Theory of Electronic and Optical Properties of Atomically Thin Films of Indium Selenide [1st ed. 2019] 978-3-030-25714-9, 978-3-030-25715-6

Table of contents :
Front Matter ....Pages i-xii
Introduction (Samuel J. Magorrian)....Pages 1-11
Tight-Binding Model (Samuel J. Magorrian)....Pages 13-33
Hybrid \(\mathbf {k\cdot p}\) Tight-Binding Theory (Samuel J. Magorrian)....Pages 35-60
Spin-Orbit Coupling Effects in InSe Films (Samuel J. Magorrian)....Pages 61-81
Conclusions (Samuel J. Magorrian)....Pages 83-84
Back Matter ....Pages 85-87

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Springer Theses Recognizing Outstanding Ph.D. Research

Samuel J. Magorrian

Theory of Electronic and Optical Properties of Atomically Thin Films of Indium Selenide

Springer Theses Recognizing Outstanding Ph.D. Research

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

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Samuel J. Magorrian

Theory of Electronic and Optical Properties of Atomically Thin Films of Indium Selenide Doctoral Thesis accepted by the University of Manchester, Manchester, UK

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Author Dr. Samuel J. Magorrian National Graphene Institute University of Manchester Manchester, UK

Supervisor Prof. Vladimir Falko National Graphene lnstitute University of Manchester Manchester, UK

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-25714-9 ISBN 978-3-030-25715-6 (eBook) https://doi.org/10.1007/978-3-030-25715-6 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Supervisor’s Foreword

This thesis provides the first comprehensive theoretical overview of the electronic and optical properties of two-dimensional Indium Selenide (InSe): atomically thin films of this compound ranging from monolayers to few layers in thickness. InSe is a new addition to the family of two-dimensional (2D) van der Waals materials, a range of substances that are layered in their bulk form, have strong covalent bonding of atoms within the layers in contrast to the weak van der Waals attraction between the layers, and retain structural stability in the form of mono- or few layer films. Opened by the discovery of monolayer graphene (in 2004) and bilayer graphene (in 2006), the list of 2D materials has been expanded by the synthesis of MoS2 (in 2011), followed by other transition metal dichalcogenides (WS2, MoSe2, WSe2, NbSe2, etc). The drive towards understanding optical properties of 2D materials is fueled by both the prospects of miniaturization of optoelectronic devices and a search for new optoelectronic functionalities. Studies of atomically thin films of InSe and GaSe started in 2013–14, prompted by the unusual valence band dispersion in the monolayers predicted by the density functional theory (DFT) modelling. While providing a generally correct dispersion of electrons and the bands, these studies could not give the correct description of the band gaps in the materials (this deficiency of DFT modelling has already been noticed in the studies of three-dimensional bulk properties of InSe and GaSe), and also, the computational cost of DFT increases with the larger number of layers that need to be included in the modelling when describing the properties of few-layer (but still atomically thin) films. As a result, at the time when Samuel Magorrian started his Ph.D. studies, there was a need in an efficient theoretical framework that would be enable one to describe electronic properties thin films (mono-, bi-, trilayers or thicker) of InSe and GaSe, their optical properties, and the performance of these materials in vertical tunneling devices. The multi-scale theory, developed by Samuel Magorrian, offers a convenient tool to model physical properties of InSe films of various thicknesses. It includes two main elements, which can be used together or, in turn, when analyzing electronic and optical spectra and describing kinetic processes in the films:

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Supervisor’s Foreword

(a) a multi-orbital tight-binding model fully parametrized using DFT modelling of band dispersions in monolayer and bulk material supplemented by the scissor correction of orbital energies for fitting the experimentally known band gap in the bulk material; and (b) a specially designed hybrid k
p tight-binding model approach (HkpTB) that combines the expansion of band parameters for the monolayer around the Brillouin zone center with the interlayer couplings, also fully parametrized using density functional theory computation. The approach developed by Magorrian and now described in this thesis enabled him to predict a crossover that InSe films undergo from weakly indirect (for monoand bilayers) to direct (for thick films) band gap semiconductor and a very strong drop (to less than half) in the size of the band gap upon increasing the number of layers in the film. These features of InSe films have been confirmed by recent angle-resolved photoemission and photoluminescence measurements. The thesis also describes the influence of spin-orbit coupling on optoelectronic properties of 2D InSe. For future studies, the material presented in this thesis gives a modelling tool to describe electronic transport properties of InSe films in lateral field-effect transistors and vertical tunneling devices, optical absorption spectra of films of InSe of all thicknesses (from monolayer to tens of layers) determined by the inter-band and intra-band transitions. Manchester, UK

Prof. Vladimir Falko

Abstract

In this thesis, I develop a theoretical overview of the optical and electronic properties of ultrathin films of layered hexagonal Indium Selenide. The analysis is carried out using a combination of atomistic tight-binding and k
p approaches, with model parameters found using density functional theory calculations, corrected for the underestimation of the band gap which would otherwise have substantial effects on the properties of the system. A tight-binding model for monolayer and few-layer Indium Selenide is developed. The model is used to explore the behaviour of the few-layer bands of InSe, showing a significant reduction in the gap for thicker crystals due to relatively strong electronic coupling (i.e. tight-binding hops) between the layers. Oscillator strengths generated by the model are used to describe the dominant polarisation character and strength of interband optical transitions, with the principal interband transition coupling primarily to out-of-plane polarised light. Taking advantage of the anisotropy of Indium Selenide we use the results of the tight-binding model to guide the development of a ‘hybrid k
p tight-binding’ model, in which the individual constituent monolayers are described in a band-basis k
p picture, while the relatively strong electronic coupling between the layers is described in a language of tight-binding hops between the monolayer bands. We use this model to describe the bands and gaps of both aligned crystals and misaligned laminate films of InSe, the latter of which exhibit increased band gaps due to reduced electronic coupling between misaligned layers. The model is applied self-consistently to the question of intersubband optical transitions involving electrons in gate-doped films. Since the band edges in Indium Selenide appear in the vicinity of the (in-plane) C-point, spin-orbit coupling does not give rise to spin-splitting at the band edges in the same manner as in, for example, the transition metal dichalcogenides. Interband mixing induced by spin-orbit coupling does however have important consequences for the polarisation of optical transitions. We analyse this effect, which is strongest in the monolayer, both using the hybrid k
p tight-binding model, and by adding atomic spin-orbit coupling to the fully atomistic tight-binding approach. The spinorbit splitting in the conduction band of few-layer Indium Selenide is found to be vii

viii

Abstract

of the Rashba type, and we show how the strength of the Rashba splitting will depend on the interplay between the symmetry-breaking of the crystal structure itself, and that provided by the electric field of gates used to dope the system. We compare the predictions of the model with magnetotransport experiments showing evidence of weak antilocalisation.

Acknowledgements

I would like to thank all those I worked with during the course of my Ph.D. In particular, special thanks must go to Viktor Zólyomi, who carried out all the density functional calculations reported in this thesis, and to Adrian Ceferino, who contributed extensively to the self-consistent analysis of electrostatic gate doping in Chaps. 3 and 4. Most importantly, I must of course thank my supervisor, Prof. Vladimir I. Fal’ko, for his ideas and guidance. Financial support for my Ph.D. research was provided by EPSRC through the Graphene NOWNANO Centre for Doctoral Training, grant EP/L01548X. Last but not least, I would like to thank my family, both Magorrians and Gales, for their love and support throughout the project. My wife, Victoria, has stuck by me through the good times and the bad. I am forever grateful.

ix

Contents

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1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Crystal Structure of Monolayer and Bulk InSe . . . . . . 1.3 Electronic and Optical Properties: Bulk Crystal . . . . . 1.4 Few-Layer and Monolayer Ultrathin Films . . . . . . . . . 1.5 Density Functional Theory Bands for Monolayer InSe 1.6 Other Layered Hexagonal III–VI Semiconductors . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Tight-Binding Model for Monolayer InSe . . . . . . . 2.1.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Parametrisation of Hamiltonian with Scissor Correction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Multilayer Films . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Bilayer Hamiltonian and Parametrisation . . . 2.2.2 Extension to Greater Numbers of Layers . . . 2.3 Interband Optical Transitions . . . . . . . . . . . . . . . . . 2.4 Effects of Scissor Correction . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Hybrid k
p Tight-Binding Theory . . . . . . . . . . . . . . . . . . . . . . . 3.1 Interlayer Tight-Binding Hops Between Monolayer Bands for Aligned and Misaligned Crystals . . . . . . . . . . . . . . . . . . . 3.2 Model Parametrisation—Bulk c-InSe . . . . . . . . . . . . . . . . . . . 3.2.1 Correction of Band Gap in Hybrid k
p Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.3 Parametrisation—Misaligned Structures . . . . . . . . . . . . . . . . 3.3.1 Change in Interlayer Distance on Misalignment . . . . . 3.3.2 Misaligned Bulk and Few-Layer Stacks . . . . . . . . . . 3.3.3 Example: Misalignment Between Two Crystals . . . . . 3.4 Multilayer Subbands and Intersubband Optical Transitions . . 3.4.1 Quantisation of Bulk InSe Bands in Ultrathin Films . 3.4.2 Intersubband Optical Transitions . . . . . . . . . . . . . . . . 3.4.3 Effects of Interlayer Screening in Gated n-Doped InSe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Dual Gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Experimental Realisation of Intersubband Transitions . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Spin-Orbit Coupling Effects in InSe Films . . . . . . . . . . . . . . . . . . 4.1 Modelling of Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Spin-Orbit Coupling in the Tight-Binding Model . . . . . . 4.1.2 Hybrid k
p Tight-Binding Theory . . . . . . . . . . . . . . . . 4.2 Rashba Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Rashba Splitting in the Hybrid k
p Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Electrostatic Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Selection Rules for Interband Optical Transitions . . . . . . . . . . . 4.3.1 Hybrid k
p Tight-Binding Theory . . . . . . . . . . . . . . . . 4.3.2 Optical Absorption Including SOC in the Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Optical Pumping of Nuclear Spin Polarisation . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

1.1 Introduction The confinement of electrons to a two-dimensional (2D) system can significantly change material properties from those of their three-dimensional counterparts. Such confinement has historically been demonstrated in semiconductor quantum wells, permitting the observation of important physical phenomena unique to 2D systems, such as the quantum hall effect [1, 2], and enabling the development of important technological applications, the quantum cascade laser being one example [3]. Another means by which confinement of electrons can be achieved is through the restriction of the dimensionality of the crystals themselves. Semiconductor crystals generally form with their constituent atoms covalently bonded in all three dimensions. This makes it difficult to limit their dimensionality by isolating ultrathin films only a few atoms thick, and limits the stability of such films once isolated. There does, however, exist a class of materials known as the layered materials, which feature strong covalent bonding within 2D layers a few atoms thick, while only experiencing relatively weak van der Waals (vdW)-type bonding between the layers, presenting the prospect of it being possible to isolate individual layers in a stable form. Graphene, a single layer of carbon atoms, was first predicted to have unusual physical properties in comparison to its 3D parent (graphite) as early as 1947 [4]. However, it is only since 2004 that it has been possible to fabricate films of graphene a single atomic layer thick, initially through mechanical exfoliation from bulk graphite crystals [5], and to demonstrate its physical properties. Since then research into 2D materials has seen a rapid expansion, with the field growing to consider 2D allotropes of many different layered crystals, and heterostructures formed from combining different such crystals in a vertical stack [6–8]. Research into 2D materials beyond graphene has been particularly focused on the transition metal dichalcogenides (TMDCs). MoS2 was identified as a layered crystal in 1923 [9], and the TMDCs were shown many decades ago to be ideal candidates for chemical and mechanical exfoliation into few-layer films [10–14]. NbSe2 , which © Springer Nature Switzerland AG 2019 S. J. Magorrian, Theory of Electronic and Optical Properties of Atomically Thin Films of Indium Selenide, Springer Theses, https://doi.org/10.1007/978-3-030-25715-6_1

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1 Introduction

in the bulk is a superconductor with Tc = 7 K, was observed to exhibit significant changes in its transport properties on the reduction of crystal thickness by mechanical exfoliation, with Tc dropping towards 3.8 K in the monolayer limit [15], while similar behaviour was observed in a study which intercalated large molecules between the crystal layers, achieving large interlayer separations which isolated the 2D layers and confined their electrons [16]. The intense interest in 2D materials that was started by the isolation of graphene, and the improvements in experimental techniques for fabrication and measurement which followed, led to a renewed interested in the TMDCs, beginning with the observation of a indirect- to direct-gap transition in the bands of MoS2 on reduction from bulk to monolayer thickness [17]. Of the properties of TMDCs explored since, particular interest has been shown in coupled spin and valley phenomena [18], which may be optically manipulated. Other 2D materials which have come to the attention of researchers include: hexagonal boron nitride (hBN), a wide-gap insulator with wide use as an encapsulator to isolate other 2D crystals from the atmosphere [19], and which can be used to form large moiré supercells due to its close lattice-matching with graphene [20]; phosphorene [21], silicene [22] and germanene [23], as elemental 2D materials near carbon in the periodic table; the chalcogenides Bi3 Te2 and Bi3 Se2 [24]; and the perovskites [25]. Of important interest now are the layered hexagonal post-transition metal (group III) chalcogenides (PTMCs, M2 X2 where M = In, Ga and X = S, Se, Te). This thesis sets out to provide a theoretical background to one particular member of this family, Indium Selenide (InSe), and to point the way to further potential properties of interest. The rest of this thesis is organised as follows: In this chapter we introduce Indium Selenide, its crystal structure, and key physical properties. In Chap. 2 we construct a tight-binding model, which we then use to describe the optical and electronic properties of monolayer and few-layer InSe. Chapter 3 introduces a hybrid k · p tight-binding model to describe aligned crystals and misaligned laminate film of InSe, which we then use to consider the case of intersubband optical transitions in n-doped crystals. In Chap. 4 we discuss the consequences of spin-orbit coupling for the optical and electronic properties of InSe, including changes to the polarisation character of the principal optical transitions and in the appearance of the Rashba effect in the conduction band, which would explain weak antilocalisation observed in magnetotransport measurements. Finally, we briefly conclude our findings in Chap. 5.

1.2 Crystal Structure of Monolayer and Bulk InSe The crystal structure of InSe is summarised in Fig. 1.1. The monolayer unit cell has a stoichiometric formula In2 Se2 , and consists of two layers of In atoms sandwiched between two layers of Se atoms. From a top-down view, the crystal exhibits a honeycomb structure in the x y plane, with A sites occupied by In ions and B sites by Se ions. The monolayer crystal has point-group symmetry D3h . This includes z → −z mirror symmetry (σh reflection), rotational symmetry centred at each atomic position (C3 rotation), and mirror symmetry (σv reflection) in the yz and equivalent

1.2 Crystal Structure of Monolayer and Bulk InSe

3

(a)

(b)

(c)

Fig. 1.1 a Side view of monolayer InSe, shaded region indicates the monolayer unit cell, containing two indium and two selenium atoms, b top view of the γ -stacked bilayer, and c a side view of the bulk crystal—shaded region is the conventional bulk unit cell, with the dashed lines dividing the cell into the one-layer primitive cells of InSe. Lattice parameters a, dIn−In , dSe−Se and az are given in Table 1.1

planes. The two-dimensional wallpaper group symmetry of the monolayer and fewlayer crystals is p3m1. The Bravais lattice is given by a1,2

a = xˆ ± 2

√ 3a yˆ , Ri = l1i a1 + l2i a2 , 2

(1.1)

where l1i and l2i are integers, with the full crystal structure then generated with the basis     dIn−In dIn−In yˆ yˆ a a xˆ + √ + zˆ , R M2i = Ri − xˆ + √ − zˆ , R M1i = Ri − 4 2 4 2 3 3 (1.2)

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1 Introduction

Table 1.1 Lattice parameters (Å) found using X-ray diffraction (XRD) [28] and density functional theory (DFT) in the local density approximation (DFT-LDA) and using the PBE functional (DFTPBE) [30] XRD DFT-LDA DFT-PBE a dIn−In dSe−Se az

R X 1i

4.00 2.77 5.28 8.32

3.95 2.74 5.30

4.09 2.83 5.38

    dSe−Se dSe−Se yˆ yˆ a a xˆ + √ + zˆ , R X 2i = Ri + xˆ + √ − zˆ . = Ri + 4 2 4 2 3 3 (1.3)

where R M(X )1/2i is the position of the upper/lower indium (selenium) atom in unit cell i. Few-layer and bulk InSe generally1 crystallise in the γ polytype [26–29], having a structure formed by stacking monolayers with a shift of each layer   successive yˆ a xˆ + √ + az zˆ . This leaves the selenium atoms in w.r.t. the layer below of 2 3 each layer directly above the indium atoms in the layer below. The indium atoms in the upper layer, however, do not lie above the selenium atoms in the lower layer, and the stacking order thus breaks the σh symmetry of the monolayer, reducing the point group symmetry of the bulk and few-layer crystals to C3v . The bulk crystal has a primitive unit cell consisting of one unit cell of the monolayer, although a conventional cell containing three monolayers is often chosen to allow the choice of the out-of-plane lattice vector to lie along z. This unit cell relates successive monolayers by 31 and 32 screw axes along z, and has space group symmetry R3m. The lattice constants found using X-ray diffraction experiments [28] and density functional theory (DFT) calculations [30] are summarised in Table 1.1.

1.3 Electronic and Optical Properties: Bulk Crystal Bulk InSe is a direct-gap semiconductor, with a quasiparticle gap of 1.26 eV at room temperature which increases to 1.35 eV at low temperature [31, 32]. An exciton binding of 14 meV gives a strong absorption peak at a fundamental optical gap of 1.335 eV. As would be expected from the highly anisotropic crystal structure, with weak vdW-type interlayer bonding compared with the covalent intralayer bonding, the electronic properties of bulk InSe are anisotropic. Conduction band electrons ε and β polytypes are occasionally reported, however the γ polytype is by far the most common, so we focus on γ -InSe in the work presented herein. 1 The

1.3 Electronic and Optical Properties: Bulk Crystal

5

are found to have an effective mass Mc⊥z = 0.138 m e (where m e is the free electron z mass) in-plane, compared with Mc = 0.081 m e out-of-plane [33]. This result for the effective mass for the out-of-plane direction is in contrast with the comparatively weak interlayer crystalline bonding, as such a light effective mass suggests a strong interlayer electronic coupling for the band-edge states. We therefore expect significant changes in the electronic properties of InSe as electrons become confined to a few layers in ultrathin 2D films. Experimental and theoretical bandstructure analyses give the band edges themselves as having a dominantly s, pz character, with the pz -type valence band edge pushed higher in energy than the deeper px,y -dominated valence bands [34–37] due to the significant anisotropy in the crystal field of InSe. InSe has been found to exhibit significant changes in electronic properties at high pressure, with the gap increasing with increasing pressure and becoming indirect due to the development of a ringshaped offset in the valence band maximum [38–43], with the change in the crystal structure with pressure being evident in the shift of Raman modes with pressure, in particular the sensitivity of the out-of-plane Raman modes to changes in the size of van der Waals gap between the layers [44]. The potential of InSe for optoelectronic applications has also been noted in studies of its photovoltaic properties, with external photovoltaic efficiencies up to 6% achieved in InSe-based solar cells [45–47], comparable with the highest efficiencies achieved for thin-film solar cell devices at the time [48]. A factor noted in these studies for the utility of InSe for solar cell device applications is that InSe cleaves easily between layers, permitting easy fabrication of very flat interfaces. Promise for applications in electronics has also been shown through in-plane electron mobilities in n-type InSe reaching μ ∼ 103 −104 cm2 /Vs [49, 50]. In agreement with the observed anisotropy and strong interlayer electronic coupling described above excitons in bulk InSe have an anisotropic but distinctly three-dimensional character [51].

1.4 Few-Layer and Monolayer Ultrathin Films The potential for InSe to undergo substantial changes in its properties on the creation of thin films of the crystal was recognised well before the advent of mechanical exfoliation in the fabrication of graphene. An angle-resolved photoelectron spectroscopy (ARPES) study [52] on ultrathin InSe epitaxially deposited on graphite observed a transition from an extended dispersive band structure for thicker films to twodimensional bands with localised states in the thinnest (monolayer to few-layer) films. In contrast, a Raman spectroscopy study [53] observed only small changes in the Raman spectra on going from the bulk crystal to 21 layers. However, on reducing the film thickness further, it became possible for spectral changes in the Raman bands to be observed at a thickness of around three layers—at these thicknesses the film thickness becomes comparable to the coherence length of bulk out-of-plane vibrations.

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1 Introduction

The short coherence length of the bulk vibrations is expected, a result of the weak crystalline bonding between the layers. The confinement effects discussed above are indeed observed in numerous experimental studies on mechanically exfoliated few-layer InSe. Room-temperature photoluminescence (PL) experiments conducted on exfoliated InSe crystals of thickness 1–8 layers show a shift in energy of the observed PL emission peaks from ∼1.25 eV for very thick crystals down towards ∼2.7 eV in the monolayer limit [54–57]. In contrast with the indirect-to-direct transition in the gap of the TMDCs on going from bulk crystals to atomically thin films [17, 58, 59], InSe is expected to develop a slightly indirect gap in few-layer crystals as a result of the appearance of an offset in the valence band maximum, which is most pronounced in the monolayer [30, 56, 60, 61]. Such an offset would give rise to a Lifshitz transition, that is a change in the topology of the Fermi surface, on hole doping, and the associated Van Hove singularity could offer opportunities for the observation of strongly correlated phenomena in monolayer and few-layer InSe crystals [30, 62], as well as the possibility of the exploitation of thermoelectric properties in films with an offset valence band maximum [63, 64]. It has, however, proved a lot easier to n-dope InSe films than it has been to introduce any significant hole populations, so experimental searches for such behaviour have been limited. The high electron mobility obtained in the bulk crystal is found also in ultrathin films, with Hall mobilities μ > 104 cm2 /Vs demonstrated [57], with the crystal and device of sufficient quality to permit the observation of the quantum Hall effect. The high mobilities obtained in an ultrathin film are exploited in studies which demonstrate few-layer InSe as a candidate material for a field-effect transistor [65, 66]. In terms of the optoelectronic properties of InSe, it has been found to show great potential in its high-sensitivity broadband photoresponse [55, 67–70], with the ultrathin films allowing the construction of bendable devices with properties tunable with crystal thickness. In a similar manner, the thinnest films can be nanotextured [71] or bent [72] to enhance their photolumiscence response.

