Tunable band structure in bilayer graphene

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Doctoral Thesis

Tunable band structure in bilayer graphene: chirality and topology Author(s): Varlet, Anastasia Publication Date: 2015 Permanent Link: https://doi.org/10.3929/ethz-a-010436438

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ETH Library

Diss. ETH NO. 22596

Tunable band structure in bilayer graphene: chirality and topology

A thesis submitted to attain the degree of DOCTOR OF SCIENCES of ETH ZURICH (Dr. sc. ETH Zurich)

presented by Anastasia Varlet Diplˆome d’Ing´enieur, Institut National Polytechnique de Grenoble born on 28.02.1988 citizen of France

accepted on the recommendation of: Prof. Dr. Klaus Ensslin, examiner Prof. Dr. Thomas Ihn, co-examiner Prof. Dr. Christian Sch¨onenberger, co-examiner 2015

Abstract After its experimental discovery in 2004, graphene and its unique properties received a lot of interest in the condensed matter community. One year later, quantum Hall effect measurements gave the experimental confirmation that charge carriers in graphene behave as massless Dirac fermions, mimicking relativistic particles. However, single-layer graphene has the disadvantage of being a gapless material and thereby reducing the possible range of applications. This will be discussed in Part I of this thesis. Bilayer graphene, a system made of two coupled layers of graphene, offers a solution to this problem. The band structure of bilayer graphene can be changed from ungapped to gapped by the application of a transverse electric field. This can be accomplished by chemical doping or gating. The resulting band gap is predicted to reach values up to 300 meV and therefore makes bilayer graphene a very appealing material for potential applications. In addition to opening a band gap, this induced layer asymmetry can qualitatively change the low energy band structure of bilayer graphene. In pristine bilayer graphene, the low energy dispersion exhibits a strong trigonal deformation. Close to the charge neutrality, valence and conduction bands do not meet at one point, but at the center of four Dirac cones. This change in the topology of the Fermi contour as a function of energy, from a unique and trigonally shaped contour to a contour broken into four pockets, is called a Lifshitz transition. This topological property can be strongly influenced by external parameters, such as strain or transverse electric fields. In Part II, we discuss how the band structure of bilayer graphene can be probed and experimentally influenced by these external parameters. In a tight binding approach, we introduce the band structure of bilayer graphene and the role of each coupling constant on the dispersion. We highlight the presence of a Lifshitz transition due to the skew interlayer hopping. To access the low energy dispersion of bilayer graphene, high quality bilayer graphene devices are required. We discuss how such high quality samples are fabricated. This subsequently enables the fabrication of dual-gated bilayer devices, which allows for the application of a displacement field and therefore of an asymmetry between the layers. We discuss the basic characteristics of these devices and investigate the experimentally induced energy gap. We focus in particular on magneto-transport experiments in a dual-gated geometry

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and we reveal how the quantum Hall effect can be used to probe the finest features of the band structure of bilayer graphene. This allowed for the first experimental observation of the presence of the Lifshitz transition in a gapped bilayer graphene system. The opening of the band gap also has consequences on the chirality of the charge carriers. While pristine bilayer graphene is predicted to exhibit anti-Klein tunneling and a Berry phase of 2π, we demonstrate in Part III of this thesis that gapped bilayer graphene has an energy-dependent transmission function, which can take any value between 0 and 1 at normal incidence, and therefore exhibits a tunable Berry phase. This is accomplished via the investigation of Fabry-P´erot interference, occurring in a dual-gated bilayer graphene region. We interpret the tunability of the transmission and of the Berry phase as being related to the breaking of the chiral properties, due to the symmetry breaking that opens the band gap. In summary, this thesis offers a thorough investigation of the consequences of an electric field-induced band gap on different transport properties, such as the quantum Hall effect, the transmission through a pn interface and the Berry phase. It provides insight into the physics derived from the rich band structure of bilayer graphene and opens the door to experiments on gate-defined nanostructures. Finally we present an attempt to realize a gate-defined quantum point contact in Part IV. We discuss experimental limitations, such as the lack of control of the gate-induced channel. We finally propose an optimized geometry, enabling for a full control of the confined region in future devices.

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R´ esum´ e Apr`es sa d´ecouverte en 2004, le graph`ene, mat´eriau aux propri´et´es uniques, a suscit´e beaucoup d’engouement dans la communaut´e de la mati`ere condens´ee. Un an plus tard, des mesures d’effet Hall quantique ont fourni la preuve exp´erimentale que les porteurs de charge du graph`ene se comportent comme des fermions de Dirac ayant une masse nulle, imitant ainsi le comportement des particules relativistes. Le graph`ene monocouche pr´esente n´eanmoins l’inconv´enient de ne pas avoir de bande interdite, ce qui r´eduit irr´em´ediablement la port´ee de ses applications potentielles. Ce sujet sera l’objet de la Partie I de cette th`ese. Le graph`ene bicouche, quant a` lui, syst`eme constitu´e de deux feuillets de graph`ene coupl´es l’un a` l’autre, offre une alternative a` cela. En effet, la structure de bande du graph`ene bicouche peut ˆetre transform´ee, de sorte qu’une bande interdite peut ˆetre induite par l’application d’un champ ´electrique transverse. Ceci peut ˆetre fait soit par dopage chimique, soit par l’utilisation d’´electrodes de grille. La bande interdite qui en r´esulte peut atteindre des valeurs allant jusqu’`a 300 meV, ce qui fait du graph`ene bicouche un excellent candidat pour des applications potentielles. En plus d’ˆetre responsable de l’ouverture d’une bande interdite, l’asym´etrie ´electrostatique g´en´er´ee entre les deux couches est aussi a` l’origine d’une transformation de la structure de bande a` faible ´energie. Dans le graph`ene bicouche naturel, la relation de dispersion `a basse ´energie arbore une forte d´eformation trigonale. Pr`es du point de neutralit´e, les bandes de conduction et de valence ne se touchent plus en un seul point mais se rejoignent au centre de quatre cones de Dirac. Ce changement dans la topologie du contour de Fermi, qui subit une transformation d’un contour unique, de forme trigonale, en un contour “´eclat´e” en quatre poches distinctes, est appel´e une transition de Lifshitz. Cette derni`ere a la particularit´e d’ˆetre influen¸cable par certains param`etres ext´erieurs, tels que les d´eformations m´ecaniques ou encore les champs ´electriques transversaux. Dans la Partie II, nous montrerons donc comment il est possible d’acc´eder a` la structure de bande du graph`ene bicouche de fa¸con exp´erimentale et comment celle-ci peut ˆetre influenc´ee par les param`etres externes ´evoqu´es ci-dessus. En appliquant le mod`ele des liaisons fortes de Wallace, nous commencerons par pr´esenter les sp´ecificit´es de la structure de bande du graph`ene bicouche et, en particulier, le rˆole jou´e par chacun des param`etres de couplage sur la-dite dispersion. L’accent sera mis sur l’existence d’une transition de Lifshitz, caus´ee par le couplage ´energ´etique

