Second-Order Photogalvanic Photocurrents in 2D Materials (SpringerBriefs in Applied Sciences and Technology) 9819706173, 9789819706174

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SpringerBriefs in Applied Sciences and Technology Nanoscience and Nanotechnology Mustafa Eginligil · Ting Yu

Second-Order Photogalvanic Photocurrents in 2D Materials

SpringerBriefs in Applied Sciences and Technology

Nanoscience and Nanotechnology Series Editors Hilmi Volkan Demir, Nanyang Technological University, Singapore, Singapore Alexander O. Govorov, Clippinger Laboratories 251B, Department of Physics and Astronomy, Ohio University, Athens, OH, USA

Indexed by SCOPUS Nanoscience and nanotechnology offer means to assemble and study superstructures, composed of nanocomponents such as nanocrystals and biomolecules, exhibiting interesting unique properties. Also, nanoscience and nanotechnology enable ways to make and explore design-based artificial structures that do not exist in nature such as metamaterials and metasurfaces. Furthermore, nanoscience and nanotechnology allow us to make and understand tightly confined quasi-zero-dimensional to two-dimensional quantum structures such as nanopalettes and graphene with unique electronic structures. For example, today by using a biomolecular linker, one can assemble crystalline nanoparticles and nanowires into complex surfaces or composite structures with new electronic and optical properties. The unique properties of these superstructures result from the chemical composition and physical arrangement of such nanocomponents (e.g., semiconductor nanocrystals, metal nanoparticles, and biomolecules). Interactions between these elements (donor and acceptor) may further enhance such properties of the resulting hybrid superstructures. One of the important mechanisms is excitonics (enabled through energy transfer of exciton-exciton coupling) and another one is plasmonics (enabled by plasmon-exciton coupling). Also, in such nanoengineered structures, the lightmaterial interactions at the nanoscale can be modified and enhanced, giving rise to nanophotonic effects. These emerging topics of energy transfer, plasmonics, metastructuring and the like have now reached a level of wide-scale use and popularity that they are no longer the topics of a specialist, but now span the interests of all “end-users” of the new findings in these topics including those parties in biology, medicine, materials science and engineerings. Many technical books and reports have been published on individual topics in the specialized fields, and the existing literature have been typically written in a specialized manner for those in the field of interest (e.g., for only the physicists, only the chemists, etc.). However, currently there is no brief series available, which covers these topics in a way uniting all fields of interest including physics, chemistry, material science, biology, medicine, engineering, and the others. The proposed new series in “Nanoscience and Nanotechnology” uniquely supports this cross-sectional platform spanning all of these fields. The proposed briefs series is intended to target a diverse readership and to serve as an important reference for both the specialized and general audience. This is not possible to achieve under the series of an engineering field (for example, electrical engineering) or under the series of a technical field (for example, physics and applied physics), which would have been very intimidating for biologists, medical doctors, materials scientists, etc. The Briefs in NANOSCIENCE AND NANOTECHNOLOGY thus offers a great potential by itself, which will be interesting both for the specialists and the nonspecialists.

Mustafa Eginligil · Ting Yu

Second-Order Photogalvanic Photocurrents in 2D Materials

Mustafa Eginligil Institute of Advanced Materials, School of Flexible Electronics (Future Technologies) Nanjing Tech University Nanjing, China

Ting Yu School of Physics and Technology Wuhan University Wuhan, China

ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISSN 2196-1670 ISSN 2196-1689 (electronic) Nanoscience and Nanotechnology ISBN 978-981-97-0617-4 ISBN 978-981-97-0618-1 (eBook) https://doi.org/10.1007/978-981-97-0618-1 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Dedicated to those who seek the truth in science and those who support the seekers in every possible way

Preface

In the context of relation between experiment and theory in physics, no one would be, in principle, against the conclusion stated by H. Dingle in his article published in 1944 in Nature 3898:65, that is, the personal genius and intuition are the main driving forces for a possible success. This applies to all branches of sciences, not only physics, but notably, it becomes more significant at the intersection of physics, chemistry, and materials science. There are several common shares of these disciplines, in particular, the topics related to symmetry and chirality. Many scientists have devoted their research lives into these two topics, and many research directions have been developed. Among them, one main field increasingly at the surge of interest for more than five decades has been “photoresponse due to lack of a center of symmetry in a medium”, or with the most common name, photogalvanic effect. Crystal symmetry is the major indicator of the photogalvanic effects at low dimensions, such as surfaces and interfaces. After the discovery of the genuinely 2D material, graphene, and subsequently 2D semiconductors, the photogalvanic effects have found their ways toward applications. Although the phenomenology of the photogalvanic effects, which can be simply seen as photoresponse nonlinear-in-electric field, has been well-established, the microscopic understanding in each material system may vary. Therefore, for those who are interested in this field, it would be useful to have a quick reference. Here, we provide a detailed roadmap starting from phenomenology and continuing with the ultimate low-dimensional materials, for which the photogalvanic effects offer a rich platform at the second-order response to an electric field. This brief was written in a sequential manner, from experimentalists’ point of view, and it is strongly suggested to start with the first chapter and continue with the other chapters, since there is prerequisite information from previous chapters. In Chap. 1, we introduced how a polarized light impinges on a sample plane, studied all possible geometries, and provided the complete general phenomenology. In Chaps. 2 and 3, we made use of the ultimate low-dimensional material—graphene, which is a model condensed-matter system, in explanation of photogalvanic effect at the secondorder, its specific phenomenology, microscopic theory, and experimental results. We particularly brought the linear-in-electric field-dependent photocurrent in graphene vii

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to attention, together with its band structure and crystal symmetry. This is essential in getting insight into the photogalvanic effect. It was mandatory to introduce the photon drag effect, which also generates a photocurrent similar to the photogalvanic effect, for which their phenomenology has some distinct features, as well as somewhat puzzling similarities. In a similar fashion, in Chaps. 4 and 5, we covered a necessary background for the photogalvanic effects in monolayer transition metal dichalcogenides (graphene’s semiconductor counterparts), with symmetry analysis, microscopic theory, and experimental results. In our discussion, we included the Berry curvature-dependent photocurrent, which can also play an important role in 2D semiconductors. The second-order photogalvanic effects that have been covered so far in graphene and monolayer transition metal chalcogenides have already excited the 2D semiconductor optoelectronic research community by several means. It seems that the interests on the photogalvanic effects will continue to escalate in near future. The persistent, diligent, and hard work on this topic would be another driving force, in addition to the “genius and intuition” comment by H. Dingle, for the advances in this interdisciplinary field. Nanjing, China October 2023

Mustafa Eginligil Ting Yu

Acknowledgements

This book brief is based on our accumulated knowledge on photogalvanic effects in 2D materials in last one decade in Nanjing Tech University, Nanjing, China; Wuhan University, Wuhan, China; and Nanyang Technological University, Singapore. There have been many contributions in several ways by theoretical discussions and experimental collaborations by co-workers and colleagues. Here, we would like to acknowledge every single effort, direct or indirect, on the production of this work. Our special thanks should be addressed to Bingchen Cao for performing helicitydependent photocurrent measurements; to Vitor M. Pereira and Fabio Hipolito for fruitful theoretical discussions; and to Cesare Soci and his group for long-lasting experimental collaborations. In realization of this work, several important financial supports have been extremely useful. We gratefully declare funding by the National Natural Science Foundation of China grant NSFC 11774170 (TY and ME); 100 Foreign Talents Project in Jiangsu Province grant JSA2016003 (ME); National Key Research and Development Program of China grant 2021YFA1200800 (TY); and Research-Oriented Teaching Exploration (Teaching Reform of Micro- and Nano-Optoelectronics Course) Fund by Nanjing Tech University (ME).

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Contents

1 Introduction to Light Polarization-Dependent Photocurrent—Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 A First Look at the Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Low-Order Non-Linear Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Stokes Parameters and Light Polarization . . . . . . . . . . . . . . . . . . . . . . 1.4 Second-Order Response and Symmetry Considerations . . . . . . . . . . 1.5 Phenomenology for a Quarter-Wave Plate . . . . . . . . . . . . . . . . . . . . . . 1.6 Photogalvanic and Photon Drag Effects . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Photogalvanic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Photon Drag Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Second Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Third-Order Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Second Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Photoconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.3 Third Harmonic Generation and Two-Photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.4 Coherent Photogalvanic Effect . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Graphene as the Model Low-Dimensional Photogalvanic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Linear Photocurrent in Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Crystal Structure and the Brillouin Zone . . . . . . . . . . . . . . . . . . . . . . . 2.3 Electronic Energy Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Graphene—1LG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Bilayer Graphene—2LG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Trigonal Warping in 2LG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Berry’s Phase in Graphene and Valley Polarization . . . . . . . . 2.3.5 Multilayer (N-Layer) Graphene (N > 2) . . . . . . . . . . . . . . . . .

1 2 4 5 6 8 12 12 13 15 15 16 16 16 17 17 18 21 23 26 27 27 29 31 32 33

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2.4 Symmetry Point Group in Graphene Layers . . . . . . . . . . . . . . . . . . . . 2.4.1 Symmetry Point Group in 1LG . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Symmetry Point Group in N-Layer Graphene (N > 1) . . . . . 2.5 Excitation by Linearly Polarized Light . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Linearly Polarized Irradiation at Normal Incidence . . . . . . . . 2.5.2 Linearly Polarized Irradiation at Oblique Incidence . . . . . . . 2.6 Excitation by Circularly Polarized Light . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Circularly Polarized Irradiation at Normal Incidence . . . . . . 2.6.2 Circularly Polarized Irradiation at Oblique Incidence . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 36 37 37 39 39 40 40 42

3 Light Helicity Dependent Photocurrent in Graphene Planes . . . . . . . . 3.1 Phenomenology of Light Helicity Dependence in Graphene Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Photogalvanic Effect in 1LG . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Photon Drag Effect in 1LG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Distinguishing the PGE from the PDE . . . . . . . . . . . . . . . . . . 3.2 Microscopic Theory of the PDE and the PGE in 1LG: Classical vs. Quantum Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Current Expression of the PDE in Classical Regime . . . . . . . 3.2.2 Discrimination of PDE Currents, EB Mechanism, and the AC Hall Effect (CPDE) . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 The PDE in Quantum Regime . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Microscopic Origin of the PGE-Based Current in 1LG . . . . 3.3 Light Helicity Dependent Photocurrent Measurements by Low Energy Terahertz Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Light Helicity Dependent Photocurrent Measurements by Midinfrared Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Light Helicity Dependent Photocurrent Measurements by Visible Radiation in 1LG and 2LG . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Photocurrent Anomaly in Transverse Geometry at Oblique Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Possibility of Berry Curvature-Dependent Photocurrents . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Influence of Spin-Valley Coupling on Photogalvanic Photocurrents in Layered Transition Metal Dichalcogenides . . . . . . . . 4.1 Lattice Structure, Symmetry, and Stacking in TMDs . . . . . . . . . . . . . 4.1.1 Lattice Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Crystal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Symmetry Reduction and the PGE in TMD Layers . . . . . . . . 4.1.4 Stacking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Symmetry around the K Points of 1L-TMDs . . . . . . . . . . . . .

43 44 45 46 47 47 49 49 51 54 57 60 64 66 67 69 71 71 71 73 73 74

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4.2 Band Structure of Mono- and Few-Layer TMDs . . . . . . . . . . . . . . . . 4.2.1 First-Principle Expectation and Photoluminescence of 1L- and Few-Layer-MoS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 TB Approximation, Spin–Orbit Coupling in 1L-TMDs, and k · p Model . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Giant Spin-Valley Coupling in 1L-MoS2 and Other TMDs . . . . . . . . 4.3.1 Spin and Valley-Dependent Optical Selection Rules . . . . . . . 4.3.2 Valley Polarization in Circularly Polarized Photoluminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Photocurrent due to Excitation by Polarized Light at Normal Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Polarized Light Excitation at Normal Incidence and the Valley Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The VHE: Similarity to the CPGE in terms of Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Photocurrent due to Excitation by Polarized Light at Oblique Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Device and Measurement Geometry . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Light Helicity Dependent Photocurrent in Layered Transition Metal Dichalcogenides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Generation of Dichroic Spin-Valley Photocurrent in 1L-TMDs . . . . 5.1.1 Band Dispersion and Anomalous Velocity . . . . . . . . . . . . . . . 5.1.2 Defining the CPGE Photocurrent for Onand Off-Resonance Excitation . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Spin-Valley Photocurrent Measurements in 1L-TMDs . . . . . . . . . . . 5.3 Bias and Gate Dependence of Helicity Dependent Photocurrent in 1L-MoS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Source-Drain (Bias) Voltage Dependence . . . . . . . . . . . . . . . . 5.3.2 Gate Voltage Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Incidence Angle, Bias, and Gate Dependence of Spectral Helicity Dependent Photocurrent in 1L-MoSe2 . . . . . . . . . . . . . . . . . 5.4.1 Spectral and Gate Voltage Dependence . . . . . . . . . . . . . . . . . . 5.4.2 Angle of Incidence Dependence at Fixed Bias and Gate Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Source-Drain Dependence as a Function of Incidence Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Light Helicity Dependent Photovoltage in an Encapsulated 1L-MoSe2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Spectral and Drain Voltage Dependence of Circular and Linear Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Spectral, Gate Voltage, and Bias Voltage Dependence of Circular Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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74 75 76 78 79 80 82 82 84 85 85 85 86 89 89 91 91 93 95 96 96 98 99 99 100 101 102 104

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5.5.3 Incidence and Azimuthal Angle Dependence . . . . . . . . . . . . . 5.5.4 Gate Voltage Dependence of Circular Contribution . . . . . . . . 5.6 Light Helicity Dependent Photocurrent in Multilayer TMDs by Excitation Above the Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Light Helicity Dependent Photocurrent in Multilayer TMDs by Excitation Below the Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 107 109 113 115

Acronyms

1LG 1L-TMD 2D 2LG 3D 3LG BZ CB CBM CPDE CPGE CVD DFG DOF E-field h-BN HC IL LPDE LPGE MLG NN PDE PGE PL PTE PV QWP SHE SHG SLG

Single-layer graphene Monolayer transition metal dichalcogenide Two-dimensional Bilayer graphene Three-dimensional Trilayer graphene Brillouin zone Conduction band Conduction band minimum Circular photon drag effect Circular photogalvanic effect Chemical vapor deposition Difference frequency generation Degree of freedom Electric field component of the electromagnetic wave Hexagonal boron nitride Hot carrier Ionic liquid Linear photon drag effect Linear photogalvanic effect Monolayer graphene Nearest-neighbor Photon drag effect Photogalvanic effect Photoluminescence Photo-thermoelectric Photovoltaic Quarter-wave plate Spin Hall effect Second harmonic generation Single-layer graphene xv

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TB TE TM TMD TNN VB VBM VHE

Acronyms

Tight-binding Transverse electric Transverse magnetic Transition metal dichalcogenide Third nearest-neighbor Valence band Valence band maximum Valley Hall effect

Chapter 1

Introduction to Light Polarization-Dependent Photocurrent—Phenomenology

Light polarization-dependent photocurrent is a non-linear (mostly second-order effects) optoelectronic response of photo-sensitive devices to irradiation that impinges on a device’s plane. The nature of polarization-dependent photocurrent is directly related with not only the polarization state of light, but also the direction of propagation of the radiation field. Such a photocurrent can be generally expressed by an electric current density, j (r, t), which has space r and time t dependency. The radiation field (or electromagnetic field), which can lead to j (r, t), can be represented by an alternating electric field, E(ω, q), with light frequency ω and light wave vector q. The origin of light polarization-dependent photocurrent relies on the geometrical constraints of materials’ crystallography, as well as the experimental geometry forced by the radiation field, E(ω, q). Macroscopically, possibility to observe polarization-dependent photocurrent of a photo-sensitive device can be comprehended by symmetry arguments, without any microscopic and electronic characteristics of the system. Major symmetry conditions, which can allow or forbid this type of photocurrent, are point group symmetries in crystal lattices. Crystal symmetries can be referenced by points (inversion symmetry), lines (rotational symmetry and rotational/inversion symmetry), or planes (reflection or mirror symmetry); and, there are 32 point groups in 7 different crystal systems with 14 different Bravais lattices, leading to 230 space group symmetries. All materials fall into one of these 230 space group symmetries; and in addition to all the mentioned symmetry conditions, time reversal symmetry can be the reason behind a polarization-dependent photocurrent upon polarized light irradiation. In regards to polarization-dependent photocurrent, low-dimensional materials systems offer a rich playground of aforementioned symmetry conditions, unlike bulk counterparts, thanks to redefined selection rules of optical transitions, which will be mentioned in later chapters. In this sense, polarization-dependent photocurrents due to photovoltaic effects are much more intriguing, compared to light momentum transfer driven photon drag effects, which are relatively less affected by symmetry

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. Eginligil and T. Yu, Second-Order Photogalvanic Photocurrents in 2D Materials, Nanoscience and Nanotechnology, https://doi.org/10.1007/978-981-97-0618-1_1

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1 Introduction to Light Polarization-Dependent …

restrictions. Indeed, polarization-dependent photocurrents have been first observed in bulk materials, such as non-perovskite [1] and perovskite oxide ferroelectric materials [2] and bulk tellurium [3], due to so-called bulk photovoltaic effect [4] (or photogalvanic effect [5], as it was called differently in the country of discovery). Regardless of how it was labeled, this effect was simply described as a quantum phenomenon taking place in non-centrosymmetric crystals [6]. It was called photovoltaic since it was found that the frequency and polarization of light leads to a generation of electric current in a fashion similar to photovoltaic effect in conventional p–n junctions. But, in this case, instead of a built-in electric field, non-centrosymmetry plays the main role to lead to a ballistic (or injection) current as a result of asymmetric distribution of charge carriers’ momenta, photocurrent due to circular photogalvanic effect [7, 8], and a shift current due to asymmetry of scattering of free carriers by defects resulting in non-zero off-diagonal matrix elements, photocurrent due to linear photogalvanic effect. Both circular and linear photogalvanic effects as well as photon drag effects have been studied both theoretically and experimentally in detail so far, in a wide range of low-dimensional materials, such as graphene and 2D direct band gap valleytronic semiconductors. In this Brief, our goal is to provide a guide for those who are interested in light polarization-dependent photocurrents in aforementioned emerging low-dimensional materials. There are already several books about to photogalvanic effects [9, 10], review on spin-related photocurrents in II–VI and III–V semiconductors-based quantum wells [11], and polarization-dependent photocurrents in graphene [12, 13], as well as recent publications, which revealed polarization-dependent photocurrents in emerging quantum [14] and spin-orbitronic materials [15]. These works have aimed to shed light on to fundamental understanding, mainly from microscopic mechanism point of view, in various physical systems, which include genuinely 2D materials, quantum materials with nontrivial surface states, metamaterials with chiral structures, and novel hybrid perovskite materials. In all these systems, there is either a surface and/or interface where symmetry conditions allow photovoltaic (or photogalvanic) along with photon drag effects-driven photocurrents. In this Brief, we will attempt to reconcile and focus polarization-dependent photocurrents, specifically, on the second-order photogalvanic effects in 2D materials, microscopic mechanisms and existing observations. But, first of all in this chapter, we will provide a general knowledge about the phenomenology of polarized light irradiation on any material system.

1.1 A First Look at the Phenomenology An experimental setup to visualize the interaction of an electromagnetic field E(ω, q) and a sample in a x–y–z coordinate system for current j x (r, t) is given in Fig. 1.1. There are three critical angles to determine polarization-dependent photocurrent, j x (r, t): the angle of photon polarization (angle to change the polarization of E(ω, q) from a given linear polarization to elliptical polarization) ϕ, the angle

1.1 A First Look at the Phenomenology

3

of incidence or direction of propagation of E(ω, q) from the z-direction, θ , and the azimuthal angle ζ . As an example, when a p-polarized E(ω, q), which can be taken as E p , is incident on a quarter-wave plate (QWP), it will change its elliptical polarization by angle ϕ, angle of photon polarization, from the fast axis of the QWP. After the light exits the QWP, it will acquire certain magnitude in E s , and E p directions—an elliptical polarization degree, where polarization is linear for ϕ = nπ/2 for n = 0, ± 1, ± 2, ± 3, ± 4, . . . and circular for ϕ = (2n + 1)π/4 for n = 0, ± 1, ± 2, ± 3, . . .. The polarized E(ω, q), will impinge on the x–y plane of the sample with a certain θ , to lead to a polarization-dependent photocurrent j x (r, t), in the x-direction for an azimuthal angle of ζ , which is defined as the angle from the plane normal to the direction of propagation of the electromagnetic wave to the x-direction.

Fig. 1.1 A p-polarized electromagnetic wave (denoted by E p ) is incident on a quarter-wave plate (QWP) changes its elliptical polarization by ϕ, angle of photon polarization, from the fast axis of the QWP (where polarization is linear for ϕ = nπ/2 for n = 0, ± 1, ± 2, ± 3, ± 4, . . . and circular for ϕ = (2n + 1)π/4 for n = 0, ± 1, ± 2, ± 3, . . .). The direction of propagation of the electric field from the z-direction is the angle of incidence, θ. Light exiting the QWP with a certain ϕ, E s , and E p impinges on to the sample for a certain θ, to lead to a polarization-dependent photocurrent in the x-direction for an azimuthal angle of ζ , which is the angle from the plane normal to the direction of propagation of the electromagnetic wave with respect to the x-direction

4

1 Introduction to Light Polarization-Dependent …

1.2 Low-Order Non-Linear Response The general form of an alternating electric field E p shown in Fig. 1.1 is E(ω, q), which represents the light irradiation. It can take the form of a plane wave, E(r, t) = E(ω, q)ei(q r−ωt) + E ∗ (ω, q)e−i(q r−ωt) ,

(1.1)

for its interaction with a sample surface plane to lead to a photocurrent of j (r, t), product of conductivity tensor, σ , and E(ω, q), which can be expanded in power series, as follows   (1) (1) ∗ jα (r, t) = σαβ E β (ω, q)ei(q r−ωt) + σαβ E β (ω, q)e−i(q r−ωt)  2 ( ) + σαβγ E β (ω, q)E γ (ω, q)e2i(q r−ωt)  ( 2  )∗ + σαβγ E β∗ (ω, q)E γ∗ (ω, q)e−2i(q r−ωt)   (2)∗ ∗ + σαβγ E β (ω, q)E γ∗ (ω, q) + · · · . (1.2) Here, Eq. (1.2) is the second-order expansion of j (r, t) in one particular dimension, and α, β, and γ are the subscript indices corresponding to Cartesian coordi(1) is the linear transport term. The following nates. The first term described by σαβ terms are the second-order current terms; they are the first and most important loworder non-linear terms. The current terms led by the third rank conductivity tensors (2  ) (2) σαβγ and σαβγ are associated with time-dependent (ac) second harmonic generation (SHG) and time-independent (dc) difference frequency generation (DFG) processes, respectively. From Eq. (1.2), we can simply write the second-order term as,     2 ( ) (2  ) ∗ (2) jα = σαβγ E β E γ e2i(q r−ωt) + σαβγ E β∗ E γ∗ e−2i(q r−ωt) + σαβγ E β E γ∗

(1.3)

As can be seen in the first and second terms of Eq. (1.3), the second-order nonlinear current is proportional to the intensity of the light beam, i.e. quadratic in electric field, I ∝ |E(ω, q)|2 . Our focus will be on the third term which is the dc term. Note that the first and second terms, the ac (or SHG) terms, are similar to the third term; therefore, operations performed for the dc term, which is the more general case, is also applicable to the SHG term. Variations of incidence angle and polarization state of the radiation are the factors determining whether second-order polarization-dependent photocurrents are allowed or forbidden and they are strongly affected by the symmetry of the system. More specifically, for an inversion symmetric material system, among the variables in Eq. (1.2), j α (r, t) will change sign, while E β (ω, q)E γ (ω, q), E β∗ (ω, q)E γ∗ (ω, q), and E β (ω, q)E γ∗ (ω, q) will not; therefore the only possible way of validity of such

1.3 Stokes Parameters and Light Polarization

5

(2  ) (2) a j α (r, t) for an inversion symmetric system is that both σαβγ and σαβγ also change sign at inversion. Otherwise, the system should have inversion asymmetry. Indeed, both q and j α (r, t) change their signs at inversion, and as j α (r, t) ∝ q|E|2 , symmetry allows the linear coupling between the current j α (r, t) and the (2  ) (2) second-order conductivities σαβγ and σαβγ . This is due to the fact that components of conductivities are proportional to components of q. From this, we can focus on the dc term (third term in Eq. (1.3)) and rewrite it considering the effect of q: (2) jα = σαβγ E β (ω, q)E γ∗ (ω, q) (2) E β (ω, q = 0)E γ∗ (ω, q = 0) = σαβγ

+ ξαβγ δ qδ E β (ω, q)E γ∗ (ω, q).

(1.4)

On the right-hand side of the equality in Eq. (1.4), the first term has the third rank tensor, as used in Eq. (1.2). But, in order to make j α symmetric at time reversal and to make second-order response allowed, there is a need for inversion asymmetry, which can be satisfied by the material system itself and without q (or elastic interaction with photons). Therefore, when the structure of a material system is non-centrosymmetric, a polarization-dependent photocurrent will be allowed due to photogalvanic effect. The second term on the right-hand side of the equality in Eq. (1.4) depends on q (or inelastic interaction with photons) and can allow secondorder response and a polarization-dependent photocurrent in both centrosymmetric and non-centrosymmetric structures. This is due to the light momentum transfer from photons to electrons in material systems, known as photon drag effect. Both photogalvanic and photon drag effect-induced photocurrents will be explained in next sections. But, before that, we will recall some fundamental knowledge necessary to evaluate Eq. (1.4) and have a better grasp of the phenomenology.

1.3 Stokes Parameters and Light Polarization In non-linear response, variations of light polarization, particularly linear and circular polarization, have a major role to determine the dichroic phenomena and phenomenology. By using dichroic optical components, such as Fresnel rhombus, half-wave plate, and quarter-wave plate, QWP (as shown in Fig. 1.1), it is possible to modulate polarization state of radiation. As a result, the current j can have various polarization-dependent contributions, which are proportional to the Stokes parameters. These parameters are widely used in description of all light polarization-dependent phenomena, including linear and circular polarization. There are four Stokes parameters to determine the intensity and polarization: I or S0 , Q or S1 , U or S2 , and V or S3 . The first parameter S0 gives the intensity; the others are related to the polarization tensor. The parameter S1 (S2 ) describes the linear polarization term, in the coordinate frame of the x–y plane, for the x-axis and

6

1 Introduction to Light Polarization-Dependent …

y-axis alone (in the coordinate frame of + 45° rotated from the x–y plane, for + 45° rotated from the + x-direction and + 135° rotated from the + x-direction). The last parameter S3 is about the degree of circular polarization. For E(ω, q) propagating in the z-direction, rotation of a polarizer with respect to the polarization plane will simply vary these parameters:  2 |E x |2 −  E y  S1 =  2 |E x |2 +  E y 

(1.5a)

E x E y∗ + E x∗ E y  2 |E x |2 +  E y 

(1.5b)

E x E y∗ − E x∗ E y  2 = Pcirc . |E x |2 +  E y 

(1.5c)

S2 = S3 = i

The Stokes parameters in Eqs. (1.5a)–(1.5c) determine the polarization state of light. For example, for linear polarization to be varied by a half-wave plate, a variation of linear polarization degree is imposed by S1 ∝ cos 2ζ , S2 ∝ sin 2ζ , and S3 = 0. For β being the relative angle between the z-direction and the plane of E, 2β = ζ for ζ being the azimuthal angle in the x–y plane from the + x-direction. On the other hand, for circular polarization to be varied by a quarter-wave plate QWP as in Fig. 1.1, a variation of circular polarization degree is decided by S1 ∝ cos2 (2ϕ), S2 ∝ sin 4ϕ, and S3 ∝ sin 2ϕ. However, it is obvious that there is no information about angle of incidence in the Stokes parameters. Therefore, the direction of propagation of E(ω, q) with respect to the z-direction (an oblique incidence) should be additionally considered, besides the given Stokes parameters. Later, we will show that by using Jones’ calculus, all the angle dependencies shown in Fig. 1.1 can be formulated.

