to the lowest state in the CB (hh3e1, see Fig. 9.3A and Fig. 9.1). Thus, MCD measurements not only evidence sp-d exchange interactions between the dopant spins and the carriers in the CB and VB, but also enable new insights into the electronic structure of higher excited states, which usually cannot be resolved via absorption spectroscopy. The synthetic degrees of freedom can be used to purposely tune the sp-d exchange interactions for specific electronic states, simply by adjusting the overlap between the charge carrier’s wavefunction and the localized dopants. As the absolute value of the hole-dopant exchange coupling constant N0 β exceeds the electrondopant one, N0 a, by typically a factor of 4-5, the magneto-optical response of different excited states is expected to be mainly controlled by the overlap of the hole wave function with the doped region in our system. Due to a small CB offset between CdS and CdSe, the electron wavefunctions are largely delocalized over core and shell, regardless of the specific architecture of the nanoplatelets. In contrast, the hole wave function overlap of a specific excited state with the doped region of a specific excited state strongly depends on the involved state of the VB. VB states close to the top of the VB (hh1 and lh1) are mainly confined within the core, while the hole states deeper in the VB (e.g. so1, hh3 and higher) are delocalized towards the shell. This allows to address the exchange interactions of each hole state individually by varying either the core or the shell thickness. Fig. 9.3B depicts how the MCD signal changes in case the shell thickness is decreased from 8 ML to 6 ML (4 ML and 3 ML on each side, respectively). The MCD signals of the excited states involving hh1 and lh1 states (features between 2.0 and 2.4 eV) are neither affected in energy nor in amplitude, as the hole states are

Figure 9.3 (A) Absorption and MCD spectra of core/shell nanoplatelets consisting of a 2 ML CdSe core surrounded by 4 ML Mn21: CdS shells on each side measured at 5 K and 1.6 T. The magneto-optical response of an undoped reference is shown in gray for comparison. Spectral ranges for the hh1e1, lh1e1 and so1e1 excited states are highlighted. (B) MCD spectra at 5 K and 1.6 T for nanoplatelets consisting of a 2 ML CdSe core and a 4 ML Mn21: CdS shell on each side (black) or of a 2 ML CdSe core and a 3 ML Mn21: CdS shell on each side (orange). (C) Calculated energies and probability density functions for the e1 and hh1 states as well as energies for the hh3 state for nanoplatelets consisting of a 2 ML CdSe core and a 4 ML Mn21: CdS shell on each side (black) or of a 2 ML CdSe core and a 3 ML Mn21: CdS shell on each side (orange). (D) effect of shell thickness on the overlap of the probability density with the doped shell for two selected VB states. A core thickness of 2 ML is assumed. (E) MCD spectra at 5 K and 1.6 T for nanoplatelets consisting of a 2 ML CdSe core and a 4 ML Mn21: CdS shell on each side (black) or of a 3 ML CdSe core and a 4 ML Mn21: CdS shell on each side (green). (F) Calculated energies and probability density functions for the e1 and hh1 states as well as energies for the hh3 state for nanoplatelets consisting of a 2 ML CdSe core and a 4 ML Mn21: CdS shell on each side (black) or of a 3 ML CdSe core and a 4 ML Mn21: CdS shell on each side (green). (G) Effect of core thickness on the overlap of the probability density with the doped shell for three selected hole states. A shell thickness of 4 ML is assumed. Adapted with permission from F. Muckel, S. Delikanli, P.L. Herna´ndez-Martı´nez, T. Priesner, S. Lorenz, J. Ackermann, et al., Sp-d exchange interactions in wave function engineered colloidal CdSe/Mn: CdS hetero-nanoplatelets. Nano Lett. 18 (2018) 2047
2053. Copyright 2018 American Chemical Society [75].

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mainly confined within the core. At the same time, the hh3e1 excited state is shifted towards higher energies (from B2.67 to 2.71 eV). This is reproduced by quantum well calculations for the hh1 and hh3 electronic states as shown in Fig. 9.3C: While the energetically lowest hole electronic state, hh1, is not affected at all by the change in shell thickness, the hh3 electronic state is pushed towards higher energies, when the shell thickness is decreased. As the spin-orbit splitting is smaller in CdS than in CdSe (B70 meV compared to B400 meV) [63], the so1 state is less confined within the core and thus more affected by a change in the shell thickness. Fig. 9.3D depicts the calculated wave function overlap with the doped shell for the hh1 and the so1 electronic states versus shell thickness. The data are normalized to the values calculated for a total shell thickness of 8 ML, where the wavefunction overlap with the shell is 0.24 for the hh1 state and 0.54 for the so1 one. With decreasing shell thickness, the overlap decreases for so1, while it remains nearly constant for hh1. This results in a decrease of the MCD intensity for the so1e1 excited state at 2.58 eV upon reduction of the shell thickness (see Fig. 9.3B), i.e. the MCD minimum at 2.55 eV is no longer resolved. Increasing the core thickness, in contrast, is expected to strongly affect the confined hh1 and lh1 states. Fig. 9.3E compares the MCD signals of samples consisting of a 2 ML and a 3 ML CdSe core, surrounded by 8 ML Mn21:CdS. With increasing core thickness, the quantization energy of the confined hole states close to the edge of the valence band decreases, which leads to a redshift in the energy for the hh1e1, lh1e1 and so1e1 excited states (hh1e1 shifts from 2.16 to 2.05 eV, lh1e1 from 2.30 to 2.20 eV and so1e1 from 2.58 to 2.53 eV, see Fig. 9.3E). Simultaneously, the increase in core thickness decreases the wave function overlap of the hh1, lh1 and so1 states with the doped shell, strongly reducing the magneto-optical response of the excited states below 2.6 eV (see Fig. 9.3E). The delocalized VB states as well as the CB states are less affected by the shell thickness, as apparent from the wave function calculations shown in Fig. 9.3F. As a result, the magneto-optical response of excited states is barely affected by the change of core thickness (compare MCD amplitudes in the energy region above 2.6 eV in Fig. 9.3E). This difference between the electronic states close to the top of the VB (hh1) and the states located deeper in the VB (hh3) is reflected in wavefunction calculations (see Fig. 9.3G). Upon increase of the core thickness, the overlap of the hh1 and hh2 states with the doped area diminishes rapidly, while it is barely affected for the hh3 state.

9.3

Zero-dimensional nanocrystals

9.3.1 Valence-band mixing in doped nanocrystal quantum dots If a nanostructure is smaller than twice the size of its exciton Bohr radius in all three dimensions, zero-dimensional systems, so-called quantum dots, are formed. Transition metal doping of spherical nanocrystals with sizes above 2 nm has been studied for about two decades, leading to a profound understanding of the luminescent and magnetic properties of this material class. Beginning in late 1990s, when

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different groups have reported on the first synthetic routes for the fabrication of doped II-VI nanocrystals [78
81], researchers have developed numerous approaches for the impurity incorporation in zero-dimensional nanostructures [5,7,17,82
86]. The nanocrystal architecture can strongly impact luminescent and magnetic properties. As an example, the energy transfer between the host material and Mn21 impurities, which finally results in Mn21 luminescence, depends on doping concentration, dopant location, excitation power, and temperature [87
90]. Recently, the Peng group reported on nearly single-exponential Mn21 PL decay dynamics in core/shell nanocrystals with PL quantum yields of ca. 70%, which have been achieved via the growth of a very uniform 7 ML thick ZnS shell on Mn21: ZnSe nanocrystals [87]. After the first report of Norris et al. on giant magneto-optical responses in Mn21: ZnSe nanocrystals [8], different aspects of sp-d exchange interactions have been investigated, such as the formation of excitonic magnetic polarons [18
20,39] as well as how the overlap between the exciton and the magnetic impurities can be used to manipulate the strength of the exchange interactions [91
93]. As explained in the introduction (see Fig. 9.1), the electronic structure of quantum dots is impacted by an effect called VB mixing. Up to now, the majority of reports exclusively addressed the influence of VB mixing on the excited state of lowest energy [94
98], while quite few studies discuss the influence of VB mixing on higher excited states in zero-dimensional semiconductor nanostructures [99,100]. Gaining a deep understanding of this effect is an important challenge in the area of nanotechnology, since VB mixing has been associated with different physical properties of nanoscaled systems
like spin dephasing [96], spin tunneling [101], polarization degree [102] or even piezo-electric effects [103,104]. As outlined above, one major advantage of MCD spectroscopy is its applicability to investigate magneto-optical responses of excited states of higher energy. This inspired us to take advantage of the enhanced magneto-optical response of exceptionally well resolved upper excited states in Mn21 doped nanocrystals to investigate an intrinsic property of zero-dimensional nanostructures: the valence band mixing. An efficient method for synthesizing Mn21-doped CdSe nanocrystals is the recently reported diffusion-doping approach, which allows the incorporation of impurities into undoped nanocrystals grown with very narrow size distributions [17]. A representative absorption spectrum obtained for 5.4 nm diameter Mn21: CdSe nanocrystals with a doping concentration of ca. 0.4% is depicted in Fig. 9.4A. Based on the size-dependent study reported by Norris and Bawendi [53], the spectrally well resolved excited states were assigned as follows: (a) 1S3/21Se; (b) 2S3/21Se; (c) 1S1/21Se; (d) 1P3/21Pe; (e) 2S1/21Se; (f) 1P5/21Se/1Pl1/21Se; (g) 3S1/21Se; (h) 1S1/21De/2S1/22Se/1S1/22Se/2S3/21De/1D5/21De/4P3/21Pe; (i) 4S3/22Se/1S1/22Se/1Pso1/21Se. As explained in the introduction (see Fig. 9.1), nhLFneLe is used as nomenclature to describe excited states in quantum dots, whereat nhLF represents the VB state and neLe the CB state involved in the excited state. 1Pl1=2 and 1Pso 1=2 are specific VB states, which are not affected by VB mixing, exhibiting pure light hole or split-off character, respectively.

Figure 9.4 (A) Absorption spectrum of Cd0.996Mn0.004Se nanocrystals measured at T 5 2 K. The observed features (labeled a to h/i) have been assigned according to the study of Norris and Bawendi [53] based on multipeak fitting (Gaussian peaks). (B) Magnetic-field dependent MCD spectra obtained at T 5 2 K. (C) Magnetic-field dependent Zeeman splitting extracted from the experimental data shown in panel (A-B). Adapted with permission from R. Fainblat, F. Muckel, C.J. Barrows, V.A. Vlaskin, D.R. Gamelin, G. Bacher, Valence-band mixing effects in the upper-excited-state magneto-optical responses of colloidal Mn21-doped CdSe quantum dots. ACS Nano 8 (2014) 12669
12675. Copyright 2014 American Chemical Society [105].

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Multiple excited states could be observed in both absorption (Fig. 9.4A) and magnetic-field dependent MCD (Fig. 9.4B) spectra, both obtained at low temperature (T 5 2 K). In general, the MCD amplitude increases with increasing applied magnetic field, however the relative amplitudes of the different excited states vary widely. As shown previously for 2D nanostructures (see chapters 2.1 and 2.2), amplitude and sign of the Zeeman splitting can give insight into the dominant hole character of an excited state. Thus, through a direct comparison among the magneto-optical response of different excited states, we expect to directly gain insight into VB mixing effects in 0D nanostructures. Due to the spectral overlap of these excitonic features, the extraction of the giant Zeeman splitting (see Fig. 9.4C) requires a complex fitting procedure (for details see Ref. [105]). The amplitudes of the extracted Zeeman splittings as well as their saturation with increasing magnetic field evidence that sp-d exchange interactions are dominant for all observed excited states. The comparison of the Zeeman splitting of different excited states allows us to identify three characteristic groups (i) the first excited state showing a large Zeeman splitting with negative g-factor (black symbols), (ii) two excited states (blue symbols) exhibiting positive Zeeman splittings, and (iii) excited states with Zeeman splittings between these two limiting cases. The first excited state (black) is, as expected [106], mainly dominated by a hh character of the involved VB state. A few excited states reveal Zeeman splittings of opposite sign (blue), thus allowing us to draw the conclusion that these excited states are mostly dominated by lh and/or so components. Further excited states (red) are intermediates between these two limiting cases, thus directly reflecting the VB mixing. These results support the theory that VB mixing effects become more important for higher excited states.

9.3.2 Going to the limit: individual dopants in single nanocrystals quantum dots A great part of research has focused on achieving large impurity concentrations in nanocrystals [17,83,86,87,107
109]. However, parallel efforts aimed the precise incorporation of a single dopant into semiconductor nanostructures [34,42
44,49,50,110
114]. Despite the fact that in singly doped nanostructures cumulative effects of multiple pairwise dopant 2 carrier exchange interactions are not present, the ability to optically or electrically manipulate a single defined spin state [34,42,43,115
117] as well as the opportunity to achieve long spin-coherence times in single magnetic impurities [112,113,118,119] are great advantages of nanostructures doped with individual transition metal ions. The ability to strongly modify electronic, optical and magnetic properties of a material by the incorporation of a single impurity forms the basis for a new research field called “solotronics” [120]. Until recently spectroscopic investigations on single zero-dimensional nanostructures doped with individual impurities have been exclusively conducted in epitaxially grown quantum dots (QDs) [34,42
44,110,117]. Due to the larger confinement compared to epitaxially

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grown analogues, effects resulting from magnetic exchange interactions are expected to be significantly larger in colloidal nanostructures. In nanocrystals doped with a large number of transition metal impurities magneto-optical effects are orders of magnitude larger than in undoped analogues [8,19,38,121]. “Standard-sized” II-VI nanocrystals exhibit a diameter of 4
6 nm. Therefore, to synthesize colloidal quantum dots doped with single magnetic impurities, the doping concentration must be reduced significantly below 1%. One approach to achieve such low doping concentrations is to take advantage of the diffusion-doping method [17], which has been applied to prepare core/shell CdSe/ ZnSe and CdSe/CdS nanocrystals containing very small amounts of Mn21. From statistical considerations one can expect that in Mn21: CdSe/ZnSe QDs with xMn 5 0.4% and a core diameter of about 5.4 nm each nanocrystal contains in average 5
6 Mn21 impurities, whereas in the case of Mn21: CdSe/CdS with xMn 5 0.03% (diameter 5.1 nm), 65% of the nanocrystals are expected to be undoped and 25% of the nanocrystals should contain a single impurity [50]. Analogous to the case of nanocrystals doped with a larger number of impurities, exchange interactions between the exciton and magnetic impurities take place also in the low dopant limit. There are two parameters defining the strength of these exchange interactions: (i) the number of impurities and (ii) the location of the impurities in the nanocrystal. Due to the inevitable statistical variation of these parameters among different nanocrystals, standard ensemble experiments are not appropriate to investigate exchange interactions in QDs containing single impurities. In contrast, single-particle spectroscopy gives insight into the coupling between an exciton and a single magnetic impurity via the observation of an exchange splitting at zero-magnetic field [22]. This interesting effect has been discovered in selfassembled CdTe/ZnTe QDs more than a decade ago [34]. In epitaxially grown QDs the first excited state is four-fold degenerated, consisting of two states with total spin 6 2 and two states with total spin 6 1. Depending on the total spin, an excited state is optically forbidden ( 6 2) or allowed ( 6 1) and called “dark” or “bright”, respectively. In undoped epitaxially grown QDs only 6 1 excited states can be probed via PL spectroscopy, resulting in the observation of a single emission feature (see Fig. 9.5A
blue curve). By introducing a single Mn21 impurity, the 6 1 excited state interacts with the six different spin projections ( 6 5/2, 6 3/2, and 6 1/2) of the Mn21 spin [34,42]. This finally leads to the observation of multiple emission features in a time-integrated experiment (see Fig. 9.5A
red curve). In contrast to the fine structure present in the excited states in epitaxially grown QDs, the first excited state 1S3/21Se in colloidal QDs is eightfold degenerated, consisting of five optically active states with total spin 6 1L, 6 1U and 0U and three optically forbidden states with total spin 6 2 and 0L (see Fig. 9.1
right panel). This fundamental difference regarding the number of excited states which can interact with a single impurity ion might lead to the observation of different spectral signatures in single atom doped colloidal QDs compared to previous studies on their epitaxial counterparts. As outlined, the PL from undoped nanocrystals should strongly differ from the excitonic emission of singly doped nanocrystals, which in turn deviates from the emission of a nanocrystal containing two or more impurities [122]. Therefore, a

Figure 9.5 (A) Scheme of ground and excited state (GS and ES) of an undoped II-VI nanocrystal (left) and of a nanocrystal doped with a single Mn21 impurity (right). The blue (red) curve schematically represents the expected single-particle PL spectrum for the undoped (singly doped) case, neglecting phonon replica and participation of additional fine structure states. (B) Representative PL spectra for an undoped, singly doped and bi- (or multi-) doped nanocrystal (blue, red and orange, respectively). (C) PL spectrum of a single Cd0.996Mn0.004Se/ZnSe nanocrystal collected at 5 K. The dots depict the experimental data, while the colored shaded areas represent the fitted peaks. The sum of the fitted peaks is represented by the black line. The red curve accounts for the broad luminescence background attributed to the multiple weak phonon replicas of the various features. (D) Energy position of fitted peaks for two different single nanoparticles, including the nanoparticle analyzed in panel (C). Adapted with permission from R. Fainblat, C.J. Barrows, E. Hopmann, S. Siebeneicher, V.A. Vlaskin, D.R. Gamelin, et al., Giant excitonic exchange splittings at zero field in single colloidal CdSe quantum dots doped with individual Mn21 impurities, Nano Lett. 16 (2016) 6371
6377. Copyright 2016 American Chemical Society [22].

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statistical investigation of the PL of a large number of nanocrystals has been performed. Basically, there are three subsets of single nanocrystal spectra from one given sample: (i) undoped QDs, which showed single sharp emission features accompanied by phonon replicas at lower energies (blue curve in Fig. 9.5B) (ii) nanocrystals containing single Mn21 impurities, whose spectra depict several spectrally well resolved emission peaks (red curve in Fig. 9.5B), and (iii) QDs containing 2 or more Mn21 dopants, which showed a multitude of spectrally overlaid, broadened features (orange curve in Fig. 9.5B). A representative spectrum of a single colloidal CdSe/ZnSe nanocrystal containing most likely a single Mn21 impurity is depicted in Fig. 9.5C. At ca. 2.04 eV, a peak of highest intensity is displayed, which is usually called zero-phonon line (ZPL) since no phonons are involved in its radiative recombination process. At lower energies, some features with constant energy spacings to the main peak are observed, whose intensities decrease with decreasing energy. At energies .2.04 eV, the energy spacing between the features is smaller and there is no obvious trend for their relative intensities. The energy positions of the individual peaks extracted from Fig. 9.5C as well as a similar analysis for an additional QD are summarized in Fig. 9.5D. The lowenergy side of the spectrum (highlighted in gray) displays features with an energy spacing of B27 meV, consistent with the longitudinal optical (LO) phonon energy in undoped CdSe [123,124]. The 11
13 meV splitting between neighboring emission peaks in the high-energy part of the spectrum (highlighted in red) originates from the exchange interaction between the exciton spin with the different projections of the Mn21 spin. Anisotropic exchange interactions arising from nanocrystal shape and crystallographic site anisotropies [125,126] may be the cause for the unequal spacing of the peaks observed in the high energy part of the spectrum. In general, the energy splitting between PL features involving different Mn21 spin projections provides a spectroscopic fingerprint of the dopant location within the nanocrystal, with larger splittings coming from dopants located closer to the core. Because the sp
d exchange interaction strongly depends on the overlap of the excitonic wavefunction and the dopant [125], one can expect significantly larger exchange splittings in nanocrystals compared to their epitaxial counterparts with larger volume. In agreement with this expectation, the exchange splittings in the nanocrystals are enhanced by more than an order of magnitude compared to previous reports in self-assembled QDs [34,42], thus paving the way for solotronic applications at elevated temperatures. Nevertheless, a quantitative analysis of the exchange splitting leads to surprising findings. For a 5.4 nm large CdSe QD containing a single Mn21 at its center, we calculated an exchange splitting of B22 meV between the extreme cases of Sz,Mn 5 6 5/2. Surprisingly, our experimental results (40 2 80 meV) are larger than theoretically predicted, which we hypothesize to be caused by anisotropic deviations from the idealized spherical QD geometry and the cubic lattice structure. Furthermore, the fact that 6 1L, 6 1U and 0U can be involved in the radiative recombination of colloidal QDs (instead of only 6 1 in the case of epitaxially grown QDs) possibly leads to a more complex behavior in the exciton-single impurity exchange interaction.

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289

At the border between quantum dots and molecules: magic sized nanoclusters

9.4.1 Smallest doped semiconductors For several years the nanocrystal community believed that the incorporation of impurities in structures with diameters below 2 nm cannot be achieved due to effects such as self-purification and surface segregation [9,127]. In contrast to this assumption, the observation of dopant-related luminescence in Mn21 doped ZnTe nanoclusters [16] as well as DFT calculations support the hypothesis that doping of ultra-small structures may be realized for specific cases [128]. Magic-sized clusters (MSCs) can be used to gain insight into fundamental limitations of semiconductor nanocrystal doping, thus contributing to this controversial discussion. The term MSCs is related to the fact that some cluster configurations are significantly more stable than their slightly smaller or larger analogues. Therefore, MSCs consist of a well-defined number of atoms [129
131], which finally results in stepwise sizedependent growth behavior. Mn21: (CdSe)13 MSCs, which are known to act as building blocks for much larger 2D nanoribbons [55], have been synthesized by adding selenocarbamate to a solution containing CdCl2 and MnCl2 dispersed in octylamine [10]. The temperature was kept at 25  C in order to prevent nanoribbon formation. A representative absorption spectrum for such nanoclusters is depicted in the top panel of Fig. 9.6A. At a first glance at least three different excited states are spectrally well resolved in the absorption spectrum. In addition, the magnetic field dependent MCD spectra plotted in the bottom panel of Fig. 9.6A consist of at least three magneto-optically active excited states. A clear mismatch, i.e. an energy shift, between the MCD zero-crossings and the maxima in absorption can been seen, which is contrary to the expectation of an ordinary A-term MCD signal of excited states related to interband transitions in semiconductor nanostructures [46]. In the extreme quantization regime of MSCs, the multiband effective mass approximation used by Efros and co-workers in the theoretical description of fine structure effects in colloidal 0D nanostructures [52] (see introduction and Fig. 9.1
right panel) does not seem to be appropriate. However, the basic concept of excited state fine structure splitting and the different spin-dependent degeneracies of each sublevel inspired a more careful analysis of the experimental data. More precisely, in larger nanocrystals (diameter .2 nm) 6 2, 6 1L, 6 1U are expected to split in presence of an external magnetic field according to their two-fold degeneracy, while 0L and 0U are non-degenerate, being magneto-optically inactive. Obviously, the excited states probed in absorption are optically active, however they can be magneto-optically active or passive, finally requiring an adjustment of the “standard” data analysis described by Eq. (9.3). According to the adapted data analysis accounting for optically active (but magneto-optically passive) excited states, the degeneracy of the first and the third excited state (schematically depicted in Fig. 9.6B as 1. ES and 3. ES, respectively) is lifted in presence of an external magnetic field, whereas the second (2. ES, see

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Figure 9.6 (A) Absorption (top) and magnetic-field dependent MCD signal (bottom) for Mn21: (CdSe)13 nanoclusters (xMn 5 4%) measured at T 5 4.5 K. (B) Scheme of magnetic field induced lift of degeneracy of excited states (ES) and the different absorption of left (green arrows) and right (red arrows) circularly polarized light. The second excited state (2.ES) is assumed to be magneto-optically inactive. For the sake of clarity, only the first three ES are depicted. (C) Magnetic field dependent giant Zeeman splitting of Mn21: (CdSe)13 nanocluster (blue: xMn 5 4%; red: xMn 5 10%) for the 1. ES and the 3. ES. Adapted with permission from J. Yang, R. Fainblat, S.G. Kwon, F. Muckel, J.H. Yu, H. Terlinden, et al., Route to the smallest doped semiconductor: Mn21-doped (CdSe)13 clusters, J. Am. Chem. Soc. 137 (2015) 12776
12779. (A and C) Copyright 2016 American Chemical Society [10].