1.5 Density Functional Theory Bands for Monolayer InSe Figure 1.2 shows the low-energy bands of monolayer InSe, calculated using density functional theory (DFT) using the local density approximation (LDA) as set out in Ref. [30]. The bands are colour-coded according to their symmetry under σh reflection, with their -point behaviour under the symmetry operations of the monolayer point group D3h summarised in Table 1.2. The band-edge states of monolayer InSe are singly-degenerate, with the conduction band (labelled c) antisymmetric under σh reflection, and transforming as A2 at . The valence band (v) meanwhile is symmetric under σh and transforms as A1 at . The two next-highest-energy occupied bands, v1 and v2 , are both twice-degenerate at , and transform under E  and E  , respectively. The LDA band gap in the monolayer is ∼1.7 eV, with the vertical gap

1.5 Density Functional Theory Bands for Monolayer InSe

7

Fig. 1.2 Low-energy bands of monolayer InSe, calculated using density functional theory in the local density approximation, neglecting spin-orbit coupling (data from Dr. V. Zólyomi [30]). Bands are classified according to symmetry under σh reflection (red bands symmetric, blue bands antisymmetric), and labelled according to their -point transformation (see Table 1.2). Bands v1 and v2 are twice-degenerate at . Zero of energy set to conduction band edge. Inset: 2D Brillouin zone of monolayer InSe with high-symmetry points marked

at  ∼ 1.8 eV, giving a slightly indirect band gap due to the offset of the valence band maximum away from . DFT calculations including spin-orbit coupling [30] show how, due to the appearance of the band edges in the vicinity of the Brillouin zone centre, spin-splitting in the band-edge states is negligible, and remains small near the K and M points. This is in contrast to, say, the TMDCs, which have their band edges at the K point in the monolayer, with substantial (up to ∼450 meV [73]) spin-orbit splitting at the top of the valence band. Despite the relatively small consequences of spin-orbit coupling for the band energies and gaps in InSe it still plays an important role in determining the selection rules for interband optical transitions in the thinnest films, and its effects are also apparent in Rashba splitting near the conduction band edge which can be observed in magnetotransport experiments. We discuss these effects in Chap. 4.

1.6 Other Layered Hexagonal III–VI Semiconductors Gallium selenide, and sulphides of both indium and gallium have an identical monolayer structure to indium selenide [30, 74], while the most stable tellurides and oxides have different (non-hexagonal) monolayer structures [75]. Of GaSe, InS and GaS, GaSe has the most extensive body of experimental work in the literature. In

8

1 Introduction

Table 1.2 Character table for irreducible representations of point group D3 h of monolayer InSe, together with -point classification of bands labelled in Fig. 1.2 D3h E 2C3 3C2 σh 2S3 3σv Band A1 A2 E A1 A2 E 

1 1 2 1 1 2

1 1 −1 1 1 −1

1 −1 0 1 −1 0

1 1 2 −1 −1 −2

1 1 −1 −1 −1 1

1 −1 0 −1 1 0

v, c1 v2 c v1

comparison with InSe, it has a larger gap, both in the bulk [76] and in the monolayer [74] while it has similarly anisotropic but significantly three-dimensional electronic properties [77–80]. The bulk crystal most commonly stacks as the ε polytype, though as in InSe the other polytypes have also been obtained [81]. A large variety of properties useful for technical applications have been reported; in particular, the non-linear optical properties of GaSe have attracted attention in the usage of GaSe in secondharmonic generation, up- and down-conversion, and as a medium for optical gain [81–85]. Since the surface of GaSe degrades quickly in air, experiments on ultrathin fewlayer films of GaSe have been fewer than those carried out on InSe. However, encapsulation using hexagonal boron nitride (hBN) to protect samples has recently started to enable more results to be obtained. In a similar manner to InSe, GaSe exhibits a notably larger band gap for few-layer films in comparison to the bulk crystal, with a slightly indirect gap developing due to the offset from  of the valence band maximum in the thinnest films [60, 74, 86]. The observation of interesting optoelectronic properties of GaSe has been possible in the few-layer case as well as in the bulk as described above, with phenomena such as non-linear second harmonic generation reported [87–89]. Proposed applications involving GaSe films have included photodetectors [90, 91] and ultrathin transistors [92, 93].

References 1. 2. 3. 4. 5.

Wakabayashi J, Kawaji S (1978) J Phys Soc Jpn 44:1839 Klitzing K, Dorda G, Pepper M (1980) Phys Rev Lett 45:494 Faist J, Capasso F, Sivco DL, Sirtori C, Hutchinson AL, Cho AY (1994) Science 264:553 Wallace PR (1947) Phys Rev 71:622 Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, Grigorieva IV, Firsov AA (2004) Science 306:666 6. Novoselov KS, Jiang D, Schedin F, Booth TJ, Khotkevich VV, Morozov SV, Geim AK (2005) Proc Natl Acad Sci 102:10451 7. Butler SZ, Hollen SM, Cao L, Cui Y, Gupta JA, Gutiérrez HR, Heinz TF, Hong SS, Huang J, Ismach AF, Johnston-Halperin E, Kuno M, Plashnitsa VV, Robinson RD, Ruoff RS, Salahuddin S, Shan J, Shi L, Spencer MG, Terrones M, Windl W, Goldberger JE (2013) ACS Nano 7:2898

References 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

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1 Introduction

52. Klein A, Lang O, Schlaf R, Pettenkofer C, Jaegermann W (1998) Phys Rev Lett 80:361 53. Schwarcz R, Kanehisa MA, Jouanne M, Morhange JF, Eddrief M (2002) J Phys Condens Matter 14:967 54. Mudd GW, Svatek SA, Ren T, Patanè A, Makarovsky O, Eaves L, Beton PH, Kovalyuk ZD, Lashkarev GV, Kudrynskyi ZR, Dmitriev AI (2013) Adv Mater 25:5714 55. Lei S, Ge L, Najmaei S, George A, Kappera R, Lou J, Chhowalla M, Yamaguchi H, Gupta G, Vajtai R, Mohite AD, Ajayan PM (2014) ACS Nano 8:1263 56. Mudd GW, Molas MR, Chen X, Zólyomi V, Nogajewski K, Kudrynskyi ZR, Kovalyuk ZD, Yusa G, Makarovsky O, Eaves L, Potemski M, Fal’ko VI, Patanè A (2016) Sci Rep 6:39619 57. Bandurin DA, Tyurnina AV, Geliang LY, Mishchenko A, Zólyomi V, Morozov SV, Kumar RK, Gorbachev RV, Kudrynskyi ZR, Pezzini S, Kovalyuk ZD, Zeilter U, Novoselov KS, Patanè A, Eaves L, Grigorieva II, Fal’ko VI, Geim AK, Cao Y (2017) Nat Nanotechnol 12:223 58. Han SW, Kwon H, Kim SK, Ryu S, Yun WS, Kim DH, Hwang JH, Kang J-S, Baik J, Shin HJ, Hong SC (2011) Phys Rev B 84 59. Ellis JK, Lucero MJ, Scuseria GE (2011) Appl Phys Lett 99:261908 60. Rybkovskiy DV, Osadchy AV, Obraztsova ED (2014) Phys Rev B 90:235302 61. Li P, Appelbaum I (2015) Phys Rev B 92:195129 62. Cao T, Li Z, Louie SG (2015) Phys Rev Lett 114:236602 63. Wickramaratne D, Zahid F, Lake RK (2015) J Appl Phys 118:075101 64. Hung NT, Nugraha ART, Saito R (2017) Appl Phys Lett 111:092107 65. Feng W, Zheng W, Cao W, Hu P (2014) Adv Mater 26:6587 66. Sucharitakul S, Goble NJ, Kumar UR, Sankar R, Bogorad ZA, Chou F-C, Chen Y-T, Gao XPA (2015) Nano Lett 15:3815 67. Tamalampudi SR, Lu Y-Y, Rajesh Kumar U, Sankar R, Liao C-D, Karukanara Moorthy B, Cheng C-H, Chou FC, Chen Y-T (2014) Nano Lett 14:2800 68. Chen Z, Biscaras J, Shukla A (2015) Nanoscale 7:5981 69. Feng W, Wu J-B, Li X, Zheng W, Zhou X, Xiao K, Cao W, Yang B, Idrobo J-C, Basile L, Tian W, Tan P, Hu P (2015) J Mater Chem C 3:7022 70. Mudd GW, Svatek SA, Hague L, Makarovsky O, Kudrynskyi ZR, Mellor CJ, Beton PH, Eaves L, Novoselov KS, Kovalyuk ZD, Vdovin EE, Marsden AJ, Wilson NR, Patanè A (2015) Adv Mater 27:3760 71. Brotons-Gisbert M, Andres-Penares D, Suh J, Hidalgo F, Abargues R, Rodríguez-Cantó PJ, Segura A, Cros A, Tobias G, Canadell E, Ordejón P, Wu J, Martínez-Pastor JP, Sánchez-Royo JF (2016) Nano Lett 16:3221 72. Ho C-H, Chu Y-J (2015) Adv Opt Mater 3:1750 73. Zhu ZY, Cheng YC, Schwingenschlögl U (2011) Phys Rev B 84:153402 74. Zólyomi V, Drummond ND, Fal’ko VI (2013) Phys Rev B 87:195403 75. Demirci S, Avazlı N, Durgun E, Cahangirov S (2017) Phys Rev B 95:115409 76. Voitchovsky JP, Mercier A (1974) Il Nuovo Cimento B 22:273 77. Tredgold R, Clark A (1969) Solid State Commun 7:1519 78. Mooser E, Schlüter M (1973) Il Nuovo Cimento B 18:164 79. Meyer F, de Kluizenaar EE, den Engelsen D (1973) J Opt Soc Am 63:529 80. Ottaviani G, Canali C, Nava F, Schmid P, Mooser E, Minder R, Zschokke I (1974) Solid State Commun 14:933 81. Fernelius N (1994) Prog Cryst Growth Charact Mater 28:275 82. Abdullaev GB, Kulevskii LA, Prokhorov AM, Savel’ev AD, Salaev EY, Smirnov VV (1972) Sov Phys J Exp Theor Phys Lett 16:90 83. Cingolani A, Ferrara M, Lugarà M, Lévy F (1981) Phys B 105:40 84. Singh NB, Suhre DR, Balakrishna V, Marable M, Meyer R, Fernelius N, Hopkins FK, Zelmon D (1998) Prog Cryst Growth Charact Mater 37:47 85. Allakhverdiev KR, Yetis MO, Özbek S, Baykara TK, Salaev EY (2009) Laser Phys 19:1092 86. ben Aziza Z, Pierucci D, Henck H, Silly MG, David C, Yoon M, Sirotti F, Xiao K, Eddrief M, Girard J-C, Ouerghi A (2017) Phys Rev B 96 87. Jie W, Chen X, Li D, Xie L, Hui YY, Lau SP, Cui X, Hao J (2014) Angew Chem Int Ed 54:1185

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Chapter 2

Tight-Binding Model

Tight-binding models have been employed extensively in descriptions of solid-state systems [1], offering a versatile means of describing systems larger or more complex than those which may be described using first-principles methods, such as density functional theory (DFT). The onsite energies and hopping parameters which make up the basic ingredients for a tight-binding Hamiltonian can be calculated directly from DFT wavefunctions [2], fitted to first-principles or experimental results, or indeed chosen to give some effective system which demonstrates an interesting physical phenomenon [3]. In the field of 2D materials tight-binding models have been used, for example, in the initial predictions of the properties of graphene, for the investigation of the response of graphene [4] and TMDCs [5] to strain, and in the calculation of the transport properties of nanoribbons [6]. In this chapter, we present a description of the electronic and optical properties of mono- and few-layer InSe in the context of a tight-binding model. Parameters for the model are found from comparison of the model with DFT calculations for monolayer and bilayer InSe. The DFT energy bands are corrected using a ‘scissor’ operator, which takes into account the underestimation of the band gap in DFT compared with experiment. The use of a tight-binding model enables the calculation of electronic bands up to a large number of layers, and, in combination with a simple single-band k · p model, we can evaluate the matrix elements for optical transitions, thus obtaining a description of the evolution of the optical properties with crystal thickness in InSe. We find a marked change in the band gap on going from the bulk (band gap ∼1.3 eV) to the monolayer (band gap ∼2.8 eV) case, in agreement with experiment [7]. We find that the principal interband optical transition is of a dominantly out-of-plane polarised character, with the oscillator strength increasing for thicker crystals, while the next-lowest energy transition couples strongly to in-plane polarised light, with an oscillator strength which is largely independent of the number of layers. © Springer Nature Switzerland AG 2019 S. J. Magorrian, Theory of Electronic and Optical Properties of Atomically Thin Films of Indium Selenide, Springer Theses, https://doi.org/10.1007/978-3-030-25715-6_2

13

14

2 Tight-Binding Model

Fig. 2.1 Schematic of InSe monolayer (left hand side shows top view, right hand side shows side view), with inequivalent hops included in the tight-binding model indicated

2.1 Tight-Binding Model for Monolayer InSe 2.1.1 Hamiltonian The basis for the tight-binding Hamiltonian for monolayer and few-layer InSe is chosen by inspection of the low-energy orbital decompositions of the DFT wavefunctions of the low-energy bands of monolayer InSe [8]. These show contributions from s and p orbitals on both In and Se, so the tight-binding basis is constructed from the s and p orbitals of the valence shells on In and Se, with the Hamiltonian then written as   H0 f + H f f + H f f  . (2.1) H= f

Here, the sum over f = 1, 2 is over the two sublayers (shown schematically in Fig. 2.1) of the monolayer unit cell, while f  = 2(1) for f = 1(2). The contributing terms are firstly the onsite energies, H0 f =

 i

 ε Ms m †f is m f is +

 α

ε pα m †f i pα m f i pα + ε X s x †f is x f is +

 α

 ε pα x †f i pα x f i pα ,

(2.2) where i is the unit cell index, running over the whole crystal. ε M(X )s are onsite energies for In(Se) s orbitals, while ε M(X ) pα are the onsite energies for p orbitals, with α = x, y, z. The operator m(x)(†) f is annihilates (creates) an electron in the s orbital of In(Se) of sublayer f , unit cell i. m(x)(†) f is is the equivalent operator for orbital pα . The next term, H f f , adds the intra-sublayer hopping terms and is written as (2M) ) + H (2X + H (3) H f f = H (1) f f + Hf f ff ff .

(2.3a)

2.1 Tight-Binding Model for Monolayer InSe

15

Here the contributing terms are the nearest-neighbour In–Se hops (H (1) f f ), hops (2M) (2X ) between the nearest neighbouring In–In and Se–Se pairs (H f f and H f f , respectively), and hops between next-nearest In–Se pairs (H (3) f f ). They are written as H (1) ff

 

=

Tss(1) x †f js m f is

M f i ,X f j  (1) − TMs−X p



Mfi X fj

Rα

x †f j pα m f is + TM(1)p−X s



α

Mfi X fj

Rα

x †f js m f i pα

α

   M X M X δαβ Tπ(1) − (Tπ(1) + Tσ(1) )Rα f i f j Rβ f i f j (x †f j pβ m f i pα ) + α,β

+ H.c., (2M)

Hf f



=



(2.3b)

(2M) Tss(2M) m †f js m f is − Tsp

M f i ,M f j 

 

+



Mfi Mfj

Rα

α

m †f j pα m f is Mfi Mfj

δαβ Tπ(2M) − (Tπ(2M) + Tσ(2M) )Rα

Mfi Mfj

Rβ





(m †f j pβ m f i pα )

α,β

+ H.c., ) H (2X = ff

  X f i ,X f j 

(2.3c)

(2X ) Tss(2X ) x †f js x f is − Tsp

 α

X

Rα f i

X fj † x f j pα x f is

   X X X X δαβ Tπ(2X ) − (Tπ(2X ) + Tσ(2X ) )Rα f i f j Rβ f i f j (x †f j pβ x f i pα ) + α,β

+ H.c.,

(2.3d)

and (3)

Hf f =





Tss(3) x †f j  s m f is

M f i ,X f j   (3) −TMs−X p

+

 

 α

M f i X f j † x f j  pα m f is

Rα

δαβ Tπ(3) − (Tπ(3) +

+ TM(3)p−X s

M f i X f j Tσ(3) )Rα



M f i X f j † x f j  s m f i pα

Rα

α

M f i X f j Rβ





(x †f j  pβ m f i pα )

α,β

+ H.c..

(2.3e)

Here the sum over M f i , X f j  is over nearest-neighbouring In–Se pairs, while the sums over M f i , M f j , X f i , X f j , and M f i , X f j   run over nearest In–In, Se–Se and next-nearest In–Se pairs, respectively. The approach taken by Slater and Koster [9] is used in relating the symmetries of s and p orbitals to their projection along the vector connecting the ions involved in a hop (hopping vector). This enables the

16

2 Tight-Binding Model

reduction of the size of the parameter set of the model by using one parameter for, say, all s − p hops between two given ions, with the actual energy of a hop to a particular p orbital found by taking a projection of the p orbital along the hopping vector. Tss(1) is a hopping integral for nearest-neighbouring s orbitals, TMs−X p and TM p−X s take into account s − p hopping, while Tπ is the component of p − p hopping where the p orbitals are parallel to each other and perpendicular to the hopping vector and Tσ is the hopping between the components of the p orbitals lying along the hopping M X vector. Rα f i f j is the projection of a pα orbital along a hopping vector, and takes the form RXfj − RMfi M X Rα f i f j = ·α ˆ (2.4) |RXfj − RMfi | where α ˆ is a unit vector along α. H f f  describes the inter-sublayer hopping as (2) (3) H f f  = H (1) f f  + H f f  + H f f ,

(2.5a)

where H (1) ff

=







Tss(1) m †f  is m f is − Tsp(1)

i

+

 



M f i M f i

Rα

m †f i pα m f  is

α 

δαβ Tπ(1)

−



(Tπ(1)

+



M f i M f i Tσ(1) )Rα

M f i M f i Rβ



(m †f  i pβ m f i pα )



α,β

+ H.c. H (2) ff =





(2.5b) 

Tss(2) x †f  js m f is

M f i ,X f  j   (2)

−TMs−X p

 α

Mfi X f j † x2 j pα m 1is

Rα

 (2)

+ TM p−X s

 α

Mfi X f j † x f  js m f i pα

Rα

   Mfi X f j Mfi X f j    δαβ Tπ(2) − (Tπ(2) + Tσ(2) )Rα (x †f  j pβ m f i pα ) + Rβ α,β

+ H.c.,

(2.5c)

and (3)

Hf f =









Tss(3) m †f  js m f is − Tsp(3)

M f i ,M f  j 

+

 







Mfi Mf j

Rα

α 

m †f j pα m f  is

Mfi Mf j

δαβ Tπ(3) − (Tπ(3) + Tσ(3) )Rα

Mfi Mfj

Rβ





(m †f  j pβ m f j pα )

α,β

+ H.c..

(2.5d)

2.1 Tight-Binding Model for Monolayer InSe

17

2.1.2 Parametrisation of Hamiltonian with Scissor Correction The DFT data to which we fit and compare the TB model of InSe are obtained as set out in Ref. [8]. As is well known [10] density functional theory systematically underestimates the band gaps of semiconductors, often severely. We therefore apply an adjustment known as a ‘scissor correction’ δ E g to the DFT bands [11–16]. Such a correction takes as its starting point the assumption that all features of the DFT bands (such as effective masses) are correct aside from the size of the gap itself. If we compare the calculated gap with that found from experiment we can then apply a rigid shift in energy of all occupied bands relative to the unoccupied bands to obtain a corrected band structure, which should be more useful for the comparison of predictions with experiment. A DFT calculation within the LDA returns a band gap for bulk InSe as 0.41 eV as compared to the bulk experimental value ∼1.35 eV at low temperature [17, 18] (1.25 eV at room temperature [19]). Hence, we subtract a rigid shift of 1 eV from the energies of all valence band states while keeping the conduction band energies unchanged for bulk, few-layer, and monolayer InSe. The underestimation of the gap by DFT is significant in InSe, particularly so for greater numbers of layers, reaching a factor of more than 3 in the bulk, and this has consequences for the behaviour of corrected models as compared to properties calculated using DFT, which will be explored further in Sect. 2.4 of this chapter concerning the optical properties, and again in consideration of the HkpTB model in Chaps. 3 and 4. Due to the crystal structure of InSe meaning that each atom has a number of nearby neighbours at similar distances, and the need to include s and p orbitals on both atomic species, the resulting parameter set is large, with 37 independent parameters to be fitted in the monolayer. As a consequence of the nature of the parameter space it is not expected to be possible to find a unique set of parameters to fit the (scissor corrected) DFT data. Instead, we aim to choose a parameter set which gives a reasonably small RMS error between the TB(-SC) bands and the (corrected) DFT data whilst paying particular attention to the symmetries and other properties of the bands near the band edges and around the high symmetry points of the Brillouin zone. Since the assumption that the bands are dominated by s and p orbitals breaks down at energies far from the band edges, and since plane-wave DFT is known to be less accurate in high-energy unoccupied bands we restrict our fitting to the five highest-energy occupied bands and the two lowest-energy unoccupied bands. We fit these 7 bands to bands 5–11 of the 16-band tight-binding model. There is no definitive strategy to fitting the bands. However, the method employed here was to perform a large number of least-squares minimisations of the RMS error between the DFT bands at only the -, K-, and M-, points (restricting the k-points used to the high-symmetry points to reduce time taken in the fit), using randomlychosen starting parameter sets. The parameter sets resulting from the minimisations with the smallest RMS errors were then taken and used as starting points for a more complete least-squares minimisation using a set of 120 k-points along the -K-M

18

2 Tight-Binding Model

Table 2.1 Fitted parameters (eV) for the TB model of InSe based on DFT data with (TB S.C.) and without (TB) scissor correction, as shown in Fig. 2.2 TB S.C.