1.4 Second-Order Response and Symmetry Considerations Now, we will revisit Eqs. (1.2)–(1.4). The complex conjugates of some of the components of conductivity tensors in Eqs. (1.2) and (1.3) can have a sign same or opposite of their complex conductivities, which means they can have symmetric and antisymmetric components, respectively. As we deal with symmetry analysis of second-order response of a material system, for the sake of clarity, we can separate the symmetric (2) (elastic processes, “el”) and ξαβγ δ (inelastic and antisymmetric components in σαβγ processes, “in”), (2) el σαβγ = Sαβγ + Ael αβγ

(1.6a)

in in ξαβγ δ = Sαβγ δ + Aαβγ δ ,

(1.6b)

1.4 Second-Order Response and Symmetry Considerations

7

el ∗ el in in in in ∗ ∗ ∗ where, Sαβγ = Sαβγ , Ael = − Ael αβγ αβγ and Sαβγ δ = Sαβγ δ , Aαβγ δ = − Aαβγ δ . el in el Basically Sαβγ and Sαβγ δ are symmetric tensors, time reversal invariant; while Aαβγ  2 ( ) and Ain αβγ δ are not. Note that the same condition holds for the SHG term σαβγ , similar (2) to the dc term σαβγ . Then, we can use the expression given for the second-order term (2) and ξαβγ δ in Eq. (1.4) with in Eq. (1.3), and rewrite it in two parts, by replacing σαβγ Eqs. (1.6a) and (1.6b):

jα = jα (q = 0) + jα (q = 0)    el ∗ jα (q = 0) = Sαβγ + Ael αβγ E β E γ + E β E γ /2  el ∗  ∗ ∗  + Sαβγ E β E γ + E β∗ E γ /2 + Ael αβγ  el   ∗ = Sαβγ + Ael αβγ E β E γ + E β E γ /2  el  ∗ ∗  ∗ + Sαβγ − Ael αβγ E β E γ + E β E γ /2, or simply

el jα (q = 0) = Sαβγ

 ⎧ ⎨ E β E γ + E β∗ E γ∗ ⎩

+ Ael αβγ

2

 +

⎭ ⎫ E β E γ∗ − E β∗ E γ ⎬ + , ⎭ 2 2



 ⎧ ⎨ E β E γ − E β∗ E γ∗ ⎩

⎫ E β E γ∗ + E β∗ E γ ⎬

2

(1.7a)

and similarly,     in in ∗ jα (q = 0) = Sαβγ δ + Aαβγ δ qβ E γ E δ + E γ E δ /2 ∗  ∗ ∗   in in ∗ + Sαβγ δ + Aαβγ δ qβ E γ E δ + E γ E δ /2  in    in ∗ = Sαβγ δ + Aαβγ δ qβ E γ E δ + E γ E δ /2  in   ∗ ∗  in ∗ + Sαβγ δ − Aαβγ δ qβ E γ E δ + E γ E δ /2, or simply

 jα (q = 0) =

in Sαβγ δ qβ

E γ E δ + E γ∗ E δ∗

 +

Ain αβγ δ qβ

2 Eγ Eδ − 2



 +

E γ∗ E δ∗

E γ E δ∗ + E γ∗ E δ 

 +



2 E γ E δ∗ − E γ∗ E δ 2

 .

(1.7b)

Equations (1.7a) and (1.7b) provide a generalized expression for all second-order terms, and the dc and the SHG processes can be distinguished by replacing the electric field with space- and time-dependent plan waves with e2i(q r−ωt) .

8

1 Introduction to Light Polarization-Dependent …

1.5 Phenomenology for a Quarter-Wave Plate For Eqs. (1.7a)–(1.7b), we can describe the phenomenology, by giving the Stokes parameters with an oblique incidence for transmission of polarized light through a QWP as in Fig. 1.1, namely the polarization state of the radiation defined by the relative angle, ϕ, between the QWP fast axis and the polarization direction of initially linearly polarized light (we take it as p-polarized in this particular case). We can use the Jones calculus [16]. In such a case, the transmitted electric field can be seen as, ⎛

⎞ s sin ζ + p cos ζ cos θ E  = E 0 ⎝ − s cos ζ + p sin ζ cos θ ⎠ei(q r−ωt) p sin θ

(1.8)

√ √ where, s = i (sin 2ϕ)/ 2 and p = (i (cos 2ϕ) − 1)/ 2. From this, one can notice that for the dc processes in Eqs. (1.7a)–(1.7b), we only need E β E γ and E β E γ∗ , which only depend on s 2 , sp, and p 2 . Hence, the resulting current is expected to be proportional to the following components: i sin 2ϕ, sin 4ϕ, cos 4ϕ, i cos 2ϕ, and a constant. But for azimuthal angle ζ = nπ/2, i cos 2ϕ should vanish. This expectation can be verified by explicitly working out the symmetric and antisymmetric combinations of the products of E β E γ . Starting with the symmetric terms, we can get, E 02  2 cos (ζ ) cos2 (θ )[cos(4ζ ) + 3] 4  + sin(2ζ ) cos(θ ) sin(4ϕ) + sin2 (ζ ) sin2 (2ϕ)

E x E x∗ + E x∗ E x =

(1.9a)

    E 02  sin(ζ ) cos(ζ ) cos2 (θ ) cos2 (ζ ) + 1 − sin2 (2ϕ) 4  1 2 1 2 (1.9b) − cos (ζ ) cos(θ ) sin(4ζ ) + sin (ζ ) cos(θ ) sin(4ζ ) 2 2

E x E y∗ + E x∗ E y =

E 02 sin(θ )(cos(ζ ) cos(θ )[cos(4ζ ) + 3] + sin(ζ ) sin(4ζ )) (1.9c) 4

E x E z∗ + E x∗ E z =

E0  2 sin (ζ ) cos2 (θ )[cos(4ζ ) + 3] 4  + sin(2ζ ) cos(θ ) sin(4ϕ) + 8 cos2 (ζ ) sin2 (2ϕ) cos2 (2ϕ) (1.9d)

E y E y∗ + E y∗ E y =

E y E z∗ + E y∗ E z =

E 02 sin(θ )(sin(ζ ) cos(θ )[cos(4ζ ) + 3] + cos(ζ ) sin(4ζ )) (1.9e) 4 E z E z∗ + E z∗ E z =

E 02 sin2 (θ )[cos(4ζ ) + 3]. 4

(1.9f)

1.5 Phenomenology for a Quarter-Wave Plate

9

There are three non-zero antisymmetric terms: − i E 02 sin(2ϕ) cos(θ ) 2

(1.9g)

E x E z∗ − E x∗ E z =

− i E 02 sin(2ϕ) sin(ζ ) sin(θ ) 2

(1.9h)

E y E z∗ − E y∗ E z =

− i E 02 sin(2ϕ) sin(ζ ) sin(θ ) 2

(1.9i)

E x E y∗ − E x∗ E y =

The results shown in Eqs. (1.9a)–(1.9i) verified the expected polarization dependence of photocurrent for dc term for both elastic processes—photogalvanic effects of Eq. (1.7a) and inelastic processes—photon drag effects of Eq. (1.7b). For transmission of polarized light through a QWP leading to a polarization(2) (Eq. 1.6a) dependent photocurrent of dc term of jα with conductivity tensors of σαβγ and ξαβγ δ (Eq. 1.6b), the most crucial experimental parameter in defining the polarization state is ϕ, and the simplified jα can be given as, jα = C1 sin(2ϕ) + L 1 sin(4ϕ) + L 2 cos(4ϕ) + D,

(1.10)

where the constants C1 , L 1 , L 2 , and D are the nonlinear-in-electric field photocurrent contributions (at the second-order level) depending on the way they are related to ϕ. Although the dependence of C1 , L 1 , and L 2 on ϕ is clear, the case of D term is more complicated; it has a similarity to L 2 (which means cos(4ϕ) related terms and Eq. (1.9b)), in addition to ϕ independent terms of Eqs. (1.9a)–(1.9f). Here, sin(2ϕ) term includes, exclusively, photocurrent due to the circular polarization state of light; while photocurrents with sin(4ϕ) and cos(4ϕ) dependencies stem from the linear polarization state of light. This could be explained by replacing Eqs. (1.9a)–(1.9i) in Eqs. (1.7a)–(1.7b). Thus, we can see the true identity of C1 , L 1 , L 2 , and D terms. Among Eqs. (1.9a)–(1.9i), the terms related to the linear polarization are Eqs. (1.9a), (1.9b), and (1.9d) (remember the Stokes parameters S1 and S2 ). Then, we can write all the possible dc jα (α = 1, 2, and 3, being any Cartesian coordinate) A ) elastic processes in Eq. (1.7a): for symmetric ( jα SDC ) and antisymmetric ( jα DC E 02  el  2 S cos (ζ ) cos2 (θ )(cos(4ϕ) + 3) 4 111  + sin(2ζ ) cos(θ ) sin(4ϕ) + 2 sin2 (ζ ) sin2 (2ϕ)  el + S122 2 sin2 (ζ ) cos2 (θ ) cos2 (2ϕ)

j1 SDC =

− sin(2ζ ) cos(θ ) sin(4ϕ) + 8 cos2 (ζ ) sin2 (ϕ) cos2 (ϕ) el + S133 sin2 (θ )(cos(4ϕ) + 3)  el + S112 sin(2ζ ) cos2 (θ )(cos(4ϕ) + 3) + sin(2ζ )(cos(4ϕ) − 1) − 2 cos(2ζ ) cos(θ ) sin(4ϕ)]



10

1 Introduction to Light Polarization-Dependent … el + S113 [2 sin(ζ ) sin(θ ) sin(4ϕ) + cos(ζ ) sin(2θ )(cos(4ϕ) + 3)]   el + 2S123 sin(ζ ) sin(2θ ) cos2 (2ϕ) − cos(ζ ) sin(θ ) sin(4ϕ) ; (1.11a)

E 02  el  2 S cos (ζ ) cos2 (θ )(cos(4ϕ) + 3) 4 211  + sin(2ζ ) cos(θ ) sin(4ϕ) + 2 sin2 (ζ ) sin2 (2ϕ)  el + S222 2 sin2 (ζ ) cos2 (θ ) cos2 (2ϕ)

j2 SDC =

− sin(2ζ ) cos(θ ) sin(4ϕ) + 8 cos2 (ζ ) sin2 (ϕ) cos2 (ϕ)



el + S233 sin2 (θ )(cos(4ϕ) + 3)  el + S212 sin(2ζ ) cos2 (θ )(cos(4ϕ) + 3) + sin(2ζ )(cos(4ϕ) − 1) − 2 cos(2ζ ) cos(θ ) sin(4ϕ)] el + S213 [2 sin(ζ ) sin(θ ) sin(4ϕ) + cos(ζ ) sin(2θ )(cos(4ϕ) + 3)]   el + 2S223 sin(ζ ) sin(2θ ) cos2 (2ϕ) − cos(ζ ) sin(θ ) sin(4ϕ) ; (1.11b)

E 02  el  2 S cos (ζ ) cos2 (θ )(cos(4ϕ) + 3) 4 311  + sin(2ζ ) cos(θ ) sin(4ϕ) + 2 sin2 (ζ ) sin2 (2ϕ)  el + S322 2 sin2 (ζ ) cos2 (θ ) cos2 (2ϕ) − sin(2ζ ) cos(θ ) sin(4ϕ)  el + 8 cos2 (ζ ) sin2 (ϕ) cos2 (ϕ) + S333 sin2 (θ )(cos(4ϕ) + 3)  el + S312 sin(2ζ ) cos2 (θ )(cos(4ϕ) + 3) + sin(2ζ )(cos(4ϕ) − 1)

j3 SDC =

− 2 cos(2ζ ) cos(θ ) sin(4ϕ)] el + S313 [2 sin(ζ ) sin(θ ) sin(4ϕ) + cos(ζ ) sin(2θ )(cos(4ϕ) + 3)]   el + 2S323 sin(ζ ) sin(2θ ) cos2 (2ϕ) − cos(ζ ) sin(θ ) sin(4ϕ) ; (1.11c)

    el el A j1 DC = − i E 02 sin(2ϕ) sin(θ ) Ael 113 sin(ζ ) − A123 cos(ζ ) + A112 cos(θ ) ; (1.11d)     el el A j2 DC = − i E 02 sin(2ϕ) sin(θ ) Ael 213 sin(ζ ) − A223 cos(ζ ) + A212 cos(θ ) ; (1.11e)     el el A j3 DC = − i E 02 sin(2ϕ) sin(θ ) Ael 313 sin(ζ ) − A323 cos(ζ ) + A312 cos(θ ) . (1.11f)

1.5 Phenomenology for a Quarter-Wave Plate

11

In Eq. (1.11), there are 18 independent symmetric elements.1 Similarly, it is possible to write all the possible dc jα (α = 1, 2, and 3, being any Cartesian coordinate) A ) inelastic processes of Eq. (1.7a). for symmetric ( jα SDC ) and antisymmetric ( jα DC in in in in There will be 54 independent terms, such as the set of “S1111 , S1112 , S1113 , S1122 , in in S1133 , and S1123 …”, which will have a form similar to Eqs. (1.11a)–(1.11c). The inelastic symmetric terms will not be explicitly written here. But, we can express the antisymmetric inelastic terms, which can be read as:     el in A j1 DC (q) = − i E 02 sin(2ϕ) sin(ζ ) sin(θ ) q1 Ain 1113 + q2 A1213 + q3 A1313    el in − cos(ζ ) sin(θ ) q1 Ain 1123 + q2 A1223 + q3 A1323    el in + cos(θ ) q1 Ain ; (1.12a) 1112 + q2 A1212 + q3 A1312     el in A j2 DC (q) = − i E 02 sin(2ϕ) sin(ζ ) sin(θ ) q1 Ain 2113 + q2 A2213 + q3 A2313    el in − cos(ζ ) sin(θ ) q1 Ain 2123 + q2 A2223 + q3 A2323    el in + cos(θ ) q1 Ain ; (1.12b) 2112 + q2 A2212 + q3 A2312     el in A j3 DC (q) = − i E 02 sin(2ϕ) sin(ζ ) sin(θ ) q1 Ain 3113 + q2 A3213 + q3 A3313    el in − cos(ζ ) sin(θ ) q1 Ain 3123 + q2 A3223 + q3 A3323    el in + cos(θ ) q1 Ain . (1.12c) 3112 + q2 A3212 + q3 A3312 We need to remind that in Eqs. (1.11) and (1.12) are general and they are valid for any incidence and azimuthal angles without symmetry point group considerations. For the case depicted in Fig. 1.1 and for the propagation in the y–z plane (α = x), we can set ζ = π/2. Then, without loss of generality, in this specific light propagation, the elastic terms of Eqs. (1.11a)–(1.11f) can be reduced to: E 02  el el 2S cos(θ ) sin(4ϕ) + S111 [1 − cos(4ϕ)] 4  112  el + S122 cos2 (θ )(cos(4ϕ) + 3)   el − 4i Ael 112 cos(θ ) sin(2ϕ) ; 2S212 cos(θ ) sin(4ϕ)  2  el el + S211 cos (θ )(cos(4ϕ) + 3) [1 − cos(4ϕ)] + S222   − 4i Ael 212 cos(θ ) sin(2ϕ) ; {0} ;

jα DC =

(1.13)

el and S el , S el and S el , S el and In fact, there are 27 terms symmetric terms. But, the terms S121 112 113 131 123 el ; S el and S el , S el and S el , S el and S el ; and S el and S el , S el and S el , S el and S el S132 121 112 113 231 223 232 321 312 313 331 323 332 are identical, which reduce them to 18 terms. 1

12

1 Introduction to Light Polarization-Dependent …

Similarly, Eqs. (1.12a)–(1.12c) of inelastic processes can be reduced to:   A jα DC (q) = |q| i Ain 1212 sin(2ϕ) cos(θ )(sin(θ ) + · · · ;    2 − i Ain 1212 sin(2ϕ) cos (θ ) + · · · ; {0} ;

(1.14)

Here, we have not applied the crystal symmetry restrictions, such as C3 symmetry el for example, in which Ael 112 = A212 = 0 of Eq. (1.13) and antisymmetric terms can vanish in elastic processes, which would mean the photogalvanic effect will be forbidden. However, Ain 1212 terms of Eq. (1.14) do not necessarily vanish, which means the photon drag effect will be allowed. The restriction by the symmetry restrictions can be better understood in the example of graphene (Chap. 2). Next, we will show the relation of the phenomenology of the photogalvanic (elastic) and photon drag (inelastic) effects.

1.6 Photogalvanic and Photon Drag Effects 1.6.1 Photogalvanic Effect As we have shown earlier in Sect. 1.2, the first term on the right-hand side in Eq. (1.4), (2) , partially describes the photogalvanic effect, which contains the third rank tensor σαβγ PGE. Since Eq. (1.4) does not have explicit symmetry information, it is valid for global non-centrosymmetric systems only, unless there is a reduced symmetry at low dimensions, such as edges, interfaces, heterostructures, and multilayer materials. In Sect. 1.4, we provided a more general case in Eq. (1.7a), by separating (2) el and antisymmetric Ael symmetric Sαβγ αβγ parts of the conductivity tensor σαβγ . In Eq. (1.7a), jα is odd under time reversal, while the products of E β E γ can be even (and invariant under time reversal) or odd. Then, there are two options for the tensors associated with the products and combinations of E β E γ : being either even or odd under time reversal, antisymmetric or symmetric, respectively. el The symmetric part Sαβγ in Eq. (1.6a) describes the linear polarization-dependent processes with elastic scattering linked to the Stokes parameters, S1 and S2 in Eqs. (1.5a)–(1.5b) and it is known as the linear photogalvanic effect (LPGE). The complete phenomenology and the non-zero products and combinations of E β E γ are given in Eqs. (1.9a)–(1.9e), which are verifying the S1 and S2 dependence, and more specifically give the dependence on the angle of photon polarization, ϕ the angle of incidence, θ , and the azimuthal angle ζ . These dependencies were exploited in detail in Eqs. (1.11a)–(1.11c), and then in Eq. (1.13), for the specific case of ζ = π/2. el tensors in Eq. (1.13) and Note that, the LPGE terms are the ones which have Sαβγ have a combination of sin(4ϕ) and cos(4ϕ) dependencies, which correspond to L 1 and L 2 , in the simplified phenomenological photocurrent formula of Eq. (1.10).

1.6 Photogalvanic and Photon Drag Effects

13

On the other hand, the antisymmetric part Ael αβγ in Eq. (1.6a) determines the circular polarization-dependent processes with elastic scattering linked to the Stokes parameters, S3 or Pcirc in Eq. (1.5c), known as the circular photogalvanic effect (CPGE). The complete phenomenology and the non-zero products and combinations of E β E γ for the antisymmetric case are given in Eqs. (1.9f)–(1.9h), which explicitly give dependencies on the angles, ϕ, θ , and ζ . The CPGE terms in Eqs. (1.11d)–(1.11f) and (1.13) have sin(2ϕ)-dependent Ael αβγ terms. Unlike the LPGE, which is allowed only in non-centrosymmetric media, the CPGE cannot be ruled out if there is a reduced symmetry at low dimensions, such as edges, interfaces, heterostructures, and multilayer materials, which can make the medium that the light propagates a gyrotropic one to allow a difference between right- and leftcircularly polarized light-induced currents. In quantum wells, such currents induced by the CPGE are included in the category of spin photocurrents along with the spin-galvanic photocurrents (which do not have the necessity of a polarized light irradiation).

1.6.2 Photon Drag Effect In Sect. 1.2, the second term on the right-hand side in Eq. (1.4), which has q dependency and the fourth rank tensor ξαβγ δ , describes the photon drag effect, PDE. While the PGE are possible only in the materials systems without an inversion center, the dc photocurrent due to the PDE is allowed in either centrosymmetric or nonel and anticentrosymmetric media. The general case of separating symmetric Sαβγ el symmetric Aαβγ parts of the conductivity tensor is the same as the PGE, as seen in Eq. (1.7b). Similar to the case of the PGE, in Eq. (1.7b), jα is odd under time reversal. But since q is also odd, the products of E β E γ and the tensors associated can be odd or even, antisymmetric or symmetric, respectively. In comparison of the PGE and the PDE, the time reversal is the major difference. On the other hand, the PDE is not allowed at normal incidence; while, the PGE is possible at low dimensions with edge effects or nanoscale structures leading to reduction of symmetry in replacement of non-centrosymmetry (such as in the case of graphene). The PDE has been long known as the transfer of momentum from photons to electrons by the propagation of light with q leading to an electric current [17]. In an earlier theoretical work, a dc current in terms of the ac Hall effect was driven by the relation j ∝ E × B [18]. It is believed that this derivation is same as the PDE, although their mechanisms seem to be different, they point at the same phenomena. wave, B(ω, q) is Indeed, although we always take E(ω, q)of the  electromagnetic  1 always coupled to (ω, q), by B(ω, q) = |q| q × E(ω, q) , and the PDE can be modified as qδ E β Bγ∗ by taking the ac Hall contribution ∝ E β Bγ∗ . In this respect the use of terminology between the PDE and the ac (or dynamic) Hall effect is determined by the microscopic treatment (which will be discussed later in polarization-dependent

14

1 Introduction to Light Polarization-Dependent …

photocurrent in graphene), in terms of the number of photons absorbed (quantum mechanical picture, the PDE) or the action of electromagnetic fields (classical picture, the ac Hall effect). in The symmetric part Sαβγ δ in Eq. (1.6b) describes the linear polarization-dependent processes with inelastic scattering linked to the Stokes parameters, S1 and S2 in Eqs. (1.5a)–(1.5b) and it is known as linear photon drag effect (LPDE). Similar to the LPGE, in the LPDE, the complete phenomenology and the non-zero products and combinations of E β E γ can be seen in Eqs. (1.9a)–(1.9e); and an extended version of Eq. (1.11) to fourth rank tensor can account for the geometrical properties of light propagation and angle dependencies. Note that, the LPDE terms are the ones which in have Sαβγ δ tensors and have a combination of sin(4ϕ) and cos(4ϕ) dependencies, which correspond to L 1 and L 2 , in the simplified phenomenological photocurrent formula of Eq. (1.10), which is exactly the same case for the LPGE. Also, both the LPDE and the LPGE can have light polarization-independent component in longitudinal geometry, where the current is in the same direction with light incidence plane, unless there is a reduced symmetry conditions which forbids the LPGE. The antisymmetric part Ain αβγ δ in Eq. (1.6b) determines the circular polarizationdependent processes with inelastic scattering linked to the Stokes parameters, S3 or Pcirc in Eq. (1.5c), known as circular photon drag effect (CPDE). The partial phenomenology and the non-zero products and combinations of E β E γ for the antisymmetric case are given in Eqs. (1.9f)–(1.9h), which are the same as the CPGE terms. However, the sin(2ϕ)-dependent Ael αβγ terms in Eqs. (1.11d)–(1.11f) and the special case Eq. (1.13) can be compared to the sin(2ϕ)-dependent Ain αβγ δ terms in Eqs. (1.12a)–(1.12c). In the case of the CPDE, in addition to momentum transfer q, the incidence angle dependency differs from the CPGE. This becomes much clear in the special case of propagation in the y–z plane (α = x) and ζ = π/2 in Eqs. (1.13) and (1.14), in which the CPGE and the CPDE have sin θ and sin 2θ dependence, respectively. In a specific symmetry reduction to C3 , unlike the CPGE, where the tensor el in elements Ael 112 and A212 can vanish, the CPDE term with tensor A1212 may not necessarily vanish. Hence, the inelastic process may account for the nonzero response to circularly polarized light for oblique incidence but not for normal incidence. The only way to make the CPGE allowed for C3 would be applying an in-plane or out-plane electric field (drain bias or gate). As it will be discussed later, this could be the source of the Berry curvature dependent polarization-dependent photocurrent. We can also draw an immediate conclusion from Eqs. (1.11)–(1.14) that the CPGE and the CPDE have opposite behavior under time reversal: being even and odd, respectively. Note that, this is completely opposite of the case of the LPGE and LPDE, which are odd and even, respectively, under time reversal. On a separate note, the CPGE can be seen as a transfer of angular momentum from photons to the directed motion of a free charge carrier in a materials system. For visualization purposes, an analogy of a screw thread that transmits rotatory motion into linear motion can be considered for the electric current due to the CPGE [11]; while, the CPDE is about the transfer of both linear and angular momenta of

1.8 Third-Order Response

15

photons to the charge carriers and reverses its sign by changing photon helicity. We will exemplify this in the next chapter when we will discuss the phenomenology in polarization-dependent photocurrents in graphene.

1.7 Second Harmonic Generation In Eq. (1.2), the second term on the right-hand side is the time-dependent (ac) second harmonic generation (SHG) contribution, which is oscillating as e−2i (ωt) , with a frequency doubling 2ω of the incident electric field’s frequency ω. It has E β E γ and E β∗ E γ∗ dependencies, which makes the βγ subscripts invariant under the permutation (2  ) for the conductivity tensor σαβγ . For this reason, even though the symmetry description of the SHG is similar to the case for the LPGE and the LPDE under unpolarized and linearly polarized irradiation, there is no SHG current for radiation helicity. This is because there are quadratic combinations which are not sensitive to elliptically polarized light, unlike the cases given for the PGE and the PDE effects in Eq. (1.9).

1.8 Third-Order Response The low-order response in Eq. (1.2) includes first- and second-order response terms, for which the current is proportional to the second power of electromagnetic field with the same frequency ω. The next term that in Eq. (1.2) is the current that is proportional to the third power of electromagnetic field and the general case can be given as,            3 jα (r, t) = σαβγ δ E β ω1 , q 1 E γ ω2 , q 2 E δ ω3 , q 3 ei (q 1 +q 2 +q 3 )r−(ω1 +ω2 +ω3 )t  

        3 ∗ + σαβγ δ E β∗ ω1 , q 1 E γ∗ ω2 , q 2 E δ∗ ω3 , q 3 e−i (q 1 +q 2 +q 3 )r−(ω1 +ω2 +ω3 )t



            3 + σαβγ δ E β ω1 , q 1 E γ∗ ω2 , q 2 E δ∗ ω3 , q 3 ei (q 1 −q 2 −q 3 )r−(ω1 −ω2 −ω3 )t 



        3 ∗ + σαβγ δ E β∗ ω1 , q 1 E γ ω2 , q 2 E δ ω3 , q 3 e−i (q 1 −q 2 −q 3 )r−(ω1 −ω2 −ω3 )t



(1.15)

    Here, there are three electromagnetic waves: E 1 ω1 , q 1 , E 2 ω2 , q 2 , and   (3 ) (3 ) E 3 ω3 , q 3 . The σαβγ δ and σαβγ δ are time-dependent fourth rank conductivity tensors, which depend on ω1 , ω2 , ω3 , q 1 , q 2 , and q 3 . The current response jα , as well as E β , E γ , and E δ , can change sign at spatial inversion, which means third-order response is allowed in centrosymmetric systems. Several optical phenomena related to a nonzero jα are possible in third-order response in given special conditions.

16

1 Introduction to Light Polarization-Dependent …

1.8.1 Second Harmonic Generation In special case of Eq. (1.15), one of the electromagnetic waves is static E(ω = 0, q = 0) can lead to electric field-induced second harmonic generation,  3    ( )  jα (r, t) = σαβγ δ E β ω1 , q 1 E γ ω2 , q 2 E δ (0, 0)ei ([q 1 +q 2 ] r−(ω1 +ω2 )t )     (3 )∗  + σαβγ δ E β∗ ω1 , q 1 E γ∗ ω2 , q 2 E δ (0, 0)e−i ([q 1 +q 2 ] r−(ω1 +ω2 )t ) . (1.16) When ω1 = ω2 = ω and q 1 = q 2 = q, Eq. (1.16) can be equivalent to photon wave vector q-induced SHG in Eq. (1.2). Simply, qδ can be replaced by E δ (0, 0), and as in the case of q-induced SHG, there is no expectation of current generation.

1.8.2 Photoconductivity In a fashion similar to Eq. (1.16), the dc term (E β E γ∗ part) of Eq. (1.15) can be expressed as follows, (3) ∗ jα (r, t) = σαβγ δ E β (ω, q)E γ (ω, q)E δ (0, 0).

(1.17)

(3) This is the photoconductivity term, for which σαβγ δ (ω1 = ω = − ω2 , ω3 = 0) is the fourth rank dc conductivity tensor. Like in the PDE current, it can have symmetric and antisymmetric terms like Eq. (1.7b), which can correspond to linear and circular photoconductivity (also called photovoltaic Hall conductivity), respectively. These two effects, again, are induced by the static electric field E δ (0, 0), unlike the case of PDE, which is induced by photon wave vector q.