Fig. 9.6B) and forth excited state are apparently magneto-optically inactive. This approach led to the conclusion that not all excited states observed in absorption are magneto-optically active, resulting in an energy shift between the MCD zero crossings and the absorption peaks. By excluding magneto-optically inactive excited states (2. and 4. ES, fitted by Gaussian peaks in Fig. 9.6A, see dashed lines), giant Zeeman splittings for the 1. ES and the 3. ES could be extracted. (CdSe)13 clusters containing a nominal Mn21 doping concentration of 4% revealed an effective g-factor of 281 ( 6 8) for the 1. ES and of 18 ( 6 1) for the 3. ES, as extracted from a linear fit of the magneticfield dependent Zeeman splittings (see Fig. 9.6C). MSCs containing higher Mn21 concentration (xMn 5 10%) exhibit slightly smaller effective g-factors of 263 ( 6 18) and 16 ( 6 2) for the 1. and the 3. ES, respectively (see Fig. 9.6C). The effective g-factors of the 1. ES are almost two orders of magnitude larger than for undoped CdSe nanocrystals [45] and of opposite sign, thus indicating substantial sp-d exchange coupling between the Mn21 ions and the charge carriers of the host. The results discussed above prove the successful incorporation of transition metal impurities into such small semiconductors.

9.4.2 Doped magic-sized alloy nanoclusters Beyond the strong quantum confinement, which allows spectrally resolving magnetooptically active and inactive states in MSCs, the small number of atoms per nanocluster also distinguishes these structures from conventional, larger nano-scaled systems.

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Due to their well-defined number of atoms, any change in composition, e.g. on the cation side, must be considered as a discrete replacement of one out of 13 atoms rather than a continuous shift in alloy composition. In bulk II-VI compounds, alloying is used to continuously tune the band gap between the values of the pure species, e.g. between 1.74 eV for CdSe and 2.7 eV for ZnSe. In (CdSe)13 and (ZnSe)13 MSC, in contrast, the bandgap changes between 3.62 eV and 4.37 eV as a consequence of the step-wise replacement of Cd atoms by Zn ones. The smallest possible modification step in composition, i.e., the replacement of one Cd in (CdSe)13 by one Zn atom is thus expected to shift the bandgap by about 60 meV. Additionally, magnetically doped alloyed (ZnxCd1-xSe)13 MSCs offers the possibility to precisely tune the energetic region of the magneto-optical response. However, this requires the combination of four different elements within one stable crystal containing only 26 lattice sites. Alloying and doping of MSC can be achieved by adding corresponding amounts of ZnCl2 and/or MnCl2 to the initial CdCl2 precursor [11]. The final proof for the magneto-optical active incorporation of magnetic dopants can be provided via MCD spectroscopy (see normalized spectra in Fig. 9.7A). With increasing Zncontent, the magneto-optical response shifts towards higher energies maintaining its characteristic shape, although with increased linewidth for alloyed MSCs. This proves the successful combination of three different atoms within 13 cation sites. In addition, it expands the spectral range of magneto-optical activity of semiconductors nanocrystals towards the high UV. For Mn21: (ZnSe)13 clusters the Zeeman splitting can be extracted as described in the previous section, leading to an effective g-factor of 281 (xMn 5 6%) for the energetically lowest excited state, similar as in Mn21: (CdSe)13. However, the MCD amplitudes of the alloy samples are significantly decreased as compared to the binary MSC (see Fig. 9.7B). For alloyed clusters, the absorption and the MCD signals of a sample with an overall Zn content represents a superposition of the

Figure 9.7 (A) Normalized MCD spectra of Mn21-doped alloy MSC with xMn 5 6% and various nominal Zn contents between 0% and 100% at 5 K and 1.6 T. (B) and (C) MCD spectra for different samples extracted from experiment and from simulations, respectively. Adapted with permission from J. Yang, F. Muckel, W. Baek, R. Fainblat, H. Chang, G. Bacher, et al., Chemical synthesis, doping, and transformation of magic-sized semiconductor alloy nanoclusters. J. Am. Chem. Soc. 139 (2017) 6761
6770. Copyright 2017 American Chemical Society [11].

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signals from different species with statistically varying numbers of Zn and Cd atoms (Mn21: ZnyCd13-ySe13, 0 , y , 13). Taking into account the superposition of MCD signals generated by the different species within the ensemble, the MCD spectra for ensembles with different overall Zn contents (xZn 5 0.2, 0.4, 0.6, 0.8) has been simulated. Hereby, it was assumed that all individual species exhibit the same giant Zeeman splitting. The simulated spectra match well with the experimental data in both, energy position of the extrema and lineshape. Moreover, the simulations predict the reduction of the MCD amplitudes (Fig. 9.7C) seen in experiment for the alloy clusters, which is apparently induced by the superposition of signals stemming from different cluster species within the ensemble. This suggests that each alloy species itself exhibits a similar amplitude for the magneto-optical response, i.e. a similar effective g-factor, as the pure Mn21: (ZnSe)13 or Mn21: (CdSe)13 clusters. These results show that the incorporation of transition metal impurities into magic-sized alloy nanoclusters is feasible.

9.4.3 “Digital” doping in nanoclusters In conventional DMS nanocrystals consisting of several hundreds to thousands of atoms, the embedded transition metal ions experience a similar local environment as in bulk DMS materials. This is completely altered in MSCs, where due to the limited number of atoms, dopants incorporated in the cluster can most likely be considered as nearest neighbors, which may imply striking consequences for their magneto-optical functionality. In bulk DMS materials the magnetic moments of two Mn21 dopants incorporated on nearest neighbors’ positions couple antiferromagnetically, finally canceling out their magnetic moment. Thus, they do not contribute to the magneto-optical response [35,47]. Hypothesizing that similar antiferromagnetic coupling between nearest neighbors takes place in MSCs, only clusters doped with an odd number of impurities are expected to exhibit sp-d exchange interactions and giant Zeeman splittings. Due to the synthesis route, one can expect Mn21 dopants to be randomly distributed among the clusters. Laser desorption/ionization time-of-flight mass spectrometry (LDI-TOF MS) of Mn21: (CdSe)13 revealed the presence of undoped clusters as well as clusters containing one (mono-doped) or two (bi-doped) dopants. No evidence of clusters with three or more dopants has been observed. With increasing doping concentration, the relative amplitude of the three peaks changes (compare inset in Fig. 9.8A). Based on the LDI-TOF MS results, the portions of the three cluster species (undoped, mono-doped, bi-doped) can be extracted among a concentration series up to xMn 5 10% (see data points in Fig. 9.8B). The theoretically expected ratio between undoped, mono-doped and bi-doped clusters are in addition simulated assuming a binominal distribution of the dopants among the three cluster species (excluding doping with three or more Mn21). These calculations (solid lines in Fig. 9.8B) fit well with the experimentally determined values. Small deviations might be due to differences in the formation entropies for different types of clusters, which are not reflected in the binominal approach. At low Mn21 concentrations,

Figure 9.8 (A) LDI-TOF mass spectra of Mn21-doped (CdSe)13 MSC with xMn 5 2%. The experimental data (black curve) are compared to simulated isotropic distributions (undoped cluster in blue, mono-doped in red and bi-doped in green). The inset depicts mass spectra for samples with different doping concentrations (xMn 5 4%, 7% and 10% from top to bottom). (B) Fraction of the different cluster species (undoped cluster in blue, mono-doped in red and bi-doped in green), as extracted from the mass spectra (symbols) and compared to simulated distributions (solid lines). (C) Upper panel: giant Zeeman splitting of the energetically lowest excited state for cluster samples with different Mn21 concentrations as extracted from MCD measurements. Data were taken at 5.2 K and 1.43 T. Lower panel: expectations for the amount of magnetically active ions xeff in Mn21-doped (CdSe)13 assuming digital doping (black line) and neglecting this effect (blue line), i.e. with and without antiferromagnetic coupling in bi-doped clusters. The case of 50% antiferromagnetic coupling in bi-doped clusters is depicted as dashed blue line. (D) Temperature dependent MCD signal of Mn: (CdSe)13 MSC with xMn 5 2%. The inset depicts the room temperature signal. Adapted with permission from F. Muckel, J. Yang, S. Lorenz, W. Baek, H. Chang, T. Hyeon, et al., Digital doping in magic-sized CdSe clusters. ACS Nano 10 (2016) 7135
7141. Copyright 2016 American Chemical Society [132].

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undoped clusters are by far the most frequent species, while for larger Mn21 content the percentage of doped clusters increases. Fig. 9.8C depicts the giant Zeeman splitting extracted from MCD measurements for the samples with different average Mn21 concentrations. The Zeeman splitting first increases up to xMn 5 6%, and subsequently decreases. Under the assumption that the mean spin per dopant is held constant, the giant Zeeman splitting in a DMS material scales proportional to the amount of magnetically active dopants xeff . It is thus instructive to compare the trend in the magneto-optical activity expressed in terms of Zeeman splitting (Fig. 9.8C
top panel) with theoretical expectations for xeff (Fig. 9.8C
bottom panel). In case that the dopants in bi-doped clusters do not interact antiferromagnetically with each other (blue curve Fig. 9.8C), literally all dopants should contribute to the MCD signal, and xeff is expected to follow xMn. In contrast, in case of antiferromagnetic coupling, the dopants in bi-doped cluster should not contribute to the MCD signal, and xeff corresponds to the portion of mono-doped clusters (black curve Fig. 9.8C). The agreement between experimental results and simulations assuming antiferromagnetic coupling between the dopants in bi-doped MSC demonstrates that only the mono-doped clusters contribute to the MCD signal. This fact combined with the existence of only three (undoped, monoand bi-doped) cluster species leads to an interesting, novel behavior, which is named “digital doping”. According to this principle, the giant magneto-optical response can be turned “on” by replacing a Cd atom by a Mn in a (CdSe)13 cluster (mono-doped MSC), and by the incorporation of a further impurity (bi-doped MSC), the magneto-optical response can be turned “off”. Investigating the temperature dependence of the magneto-optical response for the Mn21: (CdSe)13 MSC, a giant Zeeman splitting - i.e., the persistence of the sp-d exchange interaction
can be detected up to room temperature. Altogether we can conclude, that the magneto-optical functionality is generated from solitary dopants, and this single dopant functionality persists up to room temperature, making the clusters promising candidates for future spintronic or solotronic applications.

9.5

Conclusion and future trends

The breakthrough in transition metal doping of colloidal nanocrystals opened a novel and exciting field in the research area of diluted magnetic semiconductors. It has been demonstrated by several examples that size and shape engineered nanocrystals, combined with the isoelectronic replacement of lattice cations by manganese leads to unexpected and fascinating magneto-optical properties. In quasi twodimensional systems, the lowest excited states are characterized by well-defined hole states (i.e. heavy hole, light hole, splitt-off hole) leading to pronounced MCD resonances with a characteristic shape, depending on the specific hole state involved. This can be used to resolve excited states with much higher resolution compared to traditional absorption measurements. The ability to tailor the architecture of nanoplatelets with the precision of atomic monolayers is shown to allow for

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a unique control of the sp-d exchange interaction of individual excited states. In nanocrystals with spherical carrier confinement, i.e. quantum dots, strong valence band mixing occurs, which could nicely be proved by MCD spectroscopy of higher excited states. Reducing the doping concentration to values far below 1%, even single atom doped quantum dots can be realized, showing characteristic fingerprints of individual Mn21-dopants. The occurrence of a multitude of photoluminescence peaks could be related to the individual spin projections of the single Mn21-dopant interacting with the optically generated exciton. A further reduction in size of the nanocrystals leads to the formation of magic-sized nanoclusters with a well-defined number of atoms. It has been proven that individual cations of magic-size (CdSe)13 clusters can be replaced stepwise by either transition metal dopants like Mn or by Zn cations. The former leads to an effect entitled digital doping, i.e. a magneticoptical response, which is switched “on” and “off” by stepwise adding one or two magnetic dopants, and a pronounced magneto-optical functionality controlled by a single dopant is found even up to room temperature. The latter results in a stepwise change of the material’s bandgap energy, extending the applicability of magnetooptically active nanocluster up to the high UV regime. The huge quantum confinement in these ultra-small nanomaterials allows for a separation of magneto-optically active and magneto-optically inactive fine structure states of the lowest excited state via MCD spectroscopy. This rapid progress of transition metal doped nanocrystals during the last couple of years offers exciting perspectives for future research. Certainly, the ability of generating nanocrystals functionalized by individual dopants may lead to novel concepts in the fields of solotronics or even quantum technologies. Moreover, the recent demonstration of an electrically driven device showing current-induced magnetic polaron formation in Mn21-doped CdSe/CdS nanocrystals [21] might be an important step towards practical applications of transition metal doped nanocrystals in electronic and optoelectronic devices. Besides the giant magneto-optical effects present in transition metal doped nanocrystals, these materials are in addition of high relevance for further applications, as e.g., in solar energy harvesting [133
136], bioimaging [137,138] or even as biomarkers with combined optical and magnetic response [139]. Although manganese doping is by far the most explored area of research hereby, alternative transition metals like Co [12,14,24], Cu [25,26,136,140,141], Ag [142
146], Fe [147
149], etc. represent emerging dopants of interest, expanding the functionality of doped nanocrystals further. The journey just started.

Acknowledgments The authors are deeply grateful to the collaborators involved in the results presented in this book chapter. We want to thank J. Yang, J.H. Yu, W. Baek, S.G. Kwon, H. Chang, B.H. Kim, M.K. Choi, J. Lee and Prof. T. Hyeon from the Seoul National University and Institute for Basic Science (Seoul, Republic of Korea) for the development and synthesis of the nanoribbons and nanoclusters. We acknowledge C. J. Barrows, V. A. Vlaskin and Prof. D. R. Gamelin from

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the University of Washington (Seattle, USA) for the collaboration on the nanocrystal quantum dots. Additionally, we are thankful to S. Delikanli, P.L. Herna´ndez-Martı´nez, M. Sharma and Prof. H.V. Demir from the Nanyang Technological University (Singapore) and Bilkent University (Ankara, Turkey) for the synthesis of the nanoplatelets. J. Frohleiks, T. Czerny (born Priesner), J. Ackermann, S. Siebeneicher, E. Hopmann and S. Lorenz are acknowledged for experimental assistance. R. Fainblat, F. Muckel and G. Bacher acknowledge the Deutsche Forschungsmeinschaft for financial support under contracts Ba 1422/13-1, Ba 1422/13-2 and Ba 1422/16-1. R. Fainblat and F. Muckel acknowledge the German Academic Exchange Service (DAAD) for funding from the German Federal Ministry of Education and Research (BMBF) and the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/ 2007
2013) under REA grant agreement no 605728 (P.R.I.M.E.
Postdoctoral Researchers International Mobility Experience).

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[144] V. Pinchetti, Q. Di, M. Lorenzon, A. Camellini, M. Fasoli, M. Zavelani-Rossi, et al., Excitonic pathway to photoinduced magnetism in colloidal nanocrystals with nonmagnetic dopants, Nat. Nanotechnol. 13 (2018) 145
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8456.

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10

Joseph A. Hagmann Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD, United States.

10.1

Introduction

An exotic phase of quantum matter exists in certain classes of chalcogenide materials, characterized by strong spin-orbit coupling and hosting time-reversal invariant systems, is the topological insulator (TI). Topological insulators are materials with a bulk band gap and band-crossing edge states or surface states supported by the non-trivial band topology of the TI. The topologically protected states are helical, with the electron spin locked perpendicular to the momentum, protecting against direct backscattering by non-magnetic impurities. The chalcogenide topological insulators discussed in this chapter are of two types. The first is the 2D topological insulator system observed in type-III HgTe semiconductor quantum wells, which hosts topologically protected edge states with a quantized conductance arising from the quantum spin Hall effect, and the V2VI3-type bismuth- and antimonychalcogenide 3D topological insulators. The existence of a phase of exotic quantum matter in systems with timereversal symmetry was found to be rather unexpected, as prior to the discovery of symmetry-protected topological states [1,2], exotic quantum states were believed to arise from symmetry breaking according to the Landau paradigm [3]. Symmetry-breaking exotic quantum phenomena include superconductivity, which arises when the electromagnetic gauge symmetry associated to electron number is broken [4], and ferromagnetism, which arises when time-reversal symmetry is broken [5]. Unlike in the Landau paradigm, topological order, which is not attributed to symmetry breaking, is instead related to the Berry phase acquired by path-integrating through a closed adiabatic loop around a complex vector space defined by the Hamiltonian (e.g. bands in the Brillouin zone). One well-known topological effect, the quantum Hall effect [1,6], is, itself, known to arise from time-reversal symmetry breaking by a strong magnetic field, which leads to a quantized Hall conductance associated with a topological invariant called a Chern number [7]. Here, the integral of the Berry curvature, which is given by the Berry phase around a small plaquette of infinitesimal area within the enclosed parameter space region divided by the area of that plaquette, over the occupied states described by the ground state Hamiltonian plus a perturbation to account for the application of a background electric field to generate current flow in a Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00015-4 © 2020 Elsevier Ltd. All rights reserved.

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strong magnetic field leads to a nontrivial topological invariant, robust against disorder and changes in material properties and device geometry [8]. It came as a surprise to many, then, that a topologically non-trivial system was predicted and observed for a system, not arising from symmetry breaking, but rather arising from symmetry protection. Specifically, these systems have a non-trivial Z2 topological classification [9].

10.1.1 The Z2 Topological insulator The chalcogenide materials systems discussed in this chapter are distinguished by strong spin-orbit coupling associated with high Z-number of the elements composing the materials, which gives rise to band inversion at time-reversal (TR) symmetry protected point of high symmetry in the Brillouin zone (BZ). According to Kramers’ theorem, for a TR-invariant Hamiltonian for spin-1/2 particles, all eigenstates are at least twofold degenerate [10]. In the absence of spin-orbit interactions, Kramers’ degeneracy is simply the spin degeneracy. However, in the presence of spin-orbit interactions, a TR-invariant Hamiltonian, H, i.e. one that satisfies ΘH ðkÞΘ21 5 H ð 2kÞ, can be classified by one of two topological classes defined by Z2 invariant ν: ν 5 0 or ν 5 1, signifying, respectively, a topologically trivial system and a topologically nontrivial system [9,11]. At the interface between two distinct systems, depending on the details of the Hamiltonian near the edge, there may or not exist band gap-crossing states bound to the edge of the shared interface of the two systems (for 2D topological insulators) or to the surface formed by the shared interface of the two systems (for 3D topological insulators). The minimum number NK of Kramers pairs of edge modes intersecting the Fermi level, EF, within the bulk band gap is given by the difference between the topological indices of the two systems according to NK 5 Δνmod2, an effect referred to as bulk-boundary correspondence [12]. A system with an even number of Kramers pairs of edge modes that cross EF can be smoothly deformed via an adiabatic transformation of the Hamiltonian to move all band-crossing edge modes out of the band gap. These states are topologically equivalent, i.e. ν 1 5 ν 2 . A system with an odd number of Kramers pairs of edge modes that cross EF can never be smoothly transformed to reduce NK below 1 without undergoing a topological phase change, revealing the topological protection of these states arising from the topological inequivalence of the two interfacing systems, i.e. ν 1 6¼ ν 2 . The Z2 topological index, ν, can be calculated from a unitary matrix built from the occupied Bloch functions defined at four points in the bulk 2D Brillouin zone (for 2D topological insulators) or eight points in the bulk 3D Brillouin zone (for 3D topological insulators) where 1 k and k coincide. These mathematical formulations are beyond the scope of this chapter, but can be studied in the following references [9,1320]. In the following sections, the HgTe quantum well 2D topological insulator system and the V2VI3-type (Bi,Sb)2(Se,Te)3 3D topological insulator systems, hereby described as V2VI3 series systems, are introduced.

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10.1.2 Mercury telluride quantum wells The type-III HgTe semiconductor quantum well is regarded as the first topological insulator to be studied [21]. Here, type-III refers to a heterojunction with a broken band gap that results from a distinctive inversion of the HgTe conduction and valence bands due to strong spin-orbit coupling. These materials systems consist of a thin HgTe sandwiched between two CdTe layers. CdTe is a topologically trivial (or “normal”) insulator with a bulk band gap of 1.5 eV [22], whereas HgTe is topologically non-trivial due to the inversion of the bands at the k 5 0 point of the Brillouin zone driven by strong spin-orbit coupling. At HgTe quantum well thicknesses, d, below critical thickness, d , dc, where dc was found to be 6.4 nm [23], the quantum well is dominated by the electronic structure of the CdTe layers, and the system is in the normal, topologically trivial, state. At HgTe quantum well thicknesses greater than the critical thickness, d . dc, the quantum well is dominated by the inverted band structure of the HgTe, giving the highest occupied band of the electronic states within the well a Z2 topological index of ν 5 1. The Z2 topological index of the CdTe layers on either side of the HgTe quantum well, conversely, is a trivial ν 5 0. At the HgTe-CdTe interface, by the bulk-boundary correspondence described in the previous section, exists a pair of topologically protected helical edge states characterized by dissipationless spin current flow. These states describe a nontrivial topological phase called the quantum spin Hall (QSH) system, which is the hallmark topological effect in the HgTe quantum well system (Fig. 10.1). HgTe and CdTe have zincblende structure with space group Td2 (F43m), a derivative of the diamond structure. These structures contain two symmetry-independent atomic sites per unit cell; cations Hg and Te occupy the Wyckoff 4a (0 0 0) positions in the cubic cell, and the Te anion occupies the Wyckoff 4c (0.25 0.25 0.25) ˚ , corresponding to a Hg-Te bond position. The HgTe unit cell parameter is 6.462 A ˚ ˚ , corresponding length of 2.797 A [24], and the CdTe unit cell parameter is 6.477 A ˚ to a Cd-Te bond length of 2.803 A [25]. The interface between CdTe and HgTe experiences very little lattice strain, on the order of 0.2 %. High quality barrier materials are often grown with the inclusion of Hg flux during growth, producing ternary alloy HgxCd1-xTe barrier layers. For example, Konig et al. observed the QSH effect in HgTe quantum wells sandwiched between topologically trivially insulating Hg0.3Cd0.7Te [26]. The topologically nontrivial HgTe quantum well system is, necessarily, a layered heterostructure, and must therefore be synthesized in thin film form. Consequently, there is practically no bulk synthesis version of this materials system. The QSH effect has been demonstrated in HgTe quantum wells [23] grown by molecular beam epitaxy (MBE) [26]. HgTe quantum well systems have also been synthesized by plasma-enhanced chemical vapor deposition (PECVD) [27] and metalorganic chemical vapor deposition (MOCVD) [28], albeit primarily for the purpose of producing infrared photodetectors rather than QSH insulators. These thin film growth techniques are described in Section 10.3. At a sufficient HgTe quantum well thickness d . 6.4 nm, measurements show a conductance plateau of close to 2e2/h,

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Figure 10.1 The quantum spin Hall effect (QSHE) in HgTe quantum well (QW) 2D topological insulators. (A) CdTe/HgTe/CdTe quantum well structure; d is the thickness of the HgTe QW layer. (B) The QW subbands undergo an inversion from a normal regime, with a valence band with p-type character and a conduction band with s-type character, to an inverted regime, with a valence band with s-type character and a conduction band with p-type character, for HgTe layer thicknesses greater than a critical thickness, d . dc, where dc 5 6.4 nm. This effect arises from the inverted band structure of HgTe and the normal band structure of the CdTe barrier layers. (C) For d , dc, the band structure of the CdTe layers dominate within the QW, and the s-type E1 quantum well subband lies above the p-type H1 subband (normal regime); (D) for d . dc, the band structure of the HgTe dominates within the QW, and the s-type E1 QW subband lies below the p-type H1 subband (inverted regime). (E) The spin-polarized topological edge states of the QSHE. (F) In the normal regime for a HgTe QW (d , dc), the two-terminal conductance will vanish when the Fermi level is positioned within the bulk band gap of the QW. (G) The experimental signature of the QSHE effect in the inverted regime for a HgTe QW is a quantized 2e2/h two-terminal conductance when the Fermi level is positioned within the bulk band gap of the QW. (H) Quantized 2e2/h two-terminal conductance was experimentally observed for HgTe QW devices with HgTe layer thickness greater than dc (samples III and IV); sample I has a HgTe QW thickness less than dc and, consequently, demonstrates normal insulating behavior. Fig. 10.1CG are adapted from B.A. Bernevig, T.L. Hughes, S.C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 318 (2006) 1757. Fig. 10.1H is adapted from M. Ko¨nig, S. Wiedmann, C. Bru¨ne, A. Roth, H. Buhmann, L.W. Molenkamp, et al., Quantum spin hall insulator state in HgTe quantum wells, Science, 318 (2007) 766770.

where e is the elementary charge and h is the Planck constant, that is independent of sample width, indicating it is caused by the predicted topological edge states. This seminal transport experiment is further discussed in Section 10.4.