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de biais. Celle-ci prenant place `a basse ´energie, il est n´ecessaire, pour y acc´eder, de travailler avec du graph`ene bicouche de haute qualit´e. Nous discuterons ainsi des proc´ed´es de fabrication qui permettent de parvenir a` une telle qualit´e. La validation de cette ´etape permettra dans un second temps de r´ealiser des ´echantillons `a double-grille, qui rendent possible l’application d’un champ de d´eplacement, et donc d’une asym´etrie, entre les feuillets de la bicouche. Les sp´ecificit´es ´electrostatiques de ces ´echantillons seront ensuite comment´ees, et leur bande interdite sera ´etudi´ee. Nous tˆacherons en particulier de d´ecrire les ph´enom`enes physiques r´egissants nos exp´eriences en champ magn´etique et nous d´emontrerons notamment que l’effet Hall quantique peut servir de d´etecteur pour identifier les d´etails les plus fins de la structure de bande d’une bicouche. Ceci nous permettra, pour la premi`ere fois, de rendre compte de l’observation de la transition de Lifshitz dans un ´echantillon de graph`ene bicouche avec bande interdite. Mais l’ouverture de la bande interdite a aussi des cons´equences sur la chiralit´e des porteurs de charge. Alors que la th´eorie pr´edit, pour le graph`ene bicouche naturel, l’apparition de l’anti-effet tunnel de Klein et d’une phase de Berry ´egale a` 2π, nous d´emontrerons dans la Partie III de cette th`ese qu’une bicouche de graph`ene poss´edant une bande interdite pr´esente en fait une fonction de transmission qui d´epend de l’´energie et qui peut prendre, a` incidence normale, des valeurs entre 0 et 1, ce qui r´esulte en une phase de Berry ajustable. Cette d´emonstration se base sur l’´etude d’interf´erences de Fabry-P´erot se produisant dans la r´egion situ´ee sous la double-grille. Le caract`ere ajustable de la transmission et de la phase de Berry sera enfin interpr´et´e en terme de destruction de la chiralit´e, caus´ee par l’ouverture de la bande interdite. En conclusion, cette th`ese propose une ´etude d´etaill´ee des cons´equences qu’a l’ouverture d’une bande interdite sur les diff´erentes propri´et´es de transport, telles que la transmission au travers d’une interface pn ou la phase de Berry. Elle met ainsi en avant les ph´enom`enes physiques qui d´ecoulent de la riche structure de bande du graph`ene bicouche et ouvre la voie aux exp´eriences mettant en lumi`ere les nanostructures d´efinies de fa¸con ´electrostatique. Enfin, dans une derni`ere partie (Partie IV), nous pr´esenterons notre tentative de r´ealisation d’un point quantique dans un tel syst`eme. Nous mettrons en avant les limitations exp´erimentales que nous avons rencontr´ees, tel que la difficult´e d’influencer et de contrˆoler le canal. Nous finirons ainsi en proposant une g´eom´etrie optimis´ee, permettant un meilleur contrˆole de la r´egion confin´ee.

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Contents I

Introduction and motivation

1 Introduction 1.1 Graphene - the first true two-dimensional material . . . . . . . . 1.1.1 Single-layer graphene band structure: the tight-binding proach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Electronic properties of single-layer graphene . . . . . . . 1.2 Single-layer graphene nanostructures . . . . . . . . . . . . . . . 1.3 The bilayer graphene alternative . . . . . . . . . . . . . . . . . . 1.4 About this work . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Impact of this work . . . . . . . . . . . . . . . . . . . . . . . . .

1

. . ap. . . . . . . . . . . .

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3 3

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4 5 6 7 8 9

II Probing and controlling the band structure of bilayer graphene 11 2 Electronic properties of bilayer graphene 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bilayer graphene: a step by step introduction to the band structure 2.2.1 Influence of the intralayer coupling γ0 and the interlayer coupling γ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Inducing an asymmetry u: the Mexican hat . . . . . . . . . 2.2.3 Influence of the skew interlayer hopping γ3 : the trigonal warping of the band structure . . . . . . . . . . . . . . . . . . . . 2.2.4 Effect of external strain on the electronic dispersion . . . . . 2.2.5 Inducing an asymmetry u with γ3 . . . . . . . . . . . . . . . 2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 . 13 . 14 . 16 . 16 . . . .