1.8.3 Third Harmonic Generation and Two-Photon Absorption In Eq. (1.15), if all the electromagnetic waves have nonzero ω and ω1 = ω2 = ω3 = ω, which means the illumination is monochromatic for all the electromagnetic waves, then it can be rewritten as,    3 jα (r, t) = σαβγ δ E β (ω, q)E γ (ω, q)E δ (ω, q)e3i (q r−ωt)   3 ∗

+ σαβγ δ E β∗ (ω, q)E γ∗ (ω, q)E δ∗ (ω, q)e−3i(q r−ωt)

 (3) + σαβγ δ E β (ω, q)E γ∗ (ω, q)E δ∗ (ω, q)e−i (q r−ωt)  (3)∗ + σαβγ δ E β∗ (ω, q)E γ (ω, q)E δ (ω, q)ei(q r−ωt) .



(1.18)

1.9 Conclusion

17

The first bracket in Eq. (1.18) describes the third harmonic generation and its (3 ) in fourth rank conductivity tensor σαβγ δ is symmetric like the symmetric part Sαβγ δ of LPDE in Eq. (1.6b). Similar to SHG, for third harmonic generation, there is no current generation for radiation helicity. (3) The second bracket in Eq. (1.18) has conductivity tensor σαβγ δ (ω1 = − ω2 = ω3 = ω) and it describes two-photon absorption of a monochromatic light, two of the electromagnetic wave frequencies are identical, while the other is negative frequency with the same magnitude. Two-photon absorption could also take place (3) for two monochromatic waves, σαβγ δ (ω1 = ω = − ω2 , ω3  = ω), with two of the electromagnetic wave frequencies have the same magnitude but opposite sign, while the other has a different frequency,   2i (q r−ωt)+i (q r−ω3 t ) (3) ∗ ∗ 3 jα−2 ph (r, t) = σαβγ δ E β (ω, q)E γ (− ω, − q)E δ ω3 , q 3 e   (3)∗ ∗ −2i (q r−ωt)−i (q 3 r−ω3 t ) . + σαβγ δ E β (ω, q)E γ (− ω, q)E δ ω3 , q 3 e (1.19)

1.8.4 Coherent Photogalvanic Effect A special case of Eq. (1.19), ω1 = 2ω; ω2 = ω3 = − ω, can lead to a dc current, (3) ∗ ∗ jα DC (r, t) = σαβγ δ E β (2ω, 2q)E γ (− ω, − q)E δ (− ω, − q) (3)∗ ∗ + σαβγ δ E β (2ω, 2q)E γ (− ω, − q)E δ (− ω, − q),

(1.20)

which is known as coherent injection of ballistic photocurrents, or coherent photogalvanic effect. This current can take place as a result of quantum mechanical interfer(3) ence of two waves with frequency ω and 2ω. The conductivity tensor σαβγ δ (2ω, ω, ω) is symmetric and can be considered phenomenologically similar to the LPDE and the SHG. As in many other optical phenomena described in this chapter, the coherent PGE also have applications in 2D materials, such as graphene.

1.9 Conclusion In this chapter, we mainly provided a necessary background about the phenomenology of low-order non-linear response resulting in a dc photocurrent expression. Briefly, there are two major effects, photogalvanic effect—PGE and photon drag effect—PDE, both having specific conditions for linearly and circularly polarized light irradiation, leading to LPGE, CPGE and LPDE, CPDE currents,

18

1 Introduction to Light Polarization-Dependent …

respectively. The origin of current due to latter effects is straightforward to understand; namely, the LPDE and CPDE do not require non-centrosymmetry of the material system; they still are dependent on time-reversal symmetry, related to bulk rather than surfaces or interfaces, and expected to vanish at normal incidence. On the other hand, the former effects require at least local non-centrosymmetry; their dependence on time reversal is different, namely the LPGE and CPGE are odd and even under time reversal, respectively. They are rather surface and interface effects, which makes them nonvanishing for normal incidence of light in real material systems with nonnegligible field effects and imperfections. All these are related to the geometry of the material system and the light propagation. As far as the state of light polarization is concerned, the PGE and PDE are indistinguishable. However, the propagation of the polarized light can ideally show difference between the CPGE and CPDE, in terms of angle of incidence θ for a reduced symmetry, as can be seen in Eqs. (1.9g)–(1.9i), (1.11d)–(1.11f), and (1.12a)–(1.12c). More specifically, for light propagation in the y–z plane (α = x) and ζ = π/2 in Eqs. (1.13) and (1.14), the CPGE and the CPDE have sin θ and sin 2θ dependence, respectively. On the other hand, the case for symmetric terms is quite complicated; hence, it is not possible to compare angle of incidence or azimuthal angle dependencies of the LPGE and the LPDE. The above summary is applicable to, and ideally possible for, any material system fulfilling the crystal symmetry requirements and time reversal symmetry. In this Brief, we only cover 2D materials and sample planes for light irradiation on sample planes (edge effects are excluded).

References 1. Glass AM, von der Linde D, Negran TJ (1974) Appl Phys Lett 25(4):233 2. Koch WTH, Münser R, Ruppel W, Würfel P (1975) Solid State Commun 17(7):847 3. Asnin VM, Bakun AA, Danishevskii AM, Ivchenko EL, Pikus GE, Rogachev AA (1979) Solid State Commun 30(9):565 4. von Blatz R, Kraut W (1981) Phys Rev B 23:5590 5. Belinicher VI, Kanaev IF, Malinovsky VK, Sturman BI (1978) Ferroelectrics 22(1):647 6. Belinicher VI, Sturman BI (1980) Sov Phys Usp 23(3):199 (or (1980) Usp Fiz Nauk 130:415) 7. Belinicher VI (1978) Phys Lett 66(3):213 8. Ivchenko EL, Pikus GE (1978) JETP Lett 27(11):604 9. Sturman BI, Fridkin VM (1992) The photovoltaic and photorefractive effects in noncentrosymmetric materials. Gordon and Breach, New York 10. Ivchenko EL, Pikus GE (1997) Superlattices and other heterostructures—symmetry and optical phenomena. Springer, Berlin 11. Ganichev SD, Prettl W (2003) J Phys Condens Matter 15:R935 12. Glazov MM, Ganichev SD (2014) Phys Rep 535(3):101–138 13. Ivchenko EL, Ganichev SD (2017) arXiv:1710.09223v1 14. Bihlmayer G, Noël P, Vyalikh DV, Chulkov EV, Manchon A (2022) Nat Rev Phys 4:642

References 15. 16. 17. 18.

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Kepenekian M, Even J (2017) J Phys Chem Lett 8:3362 Jones RC (1941) J Opt Soc Am A 31(7):488 Danishevskii AM, Kastal’skii AA, Ryvkin SM, Yaroshetskii ID (1970) Sov Phys JETP 31:292 Barlow HM (1958) Proc IRE 46:1411

Chapter 2

Graphene as the Model Low-Dimensional Photogalvanic Material

Graphene1 can be considered as the ultimate low-dimensional material; it is genuinely two-dimensional—2D, even though its ripples can make it a 2D flexible material in 3D [1]. It was discovered by mechanical exfoliation from crystalline graphite on top of a SiO2 /Si wafer in 2004 [2]. As a member of the carbon family, graphene is the basic building block for all sp2 carbon-based low-dimensional materials [3]. As depicted in Fig. 2.1, wrapping-up graphene as a sphere gives zero-dimensional—0D fullerene, rolling-up graphene layers along a particular direction and reinforcing the carbon atoms to make bonds results in one-dimensional—1D carbon nanotube, and, as a reverse process, stacking-up graphene layers will end up in bulk—3D graphite. Graphene layers have weak van der Waals force in between, which makes peeling off and isolating one layer easy. A freestanding undoped single layer graphene has a perfect hexagonal lattice as crystal structure. As it will be explained later, this is the main reason behind its linear energy–momentum dispersion of electrons and holes. By this property, graphene can be considered as a semiconductor with zero density of states, near its Dirac point, where its conduction and valence band meet, as well as a metal, since there is no actual gap in its energy band; thus, graphene is a semi-metal. This also makes electrons and holes, near the charge neutrality point in graphene, massless Dirac fermions, which is experimentally manifested in the observation of half-integer of one quantized conductance (2e2 /h) instead of integer ones [4], as a result of specific spin and valley properties of graphene at zero density of states and the exceptional topology of the graphene band structure. Furthermore, it is also possible to observe the low temperature quantum interference effects, e.g. suppression of weak-localization in graphene at surprisingly higher temperatures, due to the additional geometric

1

Abbreviation of graphene can be an issue, since single-layer or monolayer graphene (SLG or MLG) can be confused with multilayer graphene. The up-to-date standardization is 1LG for single-layer graphene, 2LG for bilayer graphene, and 3LG for trilayer graphene (see ISO/TS 80004-3:2020(en), 3.1.13).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. Eginligil and T. Yu, Second-Order Photogalvanic Photocurrents in 2D Materials, Nanoscience and Nanotechnology, https://doi.org/10.1007/978-981-97-0618-1_2

21

22

2 Graphene as the Model Low-Dimensional Photogalvanic Material

Fig. 2.1 Real space orientation of carbon atoms: (a) graphene, (b) wrapping-up graphene as a sphere gives fullerene, (c) rolling-up graphene layers in a particular direction ends up in carbon nanotube and (d) stacking-up graphene layers form graphite (here, as ABC stacking)

phase (Berry phase) as a result of carriers’ motion along the closed path near the Dirac point [5]. There are also immediate consequences of the linear dispersion of graphene in its optical response, such as flat broadband absorption (from 300 to 2500 nm). It is possible to see graphene on a SiO2 /Si by naked eye thanks to high saturable absorption—2.3% (due to the Pauli blocking), which is directly proportional to the number of graphene layers, up to six layers. This optical contrast is limited by the Fresnel law based on the interference between (few-layer) graphene and SiO2 spacer thickness and light wavelength, where each sheet can act like separate, almost isolated layer [6]. Moreover, electrostatic doping of graphene can stimulate unexpected broadband luminescence due to hot carriers, as the Fermi energy (EF ) approaches to half of the excitation energy, 2|E F | = 1.4 eV, which is the threshold energy for hot carriers to radiatively recombine by obeying the Pauli blocking [7]. Intrinsically, strong carrier– carrier interactions can stimulate multiple excitations by absorption of just a single photon. As the time scales of excitation is shorter than relaxation, thermal equilibrium cannot be reached and carriers’ (electrons) temperature will be high, leading to formation of hot carriers. Some of the distinct optical and electrical properties of graphene, mentioned above, point at some possible unprecedented outcomes in terms of photocurrent (or photovoltage). Indeed, graphene photoresponse is quite rich in many senses, including non-linear response, such as photocurrent due to second-order effects of electromagnetic irradiation mentioned in the first chapter. Here, in this chapter, we

2.1 Linear Photocurrent in Graphene

23

will briefly review optoelectronic properties of graphene, especially photocurrent as a function of light wavelength and intensity with light being unpolarized or in fixed linear polarization state. Then, we will provide necessary knowledge about the band structure, crystal symmetry, and geometric properties of graphene related to second-order response, particularly the most important information related to photocurrent due to the photogalvanic effect together with the photon drag effect. This will include the simple phenomenology upon linearly and circularly polarized irradiation at normal and oblique incidence.

2.1 Linear Photocurrent in Graphene A direct outcome of the aforementioned linear dispersion is the broadband photocurrent response; i.e. graphene can absorb light in a wide range of wavelengths and convert it to a current. This can be taken as “linear-in-electric field transport” or linear photocurrent, as the first term of Eq. (1.2). Normally, photo-excited carriers in a freestanding graphene (without any electric field (source-drain or source-gate) effect) can only have negligible photocurrent as there is no driving force for the carriers to flow to the electrodes. However, a graphene field-effect transistor or phototransistor, usually consisting of graphene layer on a SiO2 /Si substrate, can have two major photocurrent generation mechanisms: photovoltaic (PV) and photo-thermoelectric (PTE). In the absence of a source-drain voltage, the PV photocurrent can be seen as due to photogenerated carriers being accelerated by the field-effect, − ∇Vg , in the excitation region with a density of n α and the mobility η. This is the main photocurrent source for high frequency (10 Gbps) operating graphene photodetectors [8]. The PTE photocurrent relies on p–n junctions formed in a graphene phototransistor based on hot carrier transport [9], which depends on the Seeback coefficient S, the conductivity σ and variance of the electron temperature ∇Te− . Then, the current density for an electric current dominated by the hot carriers near the Dirac point can be expressed as,      j = σ E + eη(n α (r)) ∇Vg (r) + σ S(∇Te− (r)) .

(2.1)

In Eq. (2.1), the first two terms are related to the PV effect and the last term describes the hot carrier-assisted PTE photocurrent. While the PTE current can be regarded as the energy being transferred to electrons via excitation near the Dirac point, the PV current is about the excitation area on the sample, resulting on electron– hole pair generation and carrier redistribution by an electric field E = − ∇(φ − μ/e), where μ is the chemical potential, φ the local vector potential, and driving a current outside the excitation region, reaching the contacts. If there is no applied source-drain voltage, the PTE current is expected to dominate over the PV one in magnitude, by the maximum quantum efficiency provided by carrier multiplication effect (Fig. 2.2) [10]. As S(μ) and μ change signs, there is double sign change of the PTE current;

24

2 Graphene as the Model Low-Dimensional Photogalvanic Material

Fig. 2.2 (a) Graphene phototransistor gated separately in regions 1 and 2. (b) Carrier multiplication at the p–n junction leads to maximum quantum efficiency. (c) The photo-thermoelectric current due to hot carrier (HC) effect has double sign change in (d) as the EF or chemical potential μ is increased (crossing from p to n or vice versa) both the Seeback coefficient and μ changes sign following Eq. (2.1). (e)–(f) The photovoltaic current changes sign once only due to the change in μ. Adapted with permission from Song et al. [10] © 2011 American Chemical Society

while the PV current changes sign once, when the carrier polarity changes sign—both effects confirmed experimentally [9]. Another scenario can be seen if a nonzero source-drain electric field is applied (increased lattice heating and appearance of bolometric photocurrent), as shown in a photoconductivity measurement in graphene phototransistor [11], in which the PTE current is being suppressed and the PV current dominates over the PTE current near the Dirac point. This helps to identify between the PTE and PV currents, as their signs will be opposite at low doping regime under source-drain bias. Although unbiased homogeneous graphene without p and n regions is not expected to produce a PTE, an applied drain voltage can cause variation in the chemical potential μ or the Fermi energy EF in various regions of the channel, distributing hot carriers heterogeneously and leading to a PTE (this is a different PTE mechanism than previously mentioned in [10]). Away from the Dirac point, the PTE is expected to be even less important as the EF is increased deep in the conduction band. Figure 2.3 summarizes this expectation, which was confirmed by the experiment [11]. So far, we described the linear photocurrent mechanisms for unpolarized or fixed linear polarized irradiation. As graphene is a semi-metal, the response of electrons in graphene to an electromagnetic wave can be treated classically by the Drude model. Then, the electromagnetic modes of the massless Dirac fermions in graphene can vary between transverse-electric TE or transverse-magnetic TM modes depending on the Fermi level and the photon energy (excitation). This property was suggested

2.1 Linear Photocurrent in Graphene

25

Fig. 2.3 (a)–(c) Energy versus space schematics (not to scale) of a graphene p–n junction device, in case laser is incident at the junction interface, with energy being the vertical and space as the horizontal axis: (a) for no applied bias, the dc current (Idc ) and the bolometric current (IBOL ) are zero, only thermoelectric (ITE ) and photovoltaic (IPV ) currents exist and have same sign. When there is an applied bias of − 1 V, (b) the gate voltage (VG ) (homogeneously applied) being near the Dirac point (low doping) yields a difference between the ITE and IPV in both sign and magnitude, while the experimentally measured IPC is towards the drain; and (c) away from the Dirac point (high doping) IBOL dominates. (d) Polarity diagram of photocurrent direction with conditions (b) and (c) indicated by orange and green circles, respectively. Adapted from [11] © 2012 Springer Nature

to be used in optical communications as unpolarized incident light can be turned into polarized light [12]. Photoresponse of graphene to a varying polarized light is encompassed in the second-order non-linear effects described in Eq. (1.2) as the dc photocurrent. The dependence of this photoresponse on the light propagation geometry and crystal arrangement, as described by Eqs. (1.7a) and (1.7b) for photogalvanic and photon drag effects-based photocurrents, will be also specific. In order to grasp this specific phenomenology in graphene, as well as the microscopic foundations, next we will cover related geometric, structural, and band properties of graphene.

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2 Graphene as the Model Low-Dimensional Photogalvanic Material

2.2 Crystal Structure and the Brillouin Zone Graphene has hexagonal (honeycomb) crystal lattice consisting of a single layer of carbon atoms [3]. The lattice of single layer graphene—1LG has two trigonal sublattices as shown in Fig. 2.4(i). The sublattices in the unit cell of 1LG has two types of carbon atoms depending on their positions: A and B. The positions of A and B sites are not equivalent because it is not possible to connect them with a lattice vector of the form R = n 1 a1 + n 2 a2 , where n 1 and n 2 are integers and the a1 and a2 are primitive lattice vectors in real space, which can be written as,   √  √  a1 = a/2 xˆ + 3 yˆ and a2 = a/2 xˆ − 3 yˆ ,

(2.2)

where a ≈ 2.46 Å, is the lattice constant [13]; and obviously, a = |a1 | = |a2 | in Eq. (2.2), which is the length of one side of the trigonal sublattice. The conventional unit cell is√the rhombus in Fig. √ 2.4(i) containing carbon atoms at A and B sites, (a/2, −( 3/2)a) and (a/2, ( 3/2)a), respectively. The nearest carbon– √ carbon distance between carbon atoms sitting on A and B sites is aCC = a/ 3 = 1.42 Å. The reciprocal (hexagonal) lattice of 1LG is shown in Fig. 2.4(ii). The first Brillouin zone (BZ) is a hexagon and the unit vectors b1 and b2 can be rendered as,     1 ˆ 1 ˆ 2π ˆ 2π ˆ k x + √ k y and b2 = kx − √ k y , b1 = (2.3) a a 3 3 which can be obtained from a1 · b1 = a2 · b2 = 2π and a1 · b2 = a2 · b1 = 0. In Fig. 2.4(ii), there are high symmetry points within the first BZ of 1LG. Among them, the K and K ' (− K ) are two inequivalent points called as the Dirac points,2 located at the corners of the 2D hexagonal BZ along kx , corresponding to A and B sites, with positions (4π/3a, 0) and (− 4π/3a, 0), with the ┌ point at the center. Similarly, for bilayer graphene—2LG, the primitive lattice vectors can be defined as Eq. (2.2), as shown in Fig. 2.4(iii). Here, 2LG is stacked as AB Bernal type, which is a case where a second layer with A2–B2 sites is placed on the lattice in Fig. 2.4(i) (top view) with A1–B1 sites, so that all A2 are on top of B1 sites and form dimer sites making their electronic orbitals vertically coupled by a strong interlayer coupling [14]. The unit cell contains the carbon atoms in A1 and B2 sites, as in the rhombus. The reciprocal lattice of 2LG is the same as 1LG. In Fig. 2.4(iv), the hopping between sites are described between dimer and non-dimer sites, and will be discussed later in detail. It is important to note that the positions of nearest-neighbor

2

Here, only, 2D BZ points without band energy values are shown. It will be clearer why the term “Dirac point” is used for K and K' points, when we discuss the band structure in 3D and energy versus 2D momentum.

2.3 Electronic Energy Band Structure

27

Fig. 2.4 Hexagonal (honeycomb) lattice of single- (1LG) and bi-layer (2LG) graphene and the first Brillouin zone (BZ). (i) The lattice structure of single layer graphene, with dark (light) color spheres A (B) sites; a1 and a2 are the lattice unit vectors. The rhombus is the unit cell (contains A and B atoms) with the lattice constant a. (ii) The first BZ and the reciprocal lattice of 1LG and 2LG with b1 and b2 reciprocal lattice vectors. Here, K and − K are the two non-equivalent points, called the Dirac points. (iii) Top and (iv) tilted side view of the crystal lattice of 2LG. Atoms A1 and B1 on the bottom layer are shown as dark and light black spheres; A2 and B2 on the top layer are grey and empty circles, respectively. All A2 sites are right on top of B1 sites and form dimer sites. The rhombus in (iii) indicates the unit cell containing A1, B1, A2, and B1. In (iv), dotted lines are hopping between dimer sites, dashed (dash-dot) lines between A2–A1 (B2–A1)

√ atoms √ of B with respect to A atoms √ may be given as δi , i = 1, 2, 3; (0, a/ 3), (a/2, − a/2 3), and (− a/2, − a/2 3), which are used to predict the electronic band structure.

2.3 Electronic Energy Band Structure 2.3.1 Graphene—1LG The sp2 hybridization between one s orbital and two p orbitals is the reason behind the trigonal planar sublattice structure with a formation of σ bond (strong) between nearest neighbor carbon atoms sitting at A and B sites. The σ band possess a filled

28

2 Graphene as the Model Low-Dimensional Photogalvanic Material

shell (deep valence band). On the other hand, the pz orbital can bind covalently (π bond) with neighboring atoms and form a half filled (one electron to share) π band. Electronic band continuum of such a structure can be obtained qualitatively by a tightbinding approach, which is applied to 1LG [13], and also applicable to 2LG and more  √  N i (k·R A,i )  + e φ r − R layers. For Bloch states of ϕm (k, r) = (1/ N ) B,i i=1  √  N i (k·R B,i )  (1/ N ) φ r − R B,i , where N is the number of unit cells, R A,i i=1 e (R B,i ) is the position vector of the ith unit cell and φ(r − R A,i ) (φ(r − R B,i )) is the atomic orbital of the A (B) in the ith unit cell, the wave function is

j (k, r) = ψ j, A (k)ϕ A (k, r) + ψ j,B (k)ϕ B (k, r).

(2.4)

 bands associated with this wave function can be of the form E j (k) =

The energy

j |H| j / j | j , where H is the Hamiltonian. Then, by following [13], the energy bands can be expressed as, E ± (k) = ± t f (k) − t ' , √

with f (k) = eik y a/

3

+ 2e−ik y a/2

√

3

cos(kx a/2),

(2.5)

where, t (also known as γ0 ) and t ' are nearest- (A to B and B to A) and nextnearest-neighbor hopping energies, respectively [14]. Here, t is due tohopping from

 A to B sites and can be expressed as − φ r − R A,i |H|φ r − R B,i , which has a ' positive value and that from B to A is its complex conjugate. Similarly,  t is due to 

 |H|φ r − R A,i ϵ = φ r − R hopping between A sites can be expressed as A,i

   A  and between B sites ϵ B = φ r − R B,i |H|φ r − R B,i , which are equivalent since A and B sublattices are identical. For intrinsic 1LG, t ' vanishes. In Eq. (2.5), the plus sign applies to the upper π ∗ band (above the E(k) = 0, about electrons) and the minus sign is for the lower π band (below the E(k) = 0. about holes) and they are symmetric with respect to E(k) = 0, if t ' = 0. For nonzero values of t ' , the π and π ∗ bands become asymmetric (the electron–hole asymmetry). By plotting Eq. (2.5) for nonzero t and t ' , one can get the full band structure of 1LG (Fig. 2.5). The function f (k), describing the in-plane hopping in Eq. (2.5), is zero at the corners of the BZ and there are two non-equivalent (not connected by a reciprocal lattice vector) points, which are shown in the inset of Fig. 2.5, close to E(k) = 0, called K and K ' (− K ) points (or valleys, or the Dirac points), with wave vectors K ± = ± (4π/3a, 0). Then, this conical dispersion can be expressed in terms of measured momentum vector relative to the Dirac points q: k = K + q, with |q|  |K |, and the energy,   E ± (q) = ± v F |q| + O (q/K )2 ,

(2.6)

where, ν F is the Fermi velocity ~ 1.0 × 106 m/s. Since this dispersion is linear in ν F and independent of energy and momentum, it reminisces the massless Dirac fermions from quantum electrodynamics, except the ν F is about 1/300 of photon

2.3 Electronic Energy Band Structure

29

Fig. 2.5 3D electronic band dispersion in the honeycomb lattice of 1LG. Left: energy spectrum for nonvanishing t and t ' . Right: the zoom-in image of band dispersion around one of the Dirac points. Adapted from [3] © 2009 American Physical Society https://doi.org/10.1103/RevModPhys.81.109

velocity. Although the rest mass is zero, their cyclotron mass can be determined from the magneto-transport measurements (from Shubnikov–de Haas oscillations) at low temperatures, as well as the hopping parameter, t (or γ0 ) ~ 3 eV [15]. The nonvanishing t ' can be seen as a shift in energy—shift in the position of the Dirac point leading to electron–hole asymmetry. The (q/K )2 dependence in Eq. (2.6) can be read as three-fold symmetry, away from the Dirac point energy, which is known as trigonal warping of the electronic spectrum. Considering the first part of Eq. (2.6) alone, the intralayer Hamiltonian can be also expressed as,  0 π† hˆ 1−0 = ξ v , π 0 where π = px + i p y , v =

(2.7)

√  3/2 aγ0 / and ξ = ± 1 valley index. Alternatively,

polar coordinates can be used as | p|ei φ , where φ is the polar angle and a geometrical phase factor.

2.3.2 Bilayer Graphene—2LG The tight-binding model can be also applied to 2LG, with four 2pz orbitals at A1, B1, A2, and B2 sites as shown in Fig. 2.4(iii)–(iv). The hopping parameters about these sites can be described in terms of the notation of the Slonczewski– Weiss–McClure (SWM) model [13], such as γ0 , γ1 , γ3 , and γ4 (as shown in between Fig. 2.4(iv)). Similar to 1LG, the parameter γ0 in 2LG  is intralayer  coupling   γ = − φ r − R |H|φ r − R = A and B sites, with a positive value: 0 A1,i B1,i     − φ r − R A2,i |H|φ r − R B2,i . The parameter γ1 and γ3 describe interlayer

30

2 Graphene as the Model Low-Dimensional Photogalvanic Material

  

 coupling between dimer orbitals B1–A2:  |H|φ r− R B1,i ; and

γ1 = φ r− R A2,i non-dimer orbitals A1–B2: γ3 = − φ r − R A1,i |H|φ r − R B2,i . The paramcoupling eter  interlayer   between non-dimer   A1–A2,or B1–B2: γ4 =

 γ4 describes φ r − R A1,i |H|φ r − R A2,i = φ r − R B1,i |H|φ r − R B2,i . Here, γ3 and γ4 are not strictly vertical, they have an in-plane hopping component such as γ0 . In the case γ3 , each A1 (non-dimer) atom on one layer has three equidistant nearestneighbors of B2 (non-dimer) atom on the other layer; whereas in the case of γ4 , each A1 (B1)—non-dimer—atom on one layer has three equidistant nearestneighbors of A2 (B2)—non-dimer—atom on the other layer. Similar to 1LG, the next-nearest-neighbor hopping terms in 2LG can be expressed as ϵ A1 , ϵ B1 , ϵ A2 , and ϵ B2 , from which band parameters U = 1/2[(ϵ A1 + ϵ B1 ) − (ϵ A2 + ϵ B2 )], ∆' = 1/2[(ϵ B1 + ϵ A2 ) − (ϵ A1 + ϵ B2 )], and δ AB = 1/2[(ϵ A1 + ϵ A2 ) − (ϵ B1 + ϵ B2 )] can be defined, where U and δ AB are attributed to substrate or doping effects. For intrinsic 2LG, γ0 , γ1 , γ3 , γ4 , and ∆' are the major tight-binding parameters, with values of 3.16, 0.381, 0.38, 0.14, and 0.022 eV, respectively, determined by infrared spectroscopy. The corresponding band energies in 2LG in M − K − ┌ direction are shown in Fig. 2.6(a). It can clearly be seen that there are four bands (valley degenerate energy E vs. momentum p, where p± = (k − K ± )) due to one 2pz orbital on each of the four atomic sites in the unit cell, which results in a double conduction bands and a double valence bands, each of which are split by ~ 0.5 eV at the K point. The split bands are due to γ1 the interlayer coupling of dimer sites B1 and A2. In Fig. 2.6(b), the zero-gap bands are due to the non-dimer A1 and B2 sites and they are called as low-energy bands and the Fermi level in undoped 2LG lies at this point. The dependence on momentum

is quadratic at low momentum values (near K points, √ p  (2γ1 )/( 3aγ0 ) , E = p 2 /2m); and linear E ∝ p at large momentum values [14]. It is possible to express the low-energy (due to non-dimer sites) bands alone, in an effective two-band Hamiltonian:

Fig. 2.6 (a) Energy band of 2LG in reciprocal space along M − K − ┌ and the four bands. (b) Gapless bands due to the non-dimer A1 and B2 sites known as low-energy bands, with crossing points only along high-symmetry axis. Adapted from [15]

2.3 Electronic Energy Band Structure

31

H2 = hˆ 2−0 + hˆ w + hˆ 4 + hˆ ∆ + hˆ U + hˆ AB .