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10.1.3 V2VI3-series 3D topological insulators V2VI3-series topological insulator (TI) materials Bi2Se3, Bi2Te3, and Sb2Te3 have tetradymite rhombohedral crystal structure with space group D53d (R3m) with five atoms per unit cell [29]. These materials form layered structures consisting of repeating layers of five lamellae, from which the structural identifier “quintuple layer (QL) structure” is derived. The lamellae of the QL are atomic planes arranged along the zdirection with the atomic composition of the lamella given by X1MX2M’X1’, where X1, X1’, and X2 denote lamellae containing Se or Te atoms, and M, M’ denote lamellae containing Bi or Sb atoms, with strong interatomic coupling of adjacent atoms within a QL and weaker coupling between QLs separated by van der Waals gaps. X1 and X1’ are equivalent and M and M’ are equivalent, with the atoms of the top lamellae, X1 and M, related to the atoms of the respective bottom lamellae, X1’ and M’, by an inversion operation in which the X2 atoms act as inversion centers. The rhombohedral lattice parameters, Wyckoff positions, bond lengths, and interlamellar spacing of Bi2Se3, Bi2Te3, and Sb2Te3 are shown in Table 10.1. These TI materials exhibit a bulk band gap and band-crossing surface states supported by the non-trivial band topology of the topological insulator. Note that Sb2Se3, which has an orthorhombic structure rather than a rhombohedral structure, is not a topological insulator, but is, rather, a topologically trivial normal band insulator. Bi2Se3 has a direct bulk band gap at the Γ point of approximately 240300 meV [33], Bi2Te3 has an indirect bulk band gap of approximately 150 meV [34], and Sb2Te3 has an indirect band gap of approximately 210 meV [35]. The topologically protected surface states are characterized by the electron spin locked perpendicular to the momentum (in the plane of the sample) related to the system’s time reversal invariance that protects these conducting states against backscattering by nonmagnetic impurities. Like HgTe, the topologically nontrivial nature of the occupied bands of these high Z-number V2VI3-series materials is due to band inversion at the k 5 0 point (the Γ point) of the Brillouin zone driven by strong spin-orbit coupling. The V2VI3-series 3D topological insulator is distinct from the HgTe quantum well 2D topological insulator in that the former is characterized by four topological invariants, ðν 0 ; ν 1 ν 2 ν 3 Þ 5 ð1; 000Þ, where, at sufficient thickness of the V2VI3 layer, the primary topological index ν 0 5 1 identifies the system as a strong topological insulator with symmetry protected surface states, and ν 1 5 ν 2 5 ν 3 5 0, where ðν 1 ν 2 ν 3 Þ can be interpreted as Miller indices describing the orientation of the layers, reveals that V2VI3-series 3D topological insulator will not host time-reversal invariant topologically protected one-dimensional helical modes at the sites of any dislocations in the crystal [36]. The topological surface states of the V2VI3-series materials appear in the band diagram as a band-gap-crossing surface state. Advantageously, the surface states in Sb2Te3, Bi2Se3, Bi2Te3 have been predicted to be nearly ideal single Dirac cone with near-linear energy-momentum dispersion [37], which has been experimentally demonstrated by angle-resolved photoelectron spectroscopy (ARPES) measurements to be the case for Bi2Se3 [38] and Bi2Te3 [38,39] (Fig. 10.2); due to a high level of intrinsic doping of naturally grown Sb2Te3, the Fermi level of this material lies in the bulk valence band continuum, below the surface states, and has,

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Table 10.1 Rhombohedral lattice parameters, Wyckoff positions, bond lengths, and interlamellar spacing of Bi2Se3, Bi2Te3, and Sb2Te3. M denotes Bi and Sb atoms and X denotes Se and Te atoms. ˚) (A) Lattice parameters (A Bi2Se3 [30,31]

Bi2Te3 [31]

Sb2Te3 [29]

a

4.14

4.39

4.26

c

28.6

30.5

30.5

(B) Wyckoff positions Bi2Se3 [30,31]

Bi2Te3 [31]

Sb2Te3 [29]

X1

(0 0 0)

(0 0 0)

(0 0 0)

M

(0.2109 0.2109 0.2109)

(0.2097 0.2097 0.2097)

(0.2128 0.2128 0.2128)

X2

(0.4006 0.4006 0.4006)

(0.4000 0.4000 0.4000)

(0.3988 0.3988 0.3988)

˚) (C) Bond lengths (A Bi2Se3 [32]

Bi2Te3 [32]

Sb2Te3 [29]

MX1

2.97

3.04

2.98

MX2

3.04

3.24

3.17

X1X1’

3.27

3.72

3.74

Bi2Se3 [30,31]

Bi2Te3 [31]

Sb2Te3 [29]

dAB

1.57

1.74

1.68

dBC

1.93

2.03

2.00

dCB’

1.93

2.03

2.00

dB’A’

1.57

1.74

1.68

dvdW

2.54

2.63

2.81

˚) (D) Lamella spacing (A

therefore, not been directly observed by using ARPES. A complication in studying the physics of carriers at the Dirac point of the surface states arises in Bi2Te3 as the Dirac point lies below the top of the BVB, signifying that studies of transport of the surface states at the Dirac point will be confounded by hole conduction in the BVB. Ternary and quaternary alloys of composition Bi2-xSbxSe3-yTey, with x and y tuned to achieve such alloys as BiSbTe3, Bi2Se2Te, and Bi2SeTe2, may be synthesized to both adjust the position of the Dirac cone within the band gap, the charge type of the Dirac surface state charge carriers, and position the Fermi level within the bandgap to limit charge transport to the surface states [40].

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Figure 10.2 3D topological insulators Sb2Te3, Bi2Se3, and Bi2Te3. (A) Side view of the quintuple layer structure of Sb2Te3, Bi2Se3, and Bi2Te3, with the three primitive lattice vectors t1, t2, and t3 and lattice sites M (Sb, Bi) and X1 and X2 (Se, Te) labeled. (B) Top view of the Sb2Te3, Bi2Se3, and Bi2Te3 structure along the z-direction. (C) Side view of the Sb2Te3, Bi2Se3, and Bi2Te3 structure showing the stacking of atomic lamellae along the zdirection. (DF) Energy-momentum band dispersion along K-Γ-M for Bi2Se3 (D), Bi2Te3 (E), and Sb2Te3 (F). (GH) Angle-resolved photoemission spectroscopy (ARPES)-measured band dispersion of Bi2-δCaδSe3 (G) and Bi2Te3 (H) (111) surfaces in the vicinity of the Γ point of the 2D Brillouin zone along K-Γ-M, revealing the topological surface states. Fig. 10.2A and Fig. 10.2DF are adapted from H. Zhang, C.X. Liu, X.L. Qi, X. Dai, Z. Fang, S.C. Zhang, Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface, Nat. Phys. 5 (2009) 438742. Fig. 10.2GH are adapted from D. Hsieh, Y. Xia, D. Qian, L. Wray, J.H. Dil, F. Meier, et al., A tunable topological insulator in the spin helical Dirac transport regime, Nature, 460 (2009) 11011105.

Experimentally, unlike the QSH state hosted by the HgTe quantum well system, the conductance of the topological surface states of the V2VI3-series topological insulators is not characterized by a quantized conduction value, as these states are not dissipationless 1D chiral modes, but rather are a diffusive 2D electron system. However, due to the time-reversal symmetry of the system, the topological surface states are protected against direct backscattering by non-magnetic impurities. The spin texture of the topological surface states leads to highly spin-polarized current,

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motivating research of these materials for the development of spintronic applications. Furthermore, the interaction of the Dirac-like surface states with symmetrybreaking states, including those involved in ferromagnetic exchange and the Cooper pairs hosted by a superconductor, can produce exotic topological states. The quantum anomalous Hall effect (QAHE), observed when the time reversal symmetry of topological insulator surface states is broken by magnetic ordering induced by spontaneous magnetization, supports a quantized Hall conductance of e2/h with a Chern number of 1 [41]. Unlike the quantum Hall effect (QHE) described in Section 10.2, the QAHE arises spontaneously without inducement by an external magnetic field. Another example of an exotic topological state is the Majorana state hosted in topological superconducting systems, either bound to vortices in topological twodimensional chiral p-wave superconductors [42] or as chiral Majorana modes in 1D chiral topological superconductors [43]. Experimental studies on these and other topological quantum states are discussed in Section 10.4.

10.2

Synthesis

This section briefly describes the thin-film growth techniques used to produce HgTe quantum well heterostructure materials and details the various bulk crystal growth, thin film growth, epitaxial deposition, and other techniques used to produce V2VI3-series 3D topological insulators. An introduction to the various growth methods that have been demonstrated successfully to produce topological insulator chalcogenides, along with key growth method details, will be presented. Crystal quality is largely discussed in terms of structural and compositional characterization measurements, which will be described here.

10.2.1 Mercury telluride quantum well growth The HgTe quantum well system is a trilayer CdTe/HgTe/CdTe heterostructure with an HgTe layer thickness of at least 6.4 nm. The CdTe layers in this heterostructure can be replaced with layers with composition HgxCd1-xTe, where x is sufficiently small such that the layer preserves its topologically trivial non-inverted band gap. The epitaxial growth HgxCd1-xTe with very low x has is somewhat easier to produce with good crystallinity than pure CdTe because Cd incorporation is improved when growth is performed in a high Hg vapor pressure environment, likely due to the prevention of tellurium precipitates [44]. The relative extent of Hg and Cd incorporation is tuned by the gas flow rates and substrate temperature. The HgxCd1-xTe/HgTe/HgxCd1-xTe system has been produced by means of several different layered heterostructure growth methods, including molecular beam epitaxy (MBE), vapor phase epitaxy (VPE) — including plasma enhanced chemical vapor deposition (PECVD), metalorganic VPE (MOCVD), and physical vapor deposition (PVD) — and liquid phase epitaxy (LPE). Laser assisted deposition and annealing have also been utilized to produce CdHgTe films [45]. Typically, a CdTe

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substrate is employed as a substrate, with different substrate surface conditions report to be best suited to growth by the different growth methods. A Cdterminated CdTe(111) surface, also called a CdTe(111)A surface, for example, has been shown to be optimal for LPE growth [46]. A Te-terminated CdTe(111) surface, also called a CdTe(111)B surface, has been shown to be a suitable surface for MBE growth [47], and CdTe(100) and CdTe(110) have both been shown to produce suitable CVD growth [48,49], but there is no well-established conclusion concerning the best orientation of the CdTe substrate for MBE or CVD growth. Much of the early work on the synthesis and study of Hg1-xCdxTe/HgTe heterostructures was for the purpose of producing infrared detectors, although the linear zero-energy gap edge modes had been observed as early as 1983 in an MBE-grown HgTe-CdTe quantum well system [50]. Of the methods listed above, MBE is the favored method for producing abrupt interfaces between the HgTe layer and the trivial insulating CdTe.

10.2.1.1 Molecular beam epitaxy growth of HgTe quantum wells The de facto method of producing high quality CdTe/HgTe/CdTe heterostructures with atomically abrupt CdTe-HgTe interfaces is molecular beam epitaxy, where molecular beams are generated under ultra-high vacuum conditions with beam intensities controlled by adjusting the temperatures of the effusion cells. Controlling effusion cell temperature (and thus elemental flux) and substrate temperature permits the highly controlled growth of epitaxial films with desired chemical compositions and CdTe/ HgTe/CdTe quantum well structures with abrupt, smooth interfaces. While HgTe quantum well heterostructures can be grown by MBE on a number of substrates, including GaAs [51,52], InSb [53], and Si [54], optimal growth has been shown to occur on lattice-matched Cd0.96Zn0.04Te, which ensures minimal dislocation [55]. MBE growth of these heterostructures requires low temperatures between 150  C and 220  C [56,57], with lower growth temperatures minimizing interdiffusion and demonstrating improved crystal quality. Growth occurs at very slow rates of around 1 μm/hr. The basic process entails generating molecular beams of desirable fluxes from Knudsen-type effusion cells under ultrahigh vacuum conditions by carefully controlling the temperature of the effusion cells. For the MBE growth of Hg1-xCdxTe using a metallic Hg vapor source, a metallic Cd source, and a Te2 source, the Cd concentration, x, is primarily controlled by tuning the Cd/Te2 ratio [58]. The Hg vapor during growth has been shown to both prevent tellurium precipitates and contribute the Hg that incorporates at Cd-substitutional sites in the CdTe lattice [44].

10.2.1.2 CVD growth of HgTe quantum wells The growth of HgTe-CdTe superlattices by CVD on CdTe substrates has been shown to produce material of good quality at high growth rates by using a number of precursor reactants, including Cd precursor dimethylcadmium (DMCd); Hg precursors dimethylmercury (DMHg), HgI2, and gaseous Hg in H2 carrier gas; and Te precursors dimethyltelluride (DMTe) and diethyltelluride (DETe). Note that DMCd

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and DMHg are highly toxic; extreme care should be taken when working with these gases. Hg gas has been shown to be a workable substitute for DMHg if a lowertoxicity precursor is preferred [59]. Hg1-xCdxTe deposition with x very close to 1 occurs in the presence of DMHg at suitable gas flow rates and substrate temperatures during plasma-enhanced CVD growth such that Hg incorporation occurs to a negligible extent relative to Cd, as the high background vapor pressure of Hg precursor prevents tellurium precipitates. The thermodynamics of ternary alloy Hg1-xCdxTe growth are complicated, as the growth is tuned by total pressure, substrate temperature, and the relative gas flows of DMTe/DETe, DMCd, DMHg/Hg, and H2. An example plasma-enhanced CVD growth [27] of a HgTe layer occurs with a CdTe substrate heated to 150  C, 0.5 Torr (66.7 Pa) system pressure, and carrier gas flow rates of 18 sccm (cm3/min) for DMTe and 20 sccm for DMHg. The slightly higher DMHg flow rate serves to reduce the incidence of Te precipitates. A CdTe layer is grown at a substrate temperature of 150  C with carrier gas flow rates of 2.4 sccm, 6 sccm, and 20 sccm for DMCd, DMTe, and DMHg, respectively. Note that in spite of the high flow rate of Hg precursor, Hg incorporation into Hg1-xCdxTe is limited below 400  C, as only above this temperature does Hg begin to react with the Te alkyl. There is a strong temperature dependence of the Hg/Cd ratio, x, in Hg1-xCdxTe [60].

10.2.2 V2VI3-series 3D topological insulators Substantial research on chalcogonide topological insulator synthesis has been dedicated to understanding and controlling the growth to achieve suitable bulk electronic characteristics and to realize desirable surface-state-dominated electronic behaviors in these systems [24]. Synthesis of Bi2Se3, both by bulk and thin film growth methods, often results in materials with strong n-type behavior arising from the high presence of selenium vacancies, which have a low energy of formation of approximately 500 meV and acts as a doubly positively charged vacancy contributing two conduction electrons to the system. Bi2Te3 tends to have weaker n-type behavior than Bi2Se3 due to a relatively lower concentration of tellurium vacancies, which have an energy of formation of approximately 600 meV, and a large concen0 tration of singly ionized BiTe antisite defects arising from the similar cation and anion electronegatives in Bi2Te3 that provide a single hole per antisite, somewhat offsetting the electron doping from Te vacancies [61]. Antisite defects are yet more pronounced in Sb2Te3. The highly similar electronegativities between Sb and Te lead to an antisite energy of formation of only 350 meV, resulting in a very high 0 concentration of SbTe antisite defects — so high, in fact, that the synthesis of ntype Sb2Te3 has never been reported. In addition to growth optimization, researchers have sought to produce exotic topological materials, such as magnetic topological insulators, which host the quantum anomalous Hall state described in Section 10.1.3, and topological superconductors formed by imparting superconductivity on the topological edge or surface states by means of the superconducting proximity effect or by growing superconducting doped topological insulators.

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315

10.2.2.1 Synthesis of V2VI3-series 3D topological insulator nanostructures Synthesis of chalcogenide topological insulator nanostructures, such as nanoplatelets, nanowires, and nanoribbons, with controllable thickness has been demonstrated for several wet chemical synthesis and chemical vapor transport methods. Such methods are particularly useful for producing samples that enable investigation of physical behaviors as a function of sample size and thickness, and for producing samples of high crystalline quality. Example methods include polyol wet chemical synthesis of Bi2Se3 nanoplatelets [62], high-yield solvothermal synthesis of Sb2Te3 nanoplatelets [63], wire-like Sb2Se3 by hydrothermal reactions [64], and Au-catalyzed vapor-liquid-solid (VLS) synthesis of Bi2Se3 nanowires in a horizontal tube furnace (Fig. 10.3) [65,66]. Modifications can be made to these recipes to produce certain doped and alloyed versions of these nanostructures, such as using an Fe-Au catalyst instead of Au to produce magnetically doped Bi2Se3 nanoribbons by VLS synthesis [67].

10.2.2.2 Bulk crystal growth of V2VI3-series 3D topological insulators Bismuth and antimony chalcogenide topological insulators can be grown by several bulk crystal synthesis techniques that produce single-crystal boules of material. The predominant bulk crystal growth method is the Bridgman furnace method. Bi2Se3 can be grown by a vertical Bridgman method with high-purity source materials of Bi:Se 5 2:3 mixed in an argon-filled ampoule and heated to 770  C for 15 hr,

Figure 10.3 Bi2Se3 nanoplatelets (A) and nanowires (B). Schematic of a Bi2Se3 nanowire field effect transistor. Fig. 10.3A adapted from J. Zhang, Z. Peng, A. Soni, Y. Zhao, Y. Xiong, B. Peng, et al., Raman spectroscopy of few-quintuple layer topological insulator, Nano Lett. 11 (2011) 24072414. Fig. 10.3BC adapted from D. Kong, J.C. Randel, H. Peng, J.J. Cha, S. Meister, K. Lai, et al., Topological insulator nanowires and nanoribbons, Nano Lett. 10 (2010) 329333.

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followed by a slow cooling of 1  C/hr until the temperature reaches 615  C, at which point the ampoule is quenched in room temperature water [68]. The growth of ternary alloy Bi2-xSbxSe3, optimized to achieve maximally suppressed bulk transport, was achieved by using the by sealing a mixture of pure elemental Bi, Sb, and Se with a nominal Bi:Sb:Se ratio of 52:7:130 in a quartz tube under Ar pressure; the ampoule was then heated to 740  C over a period of 14 hr, held at 770  C for 4 hr, cooled over a period of 50 hr to 550  C, then held at 550  C for 80 hr [69]. Incorporating other elements, such as Cu or Mn, during Bridgman synthesis has been demonstrated to produce, respectively, superconducting doped topological insulators [70] and magnetically doped topological insulators [71].

10.2.2.3 V2VI3-series 3D topological insulator thin films grown by molecular beam epitaxy Unlike bulk crystal growth under equilibrium conditions, such as in a Bridgman single crystal growth furnace described above, MBE growth of Bi2Se3, Bi2Te3, Sb2Te3, and related alloys allows the growth of monolithic crystalline structures with high crystalline perfection, tunable thickness, and arbitrary compositional profile, permitting careful doping, as well as the synthesis of V2VI3-series 3D TI-based heterostructures, made possible by the precise atomic-layer-by-atomic-layer growth that characterizes the MBE method. For example, by using MBE, it is possible to produce heterostructures that consist of Bi2Se3 or Bi2Te3 layers separated by layers ˚ [72], or Bi2Se3 or Bi2Te3 of ZnSe with arbitrary layer thickness as low as 10 A layers capped with a layer of Al or Se, or Bi2-xSbxSe3-yTey alloys with tuned Bi:Sb and Se:Te ratios and doped with, for example, magnetic materials. A generic MBE growth of V2VI3-series chalcogenide samples typically involves the something resembling the following sequences. A substrate is deoxidized at temperatures well above room temperature, e.g. 600  C for GaAs (100). The growth of Bi2Se3 (Bi2Te3) is initiated by the deposition of a sequence of Se-Bi-Se-Bi-Se (Te-Bi-Te-Bi-Te) atomic layers at room temperature, after which the substrate is gradually heated to 300  C to anneal the film to form the first quintuple layer (QL) of Bi2Se3 (Bi2Te3). MBE growth is performed under typical temperature, T, conditions of TSe (TTe) , Tsubstrate , TBi for the effusion cell temperatures. Notably, the V2VI3-series chalcogenide TI materials, due to the interlayer van der Waals bonding mechanism for these layered materials, demonstrate rapid strain relaxation at the interface with a substrate, permitting the layers to grow in highly parallel fashion regardless of substrate composition and substrate growth surface orientation, as shown for GaAs (111) [73] and GaAs (100) substrates [74], Si substrates [75], sapphire substrates, SrTiO3 substrates [76], InP substrates [77], etc. (Fig. 10.4). Several methods have been shown to control the defect chemistry in V2VI3series chalcogenides to tune the Fermi level to the bulk band gap (or, better yet, to the Dirac point of the surface states). These approaches fall primarily into the categories of compensation doping and optimizing the growth to eliminate the formation of vacancies and defects. Compensation doping takes advantage of the intrinsic doping tendencies arising from preferential formation of vacancies

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Figure 10.4 (A) Transmission electron microscopy cross sectional image and (B) x-ray diffraction pattern of Bi2Se3 grown by molecular beam epitaxy on GaAs (100) substrate. Adapted from X. Liu, D.J. Smith, J. Fan, Y.H. Zhang, H. Cao, Y.P. Chen, et al., Structural properties of Bi2Te3 and Bi2Se3 topological insulators grown by molecular beam epitaxy on GaAs(001) substrates, Appl. Phys. Lett. 99 (2011) 171903.

and antisites for the selected constituent anion and cation elements. For example, by growing a ternary alloy of Bi2-xSbxTe3, the n-type doping arising from naturally forming tellurium vacancies can be compensated by p-type doping 0 contributions from SbTe antisite defects. By tuning the Bi:Sb ratio, the position of the Fermi level can be correspondingly tuned. Brahlek et al. present a similar method of suppressing the bulk conducting states in Bi2Se3 by doping the alloy with Cu [78]. Optimizing the growth of V2VI3-series chalcogenide topological insulators demands the exploration of the boundless growth phase space, involving ascertainment of the optimal settings (typically specific to each individual growth system, no two ever being exactly alike) of growth parameters including growth temperature, growth chamber pressure, elemental beam flux ratios, choice of substrate, inclusion of a buffer layer prior to the main growth, and adequately sustaining low concentrations of unwanted reactants in the chamber [7981]. Wang et al. have demonstrated reliable synthesis of high-quality Bi2Se3 by beginning sample growth by depositing a layer of a trivially insulating (Bi1-xInx)2Se3 buffer layer [82]. Walsh et al. present a Bi2Se3 growth methodology that allows for the tuning of the Fermi level through native doping in the binary alloy, demonstrating their growth of highquality thin film Bi2Se3 with a mid-gap Fermi level by minimizing the Se vacancy concentration [83].

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Experimental investigations

Numerous experiments that have been devised to investigate the topological states hosted in topological insulators (TIs). These include spectroscopy measurements of the surface band structure, scanning tunneling microscopy measurements of the local density of states of the TI surface, and electrical transport measurements probing spin-polarization, time-reversal invariance, and quantization in topological edge and surface states. This section offers a brief overview of the experimental work that has revealed the novel electronic properties of topological states hosted in chalcogenide topological insulators.