18 19 21 22

3 Backgated bilayer graphene devices 23 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

v

3.2

3.3

3.4

Fabrication of backgated bilayer graphene devices . . . . . . . . . . . 3.2.1 Preparation of h-BN flakes on Si/SiO2 : substrate . . . . . . . 3.2.2 Preparation of bilayer graphene flakes on transferable substrates 3.2.3 Transfer of a graphene flake on a h-BN substrate . . . . . . . 3.2.4 Patterning and contacting the graphene flake . . . . . . . . . . 3.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Improving the quality of bilayer graphene on h-BN . . . . . . . . . . 3.3.1 Quantifying disorder in bulk graphene . . . . . . . . . . . . . 3.3.2 Thermal annealing . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Current annealing . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Mechanical cleaning . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Dual-gated bilayer graphene 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Encapsulating graphene using the dry transfer technique 4.2.2 Top gate definition . . . . . . . . . . . . . . . . . . . . . 4.2.3 Alternative process: the stacking technique . . . . . . . . 4.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Electrostatics of dual-gated bilayer graphene . . . . . . . . . . . 4.3.1 Basic electrostatic model . . . . . . . . . . . . . . . . . . 4.3.2 Environment . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Opening a band gap . . . . . . . . . . . . . . . . . . . . 4.4 Transport in dual-gated bilayer devices . . . . . . . . . . . . . . 4.4.1 Device presentation . . . . . . . . . . . . . . . . . . . . . 4.4.2 Inducing a band gap in bilayer graphene . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Magneto-transport in dual-gated bilayer graphene 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 5.2 Device presentation . . . . . . . . . . . . . . . . . . 5.3 Conductance in strong magnetic fields . . . . . . . 5.3.1 Unipolar regime . . . . . . . . . . . . . . . . 5.3.2 Bipolar regime . . . . . . . . . . . . . . . . 5.3.3 Comparison between model and experiment 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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24 24 26 27 29 30 30 30 32 33 39 45

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47 47 47 48 48 49 53 54 54 55 55 56 56 56 60

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61 61 61 64 64 66 67 68

6 Probing the Lifshitz transition of high quality BLG using large displacement fields 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Device presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Probing the Lifshitz transition using the quantum Hall effect . . . . . 6.4 Evolution of the position of the Landau level crossing as a function of the displacement field . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Possible observation of the van Hove singularity at higher displacement fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III Band gap and chirality

69 69 70 73 77 78 80

83

7 Introduction to Klein tunneling 7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Particles scattering on a potential step: introduction to the Klein paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Klein tunneling in graphene . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Sharp potential barriers . . . . . . . . . . . . . . . . . . . . 7.3.2 Smooth potential steps . . . . . . . . . . . . . . . . . . . . . 7.3.3 The true hallmark of Klein physics: the Berry phase . . . . 7.4 Conclusion: Fabry-P´erot interference in single-layer graphene . . . .

85 . 85 . . . . . .

86 87 87 89 90 92

8 Anti-Klein tunneling in bilayer graphene 93 8.1 Sharp barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.2 Observing Fabry-P´erot interference in an ideal ballistic BLG np0 n junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9 Inducing a cavity in a gapped BLG system: observation of FabryP´ erot interference 97 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.2 Motivation: Tunable transmission function . . . . . . . . . . . . . . . 98 9.3 Device characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 99 9.4 Observation of an oscillatory behavior of the transconductance signal in the pn0 p regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9.5 Magnetic field dependence . . . . . . . . . . . . . . . . . . . . . . . . 103 9.6 Bias dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

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9.7 9.8

Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . 106 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

10 Band gap and broken chirality 10.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Berry phase . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Berry phase in a two-level system . . . . . . . . . . 10.2.2 Berry phase in pristine single- and bilayer graphene 10.3 Berry phase in gapped single-layer and bilayer graphene . . 10.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

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109 . 109 . 109 . 110 . 112 . 113 . 115

IV Towards dual-gated bilayer graphene nanostructures 117 11 Defining nanostructures by backgate 11.1 Motivation . . . . . . . . . 11.2 Split gate geometry . . . . 11.3 Electrostatic limitations . 11.4 Improved design . . . . . . 11.5 Conclusion . . . . . . . . .

combining local top gates and the Si 119 . . . . . . . . . . . . . . . . . . . . . . . . 119 . . . . . . . . . . . . . . . . . . . . . . . . 120 . . . . . . . . . . . . . . . . . . . . . . . . 122 . . . . . . . . . . . . . . . . . . . . . . . . 125 . . . . . . . . . . . . . . . . . . . . . . . . 126

Conclusion

128

Appendices A A step by step process sheet to perform a safe and efficient AFM cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Setting up the workspace . . . . . . . . . . . . . . . . . . . . A.2 Measuring the deflection sensitivity s . . . . . . . . . . . . . A.3 Estimating the spring constant k . . . . . . . . . . . . . . . A.4 Evaluating the necessary DSP and force . . . . . . . . . . . A.5 Cleaning the device . . . . . . . . . . . . . . . . . . . . . . . A.6 End of the cleaning procedure . . . . . . . . . . . . . . . . . B Full Landau level spectrum at high displacement fields . . . . . . . C Additional information on the observation of Fabry-P´erot interference in dual-gated bilayer graphene . . . . . . . . . . . . . . . . . . . . . C.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132

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133 133 133 134 134 135 135 135

. 137 . 137

C.2

Experiment: density dependence of the oscillations at other backgate voltages . . . . . . . . . . . . . . . . . . . . . . . . . 141