(2.8)

The first term in Eq. (2.8) is about massive chiral electrons, which is described by, hˆ 2−0

1 =− 2m



 2    0 π† = p 2 /2m σ nˆ 2 , 2 π 0

(2.9)

where nˆ 2 is a unit vector nˆ 2 = − (cos 2φ, ξ sin 2φ, 0), φ is the polar angle of momentum or a geometrical phase factor and σ is pseudospin vector. This is similar to Eq. (2.7) of 1LG, but quadratic rather than linear. For 2LG, Eq. (2.9) describes hopping between the non-dimer sites A1–B2, starting with a hopping between A1 to B1 (a factor of vπ , as in Eq. (2.7)), followed by a transition between B1–A2 dimer sites (by interlayer coupling γ1 —related to mass, m = γ1 /2v 2 ), and finally hopping from A2 to B2 (another factor of vπ ). Simply, intrinsic 2LG has no band gap like 1LG, but the low-energy dispersion is quadratic (rather than linear as in 1LG) with massive chiral quasiparticles rather than massless ones in 1LG. In this sense the expressions in Eq. (2.7) for 1LG and Eq. (2.9) for 2LG can be considered as chiral Hamiltonians. On the other hand, in  2LG the density of states is m/ 2π 2 per spin and per valley, and the Fermi velocity v F = p F /m, unlike the Fermi velocity v of 1LG. The second term hˆ w of Eq. (2.8) is related to interlayer coupling γ3 and causes a triangular distortion or warping of the Fermi circle around each K point, which is also possible in 1LG if there is Rashba spin–orbit coupling3 [16]. The terms hˆ 4 and hˆ ∆ are the source of electron–hole asymmetry due to γ4 and ∆' ; and the terms hˆ U and hˆ AB are about band gap opening by substrate and gating effects [14].

2.3.3 Trigonal Warping in 2LG The three-fold symmetry and triangular perturbation of (q/K )2 term in Eq. (2.6) for 1LG reflect the same nature on hˆ 2−0 (Eq. 2.8) for 2LG, at energies and momenta away from K points by losing the ideal isotropic and quadratic dependence of p 2 /2m, which is basically a consequence of f (k) of Eq. (2.5). This is the standard trigonal warping in graphene and graphite. But, there is an additional contribution to the trigonal warping in 2LG, by the interlayer coupling hopping parameter of γ3 and hˆ w of Eq. (2.8)

3

Rashba spin-orbit coupling (SOC) is about the interaction of electron spin with its orbital movement as a result of an external electric field, which can break the inversion symmetry of intrinsic 1LG and lead to trigonal warping.

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2 Graphene as the Model Low-Dimensional Photogalvanic Material

  † 2  a v 0 π 3 hˆ w = v3 − √ , 4 3 π 2 0  2   √ ξ px − i p y 2 ξ px + i p y f (k) ≈ − 3a , +a 2 82 

0 π π† 0



at low energies ε21 = (v3 p)2 −

 2 ξ v3 p 3 cos(3φ) + p 2 /2m . m

(2.10a)

(2.10b)

(2.10c)

Here, the second term of Eq. (2.10a) is of the same form of hˆ 2−0 ; but, the first term is the source of the additional trigonal warping, which gets larger when the energy and momentum gets closer to the K point. For energies below the critical energy, where a topological phase transition (Lifshitz transition) takes place at ε L = (γ1 /4)(v3 /v)2 — about 1 meV [14], iso-energetic lines get distorted and form four vertices with one at zero momentum and other three at p ≈ γ1 v3 /v 2 . This sophisticated trigonal warping in 2LG may account for novel photogalvanic and photon drag effects.

2.3.4 Berry’s Phase in Graphene and Valley Polarization The Berry’s phase, which manifests as a phase shift in electrical transport, originates from an adiabatic evolution on a closed path, either in real space or, in the case of periodic systems, in reciprocal space [17]. In periodic systems with n bands, q, and the basis function u n (q), the Berry phase can be the canonicalmomentum

expressed as dq u n (q)|i∇q |u n (q) and the associated Berry curvature as,

 n (k) = ∇q × u n (q)|i∇q |u n (q) .

(2.11)

For strictly 2D materials, such as graphene, the Berry curvature is a vector normal to the 2D plane, because the wave vector is only defined in the 2D plane. Consequently, in general, the Berry curvature will only manifest itself by coupling with another out-of-plane quantity, such as the term in Eq. (2.7) for 1LG or Eq. (2.9) for 2LG. This implies that this additional geometrical factor should be added into the energy dispersion as the anomalous velocity in terms of the (k). The anomalous velocity is the product of the Berry curvature (k) and the time derivative of the quasi, where k is the Bloch wave vector. In graphene, particle wavevector’s component dk dt each K valley has their own Berry phase, which is π for 1LG and 2π for 2LG; while, their Berry curvature are identical if the symmetry of the system is preserved. It is possible to break the symmetry of the system to acquire a net non-zero (k) by placing the graphene on a substrate, apply magnetic or electric fields. A magnetic field can induce a cyclotron motion, defining a closed path in the momentum space and the Berry curvature will have opposite sign in valleys. An electric field, on the other hand, can move an electron from one Brillouin zone to another, such that the

2.3 Electronic Energy Band Structure

33

Fig. 2.7 Valley contrasting in graphene with broken inversion symmetry. In the top panel, the Dirac cones at A and B sites with K and K ' valleys, respectively, are shown, with red color conduction band and blue color valence band, with tight-binding parameters t = 2.8 eV and ∆ = 0.28 eV. In the bottom graph, the orbital magnetic moment of a n-doped graphene with broken inversion symmetry is plotted as a function of k x . At ± 4π/3a, the Berry curvature given in Eq. (2.10) is similar to the orbital magnetic moment m(k). The first Brillouin zone is outlined by the dashed lines. Adapted with permission from [18] © 2007 American Physical Society https://doi.org/10. 1103/PhysRevLett.99.236809

momentum variation is q → q + G, where G is reciprocal lattice vector to obtain Berry curvature with opposite sign in K and K ' valleys. The non-centrosymmetry or broken symmetry in graphene can induce a valley polarization via orbital magnetic moment arising from the change in the Berry curvature [18], as depicted in Fig. 2.7. In 2D, the direction of an orbital magnetic moment is perpendicular to the sample 2 ∆t 2 , where τz = ± 1 for K and K ' plane and can be written as m(k) = τz 4 ∆3ea ( 2 +3q 2 a 2 t 2 ) 2 2 valley. This is similar to the Berry curvature Ω(k) = τz 2 3a ∆t 3/2 . 2(∆ +3q 2 a 2 t 2 )

2.3.5 Multilayer (N-Layer) Graphene (N > 2) For 2LG, the energetically more stable stacking of two layers is the one, in which the A2 site is on top of the B1 site, so-called AB or Bernal stacking. The other possible option is A2 is on A1 and B2 is on B1, which is known as AA stacking, which could be prepared in the lab. For more than two layers, say three layers, there could be two different orientations of the third layer: either the same way as AB then A, which is

34

2 Graphene as the Model Low-Dimensional Photogalvanic Material

Fig. 2.8 Graphene stacking with more than two layers, top view (a) ABA stacking, Bernal, (b) ABC stacking, rhombohedral. Each layer from bottom to top labeled as 1, 2, and 3, with corresponding A1–B1, A2–B2, and A3–B3 sites. In ABA stacking, there is no other site on top or bottom of B2 site; while there is A3 site on top of the A1 site. There is B3 site on top of the A2 site and, as in typical AB stacking, the A2 site on top of the B1 site. In ABC stacking, on top of Bernal AB stacking, in which there is the A2 site on top of the B1 site, there is the A3 site on top of the B2, as well as the B3 site on top of the A1 site

ABA, or AB then the third option of C, which is ABC. There are no other inequivalent positions where a new layer can be placed, thus all the stacking for N layers can be described in terms of ABAB… (Bernal, which is again more stable) or ABCA… (rhombohedral, observed in 3LG and more layers), as described in Fig. 2.8. Graphene with ABA Bernal stacking with N layers, and N is even, has N/2 electron-like and N/2 hole-like parabolic conduction and valence bands. For odd N, such as ABA stacked 3LG, an additional pair of conduction and valence bands with linear dispersion will coexist with the parabolic one. For ABC stacked 3LG, rhombohedral system, there will be only two pair of valence and conduction bands touching at the Dirac point. The band structure of a 3LG with the ABA Bernal stacking has two touching parabolic bands, and one with Dirac dispersion, combining the features of 1LG and 2LG. The chiral Hamiltonians of Eq. (2.7) for 1LG and Eq. (2.9) for 2LG can be generalized to N layers in relation to the Berry phase N π :  hˆ N −0 = g N

 N  0 π† , πN 0

(2.12)

where g N is a hopping parameter dependent coefficient, with g1 ∝ γ0 for 1LG, g2 ∝ γ1 for 2LG, and g3 ∝ γ3 /γ1 for 3LG.

2.4 Symmetry Point Group in Graphene Layers

35

2.4 Symmetry Point Group in Graphene Layers As can be seen in Sects. 2.3 and 2.4, intrinsic graphene—1LG and graphene bilayer— 2LG have unique crystal structure and intriguing band properties, particularly, a band gap opening in 2LG, which can be varied by substrate and gating effects. The latter effects are the key factors to observe light helicity dependent photocurrents in 1LG and 2LG, not because of the band gap opening but the symmetry reduction, since the non-centrosymmetry is an essential ingredient, especially, to observe circular photogalvanic effect—CPGE, while the PDE does not require centrosymmetry. Simply, the symmetry point group determines how the unpolarized or polarized light interacts with the material leading to the PDE- or PGE-based photocurrents.

2.4.1 Symmetry Point Group in 1LG With respect to the BZ center, the inversion symmetric 1LG belong to dihedral point group D6h , formed by the addition of the horizontal mirror plane σh to the D6 point group, which is formed from C6 and C2 (where σh ⊥ C6 axis). Since 1LG is D6h , it also has six vertical mirror plane σv that contain both the C6 and the C2 axes, and six C2 axes are included in the group. As shown in Fig. 2.9(a), 1LG has a C6 axis that contains the C–C bond and six C2 axes perpendicular to C6 being origin at the centre of the carbon ring. In freestanding 1LG, the horizontal mirror plane σh is also present, which contains all six C2 axes within and is perpendicular to the C6 axis. In addition, vertical mirror planes, σv which contain both the C6 axis and each C2 axis can be seen. Since D6h is centrosymmetric, the CPGE is not allowed. But, when the graphene flake is placed on a substrate, horizontal mirror plane symmetry will be lost as well as six C2 axes, and the symmetry will be reduced to C6v , as in Fig. 2.9(b); then, this non-centrosymmetry can make the CPGE allowed. Alternatively, in 1LG with naturally asymmetric ripples or edges, the symmetry can be reduced from D6h without substrate effects to lower symmetries, which can allow the nonvanishing term related to the interaction of the electric field (E-field) component of the electromagnetic wave normal to the local surface plane. In such a case, the symmetry point group of C2v can be intact, with rotation axes perpendicular to the edge and within the sample plane. A further reduction of edge symmetry can be due to the edge orientation while its direction differs from higher symmetry, such as Cs symmetry in the case of no substrate or C1 in the case of 1LG on a substrate [19]. In either case, light (polarized or unpolarized) incident normal to the plane of the sample can be the reason behind the PGE current along the edge.

36

2 Graphene as the Model Low-Dimensional Photogalvanic Material

Fig. 2.9 Symmetry point group in graphene 1LG (a), freestanding 1LG with D6h symmetry point group (b) 1LG placed on a substrate, symmetry is reduced to C6v

2.4.2 Symmetry Point Group in N-Layer Graphene (N > 1) In addition to substrate or edge reduction, it is possible to reduce the symmetry further in multilayer and this may be source of the CPGE, particularly characteristic of that multilayer system. The symmetry point group of graphene with N layers depends on the stacking type and on the layer number N. The Bernal stacking ABAB… can be described by either D3d group (for even N) or by D3h group (for odd N > 1). For example, in 1LG, D6h as shown in Fig. 2.9(a), but, in ABA stacked 3LG, the symmetry will be reduced to D3h , since the system will have C3 symmetry together with three C2 axis while keeping horizontal plane mirror symmetry σh . On the other hand, 2LG with AB stacking, the rhombohedral stacked 3LG and four-layer graphene can be described by the D3d group, which does not contain σh and centrosymmetric. But, placing 2LG on a substrate can reduce the symmetry to C3v and the CPGE can be allowed. Therefore, in all these cases, the CPGE is only allowed by symmetry reduction by gating, bias, or substrate. In the ultimate case of bulk graphite, the symmetry point group is D6h , as in the case of 1LG; and no CPGE-based current is allowed, unless there is gating, bias, and substrate effects. A summary of symmetry point group for all graphene layers at the first BZ center (┌), and K and K ' valleys is given in Table 2.1 [20].

2.5 Excitation by Linearly Polarized Light

37

Table 2.1 The symmetry point group at the first Brillouin zone center (┌) and K valleys for all graphene layers with and without substrate (sub), gate, or bias     ┌ (no sub) K K ' (no sub) ┌ (sub/gate/bias) K K ' (sub/gate/bias) 1LG

D6h

D3h

C6v

C3v

N even layer

D3d

D3

C3v

C3

N odd layer (ABA)

D3h

C3h

C3v

C3v

Bulk

D6h

D3h

C6v

C3v

2.5 Excitation by Linearly Polarized Light The nonlinear-in-electric field-dependent photocurrent of graphene, upon excitation by linearly polarized light, can be understood in terms of symmetry arguments of Table 2.1. Here, we will provide the basic phenomenology with symmetry consideration of linearly polarized irradiation on 1LG and 2LG, discuss possible experimental expectations. Later, the full phenomenology together with microscopy understanding for light helicity dependent photocurrent, including the dependence on degree of circularly polarization, will be provided.

2.5.1 Linearly Polarized Irradiation at Normal Incidence The selection rules for carrier and light interaction for linearly polarized light in 1LG and N-layer of graphene can result in various effects, such as valley-polarized current. This can happen at normal incidence in freestanding 1LG, for the electron states in K or K ' valleys, which belong to D3h symmetry point group [21], which can lead to a current,   j y@K = − 2χ ex e y I ; jx@K = χ ex2 − e2y I,   2 jx@K ' = − χ ex − e2y I, j y@K ' = 2χ ex e y I

(2.13)

Here, in Eq. (2.13), I is the intensity, and χ is a parameter describing electron– impurity or electron–electron interaction. In this situation, the current in each valley is due to the asymmetry of optical transitions and electron scattering by impurities as a result of the trigonal warping (Fig. 2.10(a)), with corresponding direction of anisotropy of velocity, vc , optical generation, gc , and scattering rate, W p' p . The direction of this valley current will be strongly dependent on light polarization from s- to p-polarized light. For undoped graphene on a substrate, having K valley with C3v symmetry, electron–electron scattering is the source of dominating contribution to the valley current, which can be understood by electron relaxation to from k( p) to k ' ( p ' ), near the K

38

2 Graphene as the Model Low-Dimensional Photogalvanic Material

Fig. 2.10 (a) Photocurrent generation as a result of trigonal warping in K valley as a result of electron scattering by impurities, with solid, dashed, and dotted arrows shows the direction of anisotropy of velocity, vc , optical generation, gc , and scattering rate, W p' p , respectively, in p space. (b) The case scattering of electrons with momenta p and k into the states with momenta p' and k', respectively, near K point. (c) For a fixed polarized light (E-field), in momentum space, electron relaxation is depicted by dark color and red circle showing the alignment of the photo-excited electrons’ momenta by the E-field near K point leading to valley current. Adapted with permission from [21] © 2011 American Physical Society https://doi.org/10.1103/PhysRevB.84.195408

point (Fig. 2.10(b)). In this case, absorption of linearly polarized light can align the electrons’ momenta and the photo-excited electrons can anisotropically scatter by the resident electrons in the valence band, as shown for a fixed polarization state of light in Fig. 2.10(c). This leads to a net electric current if the momentum scattering and relaxation times are long enough, which can be realized for 1LG on a curved substrate with local effective out-of-plane magnetic fields that are directed oppositely for electrons in opposite valleys [21]. In order to test this theory, the experimental challenges of sample preparation should be overcome. On the other hand, for 2LG, AA stacking can manifest a symmetry point group of 1LG—D6h and AB stacking D3d , which result in isotropic conductivity tensor dependent nonlinear photocurrent upon linearly polarized irradiation at normal incidence. However, for twisted and slid 2LG, the symmetry can be reduced thanks to a twist angle and sliding vector, from which the conductivity tensor can be rendered anisotropic [22]. The broken spatial symmetries are expected to lead to a finite optical Hall conductivity. For instance, introducing a twist angle can break the translational and mirror symmetry, reducing the symmetry to D6 or D3 depending on the twist angle. On the other hand, introducing a slid angle can reduce the symmetry to C2h . These symmetries can be further reduced to C6 or C3 , and Cs , for twist and slid angle case, respectively. Consequently, the symmetric conductivity tensor elements σx x and σ yy will have same sign and magnitude for both twist and slid angle 2LG; while the asymmetric elements σx y and σ yx are expected to have same magnitude and opposite sign for twist angle and different magnitude and same sign for slid 2LG, in case there is no mirror symmetry [22]. This is expected to be observed as a net transverse (Hall) photocurrent in twist and slid 2LG.

2.6 Excitation by Circularly Polarized Light

39

Fig. 2.11 Linearly polarized irradiation at an oblique incidence in graphene. (a) Excitation by photon energy ω, with light linear momentum q being transferred through the linearly polarized E-field, leading to a photocurrent jx and j y —LPDE; there is no necessity of non-centrosymmetry and substrate. (b) Similar irradiation process as (a), except no light linear momentum transfer, but instead non-centrosymmetry and linearly polarized light-driven jx and j y —LPGE; substrate is necessary to break the centrosymmetry to observe this current

2.5.2 Linearly Polarized Irradiation at Oblique Incidence When a linearly polarized light is obliquely incident on a graphene sample plane, the phenomenology described in Chap. 1 will apply. As shown in Fig. 2.11, there will be two possibilities of obtaining current on the surface of the graphene—1LG plane in the form jx and j y : either via light linear-momentum q transfer of the linearly polarized E-field, which do not require a substrate, or asymmetry-induced interaction of linearly polarized E-field with non-centrosymmetric graphene/substrate system [23]. In either case, linearly polarized E-field can be described by S1 and S2 Stokes parameters. The former case in Fig. 2.11(a) is the linear photon drag effect—LPDE current; while, the latter one in Fig. 2.11(b) is the linear photogalvanic effect—LPGE current. This is also applicable to 2LG, by following the symmetry consideration shown in Table 2.1.

2.6 Excitation by Circularly Polarized Light The symmetry arguments of Table 2.1 also apply for nonlinear-in-electric fielddependent photocurrent of graphene upon excitation by circularly polarized. Here, we will introduce the basic phenomenology and symmetry consideration for 1LG and 2LG sample planes.4

4

Edge effects will not be discussed here.

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2 Graphene as the Model Low-Dimensional Photogalvanic Material

Fig. 2.12 The Berry curvature [∇k × Aα (k)]z for K and K' points in 1LG irradiated by light for the conduction band. Adapted with permission from [24] © 2009 American Physical Society https://doi.org/10. 1103/PhysRevB.79.081406

2.6.1 Circularly Polarized Irradiation at Normal Incidence At normal incidence, the chiral geometric phases (such as Berry-like, mentioned in Sect. 2.3) can also lead to a nonvanishing net current upon excitation by circularly polarized light, similar to the linearly polarized irradiation. Application of a nonzero dc bias (or even the existence of two electrodes) can break and reduce the symmetry, open a band gap at the K point, and as a result, it can be the source of a transverse (Hall) current [24]. This is not a result of any impurity or electron–electron scattering; but, instead, it is like a conventional photovoltaic effect, in which circularly polarized light at normal incidence acts as ac electric field and replace the electric field in Eq. (2.1). This longitudinal current can produce a Hall current upon a longitudinal dc bias (with a time-dependent (ac) gauge potential A AC (t) = E · t ∝ (cos Ωt, sin Ωt), where E is dc bias field and the frequency Ω = 2π/T), which can switch sign by changing the polarization state of the circularly polarized light (right-circular (cos Ωt, sin Ωt) and left-circular (cos Ωt, − sin Ωt)) or the sign of dc bias. This Hall current can be expressed in terms of the Berry curvature (as in Eq. 2.11) in the honeycomb lattice of 1LG irradiated by circularly polarized light, which comes from the time-dependent and periodic (Floquet) states in K and K ' points, [∇k × Aα (k)]z , as shown in Fig. 2.12 [24].

2.6.2 Circularly Polarized Irradiation at Oblique Incidence When a circularly polarized light is obliquely incident on a graphene sample plane, the phenomenology described in Chap. 1 will apply. As shown in Fig. 2.13, there will be two possibilities of obtaining current on the surface of the graphene—1LG plane in the form of transverse current j y : either via light linear-momentum q transfer of the

2.6 Excitation by Circularly Polarized Light

41

Fig. 2.13 Circularly polarized irradiation at an oblique incidence in graphene. (a) Excitation by photon energy ω, with light linear momentum q being transferred through the circularly polarized E-field, with the Stokes parameter Pcirc , leading to a transverse photocurrent j y —CPDE; there is no necessity of non-centrosymmetry and substrate. (b) Similar irradiation process as (a), except no light linear momentum transfer, but instead non-centrosymmetry and circularly polarized lightdriven transverse photocurrent j y —CPGE; substrate is necessary to break the centrosymmetry to observe this current

linearly polarized E-field, which do not require a substrate, or non-centrosymmetry induced circularly polarized light irradiation on graphene/substrate system [23]. In either case, circularly polarized E-field can be described by S3 or Pcirc Stokes parameter. The former case (Fig. 2.13(a)) is the circular photon drag effect—CPDE current; while, the latter one (Fig. 2.13(b)) is the circular photogalvanic effect— CPGE current. There is a sign difference of j y for the same state of circularly polarized radiation between the CPGE and the CPDE, since the time reversal for the CPGE (CPDE) is even (odd). As it will be shown later, microscopically, the conductivity type (n or p) of the sample and the radiation helicity determine the sign of the CPDE. On the other hand, the sign of the CPGE is merely dependent on the sample asymmetry (symmetry considerations). The centrosymmetry related effect and the non-centrosymmetry related effect have opposite signs. These features can be also observed in 2LG, by following the symmetry consideration shown in Table 2.1.

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2 Graphene as the Model Low-Dimensional Photogalvanic Material

References 1. Fasolino A, Los JH, Katsnelson MI (2007) Nat Nanotechnol 6:858 2. Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, Grigorieva IV, Firsov AA (2004) Science 306(5696):666 3. Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, Geim AK (2009) Rev Mod Phys 81:109 4. Zhang Y, Tan YW, Stormer HL, Kim P (2005) Nature 438:201 5. Tikhonenko FV, Horsell DW, Gorbachev RV, Savchenko AK (2008) Phys Rev Lett 100:056802 6. Bonaccorso F, Sun Z, Hasan T, Ferrari AC (2010) Nat Photonics 4:611 7. Chen CF, Park CH, Boudouris B, Horng J, Geng B, Girit C, Zettl A, Crommie MF, Segalman RA, Louie SG, Wang F (2011) Nature 471:617–620 8. Mueller T, Xia F, Avouris (2010) Nat Photonics 4:297 9. Gabor NM, Song JCW, Ma Q, Nair NL, Taychatanapat T, Watanabe K, Taniguchi T, Levitov LS, Jarillo-Herrero P (2011) Science 334(6056):648 10. Song JCW, Rudner MS, Marcus CM, Levitov LS (2011) Nano Lett 11:4688 11. Freitag M, Low T, Xia F, Avouris P (2013) Nat Photonics 7:53 12. Bao Q, Loh KP (2012) ACS Nano 6(5):3677 13. Wallace PR (1947) Phys Rev 71(9):622 14. McCann E, Koshino M (2013) Rep Prog Phys 76:056503 15. Latil S, Henrard L (2006) Phys Rev Lett 97:036803 16. Rakyta P, Kormanyos A, Cserti J (2010) Phys Rev B 82:113405 17. Xiao D, Chang MC, Niu Q (2010) Rev Mod Phys 52:1959 18. Xiao D, Yao W, Niu Q (2007) Phys Rev Lett 99:236809 19. Glazov MM, Ganichev SD (2014) Phys Rep 535(3):101–138 20. Malard LM, Guimarães MHD, Mafra DL, Mazzoni MSC, Jorio A (2009) Phys Rev B 79:125426 21. Golub LE, Tarasenko SA, Entin MV, Magarill LI (2011) Phys Rev B 84:195408 22. Do VN, Le HA, Nguyen VD, Bercioux D (2020) Phys Rev Res 2:043281 23. Jiang C, Shalygin VA, Panevin VYu, Danilov SN, Glazov MM, Yakimova R, Lara-Avila S, Kubatkin S, Ganichev SD (2011) Phys Rev B 84:125429 24. Oka T, Aoki H (2009) Phys Rev B 79:081406(R); (2009) Phys Rev B 79:129901(E)

Chapter 3

Light Helicity Dependent Photocurrent in Graphene Planes

In this chapter, specific phenomenology of second-order nonlinear-in-electric fielddependent photocurrents in graphene will be introduced together with the microscopic theories of the effects responsible for generation of such photocurrents, namely the photogalvanic and photon drag effects. Detailed analysis for the phenomenology and microscopic understanding will be provided with comparisons to various perspectives presented in the previous chapters, e.g. crystal symmetry and time reversal symmetry, as well as measurement geometry conditions. The nonlinearin-electric field-dependent photocurrent data of graphene photodetectors and phototransistors (excluding the edge effects) available in literature will be discussed mainly in terms of photon energy excitation dependence, starting from classical to quantum frequency range.

3.1 Phenomenology of Light Helicity Dependence in Graphene Plane Symmetry consideration, or more specifically the non-centrosymmetry, mentioned in Table 2.1, is the necessary condition for the existence of the photogalvanic effect— PGE in the case of light incident on graphene plane at an oblique incidence. The PGE photocurrent is not allowed without a substrate (or an interface) for 1LG. On the other hand, even without a substrate, it is allowed in 2LG due to inherent broken inversion symmetry. First, we will show the case for 1LG.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. Eginligil and T. Yu, Second-Order Photogalvanic Photocurrents in 2D Materials, Nanoscience and Nanotechnology, https://doi.org/10.1007/978-981-97-0618-1_3

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3 Light Helicity Dependent Photocurrent in Graphene Planes

3.1.1 Photogalvanic Effect in 1LG As described in Figs. 2.11(b) and 2.13(b), the currents due to the LPGE and the CPGE at an oblique incidence in the x–z plane can be expressed in terms of linear (symmetric) and circular (asymmetric) terms of Eqs. (1.9a)–(1.9i) as follows, jx = χl j y = χl

E y E z∗ + E y∗ E z 2

E x E z∗ + E x∗ E z , 2 ( | |2 ) + χc Pcir c eˆ x |E x |2 + | E y | .

(3.1a) (3.1b)

Here, eˆ = eˆ x + eˆ y + eˆz ≡ q/|q| is a unit vector in the direction of light propagation, the two parameters χl and χc describe the LPGE and CPGE, respectively. The linear current flows along the projection of the E-field of light impinging onto the 1LG surface; therefore, it can have components both in the x- and y-direction (along the light incidence plane). This follows Eq. (1.10), in which all the contributions as a function of angle of photon polarization can be written in terms of C1 , L 1 , L 2 , and D, for both the PDE and PGE terms. In the case of 1LG, for small incidence angle of θ0 (where sin θ0 ≈ θ0 ), Eqs. (3.1a)–(3.1b) can be simplified as, jx = L 2−P G E cos(4ϕ) + (D P G + D N P ),

(3.2a)

j y = C1−P G E sin(2ϕ) + L 1−P G E sin(4ϕ).