10.3.1 Spectroscopy Several spectroscopy techniques have been applied to the study of topological insulators. These methods, which rely on the interaction between light and matter, reveal fundamental characteristics of the studied matter, including composition, physical structure, and electronic structure. Photoemission spectroscopy experiments are principally based on the photoelectric effect discovered by Hertz [84] and described by Einstein [85]. Incident photons with energy greater than the work function φ of the material will expel electrons from the topmost atomic layers of the sample surface, with the energy of the expelled electrons given by Ekf 5 hν 2 EB 2 φ, where EB is the binding energy of the electron, hν is the known energy of the incident photon, and Ekf is the measured kinetic energy of the emitted electron. Two key photoemission spectroscopy methods that have provided key insights into topological insulator properties are x-ray photoemission spectroscopy (XPS), which elucidates the elemental and chemical state composition of the material surface, and angle-resolved photoemission spectroscopy (ARPES), which measures the density of single particle excitations in the reciprocal space of a solid, allowing for simultaneous measurement of both energy and momentum of electrons in the solid. By directly relating the kinetic energy of emitted electrons to both EB and the crystal momentum ¯hk of the solid to resolve occupied states in energy-momentum space, ARPES provides a unique capability of imaging the electronic band structures of materials [86,87]. Characterization of the electronic properties of chalcogenide 3D TI by XPS reveals significant information on surface oxidation, a common effect in chalcogenide compounds [88,89], and a generally undesirable one as oxides such as BiOx and SeOx do not support the desirable topologically non-trivial properties of Bi2Se3 and other TIs. Atuchin et al. have shown success in fine-tuning bulk sample synthesis by using XPS to guide optimal synthesis techniques to produce chemically inert, non-oxidizing Bi2Se3 (0001) surfaces of excellent crystallographic quality [90]. In addition to surface compositional information, considerable analysis of the XPS spectra can also reveal compositional information of the bulk material. For example, by the measuring certain oxidation states, such as Mn oxidation states in Mndoped Bi2Se3, and comparing the differences in the amount of Se and Bi that

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precipitates out of the material as a function of Mn inclusion reveals the degree of disruption of the thermodynamic equilibrium of Bi2Se3 growth by the inclusion of Mn during synthesis [91]. The topological surface states that had been theoretically predicted to exist at the surface of 3D TIs are depicted in the surface electronic band diagrams as helical Dirac-like band gap-crossing states with linear dispersion. Conveniently, ARPES is an ideal tool for directly measuring the surface band structure of samples of 3D TI candidate materials. Xia et al. compared first-principles calculations of the surface band structure of Bi2Se3 to ARPES measurements, matching the observation of a single Dirac cone of the surface state to that predicted as a characteristic signature of a topological insulator [33]. The position of the Dirac point of the Bi2Te3 surface state, which had been predicted by first-principles calculations to exist below the top of the bulk valence band, was confirmed by ARPES measurement to be about 0.13 eV below the top of the bulk valence band [39]. A study by Hsieh et al. showed that by combining spin-imaging with ARPES with a double Mott detector set-up [92], the spin helicity of Bi2Se3 and Bi2Te3 can be directly resolved in the band diagram by ARPES [38]. Wang et al. performed similar measurement to observe the helical spin texture in Bi2Se3 by using a combination of ARPES and magnetic circular dichroism (Fig. 10.5) [93]. In addition to probing the physics of 3D TIs as revealed in the surface band structure, is a powerful tool for assessing material quality. As discussed in the Synthesis section, the synthesis of nominally stoichiometric Bi2Se3, Bi2Te3, and Sb2Te3 is challenged by thermodynamic conditions during growth that lead to varying degrees of n-doping by the formation of Se and Te vacancies and p-doping by the formation of antisite defects. Ideally, the Fermi level for these binary alloy materials with stoichiometric compositions would be positioned within the band gap. The position of the Fermi level as measured by ARPES informs researchers the extent to which they must work to overcome undesirable n- and p-type doping to achieve the synthesis of 3D TI materials of desirable quality. Raman spectroscopy is used to identify the structural fingerprints of solids by probing the vibrational, rotational, and other resonant modes of a crystalline system as a response to the inelastic scattering of incident visible, near-infrared, and nearultraviolet monochromatic light [94]. This tool, in addition to providing structural characterization information, is useful technique for the investigation of phonons and electron-phonon interactions in systems with Dirac-like linear dispersion by means of such technique as double resonant Raman scattering, which has been used to measure the G and 2D modes in graphene [95]. This, in turn, offers a means of probing the coupling between charge carriers and Raman modes to monitor doping [96]. This technique has been employed to study how phonon properties, such as frequency and lifetime of the vibrational modes, in Bi2Se3 vary as a function of thickness from bulk to the atomically thin QL regime, revealing enhanced electronphonon coupling in the few QL regime [62]. Similar results were revealed by micro-Raman study of few-QL flakes of Bi2Se3, Bi2Te3, and Sb2Te3 exfoliated from bulk material [97]. Investigation of the Raman scattering response of Cu: Bi2Se3 exfoliated from bulk material measured in a quasi-backscattering geometry

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Figure 10.5 (A) ARPES experimental apparatus schematic. Emitted photoelectron energies are characterized by a finite acceptance angle in the angle-resolved electron energy analyzer, and parallel and perpendicular components of the momentum are determined from the polar (ϑ) and azimuthal (φ) emission angles. (B) A schematic diagram of the Bi2Se3 bulk threedimensional Brillouin zone and the two-dimensional Brillouin zone of the projected (111) surface. (CD) ARPES measurements of the Bi2Se3 (111) band dispersion, including the gapless surface bands, near the Γ point of the 2D Brillouin zone along M-Γ-M (C) and along K-Γ-K (D). (E) Time-of-flight-ARPES data for all momentum directions measured using right- and left-circularly polarized light showing the spin texture of the electronic surface bands. (F) A slice of the data in (E) showing the spin polarization of the surface states in the energy-momentum map of Bi2Se3 (111) near the Γ point of the 2D Brillouin zone along M-Γ-M. Fig. 10.5BD adapted from Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, et al., Observation of a large-gap topological-insulator class with a single Dirac cone on the surface, Nat. Phys. 5 (2009) 398402. Fig. 10.5EF adapted from Y.H. Wang, D. Hsieh, D. Pilon, L. Fu, D.R. Gardner, Y.S. Lee, et al., Observation of a warped helical spin texture in Bi2Se3 from circular dichroism angle-resolved photoemission spectroscopy, Phys. Rev. Lett. 107 (2011) 207602.

in various incident and scattered light polarization configurations reveals a strong temperature dependence of collision-dominated scattering of Dirac states at the Fermi level on bulk-valence states related to screening induced by thermally excited carriers [98].

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Other spectroscopy techniques that have been employed to study the unique characteristics of the topological surface states include terahertz spectroscopy and magneto-optical Kerr and Faraday spectroscopy. Terahertz spectroscopy using linearly polarized THz range electromagnetic radiation incident on a 3D TI of sufficiently good material quality with low crystalline disorder has been used to demonstrate a predicted Kerr rotation of the electromagnetic radiation’s polarization plane [99] that is a signature of a topological magnetoelectric effect unique to these materials [18]. The Kerr rotation of reflected linearly polarized electromagnetic radiation, as well the related Faraday rotation that occurs for linearly polarized radiation transmitted through a sample, are also predicted to be observable with techniques using low frequency electromagnetic radiation [100]. Terahertz spectroscopy has also been to reveal evidence of Dirac plasmons in Bi2Se3 micro-ribbon arrays [101].

10.3.2 Electrical transport Several key signatures of the non-trivial electronic properties of both HgTe QW 2D TIs and V2VI3-series 3D topological insulators can be probed by transport experiments. Ko¨nig et al. fabricated HgTe QW Hall bar devices and performed the transport experiment that showed the 2e2/h quantized conductance, one conductance quantum for the top and bottom edges of the quantum well, convincingly demonstrating the topological edge state transport of this 2D topological insulator [23]. For the V2VI3 3D TIs, while conventional magnetotransport measurements of resistivity and Hall mobility can be used to provide critical insight into material quality that drives improvements in synthesis techniques [81,102], distinguishing the transport of the topological surface states from that of the bulk bands is a major challenge. The synthesis of the materials leading to not quite stoichiometric composition, as discussed in the Synthesis section of this chapter, positions the Fermi level, not only away from the Dirac point, but often outside the bandgap. Furthermore, scattering events can occur during transport that cause conducting charges in the topological surface state to scatter into bulk states and vice versa. For this reason, probing the topological physics of the surface states of 3D TIs, requires synthesis of high-quality materials, clever adjustments to standard transport measurements, including fabricating device components that allow the adjustment of the position of the Fermi level by the application of a voltage to a gate terminal [103], and rigorous analysis of transport data. A notable property of the 2D surface states of the 3D TIs is the disallowance of direct backscattering by non-magnetic impurities due to time-reversal invariance. Consequently, the 2D topological surface states cannot be localized, even by strong disorder [104]. This leads to a contribution to the electrical conductivity in the form of weak antilocalization, a purely quantum mechanical phenomenon in which the destructive Aharonov-Bohm interference of two self-intersecting closed paths of scattered conducting charges reduces the probability of localized paths [105]. Results of separate studies by Chen et al. [106] and Checkelsky et al. [107] on Bi2Se3 and He et al. on Bi2Te3 [108] substantiate the predicted relationship between weak antilocalization and the topologically protected surface states (Fig. 10.6A-B).

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Figure 10.6 (AB) The contribution to conductivity due to weak localization for a 48 nm sample of Bi2Se3 (A) and a 10 nm sample of Bi2Se3 (B) grown on SrTiO3 with various voltages applied to a back gate, demonstrating very little gate-voltage dependence of the weak antilocalization signal. (C) The derivative of the Hall resistivity of a bulk-grown Bi2Te2Se cleaved single crystal sample with respect to the magnetic field as function of the perpendicular component of the magnetic field, B\ , at various angles of the magnetic field with respect to Bi2Te2Se crystalline c-axis. The 1=B\ periodicity of the oscillations indicates the existence of a well-defined two-dimensional Fermi surface. Fig. 10.6AB are adapted from J. Chen, H.J. Qin, J. Liu, T. Guan, F.M. Qu, G.H. Zhang, et al., Gate-voltage control of chemical potential and weak antilocalization in Bi2Se3, Phys. Rev. Lett. 105 (2010) 176602. Fig. 10.6C is adapted from Z. Ren, A.A. Taskin, S. Sasaki, K. Segawa, Y. Ando, Large bulk resistivity and surface quantum oscillations in the topological insulator Bi2Te2Se, Phys. Rev. B 82 (2010) 241306(R).

Another quantum transport phenomenon observed in 3D TIs is the Shubnikovde Haas (SdH) effect, which manifests as an oscillation in the conductivity at high magnetic fields associated with Landau level occupation and has a period of 1=jBj, where jBj is the magnitude of the applied magnetic field, from which the carrier concentration can be derived [109]. Additionally, analysis of the temperature dependence of the SdH oscillations reveals the effective mass of the high mobility charge carriers participating in the SdH effect [110]. Analytis et al. show that when the Fermi level lies within the Bi2Se3 bulk conduction band, the SdH oscillations are dominated by bulk transport, and the effective cyclotron mass derived from analysis of the SdH oscillations match well with the effective mass calculated from the parabolic fit to the band dispersion of the ARPESmeasured conduction band [111]. Qu et al. [112] and Ren et al. [113] show that when the Fermi level is positioned well within the bulk band gap for Bi2Te3 and Bi2Te2Se, respectively, the SdH oscillations come from the topological surface states (Fig. 10.6C).

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The spin-momentum locking of the topological edge and surface state carriers can be observed by measuring the spin-polarization of the transported charges, injected as an unpolarized bias current, at magnetic tunnel barrier surface contacts. A study by Brune et al. on HgTe QW 2D TIs, in which the HgTe QW sample was fabricated into a split-gate H-bar device, combined quantum spin Hall (topologically non-trivial) and metallic spin Hall (topologically-trivial) transport in a single device to show that the quantum spin Hall effect can be used as a spin current injector and detector for the metallic spin Hall effect, establishing the spin polarization of the helical edge states in HgTe QW 2D TIs purely by means of an electrical transport [114]. The spin-polarization of the 3D TI Bi2Se3 surface current was detected by Li et al. by using magnetic tunnel barrier surface contacts as spin detectors, where the voltage measured by the magnetic tunnel barrier surface contact depends on the orientation of the contact magnetization, showing a lower voltage when the orientations of the spin-polarization of the current and the magnetization of the magnetic tunnel barrier contact are aligned [115]. The spin-polarized 3D TI surface states have been shown to provide efficient spin-orbit induced torques on magnetic layers adjacent to the TI via the Rashba-Edelstein effect [116,117], exerting strong spin-transfer torques, even at room temperature, on adjacent ferromagnetic permalloy layers such as Ni81Fe19 [118] and Co40Fe40B20 [119]. (Fig. 10.7)

10.3.3 Exotic topological states 10.3.3.1 Quantum anomalous Hall effect The quantum anomalous Hall state occurs when the time-reversal symmetry of the topological surface state is broken by spontaneous magnetization induced by proximity effect from an interfacing ferromagnetic system or by magnetic dopantinduced magnetization in the topological insulator material, itself, hosting a topologically protected edge mode with a Chern number of one corresponding to a quantized Hall conductance of e2/h. The quantum anomalous Hall effect has been predicted for both magnetically doped HgTe [120] and for (Bi,Sb)2(Se,Te)3 systems [121] with induced magnetization, but has thus far been observed only convincingly in the latter. The challenge in observing the QAH effect in HgTe quantum wells arises largely due to the exchange field in Mn-doped HgTe quantum wells being insufficiently strong to produce ferromagnetic ordering, even at very low temperatures, preventing the spontaneous magnetization necessary to break the time reversal symmetry of the topological edge modes without a persistent external magnetic field to paramagnetically align the Mn spin moments. On the other hand, in magnetically-doped (Bi,Sb)2(Se,Te)3 systems, spontaneous magnetization is induced by a strong van Vleck mechanism [122]. Van Vleck paramagnetism produces considerable spin susceptibility without the need for itinerant charge carriers to mediate magnetic exchange, as is the case for the Ruderman-Kittel-KasuyaYosida (RKKY) exchange mechanism in conventional diluted magnetic semiconductors such GaMnAs [123]. This has the significant advantage of excluding additional conduction channels in the topological insulator bulk. In a high-quality

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Figure 10.7 Schematic (A) and top view (B) of the contact layout for a spin polarization detection method in Bi2Se3, with parallel rows of ferromagnetic Fe contacts (top row, red) and non-magnetic Ti/Au reference contacts (bottom row, yellow). The magnetic field dependence of the voltage measured at the ferromagnetic spin detector contact with the contact magnetization parallel to the topological surface state spin for bias currents of 12 mA (C) and 22 mA (D), revealing significantly lower detector voltages when the Fe contact magnetization is aligned with topological surface state spin. (E) Schematic diagram of the Bi2Se3/permalloy layer structure for measuring spin transfer torque produced by the spin-polarized topological surface states exerted on the magnetic moments of the permalloy, revealed by spin-torque ferromagnetic resonance experiment (F). Fig. 10.7AD are adapted from C.H. Li, O.M.J. van ‘t Erve, J.T. Robinson, Y. Liu, L. Li, B.T. Jonker, Electrical detection of charge-current-induced spin polarization due to spinmomentum locking in Bi2Se3, Nat. Nanotechnol. 9 (2014) 218224. Fig. 10.7EF are adapted from A.R. Mellnik, J.S. Lee, A. Richardella, J.L. Grab, P.J. Mintun, M.H. Fischer, et al., Spin-transfer torque generated by a topological insulator, Nature 511 (2014) 449451.

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Figure 10.8 The quantum anomalous Hall effect (QAHE). (A) The magnetic field dependence of the longitudinal resistivity ρxx and the Hall resistance ρyx of four-quintuplelayer of an MBE-grown (Bi0.29Sb0.71)1.89V0.11Te3 film on SrTiO3 substrate reveals, within one standard deviation the signature zero-field h/e2 quantized Hall resistance (B) and, within approximately two standard deviations, the expected zero resistance at zero magnetic field (C). Fig. 10.8AC adapted from C.Z. Chang, W. Zhao, D.Y. Kim, H. Zhang, B.A. Assaf, D. Heiman, et al., High-precision realization of robust quantum anomalous Hall state in a hard ferromagnetic topological insulator, Nat. Mater. 14 (2015) 473477.

magnetically doped (Bi,Sb)2(Se,Te)3 material system, when the Fermi level is tuned to the magnetic gap in the Dirac cone, this would limit current to flow through the quantum anomalous Hall chiral edge state. A robust QAH state has been observed in magnetically doped 3D TI alloys of several different compositions. Below the Curie temperature, with out-of-plane oriented magnetization and at zero magnetic field, the signature of the QAH effect is a Hall conductance σxy of precisely e2/h and zero longitudinal resistance. The QAH has been successfully observed in Cr-doped (Bi,Sb)2Te3 [41,124] and V-doped(Bi, Sb)2Te3, the latter of which was shown to support a Hall conductance of 0.9998 6 0.0006 e2/h and a zero-field longitudinal resistance of 0.00013 6 0.00007 h/e2 At 25 mK. (Fig. 10.8) [125]

10.3.3.2 Topological superconductors Direct proximity to an s-wave superconductor gives rise to a topological twodimensional chiral p-wave superconductor at the superconductor/topological insulator (SC/TI) interface when the Cooper pairs tunnel into the topological surface states and induce a superconducting energy gap in the Dirac cone [126]. Such experimentally accessible topological superconductors were proposed by Fu and Kane [127], and signatures of superconducting proximity effect have been observed in numerous SC/TI systems NbSe2/Bi2Se3 [128,129], NbSe2/Bi2Te3 [130], W/Bi2Se3 [131], In/Bi2Te3 [132], and others. Superconductivity has also been demonstrated in superconducting doped topological insulators such as Cu:Bi2Se3 [70,133].

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Topological superconductivity can be induced in topological surface states by the generation of supercurrent between two superconducting contacts at the edge of a 2D TI or the surface of a 3D TI via the Josephson effect [134]. The Josephson effect for a superconductor/2D TI (e.g. HgTe QW)/superconductor junction has been predicted to produce a fractional Josephson effect with a current-phase relationship with 4π periodicity, which is half the usual 2π periodicity in the nontopological Josephson effect [135]. The signature 4π periodicity of the Josephson supercurrent has been demonstrated in HgTe QW-based Josephson junction devices as missing odd-integer-index (n 5 1, 3, 5, etc.) Shapiro steps in the response of a Josephson junction to rf radiation [136]. This has also been observed in Bi2Se3based Josephson junctions as a missing n 5 1 Shapiro step. (Fig. 10.9) [137] Topological superconductivity shares with conventional superconductivity several essential signatures, the most straightforward of which is zero electrical resistance below the Curie temperature, TC. Furthermore, a superconducting gap is measurable as a pronounced dip in the density of states, which can be measured locally by scanning tunneling spectroscopy. Unfortunately, the direct measurement of evidence of the formation of the exotic superconducting condensates of the topological surface states is confounded by the fact that the bulk states of the material system are also superconducting. While superconductivity and topological order can be independently observed in a single material system [70,129], the unique signature of topological superconductivity is the existence of topologically protected zero-energy quasiparticle states called Majorana modes that exist at the physical boundary of the topological superconductor [127].

Figure 10.9 (A) Schematic of Josephson junction device with Nb contacts patterned on HgTe mesa stripes grown on CdTe substrate. (B) Shapiro steps are observable in the I-V curve for a Josephson junction system in the presence radio frequency radiation, shown here for three different frequencies measured at TC800 mK. The first Shapiro step is reduced for f 5 5.3 GHz and is fully suppressed for f 5 2.7 GHz. Fig. 10.9AB are adapted from J. Wiedenmann, E. Bocquillon, R.S. Deacon, S. Hartinger, O. Herrmann, T.M. Klapwijk, et al., 4π-periodic Josephson supercurrent in HgTe-based topological Josephson junctions, Nat. Commun. 7 (2016) 10303.

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10.3.3.3 Majorana fermions The zero energy Majorana modes exist at the physical boundaries of topological superconductors as either bound Majorana zero modes or as chiral Majorana modes [12]. The Majorana fermion can be imagined as a half of the ordinary Dirac fermion of the topological edge (surface) mode of the 2D TI (3D TI). Moreover, due to the particle-hole redundancy of these particles, they carry no charge and are their own antiparticle [138]. Majorana modes are always created in pairs, that when physically separated, define a degenerate two-level system whose overall quantum state is stored nonlocally, a property that motivates research to explore using Majorana modes for quantum information processing [139]. A pair of Majorana zero modes appear at the two ends of a 1D topological superconducting wire of finite length, which is formed by inducing superconductivity in the edge modes of a 2D TI such as an HgTe QW. The signature of the Majorana zero mode is a zero-bias differential conductance peak indicating a bound Andreev state within the superconducting gap, actual measured differential conductance of which will depend on the conditions of the experiment [140]. The signature zerobias differential conductance peak has been observed in InSb nanowires contacted by a superconducting NbTiN electrode, supporting the hypothesized existence of Majorana zero modes in this system [141]. It should be noted, however, that these differential conductance peaks also appear for topologically trivial Andreev bound states, and it can be difficult to distinguish topologically nontrivial Majorana bound states from topologically trivial Andreev bound states in a differential conductance measurement [142]. While the Majorana zero mode signature zero-bias peak has not yet been observed in HgTe QW-based systems, the 4π periodic supercurrent in HgTe QW-based topological Josephson junctions, which has been theoretically shown to be a signature of the existence of topological gapless Andreev bound states [135], has been observed in these systems [143]. Propagating chiral Majorana modes exist either in the cores of superconducting vortices [144] or along the physical edge of a quantum anomalous Hall system with induced superconductivity [43,145], observed in a quantum anomalous Hall insulator/ topological superconductor Hall bar device fabricated from a (Cr0.12Bi0.26Sb0.62)2Te3 grown on GaAs (111)B by molecular beam epitaxy with a Nb superconductor bar deposited across the center of the Hall bar [146]. The unique signature of the chiral Majorana mode, provided the two Majorana fermions are sufficiently well separated such that the dephasing length is greater than the superconducting coherence length, is a half-integer Hall conductance plateau of 1/2e2/h. (Fig. 10.10) [147]

10.4

Summary and outlook

Research on chalcogenide topological insulators has, thus far, established the basic properties of these materials systems and made measurable progress in the developing good synthesis methods and measurement techniques that probe the unique physics of topological insulators. Substantial effort is yet required, however, to

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Figure 10.10 Electrical transport measurement of chiral Majorana fermion signature. (A) Schematic of a topological superconducting device consisting of a quantum anomalous Hall insulator (QAHI), a 6 nm-thick (Cr0.12Bi0.26Sb0.62)2Te3 thin film grown on a GaAs(111) substrate by molecular beam epitaxy, and a Nb superconductor bar. A current is applied along the QAHI and the four terminal longitudinal conductance σ12 is measured from the potential drop across terminals 1 and 2. (B) σ12 as a function of magnetic field shows the half-integer e2/2 h conductance plateaus signifying the existence of single chiral Majorana edge modes. Fig. 10.10AB are adapted from Q.L. He, L. Pan, A.L. Stern, E.C. Burks, X. Che, G. Yin, et al., Chiral Majorana fermion modes in a quantum anomalous Hall insulatorsuperconductor structure, Science 357 (2017) 294299.

realize the full potential of these materials in terms of materials quality and applications. As of the publication of this text, a number of promising devices based on chalcogenide topological insulator (TI) materials are in the development pipeline. Several devices fall in the category of spintronics devices, making use of the topologically protected spin texture of the surface states to perform such functions as conduct coherent spin information along TI interconnects or control spin moments in thin magnetic films by means of the spin transfer torque (STT) effect described in Section 10.4 to control memory states. Other devices are topological quantum computing devices that utilize Majorana fermions — either bound modes or chiral modes, depending on the design of the system — as the building blocks for topological qubits.

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Going forward, researchers will continue to find a vast investigative space to study new and improved synthesis techniques to grow HgTe QW structures and (Bi,Sb)2(Se,Te)3-based materials, develop and utilize sensitive measurements to probe their rich physics, and innovate next generation sensors and electronic devices. In addition, other unique physical phenomena will emerge in topologically non-trivial systems produced from chalcogenide topological insulator-based alloys and heterostructures, requiring the exploration of a boundless parameter space of compositions, growth conditions, device designs, and field effects. Moreover, these materials offer an exciting materials platform to better understand topological aspects of physical systems. In conclusion, researchers in the intersecting fields of chemistry, materials science, physics, and engineering can look forward to many exciting advances and new directions in the study of chalcogenide topological insulators.

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[134] B.D. Josephson, Possible new effects in superconductive tunnelling, Phys. Lett. 1 (1962) 251253. [135] L. Fu, C.L. Kane, Josephson current and noise at a superconductor/quantum-spinHall-insulator/superconductor junction, Phys. Rev. B 79 (2009) 161408(R). [136] J. Wiedenmann, E. Bocquillon, R.S. Deacon, S. Hartinger, O. Herrmann, T.M. Klapwijk, et al., 4π-periodic Josephson supercurrent in HgTe-based topological Josephson junctions, Nat. Commun. 7 (2016) 10303. [137] K. Le Calvez, L. Veyrat, F. Gay, P. Plaindoux, C.B. Winkelmann, H. Courtois, et al., Joule overheating poisons the fractional ac Josephson effect in topological Josephson junctions, Commun. Phys. 2 (2019) 4. [138] E. Majorana, A symmetric theory of electrons and positrons, Nuovo Cimento 14 (1937) 171. [139] C. Nayak, S.H. Simon, A. Stern, M. Freedman, S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80 (2008) 10831159. [140] J.D. Sau, S. Tewari, R.M. Lutchyn, T.D. Stanescu, S. Das Sarma, Non-Abelian quantum order in spin-orbit-coupled semiconductors: search for topological Majorana particles in solid-state systems, Phys. Rev. B 82 (2010) 214509. [141] V. Mourik, k. Zuo, S.M. Frolov, S.R. Plissard, E.P.A.M. Bakkers, L.P. Kouwenhoven, Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices, Science 336 (2012) 10031007. [142] Spin-resolved Andreev levels and parity crossings in hybrid superconductorsemiconductor nanostructures, Nat. Nanotechnol. 9 (2014) 7984. [143] E. Bocquillon, R.S. Deacon, J. Wiedenmann, P. Leubner, T.M. Klapwijk, C. Bru¨ne, et al., Gapless Andreev bound states in the quantum spin Hall insulator HgTe, Nat. Nanotechnol. 12 (2017) 137143. [144] N. Read, D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B 61 (2000) 10267. [145] X.-L. Qi, T.L. Hughes, S.-C. Zhang, Chiral topological superconductor from the quantum Hall state, Phys. Rev. B 82 (2010) 184516. [146] Q.L. He, L. Pan, A.L. Stern, E.C. Burks, X. Che, G. Yin, et al., Chiral Majorana fermion modes in a quantum anomalous Hall insulatorsuperconductor structure, Science 357 (2017) 294299. [147] B. Lian, J. Wang, X.-Q. Sun, A. Vaezi, S.-C. Zhang, Quantum phase transition of chiral Majorana fermions in the presence of disorder, Phys. Rev. B 97 (2018) 125408.