Publications

142

Bibliography

145

Acknowledgements

153

Curriculum Vitae

156

ix

Lists of symbols physical constants -e< 0  o h = 2π~ kB

explanation electron charge dielectric permittivity vacuum dielectric constant Planck’s constant Boltzmann constant

xi

Abbreviation 2DEG AC AFM BLG BG h-BN CNP DC DG-BLG DOS E-beam FF FIRST FP FWHM HSQ PMMA PVA LL QD QPC RIE SLG TG

Explanation two-dimensional electron gas alternating current atomic force microscope bilayer graphene backgate hexagonal boron nitride charge neutrality point direct current dual-gated bilayer graphene density of states electron beam lithography filling factor frontiers in research, space and time or simply our clean room Fabry-P´erot full width at half maximum hydrogen silsesquioxane poly(methyl methacrylate) Poly Vinyle Alcohol Landau level quantum dot quantum point contact reactive ion etching single-layer graphene top gate

xii

Symbol L,W Lc B E D n α β γ EF Egap I V G R dG dVTG

vF kF λF `e m? µ ν ωc T u γ0 γ1 γ3

Explanation system size (length, width) cavity size magnetic field electric field displacement field density top gate relative lever arm backgate relative lever arm ratios of the relative lever arms Fermi energy energy gap current voltage conductance resistance normalized transconductance Fermi velocity Fermi wavenumber Fermi wavelength elastic mean free path effective electron mass of bilayer graphene mobility filling factor cyclotron frequency temperature or transmission layer asymmetry intralayer coupling interlayer coupling skew interlayer coupling

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Part I Introduction and motivation

Chapter 1 Introduction In this chapter, we introduce graphene and its unique properties. We explain why this material has raised such interest after its still recent discovery in 2004. We emphasize in particular the potential of graphene nanostructures, before explaining the limitations that were experimentally encountered. The latter will set the frame of this thesis. Faced with the limitations of etched graphene nanostructures, alternatives have to be developed. Bilayer graphene provides such an alternative with its electric field-tunable band gap. We will introduce the physical properties of bilayer graphene, which are studied intensively throughout this thesis.

1.1

Graphene - the first true two-dimensional material

Graphene, a single atom thick layer of carbon atoms arranged in a honeycomb lattice, was first studied theoretically in 1947 by P.R. Wallace [1], as the simplest building block of graphite. However, the term “graphene” only made its appearance in 1987 [2]. Since the 1970s, efforts have been made to grow such a single layer of carbon atoms, but only on a late Friday night of 2003, at the University of Manchester, Andrei Geim and Konstantin Novoselov came up with an idea to extract graphene. This idea is based on a process that was known for many years in the surface science community: using scotch tape to peel off the first layers of a thick graphite flake in order to generate cleaner surfaces. The brilliant input of Geim and Novoselov was to realize that what was left on the tape was actually much thinner that anything that could be obtained by reduction of a crystal. They therefore started looking at the graphite residues left on the tape and quickly identified single-layer flakes [3]. The “scotch tape technique” was born and with it, a completely new field of experimental physics. In 2010, Geim and Novoselov were awarded the Nobel Prize for this discovery and for the first experimental investigations carried out on this unprecedented material.

3

Chapter 1. Introduction (a)

(b)

y

γ0

ky

E

x

kx

a1 a2

A

B

K

a

K’

Figure 1.1: (a) Graphene lattice structure. A and B are the two non-equivalent sites which form the unit cell. This unit cell has a rhombus shape, spanned by the two lattice vectors a1 and a2 . The distance between carbon atoms belonging to the same sublattice is a. (b) Band structure of graphene calculated in the tight-binding model within the nearest-neighbor hopping approximation. Conduction and valence bands touch at the K and K 0 points.

In the following, we will discuss the astonishing properties of graphene. To do so, we will start by looking at its band structure, which gives rise to its unusual electronic properties.

1.1.1

Single-layer graphene band structure: the tight-binding approach

Single-layer graphene consists of a layer of carbon atoms, arranged in a honeycomb pattern. Its unit cell is defined by two carbon atoms sitting on two sites that we label A and B and that are non-equivalent. The unit cell has the shape of a rhombus and the distance between two unit cells is a = 2.46 ˚ A [4]. The lattice structure is sketched in Fig. 1.1(a). In this figure, the primitive lattice vectors a1 and a2 are shown. They are defined as: √ a1 = ( a2 , a 2 3√)T , (1.1) a2 = ( a2 , − a 2 3 )T . The relation between the lattice constant √ and the shortest distance between carbon atoms aAB is: a = |a1 | = |a2 | = aAB 3. The reciprocal lattice vectors therefore are: √ )T , b1 = ( 2π , 2π a a 3 √ )T . b2 = ( 2π , − a2π a 3

(1.2)

These vectors define a hexagonal lattice, with two non-equivalent points, K and

4

1.1. Graphene - the first true two-dimensional material

K 0 , which are called Dirac points and form two valleys. Graphene is sp2 hybridized. Each carbon atom is covalently bonded to its three nearest neighbors, forming σ-bonds in the xy-plane. The corresponding bands are located at high energies. On the other hand, the remaining pz electron leads to the formation of a half-filled π-band, which governs the electronic properties of graphene by enabling its conductivity. Thus, the band structure of graphene can be calculated taking into account one 2pz orbital per atomic site, i.e. two per unit cell. In 1947, P.R. Wallace calculated the energy bands of such a system in a tightbinding calculation [1] and found, in the next-neighbor hopping approximation: s √ kx a aky 3 kx a cos + 4 cos2 (1.3) E(kx , ky ) = ±γ0 1 + 4 cos 2 2 2 Here, the atomic sites A and B are coupled by the intralayer coupling γ0 which represents the in-plane hopping to the nearest neighbors (γ0 = 3.16 meV [5]). The plus and minus signs refer to the π ∗ - and the π-band, which respectively form the conduction and valence bands. The tight-binding model of single-layer graphene has been presented in different reviews that we recommend for further details [4, 6–9]. The result of equation (1.3) is shown in Fig. 1.1(b). As we can see, conduction and valence bands are symmetric in the nearest-neighbor approximation. In the center of the Brillouin zone, the two bands are far apart (≈ 20 eV) while they directly touch at the K- and K 0 -points. This is the reason why graphene is called a zero-gap semiconductor. In the vicinity of the K-points and in the low energy range (E < 1 eV), graphene charge carriers can be described by the Dirac-like Hamiltonian: "