(3.2b)

Here, the circular PGE current is C1−P G E = (P G)C1 |E|2 θ0 . The linear PGE current is L 1−P G E = (P G) L1 |E|2 θ0 for transverse photocurrent (y-direction) and L 2−P G E = (P G) L2 |E|2 θ0 for longitudinal photocurrent (x-direction), where the two contributions L 1−P G E and L 2−P G E are expected to have the same magnitude. Here, the component (D P G + D N P ) = ((P G) D + (N P) D )|E|2 θ0 , where (P G) D is partly related to (P G) L2 , in the sense (P G) L2 can be expressed in terms of θ independent and cos2 (θ )-dependent terms of Eq. (1.13), namely [1 − cos(4ϕ)] and [cos(4ϕ) + 3] terms, respectively; while (N P) D is the polarization-independent contribution. Further, we can deduce from Eqs. (3.1) and (3.2) that, at ϕ = 0, D P G = − L 1−P G E for s-polarized light state and D P G = 3L 1−P G E for p-polarized light state. On top of these expectations, the E-field contributions and the crystal symmetry considerations should be included. In 2D systems, such as quantum wells and interfaces, the electron wave function is extended over many atomic layers and can be easily affected by the E-field component in the z-direction and the PGE is allowed. However, in an ideal flat 1LG, the PGE should almost vanish, since the E-field component in the z-direction is negligible. As it is stated in Chap. 1, the polarization behavior of the PGE and the PDE are similar. Therefore, even the PGE exists in 1LG (due to the reduced symmetry), it is usually masked by the PDE, unless the PDE becomes weaker as the excitation energy of

3.1 Phenomenology of Light Helicity Dependence in Graphene Plane

45

incident photons increases (see Sect. 3.5). In order to understand the PGE in 1LG, a fine understanding of the PDE is necessary.

3.1.2 Photon Drag Effect in 1LG As mentioned earlier, unlike the PGE, the symmetry of the system does not restrict the photon drag effect—PDE, and can take place in centrosymmetric systems as well. As described in Figs. 2.11(a) and 2.13(a), the currents due to the LPDE and the CPDE at an oblique incidence in the x–z plane can be expressed in terms of linear (symmetric) and circular (asymmetric) terms of Eqs. (1.9a)–(1.9i) as follows, | |2 | |2 |E x |2 − | E y | |E x |2 + | E y | + T2 qx , jx = T1 qx 2 2 j y = T2 qx

E y E z∗ + E y∗ E z 2

( | |2 ) − T˜1 qx Pcir c eˆz |E x |2 + | E y | .

(3.3a) (3.3b)

Here, as the light propagation is in the x–z plane, q y = 0. There are three parameters T1 and T2 and T˜1 , which describe the LPDE and CPDE, respectively. Instead of five parameters expected from Eqs. (1.9a)–(1.9i), there are only three. Because, in flat 1LG the component of the E-field in the z-direction and qz should almost vanish, and for the sake of clarity, these two components are omitted. Note that, the major difference of this ultimate 2D system from bulk counterparts is that the symmetry is reduced to C6v at ┌ point (see Table 2.1), which leads to a difference between the zand − z-directions; this is the source of nonvanishing CPDE in the y-direction, under oblique incidence only. This broken symmetry could be provided by the presence of a substrate or gate and it is the reason of sign reversal of the CPDE when the incidence angle changes sign (odd under time reversal). In another words, the second part of Eq. (3.3b) explains the phenomenology of the CPDE in 1LG, which is described in Fig. 2.13(a) as T˜1 (and described by Pcirc )—due to transfer of both light linear and angular momenta to electrons. It experiences a sign reversal upon changing the sign of photon polarization. Photon helicity consists of circular and linear polarization, and in addition to the circular photocurrent due to the CPDE, there is also linear photocurrent in the y-direction, due to the LPDE (described by the Stokes parameter S2 (∝ sin(2ζ )) and constant T2 ). The latter is similar to the LPGE, in the y-direction (Eq. 3.1b), a component of the linear current flows along the projection of the E-field of light impinging onto the 1LG surface. On the other hand, the current in the x-direction has two contributions: the linear polarization-dependent term T2 , described by the Stokes parameter S1 (∝ cos(2ζ )) and the polarization-independent term T1 . This description of the PDE is in agreement with Eq. (1.10), in which all the contributions

46

3 Light Helicity Dependent Photocurrent in Graphene Planes

as a function of angle of photon polarization can be written in terms of C1 , L 1 , L 2 , and D; and again, for small incidence angle of θ0 (where sin θ0 ≈ θ0 ), Eqs. (3.3a)–(3.3b) can be simplified as, jx = L 2−P D E cos(4ϕ) + (D P D + D N P ),

(3.4a)

j y = C1−P D E sin(2ϕ) + L 1−P D E sin(4ϕ).

(3.4b)

Here, the circular PDE current is C1−P D E = (P D)C1 |E|2 θ0 . The linear PDE current is L 1−P D E = (P D) L1 |E|2 θ0 for transverse photocurrent (y-direction) and L 2−P D E = (P D) L2 |E|2 θ0 for longitudinal photocurrent (x-direction), where the two contributions L 1−P D E and L 2−P D E are expected to have the same magnitude. Here, the component (D P D + D N P ) = ((P D) D + (N P) D )|E|2 θ0 , where (P D) D is partly related to (P D) L2 , in the sense (P D) L2 can be expressed in terms of θ -independent and cos2 (θ )-dependent terms of Eq. (1.13), namely [1 − cos(4ϕ)] and [cos(4ϕ) + 3] terms, respectively; while (N P) D is the polarization-independent contribution. This expectation is the same as the case for the PGE and they can’t be distinguished without crystal symmetry considerations. Most specifically, the difference between (P G)C1 and (P D)C1 ; (P G) L1 and (P D) L1 ; and (P G) L2 and (P D) L2 stem from the existence (absence) of centrosymmetry in PDE (PGE).

3.1.3 Distinguishing the PGE from the PDE In brief, linear and circular PGE currents can be distinguished in terms of angle of photon polarization, ϕ, dependence of sin(4ϕ) and sin(2ϕ), respectively, as in Eq. (3.2); similarly, the LPDE and CPDE are distinct from each other phenomenologically by Eqs. (3.4a)–(3.4b). However, the separation of the PGE photocurrent from the PDE is not straightforward. There is no difference between the LPGE and the LPDE, except under time reversal and where there is a symmetry reduction. If the symmetry is reduced, the only difference can be read from the polarization dependence in transverse or longitudinal geometries. Namely, the LPDE (LPGE) is (not) allowed in transverse geometry; but in longitudinal geometry both effects can be observed. On the other hand, the difference between the CPGE and the CPDE current can be read from the excitation energy dependence, as well as the difference in symmetry and angle of incidence dependence.

3.2 Microscopic Theory of the PDE and the PGE in 1LG: Classical …

47

3.2 Microscopic Theory of the PDE and the PGE in 1LG: Classical vs. Quantum Regime The theory behind the photocurrent due the photon drag effect—PDE and the photogalvanic effect—PGE can be understood in terms of electron dynamics as a function of excitation wavelength (photon energy, ω) and scattering events (momentum relaxation time, τ ). The PDE current can be observed at classical frequency regime of ω 1 in the classical regime of ω > 1, in 1LG, there would be intraband transitions with carrier scattering by phonons or impurities. The Drude absorption of ω requires the conservation of energy and momentum law, which makes only certain transitions possible from the initial states with energy εa to final states with εb (εb = εa + ω). The matrix elements describing these transitions

50

3 Light Helicity Dependent Photocurrent in Graphene Planes

Fig. 3.1 Joint action of E and B field on charge carriers (holes in this case) for E || or E x−y for two different times, t1 and t2 , separated by half a period T /2—EB mechanism, based on the Lorentz force, FL , resulting in dc current, j (a) for linearly polarized irradiation, j is longitudinal only (b) for circularly polarized irradiation (σ+ radiation for this case), the retardation results in two components j A and jC (C1−P D E and D P D + D N P of Eq. (3.4), respectively). The dashed circle is the orbit of holes caused by the ac E-field and C1−P D E is known as current due to the circular ac Hall effect. (c) and (d) Show top view of panel (b) for σ+ and σ− radiation, respectively. Adapted with permission from [2] © 2013 Elsevier B.V.

Fig. 3.2 Possible intraband transitions (either valence band VB or conduction band CB) under the photon drag effect in the quantum frequency range ( ω ≤ |E F |) and ωτ >> 1 are summarized for the case of n-type 1LG. The carriers at an initial state with energy εa make a transition to a final state with energy εb by phonon or impurity scattering via virtual intermediate states either in the VB or CB. Solid and dashed arrows show electron–photon interaction and electron scattering caused by phonons or impurities, respectively. (a) Electron makes a transition to an intermediate state with energy εb , followed by scattering and sign change of its momentum (being odd). (b) Electron makes a transition to an intermediate state with energy higher than εb and change in its momentum, followed by relaxation to εb . (c) Electron (by conserving its momentum sign) relaxes to a virtual intermediate state in the VB (or hole makes a transition to εa ), followed by a virtual transition to εb in the CB by changing its momentum sign. (d) Electron (by changing its momentum sign) relaxes to a virtual intermediate state in the VB, followed by a virtual transition to εb in the CB by conserving its momentum sign. Adapted with permission from [2] © 2013 Elsevier B.V.

3.2 Microscopic Theory of the PDE and the PGE in 1LG: Classical …

51

can be considered within the second-order perturbation [4] and the electron–photon interactions for emission and absorption processes will be the same, resulting in a q same transition matrix of Mb,a for photon wave vector q, with intermediate states anywhere in the band (described in detail in Fig. 3.2), following the Fermi–Dirac statistics and the distribution functions of f (εa ) and f (εb ). Then, the current can be given by Fermi’s golden rule, j = (8eπ/ )

∑

| q |2 [v b τ1 (εb ) − v a τ1 (εa )]| Mb,a | [ f (εa ) − f (εb )]δ(εb − εa − ω).

a,b

(3.10) The LPDE terms of Eq. (3.9) for the allowed transitions in terms of the current of Eq. (3.10) yield, 32e3 v 4 ∑ εb , [ f (εa ) − f (εb )] ω4 b (εa + εb )2

(3.11a)

εb2 + εa2 + ( ω)2 8e3 v 4 ∑ . f − f [ (ε ) (ε )] a b ω4 a εb (εa + εb )2

(3.11b)

T1 = − T2 = −

Here, T1 and T2 have similar frequency dependence and corresponding LPDE has 1/ω2 dependence and decrease monotonically. However, the CPDE has 1/ω3 τ dependence and T˜1 is nonmonotonic; it increases first, gets its highest value at ωτ ∼ 1, then decreases.

3.2.4 Microscopic Origin of the PGE-Based Current in 1LG As mentioned earlier, non-centrosymmetry is the necessary ingredient to observe the PGE in 1LG as well as any other low-dimensional system. Also, another important ingredient is the the E-field component in the z-direction acting on the electron wavefunction in 1LG (as mentioned in Sect. 2.3), from which there would be no PGE if there was only π -orbitals. As 1LG has sp2 hybridization, there are σ -orbitals forming deep valence bands, which can interfere with the Drude-like indirect intrabands (the bands that are previously mentioned as source of the PDE current) to give rise to the PGE current. Briefly, the PGE in 1LG for ωτ >> 1 and ω ≤ |E F | is due to the quantum interference of the optical transitions responsible for the PDE effect without light momentum transfer and the indirect intraband transitions with intermediate states in distant valence bands, as depicted and described in Fig. 3.3. These transitions take place for both the LPGE and the CPGE. The PGE current due to this quantum interference consists of two different dependence on the transition matrix. First is the one similar to the PDE-like transitions

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3 Light Helicity Dependent Photocurrent in Graphene Planes

Fig. 3.3 Quantum interference mechanism of the indirect intraband transitions via distant bands, with the Drude-like indirect intraband transitions shown in Fig. 3.2 for the PDE effect, responsible for the observation of the PGE in n-type 1LG is illustrated with two possible scenarios, in the quantum frequency range ( ω ≤ |E F |) and ωτ >> 1. Solid and dashed arrows show electron– photon interaction and electron scattering events, respectively. The π -orbitals form the same band as in Fig. 3.2 with irreducible representation of P3− band and the σ -orbitals form the distant (deep valence) with P3+ band. (a) An electron at state with energy εa relaxes to an intermediate state at a distant (∆ ~ 10 eV) valence band, followed by scattering to a final state with energy εb without sign change of its momentum, (b) same transition with sign change of its momentum. Adapted with permission from [2] © 2013 Elsevier B.V. (1) q=0

(with q = 0), Mb,a , which is linearly proportional to the wave vectors b and a. This is followed by the indirect transitions via distant bands shown in Fig. 3.3, (2) Mb,a , for which (due to large energy difference ∆ ~ 10 eV from the distant bands) its dependence on b and a is almost negligible. The nonzero part of total transition |2 | | (1) q=0 (2) | + Mb,a rate for this interference Wb,a ∝ |Mb,a | can be simply expressed as the ] [ (1) q=0 (2)∗ (1) (2) momentum odd part, Wb,a ∝ 2Re Mb,a Mb,a [2], where Mb,a and Mb,a account for transitions related to E || (or E x−y ) and E z , respectively. This indicates that, even

3.2 Microscopic Theory of the PDE and the PGE in 1LG: Classical …

53

(2) the contribution due to Mb,a is quite small, it is the reason behind the PGE current. (2) The intermediate bands in Mb,a processes determined by the irreducible representa+ tion of P3 are crucial for the PGE, which is even under horizontal mirror plane σh , unlike the P3− of the conduction and valence band states in the PDE, which is odd under σh . This can be understood by the sign change of momentum in Fig. 3.3 as a result of different velocity, for linear dispersion near the Dirac points. Therefore, the quantum interference and the resulting PGE-based current is only possible for the case when there is no σh , which requires a substrate under 1LG. q The transition matrix of Mb,a in the case of the PGE is different than the PDE (which only deals with π -orbitals), as there involves quantum interference processes as a result of short-range scattering, allowing the electron transition between σ - and q,σ (2)∗ π -orbitals, which can be denoted as Mb,a (or as expressed earlier, Mb,a ). With the and q = 0 in assumption of electron dispersion is the same [for σ - and π -orbitals ] (1) q=0 (2)∗ Mb,a can be rewritten in the PGE, the total transition rate Wb,a ∝ 2Re Mb,a Eq. (3.10) to obtain the PGE current,

j = (8eπ/ )

∑

} { q=0 q=0,σ [v b τ1 (εb ) − v a τ1 (εa )]2Re Mb,a Mb,a

a,b

[ f (εa ) − f (εb )]δ(εb − εa − ω). q=0

(3.12) q=0,σ

Then, as their dispersion is the same, Mb,a and Mb,a can be taken as the same, in which the electron–photon interaction (i emAz0 cp0 ) and the electron scattering by an [ ] impurity or a phonon ( V21 1 + ei (ϕa +ϕb ) ) in a and b states are considered [4]. This makes q=0

Mb,a ≈ i

]2 ω e A z p0 V1 [ 1 + ei (ϕa +ϕb ) , 2m 0 c ∆2

(3.13)

where A z is the vector potential in the z-direction, p0 the momentum matrix element, and V1 is the scattering matrix element between the σ and π bands. Casting Eq. (3.13) into Eq. (3.12), the photocurrent expression for the CPGE can be obtained as, χc = − ev

4π w ∑ [εa τ1 (εb ) + εb τ1 (εa )] [ f (εa ) − f (εb )]δ(εb − εa − ω), εa + εb a,b (3.14)

e p0 , and V0 is the scattering matrix element within the π band. where w = 2π m 0 cω2 ∆2 The expression in Eq. (3.14) is general and can be seen as ∝ 1/ω for the classical 0 V1 > E F for interband range of ω ≤ |E F |, as it can be simplified as χc = − 8 epm00 E F . In the case of the measurecurrent, as it can be simplified as χc = − 8 epm00 ments in Sect. 3.4, with ω ~ 100 meV, |E F |/ ω ~ 3, , with the conduction band in K valley is filled more than that of the K ' valley

they are more pronounced in a semiconductor. There are several 3D semiconductors, which show valley polarization at low temperatures due to anisotropy in different crystal orientations, such as diamond and silicon. However, these materials still lack a strong coupling between the valley index and the external field, hindering manipulation of valley DOF and write-in read-out in a specific valley. For a valleytronic 2D material, the valleys are almost degenerate but still inequivalent due to non-centrosymmetry. The valley index is associated with the collective response of the system, so-called pseudospin, which acquires intrinsic magnetic moment. Therefore, it is possible to manipulate the valley DOF via external fields, manifesting itself as extrinsic strain, optical and/or magnetic response in 2D semiconductors, particularly the ones belonging to the family of monolayer group-VI transition metal dichalcogenides (TMDs) with strong SOC. Inequivalent K and K ' valleys in 2D TMDs can form discrete valley states, which can realize a valley qubit (Fig. 4.1). Therefore, the discovery of these materials was groundbreaking in the field of valleytronics. Monolayer semiconducting TMDs have been studied theoretically (for a detailed review see [3]); they were expected to exhibit electric field- and magnetic fielddependent robust valley-selective interactions, which have been observed even at room temperature. Their direct band gap at monolayers offers a unique characteristic of coupling with spin DOF, resulting in valley contrast and polarization, as observed in photoluminescence [4, 5]. In addition to the strong SOC, their lattice structure, existence or absence of centrosymmetry, and stacking play a crucial role in valleydependent properties. For this reason, photogalvanic photocurrents could offer highly efficient valleytronic device functionalities. In this chapter, we will first introduce the basic information about TMDs, such as lattice structure, symmetry, and stacking order, particularly in respect to photogalvanic effect—PGE. The phenomenology

4.1 Lattice Structure, Symmetry, and Stacking in TMDs

71

described for graphene in Chaps. 2 and 3 is also applicable to monolayer TMDs, except a few differences. However, the microscopic origin is distinct due to the spinvalley coupling. Here, the outcome of this coupling together with optical selection rules will be discussed. Finally, the expectation and influence of spin-valley coupling in photocurrent contributions nonlinear-in-electric field due to second-order response will be given with light impinging geometries.

4.1 Lattice Structure, Symmetry, and Stacking in TMDs 4.1.1 Lattice Structure The most well-known and well-studied monolayer group-VI TMD is monolayer molybdenum disulphide 1L-MoS2 . In this chapter, unless specified, we will use the example of MoS2 as the representative 2D semiconducting TMD. In fact, the formula is known as MX2 , where M can be Mo or W atoms and X can be S or Se atoms; therefore, they are also named as “MoWSeS”. Two possible orientation of 1L-MX2 with hexagonal (honeycomb) lattice are shown in Fig. 4.2(a). The unit cell of a MX2 with 2H stacking is shown in Fig. 4.2(b). Each unit cell contains a single layer of upper and lower chalcogen atoms of “X” with middle layer of “M” metal atoms. It is located in a triangular prism coordinate system with chalcogen atoms, interlocking with each other through a mixed chemical bond to form a hexagonal lattice, and the layers are coupled by a weak van der Waals force, allowing mechanical exfoliation for easy sample preparation. Since the crystals follow the 2H stacking order, in the bulk and even number of layers of MoS2 crystals, the atoms are space inversion symmetric. However, in 1L-MoS2 this spatial inversion symmetry is broken; therefore, unlike graphene, 1L-MoS2 is inherently non-centrosymmetric (Fig. 4.2(c)).

4.1.2 Crystal Symmetry For the hexagonal lattice of 1L-MoS2 , the top view in Fig. 4.2(d) is the projection of all layers onto the plane. Together with Fig. 4.2(a) either top or bottom layer can be taken as C3 symmetry together with horizontal mirror plane σh , forming symmetry point group D3h 1 . The position vectors of S atoms on the top and bottom layer with respect to the layer of Mo atoms are given as {a1t , a2t , a3t } and {a1b , a2b , a3b }, respectively, in Fig. 4.3(a). In addition, there is S3 (improper rotation) consisting of three C2 axes on the layer of Mo atoms and vertical mirror planes σv . Bulk or even layers of MoS2 has D6h 4 symmetry, such as the one shown in Fig. 4.2(a). Although the rotation center is taken as the center of the hexagon in Fig. 4.3(a), there could be other rotation center considerations, such as the Mo or the S atoms, which could be decisive in uncovering sensitive electronic band properties.

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Fig. 4.2 Schematic diagram of molybdenum disulfide (MoS2 ) crystal lattice structure. The Mo layer is in the middle of two S layers. Position of top layer consisting of S atoms coincide with the bottom layer consisting of S atoms. Here, Mo can be replaced by W atoms and S can be replaced by Se atoms. The family members of semiconducting TMDs are MoS2 , WS2 , MoSe2 , and WSe2 with similar lattice structures. (a) 2H stacked MoS2 structure, with (b) corresponding unit cell. (c) Monolayer MoS2 (1L-MoS2 ) with inherent broken inversion symmetry. (d) 1L-MoS2 top view is similar to graphene, top layer of S atoms is on the top of the bottom layer of S atoms

Fig. 4.3 (a) Schematic of hexagonal lattice of MoS2 and the crystal symmetry. Blue color plane of metal atoms Mo (the horizontal mirror plane σh ) and C3 rotation symmetry is defined with respect to the center of the hexagon, labeled as 0, together forming D3h 1 symmetry point group. Positions of chalcogen atoms S for top and bottom layers are defined as {a1t , a2t , a3t } and {a1b , a2b , a3b }. (b) The first Brillouin zone with b1 and b2 lattice vectors

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Table 4.1 The symmetry point group at the first Brillouin zone center (┌) and K valleys for all TMD layers with and without substrate (sub), gate, or bias ( ) ( ) ┌ (no sub) K K ' (no sub) ┌ (sub/gate/bias) K K ' (sub/gate/bias) 1L-TMD

D3h

C3h

C3v

C3

N even layer (AB)

D3d

D3

C3v

C3

N odd layer (ABA)

D3h

C3h

C3v

C3

Bulk

D6h

D3h

C6v

C3v

4.1.3 Symmetry Reduction and the PGE in TMD Layers In the centrosymmetric symmetries D3h (1L-MoS2 ) and D6h (bulk MoS2 ), the CPGE is not allowed. But, when they are placed on a substrate, horizontal mirror plane symmetry will be lost as well as six C2 axes, and D3h of 1L-MoS2 and D6h of bulk MoS2 can be reduced to C3v and C6v , respectively, which are symmetries allowing the CPGE. In multilayer MoS2 , the symmetry is different than that of 1L-MoS2 and bulk TMDs. Although the inversion symmetry is restored in 2L-MoS2 , because of the lack of translation symmetry along the z-direction, the point group is D3d , which holds for all even number layers with AB stacking. For odd number layers, the symmetry is the same as the 1L-MoS2 . Placing 2L-MoS2 (even number layers) on a substrate can reduce the symmetry to C3v and the CPGE can be allowed. Therefore, in all these cases, the CPGE is only allowed by symmetry reduction via gating, bias, or substrate, as in the case of graphene. A summary of symmetry point group for all MoS2 layers at the ┌ and K and valleys is given in Table 4.1. Alternatively, natural asymmetry of samples and devices can reduce the symmetry to lower symmetries, which can allow the nonvanishing term related to the interaction of the electric field (E-field) component of the electromagnetic wave normal to the sample surface plane and make PGE allowed.

4.1.4 Stacking Layered MoS2 can have three kinds of phases: 1T, 2H, and 3R where the letters stand for trigonal, hexagonal, and rhombohedral, and the leading digits are the number of S–Mo–S units in the unit cell. As shown in Fig. 4.2(a), 1L-MoS2 has 2H phase with stacking order AbA BaB, where capital letter is for S and small letter is for Mo atom. Here, the metal coordination is trigonal prismatic and 2H 1L-MoS2 is semiconducting as the d orbitals are fully occupied. Another phase for 1L-MoS2 could be 1T phase (upon intercalation) with stacking order AbC and metal coordination being octahedral and symmetry point group D3d 4 ; this phase is metallic as the d orbitals are partially occupied [6]. Multilayers and bulk MoS2 can also exhibit 3R phase with stacking sequence AbA CaC BcB and metal coordination is trigonal prismatic as in the case of 2H phase. The 2H phase is the most stable and naturally available form of MoS2 .

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Both 3R and 1T phases can transform into 2H phase by annealing and microwave radiation.

4.1.5 Symmetry around the K Points of 1L-TMDs In 2H stacked 1L-MoS2 , which is a direct band gap semiconductor, the conduction band minimum (CBM) and the valence band maximum (VBM) are both located at the corners of the first Brillouin zone, BZ. The six corners of the BZ are the corresponding group of K and K ' points or valleys, each of which has a valley index (or pseudospin) ± 1, and each group has three equivalent corners related to each other by reciprocal lattice vectors, as shown in Fig. 4.3(b). At the K points, the symmetry point group is C3h (as in the case of odd layered ABA stacked graphene), and all the Bloch electronic states at the K point are nondegenerate. Therefore, σh symmetry contained in C3h separates the five Mo d orbitals as even {dz 2 , dx 2 −y 2 , dx y } and odd {dx z , d yz }. Similarly, the p orbitals { px , p y , pz } of chalcogen atom S, which are above and below the layer of Mo atoms, can be arranged as odd and even σh symmetry and as rotation centers can be selected as either the hexagon center, or the Mo atoms, or the S atoms. There would be ideally no coupling between the even and odd orbital sets and an eigenstate of C3 can be expressed as [3], C3 αm = e−(i2mπ/3) αm , for α : p or d and m = 0, ± 1, ± 2

(4.1)

where, αm can be: ) ) 1 ( 1 ( d±2 = √ dx 2 −y 2 ± idx y , d±1 = √ dx z ± id yz , d0 = dz 2 2 2 p±1 = px ± i p y , p0 = pz .

(4.2a) (4.2b)

Here, Eqs. (4.1)–(4.2) describe transformation of orbitals under rotation about the atomic center and it is the case for the K valley. In the case of − K (or K ' ) valley, it is the complex conjugate of Eqs. (4.1)–(4.2).

4.2 Band Structure of Mono- and Few-Layer TMDs The electronic band structure of monolayer TMDs is more complicated than graphene, where tight-binding—TB approximation was able to describe the band structure. For 1L TMDs, first-principles calculations provided important information about the electronic structure, such as the energy band gap and exciton binding energy, agreeing well with experiments. But, the band-edge locations, the energy separation between ∆ (or also known as Q) and K points in the CB and between ┌

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75

and K points in the VB, based on the lattice constant determined by local density approximation functions, may not be as accurate. In order to account for this, there are several developed TB models, which can provide detailed band descriptions in the entire BZ, including edge states. But, near the K points, the most efficient approach is k · p method, as exemplified in the determination of the Berry curvature.

4.2.1 First-Principle Expectation and Photoluminescence of 1L- and Few-Layer-MoS2 First-principle calculations yielded good estimate for 1L-MoS2 energy band gap as shown in Fig. 4.4(a), agreeing well with photoluminescence data in Fig. 4.4(b) [7]. The band structure shown in Fig. 4.4(a) is the simplified version including bulk and 1L-MoS2 . Only the two bands within the VB, labeled as v1 and v2, and the band in the CB, c1 are shown. In addition to ┌ and K points, the H and A points as the corner point and the center of a hexagonal face in the BZ, respectively, are also shown. For samples with more than one-layer, electronic states are affected by interlayer coupling and quantization of out-of-plane momenta (in planes perpendicular to the ┌–A or K –H directions) favors a minimum energy gap between the VB (v1 maxima) and the CB (c1 minima) at ┌ point and ┌–K direction, respectively, which is indirect ~ 1.29 eV. For 1L-MoS2 , the out-of-plane momentum is in the same direction as the line connecting the H and A points; therefore, the v1 maxima and the c1 minima for the 2D bands occur at the H point as direct band gap (Fig. 4.4(a)), ~ 1.8 eV. The observed splitting of v1 and v2 is due to the combination of interlayer coupling and spin–orbit coupling,1 and associated direct interband transitions from the v1 and v2 to the CB are labeled as A and B, respectively (Fig. 4.4(a)). Suspended samples of 1L- and 2L-MoS2 were measured by photoluminescence (PL) spectroscopy under excitation at 2.33 eV, and 1L-MoS2 exhibited roughly three orders of magnitude larger PL intensity compared to 2L-MoS2 , as shown in Fig. 4.4(b). Much clearer features can be seen in the normalized PL spectra in Fig. 4.4(c). While the PL spectrum of 1L-MoS2 has one single narrow peak, several peaks can be clearly identified for 2L-MoS2 and other samples with more layers. For samples more than one-layer, the peak A slightly redshifts and broadens with increasing layer numbers, while the higher energy transition peak B stays about 150 meV larger than the peak for all layers. The broad feature at energies ~ 1.6 eV for 2L-MoS2 redshifting with layer numbers is the indirect transition I, approaching the bulk indirect-gap energy of 1.29 eV (Fig. 4.4(c)). The evidence of direct band gap luminescence was given by absorbance resonance [7]. The energy gap of 1LMoS2 was taken as the direct transition A, while all other layers was taken as the indirect gap transitions I (Fig. 4.4(d)). First principle calculations matched well with the experiments. 1

Here, for simplicity, the spin-orbit coupling—SOC was not included, and it does not change the transitions being direct or indirect. Its effect will be shown later.