Further reading K.T. Law, P.A. Lee, T.K. Ng, Majorana fermion induced resonant Andreev reflection, Phys. Rev. Lett. 103 (2009) 237001. N. Levy, T. Zhang, J. Ha, F. Sharifi, A.A. Talin, Y. Kuk, et al., Experimental evidence for swave pairing symmetry in superconducting CuxBi2Se3 single crystals using a scanning tunneling microscope, Phys. Rev. Lett. 110 (2013) 117001.

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A. Olin, B. Nolang, E.G. Osadchii, L.O. Ohman, E. Rosen, Bismuth compounds, Chemical Thermodynamics of Selenium, Elsevier, Amsterdam, 2005, pp. 196203. R. Nitsche, H.U. Bo¨lsterli, M. Lichtensteiger, Crystal growth by chemical transport reactions—I: Binary, ternary, and mixed-crystal chalcogenides, J. Phys. Chem. Solids 21 (1961) 199205.

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Meng An1, Han Meng2, Tengfei Luo3 and Nuo Yang2 1 College of Mechanical and Electrical Engineering, Shaanxi University of Science and Technology, Xi’an, China, 2State Key Laboratory of Coal Combustion and Nano Interface Center for Energy, School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan, China, 3Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, United States

11.1

Introduction

11.1.1 Basic theory of heat conduction 11.1.1.1 Phonons Phonons are the quantized energy levels of lattice vibrations, in analogy with the photo, which is the quantum of the electromagnetic wave [1]. Heat conduction processes involve phonons of all allowable energies, as determined by the phonon dispersion relation of each material. There are typically three acoustic branches including two transverse acoustic branches and one longitudinal acoustic branch, and 3(N-1) optical branches, where N is the number of atoms at each lattice point.

11.1.1.2 Phonon dispersion Phonon dispersion is the relationship ωp(k) between the lattice vibration frequency ωp and a phonon wave vector k. Phonon dispersion relation can be determined by the inelastic scattering of neutrons with emission or absorption of phonons. The group velocity is calculated by vp ðkÞ 5

@ωðkÞ @k

The group velocity is the velocity of the propagation of the phonon wave packet, or the velocity of the propagation of energy in the medium.

Chalcogenide. DOI: https://doi.org/10.1016/B978-0-08-102687-8.00008-7 © 2020 Elsevier Ltd. All rights reserved.

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11.1.1.3 Gru¨neisen parameter The Gru¨neisen parameter describes the overall effect of the volume change of a crystal on vibration properties, which is expressed as γG 5

V ðdP=dTÞV CV

Where V is the volume of a crystal, Cv is the heat capacity at the constant volume, and (dP/dT) is the pressure change due to temperature vibration at the constant volume.

11.1.1.4 Phonon density of states The phonon density of states G(ω) measures the number
of phonon modes of a selected frequency ω(k, p) in a given frequency interval ω 2 12 Δω; ω 1 12Δω . ÐN The phonon density of states is normalized 0 dω GðωÞ 5 1. The phonon density of states spreads from zero to the maximal phonon frequency existing in a given crystal. In simple crystals, due to a large mass difference of the constituents the vibrations could be separated to a low-frequency phonon band, caused mainly by oscillations of heavy atoms, and to a high-frequency band, occupied by light atoms, Thus such bands could be separated by a frequency gap. To describe the vibration of a specific atom μ moving a partial phonon density of states Gu, Ð N along i-direction, 1 (ω) is introduced. It is dω G ðωÞ 5 , where r is the number of degree of freeu;i i r 0 dom in the normalized to primitive unit cell.

11.1.1.5 Fourier Law The capability of a material to conduct heat energy can be described by thermal conductivity through a macroscopic expression [24], i.e. Fourier’s law, j 5 2 κrT Where j is the local heat flux and represents the amount of heat energy that flows through a unit area per unit time, and rT is the temperature gradient along the heat flux direction. It is widely accepted in non-metals, in which heat is mainly carried by phonons.

11.1.1.6 Phonon Boltzmann transport equation (BTE) In nonmetal crystalline materials, the superposition of phonon waves leads to wave pockets carrying energy at the group velocity. These wave pockets can be treated as quasi-particles, and phonon transport in a crystal is similar to that of gas molecules inside a container. Thus, the phonon gas model is generally used to describe the lattice thermal transport: phonons is treated as a quasi-particles careering a certain

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amount of energy and transporting in a certain direction with group velocity v. Under this framework, the transport of phonons can be described by the phonon Boltzamnn transport equation (BTE): [1,5,6]   @fλ dfλ 1 vg;λUrfλ 5 (11.1) @t dt scattering where f is the probability distribution function for a phonon mode labeled by λ, vg,λ is the phonon group velocity, and the right-hand side of Eq. (11.1) corresponds to the changes affected in the distribution function due to collisions which act to restore equilibrium. This scattering term that incorporates different scattering mechanisms. Under small perturbation, the nonequilibrium distribution function can be written as fλ 5 fλ0 1 f 0λ , where fλ0 is the equilibrium BoseEinstein distribution function and f 0λ is a temperature-independent small perturbation. If we further assume that the temperature gradient rT is small, the λterm rfλ can be linearized and written as rTUð@fλ0 =@TÞ. At steady state Eq. (11.1) becomes vg;λU rT

  fλ0 dfλ 5 @T dt scattering

(11.2)

Single-mode relaxation time approximation (SMRTA) is commonly used to solve this equation, in which a relaxation time τλ is assigned to each phonon mode, and the scattering term can be written as 

dfλ dt



 52 scattering

fλ 2 fλ0 τλ

 (11.3)

Different phonon scattering mechanisms, including phonon-phonon (p-p) scattering, phonon-impurity (p-i) scattering, and phonon-boundary (p-b) scattering etc., can be incorporated into the relaxation time through Matthiessen’s rule: 1 1 1 1 5 1 1 τλ τp2p τp2i τp2b

(11.4)

When applying the expression of heat current and Fourier’s law, the thermal conductivity can be evaluated as X καβ 5 c v v τ (11.5) λ λ λα λβ λ where cλ is the heat capacity (per unit volume) of mode λ and vλα is the group velocity of the phonon mode λ along the α direction.

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11.1.2 The structure characteristics of chalcogenides 11.1.2.1 The chemical composition of chalcogenides In this chapter, we review the thermal transport properties of chalcogenides. The thermal conductivity is strong related with the chemical composition of materials. For chalcogenides, Chalcogen, or the oxygen family, consist of the elements S, Se, and Te. The name is chalcos (ore)  gen (formation) [7]. Chalcogenides are a kind of compounds as shown in Fig. 11.1 including one or more chalcogen elements e.g. sulfur (S), selenium (Se) and tellurium (Te) (MaXb, M is an element of Group IV, Group III, Group VI or transition metal, and X: S, Se, Te). These chalcogenides contain an uncommonly wide range of crystal structures, which exhibit promising applications areas related to thermal properties due to an indirect- to direct-bandgap transfromation with the a reduction in the materials thickness from bulk to monolayer, such as thermoelectric materials [814], semiconducting materials [1517] and batteries [18,19]. Thus, the understanding of thermal properties of chalcogenides are of vital importance for solving the challenge of thermal management. More importantly, the emergency of graphene-like two-dimensional (2D) (such as MoS2, a transition metal layer sandwiched by two chalcogenide atomic layers [20,21]) and lower-dimensional chalcogenide materials provide perfect candidate materials to investigate the fundamental questions [22,23] of thermal conduction shown in the following: 1. Are the thermal conductivity independent of the characteristic size (dimension effect or size effect)? 2. In practical applications, these semiconductor devices based on chalcogenides are placed on substrate. How do the substrates affect thermal conductivity of chalcogenides? 3. In addition, imperfections such as defects, vacancies, and grain boundaries inevitably occur during sample preparations. How do these imperfections and mechanical strains modulate the lattice thermal conductivity? 4. The 2D chalcogenides can be functionalized and intercalated. How can the thermal conductivity of chalcogenides be manipulated by functionalization and intercalation?. Slack formula [24,25] suggested that the thermal conductivity usually is determined by four factors, including (1) average atomic mass, (2) interatomic bonding, (3) crystal structure and (4) size of anharmonicity. The first three factors determines from harmonic properties. This expression can estimate thermal conductivity κs 5 A

M n1=3 δθ3 Tγ 2

Figure 11.1 The chemical composition of chalcogenides.

(11.6)

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where A is the numerical coefficient that equals 3.13106, M is the average mass of the basis atom, n is the number of atoms in the primitive unit cell, T is temperature and γ is the Gru¨neisen parameter. The average volume per atom is denoted by δ3 and θ is the Debye temperature, defined as ¯hωD/kB, in which ωD denotes the maximum vibrational frequency of a given model in a crystal. For thermal conductivity in W/m-K, mass in amu, and δ in Angstroms. Based on the Slack derivation, the low Debye temperature and heavy mean atomic mass and complex crystal structure can result in low lattice thermal conductivity [26]. we take transition metal dichalcogenides (TMDs) as an example to analyze their thermal conductivity. Taking the general chemical formula of TMDs to be XY2, thermal conductivity is roughly proportional to the thermal average ,ω2 . of phonon frequency. Since in a diatomic linear chain the maximum frequency ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωD ~ M121 1 M221 (where M1 and M2 are the mass of the constituent species), increasing either mass should, if the stiffness of the new compound is equal to or smaller than that of the old compound, decrease the ωD and hence thermal conductivity.

11.2

Geometrical effect

Intensive experimental and theoretical efforts have also been directed toward the effect of domain size on thermal properties.

11.2.1 Dimensional effect Transitional metal chalcogenides are an important family of chalcogenides materials that have received significant interests in recent years as they have a promising potential for diverse applications ranging from use in electronics, sustainable photovoltaic to industrial lubricants. Due to their inherent 2D layered structure (taking MoS2 as an example shown in Fig. 11.2), these materials behave significant anisotropy of in-plane (i.e., basal plane) vs. out-of-plane (i.e. cross-plane) physical properties due to differences in the nature of the atomic interactions. Within a molecular layer (in-plane), these interactions are characterized by covalent bonding, while the individual layers (cross-plane) are held together by significantly weaker van der Waals forces. Experimentally, the measured in-plane thermal conductivity values in bulk natural MoS2 crystal are about 100 W/m-K [2729], while the cross-plane thermal conductivity value are more than one order of magnitude smaller, ranging from 2.0 6 0.3 W/m-K to 4.75 6 0.32 W/m-K. The Raman optothermal measurements indicated that the room temperature thermal conductivity of tantalum diselenide (2H-TaSe2) obtained via mechanical exfoliation of crystals grown by chemical vapor transport, is 16 W/m-K [30]. However, its thermal conductivity of 45-nm thick films is 9 W/m-K. The thermal conductivity of the exfoliated thin films of 2H-TaSe2 is dominated by phonon contributions and reduced substantially compared with the bulk value.

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Chalcogenide

Figure 11.2 Schematic illustration of the different dimensional MoS2 systems with the top view and side view: bulk MoS2, single-layer MoS2 sheet, and (8, 8) armchair MoS2 nanotube.

Lead chalcogenides, the widely studied thermoelectric (TE) materials, are one kind of the best TE semiconductors for power generation at intermediate temperatures [31]. Therefore, their lattice thermal conductivities have attracted numerous attention due to that is the only materials properties that can be manipulated independently to enhance thermoelectric efficiency. For PbX (PbX, X 5 S, Se, Te), their thermal conductivity are 1.66, 1.01 and 1.91 W/m-K for PbS, PbSe, and PbTe at room temperature, respectively [32]. Another first-principles calculations [33] successfully reproduced the phonon dispersion relations of PbSe and PbTe along the high symmetry lines of the experiments [34] shown in Fig. 11.3A. The calculated lattice thermal conductivity matches well with the experimental results above 400 K [35, 36] (Fig. 11. 3B). By comparison of the thermal conductivity PbSe and PbTe, the author also revealed that the optical phonons not only contribute to thermal transport around 20%, but they provide strong scattering channels for acoustic phonons. This physical insight can advance the development of reducing thermal conductivity. Recently, germanium chalcogenides (GeTe, GeSe, and GeS) from IV-VI family have been increasingly attracting researchers’ attention as promising candidates for the replacement lead-based thermoelectric materials due to the toxic nature of leadbased chalcogenides [37]. In order to gain a deep insight into the thermal conductivity and other related properties of GeTe, it is essential to understand more about the chemical bonding, crystal structure. GeTe possesses a thermal conductivity of 2.6 W/m-K at room temperature. Due to the requirement of low thermal conductivity in thermoelectric fields, large number strategies have been used to reduce thermal conductivity, such as twinning, presence of secondary phase and doping [38]. Jana et al. reported an ultralow lattice thermal conductivity 0.4 W/m-K in highquality crystalline ingots of InTe, in the temperature range of 300650 K. This is due to the presence of strongly anharmonic phonons originate from rattling

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Figure 11.3 (A-B) Phonon dispersion for PbSe and PbTe: red line-calculated results and black dots-experimental results [34]. (C-D) Temperature-dependent lattice thermal conductivity of PbSe and PbTe, red line: calculated results; black crosses: experimental data [35,36]. Reproduced with permisssion [33]. Copyright 2012, American Physical Society.

vibrations of In1 cations (along the z-axis) within the columnar ionic substructure, which couple with the heat-carrying acoustic modes and lead to an ultralow thermal conductivity [39]. When the dimension of chalcogenides decreases from bulk to single layer sheet (shown in Fig. 11.2), the thermal conductivity exhibits many interesting properties due to the increased phonon scattering originating from the large surface area-tovolume ratio. The thermal conductivity of bulk, bilayer, and single layer MoS2 are 75.37 6 2.16, 84.26 6 3.24, and 132.68 6 4.67 W/m-K [57]. The DFT-calculated lattice thermal conductivities of MoS2 sheet are about 54 W/m-K along zigzag and 57 W/m-K along armchair directions at 300 K, which suggests insignificant difference between the two directions. It is also found that the calculated thermal conductivity also slighted decreased with decreasing width [40]. More interestingly, folding effect significantly reduce thermal conductivity of over 60% relative to the flat single layer due to the increased contribution of anharmonic phonon scattering introduced by the folding. Meanwhile, the broken symmetry results in highly distorted bonds, driving the low energy-high group velocity modes to higher energies and reducing viable transverse modes along the folding axis [41].

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Chalcogenide

In addition to 2D sheet of chalcogenides, its low-dimensional structures i.e. nanoribbons and nanotube, have shown wide application potential in nanoelectronics, optoelectronic device and thermoelectric applications. In such applications, thermal conductivity directly affects the life span and performance of chalcogenide-based devices. The thermal conductivity of MoS2 nanoribbons can be modulated by several geometries parameters, such as chirality, width, length and the type of edge [42]. Though the first-principle calculations, the thermal conductivity of MoS2 nanoribbons shows a non-monotonic dependence on crystal chirality   within the chirality’s from 0 to 30 . The thermal conductivity of MoS2 nanoribbons reaches a local maximum thermal conductivity at 19.1 . Moreover, the thermal conductivity can be decreased by increasing the edge roughness due to the largely degraded longitudinal phonons [43]. For MoS2 nanoribbons, the width can modulate electronic transport. In contrast, thermal conductivity is not senstive with ribbon width. The experimental measurement and numerical simulation have revealed that the thermal conductivity of MoS2 nanoribbons decreases with weak dependence on the ribbon width and the type of edge [40, 43, 44] which are explained by the local heat flux analysis and phonon scattering mechanisms [44]. The similar trend has been demonstrated in the experimental measurement of confocal micro-Raman spectroscopy. The 2 μm and 1 μm ribbons exhibit thermal conductivity of 31.2 6 2.5 and 29.6 6 1.1 W/m-K, respectively. Both values were further decreased to 27.7 6 1.9 and 25.8 6 3.7 W/m-K for system length 500 nm and 250 nm [40]. The electron beam self-heating technique is applied to measure the thermal conductivity of MoS2 nanoribbon and its value is around 30 W/m-K [58]. The thermal conductivity of single-wall and multi-wall MoS2 nanotubes has been studied by mean of molecular simulations and experimental measurement [4547]. The thermal conductivity, 16 W/m-K for L 5 10 nm at room temperature of single-wall MoS2 nanotube is two orders of magnitude smaller than that of carbon nanotubes. Moreover, the chirality, temperature, length, diameter and strain dependences of thermal conductivity are also discussed. Interestingly, it is found that thermal conductivity of armchair nanotube slightly decreases within a small range of strain. In contrast, thermal conductivity of zigzag nanotube exhibits a significant decreasing trend with strain. Such chirality-dependent strain effect is identified and originates from the different sensitivity of phonon group velocity with strain [46]. In addition, thermal conductivity of the multi-walled MoS2 nanotubes at room temperature was estimated to be in the range of 4.8 6 0.1 to 11.2 6 0.2 W/m-K based on the measurement of the temperature-dependent Raman signals [45].

11.2.2 Length dependence Several theoretical calculations attempted to reveal the intrinsic thermal conductivity of both monolayer and few-layer MoS2 [4953]. In comparison, the experimental measurement is very rare and leaving the intrinsic thermal conductivity of MoS2 almost unclear. The experimental study of thermal transport in few-layer MoS2 prepared by chemical vapor deposition method has been reported by Sahoo et al.

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Figure 11.4 (A) The calculated thermal conductivity of MoS2 at 300 K as a function of sample size [62]. (B) 1/κ(L) as a function of 1/L, where the solid line represents a quadratic fit and the dashed line represents a linear fit to the MD data with L 5 200 to 400 nm [47]. Reproduced with permisssion [47]. Copyright 2014, American Physical Society. Reproduced with permisssion [62]. Copyright 2019, American Physical Society.

The thermal conductivity of an 11-layer sample was measured to be about 52 W/m-K at room temperature [54]. Lately, a more detailed study of temperature and laser-power dependent Raman characterization on monolayer MoS2 exfoliated from naturally occurring bulk materials yielded a thermal conductivity of 34.5 6 4 W/m-K at room temperature [55]. Jo et al [56]. utilized the micro-bridge method to measure the basal-plane thermal conductivity of MoS2 with 4-layer and 6-layer thickness across a wide temperature range. The results showed that thermal conductivity of monolayer and few-layer MoS2 are lower than that of bulk MoS2. In addition to the suspended sample, a relative higher thermal conductivity 62 W/ m-K was observed in supported monolayer MoS2 [57]. As to the thickness dependence for thermal conductivity of MoS2, a theoretical calculation has shown a decreasing trend from monolayer to three layers due to the smaller group velocity for different phonon modes and higher phonon scattering rate induced by the changes of the anharmonic force constant [58]. The measured room temperature thermal conductivity of MoS2 is around 30 W/m-K by means of electron beam selfbeam self-heating technique [59]. In addition to MoS2, other kinds of TMDs are also studied. For example, thermal conductivity of monolayer WS2 has been measured with the optothermal Raman technique, which is comparable to monolayer MoS2 in Yan’s work [54]. The thermal conductivity of a 45 nm-thick TaSe2 sample was measured as 9 W/m-K [30]. Moreover, an ultralow cross-plane thermal conductivity (0.05 W/m-K) was observed in disordered, layered WSe2 sheets by TDTR technique [60]. Thermal conductivity of 5 nm thickness polycrystalline MoS2 utilizing 2-laser Raman thermometry, is 0.73 6 0.25 W/m-K. Yan et al. [54] utilized frequency domain thermoreflectance to study the cross-plane thermal transport in mechanically exfoliated MoS2 samples supported on SiO2 and the substrate. It is observed a significant improvement in heat transport across monolayer MoS2 as compared to few layer

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Chalcogenide

MoS2. Thermal conductivity of monolayer MoS2 is 34.5 6 4 W/m-K at room temperature obtained from temperature-dependent Raman spectroscopy, which agrees well with that of the first-principles lattice dynamics simulations [55]. As for the cross-plane thermal conductivity of MoS2 film, thermal conductivity of MoS2 obtained from NEMD tends to a convergent value when the film thickness is beyond about 40 nm. The analysis of cross-plane phonon MFPs suggests that phonons with MFPs below 40 nm contribute 90% of the MoS2 cross-plane thermal conductivity at room temperature [61]. The thermal conductivity of polycrystalline MoS2 with 5 nm thick is investigated by means of the combined 2-laser Raman thermometry and theoretical method [62].

11.2.3 Single-layer sheet As a family of novel two-dimensional (2D) materials beyond graphene, monolayer TMDs and Si-based chalcogenides exhibits unique physical properties. For TMDs, there are two polymorphs for monolayer TMDs: 1T phase with D3d point group and 2H phase with D3h point group shown in Fig. 11. 5. Monolayer MoX2 (X 5 S, Se, Te) with 2H phase have extensively studied as a representive materials of TMDs. Therefore, in this chapter, we mainly discuss the thermal transport of 2H MoX2, denoted as MoX2. For monlayer Si-based chalcogenides, the structure of puckered SiX (X 5 S, Se, Te) is shown in Fig. 11. 5. The corresponding phonon dispersions are presented in Fig. 11. 5. For MoX2 and SiX, the anion (S, Se and Te), the same trend can be found that the maximum frequencies of acoustic as well as optical branches are shifted downwards due to the inverse relationship of frequency to overall mass, with the change of anion inducing the largest shift due to the heaviest mass of Te. Unlike the ultra-high thermal conductivity in graphene, the low thermal conductivity of TMDs have been reported in both theoretical and experimental studies. At room temperature, the thermal conductivity of single-layer MoS2 is about 108 W/m-K for a 10-μm-long sample obtained using the Boltzmann transport equation (BTE), with the third-order anharmonic force constants obtained from quantum-mechanical density functional theory (DFT) calculations [63]. With a similar method, Gu et al. found that the in-plane thermal conductivity of 10-10-μmlong samples in layered, naturally occurring MoS2 monotonically reduces from 138 to 98 W/m-K when the thickness increases from one to three layers. The BTE approach is widely used in predicting the thermal conductivity of materials, which has its limitation that it is assumed that the higher-order phonon-phonon interactions are unimportant and expensive computational cost. Its thermal conductivity is about 23.3 W/m, obtained by solving the nonequilibrium Green’s function (NEGF) [64], Compared to experimental value of 34.5 W/m [55]. The thermal conductivity of monolayer WSe2 is 3.935 W/m, one order of magnitude lower than that of MoS2. The is due to the ultralow Debye frequency and heavy atom mass in WSe2 [65]. Similar to the substrate effect on thermal conductivity of graphene, the flexural acoustic phonons are damped. Through optothermal Raman technique, the measured thermal conductivity of suspended monolayer MoS2 and MoSe2 are 84 and 59 W/m-K at room temperature, respectively, while

Figure 11.5 Some typical atomic structures phonon dispersion curve of 2D TMDs and Si-based chalcogenides sheet. They are MoS2, MoSe2, MoTe2, SiS, SiSe, SiTe, respectively.