H = ~vF

#

0 kx0 − iky0 , 0 kx0 + iky0

(1.4)

where we redefined the wavevector in order to be centered at a K-point (k’ = K−k). This Hamiltonian operates on the two-component wavefunction (ψA , ψB )T . In this √ aγ0 3 equation, vF is the Fermi velocity (vF = 2~ ≈ 106 m/s). This describes the following dispersion relation: E(k’) = ±~vF |k’|.

(1.5)

As we can see, the dispersion relation is linear. Particles in graphene therefore resemble massless relativistic particles. Hence, graphene is a solid state system in which particles obey relativistic physics, with the only difference that these particles move with the Fermi velocity, vF , instead of the speed of light.

1.1.2

Electronic properties of single-layer graphene

As demonstrated in the preceding subsection, the charge carriers in graphene behave as massless Dirac Fermions. They can be described by a two-component wavefunction which obeys: − i~vF σ · ∇ψ(r) = Eψ(r), (1.6)

5

Chapter 1. Introduction

where σ = (σx , σy )T is the vector of Pauli matrices. Here, the analogy with quantum electrodynamics can be made by realizing that the two sublattices A and B are playing the role of the spin up and down and that σ is not the spin but the pseudospin. In graphene, the direction of motion is therefore coupled to the pseudospin. This indicates that charge carriers in graphene are chiral. Indeed, the wavevector and the pseudospin are oriented in the same direction for electrons (chirality +1) whereas they are anti-parallel for holes (chirality −1). The chirality of charge carriers has important consequences on transport. It is responsible for a Berry phase of π [10, 11] and the suppression of backward scattering [12, 13]. These two effects are related to the Klein paradox and will be explained in more detail in Part III. The absence of backscattering represents one of the strengths of the material, by enabling room temperature mobilities much higher than in other materials used in the semiconductor industry (more than 10 times higher than in Si). However, when considering the fact that graphene is also a gapless material, the suppression of backscattering poses a large drawback for possible electronic applications. It prohibits the fabrication of transistors with large on-off ratios and prevents any confinement of charge carriers by electrostatic barriers. Alternatives have to be found to overcome this limitation. Cutting graphene into narrow ribbons is one of them and will be the topic of the next section.

1.2

Single-layer graphene nanostructures

The band structure of graphene allows for continuous tuning of the Fermi energy from the valence band to the conduction band. It was however observed that patterning graphene into narrow ribbons results in the opening of a so-called transport gap, within which transport is strongly suppressed [14–16]. These graphene constrictions were therefore adopted as tunneling barriers. This observation opened the door to the field of graphene nanostructures. This is of particular interest for potential applications in quantum information processing, which rely on the use of quantum dots (QDs). While GaAs-based QDs suffer from decoherence, originating from spin-orbit (SO) and hyperfine interactions [17– 22], carbon-based materials should offer an ideal alternative. Carbon has a small atomic mass. Hence SO effects should be strongly reduced and, due to the expected predominance of 12 C atoms, the hyperfine interaction should be weaker as well [23]. Graphene was therefore considered as a suitable candidate for engineering spin qubits and theoretical proposals for designing such structures have been made [24]. On the experimental side however, engineering a graphene quantum dot is not straightforward. The edge and bulk disorder, responsible for the localized states leading to the suppressed transport in the constrictions, have a strong effect on the patterned dots as well [25], leading to a multiple-dot behavior. It was further

6

1.3. The bilayer graphene alternative

demonstrated that reducing bulk disorder by using better substrates is not enough to solve this issue [26], indicating the relevant role of the edge disorder [27, 28]. To solve this issue, alternative ways of patterning graphene nanostructures are needed.

1.3

The bilayer graphene alternative

Bilayer graphene consists of two coupled layers of graphene on top of each other. In this thesis, we consider Bernal-stacked bilayer graphene, where one sheet is rotated by 60◦ compared to the other one. This material is particularly appealing because of its electronic tunability. Not only the Fermi level can be electrostatically tuned through the electronic spectrum as in single-layer graphene (SLG), but the band structure itself can be strongly influenced by transverse electric fields. By inducing an asymmetry between the top and the bottom layer, a band gap can be opened [29–32]. This is shown schematically in Fig. 1.2. The size of this band gap depends on the strength of the induced asymmetry u. To induce such an asymmetry, one can for example use chemical doping [33] or, for better control, external gates [34]. The latter means that in a similar way as in GaAs nanostructures, gates can be used to confine and guide electrons [35]. However, top- and bottom gates are required to control the induced asymmetry of BLG. We call such a device a “dual-gated” device. Such a technology would allow the comparison between etched graphene nanostructures and gate-defined bilayer graphene nanostructures. This would furthermore enable us to conclude whether electrostatically defined edges constitute an improvement compared to etched edges.

E (eV)

0.3

u = f(D)

Figure 1.2: Band structure of bilayer graphene (adapted from Ref. [31]). The dashed lines show the band structure of pristine bilayer graphene, whereas the solid lines illustrate the situation where an asymmetry is induced between the top- and bottom layers of the flake, resulting in the opening of a band gap.