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Fig. 4.4 (a) Simplified band structure of bulk MoS2 , with the lowest CB c1 and the highest VBs v1 and v2, from which direct transitions take place to c1, and the direct gap E g . Also, I is the indirect transition with indirect gap of E g' . (b) Photoluminescence (PL) spectra for 1L- and 2L-MoS2 samples, with strong (very small) PL intensity observed in 1L-MoS2 (2L-MoS2 ). (c) Normalized PL spectra by the intensity of peak A of few-layer MoS2 . (d) Band-gap energy of few-layer MoS2 , energies extracted from E g' , except 1L-MoS2 . Adapted with permission from [7] © 2010 American Physical Society https://doi.org/10.1103/PhysRevLett.105.136805

4.2.2 TB Approximation, Spin–Orbit Coupling in 1L-TMDs, and k · p Model The two-band k · p model describe the electronic structure quite well in the neighborhood of K points, as well as the band dispersion, the large spin–orbit splitting of the VB, the Berry curvature, and the valley-dependent optical selection rules. However, the TB models have the advantage of being applicable to the entire BZ. Among various TB models, the symmetry-based 3-band TB model (only metal d orbitals, dz 2 , dx 2 −y 2 , dx y are used) was found to be simple and have useful features: specifically, 3-band nearest-neighbor (NN) TB model agreed well with the first-principles

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77

only at the K valleys, whereas this agreement was for the entire BZ for 3-band third nearest-neighbor (TNN) model. The calculations based on 3-band NN and TNN models with comparison to the first-principle calculations for several 1L-TMDs are shown in Fig. 4.5, with and without spin–orbit coupling—SOC [8]. These results would be crucial in understanding the photogalvanic photocurrents in TMDs, such as the Berry curvature effects near K valleys for 1L-TMDs with direct interband transitions and ∆ valleys for multilayer TMDs with intraband transitions. Considering the SOC, an effective two-band k · p Hamiltonian model can be ideal to take only the effects of the VB splitting (due to product of orbital and spin angular momenta L · S) into account, since the CB splitting in this model is small. The descriptive Hamiltonian can be simply expressed as [9],

Fig. 4.5 Results of electronic band calculations from the 3-band tight-binding (TB) model (blue curves) and the first-principles (red curves and dots) for 1L-TMDs (a), (b) without ((c), (d) with) spin–orbit coupling. (a), (c) are for nearest-neighbour; NN (b), (d) are third NN (TNN) TB. Big red dots in (a) and (b) show the compositions from the dz 2 , dx 2 −y 2 , dx y orbitals. From [3]

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4 Influence of Spin-Valley Coupling on Photogalvanic Photocurrents …

( ) ∆ 2λξ (σˆ z − 1) sˆz Hˆ = at ξ k x σˆ x + k y σˆ y + σˆ z + 2 4

(4.3)

where a is the lattice constant, t is the effective hopping integral, ξ (± 1) is the valley index, σˆ is the Pauli matrices related to the CB and VB states at K points, ∆ is the energy gap, 2λ is the spin splitting in the VB, and sˆz is the Pauli matrix for spin; k x , k y and k z are the momentum components in x, y, z coordinates. The first term is about to the valley-related hopping, the second term is purely spin-dependent, and the last term describes the spin-dependent phenomena as a result of SOC in each valley. Within this simple model, the electron and hole masses are comparable; the low-energy band-edge physics in the K valleys and the band dispersion near these valleys, the large SOC splitting of the VB together with the valley-dependent Berry curvature, and the valley-dependent optical selection rules can be explained.

4.3 Giant Spin-Valley Coupling in 1L-MoS2 and Other TMDs All TMDs have a strong SOC and the VB splitting due to the d orbitals of the transition metal atoms due to the interlayer coupling [10]; and the coexistence of the inversion symmetry and the time reversal symmetry does not allow spin splitting at K points. But, in the case of 1L-TMDs, the SOC leads to large spin splitting of bands which can be understood by symmetry arguments that are described by the Hamiltonian in Eq. (4.3). There would be inherently (no) sign change of in-plane (out-of-plane) spin vector under the horizontal mirror plane σh , allowing spin splitting only in the z-direction (spins can be either spin-up or spin-down, keeping sˆz as a good quantum number). Also, the spin splitting (defined as λ∓K = (∓|E v1 − E v2 |)) at the K valleys is odd under the time reversal (same magnitude, but changes sign from K valley to − K valley). This leads to an effective coupling between the spin and valley pseudospin (or valley DOF), which is the spin-valley coupling. The role of inherent inversion symmetry breaking in 1L-MoS2 is decisive in the spin-valley coupling, as for E-field impinging on a material (excluding all other fields) with broken inversion symmetry, the only way to make two valleys, the K valley and the − K valley, distinct from each other would be the breaking of time reversal, which is the case here. This is a requirement of time T and parity (space inversion) P, PT invariance [11]. At the same time, the mirror symmetry in the z-direction allows the hybridization of dz 2 and dx 2 −y 2 orbitals, described by Eq. (4.2), which explains the band gap opening in the K valleys.

4.3 Giant Spin-Valley Coupling in 1L-MoS2 and Other TMDs

79

4.3.1 Spin and Valley-Dependent Optical Selection Rules As shown in Fig. 4.6(a), for a 1L-TMD, there are six K points in the momentum space (as in Fig. 4.3(b)), three pairs of K and − K . The CBM (which are also split by a small energy difference, with c1 and c2, c1 being the CBM) and the top VBM (v1) at each of these K valley to − K valleys are separated by the direct energy gap. Based on the aforementioned symmetry requirements, the spin states (shown by green and black color in Fig. 4.6(a) for spin-up and spin-down states) for each band in the pair of K and − K valleys have opposite sign (spin-valley locking). Large band splitting due to giant SOC in the VB (~ 0.1–0.5 eV), has the key role in valley contrasting and making these states robust against spin-flipping. Therefore, the two VB states should have opposite spins within the same valley. Once a circularly polarized light is incident on a 1L-TMD, at the ± K valleys, the chiral C3 symmetry can allow only certain interband transitions. The transition from an initial state a to a final state b, with wave functions ψa and ψb (the Bloch states in the VB and the CB, respectively, in this case), matrix elements can be defined as,
< > < > 2π ψb | Pˆ± |ψa ≡ C3 ψb |C3 Pˆ± C3−1 |C3 ψa = ei 3 (m b −m a ) ψb | Pˆ± |ψa ,

(4.4a)

2π for light σ ± momentum operator Pˆ± and C3 Pˆ± C3−1 = e∓i 3 Pˆ± .

(4.4b)

Fig. 4.6 Spin-valley coupling in 1L-TMDs. (a) The band structure for the K and − K valleys is shown with black (green) color for spin-up (-down) bands. (b) Valley- and spin-dependent optical transition selection rules. (c) The situation where the positions of the K and − K valleys are switched by rotation, in which the spins, the Berry curvatures, and the light circular polarizations remains the same. (c)–(d) summarize the optical transition rules in two decoupled 2H stacked 1L-TMD of Fig. 4.1(a). The sign of the spin splitting (a) and (c) is opposite as valleys are switched by 180° rotation. Considering the two layers together in (b) and (d), the polarized light cannot (can) distinguish the valleys (spins). From [3]

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4 Influence of Spin-Valley Coupling on Photogalvanic Photocurrents …

|< >|2 For an interband transition based on Eq. (4.4a), the quantity | ψb | Pˆ± |ψa | , which is the likelihood of a transition upon σ± radiation can be 1 or 0. In order to get 1, we need m b − m a ∓ 1 = 0 (in mod 3, m b and m a are integers and (m b , m a ) results could be (− 1, 1) and (1, − 1)). The value of m b − m a can be obtained from Eqs. (4.1)–(4.2), for specific orbitals [3]. This explains how a transition of a spindown (spin-up) hole in + K valley from the VB v1 (the VB v2) to the CB, upon σ− excitation can happen. For the − K valley, it is the complex conjugate of the + K valley, by time reversal; i.e., a transition of a spin-up (spin-down) hole in − K valley from the VB v1 (the VB v2) to the CB, upon σ+ excitation can happen. For a fixed light frequency (or a wavelength), the polarization state of the circularly polarized light determines the transition of holes from the VB to the CB, with certain spin in certain valley, by spin-valley locking, as shown in Fig. 4.6(b). By nπ/3 (n being an odd number) rotation of the band structure of Fig. 4.6(a), the positions of the K and − K valleys will be switched (Fig. 4.6(c)), while the spin state, the Berry curvature, and the polarization state of light will be the same, as can be seen in Fig. 4.6(d). If the two layers of Fig. 4.6(a) and Fig. 4.6(c) are put on top of each other forming a bilayer, the optical selection rules in the separate cases of Fig. 4.6(b), (d) will be modified and the polarized light cannot distinguish valleys, it can only distinguish spin [3]. Placing 1L-MoS2 on a substrate will reduce the symmetry to C3 at the K valleys, which is described by Eq. (4.2) and the system is non-centrosymmetric, inversion symmetry will stay broken and PT will be conserved. The outcome of this symmetry reduction will be important in the interpretation of photocurrent data, as well as the role of valley-dependent Berry curvature.

4.3.2 Valley Polarization in Circularly Polarized Photoluminescence After the theoretical treatment of the spin-valley coupling in 1L-TMDs, there is an immediate consequence of spin- and valley-dependent optical selection rules upon light absorption. The major outcome of these selection rules is valley polarization due to direct excitons, which is also related to the microscopic origin of the PGE in 1L-TMDs. The absorption/emission process and photoluminescence—PL result in the exciton formation; and circularly polarized light can be used to manipulate valley polarization without any other external field. The A and B direct transitions from the VB to the CB are excitonic and the optical selection rules are similar to the rules mentioned in Fig. 4.6. However, taking the excitons into account, the selection rules for “valley pumping” are valid over a large region around the K valleys due to the d orbital features and the large band gap (Fig. 4.7(a)). Here, only the strongest feature in PL just below 1.9 eV, which is due to the A exciton complex (including the neutral and redshifted charged excitons) are presented. Two experiments of polarization-resolved PL spectra of a naturally

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81

Fig. 4.7 Control over valley (spin) polarization in 1L-MoS2 and the first experiments. (a) Optical selection rules for the A and B excitons at the K valleys upon circularly polarized radiation. Spin states are opposite from the A excitons to the B excitons, and from K to − K (or K ' ) valley, by time reversal symmetry for the same light chirality. Polarized light can distinguish the valleys and the hole spins for specific excitonic states. (b) Polarization, P as defined in terms of intensity I, (I(σ+) − I(σ−))/(I(σ+) + I(σ−)) for σ+ (σ−) emission upon σ+ and σ− excitation yielded ~ + 32 (− 32)%. Adapted with permission from [4] © 2012 Springer Nature Limited. (c), (d) PL intensity of σ± emission for σ− excitation, demonstrating the unexpected result of strong valley polarization, P ~ 100% (see text for details). Adapted with permission from [5] © 2012 Springer Nature Limited

n-doped 1L-MoS2 , excited by a radiation ( ω = 1.96 eV (633 nm)) on-resonance with the A exciton are shown in Fig. 4.7(b)–(d). In the spectra, in addition to the A exciton emission, there is also quite weak defect-trapped exciton feature, which was found to have no obvious polarization dependence due to quite weak PL. In one experiment, excitation by chiral light results in polarized PL; i.e. σ+ (σ−) radiation produces right-handed (left-handed) PL emission, which corresponds to the intensity of circularly polarized emission, I (σ±). The degree of polarization is defined as P = [(I (σ+) − I (σ−))/I (σ+) + I (σ−)]. The data shown in Fig. 4.7(b) exhibited a polarization of ~ 30% for the same sample excited either by σ+ or σ− radiation at normal incidence [4]. The PL emission is closely related to excitonic processes: the sign of hole spins being opposite from the A exciton to the B exciton is the outcome of time reversal symmetry, large SOC (leading to valley contrast), and inversion symmetry breaking; they are all controlled by light chirality. In another experiment, the PL emission which is purely σ− polarized upon σ± radiation was plotted (Fig. 4.7(c)). While the I (σ−) was maximized, I (σ+) almost negligible, and

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P was ~ 100% (Fig. 4.7(d)), demonstrating the robustness of valley polarization and no mixing of polarized states (the same was obtained for σ + PL emission). This was not observed for a 2L-MoS2 , P was only 25%. This unexpected result in 1LMoS2 was attributed to the direct exciton lifetime and steady-state hole valley-spin population between valleys (which is much faster in 2L samples as it has indirect band gap), as well as the spin-valley coupling and related spin and valley selection rules mentioned previously [5].

4.4 Photocurrent due to Excitation by Polarized Light at Normal Incidence Phenomenology of light helicity dependent photocurrent in 1L-TMDs is similar to that of graphene and observations can be interpreted in terms of crystal symmetry reduction and time reversal symmetry; however, the microscopic origin is quite different, since 1L-TMDs are direct band gap semiconductors with strong SOC and giant spin-valley coupling. Excitations at normal incidence and oblique incidence, as well as high carrier and low carrier doping regimes should be separately considered.

4.4.1 Polarized Light Excitation at Normal Incidence and the Valley Hall Effect The optoelectronic detection of valley-dependent Hall currents, or so-called valley Hall effect (VHE), which is analogous to the spin Hall effect (SHE), and it is due to the coupling of valley DOF to the orbital motion of electrons, by applying an electric field parallel to the sample plane and polarized light at normal incidence. As mentioned in Sect. 2.3, for hexagonal structure such as graphene, there are nonzero Berry curvatures and this could lead to nonvanishing Berry curvature-dependent currents upon circularly polarized radiation at normal incidence (Sect. 2.6). The magnitude of nonlinear-in-electric field-dependent photocurrent due to Berry curvature in graphene can be small (Sect. 3.7). In the case of 1L-MoS2 , the magnitude of this current should be ~ pA for lightly n-doped samples. However, for heavily n-doped samples, it is possible to observe large transverse (Hall) currents under an electric field applied through source-drain (in-plane). This will be possible still as a result of the spin-valley locking and the reduced symmetry, and this valley-dependent current is called the valley Hall effect—VHE [12]. An applied in-plane electric field (breaking the inversion symmetry) with a charge current, leading to a valley-dependent Hall current at high carrier densities is depicted in Fig. 4.8(a) left upper panel [3]. The sign of the current would be opposite from K to − K valley, as well as the sign of the spin of these valley-dependent carriers (electrons or holes). In the case of electrons (heavily n-doped), this is a VHE by

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83

definition, as the photoexcited electrons contribute to the Hall response. In the case of two layers of 1L-MoS2 on top of each other with 180° (as in the case of Fig. 4.1(a)), in the other layer, the positions of the K and − K valleys will be switched (the left lower panel in Fig. 4.8(a)) compared to the first layer, but the spin state will stay the same. Unlike the n-doping, the VHE reduces to the SHE for p-doping due to the spin-valley locking in the VB and the fact that the holes can be trapped easily and recombine non-radiatively. Phenomenologically, the VHE can be considered as the case of circularly polarized irradiation at normal incidence of Sect. 2.6. The VHE could be due to an intrinsic effect of the Berry curvature (in addition to the extrinsic disorder effects), which reflects itself as side-jump scattering due to asymmetry by impurities or phonons. Microscopically, the Berry curvature can lead to a Hall current in a similar way shown for graphene [13]. In the case of 1L-MoS2 , following Sects. 2.3 and 2.6, the

Fig. 4.8 Mechanism and observation of the valley Hall effect (VHE) in 1L-MoS2 . (a) Left upper and bottom sketch shows the sign change of the VHE from K to − K valley as well as carrier spin. Optical selection rules for the A and B excitons at the K valleys upon circularly polarized radiation. The VHE in the right upper n-doped system reduces to the spin Hall effect in the right lower p-doped system due to the strong spin-valley locking. Adapted from [3]. (b) A heavily n-doped 1L-MoS2 device for the Hall effect measurements with σ+ excitation on-resonance with excitons. (c) The Hall voltage, at normal incidence, a fixed gate voltage and fixed angle of photon polarization ϕ = 45°, as a function of in-plane electric field (as Vx ), for right (σ+) and left (σ−) circularly polarized excitation and the difference of VH1-H2 = VH ; solid (dashed) orange line is [VH(σ+) − VH(σ−) ] ([VH(σ−) − VH(σ+) ]), which is the VHE. For linearly polarized light (s or p) in 1L-MoS2 and the case of σ± in a 2L-MoS2 device, no net VH , Vx -independent. (d) The Hall resistance has sin 2ϕ dependence for 1L-MoS2 , which is the phenomenology similar to the CPGE and the CPDE

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4 Influence of Spin-Valley Coupling on Photogalvanic Photocurrents …

expression for the valley-dependent Hall conductivity due to the Berry curvature is given by [12], σH ≈

e2 2 π ∆n v , h 2m e E g

(4.5)

where ∆n v is the difference between the carrier densities in the K and − K valleys. However, the contribution to the valley Hall conductivity from the side-jump scattering is twice as large in magnitude compared to Eq. (4.5) with a sign change, which makes the resulting valley Hall conductivity as in Eq. (4.5). For an excitation on-resonance with excitons, a device under an electric field in-plane, a source-drain voltage Vx , can have a transverse (Hall) voltage for σ+ radiation and σ− radiation separately then the Hall voltage between two contacts, VH 1−H 2 = VH = Vσ + − Vσ − , as in Fig. 4.8(b). A linear dependence on Vx (solid orange line in Fig. 4.8(c)) can be observed and reversing the polarity of the radiation would reverse the polarity (dashed orange line), as a result of time inversion, as expected. Measurements on 2L-MoS2 would have almost no difference, since there is inversion symmetry; and for linear polarization in 1L-MoS2 there would be no difference between s- and p-polarized radiation, as observed in [12].

4.4.2 The VHE: Similarity to the CPGE in terms of Phenomenology As expected from the phenomenology, at normal incidence, the Hall resistance as a function of angle of photon polarization shows sin 2ϕ dependence in Fig. 4.8(d). This is similar to the CPGE and the CPDE. But, at normal incidence, the PGE, which is an effect due to non-thermalized carriers, vanishes if there is no necessary symmetry reduction; while the PDE will not be allowed in any symmetry condition. Here, at high carrier doping and for thermalized carriers, the scattering by impurities and phonons, as well as the intrinsic Berry curvature, determines the transport. This is a Hall effect—the VHE, in which charge current can lead to a valley current, as a result of light chirality. This is analogous to the case of the spin Hall effect—the SHE. The inverse is true: a pure spin current, due to existing spin imbalance, can generate a charge current due to light chirality, as photo-induced inverse SHE, sharing the same phenomenology as the CPGE. Similarly, a pure valley current, due to existing valley imbalance, could generate a photo-induced effect due to light chirality and would be named as the photo-induced inverse VHE.

4.5 Photocurrent due to Excitation by Polarized Light at Oblique Incidence

85

4.5 Photocurrent due to Excitation by Polarized Light at Oblique Incidence Semiconductor optoelectronic research community was enticed by the promises of photocurrent nonlinear-in-electric field in 1L-TMDs excited by polarized light. Because several scientifically intriguing and technologically appealing phenomena can be directly observed, as in the example of the VHE and similar effects that become important at normal incidence. Here, the phenomenology and the microscopic origin of the incidence angle dependence (oblique incidence) will be covered for completeness and opening up new avenues.

4.5.1 Phenomenology For circularly polarized irradiation, at an oblique incidence angle θ , the phenomenology described in Chap. 1 and the specific case of graphene in Sect. 2. 6 applies for a 1L-TMD. As shown in Table 4.1, the symmetry point group of D3h 1 at the BZ center is reduced to C3 at K valley in a 1L-TMD sample placed on a substrate and additionally a source-drain bias is applied. This allows photocurrent due to the PGE, in addition to the PDE. Unlike the graphene case, here, a photocurrent with sin 2ϕ dependence will be the major photocurrent contribution; and more specifically, this current will be mainly due to the CPGE, jCPGE , if the light excitation wavelength is on-resonance with excitons.2 For this specific symmetry reduction, the angle of incidence dependence of jCPGE will be sin θ ( jCPDE ∝ sin 2θ ). For light helicities other than circularly polarized states, two photocurrent contributions with cos 4ϕ dependence, which are due to the LPGE, jLPGE , and the LPDE, jLPDE , are also expected to be large, since 1L-TMDs are inherently non-centrosymmetric. Based on these three contributions, the total light helicity dependent photocurrent can be expressed by Eq. (1.10), J = C1 sin 2ϕ + L 1 sin 4ϕ + L 2 cos 4ϕ + D, which is the phenomenological photocurrent formula. Here, for on-resonance excitation, C1 can be taken as the coefficient for the CPGE current ( jCPGE = C1 sin 2ϕ); L 1 is for the LPGE current ( jLPGE = L 1 sin 4ϕ); L 2 is for the LPDE current ( jLPDE = L 2 cos 4ϕ); and D is the polarization-independent term.

4.5.2 Device and Measurement Geometry The most common measurement and device geometry studied in the literature so far are shown in Fig. 4.9. For a 1L-TMD based phototransistor consisting of source, 2

For off-resonance excitation, there could also be a CPDE contribution, jCPDE , but for on-resonance excitation it would be much smaller in magnitude. Phenomenologically, both the CPGE and the CPDE can be related to the Berry curvature, as mentioned for graphene (Sects. 2.6 and 3.7).

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4 Influence of Spin-Valley Coupling on Photogalvanic Photocurrents …

Fig. 4.9 Two different experimental setup for light helicity dependent photocurrent measurements. (a) A linearly polarized laser directed onto a quarter-wave plate (QWP) to obtain circularly polarized light state (as well as all other elliptical polarization states) and impinging on the sample plane (placed on a SiO2 /Si substrate) with an oblique angle θ. By rotating the QWP, the angle of photon polarization ϕ can be varied. The device is back-gated and it has source-drain. Adapted from [14]. (b) A more advanced device, a Hall bar made out of a 1L-MoSe2 on a thin layer of hexagonal boron nitride (h-BN) on SiO2 /Si substrate, and encapsulated on top by a bilayer h-BN and a thicker layer of h-BN at the bottom, which allowed to perform several other electrical measurements at the same time. Adapted from [15]

drain, and gate electrodes, it is possible to collect a photocurrent signal by using a lock-in amplifier, similar to the case shown in Fig. 3.9(a) as in the graphene case. Measurement and device schematics were as in Fig. 4.9(a) for a 1L-TMD on SiO2 /Si substrate and light impinging in transverse direction (the x–z plane). This setup also allowed to vary the direction of light propagation in the y–z plane, the longitudinal geometry [14]. Here, the light can be directed in any plane between the x–z plane and the y–z plane which can allow to resolve the transverse currents as well as the longitudinal currents. In another setup, a Hall bar made out of 1L-MoSe2 was used to perform similar light helicity dependent photocurrent measurements, as well as other electrical transport measurements (Fig. 4.9(b)). The 1L-TMD was placed on a hexagonal boron nitride (h-BN) layer to minimize trapping and yield a stronger exciton absorption; encapsulated on the top by a bilayer h-BN layer as a tunnel barrier to prevent the Fermi level pinning at the interface, in addition to the bottom thicker layer of h-BN [15]. In either device design, the PGE-based currents are allowed due to reduction of symmetry; with C3 symmetry (or lower symmetries) with applied electric field, in Fig. 4.9(a), and C3v or lower symmetry (such as a single mirror symmetry) in Fig. 4.9(b).

References 1. Han W, Kawakami RK, Gmitra M, Fabian J (2014) Nat Nanotechnol 9:794 2. Schaibley JR, Yu H, Clark G, Rivera P, Ross JS, Seyler KL, Yao W, Xu X (2016) Nat Rev Mater 1:16055

References 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

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Liu GB, Xiao D, Yao Y, Xu X, Yao W (2015) Chem Soc Rev 44(9):2643 Zeng H, Dai J, Yao W, Xiao D, Cui X (2012) Nat Nanotechnol 7:490 Mak KF, He K, Shan J, Heinz TF (2012) Nat Nanotechnol 7:494 Chhowalla M, Shin HS, Eda G, Li LJ, Loh KP, Zhang H (2013) Nat Chem 5:263 Mak KF, Lee C, Hone J, Shan J, Heinz TF (2010) Phys Rev Lett 105:126805 Liu GB, Shan WY, Yao Y, Yao W, Xiao D (2013) Phys Rev B 88:085433 Xiao D, Liu GB, Feng W, Xu X, Yao W (2012) Phys Rev Lett 108:196802 Matthiess LF (1973) Phys Rev B 8:3719 Stedman GE (1985) Adv Phys 34(4):513 Mak KF, McGill KL, Park J, McEuen PL (2014) Science 344(6191):1489 Xiao D, Yao W, Niu Q (2007) Phys Rev Lett 99:236809 Eginligil M, Cao B, Wang Z, Shen X, Cong C, Shang J, Soci C, Yu T (2015) Nat Commun 6:7636 15. Quereda J, Ghiasi TS, You JS, van den Brink J, van Wees BJ, van der Waal CH (2018) Nat Commun 9:3346

Chapter 5

Light Helicity Dependent Photocurrent in Layered Transition Metal Dichalcogenides

In this chapter, we will first explain how a spin-valley dependent photocurrent can be generated in monolayers of TMDs, particularly in terms of photogalvanic photocurrents and contrast it with other effects such as the photon drag and the Berry curvature. The nonlinear-in-electric field-dependent photocurrent data of 1L-TMD phototransistors will be introduced together with detailed data analysis, under various light and measurement geometries. The phenomenology has been already covered in Chap. 4; here, we will highlight the cases that deviate from the expectation. The microscopic origin of helicity dependent photocurrent in 1L-TMD is quite distinct from graphene and other similar 2D materials. More specifically, features like being a direct band gap semiconductor with strong SOC and broken inversion symmetry can be decisive in the second-order response dependent photocurrent contributions. This can be better understood by comparing the results for thicker TMDs to those with monolayers for excitation above and below the gap.

5.1 Generation of Dichroic Spin-Valley Photocurrent in 1L-TMDs For 1L-TMD, spin-valley coupling at the K valleys can lead to a dominant jCPGE , due to interband transitions from the VB to the CB, for circularly polarized radiation with energies closer to the exciton energies (on-resonance). When spin couples to valley, the sign of the spin states of the holes in the VB, which are the components of the A (or B) excitons, in one valley opposes to that in the other valley, due to time reversal symmetry, strong SOC, and broken inversion symmetry. When a 1L-TMD sample, such as the one in Fig. 4.9(a), is excited by σ± excitations with energy equal or slightly higher than the formation/dissociation energy of the A exciton (or negatively charged exciton without a need for dissociation), an electron in the VB makes an interband transition in the K (− K ) valley, with the hole spin up ↑ (spin down ↓). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. Eginligil and T. Yu, Second-Order Photogalvanic Photocurrents in 2D Materials, Nanoscience and Nanotechnology, https://doi.org/10.1007/978-981-97-0618-1_5

89

90

5 Light Helicity Dependent Photocurrent in Layered Transition Metal …

Similarly, when excited by σ± excitations with energy equal or slightly higher than the formation/dissociation energy of B exciton, an electron in the VB makes an interband transition in the K (− K ) valley, with hole ↓ (↑), as depicted in Fig. 5.1. In order to understand this mechanism and expected photocurrent contributions and magnitude, we need to be familiar with the band dispersion.