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Chalcogenide

the thermal conductivity of that supported on SiO2 substrate decrease to 55 and 24 W/m-K [57]. Its experimental thermal conductivity of monolayer MoS2 is much lower. Liu et al. investigated thermal conductivity of monolayer MoS2 sheet and nanoribbon. In addition to low thermal conductivity, the length dependence is also quite different from that of carbon-based materials. The thermal conductivity of monolayer MoS2 is calculated to be as high as 116.8 W/m-K [44]. Phonon transport of monolayer WSe2 is found to have an ultralow thermal conductivity due to the ultralow Debye frequency and heavy atom mass. The room temperature thermal conductivity for a typical sample size of 1 m is 3.935 W/m-K, which is one order of magnitude lower than that of MoS2. In addition, it is also found that the ZA phonons have the dominant contribution to thermal conductivity, and the relative contribution is almost 80% at room temperature, which is remarkably higher than that for monolayer MoS2 [65]. As a member of the 2D chalcogenides, MoS2 has a unique sandwich structure and a natural thickness-dependent energy gap. This unique property makes MoS2 a promising candidate material for transistor as an alternative to graphene. They calculate the lattice thermal conductivity of monolayer MoS2 by an iterative solution of the phonon BTE with the help of first-principle force constants. The lattice thermal conductivity is reduced by about 10% with the introduction of isotopes. The diffusion-limited MFP of monolayer MoS2 is longer than 7 μm at 300 K [52]. Peierls-Boltzmann transport equation (PBTE) suggested that the thermal conductivity of single-layer MoS2 could even be higher than 70 W/m-K when the size of sample is larger than 1 μm [50]. Individual phonons thermalize independently under SMRTA, without collisions repopulating them. The simultaneous interaction of all phonon populations, however, can be important, and such collective behavior can lead to the emergence of composite excitation as the leading heat carriers. Such collective behavior is driven by the dominance of normal (that is, heat-flux conserving) phonon scattering events, which allow the phonon gas to conserve to a large extent its momentum before other resistive scattering mechanisms can dissipate away the heat. In thermal transport, the bonding strength, mass of the basis atoms and frequency gap plays an important role in determining their phonon dispersion relations, which in term determines the related group velocity and thermal properties [63]. In a smaller system, the difference between full iterative solution of PBTE and SMRTA methods with the systems smaller than 30 nm is negligible (less than 5%) due to the dominating role of phonon-boundary scattering over phonon-phonon scattering. However, SMRTA cannot distinguish the resistive Umklapp process and the momentum-conserving normal process, which does not directly provide the resistance to heat flow. The under-prediction of SMRTA becomes distinguishable when the scattering due to normal process is strong. The difference between SMRTA and the iterative solutions of PBTE for all single layer TMDs are larger than 10% when the sample size is 1 μm. Among the four monolayer 2H TMDs, WS2 has the highest thermal conductivity of 142 W/m-K at room temperature and then followed by MoS2 (103 W/m-K), MoSe2 (54 W/m-K), and WSe2 (53 W/m-K). The MoS2 sheet

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suspended on QUANTIFOIL exhibits thermal conductivity of 38.3 6 3.8 W/m-K, which is comparable to above reports about multilayer MoS2 sheet [54,56]. The thermal transport of MoS2/graphene heterostructure is studied. It is found that the thermal conductivity can be tuned by interlayer coupling, environment temperature, and interlayer overlap. Interestingly, the highest thermal conductivity at room temperature is achieved as more than 5 times of that of monolayer MoS2 [66]. In addition to the multilayer heterostructure, the in-plane MoS2-graphene heterostructure is widely investigated. The interfaces are connected via strong covalent bonds between Mo and C atoms, were energetically stable. Interestingly, the interfacial thermal conductance was high and comparable to those of covalently bonded graphene-metal interfaces. Each interfacial Mo-C bond served as an independent thermal channel, enabling modulation of the interfacial thermal conductance by controlling the Mo vacancy concentration at the interface [67].

11.2.4 Discussion on the overall trend from single-layer to bulk The thermal conductivity of layered 2D chalcogenide materials was reported either to be substantially lower than that of their bulk counterparts or to increase with sample thickness. This observation seems to support the argument that the classical size effect in conventional 3D materials, which assumes that boundary scattering reduces the thermal conductivity, also occur in layered 2D crystals (Table 11.1). Recent experiments showed a quite different trend for the layer thicknessdependent thermal conductivity of MoS2 by different research groups, due to differences in sample quality and experimental conditions, thermal conductivity of MoS2 increases with the number of layers (Fig. 11.6). A recent first-principles-based PBTE study [58] showed that basal-plane thermal conductivity of 10-μm-long Table 11.1 Thermal conductivity of single-layer MoS2. Experimental simulations

TC (W/m-K)

Method

Layer number

Cited work

Exp.

34.5

Raman spectroscopy

Single-layer

Exp.

52

Raman spectroscopy

Few layer

Exp.

85100

Pump-probe

Bulk

Theory

26.2

Klemens’ formula

Single-layer

Simulation

100

PBTEs

Single-layer

ACS Nano 8, 986 (2014). J. Phys. Chem. C 117, 9042 (2013) J. Appl. Phys. 116, 233107 (2014). Appl. Phys. Lett. 105, 103902 (2014). Appl. Phys. Lett. 105, 131903 (2014).

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Figure 11.6 Thermal conductivity as a function of the number of layers for MoS2 at room temperature and zero pressure. The source of reference data: Gu [58]; Fan [48]; Luo [87]; Bae [68]; Yarali [88]; Yan [55]; Li [40]; Zhang [69]; Jo [56]; Aiyiti [70]; Sahoo [54]; Liu [67]; Jiang [97]; Zhu [28].

samples reduces monotonically from 138 to 98 W/m-K for naturally occurring MoS2 when its thickness increases from one layer to three layers, and thermal conductivity of trilayer MoS2 approaches that of bulk MoS2. The reduction is attributed to both the change of phonon dispersion and thickness-induced anharmonicity. Phonon scattering for ZA mode in bilayer MoS2 is found to be substantially larger than that of monolayer MoS2, which is attributed to the fact that of single-layer MoS2, which is attributed to the fact the additional layer breaks the mirror symmetry. The measured thermal conductivity of MoS2 are presented, but the experimental results show no clear thickness dependence.

11.3

Extrinsic thermal conductivity of chalcogenide

11.3.1 Strain effect Strain engineering as a mechanical means that can regulate geometric shapes and morphologies of materials and structures down to the nanoscale is particularly attractive because of their direct coordination with the phononic mechanism of thermal transport, where mechanical strain can be introduced by external load or force. Depending on the direction of external force, it could include either tensile or compressive strain. In fact, during the synthesis and processing of nanoscale materials and structures, it is difficult to ensure a strain-free environment and nanostructured materials will commonly have residual strain. On the other hand, there exists a series of techniques to regulate the strain of materials, such as bending the flexible

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substrate, elongating the substrate, and local thermal expansion of the substrate. It is known that strain has large effects on the electronic and optical properties of nanostructure materials. For example, with about 9% cross-plane compressive strain, MoS2 exhibits a semiconductor to metal transition [71,72], with an electrical conductivity enhancement from 0.03 to 18 S/m [73,74]. Following the formation of Peierls-Boltzamnn, Bhowmick and Shenoy [75] introduced a power-law scaling of thermal conductivity on phonon frequency  αγ ω (11.5) κ 5 AT 21 ω0 where ω and ω0 are the phonon peak frequencies with and without strain, A is the constant, α is a material-dependent positive constant, and γ is the Gru¨neisen parameter. From Raman spectroscopy measurement, a redshift was observed in phonon peaks in monolayer MoS2 under tensile strain [85]. Under the same experiment, the extracted Gru¨neisen parameter for MoS2 monolayer is about 1.06. The redshift in phonon peaks and positive Gru¨neisen parameter lead to the same trend with MD simulation results that thermal conductivity of monolayer MoS2 monotonically decreases with tensile strain [77]. Ding et. al found that for defect-free and defective MoS2, the reduction of thermal conductivity results from the softened phonon vibrations and the decreased of both group velocity and specific heat under a tensile strain and the increased phonon scatterings due to the out-of-plane deformation of monolayer structure under a compressive strain [58]. In adidition, when a moderate biaxial tensile strain 2-4% is applied, the thermal conductivity of single-layer MoS2 can be reduced by 10-20%. The effect of tensile strain is more obvious than that of compressive strain. When the system size of MoS2 sample varies, the reduction rate of thermal conductivity is size-dependent due to different dominant phonon scattering mechanisms [80]. In addtion, the thermal conductivity of other 2D chalcognides also exhibits a similar decreasing trend under a tensile strain. For example, The lattice thermal conductivity of the strained ZrS2 monolayer is much smaller than that of the unstrained system. Specifically, the thermal conductivity decreases from 3.29 W/m-K to 1.99 W/m-K, a 40% reduction when the strain of 6% is applied at 300 K. In such circumstance, the phonon dispersion of transverse and longitudinal acoustic (TA and LA) modes become softened, while the out-of-plane acoustic (ZA) mode is slightly stiffened. Based on the dispersion, the phonon softening leads to the reduced group velocity, which will be a source of the reduction in the lattice thermal conductivity compared with the unstrained ZrS2 monolayer, the relaxation times of the strained system are generally increased [78]. Shafique et al. studied the effect strain on lattice thermal conductivity of monolayer 2H-MoTe2 by solving BTE based on the first principles calculations. It is found that the monolayer 2H-MoTe2 is more sensitive to strain compared with that of ZrS2, and the lattice thermal conductivity is reduced by approximately 2.5 times by applying 8% biaxial tensile [88]. These diverse dependence of thermal conductivity on strain for different 2D chalcognides materials can be attributed to different strain dependece of

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heat capacity, group velocity and phonon scatterings rate with different crystal structures in accordance with the simple kinetic theory. Besides the in-plane strain effect on thermal conductivity of chalcogenides discussed above, the out-of-plane strain also can modulate thermal conductivity of layered chalcognides. Van der Waals layered chalcogenides are very sensitive to the strain due to the weak nature of van der Waals interactions. Under 9% crossplane compressive strain created by hydrostatic pressure in a diamond anvil cell shown in Fig. 11.7, it is observed the increasement of the cross-plane thermal conductivity in multilayer MoS2. This enhancement is due to the greatly strengthened interlayer interaction and heavily modified phonon dispersion along crossplane direction [81]. In experiment, the picosecond transient thermoflectance integrated with DAC device are used to study strain-tuned cross-plane thermal conductivity in bulk MoS2 over 9% cross-plane strain. Specifically, it is observed

Figure 11.7 Experimental setup, total, and electronic thermal conductivity under high pressure. (A) Schematic of thermal conductivity measurement with a diamond anvil cell integrated with a ps-TTR system. (B) Experimental data and fitting of ps-TTR measurements at two selected pressures, with 20% confidence interval shown. (C) Extracted cross-plane thermal conductivity (both lattice and electronic) as a function of pressure. The red curve is included only as a guide to the eye. Semiconducting and intermediate regions are labeled based on Ref [29]. (D) Electronic thermal conductivity of MoS2 against pressure, determined from measured electronic conductivity via the Wiedemann-Franz law. Three regions of the semiconductor to metal transition are labeled [81]. Reproduced with permisssion [81]. Copyright 2019, American Physical Society.

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roughly a 7 3 increase of cross-plane thermal conductivity, from 3.5 W/m-K at ambient pressure to about 25 W/m-K at 9% strain. First-principle and electrical conductivity measurements reveal that this drastic change originate from the substantially strengthened interlayer force and heavily modified phonon dispersions along the cross-plane direction. The group velocity of coherent longitudinal acoustic phonons, increase by a factor of 1.6 at 9% strain due to phonon stiffening, while their lifetimes decrease due to the phonon unbundling effect [81].

11.3.2 Effect of atomic disorder and defect Most of the phonon scattering mechanisms are prevailing toward a certain range of phonon frequencies, point defect scattering targets high frequency phonons, 2D interfacial scattering, grain boundaries, or fine nanoprecipitates dominantly scatter low frequency phonons. To achieve the minimum thermal conductivity of chalcogenides-based thermoelectric materials, many phonon scattering mechanisms are applied to suppress the entire scale hierarchical phonon transport in practical applications. For example, the lattice disorder and large number of grain boundary are present in fabricated thin film samples. The cross-plane thermal conductivity of bulk polycrystalline MoS2 is around 1.1-5.8 W/m-K [82] and the in-plane thermal conductivity is 85-110 W/m-K [27]. Generally, heat in dielectric materials mainly is transported by phonons. The thermal conductivity of layered two-dimensional TMDs alloys plays a critical role in the reliability and functionality of TMDs-enabled devices. In thermoelectric areas, the thermal conductivities of chalcogenides are manipulated based on different promising strategies, such as introduction of doping and defect. The strategy of alloying/doping engineering aim to introduce atomic disorder in the crystal lattices. Alloying/doping leads to the mass contrast between foreign atoms and regular lattice sites, which is generally considered as point defects, indeed enhancing the phonon scattering rate. Frequency dependence of the point defect phonon scattering relaxation time is given by the following equation [83]: τ21 PD 5

Vω4 Γ 5 Aω4 4πv3s

Where V is the average volume per atom, is the phonon frequency, Γ is the disorder scattering parameter which denotes Γ 5 ΓMF 1 ΓSF. The subscripts are mass fluctuation term and strain field term, respectively. Therefore, the higher the mass and size mismatched between the host and the foreign atom, the higher the phonon scattering and lower thermal conductivity. Aiyiti et. al. introduced the defects by mid oxgen plasma and observed the oxotic crystllineamorphous transition evidenced by typical characteristic of crystalline and amorphous phases of the samples shown in Fig. 11. 8A and 8B. In the typical crystalline phase, thermal conductivity increases due to the activation of more phonon modes and further decreases due to the increased Umklapp scattering

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Figure 11.8 Characterization of samples in crystalline and amorphous phase with TEM and thermal measurements. (A) HRTEM image and diffraction pattern of the intrinsic sample indicative of crystalline phase. (B) HRETEM image and diffraction pattern of the tailored sample indicative of amorphous phase. (C) the derived thermal conductivity of the samples as a function of temperature. (D) the measured thermal conductivity in the range of 20300 K after the oxygen plasma process. The blue dash lines denotes the theoretical low limit of amorphous thermal conductivity in MoS2. Reproduced with permission [84]. Copyright 2018, Royal society of Chemistry.

with temperature (Fig. 11. 8C). In contrast, thermal conductivity of MoS2 sample after oxygen plasma treatment is two orders of magnitude smaller than that of pristine samples (Fig. 11. 8D). Mo, as the heavier atom, contributes to the eigenvectors of low frequency phonon modes more than the S atom. Therefore, the largest thermal conductivity reduction when Mo isotopes are introduced is B 20%, which is larger than the 7%B 10% achieved using natural isotopes for the sample length [82]. In TMDs, the isotopes of metal atoms generally have a larger impact on the thermal conductivity than the chalcogen isotopes. This is due to the fact that Mo or W are heavier than S and thus contribute more to the eigenvectors of low-frequency acoustic phonon modes. For example, S isotopes can influence optical phonons from 8 to 14 THz, while Mo isotopes impact acoustic phonons in the frequency range of 27 THz that dominate thermal transport at room temperature [85].

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In phonon transport, the phonon-point defect scattering, such as isotope, vacancy, and substitution can suppress thermal conductivity. Wang et al [86]. studied the point effect (sulfur vacancies and oxygen substitution to sulfur) effect on thermal conductivity of MoS2 nanoribbon. They found that the suppression of thermal conductivity by vacancies is stronger than that by substitution. Wu et al [87]. use molecular dynamics simulations with first-principles force constants to study the isotope effect on thermal transport of single layer MoS2. It is found the isotope scattering in MoS2 strongly scatter phonons with intermediate frequency, in which compared with S isotopes, the Mo isotopes have stronger impact on thermal conductivity. Milad et al [88]. explored the effect the lattice defects on thermal conductivity of the suspended MoS2 monolayer grown by chemical vapor deposition (CVD). The measured room temperature thermal conductivity are 30 6 3.3 and 35.5 6 3 W/ m-K for two samples, which are more than two times smaller than that of their exfoliated counterpart. In addition, they also explore the effect of lattice vacancies and substitution tungsten (W) doping on the thermal transport of the suspended MoSe2 monolayers grown by chemical vapor deposition (CVD). The results suggest that a Se vacancy concentration of 4% results in thermal conductivity reduction up to 72% [89]. Using a semi-ab initio method, they have computed the thermal conductivity of MoS2, WS2, MoTe2. The results suggests that for a TMDs XY2 where one constituent species is fixed, more pronounced charged to the thermal conductivity will be changed by changing the masses of Y rather than X [90]. The thermal properties of 2D TMDs can be tailored through isotope engineering. Monolayer crystals of MoS2 were synthesized with isotopically pure Mo and Mo by chemical vapor deposition employing isotopically enriched molybdenum oxide precursors. The in-plane thermal conductivity of the MoS2 monolayers, measured using a non-destructive, optothermal Raman technique, is found to be enhanced by 50% compared with the MoS2 synthesized using mixed Mo isotopes from naturally occurring molybdenum oxide. The boost of thermal conductivity in isotopically pure MoS2 monolayers is attributed to the combined effects of reduced isotopic disorder and a reduction in defect-related scattering, consistent with observed stronger photoluminescence and longer exciton lifetime. The in-plane thermal conductivity of the suspended 100MoS2 and 50% 100MoS2 monolayers are 61.6 6 6.0 W/m-K and 52.8 6 2.4 W/m-K [91], respectively, showing a B50% and a B30% enhancement compared with the NatMoS2 (40.8 6 0.8 W/m-K) shown in Fig. 11.9. Both of these values may be underestimated due to the assumption that all phonon modes by Raman thermal measurement are in equilibrium. The in-plane thermal conductivity of 92MoS2, 96MoS2, 100MoS2 and NatMoS are calculated using self-consistent Boltzmann transport equation (BTE) by considering scattering mechanisms, such as boundary scatterings and isotopes scatterings. These conductivities of four samples do not show significant difference (B100 W/m-K at 1 μm). Qian et al. investigated the temperature-dependent anisotropic thermal conductivity of the phase-transition 2D TMDs alloys WSe2(1-x)Te2x in both the in-plane direction and the cross-plane direction using time-domain thermoreflectance measurement [92]. In addition, the thermal conductivity of lead selenide (PbSe) and

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Figure 11.9 (A) A sketch of different Mo isotoped-MoS2. (B) The sample length-dependent thermal conductivity of natural MoS2, 92MoS2, 96MoS2 and 100MoS2. Reproduced with permisssion [91]. Copyright 2019, American Chemical Society.

lead telluride (PbTe) and their alloys (PbTe1-xSex) are investigated by firstprinciples calculations [33]. Compared with the thermal conductivity of PbSe and PbTe, they found that the contribution of optical phonons is not negligible based on their direct contribution and strong scattering channels for acoustic phonons. Besides, the nanostructure of less than B10 nm is effective to reduce thermal conductivity of pure PbSe and PbTe. Importantly, the alloying is a relatively effective way to reduce the lattice thermal conductivity.

11.3.3 Anisotropy The thermal anisotropy of layered chalcogenides-based materials is fundamentally associated with their crystal structures. In most materials, the thermal conductivity along the cross-plane direction is much lower than the thermal conductivity within plane direction primarily because of the much weaker van der Waals interactions between layers compared to the stronger covalent bonds between atoms in plane. For layered van der Waals chalcogenides, heat transport is strongly anisotropic, featuring high thermal conductivity in plane and low conductivity across the layers. For example, the reported in-plane thermal conductivity of MoS2 ranges from 35 to 85 W/m-K [27,54,56], more than 10 3 higher than the cross-plane thermal conductivity (24.5 W/m-K) [27,88,89] shown in Fig. 11.10. For polycrystalline MoS2 thin films (50150 nm thick), the thermal conductivity was found to be approximately 1.5 W/m-K in-plane and 0.25 W/m-K out-of-plane [90], which demonstrates the importance of thermal boundary scattering as the limiting factor for thermal conductivity in nano-crystalline MoS2 thin films. They demonstrate that 2D nanoplates of vertically grown MoS2 can have anomalous thermal anisotropy, in which in-plane thermal conductivity 0.83 W/m-K at 300 K is B1 order of magnitude lower than out-of-plane thermal conductivity about 9.2 W/m-K at 300 K. Lattice dynamics analysis reveals that this anomalous thermal

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Figure 11.10 Measured in-plane and out-of-plane thermal conductivity (solid symbols) of MoS2, WS2, MoSe2, and WSe2 as a function of temperature, compared with literature values, both numerically and experimentally. The solid curves are calculated in-plane and out-ofplane thermal conductivity of natural, bulk MX2 from Ref. [84]. The dashed curved are outof-plane thermal conductivity of natural WS2 and WSe2 with boundary scattering length of 150 nm from Ref. [84]. The dashed-dotted curves are the calculated in-plane and out-ofplane thermal conductivity of natural bulk MoS2 from Ref. [57]. Measurement from literature are synthetic MoS2 by Pisoni et al [85]. natural MoS2 crystal by Liu et al. [27]. synthetic WS2 by Pisoni et al [86]. single crystal WSe2 by Chiritescu et al. [59]. and single crystal MoS2 and WSe2 by Murato et al. [87]. Reproduced with permisssion [88]. Copyright 2017, Wiley.

anisotropy can be attributed to the anisotropic phonon dispersion relations and the anisotropic phonon group velocity along different directions. The low in-plane thermal conductivity to the weak phonon coupling near the x-y plane interface [91]. The thermal conductivity anisotropy can be modulated by two orders of magnitude by mean of lithium intercalation and cross-plane strain. Specifically, the inplane and out-of-plane thermal conductivity can be tuned over one and two orders of magnitude, respectively. For LiMoS2, lithium intercalation leads to a seven-fold reduction of in-plane thermal conductivity, and two-fold reduction in cross-plane thermal conductivity. The two-fold reduction of PBTE calculation is consistent with the experimental measurement [28]. The effect of lithium intercalation on inplane phonon modes to reduce the frequency range of acoustic modes with significant group velocity, as well as to reduce the overall lifetime and mean free path. The two effect contribute to the consequence of major reduction from 93.7 to 12.2 W/m-K. However, for pristine MoS2, strain reduces the MFPs of ZA modes through two different mechanisms. In details [92]. Thermal conductivity of molybdenum disulfide can be modified by electrochemical intercalation. Distinct behavior for thin film with vertically aligned basal planes and natural bulk crystals with basal planes aligned parallel to the surface is observed by Gaohua, et al. The change of thermal conductivity correlates with the lithiation-induced structural and compositional disorder. The ratio of the in-plane to

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through-plane thermal conductivity of bulk crystal is enhanced by the disorder [28]. These results suggest that stacking disorder and mixture of phases is an effective mechanism to modify the anisotropic thermal conductivity of 2D materials. The thermal conductivity of LixMoS2 samples with different degrees of electrochemical interaction of lithium ions were measured by time-domain thermoreflectance (TDTR). We show that lithium ion intercalation has drastically different effects on thermal transport in these different forms of MoS2 due to the differences in crystalline orientation and initial structural disorder. Our most striking observation is that the thermal anisotropy ratio in bulk LixMoS2 crystals increases from 52 (x 5 0) to 110 (x 5 0.34) as a result of lithiation-induced stacking disorder and phase transitions. In addition, Na-intercalating in molybdenum disulfide leads to slightly structural modification and reduced thermal conductivity, which results from the enhanced anharmonicity of low-frequency phonons and the increased several quasi-local modes of vibration in the range of moderate frequency, which increases the scattering channel of phonon-phonon interactions [93]. The intercalates could influence dramatically the structural properties between layers, while slightly within layers, thus Na-intercalating could make more influence on the in-plane phonon transport than on that along interlayer (cross-plane) direction.

11.4

Fundamental insight into thermal transport

Understanding thermal transport properties of chalcogenides is critical to the development of better chalcogenides-based thermoelectric and phase-change materials. In particularly, the perspective from lattice dynamics and chemical bonding point have been extensively investigated. Based on previous studies, we will review several strategies of yielding anharmonicity including lone pair electron, resonant bonding and rattling mode.