-0.3

k

7

Chapter 1. Introduction

1.4

About this work

In this thesis, we investigate transport through dual-gated bilayer graphene devices. In previous studies [34, 36–38] carried out on such devices, the opened band gaps were far from perfect: localizations under the gates lead to hopping transport, preventing efficient confinement. However, confinement was still achieved in some rare high quality devices [35, 39]. The main motivation for this thesis is to proceed to obtaining and understanding high quality bilayer graphene devices for potential applications to nanostructure patterning. To do so, we focused on investigating the process of the band gap opening and on better understanding how the presence of the gap influences the transport properties. This thesis is composed of three main parts: In Part II, we will present how the band structure of bilayer graphene can be probed and controlled. This band structure will be presented in detail and the influence of external parameters will be emphasized. Subsequently, we will highlight that obtaining high quality bilayer graphene flakes is a requirement, in order to probe the smallest features of the band structure, such as the Lifshitz transition (a change in the topology of the Fermi contour, as a function of energy). We will therefore describe the fabrication process and most particularly the different cleaning procedures required for obtaining high quality backgated bilayer graphene on hexagonal boron nitride substrates. The quality will be confirmed by intermediate transport measurements. The fabrication process will then be continued to produce dual-gated bilayer graphene devices. In order to assess the possibility of opening a band gap in bilayer graphene, uniform top gates have been designed. Such a top gate, called a barrier, covers the full width of the graphene flake, such that when a high voltage asymmetry is applied between the top- and the backgate and a band gap is opened, the current from source to drain is suppressed. Basic experiments have been carried out using these devices, with and without magnetic fields, which will be presented here. Finally, we will demonstrate that the Lifshitz transition of bilayer graphene can be probed using the quantum Hall effect as a tool to identify the different degeneracies of the band structure. Our result is the first experimental observation of this change of topology as a function of energy in a gapped bilayer graphene system. Part III carries on with the investigation of these barrier-like structures, but addresses the question of the effects of the presence of a band gap on the chiral properties of bilayer graphene. Anti-Klein tunneling is predicted to occur in pristine bilayer graphene and the Berry phase is expected to be 2π. We will demonstrate that opening a band gap breaks these two properties and allows for tunability of the transmission through a pn interface and of the Berry phase with the displacement field strength. Most interestingly, we will show that, depending on the gap size and the energy, Klein

8

1.5. Impact of this work

tunneling can be recovered, together with a Berry phase of π, as in single-layer graphene. We further conclude that these observations are related to the breaking of the chirality due to the existence of the band gap. Finally, in Part IV, we will present preliminary results on a dual-gated device with split top gates, expected to induce a one-dimensional confinement, similar to a quantum point contact. We will discuss the limitations of the designed geometry and will finally propose an improved version of the device, which allows for better control of the induced channel.

1.5

Impact of this work

Changing the topology of an object can significantly change its properties. A wellknown example is changing a mug without handle (topology of a sphere) into one with an arched handle (topology of a doughnut). In electronic materials it is the more abstract topology of constant-energy surfaces for electrons that determines their potential for functional uses. Usually, objects of one topological class cannot be smoothly transformed into those of another class. Bilayer graphene turns out to be a very special case: electrical voltages allow us to change the topology of the constant energy surfaces with external knobs. The unique opportunity offered by this material is demonstrated all along this thesis, and the consequences of this high tunability are investigated through transport measurements. In particular, we report here on the first observation of the Lifshitz transition in a gapped bilayer graphene device and of Fabry-P´erot interference in such a gapped system. Our investigation of Fabry-P´erot interference allowed us to consider in detail the role played by the Berry phase and revealed the importance of this parameter in the case of gapped bilayer graphene. The tunability of the Berry phase enabled us to gain further understanding of the consequences of the opening of the band gap on the chirality and on the pseudospin. In summary, this thesis investigates the effects on transport of band gap engineering in bilayer graphene. Our conclusions allow for a better understanding of the system and therefore prepare the way for future better controlled gate-defined nanostructures in bilayer graphene.

9

Part II Probing and controlling the band structure of bilayer graphene

Chapter 2 Electronic properties of bilayer graphene In this chapter, we introduce the bilayer graphene system and its band structure. We present the influence of each coupling parameter between carbon atoms on the energy dispersion and further explain how this dispersion can be influenced by external parameters, such as strain or displacement fields. The following introduction to the band structure of bilayer graphene has been partially presented in the review (currently under consideration):

Tunable Fermi surface topology and Lifshitz transition in bilayer graphene A. Varlet, M. Mucha-Kruczy´ nski, D. Bischoff, P. Simonet, K. Watanabe, T. Taniguchi, V. Fal’ko, T. Ihn and K. Ensslin Submitted to Synthetic Metals, Special Issue on Graphene (2015)

2.1

Introduction

Bilayer graphene possesses charge carriers that can be described as massive chiral quasiparticles. An interesting aspect of this material lies in its high tunability: not only the Fermi energy can be tuned within a wide range of energy using electric fields, as in SLG, but the shape of the band structure itself can be changed under the influence of external parameters. It was indeed predicted in Ref. [29] that inducing an asymmetry between the on-site energies of each of the two layers would open a band gap. Furthermore, the band structure presents another interesting feature: close to the charge neutrality point, a change in the Fermi contour topology occurs. This is called a Lifshitz transition. Under the influence of strain or of displacement fields, this phenomenon can be tuned as well. In this chapter, we will try to understand the role of each coupling parameter on the energy dispersion, as well as the role played by external parameters.