TB

ε Ms

−7.174

−7.595

ε M px = ε M py

−2.302

−3.027

ε M pz

1.248

0.903

εXs

−14.935

−15.188

ε X px = ε X py

−7.792

−8.045

ε X pz

−7.362

−7.615

0.168

0.331

(1)

Tss

(1)

TMs − X p

2.873

2.599

−2.144

−2.263

1.041

0.977

1.691

1.342

(2M)

−0.200

−0.248

(2M)

−0.137

−0.113

(2M)

−0.433

−0.561

Tσ (2X ) Tss

(2M)

−1.034

−1.130

−1.345

−1.451

(2X )

−0.800

−0.843

(2X )

−0.148

−0.110

(2X )

−0.554

−0.613

0.821

0.793

(1)

TM p − X s (1)

Tπ

(1)

Tσ

Tss

Tsp Tπ

Tsp Tπ

Tσ

(3)

Tss

(3)

TMs − X p (3)

TM p − X s (3)

0.156

0.179

−0.294

−0.323

0.003

−0.015

−0.455

−0.477

(1)

−0.780

−0.518

(1)

−4.964

−4.644

(1)

−0.681

−0.769

−4.028

−4.052

Tπ (3) Tσ Tss

Tsp

Tπ (1) Tσ (2)

Tss

0.574

0.472

(2)

−0.651

−0.544

(2)

−0.148

−0.138

0.100

0.082

TMs − X p TM p − X s (2)

Tπ

(2)

Tσ (3) Tss

0.343

0.373

−0.238

−0.187

(3)

−0.048

−0.065

(3)

−0.020

−0.052

(3)

−0.151

−0.168

Tsp Tπ

Tσ

2.1 Tight-Binding Model for Monolayer InSe

19

Fig. 2.2 Fitted tight-binding bands for monolayer InSe. Solid lines are for fitting to scissor-corrected DFT (dots), with energies on the left-hand axis. Dashed lines: fitted to uncorrected DFT, right-hand axis. Zero of energy is set to the bottom of the conduction band, with the bands colour-coded according to their symmetry under σh reflection (red even, blue odd), with the bands labelled according to their -point symmetry classification in Fig. 1.2. The vertical lines marked A and B denote two important optical transitions

line plus a grid of 19 k-points covering the irreducible portion of the Brillouin zone away from the -K-M line. From these expanded minimisations a smaller set of the best parameter sets were again taken, and additional weight was applied to the valence and conduction bands near  as particular importance was attached to obtaining good values for the band gap and effective masses near the band edges. Finally, using the remaining parameter sets a qualitative comparison was taken by eye between the predicted eigenvectors of the bands and the orbital decomposition of the projected DFT wavefunctions (see Fig. 2.3), which, combined with checks on the momentum matrix elements (see Sect. 2.3) and the physical reasonableness of the model parameters, was used to choose a final parameter set. The parameters obtained in the fit are given in Table 2.1 for the fit with scissor correction taken into account (TB S.C.), together with the result of fitting without scissor correction (TB). The latter parameter set was obtained from a single leastsquares minimisation using the TB S.C. parameter set as a starting point. Figure 2.2 shows the fitted bands together with corrected and uncorrected DFT bands. Figure 2.3 shows the results of a comparison of the orbital decompositions of the bands from the model Hamiltonian with those given by DFT results. Defining Cnk (λ) as the coefficient of the eigenvector of band n, orbital λ, at momentum k we plot |Cnk (λ)|2 alongside the modulus square of the normalised overlap between the DFT wavefunction and the spherical harmonics centred on each atom. Larger markers indicate a more dominant contribution. Table 2.2 gives numerical decompositions for the low-energy -point bands labelled in Fig. 2.2. A generally good agreement of

20

2 Tight-Binding Model

Inpz

Sepx

Ses

Sepy

Sepz

DFT

−2 −4 Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ

0

TB S.C.

Energy(eV)

Inpy

0

2

−2 −4 2

Energy(eV)

Inpx

Ins

Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ

0

TB

Energy(eV)

2

−2 −4 Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ Γ KM Γ

Fig. 2.3 Orbital decomposition for fitted bands for InSe from both DFT data, and fitted model with (TB S.C.) and without (TB) scissor correction. Marker size is proportional to normalised contribution Table 2.2 The relative spherical harmonic character of the plane-wave wave function (modulus square of the overlap integral between the DFT wave function and the spherical harmonics centered on each atom) on the valence s and p orbitals of In and Se atoms in monolayer InSe at the -point, in the conduction band (c), the valence band (v), and the two twice degenerate bands just below the valence band (v1 and v2 ), as labelled in Fig. 2.2. The equivalent contribution found in the scissor corrected TB model is given in square brackets. Atoms are listed in order of increasing z coordinate

Se1 In1 In2 Se2

c

v

v1

v2

0.09[0.00]s 0.17[0.22] pz 0.23[0.16]s 0.01[0.12] pz 0.23[0.16]s 0.01[0.12] pz 0.09[0.00]s 0.17[0.22] pz

0.00[0.01]s 0.35[0.36] pz 0.03[0.10]s 0.12[0.02] pz 0.03[0.10]s 0.12[0.02] pz 0.00[0.01]s 0.35[0.36] pz

0.22[0.24] px(y)

0.21[0.24] px(y)

0.03[0.01] px(y)

0.04[0.01] px(y)

0.03[0.01] px(y)

0.04[0.01] px(y)

0.22[0.24] px(y)

0.21[0.24] px(y)

2.1 Tight-Binding Model for Monolayer InSe

21

the character of the wavefunctions is obtained, the main difference from DFT being the division of the In contribution to the conduction band wavefunction at  between s and pz orbitals.

2.2 Multilayer Films 2.2.1 Bilayer Hamiltonian and Parametrisation We now extend the model beyond the monolayer case, and describe the effect of the addition of interlayer hopping terms to the Hamiltonian. First we consider the case of bilayer γ-InSe. With the hops included as shown in Fig. 2.4 the Hamiltonian can be written as (2.6a) H = H1 + H2 + H1,2 where H1 and H2 describe the individual monolayers comprising the bilayer structure, and H1,2 describes the interaction between them and can be written as H1,2 = H X 1 ,M2 + HM1 ,X 2 + H X 1 ,X 2

(2.6b)

where each term corresponds to the hops labelled in Fig. 1.1d. The vertical In–Se contribution is  M) tss(X M) m †(2)1is x(1)2is + t X(Xs −M m †(2)1i pz x(1)2is H X 1 ,M2 = p i M) −t X(Xp −M m †(2)1is x(1)2i pz + tπ(X M) s

−tσ(X M) m †(2)1i pz x(1)2i pz



 α=x,y

m †(2)1i pα x(1)2i pα

+ H.c.,

(2.6c)

where the operators now have an additional layer index—for example, m (†) (n)1is annihilates (creates) an electron in the s orbital of the In atom in layer n, sublayer 1. For the other (non-vertical) In–Se interlayer hop we have HM1 ,X 2 =

 (X X ) † tss x(2)1 js m (1)2is

i, j

+



M

Rα (1)2i

α

X (2)1 j



(M X ) (M X ) † † tM p−X s x (2)1 js m (1)2i pα − t Ms−X p x (2)1 j pα m (1)2is



⎤   M(1)2i X (2)1 j M(1)2i X (2)1 j † (M X ) (M X ) (M X ) + − (tσ + tπ )Rα Rβ δαβ tπ x(2)1 j pβ m (1)2i pα ⎦ α,β

+ H.c.,

(2.6d)

22

2 Tight-Binding Model

Fig. 2.4 Schematic of InSe bilayer, with inequivalent interlayer hops included in the tight-binding model indicated

while Se–Se hoppings are included as H X 1 ,X 2 =

 i, j

 (X X ) † x(2)1 js x(1)2is + tss

+

 α,β

+ H.c.,

 α

X

Rα (1)2i

X (2)1 j (X X ) tsp



† † x(2)1 js x (1)2i pα − x (2)1 j pα x (1)2is

X X δαβ tπ(X X ) − (tσ(X X ) + tπ(X X ) )Rα (1)2i (2)1 j

X X Rβ (1)2i (2)1 j





⎤ † ⎦ x(2)1 j pβ x (1)2i pα

(2.6e)

with the sums over i, j going over next-nearest-neigbhouring In–Se pairs and nearest-neighbouring Se–Se pairs in adjacent layers, respectively. The additional 14 parameters added by the interlayer hops were determined by fitting to (scissor-corrected) DFT calculations for the bilayer case. Since DFT does not handle van der Waals type interactions well in crystal structure relaxation calculations it was decided to combine the monolayer geometry relaxed within the LDA [8] with the experimentally known interlayer separation az = 8.32 Å[20]. The convergence parameters for the bilayer calculation were the same as those for the monolayer case (600 eV plane wave cutoff, 12 × 12 k-point grid) [8]. The bilayer

2.2 Multilayer Films

23

Table 2.3 Fitted interlayer hops (eV) for γ-InSe as defined in the Hamiltonian (Eq. (2.6b)). Italicised values are those for the fit without scissor correction (X X ) tss

(X X ) tsp

tπ(X X )

tσ(X X )

−0.647 −0.731

−0.626 −0.461

−0.137 −0.119

−0.830 −0.761

(M X )

tss

−0.397 −0.152 (X M)

tss

−0.238 −0.332

(M X )

(M X )

t X s −M pz

t X pz −Ms

0.112 0.072

−0.734 −0.504

(X M)

(X M)

(M X )

tπ

0.193 0.198 (X M)

(M X )

tσ

0.011 0.015 (X M)

t X s −M pz

t X pz −Ms

tπ

tσ

0.042 0.042

−0.233 −0.208

−0.398 −0.393

0.450 0.347

Fig. 2.5 Fitted tight-binding bands for bilayer InSe. Solid lines: fitting to scissor-corrected DFT, energies on left-hand axis. Dashed lines: fitted to uncorrected DFT, right-hand axis. Zero of energy is set to the bottom of the conduction band. The vertical lines marked A and B denote two important optical transitions

bands can be considered to consist of two ‘sub-bands’ for each monolayer band, so of the 32 bands produced in the bilayer bands 9 to 20 are included in the fitting. In a similar manner to the monolayer, the interlayer hops are chosen to achieve a good agreement between the model bands and (scissor-corrected) DFT results. The resulting parameters are given in Table 2.3, with a plot of the fitted bands against (scissor-corrected) DFT shown in Fig. 2.5.

24

2 Tight-Binding Model

2.2.2 Extension to Greater Numbers of Layers For greater numbers of layers the model is extended by making the assumption that the tight-binding parameters will be the same for all N . This gives a Hamiltonian of the form N N −1   Hn + Hn,n+1 (2.7a) H= n=1

n=1

where N is the total number of layers, Hn is the monolayer Hamiltonian on layer n as set out above, and Hn,n+1 takes into account inter-layer interactions between adjacent layers n and n + 1. It has the form Hn,n+1 = H X n ,Mn+1 + HMn ,X n+1 + H X n ,X n+1 .

(2.7b)

The vertical In–Se contribution is  M) tss(X M) m †(n+1)1is x(n)2is + t X(Xs −M m †(n+1)1i pz x(n)2is H X n ,Mn+1 = p i M) −t X(Xp −M m †(n+1)1is x(n)2i pz + tπ(X M) s

−tσ(X M) m †(n+1)1i pz x(n)2i pz



 α=x,y

m †(n+1)1i pα x(n)2i pα

+ H.c..

(2.7c)

For the other In–Se inter-layer interaction we have H Mn ,X n+1 =

 (X X ) † tss x(n+1)1 js m (n)2is

i, j

+ +

  M(n)2i X (n+1)1 j  (M X ) † (M X ) † Rα t M p−X s x(n+1)1 js m (n)2i pα − t Ms−X p x(n+1)1 j pα m (n)2is α



(M X )

δαβ tπ

α,β (M X )

−(tσ

(M X )

+ tπ

M(n)2i X (n+1)1 j

)Rα

M(n)2i X (n+1)1 j

Rβ

+ H.c.,

while Se–Se hoppings are included in the form



† x(n+1)1 j p m (n)2i pα



β

(2.7d)

2.2 Multilayer Films

H X n ,X n+1 =

25

 † tss(X X ) x(n+1)1 js x (n)2is i, j

+



X

Rα (n)2i

X (n+1)1 j (X X ) tsp

α



† † x(n+1)1 js x (n)2i pα − x (n+1)1 j pα x (n)2is



 + δαβ tπ(X X ) α,β

−(tσ(X X ) + tπ(X X ) )Rα (n)2i X

X (n+1)1 j

X

Rβ (n)2i

X (n+1)1 j

+ H.c.,



† x(n+1)1 j pβ x (n)2i pα



(2.7e)

where the sums over i, j are over nearest-neighbouring Se–Se pairs and nextnearest-neighbouring In–Se pairs in adjacent layers. Figure 2.6 shows the bands found from diagonalisation of the tight-binding model for N = 3, 4, and 5. The results show a significant decrease in the gap as the valence and conduction bands of the monolayer split strongly at , due in a large part to interlayer hopping between pz orbitals on the Se atoms, which gives a strong contribution due to the large value of tσ(X X ) . The conduction band minimum remains at , while the splitting in the valence band raises the centre of the band at  more than the band edge, moving the valence band maximum towards  for larger numbers of layers, towards the direct-gap limit in the bulk. The bottom panel of Fig. 2.6 shows the vertical band gaps at  according to the tight-binding model for N = 1−15 layers, together with a fit to a bulk quantisation model, which is developed to show how the change in the gap with number of layers can be understood in the context of confinement from a bulk picture as well as in the splitting of monolayer bands which the tight-binding model shows. The general boundary conditions for a wavefunction at a crystal surface can be written as ψ ± νaz ∂z ψ = 0,

(2.8)

where ν is a dimensionless constant ∼1, and characterises the distance to which the wavefunction extends beyond the crystal surface, and +/− is chosen for the upper/lower surface of the slab. If we substitute a general plane-wave wavefunction, ψ = uei pz z + ve−i pz z , where u and v are constants and pz is a small expansion term in out-of-plane momentum (k z ) around the bulk band-edges, we obtain the relation N pz az + 2 arctan(ν pz az ) = nπ,

(2.9)

where n is an integer, which, when expanded for small pz , gives the quantisation relation nπ . (2.10) pz = (N + 2ν)az

26 Fig. 2.6 Band structures for N = 3−5 layer γ-InSe found using scissor-corrected tight-binding Hamiltonian. Zero of energy is set at the conduction band edge in each case. The lowest panel shows the dependence of the vertical gap at the -point on the number of layers in N -layer InSe, compared with a fit to the bulk quantisation model, Eq. (2.11)

2 Tight-Binding Model

2.2 Multilayer Films

27

If we write the conduction (valence) band expansions as E c(v) ( pz ) = E c(v) (0) + 2 pz2 z , where Mc(v) is the bulk conduction (valence) band out-of-plane effective z 2Mc(v) mass, we can obtain a k z size quantisation formula for the band gap as a function of the number of layers, E g (N ) = E g (∞) +



2 z

2Mr

π 1 az (N + 2ν)

2 ,

(2.11)

z

z

with the reduced mass Mr =

Mc Mvz

= 0.032 m e and ν = 1.14 obtained from z  Mc + Mv fitting to vertical gaps (i.e. gaps at the -point) from the tight-binding model. The behaviour described by Eq. (2.11) is shown by a solid line in the lowest panel of Fig. 2.6. It should be noted that the limiting behaviour of the gap on approaching the bulk is E g (∞) = 1.27 eV, rather than the ∼1.4 eV expected from the scissor correction. We shall offer a potential means of overcoming this unexpected behaviour in Chap. 3.

2.3 Interband Optical Transitions To calculate the response of the system to light, and to evaluate the strength of interband optical transitions, we construct a k · p model of the monolayer consisting of single-band k · p models of the low-energy bands near , with the bands coupled by the perturbations expected from incident photons; ⎛

Hc E z dz ⎜ E z dz Hv H =⎜ ⎝ e βA 0 cm e 0 0

e βA cm e

0 Hv1 0

⎞ 0 0 ⎟ ⎟. 0 ⎠ Hv2

(2.12)

Of the single-band k · p expansions, the conduction band is described by an effective mass expansion, Hc = 2 k 2 /2m c , with the conduction band effective mass m c = 0.18 m e , while on the scale of the band gap and the dispersions of the other bands we approximate the valence band as flat (Hv = E v ).1 Bands v1 and v2 are twicedegenerate at , and must be described by two-component Hamiltonians, which we write as

· p expansion up to 8th order, necessary to obtain the anisotropy responsible for the expected Lifshitz transition, but as we neglect spin-orbit coupling, which changes the depth of the offset, we do not take that approach here.

1 Reference [8] demonstrates the k

28

2 Tight-Binding Model

Hv1(2) = E v1(2) 1σ +

2 (k x2 − k 2y ) 2 2k x k y 2 k 2 1σ + σ + σy x 2m 1(2) 2m 1(2) 2m 1(2)

(2.13)

are Pauli matrices, and 1σ is an identity matrix, in a space of p ± ≡ where σx,y √ ( px ± i p y )/ 2 orbitals. This gives the -point v1 , v2 bands a basis with orbital angular momentum projection L z = ±1. The masses m 1 = m 2 = −0.30 m e and m 1 = m 2 = −0.45 m e are contributions to the effective mass of the light and heavy hole branches of v1 and v2 . The valence and conduction bands are singly-degenerate at , and therefore (due to a lack of angular momentum difference between the bands, which both have orbital angular momentum projection L z = 0) the principal interband optical transition will have an out-of-plane dipole character. Furthermore, in the monolayer, c and v have opposite parity under σh reflection, so the interband matrix element of the in-plane momentum operator is zero even away from , while although for few-layer InSe σh symmetry is broken, the effect is weak, so the momentum matrix element for the transition grows extremely slowly away from . The appropriate means of describing coupling to applied light is therefore E z dz , where E z is the out-of-plane electric field associated with the light, and dz is the out-of-plane interband dipole matrix element which since the crystal is finite in z we calculate directly as dz (k) = ec |z| v(k) = e



∗ Cck (λ)Cvk (λ)z(λ),

(2.14)

λ

where Cc(v)k (λ) is the eigenfunction coefficient for the conduction (valence) band, orbital λ at k. z(λ) is the z-coordinate, relative to the mean plane of the crystal, of the atom on which orbital λ sits. In contrast, v1 has the same parity under σh reflection as c, and is twice degenerate, with L z = ±1, so the transition between v1 and c couples strongly to in-plane polarised light. We couple c and v1 in Eq. (2.12) by cme e βA, where m e is the free electron mass and A is the vector potential of in-plane polarised light, in a basis of σ ± polarisation. β ≡ |c |p| v1 |, where the momentum operator is found in a tight-binding context from the k-space gradient of the Hamiltonian as [21] p=

me ∇k H, 

(2.15)

which enables the calculation of the matrix element of momentum directly from the tight-binding model, with no additional parameters required. For example, β is found from the model as     me  ∗    (2.16) Cck=0 (λ)Cv1 k=0 (λ )∇k Hλ,λ (k = 0), β=   λ,λ

2.3 Interband Optical Transitions

29

with the sum over λ, λ running over all orbitals in the model. Cv1 k=0 (λ) is the -point eigenfunction coefficient2 of v1 , orbital λ, and Hλ,λ = λ |H | λ. Writing the flux of the Poynting vector of in-plane polarised light across the crystal surface as cE 2 , (2.17) W = 4π and treating the coupling of the B-line transition to in-plane polarised light, cme e βA, as a perturbation we write the rate of energy absorption using the Fermi golden rule as  2π δ(E c − E v1 − ω), (2.18) W = ω |A|2 β 2  k where ω is the energy of the transition, which gives an absorption coefficient at , gB =

m ∗c,v1 W e2 , = 8π β 2 W c ωm 2e

(2.19)

where m ∗c,v1 is the reduced effective mass of c and v1 , which enters into the joint density of states of the transition. As the -point v1 state is a degenerate pair of heavy and light holes an appropriate treatment of the contribution of m v1 to the reduced mass must be carried out. For small displacements along k x away from  the heavy holes are composed of p y orbitals, with c |p| v1  pointing along y, while the light holes have px orbitals and the matrix element points along x. For a given in-plane polarisation the appropriate effective mass will vary from the light to heavy holes dependent on the angle of k. Integrating round all angles of k we obtain m ∗c,v1 =

 −1  

1  −1 −1 −1 −1 −1 −1 m c + m −1 . + m + m + m − m c 1 1 1 1 2

(2.20)

The absorption coefficient for the B-line for in-plane polarised light is plotted in Fig. 2.7b. As a transition from bands which couple only weakly between layers, the absorption coefficient is nearly independent of the number of layers. To find a similar quantity for the out-of-plane dipole coupling for the principal interband transition, we take our perturbation as E z dz , which for a photon incident at 45◦ to the crystal surface yields an absorption coefficient gA =

  e2  dz 2 ωm c W = 8π   , W c e 2

(2.21)

where the reduced effective mass is approximated as the conduction band effective mass, since we approximate the valence band as flat. 2 As

we take the modulus of p it is sufficient to choose either component of the twice-degenerate band.

30

2 Tight-Binding Model

(a)

(b)

(c)

(d)

Fig. 2.7 Variation with N for N -layer InSe of a interband in-plane momentum matrix element for ‘B-line’ absorption from twice-degenerate px,y -dominated bands to the conduction band and b absorption coefficient for the transition, c principal interband out-of-plane dipole matrix element and d absorption coefficient for light incident at 45◦ to the crystal surface

The interband matrix elements β and dz are plotted in Fig. 2.7, together with the absorption coefficients resulting from Eqs. (2.19) and (2.21). These show that dz is strongly dependent on crystal thickness, while β, being related to absorption from in-plane ( px,y -dominated) bands which couple only weakly between layers, has only a very slight dependence on N .

2.4 Effects of Scissor Correction Figure 2.8 shows the dz and px momentum matrix elements between c and v, described above, calculated for few-layer InSe with and without scissor correction taken into account. In contrast to the results for monolayer band dispersions and orbital decompositions (Figs. 2.2 and 2.3), which show negligible changes on scissor correction, we see significant changes to the results obtained when applying the scissor correction. With the gap in the uncorrected model close to the DFT gap we see the DFT and tight-binding interband matrix elements taking on similar values.