11.4.1 Resonant bonding Resonant bonding [9497] has been appreciated as an important feature in some electron-deficient chalcogenides (i.e., compounds that contain one of the chalcogen elements S, Se, or Te, such as lead chalcogenides). Resonant bonding can be understood as resonance or hybridization between different electronic configurations: three valence p-electron alternate their occupancy of six available covalent bonds that exist between a given atom and its octahedral neigh-bonds that exist between a given atom and its octahedral neighbors. The established of resonant bonding can significantly delocalize the electrons and shrink the band gap, leading to soft optical phonons. Such behavior reduces thermal conductivity through two mechanisms: strong anharmonic scattering and a large scattering phase-space volume [94]. For PbTe of a typical lead chalcogenides, due to the fact that sp-hybridization is small and the s-band is lower than the p-band by 1.5 eV, it is considered only

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p-electrons for valence states and each atom has three valence electrons on average. Given PbTe’s octahedral structure and its three valence electrons per atom, the choice of bond occupation is not unique. This induces hybridization between all the possible choices for the three electrons forming the six bonds. This description of resonant bonding is based on IV-VI compounds, but the resonant bonding exists in even more complicated materials. In general, the unsaturated covalent bonding by p-electrons with rocksalt-like crystal structure can be regarded as a resonant bonding. The main feature of resonant bonding is the long-ranged interaction and energy density distribution due to the resonant bonding. For example, for the harmonic force constants of typical chalcogenides PbTe, it is obviously found that these compounds is the presence of long-ranged interactions along the ,100. direction of rocksalt structure. The long-ranged interactions of resonant bonding, different from monotonically decreasing of the long-ranged Coulomb interaction, are that fourthnearest neighbor interactions are stronger than second-or third-nearest neighbor interactions due to the long-ranged electronic polarizability. Moreover, compared with ground-state electron density distribution of NaCl, it is clearly observed that PbTe has largely delocalized electron density distribution due to the resonant bonding. In PbTe, the valence p-electrons form highly directed networks of resonant bonds that lead to long-range interatomic interactions along particular directions, which in turn causes soften transverse optical (TO) phonons and large lattice anharmonicity, both contributing to a low thermal conductivity, which was predicted and experimentally observed. By controlling resonant bonding in chalcogenides, the thermal transport properties can be modified for various applications in extreme conditions. The resonance character can be weaken by increasing hybridization and iconicity. Strong hybridization increases the number of covalent bond and iconicity tends to localize the electrons. Therefore, the material with weak hybridization and iconicity (some state-of-the-art thermoelectric materials such as PbTe, SnTe, PbSe and SnSe), which possess stronger resonant bonding [96]. In addition, resonant bonding can also be tuned by various means, including thermal excitations, changes in composition, and large hydrostatic-like pressure. Recently, by means of Synchrotron X-ray diffraction and density functional theory, Xu et al. found that at high pressure orthorhombic lattice of GeSe appears to become more symmetric and the Born effective charge has significantly increased, indicating that resonant bonding has been established. In contrast, the resonant bonding is partially weakened in PbSe at high pressure due to the discontinuity of chemical bonds along a certain lattice [95].

11.4.2 Lone pair electron Anharmonicity can be significantly amplified by the presence of stereochemically active lone pair electrons (LPEs). A lone electron pair effect refers to a pair of charge electrons that are not shared with another atom and is also called a nonbonding pair [96]. The lone pair is formally from the s-valence electron pair (s2), which tends to be more and more difficult to remove from the metal as we move

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down the respective group in the p-block elements. For example in group 13, the lone pair becomes increasingly stable as the element become heavier. Thus it is more stable configuration for the heaviest. The large stability of the s2 pair in the heavier elements of the main group is attributed to relativistic effects that contract the size of the s-orbital and bring its elements closer to the nucleus. The Cu-Sb-Se ternary system presents a unique opportunity to study the effect of LEPs on thermal conductivity [98]. In CuSbSe2 and Cu3SbSe3, however, Sb has the same coordination yet the average Se-Sb-Se angle is quite different. Wang and Libau studied this effect, and found that the change in X-Sb-X bond angle (where X denotes a chalcogen atom) correlates to the stereochemical activity of the LEP, or the delocalization of the Sb 5 s LEP away from the Sb nucleus (shown in Fig. 11.11). The morphology of LEP is directly related to lattice anharmonicity, and propensity of a given crystal to exhibit. The interaction of lone-pair electrons with

Figure 11.11 (A) Schematic representation of the local atomic environment of Sb in Cu3SbSe4, Cu3SbSe3, and CuSbSe2. Shaded lines represent Sb-Se bonds, dashed lines illustrate the approximate morphology of the Sb lone-pair 5 s electron oribial. (B) Temperature of the lattice thermal conductivity of Cu3SbSe4, Cu3SbSe3, and CuSbSe2. Reproduced with permisssion [97]. Copyright 2011, American Physical Society.

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neighboring atoms can produce minimum lattice thermal conductivity in group VA chalcogenide compounds. Both the morphology of the lone-pair electron orbital and the coordination environment of the group VA atom affect the extent to which the LEPs induce anharmonicity in the crystal lattice. The s2 lone pair behaves in unique ways depending on the local coordination environment. It can either stereochemically express itself by occupying its own distinct space around the metal atom or it can effectively from view. The relationship between anharmonicity and stereochemically active LPEs is that there exist a nonlinear repulsive electrostatic force between LPEs and neighboring bonds, which lowers the lattice symmetry and hinders lattice vibration [98,99]. A microscopic understanding of the lone-pair s2 electrons and the mechanisms responsible for enhanced anharmonicity is missing. Our focus is on the role of the stereochemically active lone-pair s2 electrons, can have different thermal transport properties.

11.4.3 Rattling modes The “Phonon-glass and electron-crystal” is an ideal strategy for high performance thermoelectric (TE) energy generator, where electron transport through the regular crystal lattice freely, while the phonon are seriously scattered. Rattling model refers to large amplitude vibrations of specific atoms or atom clusters in materials, in which phonons are seriously scattered. The typical ratting models are in clathrates and skutterudites with cage-like structure [100]. In these materials, guest atoms reside in cage-like structures. The large space and weak bonding make the guest atoms vibrate with larger displacement and different frequency compared with atoms in the host framework. All this kind of vibration yields additional scattering of phonons depending on the structure of the cage and the guest atoms induced. In term of phonon dispersion, the rattling model results in a downward shift of acoustic branch due to an avoided crossing between transverse optical branch and longitudinal acoustic branch. It is noteworthy that the coupling between the optical branch and the acoustic branch is different from that in resonant bonding. In resonant bonding, the coupling happened mainly due to the softening of the optical branch, where the acoustic branch in rattling model is also soften together with optical branch [94]. The potential energy of rattling atom shows a flatter vibration around the equilibrium position, which means that the rattling atoms vibrate with larger amplitude compared with the normal atoms when the potential energy is fixed. Moreover, the difference in the amplitude is amplified with increasing temperature [101]. It is noteworthy that the coupling between the optical branch and the acoustic branch is different from that in resonant bonding. In resonant bonding, the coupling happened mainly due to the softening of the optical branch, whereas the acoustic branch in ratting model is also soften together with optical branch. Another signal is the large atomic displacement parameter (ADP). ADP can be obtained from the inelastic neutron scattering (INS) and the refinement of rattling atom (weak bonded) and normal atoms (strong bonded) as a function of ADP in crystal lattice. The potential energy of rattling atom exhibits a flatter vibration around the equilibrium position, which is

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when the potential energy is fixed the ratting atoms vibrate with larger amplitude compared with the normal atoms. In these compounds, a guest atom rattles within oversized structural cages and scatters the heat-carrying acoustic phonons, thereby significantly lowering thermal conductivity. The exploration of new materials with intrinsically low along with a microscopic understanding of the underlying correlations among bonding, lattice dynamics, and phonon transport is fundamentally important towards designing promising thermoelectric materials. The presence of strongly anharmonic phonon originating from rattling vibration of In1 cations (along the z-axis) within the columnar ionic substructure, which couple with the heat-carrying acoustic modes and lead to an ultralow. By intentional p-type doping through creation of Indeficiencies, the power factor. In the nominal InTe sample, which is significantly higher than that of pristine InTe.

11.5

Conclusion and outlook

In this chapter, we summarized the recent development the emerging theory and many remarkable achievements and progress on phonon and thermal properties of chalcogenides materials including lead chalcogenides, transition metal chalcogenides, etc. In order to satisfy the diverse application of chalcogenides, several strategies of engineering thermal conductivity including isotopes, defects, strain and disorder are discussed. Finally, three fundamental physical mechanisms of thermal transport are concluded. Therefore, we hope this chapter can provide some inspirations to the researchers and will be beneficial for further progress in this field of thermal transport properties of chalcogenides.

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Index

Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively. A Absorption, 68 coefficient, 67 68 Lorentz theory of, 85 86 spectroscopy, 257 258 ADP. See Atomic displacement parameter (ADP) AFM. See Atomic force microscopy (AFM) AHE. See Anomalous Hall effect (AHE) Angle-resolved magneto-Raman scattering, 194 Angle-resolved photoemission spectroscopy (ARPES), 7 8, 255, 309 310, 318 319, 320f Anisotropy, 358 360, 359f Annealing, 312 313 Anomalous Hall effect (AHE), 196 197 Anti-bonding levels, 153 Antiferromagnetic coupling, 10 Antimony chalcogenide topological insulators, 315 316 Arizona State University (ASU), 59 ARPES. See Angle-resolved photoemission spectroscopy (ARPES) ASU. See Arizona State University (ASU) Asymmetric QD double-layer structures, 178 Atomic disorder and defect, effect of, 355 358 TEM and thermal measurements, 356f Atomic displacement parameter (ADP), 363 364 Atomic force microscopy (AFM), 247 248 Average band gap, 85 86 Average transition energy, 99 101 B Band alignments, 236 Band gap, 153

energy of II-V semiconductors, 67 Band structure, 155 157 engineering, 43 44 resonant level doping, 43 of superlattice with sinusoidal energy profile, 162 163, 163f “Barrier-modulation” model, 58 59 Bessel functions, 274 275 Bethe-Salpeter Equation (BSE), 125 126 Bhowmick, 353 Bi2Se3, 12 14, 18 Bi2Te3, 12 14 Bilayer (BL), 254 Bismuth chalcogenide topological insulators, 315 316 Bohr radius, effective, 156 157 Boltzmann transport equation (BTE), 348, 357 358 Bonding levels, 153 Bottom-up fabrication, 34 35 Bowing parameters for ZnMnTe and ZnMgTe, 74 75 BPVEs. See Bulk photovoltaic effects (BPVEs) Bridgman furnace method, 315 316 Brillouin function, 198 199 Brillouin zone (BZ), 70, 153, 155, 306 Brillouin-type function, 273 BSE. See Bethe-Salpeter Equation (BSE) BTE. See Boltzmann transport equation (BTE) Bulk crystal growth of V2VI3-series 3D topological insulators, 315 316 Bulk photovoltaic effects (BPVEs), 132 Bulk-boundary correspondence, 306 BZ. See Brillouin zone (BZ)

372

C Carrier spin relaxation, 169 Cauchy relation, 72 CBD. See Chemical bath deposition (CBD) CBM. See Conduction band minima (CBM) CBO. See Conduction band offset (CBO) CBs. See Conduction bands (CBs) (Cd,Mn)Se, 3 4 CdCr2S4, 9 CdMnSe/ZnSe, 21 CdMnTe, 17 QWs, 194 CdMnTe/ZnTe, 21 CdSe, 153 154 CdSe/ZnMnSe, 21 CdSe/ZnSe system, 21 22 CdTe, 153 155, 307 CdTe/ZnMnTe QD system, 21 22 CdTe/ZnTe system, 21 22 CE-SPS. See Chemical exfoliation and spark plasma sintering (CE-SPS) CF. See Composite fermion (CF) Chalcogenide multilayers, novel topological phases in, 211 220 domain walls and non-Abelian excitations, 212 217 quantum anomalous Hall effect in HgTe QWs, 219 quantum spin Hall effect in HgTe QWs, 217 219 topological phases in IV-VI materials, 220 wireless Majorana bound states, 217 Chalcogenide TI, 305 experimental investigations, 318 327 electrical transport, 321 323 exotic topological states, 323 327 spectroscopy, 318 321 mercury telluride quantum wells, 307 308 growth, 312 314 synthesis, 312 317 V2VI3-series 3D topological insulators, 309 312 Z2 topological insulator, 306 Chalcogenides, 1 beyond 2D, 19 22 in 3D form, 2 9 materials, 153 QWs, 189

Index

thermal conductivity, 352 360, 358f anisotropy, 358 360 effect of atomic disorder and defect, 355 358 strain effect, 352 355 thermal transport, 342, 360 364 categories, 342f chemical composition, 342 343 geometrical effect, 343 352 heat conduction theory, 339 341 structure characteristics, 342 343 topological insulator nanostructures, 315 two-dimensional chalcogenide structures, 9 19 Chalcogens, 153, 342 Chemical bath deposition (CBD), 57 58 Chemical exfoliation and spark plasma sintering (CE-SPS), 49 Chemical vapor deposition (CVD), 137, 235 236, 357 growth of 2D nanostructures, 137 140 growth of HgTe quantum wells, 313 314 Chemical vapor transport (CVT), 235 236 Chern insulator, 12 14 Chern number, 305 306 Chiral Majorana modes, 327 Cluster formation probability, 201 202 CMOS technology. See Complementarymetal-oxide-semiconductor technology (CMOS technology) CMR. See Colossal magnetoresistance (CMR) Co-based DMS alloys, 6 Colloidal atomic layer deposition, 279 Colloidal nanostructures, 285 286 Colossal magnetoresistance (CMR), 9 Commensurability oscillations, 206 207 Complementary-metal-oxide-semiconductor technology (CMOS technology), 235 Composite fermion (CF), 204 Composition modulated ZnSeTe sinusoidal superlattice, 161 166 band structure of superlattice with sinusoidal energy profile, 162 163, 163f growth of ZnSeTe superlattices with sinusoidal composition modulation, 163 165

Index

optical transitions in ZnSeTe sinusoidal superlattices, 165 166 Conduction band minima (CBM), 123 125, 255 Conduction band offset (CBO), 9 Conduction bands (CBs), 127 129, 155, 271 Conduction electron g-factor, 170 171 Confinement, 191 Consolidation method, 35 36 Conventional DMS nanocrystals, 292 Core/shell Zn1-xMnxTe/Zn1-xMgxTe nanowires, 19 Coulomb screening, 15 CP. See Critical points (CP) Cr-based DMS alloys, 6 7 Critical points (CP), 70 Crystal structure engineering, 44 49 band structure, 46f complex structure, 44 45 degree of orientation, 48 49 layered structure, 46 48 Peierls distortion structure, 45 46 thermoelectric transport properties, 47f Cu-Sb-Se ternary system, 362 Cu2X system, 40 41 Curie temperature (TC), 326 CVD. See Chemical vapor deposition (CVD) CVT. See Chemical vapor transport (CVT) D d-shell electrons, 104 DBRs. See Distributed Bragg reflectors (DBRs) De facto method, 313 Defect engineering, 38 43 approaches, 41 43 element deficiency, 40 41 normal doping, 38 39 point defect, 40 Dember effect, 136 Density function theory (DFT), 255 Density functional theory calculations (DFT calculations), 348 Density of states (DOS), 42 44 Destructive Aharonov-Bohm interference, 321 DETe. See Te diethyltelluride (DETe) DFT. See Density function theory (DFT)

373

DFT calculations. See Density functional theory calculations (DFT calculations) Diabatic Landau-Zener transitions, 207 Dielectric materials, 67 Dielectric medium, 68 Diffusion-doping approach, 283 Digital alloy, 191 192 Digital doping in nanoclusters, 292 294 Diluted magnetic II-VI-based semiconductors, 8 Diluted magnetic semiconductors (DMSs), 154, 174 175, 189, 272 multilayer design, 11 spin polarization enhancement in nonDMS and DMS coupled QDs, 178 181 II-VI quantum structures involving, 173 178 Dimensional effect, 343 346 MoS2 systems, 344f phonon dispersion for PbSe and PbTe, 345f Dimethylcadmium (DMCd), 313 314 Dimethyltelluride (DMTe), 313 314 Dirac cone, 192 Dirac point, 309 310, 321 Dirac states, 11 Direct-exciton, 156 157 Discrete LLs, 198 Disordered alloy, 191 192 Dispersion, 68 energy, 105 107 Lorentz theory of, 85 86 Distorted quasi-one-dimensional chain-like structure, 45 Distributed Bragg reflectors (DBRs), 67, 95 96 DLQD system. See Double layer QD system (DLQD system) DMCd. See Dimethylcadmium (DMCd) DMHg. See Hg precursors dimethylmercury (DMHg) DMSs. See Diluted magnetic semiconductors (DMSs) DMTe. See Dimethyltelluride (DMTe) Domain walls (DWs), 212 217 Doped magic-sized alloy nanoclusters, 290 292

374

Doped magnetic semiconductors, 3 4 Doped nanocrystal quantum dots, valenceband mixing in, 282 285 Doping, 271 digital, 292 294 normal, 38 39 QWs, 192 193 resonant level, 43 with transition metals, 271 272 DOS. See Density of states (DOS) Double layer QD system (DLQD system), 171, 178 Double quantum well (DQW), 154 Double resonant Raman scattering, 319 320 DQW. See Double quantum well (DQW) “Dual emitters”, 271 272 DWs. See Domain walls (DWs) Dysprosium stripes (Dy stripes), 207 209 E e-beam evaporator, 242 E2 peak for photo-energies, 70 Edge channel picture, 198 Edge states, 211, 218 219, 353 EDX. See Energy dispersive x-ray spectrum (EDX) EFE. See Energy filter effect (EFE) Effective refractive index, 67 68 Effective Rydberg, 156 157 Electrical transport, 321 323 Electromotive force (EMF), 32 Electron-hole correlation, 161 Electron-LO phonon interaction, 161 Electronic band gap, 253 energies, 125t band structure, 123 126, 124f, 126t and dispersion, 69 71 and optical properties, 126 131 angular-dependent spectra, 128f masses and mobility values, 130t polarized micro-photoluminescence spectra, 129f structure of 2D materials, 253 255 ARPES, 255 STM, 253 255 Element deficiency, 40 41 thermoelectric properties, 42f, 44f EMF. See Electromotive force (EMF)

Index

Empirical models, 71 72 Energy dispersive x-ray spectrum (EDX), 247 248 Energy filter effect (EFE), 36 Energy gap, determination of, 73 76 Epilayer, 153 154 Epitaxial crystal growth techniques, 161 162 Epitaxial graphene, 249 Epitaxial II-VI semiconductor quantum structures composition modulated ZnSeTe sinusoidal superlattice, 161 166 Landau level transitions and magnetopolaron effect, 158 161 magneto-optical properties of ZnSe and ZnTe epilayers, 155 158 spin polarization enhancement in nonDMS and DMS coupled QDs, 178 181 II-VI quantum structures involving DMSs, 173 178 II-VI-based zero-dimensional structures, 166 173 Epitaxially-formed chalcogenides, 9 11 epitaxial II1-xMnxVI magnetic semiconductor quantum structures, 11 magnetic properties, 9 10 tunable Dirac interface states in topological superlattices, 11 II VI quantum cascade emitters, 9 Europium chalcogenides, 3 4 EuS (ferromagnetic insulator), 18 Exciton, 70, 155 157, 275 276 magnetic polarons, 21 mapping of exciton localization in QDs, 174 178 transitions in absence of magnetic field, 157 158 Exotic topological materials, 314 Exotic topological states, 323 327 Majorana fermions, 327 QAHE, 323 325, 325f topological superconductors, 325 326 External magnetic field, 210 F f-sum-rule integral, 104 Fabry-Perot maxima and minima, 72 73

Index

Faraday rotations, 4 5 Fast Fourier transform (FFT), 245 247 Fe-based DMS alloys, 7 Fermi energy, 201f Ferromagnetic monolayer (FM monolayer), 15 Ferromagnetism, 14 FFT. See Fast Fourier transform (FFT) Filling factor, 198, 203 FM monolayer. See Ferromagnetic monolayer (FM monolayer) Fourier law, 340 IV-VI compounds, 7 8 IV-VI materials, topological phases in, 220 IV-VI QWs, magnetotransport in, 206 Fractional filling factors, 204 205 Fractional quantum Hall effect (FQHE), 15 17, 197, 204 205 Full width at half maximum (FWHM), 244 245 G (Ga, Mn)As, 3 4 Geometrical effect, 343 352 dimensional effect, 343 346 length dependence, 346 348 single-layer sheet, 348 351 trend from single-layer to bulk, 351 352 Germanium selenide (GeSe), 119 120, 122t, 123 125 Germanium sulphide (GeS), 119 120, 122t, 123 125 crystal, 127 129 GI-XRD. See Grazing incidence X-ray diffraction (GI-XRD) Giant magneto-optical response in Mn21-doped CdSe nanoribbons, 275 279 nanoribbon bundle including miller indices, 277f Giant Zeeman effect, 11, 17, 19 20, 272 Giant Zeeman splitting, 11, 17, 20f, 154, 174 175, 178, 280, 285, 290 Graphene, 15 Graphene-like chalcogenide 2D systems, 1 2 Grazing incidence X-ray diffraction (GIXRD), 244 245 Group-II-based chalcogenides, 17

375

Group-IV monochalcogenides, 119 121, 123, 129 131 crystal lattice and band structure, 120 123, 120f crystal structure parameters for, 124f double-well potential of, 123f electronic band structure, 123 126 fabrication, 137 140 imaginary part of dielectric function of, 134f injection current of, 133f monolayer, 127f nonlinear optical properties, 131 137 Gru¨neisen parameters, 339 340 H Hall conductance, 15 17 Hall mobility, 321 HE-SPS. See Hydrothermal exfoliation and SPS process (HE-SPS) Heat conduction theory, 339 341 Fourier law, 340 Gru¨neisen parameters, 339 340 phonon BTE, 340 341 phonons, 339 Heavy-hole (hh) hole state, 273 274 Heavy-hole (hh) valence bands, 70 Helical spin states, 12 14 Heterojunctions, 154 Heyd-Scuseria-Ernzerhof (HSE06) exchange-correction term, 123 125 Hg precursors dimethylmercury (DMHg), 313 314 High-resolution x-ray diffraction (HR-XRD), 242 Highly monodisperse II-VI nanocrystals, 271 Highly-ordered pyrolytic graphite (HOPG), 236, 249 HOPG. See Highly-ordered pyrolytic graphite (HOPG) HR-XRD. See High-resolution x-ray diffraction (HR-XRD) Hydrothermal exfoliation and SPS process (HE-SPS), 49 Hypothetical materials, 92 I In-plane Zeeman effect, 193 194 Indices of refraction, 76 83

376

Individual dopants in single nanocrystals quantum dots, 285 288 Inelastic neutron scattering (INS), 363 364 Infrared (IR), 57 Injection current, 132 133 Injection dc-current, 132 INS. See Inelastic neutron scattering (INS) Integer quantum Hall effect (IQHE), 197 Interband optical transitions, 168 Interface phenomena in chalcogenide structures, 15 19 magnetic proximity effects at interfaces, 18 19 2DEGs in chalcogenide multilayers, 15 17 Interfacial exchange coupling, 18 Inversion symmetry, 14 Ionicity, 105 107 IQHE. See Integer quantum Hall effect (IQHE) IR. See Infrared (IR) K k-dependent band polarization difference, 133 Ketteler-Helmholz relation, 72 Kramers-Kronig relations (K-K relations), 68 69, 71 72, 102 104 Kramers’ theorem, 306 L Landau levels (LLs), 197 198 splitting, 17 structure, 11 transitions, 158 161 Landau paradigm, 305 306 Landau quantization, 197 198 of electronic levels, 15 17 Landau subbands, 157 Landau-Ginsburg Hamiltonian, 121 Landauer-Bu¨ttiker formalism, 198, 213 214 Laser assisted deposition, 312 313 Laser Components DG (LCDG), 57 Laser desorption/ionization time-of-flight mass spectrometry (LDI-TOF MS), 292 294 Lattice constants, 96 98, 105 for binary zinc blende compounds, 75 of ZnTe, MnTe and MgTe, 72

Index

Layered chalcogenides, 20 2D, 1, 255 Layered heterostructures, 247 249 growth methods, 312 313 Layered selenide and telluride films growth and heterostructures, 242 249 layered heterostructures and SLs, 247 249 molybdenum selenide, 245 247 molybdenum telluride, 244 245 tin selenide, 242 244 Layered structure, 46 48 LCDG. See Laser Components DG (LCDG) LDI-TOF MS. See Laser desorption/ ionization time-of-flight mass spectrometry (LDI-TOF MS) Lead salt photoconductor, 60 61, 64 Lead salt photodetectors, 57 characterization, 60 64 fabrication, 59 lead salt testing circuit, 62f photograph of patterned and metallized PbSe film, 61f Lead selenide (PbSe), 7 8, 58, 357 358 Lead sulfide (PbS), 7 8, 58 Lead telluride (PbTe), 7 8, 357 358 Lewis acid, 276 Light-hole (lh) hole state, 273 274 Light-hole (lh) valence bands, 70 Liquid phase epitaxy (LPE), 312 313 LLs. See Landau levels (LLs) LO phonon energy. See Longitudinal optical phonon energy (LO phonon energy) Localized magnetic ions in DMSs, 174 Lone pair electrons (LPEs), 361 363 Longitudinal optical phonon energy (LO phonon energy), 288 Lorentz theory of absorption and dispersion, 85 86 Low-dimensional heterostructures, 190 193 Low-dimensional structures, 344 LPE. See Liquid phase epitaxy (LPE) LPEs. See Lone pair electrons (LPEs) M M1-type critical points, 70 M2-type critical points, 70 Magic sized nanoclusters, 289 294 Magic-sized clusters (MSCs), 289