13

Chapter 2. Electronic properties of bilayer graphene

2.2

Bilayer graphene: a step by step introduction to the band structure

Bilayer graphene consists of a stack of two single-layer graphene sheets that are coupled. We consider here a Bernal-stacked bilayer graphene system, where half the atoms of one layer lie on top of half of the atoms in the second layer, as shown in Fig. 2.1(a). The unit cell now contains four sites, two of them being on the bottom layer, labeled 1, and the two others on the top layer, labeled 2: A1 , B1 , A2 , B2 . The full description of the tight-binding approach can be found in Refs. [7, 40–42]. Close to the K-points, the bilayer system can be described by an effective four-band Hamiltonian, in the basis A1 , B1 , A2 , B2 [31, 42]:

H=

 u 2  vπ    0

v3 π †

vπ † u 2

γ1 0

 0 v3 π  + ipy   π=p √x γ1 0   3 aγ0 v = 2√ ~ ∼ 106 m/s  , − u2 vπ †     v = 3 aγ3 ∼ 0.1 v 3 2 ~ vπ − u2 

(2.1)

In these equations, the γi terms represent the tight-binding parameters, defined based on the notation of the Slonczewski-Weiss-McClure model for graphite [5, 43– 45]. They illustrate the different couplings between different pairs of orbitals: γ0 is the coupling between neighboring orbitals within the same layer (γ0 = γA1 B1 = γA2 B2 ); γ1 represents the coupling between the orbitals belonging to the two sites directly sitting on top of each other which form dimers (γ1 = γA2 B1 ); γ3 couples the two non-dimer sites (A1 and B2 ) which are not vertically aligned, but still contains an in-plane component. For the purpose of this thesis, we will leave out higher order couplings, such as γ4 , coupling the A1 and the A2 sites, as well as the B1 and B2 sites, as their influence on the band structure is small. The strength of these couplings has been determined by the use of infrared spectroscopy [46]: γ0 = 3.16 meV, γ1 = 0.381 meV, γ3 = 0.38 meV.

(2.2)

These parameters give rise √ to effective velocity √ terms, directly proportional to the 3 aγ0 coupling parameters (v = 2 ~ and v3 = 23 aγ~3 ). The parameter u appears in the Hamiltonian when different on-site energies are applied to the two layers and is called the interlayer asymmetry. Experimentally this can be accomplished by using a top- and a backgate and generating a vertical electric field. In the following, we will focus on the effect of each of these terms on the band structure of bilayer graphene.

14

2.2. Bilayer graphene: a step by step introduction to the band structure

(a)

(b) 0,8

γ0

γ3

B2

γ1

A1

E [eV]

A2

B1 γ0 (c) 0,8

-0,8 (d) -45

E [eV]

E [meV]

py

∆u = 1.6meV

py

py -0,8

px

px

-55

px

Figure 2.1: (a) Bernal-stacked bilayer graphene: two layers of graphene (labeled 1 and 2) sit on top of each other. Each layer is made of carbon atoms, arranged in a honeycomb lattice. The unit cell defines four non-equivalent sites, labeled A1 , B1 , A2 and B2 . The atoms A2 and B1 are sitting directly on top of each other, forming dimers. The coupling terms between the atoms are highlighted with arrows. (b) Low energy band structure of bilayer graphene in the proximity of the K-point, calculated taking into account only the in-plane coupling γ0 and the interlayer coupling between dimers γ1 . It exhibits two parabolic bands that are touching at the K-point and two split bands. (c) Effect of the interlayer asymmetry on the band structure: a band gap is opened (here u is set to 100 meV). (d) Zoom on the top of the valence band: the so-called “Mexican hat” shape is visible (upside-down, for the valence band). For u = 100 meV, the amplitude of the kink is ∆u = 1.6 meV.

15

Chapter 2. Electronic properties of bilayer graphene

2.2.1

Influence of the intralayer coupling γ0 and the interlayer coupling γ1

To understand the role of each term of the Hamiltonian H described in Eq. (2.1), we can start by neglecting the influence of the skew interlayer hopping γ3 and of the interlayer asymmetry u by setting them to zero. Only the terms containing v (∝ γ0 ) and γ1 are left. Solving this Hamiltonian provides the band structure displayed in Fig. 2.1(b). One layer of graphene, in which the orbitals are coupled only through γ0 , gives rise to a linear dispersion around the K-points. As we can see in Fig. 2.1(b), introducing γ1 further transforms this linear energy-momentum relation into a parabolic one: the charge carriers are not massless anymore [29, 47]. The mass term is given by: m∗ = γ1 /2v 2 = 0.034 m0 , where m0 is the free electron mass. The other effect of γ1 is the formation of two split bands, resulting from the dimers created out of the A2 − B1 orbitals [48]. These bands are positioned at an energy E = ±γ1 from the Fermi energy. Such massive chiral fermions exhibit a different quantum Hall effect behavior than the massless Dirac fermions of single-layer graphene. In bilayer graphene, instead of being fourfold degenerate as in single-layer graphene, the lowest Landau level exhibits an additional degeneracy due to the doubled chirality (J = 2) [29, 47]: the n = 0 and n = 1 levels are degenerate and sit at the same energy in both valleys. This eightfold degeneracy leads to a step of 8 e2 /h in the measured Hall conductance around the charge neutrality point. The higher LLs are still fourfold degenerate due to spin and valley degeneracies, and the LL spectrum evolves as: q

n = ±~ωc n(n − 1).

(2.3)

The magnetic field dependence of the LLs therefore leads to an equidistant conductance staircase, with the exception of the larger conductance step across the charge neutrality. The LL spectrum is shown in Fig. 2.2 in red. In high quality bilayer graphene, strong electron-electron interaction can however lift these degeneracies, as will be discussed later. Another consequence of this unusual quantization at zero-energy is that it gives rise to a Berry phase of 2π [29, 47], instead of π for single-layer graphene [13, 49]. The Berry phase of pristine bilayer graphene as well as the one of gapped bilayer graphene will be studied in detail in Chapter 10.