2.4 Effects of Scissor Correction

31

(a)

(b)

Fig. 2.8 Effects of scissor correction on variation with N of values of a z-component of interband dipole matrix element at  and b in-plane interband momentum matrix element near  (N = 2, 3, 4, 5 in order of increasing magnitude)

On correcting the gap, the increase in the magnitude of the matrix elements with N slows greatly, and saturation is approached at a much smaller value. We can gain insight into the reasons for this behaviour by comparing the wavefunctions across the slab for the corrected and uncorrected models. In Fig. 2.9 we compare the coefficients from the TB S.C. and TB models of the selenium pz orbitals in the -point valence and conduction band wavefunctions of 19-layer InSe. The basic form of the wavefunction across the slab resembles that of a ‘particle-in-a-box’ for a confining potential. However, superimposed on this is a substantial sublayer-by-sublayer

32

2 Tight-Binding Model

Fig. 2.9 Coefficients of Se- pz components of -point conduction and valence band tight-binding wavefunctions across the 38 sublayers of 19-layer InSe, calculated using parameter sets a with and b without scissor correction

oscillation in the magnitude. The similarity between the orbital decompositions of the two parameter sets in the monolayer case (Fig. 2.3) does not carry over to the few-layer model, with the oscillations being much larger when the scissor correction is not considered. It is these oscillations which give the long-range structure needed to see an increase in dz with N , and it is the slight difference in the magnitude of the oscillations between the surfaces of the crystal that breaks σh symmetry and allows for a finite px,y in few-layer crystals. When the gap is underestimated, the intralayer energy scale on which the interlayer hops act is much reduced, and their effect on the sublayer wavefunctions is overstated. This effect will be discussed further in the context of modelling few-layer InSe in a basis of the monolayer bands, with interlayer hops between the bands, in Chaps. 3 and 4. In this chapter we have constructed a tight-binding model to describe the electronic and optical properties of the bands of mono- and few-layer InSe, including s and p orbitals on both In and Se, to reproduce the features of the low-energy bands. The model has been parametrised by fitting to DFT calculations, with a rigid ‘scissor-correction’ applied to the bands to correct for the underestimation of the gap compared with experiment in the bulk crystal. The calculations of optical properties show that the principal interband optical transition is of an out-of-plane polarised character, while the next-lowest-energy transition couples to light polarised in-plane. Spin-orbit coupling, which we discuss in Chap. 4, modifies the absolute restrictions in the monolayer and at , with important consequences for experiments, but the dominant polarisation character of the transitions remains. Finally, we note that the underestimation of the gap by DFT, being significant especially in few-layer InSe, has consequences for the few-layer wavefunctions, and hence for the oscillator strengths of the optical transitions, as well as their energies.

References

33

References 1. Goringe CM, Bowler DR, Hernández E (1997) Rep Prog Phys 60:1447 2. Frauenheim T, Seifert G, Elstner M, Niehaus T, Köhler C, Amkreutz M, Sternberg M, Hajnal Z, Carlo AD, Suhai S (2002) J Phys Condens Matter 14:3015 3. Haldane FDM (1988) Phys Rev Lett 61:2015 4. Pereira VM, Neto AHC, Peres NMR (2009) Phys Rev B 80:45401 5. Pearce AJ, Mariani E, Burkard G (2016) Phys Rev B 94:155416 6. Hancock Y, Uppstu A, Saloriutta K, Harju A, Puska MJ (2010) Phys Rev B 81:245402 7. Bandurin DA, Tyurnina AV, Geliang LY, Mishchenko A, Zólyomi V, Morozov SV, Kumar RK, Gorbachev RV, Kudrynskyi ZR, Pezzini S, Kovalyuk ZD, Zeilter U, Novoselov KS, Patanè A, Eaves L, Grigorieva II, Fal’ko VI, Geim AK, Cao Y (2017) Nat Nanotechnol 12:223 8. Zólyomi V, Drummond ND, Fal’ko VI (2014) Phys Rev B 89:205416 9. Slater JC, Koster GF (1954) Phys Rev 94:1498 10. Jones R (2015) Rev Mod Phys 87:897 11. Fiorentini V, Baldereschi A (1995) Phys Rev B 51:17196 12. Johnson KA, Ashcroft NW (1998) Phys Rev B 58:15548 13. Bernstein N, Mehl MJ, Papaconstantopoulos DA (2002) Phys Rev B 66:075212 14. Parashari SS, Kumar S, Auluck S (2008) Phys B 403:3077 15. Thilagam A, Simpson DJ, Gerson AR (2010) J Phys Condens Matter 23:025901 16. Babu KR, Lingam CB, Auluck S, Tewari SP, Vaitheeswaran G (2011) J Solid State Chem 184:343 17. Camassel J, Merle P, Mathieu H, Chevy A (1978) Phys Rev B 17:4718 18. Millot M, Broto JM, George S, González J, Segura A (2010) Phys Rev B 81:205211 19. Mudd GW, Svatek SA, Ren T, Patanè A, Makarovsky O, Eaves L, Beton PH, Kovalyuk ZD, Lashkarev GV, Kudrynskyi ZR, Dmitriev AI (2013) Adv Mater 25:5714 20. Rigoult J, Rimsky A, Kuhn A (1980) Acta Crystallogr B 36:916 21. Lew Yan Voon LC, Ram-Mohan LR (1993) Phys Rev B 47:15500

Chapter 3

Hybrid k · p Tight-Binding Theory

In Chap. 2 we described the electronic and optical properties of InSe using an atomistic tight-binding model, itself parametrised to an atomistic density functional theory calculation. In describing the optical properties, we briefly introduced a k · p description of the monolayer bands, constructing a model consisting of single-band k · p expansions around the  point coupled by perturbative electromagnetic fields due to incident light. Here we take advantage of the significant anisotropy of the 2D materials and InSe, which feature strong covalent bonding within crystalline layers, whilst the bonding between the layers has a comparatively weak van der Waals character. Each layer is reduced to a basis of its monolayer k · p bands, then in multilayer films we couple successive layers with tight-binding hops—which are between monolayer band states rather than atomic orbitals. A model developed using this approach is here described as a hybrid k · p tight-binding (HkpTB) model.

3.1 Interlayer Tight-Binding Hops Between Monolayer Bands for Aligned and Misaligned Crystals The HkpTB modelling technique is first employed here in a description of the band edges and gaps of aligned crystals and misaligned films of InSe. Basing the model on results developed in Chap. 2, two features of monolayer and few-layer InSe are exploited: first, that the conduction and valence band edges of InSe appear in the vicinity of the -point with an almost isotropic dispersion, and second, that the interlayer hopping of electronic states close to the band edges is strong, as seen in the strong subband splitting in few-layer InSe. These two features of InSe simplify an analytical description of misaligned multilayer films, compared with graphene and transition metal dichalcoegnides where the monolayer band edges appear in the corners of the Brillouin zone, or phosphorene, which has a strongly anisotropic dispersion near the -point [1–3]. © Springer Nature Switzerland AG 2019 S. J. Magorrian, Theory of Electronic and Optical Properties of Atomically Thin Films of Indium Selenide, Springer Theses, https://doi.org/10.1007/978-3-030-25715-6_3

35

3 Hybrid k · p Tight-Binding Theory

36

To determine which monolayer bands need to be included in the model we consider the splittings of the monolayer tight-binding bands into few-layer subbands as the number of layers is increased (Figs. 2.5 and 2.6). Of the seven monolayer bands fitted in the tight-binding model, only two, c and v, exhibit significant interlayer-hoppinginduced splitting due to s and pz contributions from Se atoms on the outside of the monolayer. Of the others, c1 is dominantly on the In atoms, which being on the inside of the monolayer crystal couple only weakly to neighbouring layers, while v1 and v2 are dominated by in-plane px,y orbitals and again exhibit only a small change on going from the monolayer to the few-layer case. We therefore construct a two-band model from the monolayer band-edge states, c and v. To construct tight-binding hops in a basis of the -point monolayer bands in an arbitrarily stacked structure, we represent the Wannier-Bloch states of -point electrons in band α = c, v of monolayer InSe in a plane-wave expansion as  α (r, z) =



αG (z)eiG·r

G

= α0 (z) + α1 (z)



eiG1(j) ·r + α∗ 1 (z)



eiG1(j) ·r + · · ·

(3.1)

j=2,4,6

j=1,3,5

where r is the 2D in-plane position vector, and z is the out-of-plane co-ordinate. Since, at , c and v are totally in-plane symmetric, we have expanded the sum over reciprocal lattice vectors G into terms corresponding to each ring of reciprocal lattice vectors with common |Gi(j) | = |Gi(j ) | (including the constant ‘0-th’ term at the origin, G0 = 0). The Fourier coefficients, G (z), then take the same magnitude for all reciprocal lattice vectors in a given ring. Since we expect the Fourier coefficients G (z) to decay faster as a function of |z| for larger |G| we retain only the 0-th and first-order terms in our sum when using the expansion to calculate interlayer couplings. In a similar manner, we can write the potential arising from a layer, V (r, z) in a plane-wave expansion as V (r, z) =



vG (z)eiG·r = v0 (z) + v1 (z)

G

 j=1,3,5

eiG1(j) ·r + v1∗ (z)



eiG1(j) ·r + · · ·

j=2,4,6

(3.2) Writing the integral for the onsite energy shift induced in layer n, band α by the potential from layer n + 1, α (u, az ), as  α (u, az ) =

dzdrV (r − u, z − az )| α (r, z)|2 ,

(3.3)

and the interlayer tight-binding hops between band α in layer n and band β in layer n + 1, tα,β (u, az ), as

3.1 Interlayer Tight-Binding Hops Between Monolayer …

 tα,β (u, az ) =

37

  dzdr V (r, z) + V (r − u, z − az )  α (r, z) ∗β (r − u, z − az ), (3.4)

we can substitute the plane wave expansions for  α and V , Eqs. (3.1) and (3.2) respectively, to obtain the dependence of α and tα,β on u and az for aligned crystals as ⎤ ⎡  α (u, az ) = η0α (az ) +  ⎣η1α (az ) (3.5) eiG1(j)·u ⎦ j=1,3,5

⎡

and α,β

α,β

tα,β (u, az ) = τ0 (az ) +  ⎣τ1 (az )



⎤ eiG1(j)·u ⎦ .

(3.6)

j=1,3,5

Here the plane-wave expansion coefficients are  η0 (az ) = η1α (az ) 

α,β

 =2

dzv0 (z − az )20 (z),

(3.7)

  α 2 dzv1 (z − az ) α0 (z)∗α 1 (z) + 1 (z) ,

(3.8)

β

dzα0 (z)0 (z − az ) (v0 (z) + v0 (z − az ))

 ∗β β + 6 v1 (z − az )α0 (z)1 (z − az ) + v1 (z)∗α 1 (z)0 (z − az ) ,

τ0 (az ) =

(3.9) and α,β τ1 (az )

 =2

∗β

dzα1 (z)1 (z − az ) (v0 (z) + v0 (z − az )) ∗β

β

+ v1 (z)α0 (z)1 (z − az ) + v1∗ (z − az )α1 (z)0 (z − az )

 ∗β β α + 2 v1∗ (z)∗α (z) (z − a ) + v (z − a ) (z) (z − a ) . (3.10) z 1 z z 1 1 1 1 In the case where the interface is between two misaligned layers (and where the misalignment angle does not carry the set of reciprocal lattice vectors into each other, i.e. where θ is not a multiple of 60◦ ) there is no dependence on the relative in-plane shift between the two layers, and the parameters reduce to ˜ α (az ) = η0α (az ) 

(3.11)

38

and

3 Hybrid k · p Tight-Binding Theory

α,β t˜α,β (az ) = τ0 (az ).

(3.12)

When α = β, then tα,α (u, az ) ≡ tα (u, az ) is an even function of both u and az (which can be easily demonstrated by applying co-ordinate transformations r = −(r − u) and z  = −(z − az ) to either Eq. (3.4) or Eq. (3.10)). For interband (α = β) interlayer hops, we have tβ,α (u, az ) = ±tα,β (−u, az )

(3.13)

where the + (−) sign is chosen when the monolayer bands α and β have the same (opposite) symmetry under σh reflection. Motivated by the domination of the interlayer hops by Se-Se pairs on the outside of the layers, which on their own would form a triangular lattice with D6h symmetry, we make the approximation that the coefficients of the harmonics in Eqs. (3.1) and (3.2) are real which leads to the relation tβ,α (u, az ) = ±tα,β (u, az ).

(3.14)

The process for finding parameters for the model for misaligned crystals has two main steps: (i) the fitting of the model to aligned first-principles calculations for various u, from which the contributions of the various terms can be decomposed, and α,β η0α (az ) and τ0 (az ) extracted; and (ii) the determination of an appropriate interlayer separation, az , at which to carry out the fitting. To demonstrate the fitting procedure and properties of the band-based tight-binding model, we first consider the model in the context of the bulk crystal, γ-InSe.

3.2 Model Parametrisation—Bulk γ-InSe The γ polytype of InSe, shown again in Fig. 3.1, is constructed with √ successive layers aligned with a relative shift between each pair of layers of u = 3a/2ˆy. This shift retains the C3 symmetry of the crystal, but the lack of inversion symmetry in the monolayer means that the shift breaks the σh reflection symmetry of the monolayer, reducing the point group symmetry of the crystal from D3h to C3v . The experimental interlayer separation, az , is 8.32 Å [4]. With bands c and v included in the model there are 6 independent parameters—E c − E v is the difference between the γ on-site energies of the monolayer bands (i.e. the -point monolayer gap) and δc(v) is the shift to the onsite energy of the conduction (valence) states due to potentials γ γ from adjacent layers. The hops tc , tvγ , tcv are the conduction-conduction, valenceγ γ valence, and conduction-valence hops, respectively. We set tvc = −tcv following the approximation made in Eq. (3.14).

3.2 Model Parametrisation—Bulk γ-InSe

39

Fig. 3.1 Structure of bilayer γ-InSe, together with schematic of HkpTB model—empty/filled circles denote the monolayer k · p valence/conduction bands of each layer. These are coupled by interlayer intraband (solid lines) and interband (dashed lines) tight-binding hops

The model as constructed then has a Hamiltonian given by H=

N 

E c cn† cn + E v vn† vn

n

+

N −1

     † † γc cn† cn + c(n+1) c(n+1) + γv vn† vn + v(n+1) v(n+1) n

N −1 

   † † † † γ tcγ c(n+1) v(n+1) cn + tvγ v(n+1) vn + tcv cn − c(n+1) vn + H.c. , (3.15) + n

which we can write in matrix form for a periodic bulk crystal as  H (k z ) =

 γ γ γ E c + 2c + 2tc cos(k z az ) 2itcv sin(k z az ) . γ E v + 2γv + 2tvγ cos(k z az ) −2itcv sin(k z az )

(3.16)

Diagonalisation yields 2 bands with k z dispersion given by γ

E 1,2 (k z ) =

 1 (E c + 2γc + E v + 2γv ) + 2(tcγ + tvγ ) cos(k z az ) 2    2 γ γ γ γ γ 2 ± (E c + 2c ) − (E v + 2v ) + 2(tc − tv ) cos(k z az ) + 4tcv sin2 (k z az ) .

(3.17)

40

3 Hybrid k · p Tight-Binding Theory

Fig. 3.2 Low energy DFT bulk dispersion (dots) for γ-InSe, interlayer separation az = 8.32 Å, along k z for k x = k y = 0. Bands v1 and v2 are twice-degenerate. The left hand axis shows scissorcorrected (S.C.) energies. The lines show the bands predicted by the two-band model, fitted to DFT data without correction. Compared with the dispersions of c and v, the other low-energy bands plotted are nearly flat, so interlayer hops involving these bands are neglected in the model

We choose E c − E v = 1.80 eV, the DFT monolayer gap, and fit the remaining parameters to bands from a DFT calculation for the bulk crystal, with the resulting bands shown alongside the DFT bands in Fig. 3.2 and the fitted parameters given in Table 3.1. Also shown in Fig. 3.2 are the other low-energy bands which were not included in the model. In agreement with the very weak subband splitting observed in the few-layer case, their bulk counterparts are only very weakly dispersive, justifying their being left out of the two-band model. The valence band actually crosses v1 and v2 , but, in the absence of spin-orbit coupling, does not hybridise with them in any way.

3.2.1 Correction of Band Gap in Hybrid k · p Tight-Binding Model As discussed in Chap. 2, the underestimation of the band gap by DFT in the local density approximation for bulk InSe is significant, with the calculation returning a bulk gap of 0.411 eV against experimental measurements which find a low-temperature band gap ∼1.4 eV. In Chap. 2 we corrected for this by assuming that DFT returns bands and dispersions which can be taken as correct aside from an underestimation of the band gap itself, and could thus be corrected for by a rigid shift of the occupied bands w.r.t. the unoccupied bands—the ‘scissor correction’. Although such an approach had a minimal effect on the orbital decomposition and matrix elements

3.2 Model Parametrisation—Bulk γ-InSe

41

in the monolayer, the changes w.r.t. the DFT calculations in the dependence of the wavefunction structure and optical transition matrix elements on the number of layers were significant. If we consider the off-diagonal term in Eq. (3.16) in the context of perturbation theory, it is clear that a 3-fold underestimation of the band gap will lead to an overestimation of the effect of the interband hopping, tcv , on the system. Such a change, as well as having consequences for the wavefunctions and matrix elements, will also have consequences for the dispersion of the bands. Consider the expression for the band-edge out-of-plane electron effective mass: Mc z

  γ2 −1 2 4t cv = 2 tcγ + , 2az Eg

γ

(3.18)

γ

where E g = (E c + 2c ) − (E v + 2γv ) − 2(tc − tvγ ) is the bulk band gap. If we keep the interlayer hopping parameters the same and simply change the band gap by increasing E c − E v then we will obtain an increase in the effective mass. Using parameters straight from a fit to uncorrected DFT data, we find an effective mass

z

z of Mc = 0.048 m e , close to the DFT value, Mc = 0.041 m e , but about half the

z mass found by experiment (Mc = 0.09 m e ). If we correct the gap by increasing the difference between the monolayer onsite energies by 1 eV, while leaving the interlayer hops unchanged (parameter set labelled ‘G.C.’ in Table 3.1), the effective

z mass in the model increases to Mc = 0.092 m e . An attempt to fit to rigidly scissorcorrected bands (parameter set labelled ‘S.C.’ in Table 3.1) results in a lower quality fit with the model unable to replicate the very light out-of-plane effective mass once the gap is increased.

Table 3.1 Model parameters (eV) for γ-InSe, with interlayer separation 8.32 Å, for fits to DFT data (fit) and scissor-corrected DFT data (S.C.). Also given is the parameter set from a fit to the DFT data, with a retrospective gap correction applied to the underlying monolayer band energies (G.C.). Out-of-plane electron effective masses (in units of m 0 , the free electron mass) and the bulk band gap are also given, with experimental and DFT results quoted for comparison. The values given for the experimental quasiparticle gap and the out-of-plane electron effective mass can be found in Refs. [5, 6], respectively DFT Fit S.C. G.C. Experiment Ec − Ev γ tc γ tv γ tcv γ c γ v E g (eV) m ∗e (m 0 )

1.80

0.41 0.043

1.80 0.34 −0.42 0.29 −0.03 0.03 0.41 0.048

2.80 0.33 −0.42 0.36 −0.03 0.03 1.43 0.076

2.80 0.34 −0.42 0.29 −0.03 0.03 1.41 0.092

1.35 (0.081 ± 0.009)

3 Hybrid k · p Tight-Binding Theory

42

We have presented two options to be chosen between when considering the effect of tcv on the band dispersion. We can either fit to the uncorrected DFT data, assuming the hops are correct but the underlying onsite energies are not, then correct the monolayer gap and obtain a dispersion different from that calculated using DFT, or we can attempt to replicate the DFT dispersion but with a larger gap. The latter option is consistent with the assumptions underlying the scissor correction—that DFT gives good bandwidths and dispersions, with only the gap itself incorrect. However, the former means of correction produces a better initial fit to DFT, and a much improved agreement with the experimental out-of-plane electron effective mass. To inform our choice in considering a departure from the traditional method of correcting the band gap we carry out an additional DFT calculation on the bulk using the HSE06 hybrid exchange-correlation functional [7, 8]. Hybrid functional approaches attempt to counter the underestimation of the band gap due to the neglect of the exchangecorrelation discontinuity by mixing in exact exchange into the exchange-correlation energy, with the proportion chosen empirically to achieve a good accuracy for a range of molecules and crystals. For bulk InSe use of the HSE06 hybrid functional

z returns an out-of-plane electron effective mass Mc = 0.098 m e , much larger than the LDA mass and in far closer agreement with the G.C. model and experimental values. We therefore adopt the ‘G.C.’ parameter set for our model, and carry out a similar procedure for finding model parameters for all other stackings—that is, setting E c − E v = 1.80 eV, fitting tc , tv , tcv , c , v to uncorrected LDA DFT data, then increasing E c − E v to 2.80 eV in the final parameter set. Figure 3.3 summarises the few-layer (up to 10 layers) gaps obtained from scissorcorrected DFT calculations, from the HkpTB model with scissor and gap correction methods as described above, and from the tight-binding model of Chap. 2. The scissor-corrected DFT and HkpTB model are almost in agreement, with a small discrepancy arising between them as the HkpTB model struggles to replicate the light band-edge effective mass from DFT in the bulk, and consequently does not exactly reproduce the discretised few-layer bands and gaps. The gap correction method

2.8

Vertical gap (eV)

Fig. 3.3 Comparison of band gaps in monolayer to 10-layer InSe obtained by DFT, tight-binding and HkpTB methods, using various means of correction of underestimation of band gap by DFT

S.C. DFT HkpTB G.C. HkpTB S.C. TB G.C. TB S.C.