Index

Magnetic field-dependent photoluminescence experiments, 276 278 impurities, 196 materials, 47 ordering, enhancement of, 18 polarons, 271 proximity effects at interfaces, 18 19 QD systems, 21 quantum ratchet effects, 210 211 semiconductors, 3 4 topological insulators, 314 Magnetic circular dichroism (MCD) spectroscopy, 276 278, 280, 289, 291 292 Magnetic tunnel junctions (MTJs), 217 Magnetically doped II-VI nanocrystals border between QDs and molecules digital doping in nanoclusters, 292 294 doped magic-sized alloy nanoclusters, 290 292 smallest doped semiconductors, 289 290 theoretical background, 272 275 2D colloidal nanocrystals, 275 282 valence band states, 275 zero-dimensional nanocrystals, 282 288 Magnetically doped nanoplatelets, 279 Magnetically doped semiconductors, 272 Magnetism, 14 Magneto-optical Kerr and Faraday spectroscopy, 321 Magneto-optical properties of ZnSe and ZnTe epilayers, 155 158 band structure and exciton, 155 157 exciton transitions in absence of magnetic field, 157 158 Magneto-polaron effect, 158 161 Magnetoresistance (MR), 191 192, 195 196 Magnetotransport in chalcogenide QWs, 195 206 Majorana fermions, 211 Majorana quasi-particles, 215 216 Majorana zero modes, 327 Manganese (Mn), 189, 193 Matthiessen’s rule, 341 MBE. See Molecular beam epitaxy (MBE)

377

MDCs. See Momentum distribution curves (MDCs) MDs. See Metal dichalcogenides (MDs) Melt-solidification (MS), 48 49 Mercury telluride quantum wells (HgTe quantum wells), 307 308 growth, 312 314 CVD growth, 313 314 molecular beam epitaxy growth, 313 V2VI3-series 3D topological insulators, 314 317 Mercury telluride quantum wells (HgTe QWs), 189 190, 192, 196 197 quantum anomalous Hall effect in, 219 quantum spin Hall effect in, 217 219 Mermin-Wagner theorem, 15 Metal chalcogenides, 31 Metal dichalcogenides (MDs), 254 Metal-chalcogenide-based thermoelectric materials, 31 Metal-organic chemical vapor deposition (MOCVD), 166 167, 307 308, 312 313 Mid wavelength infrared radiation (MWIR radiation), 57 MLs. See Monolayers (MLs) Mn21-doped CdSe nanoribbons, 275 279 MOCVD. See Metal-organic chemical vapor deposition (MOCVD) Modified single effective oscillator method (MSEO method), 84 85 Molecular beam epitaxy (MBE), 3 4, 153 154, 157 158, 161 162, 166 167, 191, 235 236, 307 308, 312 313 growth of 2D materials, 237 253, 238t advantages, 237 242 challenges, 252 253 cross between 2D and 3D structures, 249 252 growth of layered selenide and telluride films and heterostructures, 242 249 growth of HgTe quantum wells, 313 motivation, 235 236 physical characterization of 2D materials, 237 253 survey of 2D chalcogenides, 237 V2VI3-series 3D topological insulator thin films grown by, 316 317

378

Molybdenum selenide (MoSe2), 245 247, 255 257 Molybdenum telluride, 244 245, 257 Momentum distribution curves (MDCs), 255 Monocrystalline CdTe solar cells, 2 Monolayers (MLs), 235, 279 280 transition metal dichalcogenides, 18 19 MQW. See Multiple quantum wells (MQW) MR. See Magnetoresistance (MR) MS. See Melt-solidification (MS) MSCs. See Magic-sized clusters (MSCs) MSEO method. See Modified single effective oscillator method (MSEO method) MTJs. See Magnetic tunnel junctions (MTJs) Multiple quantum wells (MQW), 154 MWIR radiation. See Mid wavelength infrared radiation (MWIR radiation) N N-methylformamide (NMF), 279 Nanocrystal glue, 34 Nanoprecipitates, 37 38 maximum achievable ZT values, 39f Nanostructure engineering, 34 38 bottom-up and top-down fabrication, 34 35 consolidation method, 35 36 nanoprecipitates, 37 38 nanostructures, 36 37 Nanostructures, 36 37 energy filtering effect, 36f Near infrared radiation (NIR radiation), 57 Nearly-free-electron (NFE), 162 NEGF. See Non-equilibrium Green’s function (NEGF) NEMD simulation. See Non-equilibrium molecular dynamics simulation (NEMD simulation) NFE. See Nearly-free-electron (NFE) NI. See Normal insulator (NI) NIR radiation. See Near infrared radiation (NIR radiation) NMF. See N-methylformamide (NMF) Non-Abelian excitations, 212 217 Non-bonding pair, 361 362 Non-DMS multilayer design, 11 Non-equilibrium crystal growth processes, 3 4

Index

Non-equilibrium Green’s function (NEGF), 348 350 Non-equilibrium molecular dynamics simulation (NEMD simulation), 346 Non-uniform flux distribution, 163 164 Non-uniform magnetic fields, 207 Nonlinear optical properties, 131 137 SGH tensor, 132f THz emission spectroscopy experiment, 135f Normal doping, 38 39 Normal insulator (NI), 11 “Numbers” model, 58 O One-dimensional chalcogenides, 19 20 Optical devices, 82 83 Optical dispersion of ternary II VI semiconductor alloys, 72 83 classical picture of dispersion, 67 69 data analysis, 91 98 experimental results for ternary II-VI alloys, 91 94 electronic band structure and dispersion, 69 71 energy gap determination, 73 76 indices of refraction, 76 83 phenomenological dispersion model, 71 72 physical interpretation, 99 107 optical dispersion and ionicity, 105 107 physical meaning of fitting parameters, 99 105 theoretical model, 83 90 Optical reflectivity, 74 75 Optical transitions in ZnSeTe sinusoidal superlattices, 165 166 Optoelectronic characterization, 60 64 Orthorhombic crystal structure, 45 Oscillator energy, 99 101 Oscillator-model formula, 85 P p-like valence bands, 155 Pb1-x-ySnxMnxTe alloy, 8 Pb1-xSnxSe alloy, 7 8 Pb1-xSnxTe alloy, 7 8

Index

PBTE. See Peierls-Boltzmann transport equation (PBTE) PECVD. See Plasma-enhanced chemical vapor deposition (PECVD) Peierls distortion structure, 45 46 Peierls-Boltzmann transport equation (PBTE), 350, 353 Peltier effect, 32 Permittivity, 67 Persistent photoconductivity (PPC) effect, 192 193 Phenomenological dispersion model, 71 72 Phonon Boltzmann transport equation (Phonon BTE), 340 341 Phonons, 339 density of state, 340 dispersion, 339 phonon-assisted transitions, 161 phonon-glass and electron-crystal, 363 properties of 2D materials molybdenum selenide, 255 257 molybdenum telluride, 257 unbundling effect, 354 355 Photoconductivity effect, 59 Photoemission spectroscopy, 318 Photoluminescence (PL), 72 73, 279 method, 257 258 peaks, 21 spectra, 165, 168, 168f, 174 Photon-energy, 68 Physical vapor deposition (PVD), 312 313 Physical vapor transport, 137 Pidgeon-Brown model, 157 PL. See Photoluminescence (PL) Planck constant, 307 308 Plasma-enhanced chemical vapor deposition (PECVD), 307 308, 312 313 PLEC materials, 40 41, 47 Point defect, 40 Polarization dependent excitation mechanism, 209 210 Potential energy, 363 364 Prism coupler method, 72 73 Proximity effect, 323 325 superconducting, 314, 325 Proximity-induced long-range ferromagnetism, 18 Pseudo-binary system, 37 PVD. See Physical vapor deposition (PVD)

379

Q QAHE. See Quantum anomalous Hall effect (QAHE) QC. See Quantum cascade (QC) QDs. See Quantum dots (QDs) QHE. See Quantum Hall effect (QHE) QL. See Quintuple layer (QL) QSHE. See Quantum spin Hall effect (QSHE) Quantum anomalous Hall effect (QAHE), 12 14, 14f, 189 190, 311 312, 323 325, 325f in HgTe QWs, 219 Quantum cascade (QC), 9 Quantum dots (QDs), 154, 166 167, 282 283, 286 border between molecules and, 289 294 individual dopants in single nanocrystals, 285 288 mapping of exciton localization in, 174 178 spin polarization and relaxation of exciton in, 167 171 valence-band mixing in doped nanocrystal, 282 285 Quantum Hall effect (QHE), 15 17, 191 192, 206, 305 306, 311 312 plateus, 200 201 in 2D systems, 197 199 Quantum Hall ferromagnetic (QHF) transitions, 202 204, 212 213, 214f domain structure during, 212f QHF resistance spikes, 213 Quantum mechanical effects, 154 Quantum spin Hall (QSH) system, 307 Quantum spin Hall effect (QSHE), 189 190, 218, 307, 308f in HgTe QWs, 217 219 Quantum wells (QWs), 154, 161 162, 174, 189, 191 192 asymmetric, 194 doping, 192 193 Quasi-one-dimensional chalcogenides, 19 20 Quintuple layer (QL), 309, 316 QWs. See Quantum wells (QWs) R Radiation induced spin currents, 209 210 Raman spectroscopy, 255 257, 319 320, 348 350

380

Rattling modes, 363 364 Reflection-high-energy electron diffraction (RHEED), 242, 244, 245f Refractive indices of ZnTe, 77, 77t, 89, 89f Resistivity, 321 Resonant bonding, 360 361 Resonant level doping, 43 Resonant Raman scattering investigations, 194 Resonant tunneling diode (RTD), 189 RHEED. See Reflection-high-energy electron diffraction (RHEED) RKKY exchange mechanism. See Ruderman-Kittel-Kasuya-Yosida exchange mechanism (RKKY exchange mechanism) Room temperature (RT), 257 258 Rotational coefficients, 196 RT. See Room temperature (RT) RTD. See Resonant tunneling diode (RTD) rth moment of optical spectrum, 84 Ruderman-Kittel-Kasuya-Yosida exchange mechanism (RKKY exchange mechanism), 323 325 S s-d exchange interaction, 191 192 s-like conduction band, 155 SAED. See Selected area electron diffraction (SAED) SBZ. See Surface Brillouin zone (SBZ) Scanning tunneling microscopy (STM), 245 247, 249 255 Scanning tunneling spectroscopy (STS), 7 8, 249 252 Schrodinger equation, 162 SdH oscillations in MR, 200 Second harmonic generation (SHG), 119 120, 131 tensor, 132f Seebeck ratchet. See Thermoratchet Selected area electron diffraction (SAED), 138 139 Selenides (Se), 153 Sellmeier-type relation, 72 73 Semi-empirical fits for ZnTe, 88 90 Semi-empirical model, 83 85 Semiconducting spinels, 3 4 Semiconductor

Index

heterostructures, 161 162 interfaces, 15 17 solder, 34 SEO method. See Single-effective-oscillator method (SEO method) Severe plastic deformation (SPD), 34 35 SHE. See Spin Hall effect (SHE) SHG. See Second harmonic generation (SHG) Shift current, 132 133 Shubnikov-de Haas (SdH) oscillations, 200 202, 322 SiC. See Single crystal substrate (SiC) SILAR. See Successive ionic layer adsorption and reaction (SILAR) Single crystal growth and exfoliation, 137 140 Single crystal substrate (SiC), 249 Single nanocrystals quantum dots, 285 288 Single quantum well (SQW), 154 Single-effective-oscillator method (SEO method), 83 improvements, 85 88 Single-layer sheet, 348 351 atomic structures of 2D TMD, 349f thermal conductivity of single-layer MoS2, 351t Sinusoidal superlattice (SSL), 164 165 SLs. See Superlattices (SLs) Smallest doped semiconductors, 289 290 so hole state. See Split-off hole state (so hole state) SOC. See Spin-orbit coupling (SOC) Solid state low dimensional structures, 20 Solotronics, 285 286 SOT. See Spin-orbital toque (SOT) sp-d exchange interaction, 4 5, 271, 288 Spark plasma sintering (SPS) techniques, 35 36 SPD. See Severe plastic deformation (SPD) Spectroscopy, 318 321 Spherical harmonics, 274 275 Spin Hall effect (SHE), 196 197, 209 210 Spin interactions in chalcogenide DMS QWs, 193 195 Spin lifetimes determination, 11 Spin polarization of carriers, 171

Index

enhancement in non-DMS and DMS coupled QDs, 178 181 and relaxation of exciton in QDs, 167 171 Spin split-off valence bands, 70 Spin superlattices, 11 Spin transfer torque (STT), 15 Spin-down carriers, 169 Spin-flip time in CdSe, 170 Spin-glass behavior, 8 Spin-orbit coupling (SOC), 194, 255 coupled massless Dirac fermions, 12 14 Spin-orbit interaction, 155 Spin-orbit split-off valence band, 155 Spin-orbital toque (SOT), 15 Spin-polarized 3D TI surface states, 323 Spin-resolved LLs, 197 198 Spin-spin interaction between coupled QDs, 171 173 Spintronics, 167 Split-off (so) hole state, 273 274 SQW. See Single quantum well (SQW) SSL. See Sinusoidal superlattice (SSL) “Standard-sized” II-VI nanocrystals, 286 STM. See Scanning tunneling microscopy (STM) Strain effect, 352 355 STS. See Scanning tunneling spectroscopy (STS) STT. See Spin transfer torque (STT) Subbands, 197 198 Successive ionic layer adsorption and reaction (SILAR), 138 Sulfides (S), 153 Superconductivity, 305 306 Superlattices (SLs), 154, 161 162, 247 249 Surface Brillouin zone (SBZ), 255 Surface state, 7 8, 12 14, 217, 220, 309 310 Symmetry-breaking exotic quantum phenomena, 305 306 T TA and LA modes. See Transverse and longitudinal acoustic modes (TA and LA modes) TaFe11xTe3, 20

381

TCI. See Topological crystalline insulators (TCIs) TDTR. See Time-domain thermoreflectance (TDTR) Te diethyltelluride (DETe), 313 314 Tellurides (Te), 153 TEM. See Transmission electron microscopy (TEM) Temperature-dependent Brillouin function, 278 Terahertz spectroscopy, 321 Ternary (Sn,Mn)Se alloy, 249 252, 251f Ternary alloys, 67 68, 72 Ternary II-VI alloys, experimental results for, 91 94 TFET. See Tunnel field effect transistors (TFET) Thermal conductivity, 37 39, 341, 346, 352 360, 358f Thermal transport, 360 364 lone pair electron, 361 363 rattling model, 363 364 resonant bonding, 360 361 Thermoelectric (TE) applications of chalcogenides band structure engineering, 43 44 crystal structure engineering, 44 49 defect engineering, 38 43 nanostructure engineering, 34 38 thermoelectric effect, 32 thermoelectric efficiency, 32 34 effect, 32 efficiency, 32 34 materials, 32 33, 38 39, 342, 344, 361, 363 Thermoratchet, 210 211 Thin-film growth techniques, 312 Thomson effect, 32 Three-dimension (3D), 1 2 chalcogenides in 3D form, 2 9 electronic and optical effects in II1xMnxVI alloys, 4 5 lead salts, 7 8 miscellaneous II-VI-based diluted magnetic semiconductors, 5 7 monocrystalline CdTe solar cells, 2 spinels, 9 II-VI magnetic semiconductors, 3 4

382

Three-dimension (3D) (Continued) semiconductor system, 236 topological insulators (TIs), 12 14 Time-domain thermoreflectance (TDTR), 359 360 Time-resolved Faraday rotation measurements, 194 Time-reversal symmetry (TR symmetry), 305 306 breaking, 14 Tin selenide (SnSe), 119 120, 122t, 123 125, 242 244 Tin sulfide (SnS), 119 120, 122t, 123 125 Tin telluride (SnTe), 7 8 TIs. See Topological insulators (TIs) TISs. See Topological interface states (TISs) TMD. See Transition metal dichalcogenides (TMDCs) TMDCs. See Transition metal dichalcogenides (TMDCs) TO. See Transverse optical (TO) Top-down fabrication, 34 35 Topological enhanced interface magnetism, 18 equivalent states, 306 materials, 7 8 nontrivial HgTe quantum well system, 307 308 nontrivial system, 305 306 order, 305 306, 326 phases in IV-VI materials, 220 superconductivity, 326 superlattices, 11 Topological crystalline insulators (TCIs), 7 8, 220 Topological Hall effect, 209 Topological insulators (TIs), 12 14, 189 190, 211, 217, 305, 309, 318 Topological interface states (TISs), 11 Topological quantum wells (TQWs), 11 Topological superconductors (TSCs), 189 190, 325 326 TQWs. See Topological quantum wells (TQWs) TR symmetry. See Time-reversal symmetry (TR symmetry) TR-invariant Hamiltonian, 306 Transistor-like semiconductor heterostructures, 189

Index

Transition metal dichalcogenides (TMDCs), 15, 46 48 Transition metal ion spins, 271 Transmission electron microscopy (TEM), 236, 242 Transverse and longitudinal acoustic modes (TA and LA modes), 353 354 Transverse optical (TO), 361 Triangular lattice, 209 TSCs. See Topological superconductors (TSCs) Tunable Dirac interface states in topological superlattices, 11 Tuning magnetic exchange interactions, 279 282 Tunnel field effect transistors (TFET), 236 II1-xCoxVI alloys, 6 7 II1-xFexVI alloys, 7 II1-xMnxVI alloys, electronic and optical effects in, 4 5 II1-xMnxVI DMS epilayers, Zeeman splitting in, 173 174 Two-dimension (2D) chalcogenides, 235 236 survey, 237 systems, 1 2 electron system, 7 8 layered chalcogenides, 235 layered compounds, 15 layered structure, 343 optical properties of 2D materials PL and absorption spectroscopy, 257 258 XPS, 258 260 sheet, 342, 346 superlattice structure, 45 systems, 235 topological insulator system, 305 transition-metal dichalcogenides, 357 Two-dimensional chalcogenide structures, 9 19 epitaxially-formed chalcogenides, 9 11 interface phenomena in chalcogenide structures, 15 19 2D “van der Waals” chalcogenides, 12 15 magnetism, 14 topological insulators (TIs), 12 14 2D dichalcogenide systems, 15

Index

two-dimensional ferromagnetic materials, 15 Two-dimensional colloidal nanocrystals (2D colloidal nanocrystals), 275 282. See also Zero-dimension (0D)— nanocrystals giant magneto-optical response in Mn21doped CdSe nanoribbons, 275 279 tuning magnetic exchange interactions, 279 282 Two-dimensional electron gas (2DEG), 15 17, 189 in chalcogenide multilayers, 15 17 in DMS-based QWs, 189 in magnetically doped QWs DMS QW in inhomogeneous magnetic fields, 206 209 DMS QWs under terahertz and microwave radiation, 209 211 low-dimensional heterostructures, 190 193 magnetotransport in chalcogenide QWs, 195 206 spin interactions in chalcogenide DMS QWs, 193 195 novel topological phases in chalcogenide multilayers, 211 220 domain walls and non-Abelian excitations, 212 217 quantum anomalous Hall effect in HgTe QWs, 219 quantum spin Hall effect in HgTe QWs, 217 219 topological phases in IV-VI materials, 220 wireless Majorana bound states, 217 2D materials, 119 120, 125 126, 129, 131 materials, 273 MBE growth, 238t advantages, 237 242 challenges, 252 253 cross between 2D and 3D structures, 249 252 growth of layered selenide and telluride films and heterostructures, 242 249 physical characterization electronic structure, 253 255 optical properties of 2D materials, 257 260 phonon properties, 255 257

383

II-VI compounds, 153 DMSs, 154 energy gap in, 153 154 II-VI magnetic semiconductors, 3 4 II-VI quantum cascade emitters, 9 II-VI quantum structures involving DMSs, 173 178 mapping of exciton localization in QDs, 174 178 Zeeman splitting in II1-xMnxVI DMS epilayers, 173 174 II-VI semiconductor compounds, 68 II-VI-based alloys, 20 II-VI-based zero-dimensional structures, 166 173 spin polarization and relaxation of exciton in QDs, 167 171 spin-spin interaction between coupled QDs, 171 173 II-valent metal chalcogenides, 137 “Type-2” antiferromagnetic order, 10 Type-III HgTe semiconductor quantum well, 307 U Ultrahigh-vacuum (UHV), 249 252 Undoped magnetic semiconductors, 3 4 V V2VI3-series 3D topological insulators, 309 312, 311f, 314 317 bulk crystal growth of, 315 316 nanostructures synthesis, 315 thin films grown by MBE, 316 317 Valence band maxima (VBM), 123 125, 255 Valence bands (VB), 155, 271 mixing, 283 in doped nanocrystal quantum dots, 282 285 effect, 274 275 states, 275 Valleytronics, 18 19 Van der Waals (vdW), 236, 249 epitaxy, 241 242 heterostructures, 18 19 magnets, 15 materials, 119 120, 139 140 Van Vleck paramagnetism, 323 325 Vapor phase epitaxy (VPE), 312 313

384

Vapor-liquid-solid (VLS), 137, 315 VASP. See Vienna ab initio simulation package (VASP) VB. See Valence bands (VB) VBM. See Valence band maxima (VBM) vdW. See van der Waals (vdW) Vegard’s law interpolation, 72 Vegard’s-law-like relationship, 83 Vienna ab initio simulation package (VASP), 123 125 VLS. See Vapor-liquid-solid (VLS) VPE. See Vapor phase epitaxy (VPE) W Wannier exciton, 156 Wave equation, 67 Wave function engineering in core/shell nanoplatelets, 279 282 mapping, 11 Wavelength dependence of refractive index, 71 72 Weak antilocalization, 206 Wide HgTe QWs, magnetotransport in, 205 206 Wireless Majorana bound states, 217 X X-ray diffraction (XRD), 72 X-ray photoemission spectroscopy (XPS), 258 260, 318 Z Z2 topological index, 306 Z2 topological insulator, 306 Zeeman effect, 11 Zeeman energy, 17, 201f Zeeman shift of CdSe QD, 176 177 Zeeman splitting, 197 198, 276 278, 285 of electronic levels, 4 5 in II1-xMnxVI DMS epilayers, 173 174 of transitions, 154 Zeeman tuning, 11 Zero-dimension (0D) chalcogenide structures, 20 22 nanocrystals, 282 288. See also Twodimensional colloidal nanocrystals (2D colloidal nanocrystals)

Index

individual dopants in single nanocrystals quantum dots, 285 288 valence-band mixing in doped nanocrystal quantum dots, 282 285 nanostructures, 274 275 spin structures, 20 Zero-phonon line (ZPL), 288 Zinc blende type II-VI semiconductor compounds, 96 98 Zinc telluride (ZnTe), 153 154 fitting formulas and parameters for, 91t magneto-optical properties, 155 158 refractive indices, 77, 77t, 89f semi-empirical fits for, 88 90 spectrum, 157 158 Zinc-blende BeSe, 104 105 Zn1-xBexSe films fitting results for dispersion, 93t refractive index n vs. Be concentration for, 111f Zn1-xCdxSe films fitting results for dispersion, 92t refractive index n vs. Cd concentration for, 109f Zn1-xCdxTe films fitting results for dispersion, 92t refractive index n vs. Cd concentration for, 110f Zn1-xCoxS alloy, 6 Zn1-xCoxSe alloy, 6 Zn1-xMgxSe films fitting results for dispersion, 93t refractive index n vs. Mg concentration for, 112f Zn1-xMgxTe films, 72, 74 75 fitting results for dispersion, 94t refractive index n vs. Mg concentration for, 114f Zn1-xMnxSe films fitting results for dispersion, 93t refractive index n vs. Mn concentration for, 113f Zn1-xMnxTe films, 72, 74 75 fitting results for dispersion, 94t refractive index n vs. Mn concentration for, 115f

Index

Zn1-xSexTe1-x films fitting results for dispersion, 94t refractive index n vs. Se concentration for, 116f Zn1 xBexSe barriers, 189 ZnBeSe, 92 ZnCdSe, 92 ZnCdSe/ZnCdMgSe heterostructure, 9 ZnCdTe, 92 ZnCr2Se4, 9 ZnMgSe, 92 ZnMgTe, 92

385

ZnMnSe, 92 ZnMnTe, 92 ZnS, 153 154 ZnSe, 153 154 magneto-optical properties, 155 158 ZnSe1-xTex sinusoidal SLs, 165 ZnSeTe, 92 sinusoidal superlattices, optical transitions in, 165 166 superlattices growth with sinusoidal composition modulation, 163 165 ZPL. See Zero-phonon line (ZPL)