Fig. 5.1 Band diagram near the K and − K valleys of a 1L-TMD and the PGE-based photocurrent generation mechanism. For σ+ and σ− excitations, which are on-resonance with the A exciton, a hole in the upper valence band (VB) (black solid (dash) curve in (a) and (b) for K (− K ) valley) and an electron in the conduction band (CB) (black solid curves in (a) and (b)) lead to CPGE photocurrent generation in the K and − K valleys, as jCPGE (ω)K and jCPGE (ω)−K , respectively. Similarly, for excitation on-resonance with the B exciton, a hole in the lower VB (black solid (dash) curve in (a) and (b) for the K (− K ) valley) and an electron in the CB lead to CPGE photocurrent generation. The blue horizontal dashed lines in (a) and (b) are the Fermi level EF which can be tuned by gate voltage. For on-resonance σ+ and (σ−) excitation, the K (− K ) valley is coupled with hole spin ↑ (↓) due to the spin-valley coupling for A excitons. The photocurrent jCPGE (ω)K due to the holes in the upper VB (i), with velocity υv(i)↑ , making transitions represented by purple vertical solid arrows in (a), by σ+ excitation will be larger (shown by thicker purple right arrow in (c)) than the jCPGE (ω)−K due to the υv(i)↓ at σ− excitation (shown by thinner purple right arrow in (d)). This difference can lead to a photocurrent polarization. Similar but opposite scenario in the lower VB (ii), for υv(ii)↓ and υv(ii)↑ , by exchanging valleys and spins applies for the B excitons (orange right arrows). For off-resonance σ+ and σ− excitations (cyan), there would be no difference between photocurrents due to simultaneous populations of both valleys, as shown by equivalent cyan color right arrows in both (c) and (d). Adapted from [1]

5.1 Generation of Dichroic Spin-Valley Photocurrent in 1L-TMDs

91

5.1.1 Band Dispersion and Anomalous Velocity The velocities υv of the holes in the VB compose of two contributions: one due to to the band dispersion δε(k)/δk for energy ε(k) in k space and the other one(is due ( 1 ) δε(k) ) − the anomalous velocity which appears as the second term in υv (k) = δk (e/ )E × Ω, where is the Planck constant, E is the electric field vector, and Ω is the Berry curvature that has the same magnitude but different sign in the K and − K valleys, Ω K (−K ) = +(−)Ωˆz . Then, for excitation that is on-resonance with the A excitons (the VB labelled v(i )), we can write the magnitude of υv (k) at the K (− K ) | | ( 1 )( δεv(i ) ) − |(e/ )E × Ω K (−K ) | and for excitation onvalley, as υv(i)↑(↓) = δk ↑(↓) ( )( δεv(ii ) ) resonance with the B excitons (the VB labelled v(ii )), υv(ii )↓(↑) = 1 − δk ↓(↑) | | |(e/ ) E × Ω K (−K ) |. The changes in the band dispersion contributing to the velocity, as well as the Berry curvature dependent velocity are the determining factors leading to a net photocurrent.

5.1.2 Defining the CPGE Photocurrent for Onand Off-Resonance Excitation For σ± excitations, the main photocurrent will be due to C1 coefficient ∝ sin 2ϕ, and for on-resonance excitation, C1 is mainly the CPGE. At σ + excitation, the photocurrent due ( to the ) hole velocity in the upper VB (i) with spin ↑ (in the lower VB (ii)) υv(i)↑ υv(ii )↓ , will be larger than( the photocurrent due to the hole velocity in the ) upper VB (i) with spin ↓, υv(i)↓ υv(ii)↑ . As an example, for the A excitons, the CPGE current jCPGE (ω), is given by jCPGE (ω)σ +(σ −) or ( jCPGE (ω) K (−K ) =

8eπ

( ) ) ∑ ( ) (e) ( ) δεv(i ) υc τ (εc ) − E × Ω K (−K ) τ εv(i ) − δk ↑(↓) v(i )→c

|| | )| ( ) ( | Mv(i)→c || f (εc ) − f εv(i) |δ εc − εv(i) − ω

(5.1a)

and ( υc τ (εc ) >

) (e) ( ) δεv(i ) E × Ω K (−K ) τ εv(i ) , − δk ↑(↓)

(5.1b)

where c is the final CB states, Mv→c is the transition matrix from the upper VB states to CB states, υc is the electron velocity in the CB states, τ (εv ) and τ (εc ) are the momentum relaxation times in the VB and CB, respectively. The Fermi–Dirac functions f (εc ) and f (εv ) can be varied by gate voltage [1]. Here, we remark at

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5 Light Helicity Dependent Photocurrent in Layered Transition Metal …

the condition in Eq. (5.1b), which is important in determination of the sign of jCPGE δε (ω) K (−K ) , which have three components: (1) ∝ υc τ (εc ); (2) ∝ δkv(i ) K ↑(−K ↓) ; and (3) ∝ (e/ ) E × Ω. The second part (band dispersion) and the third part (anomalous part) are the ones that differ for on-resonance excitation, due to the VB edge and the Berry curvature, respectively. There is an imbalance between photocurrents due to δεv(i ) δε and δkv(i ) −K . Also, the anomalous parts of the hole velocities in the upper VB δk K states for the K and − K valleys for excitation on-resonance with the A excitons δε δε are υv(i )↑ − δkv(i ) K and υv(i )↓ − δkv(i ) −K , respectively, which are equal in magnitude but opposite in sign, based on the valley selection rules shown in Sect. 4.3. For excitations on-resonance with the B excitons, again, there is an imbalance between δεv(ii ) δεv(ii) and δk and the anomalous parts of the hole velocities photocurrents due to δk K −K δε

δε

v(ii) v(ii ) in the lower VB states for K and − K valleys are υv(ii)↓ − δk and υv(ii)↑ − δk , K −K respectively, and which are equal in magnitude but opposite in sign. However, the overall photocurrent jCPGE (ω)K and jCPGE (ω)−K should be in the same direction as shown in Fig. 5.1, because of Eq. (5.1b). In brief, for σ± excitations on-resonance with either the A or B excitons, the jCPGE (ω)σ +(σ −) is determined by the contributions due to the band dispersion, which varies only in magnitude between the valleys, and the Berry curvature, which opposes in sign between valleys. Since light chirality is coupled to the valleys, the jCPGE (ω)σ +(σ −) can be read as jCPGE (ω) K (−K ) . In the scenario described in Fig. 5.1, spin-coupled valleydependent currents are expected to be due to excitons formed (or negatively charge excitons) after light chirality-selective transitions of carrier at oblique incidence. An excitation off-resonance with the A and B excitons means a photon energy ω much larger than the energy between the CB and the VB, where the holes of B excitons are. In this case, for σ± excitations, it is not possible to assign the C1 contribution to only the CPGE photocurrent. For this excitation energy, it is possible to expect the CPDE, as well as the CPGE. Taking only the CPGE into account due to the spin-valley coupling around the K valleys, the jCPGE as a result of interband transitions from the VB to the CB, can be expressed as,

( jCPGE (ω) K (−K ) =

8eπ

) ) )∑( ( (e) δεv υc τ (εc ) − E × Ω K (−K ) τ (εv ) − δk ↑ or ↓ v→c

|Mv→c || f (εc ) − f (εv )|δ(εc − εv − ω),

(5.2)

where v can be v(i) (upper VB) or v(ii ) (lower VB). In this case, similar current values for the K and − K valleys are expected, since the off-resonance excitation can simultaneously populate both valleys. In this respect, the same velocities shown in Fig. 5.1 implies the only variation between the K and − K valleys would be due to the Berry curvature, and according to the spin-valley selection rules, jCPGE (ω) K and jCPGE (ω)−K would be identical for off-resonance excitation. On the other hand, in the case of light moment transfer, we can expect that, jCPDE (ω)σ + and jCPDE (ω)σ − can yield a difference due to the Berry curvature, for off-resonance excitation, but still low in magnitude.

5.2 Spin-Valley Photocurrent Measurements in 1L-TMDs

93

5.2 Spin-Valley Photocurrent Measurements in 1L-TMDs Monolayer TMD samples grown by chemical vapor deposition (CVD) can yield strong PL excitonic peaks, such as in 1L-MoS2 with high valley polarization at room temperature [2], which are comparable to the valley polarization in mechanically exfoliated samples [3, 4]. In addition, thanks to modification possibilities in the CVD technique for 1L-TMD growth [5], it is possible to manipulate excitonic luminescence. Also, although non-radiative recombination processes (such as the ones due to dark excitons) can hamper high valley polarization in PL, in photocurrent this could turn to an advantage for light chirality dependent measurements, by geometrical asymmetry such as an oblique incidence, and could lead to even higher valley polarization to be probed in photocurrent. For the above purpose, a naturally n-doped 1L-MoS2 , with strong SOC (150 meV) grown by CVD method was fabricated into a phototransistor device and tested in a longitudinal photocurrent setup with light propagating in transverse direction (Fig. 4.9a), by on- and off-resonance σ± excitation [1]. This was the first light helicity dependent photocurrent measurement at an oblique incidence, at the off-state (by an applied gate voltage of − 40 V) of a 1L-TMD phototransistor, and at low excitation intensities. The sample exhibited a dominating A exciton complex at ~ 1.84 eV, with a weak B exciton peak about ~ 2 eV [1]. A laser with photon energy 1.96 eV was used and the device’s gate-dependent photocurrent was measured at high and low laser intensity (with a linear dependence of photocurrent on laser intensity), as well as high and low source-drain voltage (bias). In order to eliminate any photothermal effect, the low intensity and low drain voltage at the minimum gate voltage value of − 40 V was chosen to conduct light helicity dependent photocurrent measurements. The results shown in Fig. 5.2 were obtained by applying a constant sourcedrain voltage of 0.7 V (also necessary bias to obtain a PGE current by reducing the symmetry) and 0.06 W cm−2 laser intensity. The data points with varying photon polarization (angle of the QWP, from linear to elliptical, circular and back to elliptical, then back to linear) at oblique and normal incidence for both on- and offresonance excitation were plotted in Fig. 5.2. Here, the phenomenological photocurrent formula (Eq. 1.10), with the circular C1 and linear L 1 and L 2 contributions, J = C1 sin 2ϕ + L 1 sin 4ϕ + L 2 cos 4ϕ + D, was used for fittings and all the contributions were determined and tabulated in Table 5.1. As mentioned earlier, the circular contribution C1 for on-resonance excitation is due to mainly the CPGE as a result of spin-valley coupling; whereas, for off-resonance excitation, there could be both the CPDE and the CPGE at oblique incidence, since there are no crystal symmetry constraints for PDE but only radiation dependence. However, at normal incidence, the only possible source of C1 could be the CPGE due to symmetry reduction and the Berry curvature-dependent photocurrent either due the PGE or light momentum transfer. Among the linear contributions, L 1 due to the LPGE followed a behavior similar to that of C1 , except it became larger in magnitude at normal incidence for on-resonance excitation. This increase in L 1 can

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5 Light Helicity Dependent Photocurrent in Layered Transition Metal …

Fig. 5.2 A CVD-grown 1L-MoS2 phototransistor, for light helicity dependent photocurrent measurements with lowest laser intensity and lowest drain voltage at − 40 V gate voltage. (a), (b) At oblique incidence, on- (633 nm) and off- (532 nm) resonance excitation yielded different values of photocurrent at σ± light polarization states. A dichroic spin-valley photocurrent for onresonance excitation was observed. (c), (d) At normal incidence, there is no difference between onand off-resonance excitation. The fitting curves in (a)–(d) are the phenomenological photocurrent formula from which the CPGE contribution (C1 ) and other linear and polarization-independent contributions can be extracted. From [1] Table 5.1 Photocurrent contributions C1 , L 1 , L 2 , and D, for the CVD 1L-MoS2 sample excited by circularly polarized light, determined by the phenomenological photocurrent formula fitting to the data of Fig. 5.2, at oblique (/) and normal (⊥) incidence C 1 (pA)

L 1 (pA)

L 2 (pA)

D (pA) 145.1

On-resonance /

− 3.5

− 1.5

1.8

Off-resonance /

− 0.2

− 0.7

3.2

99.5

On-resonance ⊥

− 0.2

− 1.1

5.4

146.7

Off-resonance ⊥

− 0.3

0.3

4.1

107.0

5.3 Bias and Gate Dependence of Helicity Dependent Photocurrent …

95

be understood as relatively increased asymmetric scattering from defects and impurities. On the other hand, the other linear contribution L 2 due to the LPDE exhibited a different sign and character. It is meaningful that L 2 gets larger for off-resonance excitation at oblique incidence, as the expectation from the PDE; however, it gets even much larger for off-resonance normal incidence, unlike the expectation from the PDE from circularly polarized excitation. This suggests there should be a linear photocurrent at normal incidence, such as in the case of bilayer graphene [6], as mentioned is Sect. 2.5, and expected to be more pronounced for on-resonance excitation in 1L-MoS2 [7], as observed in the experiment (Table 5.1). The polarizationindependent D term showed larger values for on-resonance excitation than offresonance excitation, which indicated that the D term can have components related to the PGE as well as strong absorption near the band edge. Here, it is possible to define a polarization of photocurrent due to the CPGE, at the K and − K valleys, PCPGE = ( jCPGE@K − jCPGE@−K )/( jCPGE@K + jCPGE@−K ), and it yielded 0.8 ± 0.4 and 60 ± 30% for off- and on-resonance excitation, respectively. Since both the CPGE and the LPDE terms share the same time reversal, a photocurrent polarization can be defined as |C1 |/|L 2 | as a figure of merit, which is 0.06 and 1.94 for off- and on-resonance excitation, respectively. These measurements on 1L-MoS2 at the transverse geometry, at a fixed oblique angle and normal incidence, for a fixed drain-source voltage and gate voltage, and for on- and off-resonance excitation were reported as the first observation of CPGE photocurrent in a 1L-TMD grown by CVD method. Later, there were three more detailed reports on mechanically exfoliated 1L-TMDs. In one of these works, gate voltage and source-drain bias dependence of helicity dependent photocurrent for onresonance excitation in a 1L-MoS2 for a measurement geometry similar to the one in Fig. 4.9(a) were studied [8]. In the other two works [9, 10], a set of detailed spectroscopic helicity dependent photocurrent measurements with varying angle of incidence, azimuthal angle, source-drain, and gate voltage were performed for mechanically exfoliated 1L-MoSe2 in device and measurement geometries like in Fig. 4.9(a), as well as Fig. 4.9(b).

5.3 Bias and Gate Dependence of Helicity Dependent Photocurrent in 1L-MoS2 The helicity dependent photocurrent of 1L-MoS2 grown by CVD was believed to be due to low density of excitons and strain. Also, photocurrent fluctuations in the CVD sample was explained in terms of non-uniform emission–absorption processes, leading to varying valley currents for on-resonance excitation, which was not observed for off-resonance excitations [1]. On the other hand, a well-defined helicity dependent photocurrent for on-resonance excitation was observed in mechanically exfoliated 1L-MoS2 sample, in the same work. The values of C1 and L 2 , at the same condition with the CVD sample, was found to be ~ 5 and ~ 8 times larger than

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5 Light Helicity Dependent Photocurrent in Layered Transition Metal …

the CVD counterpart, while the L 1 and D terms stayed about the same magnitude. This was confirmed by the subsequent work [8], in which they studied drain bias and gate voltage dependence of the helicity dependent photocurrent.

5.3.1 Source-Drain (Bias) Voltage Dependence In regards to photocurrents nonlinear-in-electric field in a phototransistor, an applied source-drain voltage across a device has the role of reducing the symmetry to allow and manipulate the CPGE photocurrent. Unless it is heavily n- or p-doped system, there would be no expectation of variation of CPDE photocurrent. In a 1L-MoS2 phototransistor prepared by exfoliation, the bias dependence of helicity dependent photocurrent at zero gate voltage (naturally n-doped sample) was studied, as presented in Fig. 5.3(a). Excitation was on-resonance with excitons and the CPGE is expected to dominate over the CPDE. The fitting values for C1 , L 1 , L 2 , and D were obtained from the phenomenological photocurrent formula (Eq. 1.10) at various source-drain bias voltages and were plotted in Fig. 5.3(b), (c). While, the PGE based currents C1 and L 1 , and D had same linear dependence with positive slope, the LPDE term L 2 showed a negative slope (which can be explained as linear light momentum transfer). As mentioned before, the D term can have dependence on PGE and PDE as well; a similar behavior with C1 , indicate that there could be a large PGE contribution in the D term. For simplicity, a “spin-dependent” photocurrent polarization was defined as P = |C1 |/| j|. By using this definition of photocurrent polarization, the CPGE contribution out of all the helicity dependent photocurrent was plotted as a function of bias voltage in Fig. 5.3(d). A maximum in P of ~ 9% was observed for a source-drain bias for which device current saturates, which is simply the most important limiting factor for a transistor.

5.3.2 Gate Voltage Dependence An application of gate voltage (moving the Fermi energy) can increase/decrease the density of carriers and may change type of carriers taking part in photocurrent nonlinear-in-electric field. At low gate voltages (transistor being in its off-state), it is possible to observe helicity dependent photocurrent in zero or narrow-gap semiconductors at zero source-drain voltage as a result of photovoltaic effect. However, for 1L-TMDs with spin-valley coupling, gate voltage dependence would be effective only by the company of a nonzero source-drain bias (to further reduce the symmetry, in addition to be on a substrate). An exception to this would be the case of an isolated sample with higher symmetry in which the only source of breaking the symmetry is σ± radiation at an oblique incidence [10]. Gate voltage dependence of helicity dependent photocurrent at various sourcedrain bias of the sample of Fig. 5.3 was studied. As shown in Fig. 5.4, P and C1 (and

5.3 Bias and Gate Dependence of Helicity Dependent Photocurrent …

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Fig. 5.3 (a) Helicity dependent photocurrent of a 1L-MoS2 phototransistor, prepared by mechanical exfoliation, and its bias dependence. The fitting curve is the phenomenological photocurrent formula from which the circular current C1 with sin 2ϕ and linear currents L 1 and L 2 with sin 4ϕ and cos 4ϕ dependence, and the polarization-independent D term can be extracted. (b), (c) Bias voltage dependence of C1 , L 1 , L 2 , and D. (d) Drain current versus drain voltage (bias) compared with polarization P, which is defined as |C1 |/| j|, where j is the total helicity dependent photocurrent. Adapted with permission from [8] © 2018 American Chemical Society

the overall helicity dependent j) monotonically increased as a function gate voltage for all source-drain bias, under the drain current saturation, as well as the LPDE term L 2 , except a sign change. However, L 1 showed almost no change as a function of gate voltage for source-drain bias under saturation. The latter was attributed to the behavior and response of existing defects in the sample and interaction of the sample with the substrate to light field. The asymmetric scattering by defects and phonons, which is signature of the LPGE-based current, wouldn’t be much altered by gate voltages due to ineffective recombination processes. Here, it is worth to mention that, at zero gate voltage, in another device with saturation current at much lower bias, P was ~ 5% [8]. But, in the earlier work on a 1LMoS2 prepared by exfoliation [1], and with no saturation up to 10 V source-drain bias, P was ~ 15%, at the lowest gate voltage (in the off-state of the transistor). Therefore, the data of Fig. 5.4(a), (b) explain the role of drain current saturation in CPGE values and photocurrent polarization in the off-state of the phototransistor. Lowering the gate voltage caused a linear decrease in P, as can be seen in Fig. 5.4(b) and

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Fig. 5.4 Gate voltage dependence of photocurrent contributions in a 1L-MoS2 phototransistor, prepared by mechanical exfoliation, at 2, 5, and 10 V drain bias voltages. All the data are from the same sample and extracted from the same formula in Fig. 5.3. (a)–(e) The circular current C1 and the polarization P defined as |C1 |/| j|, where j is the total helicity dependent photocurrent, and linear currents L 1 and L 2 , and polarization-independent D term as function of gate voltage, respectively. (f) Drain current versus gate voltage dependence at 2, 5, and 10 V drain bias voltage. Adapted with permission from [8] © 2018 American Chemical Society

expectedly, non-radiative recombination probability decreased due to the increase in defect density. This sample was n-doped at zero gate voltage (Fig. 5.4(f)), and as shown in the PL data [8], the defect density gradually decreased and free electron density increased at higher gate voltages, leading to higher P even after current saturation. This indicates at high gate and drain voltages there could be contribution due to the CPDE, in addition to the CPGE, even for on-resonance excitation.

5.4 Incidence Angle, Bias, and Gate Dependence of Spectral Helicity Dependent Photocurrent in 1L-MoSe2 There could be a variety of consequences of spin-valley coupling on helicity dependent photocurrent of 1L-TMDs from excitation wavelength, optical transitions, light impinging geometries, as well as source-drain and source-gate voltage dependence of phototransistors. Some of them were presented for 1L-MoS2 phototransistors; but spectral resolution and light impinging geometries were not studied. In phototransistors made out of monolayer molybdenum diselenide (1L-MoSe2 ), another member of 1L-TMDs, various aspects of helicity dependent photocurrents were systematically studied [9, 10].

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5.4.1 Spectral and Gate Voltage Dependence In a 1L-MoSe2 , the excitonic A and B peaks complexes are located at ~ 1.58 eV (785 nm) and 1.74 eV (713 nm), respectively. Unlike the 1L-MoS2 , negatively charged excitons (especially the strong luminescent A excitons) are in minority in 1L-MoSe2 and the sample in Fig. 5.5 was intrinsically less n-doped compared to 1LMoS2 samples presented earlier; it had a threshold voltage of ~ 40 V. The photocurrent data as a function of angle of photon polarization (or QWP angle) upon several excitation wavelengths, between 720 and 840 nm are shown in Fig. 5.5(a). In agreement with the expectation of spin-valley coupling in 1L-TMDs (Sect. 5.1), excitation at ~ 790 nm, on-resonance with the A exciton complex, at oblique incidence, leads to the most pronounced helicity dependent photocurrent in Fig. 5.5(a). In addition to that, by fitting all the photocurrent data of Fig. 5.5(a) (~790 nm and all other laser excitation wavelengths) to the phenomenological photocurrent formula, all photocurrent contributions, namely, circular C1 , linear, and polarization-independent D contributions were determined and plotted in Fig. 5.5(b). The ratio of |C1 |/|D| has a maximum at about 790 nm excitation (~ 16%) and it gets lower for higher and lower excitations (e.g. ~ 7% at 750 nm and ~ 11% at 810 nm). At a fixed large drain bias voltage of 10 V, the gate voltage dependence of C1 and D contributions were plotted in Fig. 5.5(c). The phototransistor didn’t yield any helicity dependent photocurrent contribution until ~ 10 V gate voltage, following the transistor characteristics of drain current versus gate voltage. For all the helicity dependent photocurrent above that the ratio of |C1 |/|D| was ~ 10%.

5.4.2 Angle of Incidence Dependence at Fixed Bias and Gate Voltage The expectation of a nonzero C1 at normal incidence would be the CPGE due to symmetry reduction or the Berry curvature-dependent photocurrent either due the PGE or light momentum transfer, as mentioned earlier. The circular contribution of C1 of the sample in Fig. 5.5 showed an unexpected behavior of a large nonzero value at normal incidence, which was followed by nonmonotonic parabolic-like angle of incidence θ dependence, as shown in Fig. 5.6. These data were taken for excitation on-resonance and fixed drain voltage of 10 V and gate voltage of 50 V. This behavior was explained as Berry curvature-dependent CPGE [9]. As the CPGE would already have a sin θ dependence (and CPDE would have sin 2θ ), we suggest that there are two circular contributions here. One is the CPGE as the major circular current with negative sign, since there is no obvious saturation about θ = 45°. The other is a relatively smaller in magnitude contribution, with opposite sign—a Berry curvaturedependent photocurrent due to, most likely, the PGE. The latter has a maximum at θ = 0° and decreases either as sin θ or sin 2θ ; but it is expected to be as sin θ —as the

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Fig. 5.5 Spectral dependence of photocurrent in a 1L-MoSe2 phototransistor, prepared by mechanical exfoliation, with an on-resonance excitation ~ 785 nm at oblique incidence. (a) By using a laser on-resonance with the A exciton, largest helicity dependent photocurrent was observed (shown by an orange rectangle). The photocurrent data was fit by the phenomenological formula to determine circular C1 , linear, and polarization-independent D contributions at all laser excitations. (b) The circular current C1 and the polarization-independent D values extracted from fittings were plotted as a function of excitation wavelength at a source-drain voltage of 10 V and gate voltage of 50 V. (c) The circular current C1 and the polarization-independent D extracted from fittings were plotted as a function of gate voltage. The inset in (c) shows dependence of C1 on D. Adapted from [9]

CPGE’s dependence, since for excitation on-resonance the CPDE is expected to be negligible.

5.4.3 Source-Drain Dependence as a Function of Incidence Angle In Sect. 5.3, source-drain dependence of helicity dependent photocurrent in 1LMoS2 at a fixed oblique incidence was presented. For 1L-MoSe2 , for a negative and a positive oblique (incidence) angle, both the circular and linear photocurrent contributions were compared, as shown in Fig. 5.7. It was concluded that both the circular (C1 ) and linear (includes both L 1 and L 2 ) photocurrent can be modulated differently by the source-drain voltage for different sets of contacts and this was attributed to the presence of nonhomogeneous Schottky barriers

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Fig. 5.6 The data points are circular photocurrent extracted from the phenomenological photocurrent fittings for the 1L-MoSe2 phototransistor shown in Fig. 5.5 at bias 10 V and gate 50 V, with the unusual angle of incidence θ dependence of C1 . The dotted line is a parabolic fitting as a guide to the eye. Adapted from [9]

between 1L-MoSe2 and the contacts [9]. Variation of symmetry at the contacts from sample to sample was also observed in graphene devices in helicity dependent photocurrent measurements. Here, as it can be seen in Fig. 5.7(a), (b), for contacts 1–2, there is an obvious change of sign of the circular photocurrent at the negative bias between ± oblique incidence, implying existence of a CPGE, while the linear current didn’t change sign. However, for contacts 14– 15 in Fig. 5.7(c), there is no change of sign of the circular photocurrent between ± oblique incidence and the values are quite close, implying existence of another mechanism in addition to the CPGE (could be more sizable contact-induced symmetry reduction and the Berry curvature-dependent CPGE, as shown in Fig. 5.6); while the linear current in Fig. 5.7(d) did change sign for negative bias, strengthen the fact that there is an additional effect. Given the difference between contacts 1–2 (over the flakes) and contacts 14–15 (covering only sides), the difference in photocurrent behavior is supporting the claim of contact and Schottky barrier dependence of the helicity dependent photocurrent.

5.5 Light Helicity Dependent Photovoltage in an Encapsulated 1L-MoSe2 The device geometries shown in Fig. 4.9(a) and in the inset of Fig. 5.5(c) have the advantage of symmetry reduction due to bias voltage and contacts, which provide a rich platform of helicity dependent photocurrent and photovoltage measurements. However, in order to identify helicity dependent contributions, the device geometry described in Fig. 4.9(b) is ideal. The 1L-MoSe2 tunneling device consisted of h-BN top and bottom layer and photovoltage was used for helicity dependent measurements

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Fig. 5.7 The data points are circular (C1 ) and linear (L 1 and L 2 together) photocurrent contributions extracted from the phenomenological photocurrent fittings for the 1L-MoSe2 phototransistor shown in Fig. 5.5 at gate 50 V and ± oblique incidence θ dependence of C1 . The device used in this experiment shown at the center. (a), (b) For source-drain voltage applied between contacts 1–2. (c), (d) For source-drain voltage applied between contacts 14–15. Behavior of circular and linear photocurrents are explained in the text. Adapted from [9]

[10]. The largest magnitude was obtained for azimuthal angle of 45°; therefore, in all measurements the azimuthal angle was set to this angle instead of longitudinal or transverse light impinging geometries. The device was again lightly n-doped and the threshold voltage was ~ 20 V.