2.4 2.0 1.6 1.2

1

5 N

10

3.2 Model Parametrisation—Bulk γ-InSe

43

gives slightly smaller gaps for few-layer InSe, as expected from the increased effective mass in the bulk resulting from this method. Finally, motivated by the corrections applied here, we revisit the question of how to fit the tight-binding model in Chap. 2. The finding here is that while it is better to use a corrected monolayer parameter set, attempting to fit interlayer hopping parameters to scissor-corrected DFT—as was done for the tight-binding model—can give misleading results. We therefore compare the scissor-corrected tight-binding as presented in Chap. 2 with a ‘gap-corrected tight-binding model’—i.e. using the ‘S.C.’ monolayer tight-binding parameters from Table 2.1, but with the interlayer hops taken as those fitted to uncorrected DFT in Table 2.3. We see in Fig. 3.3 that while the scissor-corrected tight-binding model gives notably smaller gaps at larger numbers of layers, the gapcorrected tight-binding model very closely tracks the chosen HkpTB model result. While the scissor-corrected tight-binding model of Chap. 2 does not reproduce the expected gap for N  1, the gap-corrected tight-binding model gets much closer, with a fit to Eq. (2.11) giving a limiting value E g (∞) = 1.39 eV as well as an

z increased out-of-plane reduced mass of Mr = 0.040 m e .1 The ‘TB G.C.’ parameter set may therefore be considered a useful choice.

3.3 Parametrisation—Misaligned Structures When crystals are extracted from a bulk material by mechanical exfoliation [9] or grown using chemical vapour deposition [10], they have very well-defined electronic band structures prescribed by their lattice and chemical composition. In contrast, multiple exfoliation of individual layers from bulk parent crystals—either by a sequence of mechanical transfers [11] or by liquid phase exfoliation [12] aimed at developing printable electronics—produces much more complex disordered-looking structures known as laminates in which successive layers of 2D materials (graphene [13], transition metal dichalcogenides [14], etc.) are stacked with arbitrary or random relative orientation of 2D lattices. However, a DFT calculation of laminate films can be very difficult, due to the added complexity of a misalignment between the layers. This requires the use of very large supercells containing hundreds or thousands of atoms, and it can be necessary to study multiple such structures to capture the orientation dependence of the phsyical properties of laminate films. In a similar manner the construction of an atomistic tight-binding model for misaligned structures requires many hopping parameters, and/or a means of describing the dependence of the hops on interatomic distances. For this reason mesoscale and phenomenological theories have often been employed when describing band-edge states of electrons in misaligned heterostructures of 2D materials [15–18].

parameter ν, which takes account of the extension of the wavefunction beyond the crystal sufrace, is reduced slightly in the fit to ν = 1.01.

1 The

3 Hybrid k · p Tight-Binding Theory

44

To apply the model to misaligned structures, we follow Eqs. (3.5) and (3.6) and write the interlayer hops and onsite energy shifts as α (u, az ) = η0α (az ) + η1α (az ) f (u) and

α,β

α,β

tα,β (u, az ) = τ0 (az ) + τ1 (az ) f (u),

(3.19)

(3.20)

where ⎡ f (u) =  ⎣



⎤ eiG1(j)·u ⎦

j=1,3,5

       uy uy 2π 2π 2u y 2π + cos + cos , ux + √ −u x + √ = cos √ a a a 3 3 3 (3.21) 

with u x and u y the x- and y-components of u, respectively. We have used here the fact α=β α=β is real as a consequence of the approximation that η1α and τ1 are real, while τ1 employed in Eq. (3.14). If we define α,β

and

α,β

AA (az ) ≡ tα,β (0, az ) = τ0 (az ) + 3τ1 (az ) tα,β

(3.22)

√ 3 α,β γ α,β tα,β (az ) ≡ tα,β (a/ 3ˆy, az ) = τ0 (az ) − τ1 (az ), 2

(3.23)

then we can express the hops for a misaligned interface as γ AA (az ) + 2tα,β (az ))/3. t˜α,β (az ) = (tα,β

(3.24)

In a similar manner the on-site energy shifts due to an adjacent misaligned layer take the form γ ˜ α (az ) = (AA  (3.25) α (az ) + 2α (az ))/3.

3.3.1 Change in Interlayer Distance on Misalignment Following the procedure outlined for the description of γ-stacked InSe with an interlayer distance of az = 8.32 Å, we now calculate the hoppings for both AA and γ-stacked InSe but with a different interlayer separation. The reason for this is that the interlayer distance az depends on the stacking order. In the case of large misalignment angles expected in laminates, the local stacking order will vary quickly between AA-like (interlayer Se pairs nearly opposite) and γ-like (Se in one layer

3.3 Parametrisation—Misaligned Structures

45

above the centre of the triangle of Se atoms in the layer below). The high bending rigidity of InSe [19] will mean that az will vary very little, and can therefore be chosen as a constant value, one which falls between the optimal separations of γ-InSe and AA-InSe. To find an appropriate interlayer distance for the rotated interface we apply the assumptions adopted in the parametrisation of the interlayer hoppings to the interlayer binding energy, giving an expression for the interlayer binding energy of the form (3.26) E b (az , u) = E b0 (az ) + E b1 (az ) f (u), so we can write

γ E˜ b (az ) = (E bAA (az ) + 2E b (az ))/3,

(3.27)

γ

where E˜ b (az ), E b (az ), and E bA A (az ) are the interlayer binding energies of the misaligned, γ and AA stacking interfaces, respectively. We calculate these energies using the optB88 van der Waals DFT functional [20]. In these calculations we make the approximation that the monolayer structure remains unchanged. We find that the lowest-energy interlayer separation in bulk AA-InSe occurs at around 9.12 Å (that is, 0.8 Å greater than the experimental interlayer separation of γ-InSe). As summarised in Fig. 3.4, the DFT LDA band structure calculation for the AA-stacked bulk InSe at an interlayer separation of 9.12 Å gives an uncorrected bulk gap of 0.852 eV, against 0.411 eV for γ-InSe at an interlayer separation of 8.32 Å, which further emphasises the importance of finding the correct interlayer distance to use for laminates. The energy calculated for rotated interfaces from Eq. (3.27) has a minimum at around az = 8.72 Å, which we adopt as our az for the rotated interface.

3.3.2 Misaligned Bulk and Few-Layer Stacks Proceeding with the parametrisation process we now calculate the band structure of γ-InSe and AA-InSe with the above discussed interlayer separation of az = 8.72 Å, with the resulting parameters shown in Table 3.2. The magnitudes of the hoppings are smaller than for γ-InSe with az = 8.32 Å, since the interlayer hopping arises primarily from a large contribution of pz orbitals on Se atoms to the band edge wave functions, giving a strong dependence on az to the hoppings. With hoppings for two stackings at our disposal, we then extract the parameters for the misaligned interface, τ0 (η0 ) in Table 3.2. Having extracted the parameters, we use Eq. (3.16) to calculate bulk band-gaps and band-edge out-of-plane effective masses, which are presented in Table 3.3. The increased interlayer separation resulting from repulsion of vertically opposite Se-Se pairs results in heavier effective masses and an increased band gap. We also use the calculated parameters to calculate band gaps for finite-thickness misaligned laminate films, with a comparison of the gaps to those calculated for crystalline γ-InSe shown in Fig. 3.5b.

3 Hybrid k · p Tight-Binding Theory

46

(a)

(d)

(b)

(e)

(c)

(f)

Fig. 3.4 Ordered stackings of InSe films and their band structures. Panels a, b, and c depict the top view structure of InSe films in γ stacking, AA stacking, and a stacking with interlayer shift equivalent to misalignment (R-InSe), respectively; atoms in each top layer are in colour, those in the bottom layer in grayscale (darker atoms In, lighter atoms Se). Panels d, e, and f show the band structures in the bulk phases of γ-InSe (az = 8.32 Å), AA-InSe (az = 9.12 Å), and R-InSe (az = 8.72 Å), respectively, comparing the model with DFT data; the left hand axis shows the energy of the model bands, while the right hand axis shows the uncorrected DFT energy Table 3.2 Model parameters (eV) for γ-InSe, AA-InSe, and predicted parameters for the misaligned interface calculated for interlayer separation az = 8.72 Å γ AA τ0 (η0 ) τ1 (η1 ) tc tv tcv (δv ) (δc )

0.258 −0.326 0.231 −0.033 0.032

0.259 −0.433 0.252 −0.012 0.011

0.258 −0.362 0.238 −0.026 0.026

0.000 −0.024 0.005 0.005 −0.005

3.3 Parametrisation—Misaligned Structures

47

Table 3.3 Comparison of gaps (eV) and effective masses (m 0 , the free electron mass) between crystalline γ-InSe and arbitrarily stacked layers (R-InSe) γ-InSe R-InSe E g (eV)

1.27

Mc (m 0 )

0.092

0.133

Mv (m e )

−0.080

−0.107

z

z

1.47

(a)

(b)

Fig. 3.5 a Top view structure of two randomly oriented InSe layers, b vertical (i.e. –) energy gap as a function of the number of layers N in laminates comprising randomly misaligned monolayers of InSe compared to the energy gaps in crystalline γ-InSe. The misaligned stacking leads to an increase in the gap compared with the ordered crystalline stacking of γ-InSe

3 Hybrid k · p Tight-Binding Theory

48

For comparison, we also carry out DFT calculations on bulk InSe with aligned layers, but with a relative shift corresponding to a root of f (u), Eq. (3.21). For relative shifts where f (u) = 0 the hops should be equivalent to what an arbitrary rotation would yield. Hence band structures obtained for a bulk crystal where the layers are shifted by a distance corresponding to one of the roots of f (u) should give the same results for the band gap as the model relying on the shift-independent hopping parameters obtained from the bulk γ and AA stackings. f (u) has roots for u satisfying ⎡  ⎤ 21

√ 2 2πu x + 2 − 3 2 cos a 3 a ⎦ arctan ⎣ (3.28) uy = ± x π −1 2 cos 2πu a for |u x | ≤ a/3. For u x = 0, u y  0.3297a = 1.303 Å. Figure 3.4f shows a comparison of the bulk k z dispersions between the DFT calculations and the parametrization in Table 3.2, obtained from fits to the γ- and AA stackings. There is good agreement for the gap and bands between the model and DFT calculations aside from (i) the expected change in dispersion near the A point due to the gap correction and (ii) the close vicinity of the avoided crossing between the valence band and the monolayer bands v1 and v2 , where the breaking of C3 symmetry by the shift means that this crossing is no longer protected (and the double degeneracy of bands v1 and v2 is lifted). The coupling is, however, weak, so away from the immediate region of the crossing these bands can still be neglected.

3.3.3 Example: Misalignment Between Two Crystals As an example of a way in which the model may be applied, we consider a structure constructed from two few-layer InSe crystals, with n 1 and n 2 layers respectively, placed one on top of the other at an arbitrary angle. The Hamiltonian then takes interlayer hops within each crystalline slab as those for γ-InSe with az = 8.32 Å, while at the misaligned interface between the slabs we use interlayer hops for the misaligned case with az = 8.72 Å. The system as a whole then has a Hamiltonian given by H=

N    (E c + 2δcγ )cn† cn + (E v + 2γv )vn† vn n=1

 

† ˜ c − γc ) cn† cn 1 + c(n − γc c1† c1 + c†N c N + ( c (n +1) 1 1 1 +1)

 

† † † γ γ † − v v1 v1 + v N v N + (˜v − v ) vn 1 vn 1 + v(n 1 +1) v(n 1 +1)  n 1 −1

  † † † † γ + tcγ cn+1 vn+1 cn + tvγ vn+1 vn + tcv cn − cn+1 vn n=1

3.3 Parametrisation—Misaligned Structures

49

Fig. 3.6 Vertical gaps for a 10-layer structure comprising an n 1 -layer crystal placed at a large arbitrary angle on top of a (10 − n 1 )-layer crystal

+

N

   † † † † γ tcγ cn+1 vn+1 cn + tvγ vn+1 vn + tcv cn − cn+1 vn n=n 1 +1

   † † † † ˜ ˜ v + H.c. , (3.29) + t˜c c(n c + t v v + t c − c v n v n cv n n 1 (n 1 +1) 1 (n 1 +1) 1 (n 1 +1) 1 1 +1) where N = n 1 + n 2 . Figure 3.6 shows the results of such an exercise for N = 10 for varying n 1 . The effect of weaker interlayer hoppings for the misaligned case is greatest when the interface is placed in the middle of the structure, as this is where the conduction and valence band-edge wavefunctions are most localised.

3.4 Multilayer Subbands and Intersubband Optical Transitions In Chap. 2 we showed how the energy of the principal interband optical transition varied significantly with the crystal thickness, with photon energies covering a substantial portion of the visible spectrum, from around 2.8 eV in the monolayer to 1.3 eV in the bulk case. Now we consider the subbands of the conduction band— that is, the bands into which the monolayer conduction band splits on going to the few-layer case (or which can be understood as the discretisation of the bulk k z conduction band dispersion on confinement in a few-layer crystal). As was historically found in semiconductor quantum well heterostructures [21–24], the splitting between subbands can be typically a lot smaller than the band gap, and therefore optical transitions between subbands can be very useful for accessing wavelengths longer than those typically associated with interband transitions, including in the technologically important far-infrared and THz ranges.

3 Hybrid k · p Tight-Binding Theory

50

For the remainder of this chapter, and in Chap. 4, we return to considering the crystalline form of InSe in the γ polytype. We therefore drop the γ superscript from notation for the interlayer hops and onsite energy shifts in the HkpTB model. To describe the conduction band subbands of few-layer γ-InSe we extend the Hamiltonian of Eq. (3.15), expanding for finite k around  Hˆ =

 N   2 k 2 † anc c (2 − δn,1 − δn,N ) + anc 2m c n  † anv + (E v + v (2 − δn,1 − δn,N ))anv

+

N −1  

(3.30)

† † (tc + tc k 2 )a(n+1)c anc + tv a(n+1)v anv

n

   † †   2 + (tcv + tcv k ) a(n+1)v anc − a(n+1)c anv + H.c. . Here we use the -point parameters fitted above for γ-InSe with az = 8.32 Å, listed again in Table 3.4 for completeness. For the expansion, we use the isotropy and non  + tc(cv) k2. degeneracy of the bands near  to expand the hops as tc(cv) (k) = tc(cv) Since we are interested in the conduction band subbands and transitions between them, we note that any dispersion in the valence band will be negligible on the scale of the gap, and approximate the valence band as flat and the valence-valence hop as independent of k. Meanwhile, the conduction band onsite energy itself is expanded using an isotropic effective mass description using the monolayer conduction band mass, m c .  , are fitted to reproduce the DFT subband The additional parameters, tc and tcv effective masses for 2- and 3-layer InSe, with the fitting carried out using the DFT monolayer gap, i.e. with E v set to −1.80 eV, before it is then returned to the corrected value, E v = −2.80 eV for calculation of results, in a similar manner to the means used to obtain the -point hops described above. On diagonalisation for N layers, the model generates N subbands. Figure 3.7 shows the subband dispersion for N = 1−4 layers, and we note that the lowest subbands have lighter effective masses (a result of the decreased splitting for higher k). The -point splitting between the two lowest Table 3.4 HkpTB theory parameters in Eq. (3.15), and -point transition energies between two lowest subbands Ev −2.80 eV tc 0.34 eV N E 2|N − E 1|N (meV) mc 0.18 m e tv −0.41 eV 2 680  c 0.03 eV tcv 0.29 eV 3 490 v −0.03 eV tc −5.91 eVÅ2 4 360  tcv −5.36 eVÅ2 5 280

3.4 Multilayer Subbands and Intersubband Optical Transitions

51

Fig. 3.7 Conduction band subbands near  for N = 1−4-layer InSe, from Eq. (3.15). Intersubband transitions allowed by the model are marked by arrows, together with their oscillator strength (intersubband out-of-plane electric dipole moment)—transitions between non-adjacent subbands are expected to be much weaker

energy subbands is shown in Table 3.4 up to 5 layers, and we plot the energy for transitions from the lowest subband to higher subbands in Fig. 3.8 up to N = 15 layers. From the lowest subband to the second lowest subband the transition energy covers a range from ∼700 meV in the bilayer down to ∼50 meV by N = 15 layers. The transitions to higher subbands are expected to have significantly weaker oscillator strengths, which we discuss further in Sect. 3.4.2. It is worth noting that the underestimation of the gap by DFT has consequences even for the 2D subband effective masses. The differing effective masses of successive subbands gives rise to thermal broadening of the intersubband transitions, to a m a|N where m a(b)|N is the effective mass of subband measure proportional to 1 − m b|N m 1|6 a(b) in N-layer InSe. For 6-layer InSe we find that with correction 1 − = 0.11, m 2|6 m 1|6 while without correction 1 − = 0.15. Meanwhile, the underestimated effective m 2|6

3 Hybrid k · p Tight-Binding Theory

52 Fig. 3.8 Interbsubband transition energies for excitation from the lowest conduction band subband to higher subbands in N -layer InSe. The oscillator strength for transitions to subbands other than the second-lowest energy subband are expected to be weak. The energies plotted as a solid line are derived from a quantisation of an effective-mass bulk dispersion (Eq. (3.34))

mass in the bulk leads to overestimation in the discretised splitting between the lowest subbands, with correction of the interband gap reducing the splitting between the two lowest subbands in 6-layer InSe from 250 to 220 meV.

3.4.1 Quantisation of Bulk InSe Bands in Ultrathin Films As discussed above, the conduction band subbands in InSe can be understood both in a ‘bottom-up’ approach, as originating from the splitting of a parent monolayer band, or in a ‘top-down’ picture, as the discretised counterparts of a bulk k z spectrum. In the latter method expansion of the bulk dispersion in vicinity of the band edge gives  E c (p, pz ) =



2 2Mc⊥z

+

η pz2 az2

k2 +

2 pz2

z

2Mc

(3.31)

where k = |k| = |(k x , k y )| and pz = k z − π/az . The x y-plane and z-axis bulk con z duction band effective masses, Mc⊥z and Mc , are 1 Mc⊥z

=

1 4t  − 2c , mc 

1

z

Mc

=

 2  2az2  4tcv t , + 2 c Eg

(3.32)

respectively, where E g = 2c − (E v + 2v ) − 2(tc − tv ) is the bulk band gap.

z These expressions give Mc⊥z = 0.12 m e and Mc = 0.09 m e , respectively, close to

z the experimentally known values of Mc⊥z = 0.14 m e and Mc = 0.08 m e [6]. The parameter η, given by η = tc −

2 22 tcv 8t  t  2 + cv cv  −0.63 , ⊥z E 2 Eg Mc 2Mc⊥z g

(3.33)

3.4 Multilayer Subbands and Intersubband Optical Transitions

53

accounts for variation in the in-plane effective mass as one expands in pz . Following the analysis leading to Eq. (2.11) the 2D -point energy of subband n in N-layer InSe (denoted n|N ) can then be expressed as E n|N (n  N ) ≈

2 π 2

n2 .

z 2Mc az2 (N + 2ν)2

(3.34)

Using subband energies calculated from the HkpTB model we find that ν = 1.42, as fitted to the intersubband transition energies for the transition from subband 1 to 2, E 2|N − E 1|N . The energies obtained from Eq. (3.34) are plotted in Fig. 3.8 (solid line) alongside those obtained from the few-layer HkpTB model (Eq. (3.15)). In a similar manner we can show how the higher-energy subbands have heavier effective masses,   6.2n 2 1 1 , (3.35) ≈ ⊥z 1 − m n|N (N + 2ν)2 Mc which produces the difference between the effective masses in the two lowest subbands shown by the red line in Fig. 3.9.

Fig. 3.9 Thermally broadened lineshapes (normalised to -point bilayer intersubband absorption strength) for a very light doping at room temperature (solid lines) and a high carrier density at low temperature (dashed lines). Inset—difference between effective masses of subbands and resulting thermal linewidth. Red line is the quantisation of the bulk band-edge expansion described in Eq. (3.41)

3 Hybrid k · p Tight-Binding Theory

54

3.4.2 Intersubband Optical Transitions For intersubband optical transitions to be active, the system must be n-doped, in which case the hole population will be negligible. Therefore, excitonic effects may be neglected and the energy of an intersubband optical transition taken as that of the subband splitting. All conduction band subbands are singly degenerate, and as we neglect SOC the optical transitions will have an out-of-plane dipole character. The oscillator strength for coupling of the transitions to out-of-plane polarised light will determined by the intersubband electric dipole matrix element, which is calculated as N  † † 1|N |z(n)(anc anc + anv anv )|b|N , (3.36) dz (1|N , b|N ) = e n

where z(n) = az (n − (N + 1)/2) is the z co-ordinate of the mean plane of each layer, w.r.t. the mean plane of the whole crystal. The assumptions made leading to Eq. (3.14) give the model system σh reflection symmetry, which does not exist in the true crystal, and this gives consecutive subbands alternating symmetry under σh reflection. Consequently dz (1|N , b|N ) = 0 for odd b. To test the validity of the assumption in the context of the energetics and optics of the subbands, we use values |dz (1|3,3|3)|2 −4 , so of dz from a DFT calculation for trilayer InSe—these give |d 2 ∼ 10 z (1|3,2|3)| 2 the transitions forbidden by the HkpTB model can continue to be neglected. In Fig. 3.7, the non-zero intersubband dipole matrix elements are labelled alongside their respective transitions, and we note that the matrix element for transitions between adjacent subbands is much larger than that for transitions between more distant subbands. The oscillator strengths, effective masses, and band energies described above can then be used to approximate a lineshape for intersubband absorption from the lowest (n = 1) to the next-lowest (n = 2) subband as g(ω) ∝ |dz (1|N , 2|N )|2 ω × DoS × FT ,

(3.37)

The joint density of states of the intersubband transition is

−1  × (E 2|N − E 1|N − ω), DoS(ω) = π2 1/m 1|N − 1/m 2|N

(3.38)

while the Fermi-Dirac occupancy of the lowest subband is 



1 FT = exp kB T

2 Especially



E 2|N − E 1|N − ω − EF m 1|N 1 − m 2|N



−1 +1

,

(3.39)

so since any symmetry breaking will come from the interband hops, as we explore in Chap. 4—therefore, when we correct the gap the effect will be even smaller.

3.4 Multilayer Subbands and Intersubband Optical Transitions

where

    π2 n e −1 E F = k B T ln exp m 1|N k B T

55

(3.40)

is the Fermi energy relative to the band edge for n-type carrier density n e . It is assumed that E 2|N − E 1|N − E F  k B T (that is, the occupancy in the second subband is negligible). The thermally broadened linewidth is then estimated as  ωFWHM ≈ max

  m 1|N k B T ln 2, E F . 1− m 2|N

(3.41)

The thermally broadened lineshapes determined using Eq. (3.37) are plotted in Fig. 3.9 for a carrier density n e = 5 × 1012 cm−2 at low temperature, assuming a high dielectric constant (see Sect. 3.4.3 below), and for a very small carrier density at room temperature. The linewidths found using Eq. (3.41) for a low carrier density are shown in an inset.

3.4.3 Effects of Interlayer Screening in Gated n-Doped InSe For the intersubband optical absorption described above to be possible there must be electrons present in the lowest conduction band subband—the system must be n-doped. To allow for easy tuning of the carrier density, such doping is often achieved by means of electrostatic gating [5]. In quantum wells, this induces the accumulation of electrons near the surface, with the quantum well potential and subbands changed by the confined electrons [25]. To understand the effects of doping on the subband splitting (and hence transition energies) in few-layer InSe we self-consistently analyse the effect of the electric field of the gate and the screening by the induced electrons on the bands produced by the Hamiltonian.3 The excess charges on layer n of the N -layer film are calculated as n e (n) =

 1  |c jn (α, Hˆ  , k)|2 FT j ( Hˆ  , k)kdk π j,α=c/v

(3.42)

where c jn (α, Hˆ  , k) is coefficient of the contribution to the wavefuction of subband j from monolayer band α on layer n at momentum k, and FT j ( Hˆ  , k) is the FermiDirac occupation factor of the subband. Hˆ  is the Hamiltonian of the gated system, with additional on-site potentials added to Eq. (3.30), Hˆ  = Hˆ +



† † Un (anc anc + anv anv )

n

3A

significant contribution to the analysis in this section was made by A. Ceferino.

(3.43)

3 Hybrid k · p Tight-Binding Theory

56

where the potential on layer n, Un , is determined for a single gate by the induced electron density as n  Un>1 = U1 + eaz E n  −1,n  , (3.44) n  =2

with the field between two layers given by E n−1,n =

N e  n e (n  ) εε0 n  =n

(3.45)

electron density on layer n. The total carrier density where n e (n) is the induced  is then simply n e = n n e (n), with the total density required to screen the charge density of the gate, giving the electric field at the film surface due to the gate as E ext =

e ne . εε0

(3.46)

For a given U1 the potential distribution {Un } is then determined self-consistently, with a final carrier density n e (U1 ) obtained at convergence. The required information, that is, a potential distribution for a desired total carrier density, is obtained by varying the choice of U1 . The results of the self-consistent calculation are shown in Fig. 3.10 for the films with 2–6 layers, over the density range where only states in the lowest subband are filled, for choices of dielectric constant ε = 1, 5, 10. For the case where we set the

Fig. 3.10 Dependence of intersubband transition on gate-induced charge density (n e ) for 2–6 layer InSe, with dielectric constant ε = 1, 5, 10

3.4 Multilayer Subbands and Intersubband Optical Transitions

57

dielectric constant ε = 1, we observe a slight decrease in the subband spacing at small carrier densities—here the wavefunction of the lowest subband still has its greatest contribution in the centre of the film, so adding electrons to it increases its energy relative to the other subbands which have smaller contributions near the centre of the film. For higher carrier densities we find a steady increase in the intersubband transition energy, as an accumulation layer develops at the surface of the crystal exposed to the electric field to the gate voltage, giving a lower-energy band detached from the rest of the subband spectrum, which is screened from the electric field by the accumulation layer. The case ε = 1 has value as a limiting case, but given its effect is to simulate the crystal as a series of infinitely thin layers, a dielectric constant ε > 1 is likely to better represent the true physical picture. We must therefore choose an appropriate dielectric constant for the system. Values reported in the literature for the out-ofplane dielectric constant of bulk InSe range from ε ∼ 5.4 [26] at room temperature to ε = 9.9 ± 1.3 [27] at low temperature. The temperature dependence, the fact that the experimental means used to determine the dielectric constant themselves depend on the degree of doping, and the expectation that the dielectric constant will decrease for the thinnest films [28] means that we cannot know a precise dielectric constant, so we plot for a range of dielectric constants in Fig. 3.10. We note that changes in the intersubband transition energy with doping are much smaller for ε > 1. We must note here that, for the moment, we have not considered the effect of a constant intralayer interband mixing due to the electric dipole matrix element E z dz of Eq. (2.12), since for the carrier densities considered here its effect on the subband energies would be negligibly small.4 We will, however, include it in Chap. 4, where it plays an important role in the symmetry-breaking effects of the applied potential.

3.4.4 Dual Gating A technique in gating 2D systems is to apply two gates to the device—a back gate via the substrate and a top gate on top of the device. If the top gate is close to the device, the application of a small voltage to this gate will give rise to a large carrier density in the system, which can then be fine-tuned by the application of a larger magnitude voltage to the back gate on the substrate. Doping by such means will clearly give rise to a different potential profile within the crystal, and hence a changed dependence of the intersubband transition energy on n-doping. To examine the effect of dual gating on the system, we begin with the setup in Ref. [5]—that is, the fixed top gate voltage (8V) gives rise to a carrier density of 4 × 1012 cm−2 , with a positive back gate voltage giving higher carrier densities, and a negative back gate required for lower carrier densities. We then amend Eqs. (3.43)–(3.45) to read 12 −2 in vacuum would have an energy ∼75 meV, z dz due to a plane of charge with n e = 5 × 10 cm which can be neglected when considering subband energetics, given the scale of the gap between the monolayer states of 2.8 eV.

4E

3 Hybrid k · p Tight-Binding Theory

58

Fig. 3.11 Intersubband transition energies as a function of total dual-gate-induced carrier density (n e ) for 2–6 layer InSe, with dielectric constant ε = 5, 10 (the case ε = 1 is not shown here since the poorly-screened and strong field gives splittings >1 eV for all N as n e → 0). The top gate voltage is fixed to give a carrier density n e = 4 × 1012 cm−2 , while the back gate voltage is varied to tune the carrier density

Un>1 = U1 + eaz

n  n  =2

(E n  −1,n  − E tg ), E n−1,n =

N e  en tg n e (n  ), E tg = , εε0 n  =n εε0 (3.47)

where n tg = 4 × 1012 cm−2 is the electron density induced by the top gate. Selfconsistent solution of the model then yields the results shown in Fig. 3.11. In contrast to the single-gated case, the intersubband transition energy increases for smaller n e , since the potential profile due to the gates becomes more asymmetric as the carrier density is decreased (the field due to the gates would be symmetric at a carrier density of n e = 8 × 1012 cm−2 ). For very light doping there is a strong electric field in the film which is poorly screened.

3.4.5 Experimental Realisation of Intersubband Transitions Observation of the intersubband transitions in few-layer InSe films presents an experimental challenge since the transitions will be active with an out-of-plane polarisation. Emitted photons will therefore travel in the plane of the crystal, making them difficult to observe. A recent experiment on WSe2 and MoS2 [29] overcomes this limitation by fabricating a device consisting of terraces of varying numbers of layers. The intersubband transitions are then observed in the near-field by excitation from a concentrated infrared field underneath an atomic force microscope (AFM) tip, which contains out-of-plane polarised components. Another proposal envisages

3.4 Multilayer Subbands and Intersubband Optical Transitions

59

an hBN-encapsulated few-layer transition-metal dichalcogenide crystal between two ABC-stacked few-layer graphene electrodes [30]. ABC-stacked few-layer graphene has a Van Hove singularity in its density of states, and the singularity for each electrode can be aligned with a subband of the TMD crystal. One can then favourably inject electrons into a higher subband with one electrode and collect them from a lower subband with the other once an intersubband transition has taken place, thus maintaining a population inversion between the two subbands.

References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

Wallace PR (1947) Phys Rev 71:622 Mak KF, Lee C, Hone J, Shan J, Heinz TF (2010) Phys Rev Lett 105:136805 Liu H, Neal AT, Zhu Z, Luo Z, Xu X, Tománek D, Ye PD (2014) ACS Nano 8:4033 Rigoult J, Rimsky A, Kuhn A (1980) Acta Crystallogr B 36:916 Bandurin DA, Tyurnina AV, Geliang LY, Mishchenko A, Zólyomi V, Morozov SV, Kumar RK, Gorbachev RV, Kudrynskyi ZR, Pezzini S, Kovalyuk ZD, Zeilter U, Novoselov KS, Patanè A, Eaves L, Grigorieva II, Fal’ko VI, Geim AK, Cao Y (2017) Nat Nanotechnol 12:223 Kress-Rogers E, Nicholas R, Portal J, Chevy A (1982) Solid State Commun 44:379 Heyd J, Scuseria GE, Ernzerhof M (2003) J Chem Phys 118:8207 Heyd J, Scuseria GE, Ernzerhof M (2006) J Chem Phys 124:219906 Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, Grigorieva IV, Firsov AA (2004) Science 306:666 Jacobberger RM, Machhi R, Wroblewski J, Taylor B, Gillian-Daniel AL, Arnold MS (2015) J Chem Edu 92:1903 Na SR, Suk JW, Tao L, Akinwande D, Ruoff RS, Huang R, Liechti KM (2015) ACS Nano 9:1325 Haar S, Bruna M, Lian JX, Tomarchio F, Olivier Y, Mazzaro R, Morandi V, Moran J, Ferrari AC, Beljonne D, Ciesielski A, Samor P (2016) J Phys Chem Lett 7:2714 Chen H, Müller MB, Gilmore KJ, Wallace GG, Li D (2008) Adv Mater 20:3557 Liu Y-T, Zhu X-D, Duan Z-Q, Xie X-M (2013) Chem Commun 49:10305 Abergel DSL, Wallbank JR, Chen X, Mucha-Kruczy´nski M, Fal’ko VI (2013) New J Phys 15:123009 Wallbank JR, Patel AA, Mucha-Kruczy´nski M, Geim AK, Fal’ko VI (2013) Phys Rev B 87:245408 Sevik C, Wallbank JR, Gülseren O, Peeters FM, Çakır D (2017) 2D Materials 4:035025 Danovich M, Ruiz-Tijerina DA, Hunt RJ, Szyniszewski M, Drummond ND, Fal’ko VI (2018) Phys Rev B 97:195452 Demirci S, Avazlı N, Durgun E, Cahangirov S (2017) Phys Rev B 95:115409 Klimeš J, Bowler DR, Michaelides A (2009) J Phys Condens Matter 22:022201 West LC, Eglash SJ (1985) Appl Phys Lett 46:1156 Levine BF, Choi KK, Bethea CG, Walker J, Malik RJ (1987) Appl Phys Lett 50:1092 Faist J, Capasso F, Sivco DL, Sirtori C, Hutchinson AL, Cho AY (1994) Science 264:553 Köhler R, Tredicucci A, Beltram F, Beere HE, Linfild EH, Davies AG, Ritchie DA, Iotti RC, Rossi F (2002) Nature 417:156 Ando T, Fowler AB, Stern F (1982) Rev Mod Phys 54:437 Allakhverdiev KR, Babaev SS, Salaev EY, Tagyev MM (1979) Phys Status Solidi (b) 96:177 Kuroda N, Nishina Y (1980) Solid State Commun 34:481 Wu D, Pak AJ, Liu Y, Zhou Y, Wu X, Zhu Y, Lin M, Han Y, Ren Y, Peng H, Tsai Y-H, Hwang GS, Lai K (2015) Nano Lett 15:8136

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29. Schmidt P, Vialla F, Latini S, Massicotte M, Tielrooij K-J, Mastel S, Navickaite G, Danovich M, Ruiz-Tijerina DA, Yelgel C, Fal’ko V, Thygesen KS, Hillenbrand R, Koppens FHL (2018) Nat Nanotechnol 13:1035 30. Ruiz-Tijerina DA, Danovich M, Yelgel C, Zólyomi V, Fal’ko VI (2018) Phys Rev B 98:35411

Chapter 4

Spin-Orbit Coupling Effects in InSe Films

In the monolayer transition metal dichalcogenides (TMDCs), the band edges are found in the corner of the Brillouin zone at the K-point, and exhibit strong spin-orbit splitting of the band-edge states. In contrast, in InSe the band edges appear close to the centre of the Brillouin zone near , and hence the effects of spin-orbit splitting on the energetics of the bands are generally negligible [1]. Even towards the Brillouin zone corners the spin-orbit splitting is still generally a lot smaller than that found in the TMDCs. Spin-orbit coupling (SOC) does however retain a physically significant role in InSe. Here we explore two important consequences of SOC near the band edges of ultrathin InSe films. Firstly, we show how conduction band electrons in few-layer InSe experience the Rashba effect, which can give rise to anomalous magnetotransport effects arising from weak antilocalisation. SOC is added to the self-consistent HkpTB model perturbatively, and the variation of Rashba SOC with the strength and profile of doping can therefore be investigated. We find that the strength of the observed weak antilocalisation depends on the interplay between the symmetry breaking provided by the crystal stacking, and that provided by the asymmetric potential of the electrostatic gates used to dope the system. These effects can add either constructively or destructively. Secondly, in Chap. 2 we showed how the non-degeneracy of the -point valence and conduction bands of InSe, combined with their opposite symmetry under σh reflection in the monolayer, gave an out-of-plane dipole character to the principal interband optical transition. In contrast, the deeper occupied band v1 is twice degenerate and of the same parity under σh as the conduction band, and thus the ‘B-line’ transition was found to couple strongly to in-plane polarised light. Such selection rules are important in the interpretation of experimental observations of these transitions, as a photon which is absorbed/emitted with out-of-plane polarisation must travel in the plane of the 2D crystal, presenting challenges for exciting and observing such transitions. However, as we show here, spin-orbit coupling (SOC) lifts the -point orbital degeneracy of bands v1 and v2 and, crucially, hybridises v1 with the band-edge valence states, satisfying the angular momentum and symmetry © Springer Nature Switzerland AG 2019 S. J. Magorrian, Theory of Electronic and Optical Properties of Atomically Thin Films of Indium Selenide, Springer Theses, https://doi.org/10.1007/978-3-030-25715-6_4

61

62

4 Spin-Orbit Coupling Effects in InSe Films

requirements to allow the principal interband optical transition to couple to in-plane polarised light. The relaxation of the selection rule for the polarisation of the principal interband optical transition is strongest in the monolayer, and becomes very weak in the bulk limit.

4.1 Modelling of Spin-Orbit Coupling Atomic spin-orbit coupling1 takes the form λL · s, where L and s are the orbital and spin angular momentum vectors of an electron, respectively. It acts between components of atomic orbitals with L = 0—in our basis it therefore acts within the p orbitals. We can understand it either to act between a basis of px , p y , and pz orbitals, or diagonally within the orbital angular-momentum projection L z = ±1 states together with spin-flip hybridisation with the L z = 0 pz orbitals as L · s = L z sz + L + s − + L − s + , where L(s)± are orbital (spin) angular momentum raising/lowering operators. We use the former description in adding SOC to the tight-binding model, while it is more convenient to use the latter for the HkpTB picture of the system.

4.1.1 Spin-Orbit Coupling in the Tight-Binding Model To include SOC in the tight-binding model we double the basis of the tight-binding Hamiltonian (H0 , Eq. (2.7a)) to include both spin states (sz = ± 21 ) in each atomic orbital, and add atomic SOC between the p orbitals on the same site to the tightbinding Hamiltonian, H0 , as H = H0 ⊗ 1s     μ† μ μ† −μ μ† −μ λIn −2iμm (n) f j px m (n) f j p y + 2μm (n) f j px m (n) f j pz − im (n) f j p y m (n) f j pz + j,n, f,μ

+ λSe



μ† μ −2iμx(n) f j px x(n) f j p y

μ† −μ + 2μx(n) f j px x(n) f j pz

μ† −μ − i x(n) f j p y x(n) f j pz



(4.1)

+ H.c..

Here, the sum runs over unit cell j, layer n, sublayer f and spin projection μ(†) μ = s · eˆ z = ± 21 . The operator m(x)(n) f i pα annihilates (creates) an electron in orbital pα with spin projection μ on an In(Se) atom in unit cell i, layer n, sublayer f of the crystal. We set the atomic spin-orbit coupling constants, λIn and λSe , as λIn = 0.17 eV, λSe = 0.15 eV based on the splitting of In px,y -dominated and Se px,y -dominated DFT bands on taking SOC into consideration [1]. The bands resulting from the diagonalisation of Eq. (4.1) are shown in Fig. 4.1 for N = 1 and N = 5 layers. Spin-splitting neglect inter-atomic SOC, i.e. finite values of, say, In |L · s| Se, which would appear due to finite overlap within the chosen basis.

1 We

4.1 Modelling of Spin-Orbit Coupling

63

Fig. 4.1 Tight-binding model bands for N = 1 and N = 5-layer InSe with SOC taken into account (Eq. (4.1))

within the bands is generally small, especially near the band edges (as a result of Kramers degeneracy), however there are several features worth noting. Firstly, repulsion between v1 and v is greatest at , so the centre of the offset valence band maximum is pushed up more than the rim, reducing its depth and making the gap more direct. Secondly, in the monolayer case there is no spin-orbit splitting along -M due to the σh and σv symmetry of the monolayer crystal. Finally, the effect of the increased distance of the valence band edge from the px,y -dominated bands for greater numbers of layers is clear in the much-reduced spin-orbit splitting of the valence band for N = 5. While the model generally reproduces the spin-orbit effects in the valence bands well, the spin-orbit splitting in the conduction band near the K point in the monolayer is greater than that given in DFT [1], which is likely due to the combination of p-orbitals in the tight-binding orbital decomposition not being correct. The energies of these bands proved the most challenging to fit in the parametrisation of the model in Chap. 2, and the conduction bands at the K-point can be traced to -point bands at a much higher energy which begin to be less well described as a superposition of atomic orbitals, an approximation on which the tightbinding model heavily relies. Nevertheless, it may be possible, by including SOC at the parametrisation stage, to find a tight-binding parameter set which would more closely reproduce the conduction-band K-point splitting given by DFT.

4.1.2 Hybrid k · p Tight-Binding Theory In considering the effect of SOC on the band-edge states in the vicinity of the -point in monolayer InSe it is instructive to also see its effect in the band-basis of the k · p model considered in Chap. 2. Since SOC will change the band dispersions one must either change the k-dependent basis of each of the bands in the single-band k · p model of Chap. 2, or re-write the model as a multiband model with the band basis fixed as that at , including off-diagonal terms for finite k [2].

64

4 Spin-Orbit Coupling Effects in InSe Films

To describe the properties of optical excitations near the band edge it is, however, sufficient to consider the model’s behaviour at  on inclusion of SOC. We therefore take the k · p Hamiltonian of Chap. 2 (Eq. 2.12) for k = 0, and double the basis to consider both spin-up and spin-down states (μ ≡ s · eˆ z = ± 21 ) of each band in the model, adding atomic SOC, λL · s, ⎛

E c 1s ⎜ E d1 z z s ⎜ Hˆ = ⎜ eβ1 ⎝ cm e 1s ⊗ AT T λc,v2 sˆ

⎞ eβ1 E z dz 1s 1 ⊗A λc,v2 sˆ cm e s eβ2 ⎟ E v 1s λv,v1 sˆ 1 ⊗A ⎟ cm e s ⎟. T λv,v1 sˆ E v1 1s + λv1 sz ⊗ σz 0 ⎠ eβ2 T 1 ⊗A 0 E v2 1s + λv2 sz ⊗ σz cm e s (4.2)

eβ1(2) couple the v1 → c interband transition (B-line) cm e and the transition between v and v2 , respectively, to in-plane polarised light with vector potential written as A = (A+ , A− ), in a σ ± polarisation basis. β1(2) = |c(v) |P| v1 (v2 )| (Eq. (2.15)) is the magnitude of the interband momentum matrix element between c(v) and v1 (v2 ). β2 is, of course, irrelevant in the absence of SOC, since it describes an optical transition between two occupied states. It is included here for completeness as it would contribute to the effect of SOC on the principal interband transition, if the mixture between c and v2 by SOC were significant. To add intra- and inter-band SOC, we write atomic SOC, λL · s, as The off diagonal factors

λL · s = λ(L z sz + L + s − + L − s + )

(4.3)

where L(s)± = L(s)x ± i L(s) y are orbital (spin) angular momentum raising/ lowering operators, and λ is a SOC strength parameter. In the chosen basis L z sz is diagonal, lifting the degeneracy between the L z = ±1 components of v1 and v2 as ±↑

±↑

∓↓

∓↓

v1(2) |λL · s| v1(2)  = v1(2) |λL · s| v1(2)  = ±λv1(2) ,

(4.4)

where the arrows indicate the spin up/down states of the bands. The parameters λv1 and λv2 are the strengths of atomic SOC in a basis of the monolayer -point bands. The ‘spin-flip’ terms, L ± s ∓ , hybridise pairs of states with orbital angular momentum difference L z = ±1 and opposing spin projections. This gives four +↓(−↑) +↓(−↑) and v ↑(↓) /v1 , candidate pairs of bands which we reduce to two, c↑(↓) /v2 due to the requirement that the bands involved have opposing symmetry under σh , as the operator L ± is σh antisymmetric. The spin-flips have matrix elements ±↓

∓↑

c(v)↑ |λL · s| v2(1)  = c(v)↓ |λL · s| v2(1)  = ±λc(v),v2(1) ,

(4.5)

where λv,v1 and λc,v2 are SOC strengths for the interband hybridisation. The bandbasis SOC parameters are determined by the atomic SOC strengths for the valence

4.1 Modelling of Spin-Orbit Coupling

65

p orbitals of In and Se (largely Se, see orbital decompositions in Chap. 2), and the orbital decompositions of the bands. T + − Eq. (4.2), 1s is an identity matrix in spin space, while sˆ ≡ (s , s ) and sˆ ≡

In + s . The SOC-strength parameters are found by comparing the band energies s− given by Eq. (4.2) with scissor-corrected DFT calculations with SOC taken into account (local spin density approximation).2 The orbital decomposition of the DFT conduction band wavefunction shows negligible changes on consideration of SOC,3 so the effects of λc,v2 (and hence of β2 ) may be neglected. The conduction band energy can then be used as a reference, which is set at 0 eV. The other parameters are all significant, and we find λv1 = λv2 = 0.30 eV, which split v1 and v2 into pairs with total angular momentum projections Jz = ± 23 and Jz = ± 21 , while λv,v1 = 0.36 eV hybridises the pz orbitals of v with the px,y orbitals of the lower branch of v1 (Jz = ± 21 ), slightly reducing the band gap. The -point bands before and after taking SOC into account are shown in Fig. 4.2, and we find good agreement between the DFT energies and those found using Eq. (4.2). The v ↔ v1 hybridisation is sufficiently strong that perturbation theory overestimates the weight, δCv (v1 ), of v1 orbitals ( px,y ) mixed into v, finding 2  1   λ 2 v,v1  ∼ 0.2,  |δCv (v1 )| =  E v − (E v1 − λv1 /2)  2

(4.6)

compared with |δCv (v1 )|2 = 0.13 found from numerical diagonalisation of the Hamiltonian in Eq. (4.2). As demonstrated in Chap. 3, near the  point bands c and v hybridise strongly between layers, while v1 and v2 exhibit only weak subband splitting. The combined effect is to move the valence band edge further away from v1 , reducing the effect of SOC on the band edge states. We can obtain a quantitative estimate of the magnitude of this decrease by employing a model combining the monolayer Hamiltonian of Eq. (4.2) with the interlayer hops between c and v described in Chap. 3, Hˆ (N ) =

N   α,α μ,μ

n

+

† anα μ + Hˆ αμ,α μ anαμ

N −1   n

α,μ

  † † α anαμ anαμ + a(n+1)αμ a(n+1)αμ

N −1    † † tc a(n+1)cμ ancμ + tv a(n+1)vμ anvμ n

μ

   † † + tcv a(n+1)vμ ancμ − a(n+1)cμ anvμ + H.c. .