5.5.1 Spectral and Drain Voltage Dependence of Circular and Linear Contributions The device was suitable for both longitudinal and Hall voltage measurements. In both measurement geometry, the azimuthal angle was set to 45° and oblique angle of 20°. The helicity dependent photovoltage data points were taken and fit to the phenomenological formula of Eq. (1.10), in a same manner as the photocurrent measurements, and contributions, C1 , L 1 , L 2 , and D were determined. The spectral dependence of the photovoltage contributions in transverse and Hall voltage measurements geometry were plotted in Fig. 5.8 [10]. In the transverse (Hall) measurement geometry with no bias and no gate voltage, the spectral data for C1 suggested excitonic effects in light helicity dependent measurements in 1L-TMDs and agreed with the 1L-MoS2 results. As seen in

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Fig. 5.8 Spectral dependence of the circular (C1 ) and linear (L 1 and L 2 ) contributions extracted from the phenomenological formula for the encapsulated 1L-MoSe2 device shown in Fig. 4.9(b). (a) In transverse (Hall) measurement geometry, the data at no gate, no bias, and oblique angle of 20° were shown. The excitonic nature of the circular photovoltage (C1 ) was confirmed. (b) In longitudinal measurement geometry, the data at no gate and oblique angle of 20° were compared for C1 at no bias and 1 V bias. (c) Bias voltage dependence of the circular (C1 ) and linear L (L 1 and L 2 together) contributions at zero gate voltage and oblique angle of 20°, for 785 nm excitation. Adapted from [10]

Fig. 5.8(a), the character of C1 and L 2 looked alike near the A excitonic peak (for on-resonance excitation), but a change of sign near the B exciton peak was observed, supporting the claim of valley current sign change from excitation on-resonance with the A and B excitons in Sect. 5.1. On the other hand, the spectral dependence of the linear contribution L 1 exhibit the same positive sign with local maxima around the excitonic peaks. This behavior could be thought as the interaction of the sample with the h-BN substrate and yielding excitation independent asymmetric scattering by defects and phonons, which is the LPGE as L 1 term. In the longitudinal measurement geometry, the spectral data at zero gate voltage for C1 also suggested excitonic features, as in the same geometry measurements in the 1L-MoS2 devices. But, here, as shown in Fig. 5.8(b), the 0 V bias voltage data showed a C1 photovoltage peak at an excitation wavelength higher than the 1 V bias data which was attributed to trion (negatively charged exciton) to exciton

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transition, which has higher excitation energy than the trion [10]. Trion domination in the helicity dependent data at zero or low source-drain voltage and the exciton domination at high drain voltage are plausible, since exciton dissociation would require in-plane applied electric field. Furthermore, the source-drain voltage dependence of the circular contribution C1 and the linear contributions in Fig. 5.8(c), at 785 nm excitation at zero gate voltage, suggested that at zero or low (below 0.4 V) drain bias the circular contribution had positive sign, while at high drain voltage of 1 V, there was a sign change and got large values. On the other hand, the linear contribution, which had low positive values at zero and low (below 0.4 V) drain bias, got higher positive values at high drain bias [10]. There was a critical voltage, above which the behavior of the circular contribution’s direction was changed, while the linear contribution was not affected. Together with the above knowledge of trion and exciton effects on the helicity dependent photoresponse, this is a powerful demonstration of how an in-plane electric field can be decisive in the behavior of the circular contribution.

5.5.2 Spectral, Gate Voltage, and Bias Voltage Dependence of Circular Contribution The spectral dependence of the helicity dependent photovoltage of the isolated sample of 1L-MoSe2 at selected gate voltages and drain bias voltages, in the longitudinal geometry with the azimuthal angle of 45° and oblique incidence angle of 20° shed more light on the behavior of circular contribution, as shown in Fig. 5.9. Above and below 0 V gate voltage, the sample behaved differently for both 0 and 1 V bias voltage, while the spectral shift, which was attributed to the excitonic/trionic effects, were still observed at 0 and 1 V bias voltage. At high gate voltage of + 40 V, the circular contribution C1 had a peak for excitation on-resonance with the A exciton for 1 V bias, which was negligible for 0 V bias. Although the device was not heavily doped and was believed that the valley Hall effect—VHE shouldn’t be observed [10], at this regime with this geometry the circular contribution C1 could be reminiscent of the VHE (expected to be maximum for normal incidence θ = 0 and decays as cos θ ). This is in fact not a Hall bar geometry; however, there is always a relation between the Hall (transverse) resistivity and the longitudinal resistivity by skew scattering and/or side-jump scattering [11]. Besides, the sign change of C1 from − 40, − 20, and 0 V (Fig. 5.8(b)) being negative, to + 40 V turning to positive suggest that there are two mechanisms at competition: the one at low gate voltages is either due to the CPGE or CPDE and the one at high gate voltages with an opposite sign, most likely, is due to the VHE with source either intrinsic Berry curvature or side-jump scattering by impurity or phonons (Sect. 4.4). Strikingly, the change in sign from 0 to 1 V bias persisted for gate voltages − 40, − 20, and 0 V gate (device in off-state), indicating the bias effect was strongly related with excitonic and trionic effect, giving a negative or positive C1 , respectively. Including the data of Fig. 5.8(b), for excitonic

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Fig. 5.9 Extension of the data in Fig. 5.8(b), spectral dependence of the circular contribution C1 at − 40, − 20, and + 40 V gate voltages (a) for 1 V source-drain voltage and (b) 0 V source-drain voltage. See text for explanation and comments. Adapted from [10]

effect (bias 1 V), the magnitude of C1 , which was positive at + 40 V gate crossed over zero, became positive in magnitude at 0 V gate, then dropped in magnitude gradually at − 20 and − 40 V gate voltage; while, for trionic effect (bias 0 V), the magnitude of C1 , which was negligibly small at + 40 V gate monotonically increased at 0, − 20, and − 40 V gate voltage. In brief, at low drain voltages, there seems to be one mechanism due to the CPGE or the CPDE, becoming larger at lower gate voltages; whereas, at high drain voltages, in addition to the effect at play at low drain voltage, there is an additional effect. The latter effect, which had an opposite sign and was competing with the former effect, is most likely due to the Berry curvature or side-jump scattering (the VHE-related); it got larger at high gate voltage and lost it strength when the device’s Fermi level was lowered and became more intrinsic.

5.5.3 Incidence and Azimuthal Angle Dependence The results on spectral and drain voltage dependence were useful to distinguish the excitonic/trionic effects as well as circular/linear contributions. Additionally, the origin of circular contribution, either due to the CPGE and the CPDE, can be identified by angle of incidence measurements, namely as sin θ or sin 2θ dependencies, respectively. However, in some setups it could be a challenge to set the incidence angle to values higher than 45° for this identification process. But still, only two values of incidence of ± 20° was informative [10]. The data at 1 V bias with 0 V gate voltage exhibited a sign change upon reversal of the incidence angle from 20 to − 20°, which is due to the circular contribution; while the sign and magnitude stayed the same upon reversal of the incidence angle from 20 to − 20° for 0 V bias and 0 V gate voltage. Although the latter results showed a distinct feature, it wasn’t conclusive whether the circular contribution is due to the CPGE or the CPDE. As it is presented in Fig. 5.10, the helicity dependent photovoltage data taken at several azimuthal angles between − 150 and 150° at 0 and 1 V bias provided a

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clue about discrimination of the CPGE and the CPDE [10]. The insets in Fig. 5.10 are the plots of the circular contribution C1 determined from the fitting of the data in the figure, as a function of the azimuthal angle α and fitted well as ∝ sin 2α in Fig. 5.10(a) and ∝ sin 3α in Fig. 5.10(b), in the case of 0 V and 1 V bias, respectively. The azimuthal angle sin 2α dependency is within the phenomenology described by Eqs. (1.12a)–(1.12c) and was taken as a consequence of the CPDE [10]. On the other hand, the sin 3α dependency could be due to the higher-order effects, as it was not expected in the second-order effects in Chap. 1. However, most likely, it could be due the CPGE. In brief, based on Fig. 5.10, at low bias, a CPDE contribution can be dominant, implying that the data in Fig. 5.8(c) at low bias with C1 having positive sign, changed its sign at large bias due to the CPGE, most likely due to the system’s symmetry reduction to C3v or lower symmetries, which could only have restriction for the CPGE. As mentioned before, the sign inversion upon reversal of angle of incidence does not discriminate two effects but bias dependence was shown to indirectly distinguish two effects, as a result of the azimuthal angle dependence data.

Fig. 5.10 Longitudinal photovoltage measured between contacts A and C for an electric field (source-drain bias) applied between contacts 1 and 2 in the setup of Fig. 4.9(b). Angle of photon polarization (ϕ) data for the same sample in Fig. 5.8 at several azimuthal angle (α) taken at (a) 0 V bias voltage, (b) 1 V bias voltage are shown. The circular contribution C1 determined from the phenomenological formula of Eq. (1.10), by fitting it to these data, and plotted in the insets showed a dependence ∝ sin 2α in (a) and ∝ sin 3α in (b). This analysis was useful to discriminate between the CPGE and the CPDE. Adapted from [10]

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5.5.4 Gate Voltage Dependence of Circular Contribution In the isolated sample of 1L-MoSe2 , two regimes were identified at low and high drain bias, as trionic effect and excitonic effect, respectively. Also, at gate voltages for the off-state of the device, it was shown that only the CPGE or the CPDE is at play, and at high gate voltage there are two competing effects: the one due to the CPGE or the CPDE and the one due to the due to the Berry curvature or side-jump scattering. Previous attempts to determine the angle of incidence dependence were not successful since it exclusively aimed to distinguish the CPGE and the CPDE. Mapping the contribution C1 at a fixed wavelength (on-resonance excitation with excitons and trions), as a function of the bias and gate voltage simultaneously, for ± 20°, could help to visualize the range of each effect, as can be seen in Fig. 5.11(a), (b). When the sign of the incidence angle θ is changed from + 20° (Fig. 5.11(a)) to − 20° (Fig. 5.11(b)) , the sign of contribution C1 switched from negative to positive for large drain voltage. For gate voltage below 0 V, the change in the contribution C1 was much weaker than that for gate voltage above 0 V. For gate voltages above 0 V, a crossover of negative to positive gate-dependent C1 took place at gradually increasing drain bias voltages. This was attributed to an increased trion population at higher charge density (higher positive gate voltages) above the CBM, as well as the exciton and trion momentum lifetimes. Interestingly, above a certain gate voltage (40 V for negative high drain voltage and > 40 V for positive high drain voltage seen in Fig. 5.11(a)), there is an abrupt decay of the contribution C1 , which was believed to be due to electron–electron scattering as well as lowered probability of available unoccupied states in the CB. This explanation is in the same line as the coexistence of two mechanisms: one due to the CPGE (or CPDE) at low gate voltages and the other one due to high carrier density dependent Berry curvature or side-jump scattering dependent one dominating over the former one at high drain voltages. On the other hand, at zero or low drain bias, the sign was preserved, as well as for all drain bias and gate voltages at θ = 0°, normal incidence. Therefore, the expectation of an additional contribution other than the band dispersion-driven CPGE current, namely a Berry curvature-dependent contribution is not compatible for the current case, since the data at normal incidence didn’t show a signature of such a current and the C3v is not compatible for a Berry curvature-dependent contribution [10]. However, a side-jump scattering-based contribution (VHE-related) could be possible, as its phenomenology is similar to that of the CPGE (Sect. 4.4) and they could coexist with opposite sign at normal incidence. At oblique incidence, this contribution can get weaker but could be still there and compete with the CPGE contribution. In another experiment on an encapsulated 1L-MoSe2 , the photocurrent contribution C1 was determined in a device similar to the geometry shown in Figs. 4.9(a) and 5.5(c). The gate of voltage dependence of the circular current C1 and polarizationindependent current D upon an excitation on-resonance with oblique incidence was plotted in Fig. 5.11(c). Unlike the case with no encapsulation, where there was a symmetry reduction from C3v to C3 and no circular photocurrent C1 at low gate

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Fig. 5.11 Mapping of the circular contribution C1 as a function of drain-source and gate voltage of the encapsulated 1L-MoSe2 device shown in Fig. 4.9(b) and data in Figs. 5.8, 5.9 and 5.10. (a) At an incidence angle θ of 20°, (b) at an incidence angle θ of − 20°. The difference is explained in the text. Adapted from [10]. (c) Another encapsulated 1L-MoSe2 device with the circular photocurrent contribution C1 and polarization-dependent photocurrent D as a function of gate voltage, at fixed source-drain bias of 1.5 V, oblique angle, and excitation 785 nm. Even at low gate voltages C1 didn’t vanish, unlike Fig. 5.8(c) and |C1 |/|D| ∼ 6%, much lower than the non-encapsulated sample [9]

voltage, here an almost constant circular photocurrent C1 was observed at low gate voltages [9]. In this case, the ratio of |C1 |/|D| was ~ 6%, much lower than the ratio of the non-encapsulated device, which implies that the nonhomogeneous Schottky barriers between 1L-MoSe2 and the contacts in the non-encapsulated device had an important role in obtaining a large value of C1 by symmetry reduction. In the encapsulated device there is mainly one major mechanism at low gate voltages, which can be clearly seen in Fig. 5.11(c); however, at high gate voltages, the CPGE competes with the VHE-related contribution.

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5.6 Light Helicity Dependent Photocurrent in Multilayer TMDs by Excitation Above the Gap The spin-valley physics in multilayer TMDs is different than that of the monolayers mainly due the energy band discussed in Sect. 4.2 and the symmetry considerations mentioned in Table 4.1. The topic of direct band gap at monolayers and large spin splitting at the K valleys in the VB is replaced by the topic of indirect band gap and Rashba-like spin splitting in the CB. Also, for layers > 1, the broken inversion symmetry is restored and the influence of giant spin-valley coupling on helicity dependent photocurrent changes character. Namely, for 2H stacked multilayer TMDs, the circular component C1 of Eq. (1.10), which can be read as the source of the CPGE for reduced symmetry may not be nonzero (unlike the case for monolayers) for excitations on-resonance with the direct gap energy at the K valley (and the indirect gap for other valleys) for multilayers. This is a direct consequence of equivalent K and − K valleys in the VB, which are even at time reversal for multilayers. Therefore, although the system has still strong SOC and reduced symmetry due to substrate or in-plane electric field, a CPGE is not expected. This was confirmed experimentally on a multilayer MoS2 phototransistor [12]. In the same experiments, an ionic liquid (IL) gate (electric double layer) was used to vary the carrier density [12]. Even without application of any external electric field, the IL gate on top of the device meets the necessary condition of breaking the spatial inversion for a multilayer TMD phototransistor, and a CPGE current can be generated at the interface between the IL gate and the top layer of a multilayer TMD. Simply, a few nm thin extremely large capacitance can be formed and a local high carrier density due to band bending can be the source of an electric field in the z-direction leading to an effective inversion asymmetry at the surface of a multilayer TMD. The band structure shown in Fig. 4.4(a) has the CBM for bulk or few-layer MoS2 with an indirect gap of 1.2 eV. The direct band gap is only at the K valley. In a 1L-MoS2 the direct excitonic transition at the K valley is energetically favourable and there is giant spin-valley coupling. In a few-layer MoS2 the indirect transition is favourable; but, for excitation on-resonance with excitons and sample with broken inversion symmetry provided by an IL gate, a valley current due to the CPGE can take place, at the surface layer of the multilayer MoS2 . This mechanism is shown in Fig. 5.12(a); it has the same coupling of σ+ to K and σ− to − K (or K ' ), as in the case of 1L-MoS2 , except there is no necessity of spin-valley coupling. The CPGE current here can be considered as only valley-selected dichroism and the source of this current would be due to the momentum-dependent asymmetric filling of CB states in the K and − K (or K ' ) valleys as in Fig. 4.1, which can be controlled by σ± excitation. For excitation energies higher than the direct band gap, i.e. off-resonance excitation in multilayer MoS2 , it would still be possible to observe the CPGE (as well as the CPDE) up to a point until all the states in the in the K and − K (or K ' ) valleys are

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Fig. 5.12 Representative sketch of band diagrams and photocurrent generation mechanisms by various excitation energies in a multilayer TMD by interband transitions. (a) For excitation onresonance with transitions at the K valleys, valley current can be controlled via light chirality by the CPGE. (b) For off-resonance excitation with energies higher than the transitions at the K valleys, but low enough to leave some available states in the K valley (direct gap), valley current can be observed as in the case of on-resonance excitation but the magnitude of the CPGE current is expected to be lower. (c) In the case of even higher excitation energies, in which there will be equal population of states in the K valleys, no net CPGE current is expected for σ+ excitation (the same scenario for σ−). (d) For excitation below the direct gap and above the indirect gap, only interband transitions can be from the ┌ to ∆ valley, which yield no CPGE current for σ+ excitation (the same scenario for σ−). Adapted from [12] © American Physical Society https://doi.org/10.1103/PhysRevB.96. 241304

filled. This is unlike the case of 1L-MoS2 in which spin-valley locking wouldn’t allow transitions and only negligible polarization-dependent contributions can be expected. In fact, for a multilayer MoS2 , ideally the polarization-dependent photocurrent for on- and off-resonance excitations wouldn’t have differed much up to the point until all the states in the in the K and − K (or K ' ) valleys are filled in Fig. 4.1. But, since there would be inelastic scattering and relaxation of carriers to the ∆ and − ∆ (or ∆' ) valleys (Fig. 5.12(b), shown only for the σ+ excitation and it is the same for σ− excitation), the CPGE magnitude, along with other polarization-dependent contributions, is expected to reduce the polarization-dependent photocurrent. Furthermore, after all the available states in the K and − K (or K ' ) valleys are filled, there would be no transitions and the carriers would equally relax to the ∆ and − ∆ (or ∆' ) valleys (Fig. 5.12(c), shown only for the σ+ excitation, it has the same nature for σ− excitation), which means no expectation of the CPGE and other helicity dependent contributions.

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For excitation energies lower than the direct gap but higher than the indirect gap, only transitions that can take place would be from ┌ valley to the ∆ valley, which would be equally populated as shown only for the σ+ excitation in Fig. 5.12(d). Similarly, there would be indirect transitions from ┌ valley to the − ∆ (or ∆' ) valley for σ− excitation. Several excitation energies were used to study the helicity dependent photocurrent of a 50 nm thick MoS2 phototransistor [12]. For excitation on-resonance with the direct transitions taking place at the bottom of the CB of the K point, 635 nm (1.96 eV excitation), the polarization-dependent contributions, such as circular (only CPGE) and linear, at 0 V IL gate voltage yielded larger values (Fig. 5.13(a)) compared to that of off-resonance, 532 nm—2.33 eV (Fig. 5.13(b)), which agrees with the aforementioned expectation. The photocurrent data was fit to the phenomenological formula of Eq. (1.10); the extracted C1 together with linear terms were plotted based on their dependency of sin 2ϕ and sin 4ϕ + cos 4ϕ, as well as the polarizationindependent contribution in Fig. 5.13. For on-resonance excitation of 635 nm, according to the IL gate voltage dependence (not shown here), the CPGE current became much larger at more negative gate voltages, since the Fermi level was lowered deep into the gap and photogenerated carriers alone were at play without carrier–carrier interactions. At the other end of positive gate voltages, the CPGE current became much lower compared to 0 V gate, since the Fermi level moved into the CB and electron scattering by impurities and phonons hampered the CPGE current generation. For off-resonance excitation of 532 nm, a negative IL gate voltage didn’t change the CPGE current much (not shown here). Because, although the Fermi level was lowered deep into the gap, the carriers making direct transition experienced scattering from K to ∆ valleys with sizable densities. Therefore, most carriers making the interband transitions couldn’t participate in the valley current at the K point. At the other end of positive gate voltages, similar to the 635 nm excitation, electron–phonon and electron–electron interactions worked against the CPGE current generation and it decreased with increasing gate voltage. In the case of off-resonance excitation by 405 nm (3.06 eV), there was only negligible helicity dependent photocurrent contributions (both polarizationdependent and polarization-independent), as seen in Fig. 5.13(c), since all the states were equally populated and there was no imbalance which could lead to a net CPGE current. In the case of excitation by 700 nm (1.77 eV), there was no signature of circular photocurrent and the CPGE in Fig. 5.13(d), since the only possible band to band transitions are from ┌ valley to the ∆ valley, as explained in Fig. 5.12(d) [12]. On a final note, the magnitude of linear contribution monotonically decreased as a function of excitation energy in Fig. 5.13 from above the indirect gap to above the direct gap energy. This was unlike the CPGE, which was absent until the direct gap energy, then decreased with increasing excitation energy. The behavior of the linear contribution can be attributed to a LPGE valley current, which takes place as a result of an interference of transitions by the in-plane and out-of-plane components of the E-field coupled to the electric dipole moment in the z-direction in multilayer TMDs [13], which may be related to Berry curvature. On the other hand, the spectral dependence of the polarization-independent contribution seems to be non-monotonic. It

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Fig. 5.13 Helicity dependent photocurrent of a multilayer MoS2 phototransistor with ionic liquid gate, at 0 V gate voltage by various excitation energies. The photocurrent data was fit to the phenomenological formula to obtain the circular contribution due to the CPGE, the linear, and the polarization-independent contributions. (a) For excitation on-resonance with the direct gap, large CPGE was observed. (b) For off-resonance excitation above the CBM at the K valley, the polarization-dependent (independent) contributions got smaller (larger) values compared to onresonance. (c) For much larger off-resonance excitation, no CPGE was observed, linear current became the smallest. (d) Excitation below the direct gap, above the indirect gap, no CPGE but largest linear and polarization-independent current was observed. Adapted from [12] © American Physical Society https://doi.org/10.1103/PhysRevB.96.241304

was relatively larger below the direct gap, above the indirect gap; but, it got smaller for excitation on-resonance with the CBM at the K valley, then it got larger for offresonance excitation (still smaller than the value below the gap). Finally, at large off-resonance excitation it tended to extinct together with polarization-dependent contributions. The behavior of polarization-independent photocurrent can be basically understood in terms of filling of the available states leading to a photocurrent by photogeneration of carriers, which would be more efficient at lower energy CB states at the ∆ valleys. At the K valleys, around the CBM, filling of the energy states leading to the photocurrent would be relatively weaker than higher energy states, which could be done by an off-resonance excitation, up to a point all the states are filled and no net current can be observed.

5.7 Light Helicity Dependent Photocurrent in Multilayer TMDs …

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5.7 Light Helicity Dependent Photocurrent in Multilayer TMDs by Excitation Below the Gap In monolayers of TMD, an excitation below the gap didn’t yield any significant helicity dependent photocurrent contribution as shown in Sects. 5.4 and 5.5, as there are no available states to make direct interband transitions in pristine MoS2 . Only observation of intraband transitions in the CB leading to a net polarization-dependent (valley) photocurrent and the CPGE was by designing asymmetric plasmonic structures on a 1L-MoS2 phototransistor, stemming from light chirality enhancement and symmetry reduction [14], similar to the work on manipulation of CPGE in topological insulator metamaterials via interband transitions [15]. In the case of multilayer TMDs, there is an indirect gap below the direct gap in energy and, as shown in the previous section, it was possible to observe the CPGE and helicity dependent photocurrent contributions by IL gating of a multilayer MoS2 via interband transitions. Furthermore, the broken symmetry by IL gate can also generate helicity dependent photocurrents in a multilayer TMD via intraband (or inter-subband transitions), due to the band structure of the CB of 2H stacked multilayer TMD with the non-symmetric ∆ point along the ┌–K direction [16]. Unlike 1L-TMD, there is non-symmorphic symmetry in multilayers, i.e. a fractional translational and glide mirror symmetry [17]. This leads to the modification of orbitals defined in Eq. (4.2a) to dz 2 to d3z 2 −r 2 and the band crossing of two lowest energy bands F1 and F2 with respect to the CBM in Fig. 5.14(a) (the special case of bulk WSe2 ) [16], where carriers are sitting near the ∆ valley (denoted by I ). The initial| state > that| carriers > sitting at I near the ∆ valley along k are of the form |I, ∆> = ia0 |dx y + b0 |dx 2 −y 2 + c0 |d3z 2 −r 2 >, with a0 , b0 , and c0 being the coefficients for this initial state. They can make direct transition to the final states with coefficients ai , bi , and ci at higher energy bands of Fi , where i = 1, 2, and 3, which are defined as, | > | > |F1 , ∆> = ia1 |dx y + b1 |dx 2 −y 2 + c1 |d3z 2 −r 2 >,

(5.3a)

| > |F2 , ∆> = |d yz ,

(5.3b)

| ) > ( | > |F3 , ∆> = − ia3 |dx y + b3 |dx 2 −y 2 + c3 |d3z 2 −r 2 > .

(5.3c)

In the case when an out-of-plane electric field is applied to the multilayer TMD with IL gating, leading to broken inversion and lifting the spin degeneracy (see below), there would be mixing of F1 and F2 bands. Then, the transitions to these bands can lead to net nonzero valley photocurrent, namely a light chirality-dependent transverse photocurrent, that can be expressed as, j ∝ χ ϒ Vg sin θ sin 2ϕ,

(5.4)

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Fig. 5.14 The energy band structure of a multilayer TMD along ┌–K and the CPGE current generation mechanism. (a) For specifically WSe2 , at the bottom of the conduction band, moving the Fermi level just above the band I near non-symmetric ∆ valley can provide a necessary condition for a transition of an electron from initial state at I to higher energy bands F1 , F2 , and F3 . An applied out-of-plane electric field induces mixing of F1 and F2 , and transitions to these states result in a net photocurrent. (b) A schematic band diagram for spin-orientation induced CPGE in a 2DES with Rashba spin splitting is shown. Similarly, in bulk WSe2 , σ+ excitation results in inter-subband transition in the CB, and a net photocurrent due to imbalance between + k y and − k y can be obtained. Adopted with permission from [16] © 2014 Springer Nature

where ϒ is the intensity of incident light, χ is the transition matrix element for transitions from I to F1 and from I to F2 , and Vg is about IL gate voltage [16]. The CPGE current described in Eq. (5.4) is similar to the first part on the right-hand side of Eq. (3.15). In fact, this current is the general form of the spin-orientation induced CPGE in 2D electron systems with Rashba spin splitting, and the mechanism is described in Fig. 5.14(b). The surface band bending and electron confinement in the accumulation layer by the IL gate can reach deep into the top layer of the multilayer TMD (about 2 nm by 1–2 V gate) leading to the breaking of inversion symmetry and time reversal degeneracy. This distinguishes the valleys along ┌–K line in the neighborhood of ∆, by distinct Berry phases. Consequently, this allows an electron, possessing spin with one direction at the bottom of the CB with Rashba splitting, make an inter-subband transition to a state where it gets a spin with opposite direction, upon σ± excitation, which leads to a net CPGE current, as in a Rashba 2D electron system (Fig. 5.14(c)). The SOC in the CB would not have much influence on the CPGE observed in this case and it is negligible. This expectation was applied to a multilayer WSe2 phototransistor with IL gating and polarization-dependent photocurrent was studied in transverse geometry [16]. The angle of photon polarization ϕ dependence of the photocurrent at an incidence angle θ of 60° was plotted as in Fig. 5.15(a). By fitting these data to the phenomenological photocurrent formula, the CPGE (∝ sin 2ϕ) and the linear contribution (∝ sin 4ϕ) were identified. The data at 60° exhibited the largest measured CPGE value, as it can be seen in Fig. 5.15(b). This result is similar to spin-orientation induced CPGE

References

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Fig. 5.15 Helicity dependent photocurrent measurements and the CPGE in a multilayer WSe2 with ionic liquid gate. (a) Photocurrent data (green empty circles) at 60° for biased WSe2 fitted to the phenomenological formula to determine the C1 contribution (here solely attributed to the CPGE) denoted by red spheres and the linear contribution denoted by blue spheres. (b) The CPGE contribution extracted from the measurements at various angles of incidence θ were plotted. (c) The gate voltage Vg dependence of the CPGE was plotted for three different samples. Adopted with permission from [16] © 2014 Springer Nature

observed in 2D electron system [18]. Most likely a contribution due to the CPDE with opposite sign could be the reason of this peak at 60°, since the CPGE (CPDE) alone would have sin θ (sin 2θ ) dependence. Also, as it is included in Eq. (5.4), the gate voltage Vg dependence of the CPGE photocurrent suggests that the direct transitions to higher energy bands also depend on Vg , which makes a superlinear dependence (second-power dependence or higher). This agreed well with the experimental observation with several samples measured in Fig. 5.15(c). While the CPGE photocurrent was explicitly elaborated in the multilayer WSe2 and mainly claimed a CPGE current such as the one in 2D electron gas Rashba systems due to surface layers and interface effect; the linear photocurrents were attributed to the LPGE, which is of the form χl term of Eq. (3.1b) and changed sign (odd) unlike the CPGE (even) upon time-reversal. Although it was not mentioned in this work, a current due to the CPDE and the LPDE shouldn’t be ruled out in inter-subband transitions in the CB.

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