2 The

(4.7)

plane-wave basis cutoff energy was again 600 eV, while the Brillouin zone was sampled by a 24 × 24 × 1 k-point grid. 3 One would expect the s, p -dominated wavefunction to pick up some p z x,y contributions.

66

4 Spin-Orbit Coupling Effects in InSe Films

Fig. 4.2 Schematic of low-energy -point bands for monolayer InSe, with (righthand side) and without (left-hand side) spin-orbit coupling (SOC) taken into account. Allowed optical transitions are marked, with their polarisation character (in-plane (σ ± ) or out-of-plane) indicated. On the right-hand side, superscripts indicate the z-component of total angular momentum (Jz ) of the bands. The conduction band edge is used as a reference energy, and is set at 0 eV, with the band energies found using Eq. (4.2). Band energies in parentheses are from a scissor-corrected DFT calculation with SOC taken into account

(†) Here, the operator anαμ annihilates (creates) an electron in band α, spin state μ = ± 21 , in layer n of the N -layer crystal. The sum over α, α runs twice over the bands included in the monolayer Hamiltonian Hˆ in Eq. (4.2), and Hˆ αμ,βμ are the matrix elements of Hˆ between band α with spin projection μ and band α with spin projection μ . The hopping parameters tc , tv , and tcv , and the onsite energy shifts, c and v , are those determined in Chap. 3 (Table 3.2). Diagonalisation of Eq. (4.7) allows us to track the dependence of the effect of SOC on the band-edge states on the number of layers N , which we characterise using the parameter |δCvN (v1 )|2 = nN |Cv (v1,n )|2 ,

4.1 Modelling of Spin-Orbit Coupling Table 4.1 Proportion to which v1 -band px,y orbitals are mixed into the valence band edge wavefunction by spin-orbit coupling, according to Eq. (4.7)

67

N

|δCvN (v1 )|2

1 2 3 4 5

0.133 0.040 0.027 0.023 0.021

which is the total weight of all px , p y orbitals from sub-bands of v1 mixed into the highest energy valence sub-band by SOC. We give values of |δCvN (v1 )|2 in Table 4.1 for N = 1−5 layers, showing the rapid drop in the effect of SOC on the valence band edge for thicker crystals.

4.2 Rashba Spin-Orbit Coupling As we have discussed above, the conduction band edge is a singly-degenerate band with orbital angular momentum projection L z = 0. Due to time-reversal symmetry, it therefore exhibits no spin-orbit splitting at the  point. Away from , however, the bands may split. In this section we explore how such splitting can take the RashbaBychkov form, and how this splitting depends on the interplay between the intrinsic asymmetry of the few-layer γ-InSe crystal, and the symmetry-breaking provided by electrostatic gates used to dope the system. One such phenomenon in which the Rashba effect may be observed is in weak antilocalisation—an anomalous magnetotransport phenomenon in which conductivity first decreases on application of an out-of-plane magnetic field, before increasing again at higher fields [3]. It arises from an interplay of two effects. First, there is a zero field quantum correction due to constructive interference between the two different directions an electron can take around a self-intersecting scattering path. This serves to increase the resistivity (i.e. to decrease the conductivity). The application of a magnetic field causes the electron to acquire a phase shift, with opposing sign for opposing direction of propagation—so on reaching the intersection constructive interference is less likely, which manifests itself in a reduction of the zero-field correction, and therefore an increase in conductivity at higher magnetic fields: weak localisation. The second effect,4 necessary for weak antilocalisation, is a precession of spins due to Rashba spin-orbit coupling. Such spin-orbit coupling can be understood to act on electrons as an effective in-plane magnetic field perpendicular to the momentum direction, so electrons travelling in opposite directions around a loop will see 4 Described here is the D’yakonov-Perel’ (DP) spin-relaxation mechanism [4], which arises in few-

layer InSe due to the spin-splitting of the conduction band. Another means of spin relaxation, the Elliot-Yafet mechanism, relies on scattering from impurities [5, 6].

68

4 Spin-Orbit Coupling Effects in InSe Films

their spins precess in opposite directions. This manifests itself in a positive correction to the zero-field conductivity. The application of a magnetic field will again lift the correction, but at a different rate to the correction leading to weak localisation. Depending on the interplay between the two rates, one can observe a negative magnetoconductivity at low fields, giving a local minimum in conductivity at a finite magnetic field, which is a sign of weak antilocalisation. Such behaviour has been observed in recent experiments on few-layer InSe [7, 8] and GaSe [9].

4.2.1 Rashba Splitting in the Hybrid k · p Tight-Binding Model To understand the effect of spin-orbit coupling on the WAL to be observed in fewlayer InSe we must determine the functional form and magnitude of the spin-orbit splitting in the lowest subband of the conduction band at small k. Since the system must be doped for magnetotransport experiments to be possible, it would be useful to be able to employ the two-band HkpTB model of Chap. 3 used for the intersubband transitions. This model, however, along with that with SOC added above, implicitly assumes a k-dependent basis to the bands, while intra- and inter-band SOC is included at the approximation of being independent of small k, so the model as presently constructed will give no spin-orbit splitting. One could, instead of allowing the band basis to vary with k, construct a model in a basis of the -point bands, and introduce interband mixing at finite k, as was done in a recent 7-band k · p model for the monolayer [2]. The addition of interlayer hops between the 7 bands would lead to a large model, so here we obtain the form and magnitude of the conduction band spin-orbit effects using a perturbative approach. In a basis of the -point bands the lowest-order interband mixing at small k is between c and v1 , and between v and v2 (the latter being responsible for the offset in the valence band maximum [10]). The mixing is linear in k, between c(v) and a ˆ Defining k ± ≡ k x ± ik y the elements p-orbital component of v1(2) pointing along k. of the perturbation between the bands, Hk , can be written as ±  = iζ1(2) k ± , c(v) |Hk | v1(2)

(4.8)

where ζ1(2) defines the extent to which px,y orbitals are mixed into the conduction (valence) band at finite k. Using second-order perturbation theory this leads to expansions of the conduction and valence bands at finite k as ± c(v) = c (v  ) + iζ1 ± (2± ) k ± v1(2)

where ζ1 ± (2± ) =

ζ1(2) ± E c(v) − E v1(2)

(4.9)

(4.10)

4.2 Rashba Spin-Orbit Coupling

69

± with E c(v) the -point conduction (valence) band energy and E v1(2) the -point band energies of the L z = ±1 components of the spin-orbit split v1 and v2 bands. The operator to create an electron at  in the lowest subband of the conduction band can be written as N  † = (ηcn cn† + ηvn vn† ) (4.11) c1|N

n

where ηc(v)n is the coefficient of the wavefunction on the monolayer conduction (valence) band of layer n. Perturbed, this becomes † c1|N =

N  +† −† +† −† (ηcn (cn† + iζ1 (v1n − v1n )) + ηvn (vn† + iζ2 (v2n − v2n ))). n

(4.12) We can now consider the spin-flip interband mixing. This is present between v and v1 , and between c and v2 , with matrix elements between the monolayer bands of the spin-orbit addition to the Hamiltonian HSO given by +↓(−↑)

v ↑(↓) |HSO | v1 and

+↓(−↑)

c↑(↓) |HSO | v2

λv,v1 2

(4.13)

λc,v2 . 2

(4.14)

 = +(−)

 = +(−)

If we then look for the matrix element of the spin-flip between the two components of the lowest conduction band subband we find ↑†

↓†

c1|N |HSO | c1|N  = ik + (ζ1 λv,v1 + ζ2 λc,v2 )

N 

ηcn ηvn .

(4.15)

n

Writing this in matrix form in a basis of the spin up/down states of the lowest subband of the conduction band we find

  0 itk + 0 ik x − k y HSO = α R = α = α R (σx k y − σ y k x ) R −itk − 0 −ik x − k y 0 (4.16) which is the Rashba Hamiltonian (σx,y are Pauli matrices in spin space), with the Rashba coefficient given by α R = (ζ1 λv,v1 + ζ2 λc,v2 )

N 

ηcn ηvn .

(4.17)

n

In a model in which the interlayer hops retain the σh symmetry of the monolayer, then nN ηcn ηvn = 0 and the Rashba effect will not be seen in the model. We must

70

4 Spin-Orbit Coupling Effects in InSe Films

therefore relax the approximation in the interlayer hops that tvc = −tcv and write amended hops as

= tcv + δcv , tvc = −tcv + δcv , (4.18) tcv where δcv will be antisymmetric as a function of the relative in-plane shift between the layers, u. Since we have so far been able to treat δcv as negligible for its effect on the energies of the bands, and on the oscillator strength for the intersubband optical transitions, adding it to the fit and determining it in the same manner as tc , tv and tcv would be likely to prove difficult, as the RMS error in the fit to bulk γ-InSe DFT is ∼20 meV, which could be of a similar order of magnitude to δcv . To isolate the effect δcv in order to better determine its magnitude, we consider the case of ε-InSe. Bulk ε-InSe is constructed by stacking InSe monolayers in a similar manner to the γ polytype, but instead of each layer being shifted in the same direction w.r.t. the one below, the direction of the shift alternates for successive layers. The bulk unit cell therefore has 2 layers, with a lattice vector along z. The bulk crystal and fewlayer crystals with an odd number of layers retain σh reflection symmetry, and have ¯ An point group symmetry D3h . The space group symmetry of the ε polytype is P6m2. individual interface between two layers is identical to that of the γ stacking, and in the HkpTB model with σh -symmetric hopping the two polytypes are indistinguishable. A distinguishing feature is introduced when we make the relaxation in Eq. (4.18)— since δcv is antisymmetric under u then the interband interlayer hops become ε ε = tcv ± δcv , tvc = −tcv ± δcv , tcv

(4.19)

with the ± signs alternating at successive interfaces. To find δcv we write Hamiltonians for bulk ε-InSe and γ-InSe (with a two-layer unit cell for comparison) in matrix form with k z az = π/2 ⎞ 0 0 2i(tcv [+δcv ]) E c + 2c ⎟ ⎜ 0 0 E v + 2v −2i(tcv [−δcv ]) ⎟ . (4.20) =⎜ ⎠ ⎝ 0 0 2i(tcv [−δcv ]) E c + 2c 0 0 E v + 2v −2i(tcv [+δcv ]) ⎛

Hˆ γ[ε]

If we consider the off-diagonal elements perturbatively, we can see that while for the γ case we obtain two twice-degenerate bands, the ε polytype will yield a small gap in each of the conduction and valence band dispersions, of size E 

4(|tcv + δcv |2 − |tcv − δcv |2 ) . E c + 2c − (E v + 2v )

(4.21)

A comparison of DFT calculations for γ- and ε-InSe should therefore yield a value for δcv . This is plotted in Fig. 4.3, which shows two nearly identical spectra, justifying the neglect of δcv in the analysis so far, and the expected small gaps. The gap in the conduction band, however, is larger than that in the valence band, which can be

4.2 Rashba Spin-Orbit Coupling

71

Fig. 4.3 Comparison of DFT-LDA calculations of bulk InSe for γ and ε polytypes; the γ dispersion is folded to correspond to a two-layer unit cell. 0 of energy is set to conduction band edge. Expanded regions show small gaps opening in ε dispersion, which can be assigned to the symmetry-breaking effect of δcv

attributed to small hops between c and c1 . To further isolate the contributions, we add band c1 to the model via interlayer hops between it and c, and write a Hamiltonian for the new system at k z az = π/2, Hˆ γ[ε],c1 =



(E c + 2c )ac†(n)π/2 ac(n)π/2 + (E v + 2v )ac†(n)π/2 ac(n)π/2 + E c1 ac†1(n)π/2 ac1(n)π/2

n=1,2

+ 2i(tcv [+δcv ])ac†(1)π/2 av(2)π/2 + 2i(tcv [−δcv ])av†(1)π/2 ac(2)π/2 + 2i(tc,c1 [+δc,c1 ])ac†(1)π/2 ac1(2)π/2 + 2i(tc,c1 [−δc,c1 ])ac†1(1)π/2 ac(2)π/2 + H.c.,

(4.22)

where the operators c(n)π/2 , v(n)π/2 , and c1(n)π/2 annihilate electrons with k z az = π/2 in layer n of the two-layer bulk unit cell in bands c, v, and c1 , respectively. When fitted to the DFT bands for ε-InSe these parameters are tc,c1 = 0.025 eV, δc,c1 = 0.051 eV and δcv = 0.017 eV. As it is the symmetry-breaking in δcv that will provide the necessary finite values for the Rashba coefficient, c1 is not included in the final amended model, and the Hamiltonian for 2D σh -asymmetric N -layer γ-InSe at  is written as H=

N   n

N −1      † c cn† cn + c(n+1) E c cn† cn + E v vn† vn + c(n+1) n

N −1    † † † † +v vn vn + v(n+1) v(n+1) + tc c(n+1) cn + tv v(n+1) vn n

     † † † † + tcv v(n+1) cn − c(n+1) vn ± δcv v(n+1) cn + c(n+1) vn + H.c. ,

(4.23)

72

4 Spin-Orbit Coupling Effects in InSe Films

where the choice of + or − will define a ±z direction for the crystal, which will be relevant for the effect of gate-doping the system. We must also determine the quantity (ζ1 λv,v1 + ζ2 λc,v2 ) from Eq. (4.17). Of the parameters required, only λv,v1 is already known. Furthermore, any attempt to determine ζ1 and ζ2 must take into account that any cross-gap k · p interband mixing will be sensitive to the size of the gap. The light conduction band effective mass has been attributed [2, 10] to strong interband c − v1 mixing (and can be understood in the tight-binding model as coming from nearest-neighbour Ins, pz -Se px,y hops), and if one parametrises the interband mixing parameters from DFT data then corrects for the underestimation of the gap, the implied values would then give a much heavier conduction band effective mass than is the case. Given that the LDA and HSE in-plane conduction band effective masses (both ∼0.18m e in the monolayer, ∼0.12m e in the bulk) are found to be similar and in reasonable agreement with experiment [11] then although the values of the parameters obtained from DFT cannot themselves be assumed to be correct, their effects on the wavefunction and energetics of the conduction band can be taken as reasonable—that is, while (for example) a value for ζ1 from DFT cannot be relied upon, that of ζ1 may be. We therefore fit the value of (ζ1 λv,v1 + ζ2 λc,v2 ) as a single parameter, calculating ηcv from the modified HkpTB model (Eq. (4.23)) (with DFT gap) and comparing it with the gradient of the linear spin-orbit splitting in the conduction band calculated using DFT for bilayer and trilayer γ-InSe. The DFT results are plotted in Fig. 4.4 with the results of the comparison given in Table 4.2, from which we find

Fig. 4.4 Spin-orbit splitting, calculated using DFT, near the conduction band edge in monolayer, bilayer and trilayer γ-InSe, showing cubic splitting in k in the monolayer, and linear splitting for the few-layer cases. k F for a carrier density n e ∼ 1.5 × 1012 cm−2 would be ∼0.02 Å−1 . The gradient of the linear component of the splitting in bilayer and trilayer is 2|α R | (Table 4.2)

4.2 Rashba Spin-Orbit Coupling

73

Table 4.2 DFT-calculated α R for bilayer and trilayer, together with ηcv calculated for the uncorrected model (up to 5 layers), which is used to obtain the constant of proportionality in Eq. (4.24). Values for ηcv and α R are also given for the gap-corrected model, showing the reduction of the effect when the band gap is increased N α R (DFT) (meVÅ) ηcv (DFT gap) ηcv (G.C.) α R (G.C.) (meVÅ) 2 3 4 5

19 35

0.0134 0.0241 0.0325 0.0392

0.0076 0.0124 0.0154 0.0173

11 18 23 25

α R = (1.46 eVÅ)ηcv .

(4.24)

Table 4.2 also shows a significant reduction in ηcv , and hence in α R , on correction of the gap, a result of the reduction of the effect of the symmetry-broken tcv + δcv . It was not possible to carry out an HSE DFT calculation for few-layer InSe with SOC taken into account to verify this, but the effect of the underestimation of the gap by DFT on the interband interlayer mixing has been shown to be significant previously, so the final values for α R in the gap-corrected model are expected to be closer to reality than those found from the DFT calculations.

4.2.2 Electrostatic Doping The final ingredient necessary here for using the model to understand the contribution of Rashba SOC to magnetoconductivity is a description of the response to electrostatic doping, which is required to enable magnetotransport experiments to be carried out. While in Chap. 3 we neglected the intralayer matrix element E z dz in calculating the response of the system to applied electric fields, as its effect on system band energies is negligible, it is important when considering the symmetry-breaking effects of the gates. To include this effect we amend Eq. (3.43) to read Hˆ = Hˆ +

  † † † † Un (anc anc + anv anv ) + E z (n)dz (anc anv + anv anc )

(4.25)

n

where Un is described as above (Eqs. (3.44) and (3.47) for the single and dual gated cases, respectively). We approximate the electric field across a layer, E z (n), as E z (n) =

E n−1,n + E n,n+1 . 2

(4.26)

So far we have been able to concern ourselves only with the magnitude of dz , and not its sign, since the sign of dz = ec |z| v in a σh -symmetric monolayer is merely

74

4 Spin-Orbit Coupling Effects in InSe Films

a result of an arbitrary choice in the phase of the eigenfunctions of bands c and v. A similar argument applies when one considers the interlayer interband hop tcv . In using a model which employs both terms, however, greater care must be taken, since while dz and tcv alone do not have a definite sign, the product dz tcv does. The sign will depend on the precise physical details of the integrals involved, and as we include dz as a symmetry-breaking term, along with δcv , it makes sense to find the sign of the products by considering a quantity which would be zero in the absence of such symmetry breaking—the intraband position operators of the band edge bilayer subbands, v |z| v and c |z| c, the signs of which we examine using DFT in the bilayer. In the context of Eq. (4.23) a calculation for the bilayer treating the interband hops tcv and δcv perturbatively yields  v |z| v = δcv

dz tcv az +

E g2L 2E g2L E g2L



and c |z| c = −δcv



dz tcv az +

E g2L 2E g2L E g2L

(4.27)  (4.28)



= E c − E v + tc − tv , and E g2L = Ec − where E g2L = E c − E v − (tc − tv ), E g2L E v − (tc + tv ) are energy differences between the bilayer bands in the absence of interband hoppings. From these we find that if the sign of tcv is chosen such that tcv = +0.29 eV, then dz = −1.68 eÅ, and δcv = +0.017 eV, where the +z direction is chosen such that of the vertical In-Se pair, the Se atom lies above the In atom. To see the effects of electrostatic gating on the Rashba effect in few-layer γ-InSe, we consider first the single-gated case. We choose a dielectric constant ε = 7 and self-consistently calculate the dependence of ηcv (and hence α R ) on the single-gateinduced carrier density, as set out in Sect. 3.4.3 of Chap. 3. The results are set out in Fig. 4.5, in which there are two curves, corresponding to the placement of the gate on one surface of the crystal or on the other. For one case, the Rashba coefficient increases with carrier density, while for the other it decreases. This can be understood as being due to the symmetry-breaking effect of the gates acting with or against the intrinsic symmetry breaking of the γ stacking, provided in the model by the δcv term. To show further the importance of the symmetry of the potentials due to electrostatic gates, we consider the dual-gated device with 6 layers of Bandurin et al. [12]. Modifying the self-consistent calculation for the dual-gated case using the method set out in Sect. 3.4.4 of Chap. 3, we again self-consistently calculate ηcv for a range of n e , fixing one gate to give a carrier density n e = 4 × 1012 cm−2 and using the other to tune the carrier density above and below that level. The resulting dependence of |α R | on carrier density, for dielectric constant ε = 7, is plotted in Fig. 4.6. In this case, at n e = 8 × 1012 cm−2 the effect of the gates is symmetric, and thus |α R | is provided entirely by the γ stacking, and has the same magnitude for whether the fixed gate is placed on top of or below the InSe. As the carrier density

4.2 Rashba Spin-Orbit Coupling

75

Fig. 4.5 Rashba coefficients from HkpTB model for N = 9 layers, showing dependence of α R on carrier density, increasing or decreasing depending on the direction of the electric field due to the gates used to dope the system. Inset: definition of electric field directions (at low carrier densities) relative to crystal stacking direction

Fig. 4.6 Rashba coefficients from HkpTB model for N = 6 layers with doping provided by two gates—one gate voltage giving a carrier density n e = 4 × 1012 cm−2 , then the other tuning the carrier density above or below the n e provided by the top gate, showing identical α R at n e = 8 × 1012 cm−2 when the potential due to the gates is symmetric. Inset: definition of electric field directions (at low carrier densities) relative to crystal stacking direction (fixed gate at bottom for E z +, at top for E z −)

76

4 Spin-Orbit Coupling Effects in InSe Films

is reduced from n e = 8 × 1012 cm−2 , the potential profile due to the gates becomes more asymmetric—in one case this effect acts in the same direction as the crystal asymmetry, increasing |α R |; in the other the gate potential and crystal asymmetries partially cancel, reducing |α R |. At small carrier densities we have a large and poorly-screened electric field, giving a significant effect. The dual gate configuration provides a means of investigating experimentally the effect of reversing the gates on the same device—if the contributions to the carrier densities of the gates were reversed this would be expected to appear in the dependence of conductivity on magnetic field. This dependence of Rashba splitting, and hence of weak antilocalisation, on the asymmetry of the confining system has previously been explored in a similar manner to the method employed above in a semiconductor quantum well context [13], with a k · p approach used to quantify the dependence of weak antilocalisation on the asymmetry of the confining potential, which may then be compared with experiment.

4.3 Selection Rules for Interband Optical Transitions 4.3.1 Hybrid k · p Tight-Binding Theory The hybrisation of v with v1 via the ‘spin-flip’ operator, L ± s ∓ enables the principal interband transition to couple to in-plane polarised light, providing the necessary orbital angular momentum difference between c and v in exchange for a change in spin projection, and via the antisymmetry of L ± under σh overcoming the parity restriction on the transition. The transitions will be between total angular momentum 1 1 states v ± 2 ↔ c∓ 2 , coupling to σ ∓ -polarised photons. Since c changes negligibly on application of SOC, we can consider the spin-projection μ to remain a good quantum number in the conduction band, and therefore note that σ ± -polarised light will excite electrons into spin up/down states in the conduction band. We can estimate the strength of the spin-orbit-induced in-plane polarised component of the principal interband optical transition, by simply combining the extent to which v1 is hybridised with v and the strength of the ‘B-line’ transition as aS O =

e|Cv (v1 )|β1 ± eβsf ± A ≡ A , cm e cm e

(4.29)

√ where A± = A(ˆx ± i yˆ )/ 2 corresponds to the vector potential of σ ± circularly polarised light, and |Cv (v1 )|2 is the weight of px , p y orbitals from v1 introduced by SOC into the dominantly pz orbital based v-band wavefunction. The oscillator strength of such transitions can be estimated as βsf = |Cv (v1 )|β1 ∼ 0.4 /Å leading to the absorption coefficient for in-plane polarised light5 5 The

expression here is closely related to Eq. (2.19), with a factor of 2 removed due to the lifting of spin degeneracy, as the polarisation of the absorbed light will select a single spin state.

4.3 Selection Rules for Interband Optical Transitions

g A(σ± ) = 4π

77

mc e2 |βsf |2 ∼ 0.7%. ± 21 c |E |m 2 v

(4.30)

e

For comparison (Chap. 2), a photon incident at an angle θ ≈ 45◦ to the surface and coupled to the the principal interband transition via the interband out-of-plane electric dipole moment, dz , is absorbed with g A(Ez ) [θ ≈ 45◦ ] ∼ 3.7%, while for the B-line g B(σ± ) ∼ 7%. For multilayer films, we again determine the absorption coefficient for the principal interband transition, coupling to in-plane polarised light, from the extent to which orbitals from v1 are mixed into the valence band, g A(σ± ) (N ) = 4π

m c (N ) e2 |δCvN (v1 )β1 |2 , c ω(N )m 2e

(4.31)

where m c (N ) and ω(N ) are the N -layer  conduction-band effective mass and band gap, respectively, and |δCvN (v1 )|2 = n |Cv (v1n )|2 is the total weight of all px , p y orbitals from sub-bands of v1 admixed by SOC into the highest energy valence sub-band. As expected from the reduction in spin-flip mixing between v1 and the band-edge states it is found for large N that g A(σ± ) (N  1) → ∼ 0.1%,

(4.32)

see Table 4.3 and Fig. 4.7. This compares with an increase in g A(Ez ) (N )[θ ≈ 45◦ ] from 3.7% in the monolayer to ∼15% for large N , and with a roughly constant g B(σ± ) (N ) ∼ 6% absorption for the B-line (Chap. 2), as shown in Fig. 4.7, which shows that coupling to in-plane polarised light of the principal interband transition has a much weaker strength than the other transitions shown, with the transitions largely retaining the dominant polarisation characters described in Chap. 2. The fact that it is non-zero, however, has significance for experiment, as a photon emitted with out-ofplane polarisation will travel in the crystal plane, and thus be much harder to observe than a photon with in-plane polarisation, which will be emitted perpendicular to the

Table 4.3 Band gaps (ω) and absorption coefficients as a function of number of layers, N , for coupling of A-line transition to in-plane polarised light after application of SOC (g A(σ± ) ), in comparison with absorption coefficients for the B-line (g B(σ± ) ) and for coupling to the out-of-plane dipole transition for a photon incident at an angle θ ≈ 45◦ to the crystal (g A(Ez ) ) (Chap. 2) N

ω (eV)

g A(σ± ) (%)

g A(Ez ) [θ ≈ 45◦ ] (%)

g B(σ± ) (%)

1 2 3 4 5

2.72 2.00 1.67 1.50 1.40

0.75 0.24 0.18 0.15 0.14

3.7 6.4 8.2 9.6 10.7

6.8 5.5 5.9 6.0 6.1

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4 Spin-Orbit Coupling Effects in InSe Films

16

gA(SOC) gB gA (dz )

Absorption (%)

12

8

4

0

1

5

N

10

15

Fig. 4.7 Comparison of dependence of -point absorption coefficient on number of layers between ‘spin-flip’ principal interband transition, polarised in-plane (dots), and absorption coefficients calculated in Chap. 2 for light incident at 45◦ to the crystal surface (diamonds) and for the B-line transition (in-plane polarised, squares)

crystal surface. Spin-orbit coupling is therefore required to explain the observation of emission across the principal interband transition in photo-luminescence (PL) experiments on few-layer InSe [12]. In relation to the PL experiments, it is worth noting that the increase in PL strength with the number of layers (despite the decrease in the absorption coefficient shown in Fig. 4.7) is likely to be due to the decrease in the offset of the valence band maximum from  (see Chap. 2, and Ref. [14]), which leads to the gap becoming more direct for thicker crystals.

4.3.2 Optical Absorption Including SOC in the Tight-Binding Model Having taken SOC into account in the tight-binding model we can now more reasonably calculate an absorption coefficient for in-plane polarised light for a wider range of energies, and for k-points covering the whole Brillouin zone. We adapt6 Eq. (2.19) to read g(ω) =

6 The

8π 2 e2  |c |∇k H | v|2 δ(ω − E cvk ), Auc Nk c c,v,k ω

(4.33)

joint density of states, which included a factor of 2 due to (now lifted) spin degeneracy, is replaced by a sum over the Brillouin zone, normalised by the total system area.

4.3 Selection Rules for Interband Optical Transitions

79

Fig. 4.8 Single-particle optical absorption spectra for 1–4 layer InSe, calculated from the tightbinding model using Eq. 4.33 with (solid lines) and without (dashed lines) SOC taken into account. A small Gaussian broadening (5 meV) is applied to the result to suppress numerical noise

where Auc = 13.53 Å2 is the area of a unit cell, and Nk is the number of k-points in the sum over the Brillouin zone. The sums over v and c run over all occupied and unoccupied bands, respectively. Figure 4.8 plots the results, for N = 1 to N = 4layer InSe, with the dashed lines indicating the calculation without SOC taken into account, while the solid lines indicate calculations with SOC included. We see how SOC enables a non-zero absorption coefficient at the band gap energies, which is strongest in the monolayer, whilst the structure of the spectra becomes more complex for increasing numbers of layers with various transitions between the increasing number of subbands becoming available. It should be noted that the absorption spectra are calculated in a single-particle picture, neglecting excitonic effects, which would be expected to generate bound resonances at band edges, and to shift the spectrum lower in energy due to the attraction between electron and hole. Excitonic effects are expected to be especially important in the thinnest films due to reduced screening of the Coulomb interaction when the exciton is confined to a 2D system [15]—while excitons in 3D systems typically have binding energies ∼10s of meV the binding energy for excitons in TMDCs has been reported to be as high as 700 meV [16].

80

4 Spin-Orbit Coupling Effects in InSe Films

Fig. 4.9 Optical pumping of In nuclear spins: an electron excited from the valence band by a σ + -polarised photon into a spin-up state of the conduction band can recombine by emitting another σ + photon. If, however, the electron flips its spin by the hyperfine interaction with the In nuclei, then it may recombine by emission of an out-of-plane polarised photon, leaving the angular momentum of the incoming photon in the spins of the In nuclei

4.3.3 Optical Pumping of Nuclear Spin Polarisation We can consider a mechanism (shown in Fig. 4.9) in few-layer InSe by which the optical pumping of the nuclear spins of In atoms could be acheived. If we optically excite an electron into the μ = ± 21 conduction band state we leave behind a hole in the 1 Jz = ∓ 21 valence band (v ∓ 2 ). If the electron then flips its spin by transferring angular momentum to an In nucleus via the hyperfine interaction it can then recombine with the photoexcited hole through emission of an out-of-plane polarised photon, leaving the angular momentum imparted by the incident σ ∓ -polarised photon in the spin of the In nucleus. Since the Indium s orbital contributes strongly to the decomposition of the conduction band wavefunction (Chap. 2, Ref. [1]) we consider In as the candidate for hyperfine coupling in the conduction band. The most abundant isotope of In is 115 In, which has a nuclear spin I = 29 , and an atomic hyperfine coupling constant AIn ≈ 60 μeV [17].7 An effective hyperfine coupling constant can be estimated for the conduction band using the orbital decomposition found in Chap. 2, which gives ef f Ac = |Cc (Ins )|2 A ≈ 15 μeV, where Cc (Ins ) is the coefficient of In s orbitals in the conduction band wavefunction near the -point.

7 For

comparison the only stable Selenium isotope with non-zero nuclear spin is 77 Se, which has spin I = 21 , an abundance of 7%, and a much smaller hyperfine constant ASe ≈ 2 μeV [18].

References

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References 1. Zólyomi V, Drummond ND, Fal’ko VI (2014) Phys Rev B 89:205416 2. Zhou M, Zhang R, Sun J, Lou W-K, Zhang D, Yang W, Chang K (2017) Phys Rev B 96:155430 3. Knap W, Skierbiszewski C, Zduniak A, Litwin-Staszewska E, Bertho D, Kobbi F, Robert JL, Pikus GE, Pikus FG, Iordanskii SV, Mosser V, Zekentes K, Lyanda-Geller YB (1996) Phys Rev B 53:3912 4. Dyakanov MI, Perel VI (1971) Sov Phys J Exp Theor Phys Lett 33:1053 5. Elliott RJ (1954) Phys Rev 96:266 6. Yafet Y (1963) Solid State Phys 14:1 7. Premasiri K, Radha SK, Sucharitakul S, Kumar UR, Sankar R, Chou F-C, Chen Y-T, Gao XPA (2018) Nano Lett 18:4403 8. Zeng J, Liang S-J, Gao A, Wang Y, Pan C, Wu C, Liu E, Zhang L, Cao T, Liu X, Fu Y, Wang Y, Watanabe K, Taniguchi T, Lu H, Miao F (2018) Phys Rev B 98:125414 9. Takasuna S, Shiogai J, Matsuzaka S, Kohda M, Oyama Y, Nitta J (2017) Phys Rev B 96:161303(R) 10. Li P, Appelbaum I (2015) Phys Rev B 92:195129 11. Kress-Rogers E, Nicholas R, Portal J, Chevy A (1982) Solid State Commun 44:379 12. Bandurin DA, Tyurnina AV, Geliang LY, Mishchenko A, Zólyomi V, Morozov SV, Kumar RK, Gorbachev RV, Kudrynskyi ZR, Pezzini S, Kovalyuk ZD, Zeilter U, Novoselov KS, Patanè A, Eaves L, Grigorieva II, Fal’ko VI, Geim AK, Cao Y (2017) Nat Nanotechnol 12:223 13. Koga T, Nitta J, Akazaki T, Takayanagi H (2002) Phys Rev Lett 89:46801 14. Rybkovskiy DV, Osadchy AV, Obraztsova ED (2014) Phys Rev B 90:235302 15. Olsen T, Latini S, Rasmussen F, Thygesen KS (2016) Phys Rev Lett 116:56401 16. Zhu B, Chen X, Cui X (2015) Sci Rep 5:9218 17. Braun P-F, Urbaszek B, Amand T, Marie X, Krebs O, Eble B, Lemaitre A, Voisin P (2006) Phys Rev B 74:245306 18. Zijlstra W, Henrichs J, Voorst JV (1972) Chem Phys Lett 13:325

Chapter 5

Conclusions

In this thesis we have explored theoretically the electronic and optical properties of Indium Selenide, using effective k · p and tight-binding approaches. We developed a comprehensive atomistic tight-binding model for monolayer and few-layer InSe, parametrised from DFT calculations. The model was used to show how the gap increases markedly on going from the bulk to ultrathin films, with the optical transition energies covering a range ∼1.3 to ∼2.8 eV—from the near infrared to violet in the visible spectrum. The principal interband optical transition has a dominantly out-of-plane dipole character which increases in strength with the number of layers, while the next-lowest energy transition couples dominantly to in-plane polarised light. We note that the effect of the scissor-correction has consequences beyond a need to correct the gaps predicted by the model, with the underestimation of the monolayer gap leading to an overestimation of the effects of the interlayer hops on few-layer wavefunctions and interband matrix elements. The highly anisotropic crystal structure and electronic properties of InSe permit the introduction of a model which describes InSe using k · p theory for the monolayer bands near the  point, while using tight-binding hops to describe electronic coupling between the monolayer bands for the few-layer cases. This hybrid k · p tight-binding (HkpTB) model was used to describe the band-gap behaviour of misaligned few-layer films, with repulsion of vertically opposite Se-Se pairs increasing the gap. Describing optical transitions between the subbands of crystalline γ-InSe we showed how these intersubband transitions may be used to extend the range of optical activity in InSe down into the mid- and far-infrared range. Since the activation of these transitions requires doping of the system, we explored the consequences of the application of a gate potential to the system. The underestimation of the gap by DFT must again be corrected for, as it has consequences for the bulk out-of-plane effective masses and few-layer subband dispersions, and we found that the best means of correction is to take the interlayer hops as those fitted straight to DFT, and correct the gap at the monolayer level. Finally, we considered the effects of spin-orbit coupling (SOC) on the transport properties and optical properties of InSe. The HkpTB model was used to calculate the © Springer Nature Switzerland AG 2019 S. J. Magorrian, Theory of Electronic and Optical Properties of Atomically Thin Films of Indium Selenide, Springer Theses, https://doi.org/10.1007/978-3-030-25715-6_5

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84

5 Conclusions

coefficient of Rashba spin-orbit splitting in the conduction band of few-layer InSe, self-consistently taking into account the potential profile induced by gates used to n-dope the system. The Rashba coefficient found in this way compares reasonably with magnetotransport measurements showing evidence of weak antilocalisation. SOC relaxes the selection rule which restricts the polarisation of the principal interband optical transition, allowing it to couple to in-plane polarised light. We evaluated the dependence of the transition strength on the number of layers by using the HkpTB approach developed in Chap. 3. The addition of atomic SOC to the tight-binding model allows for the consideration of its effect on the bands of InSe more generally, and we plot the optical absorption as a function of transition energy for the whole Brillouin zone. The work presented in this thesis lays the foundation for further investigations into the properties of the hexagonal III–VI semiconductors. The tight-binding and HkpTB models can be easily adapted to suit GaSe and the other III–VI materials, the most significant differences being an increase in the gap and the occurrence of other polytypes in multilayer crystals. The HkpTB model is well-suited to a description of heterostructures including more than one III–VI crystal, and other 2D materials. A full description of the optical properties of any 2D material would obviously benefit from an inclusion of excitonic effects, which are important due to the reduction of screening compared with the 3D case. For the monolayer excitons may be included by solving the Bethe-Salpeter equation in the context of a tight-binding model [1]. The offset in the valence band maximum in the thinnest films will be of great interest if the necessary hole-doping becomes experimentally possible, and accurate band descriptions including SOC will be necessary for the analysis and prediction of strongly-correlated phenomena. Finally, the III–VI semiconductors have been shown to exhibit a greatly reduced band gap in response to strain [2, 3],1 a problem to which the HkpTB model would be well suited.

References 1. Ridolfi E, Lewenkopf CH, Pereira VM (2018) Phys Rev B 97:205409 2. Li Y, Wang T, Wu M, Cao T, Chen Y, Sankar R, Ulaganathan RK, Chou F, Wetzel C, Xu C-Y, Louie SG, Shi S-F (2018) 2D Materials 5:021002 3. Song C, Fan F, Xuan N, Huang S, Zhang G, Wang C, Sun Z, Wu H, Yan H (2018) ACS Appl Mater Interfaces 10:3994 4. Zhu Z, Cheng Y, Schwingenschlögl U (2012) Phys Rev Lett 108:266805

1A

study in 2012 predicted a topological phase transition in GaSe under 2% strain [4], however as a DFT study it has a much-underestimated band gap to start with, so a much greater strain would be required to close the gap. Inspection of the gradient of the gap against strain suggests InSe would undergo the transition at around 4% strain.

Curriculum Vitae

Dr. Samuel J. Magorrian Research Associate, National Graphene Institute, University of Manchester Research Interests My interests are in the area of theoretical condensed matter physics, in particular the calculation of the electronic structure and properties of semiconducting systems. My current research focuses on the optical and electronic properties of two-dimensional semiconductors, particularly the hexagonal III–VI monochalcogenides such as InSe and GaSe. I use effective models, such as continuum and tight-binding models, to interpret and extend first-principles results, and to understand and predict results of experimental work. Education 2014–Oct. 2018 Ph.D., School of Physics and Astronomy, University of Manchester. Supervisor: Prof. V. I. Fal’ko. Thesis Title: Theory of electronic and optical properties of atomically thin films of indium selenide. 2010–2014 Master of Physics, University of Oxford (The Queen’s College). MPhys project: Investigations of the multiferroic material CuFeO2 , supervisor Prof. P. G. Radaelli. Summer project: Turbulent plumes from a glacier terminus melting in a stratified ocean, supervisor Dr. A. J. Wells. Employment Oct. 2018–present: Research Associate, University of Manchester. 2016–2018: Graduate Teaching Assistant, University of Manchester.

© Springer Nature Switzerland AG 2019 S. J. Magorrian, Theory of Electronic and Optical Properties of Atomically Thin Films of Indium Selenide, Springer Theses, https://doi.org/10.1007/978-3-030-25715-6

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Curriculum Vitae

Research Skills and Experience • Theoretical calculations of electronic structure, and optical and electronic properties, of condensed matter systems using effective continuum and tight-binding methods, informed by first-principles calculations. • Extensive collaboration with experimental groups, both in Manchester and externally. • Programming in Python including parallelisation, use of MATLAB/Mathematica. • Understanding and regular use of local Linux computing systems (as administrator) and clusters. Compilation and use of codes (e.g. QUESTAAL) for first-principles many-body electronic structure calculation. Teaching experience • Demonstrator in 1st year undergraduate teaching labs—supervising student lab work, conducting assessment interviews and marking lab reports. Tutor for 1st year maths/physics tutorials marking tutorial work, delivering tutorials and answering questions from students. • Assisting in the supervision of a 2nd year Ph.D. student. Publications 7. Z. Ben Aziza, V. Zólyomi, H. Henck, D. Pierucci, M. G. Silly, J. Avila, S. J. Magorrian, J. Chaste, C. Chen. M. Yoon, K. Xiao, F. Sirotti, M. C. Asensio, E. L’huillier, M. Eddrief, and V. I. Fal’ko Valence band inversion and spin-orbit effects in the electronic structure of monolayer GaSe, Physical Review B 98 115405 (2018) 6. D. J. Terry, V. Zólyomi, M. Hamer, A. V. Tyurnina, D. G. Hopkinson, A. M. Rakowski, S. J. Magorrian, N. Clark, Y. M. Andreev, K. Novoselov, S. J. Haigh, V. I. Fal’ko and R. V. Gorbachev Infrared-to-violet tunable optical activity in atomic films of GaSe, InSe, and their heterostructures, 2D Materials 4, 041009 (2018) 5. S. J. Magorrian, A. Ceferino, V. Zólyomi and V. I. Fal’ko, Hybrid k · p tightbinding model for intersubband optics in atomically thin InSe films, Physical Review B 97 165304 (2018) 4. S. J. Magorrian, V. Zólyomi and V. I. Fal’ko, Spin-orbit coupling, optical transitions, and spin pumping in monolayer and few-layer InSe, Physical Review B 96 195428 (2017) 3. S. J. Magorrian, V. Zólyomi and V. I. Fal’ko, Electronic and optical properties of two-dimensional InSe from a DFT-parameterized tight-binding model, Physical Review B 94 245431 (2016) 2. J. Beilsten-Edmands, S. J. Magorrian, F. R. Foronda, D. Prabhakaran, P. G. Radaelli and R. D. Johnson, Polarization memory in the nonpolar magnetic ground state of multiferroic Cu FeO 2 , Physical Review B 94, 144411 (2016) 1. S. J. Magorrian and A. J. Wells, Turbulent plumes from a glacier terminus melting in a stratified ocean, Journal of Geophysical Research: Oceans 121, 4670 (2016)

Curriculum Vitae

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Conference Presentations 8. Rank Prize Funds meeting (2D materials for optoelectronics applications), Grasmere, UK—invited speaker (April 2019) 7. Theoretical Physics Seminar, University of Birmingham, Birmingham, UK— invited speaker (February 2019) 6. International Renewable and Sustainable Energy Conference, Rabat, Morocco— invited speaker (December 2018) 5. New Horizons in Atomistic Simulation, York, UK (January 2018) 4. Condensed Matter Theory Seminar, Lancaster University, Lancaster, UK— invited speaker (November 2017) 3. Flatlands beyond Graphene 2017, Lausanne, Switzerland (September 2017) 2. Euro TMCS II, University College Cork, Cork, Ireland (December 2016) 1. Graphene NowNANO CDT student conferences, 2017 & 2018