2.2.2

Inducing an asymmetry u: the Mexican hat

One of the most interesting things about bilayer graphene as a material is the fact that one could, in principle, address each layer independently. To do so, one can tune the bottom layer to a potential u/2 and the top layer to a potential −u/2: this way, an asymmetry of amplitude u is induced between the two layers. To induce this asymmetry, chemical doping [33] or external gates [34] can be used. The relation

16

2.2. Bilayer graphene: a step by step introduction to the band structure

30 20

E [meV]

10 0 -10 -20 -30 0

0.2

0.4

0.8

0.6

Figure 2.2: Comparison of the Landau level spectra of gapless bilayer graphene. In red, the calculation is done without taking into account the γ3 coupling (solid red lines) and in black, the corresponding spectrum, with γ3 , is displayed. The only noticeable difference occurs below B = 0.1 T. This figure is a courtesy from nski. 1 M. Mucha-Kruczy´

B [T]

between u and the applied voltages has been investigated in Refs. [30, 50]. This induced asymmetry results in the opening of a band gap, as shown in Fig. 2.1(c) [29–32]. The shape of this gap is particular because of the presence of the two split bands: they induce a kink at the K-points, which gives rise to the nickname “Mexican hat” for the shape of the gapped region. This is shown in Fig. 2.1(d): for an induced asymmetry u = 100 meV, the amplitude of this kink is ∆u = 1.6 meV. This asymmetry also has consequences for the magnetic field behavior. In symmetric bilayer graphene, as mentioned in the previous part, there is an extra degeneracy for the zero-energy Landau levels. Applying an interlayer asymmetry lifts this degeneracy. This is a consequence of the distribution of the electron wave-function between the lattice sites [29]: electronic states in the valley K are formed by orbitals that sit on the A1 –sites, while in the valley K 0 , these states sit on the B2 sublattice. Thus, since these sites are set to different potentials, the levels split in energy. Opening a band gap has also consequences for the higher LLs: they become weakly valley-split. Applying different on-site energies therefore lifts the valley degeneracy. However, in experiments, even though the lifting of all the degeneracies has been observed for the lowest LL [37], temperature and disorder broadening usually hinder the observation of the splitting for the highest LLs. The opening of the band gap induces a change for the Berry phase: it is not trivially a multiple of 2π anymore, but can take any value between 0 and 2π [51]. The Berry phase will be discussed in more detail in the Part III of this thesis.

17

Chapter 2. Electronic properties of bilayer graphene

3

(b)

0

E [meV]

(a)

py py

-3 px

px

-20

Figure 2.3: (a) Effect of the skew interlayer coupling γ3 on the band structure, close to the K-point (the K 0 -point is not shown but exhibits the same shape, only rotated by π/3 in the momentum plane). On a large energy scale, the overall band structure shape is preserved. However, in the range of a few meV, a dramatic change is observed: the bands do not touch at one single point anymore, but the low energy dispersion exhibits four Dirac cones. (b) The evolution of the shape of the Fermi contour as a function of energy is highlighted for the valence band with constant energy contours: one can notice that going towards the charge neutrality, the contour gets broken, from one unique contour (solid back line) into four pockets (black dashed lines), illustrating the Lifshitz transition.

2.2.3

Influence of the skew interlayer hopping γ3 : the trigonal warping of the band structure

Even though γ0 and γ1 have the strongest effect on the band structure, the influence of the skew interlayer hopping γ3 cannot be neglected anymore when entering the low energy range. If included in the Hamiltonian, with a zero interlayer asymmetry, the low energy dispersion is modified as shown in Fig. 2.3(a): the band structure undergoes a trigonal deformation called “trigonal warping” [29, 31]. The two lowest bands do not only touch at the K-points, but meet now, at zero-energy, at the center of four “mini” Dirac cones: the three outermost cones are slightly asymmetric (with respect to their local Dirac point), and carry a Berry phase of +π, while the central one is perfectly symmetric and carries a Berry phase of −π, conserving the overall 2π Berry phase known for bilayer graphene. In this range, the linear term appearing in the Hamiltonian becomes as relevant as the quadratic one and therefore, for small momenta, the shape and the symmetry of the band structure is dramatically changed. The consideration of γ3 leads to two important consequences. Firstly, it induces the trigonal deformation of the Fermi contour, which previously exhibited a circular shape. This deformation takes place along the momentum directions

18

2.2. Bilayer graphene: a step by step introduction to the band structure

φ = 0, 2π/3, 4π/3 at the K-point and the same with a π/3 offset at the K 0 -point (not shown in the figure). In the second place, the topology now changes as a function of energy: if one draws the shape of the Fermi contour at constant energy, as done in Fig. 2.3(b) for the valence band, one goes from a situation with one unique trigonally shaped contour at positive or negative energy to a situation, beγ1 v32 tween ELT = ± 4v 2 = ±1 meV [52], where the Fermi contour is broken into four distinct pockets [shown in bright yellow in Fig. 2.3(b)]. This change of topology as a function of energy is called a Lifshitz transition [53]. The energy at which this phenomenon occurs is small compared to the usually observed disorder in bilayer graphene. This transition is therefore hard to access experimentally: the quality of the graphene device has to be astonishing. How such high sample qualities can be obtained will be discussed in Chapter 3. This topology change leads to a change of the degeneracies and therefore to a different quantum Hall effect quantization than the one previously presented. The high magnetic field behavior is similar to the one discussed in Section 2.2.1, without any skew interlayer hopping included: the evolution of Landau levels (LLs) as a function of magnetic field is linear and exhibits an eightfold degenerate zero-energy LL and fourfold degenerate higher LLs. The corresponding energy spectrum is illustrated in Fig. 2.2 with black lines. These black lines are to be compared with the red lines which correspond to the case where no skew interlayer hopping is included: one can clearly see that the difference between red and black lines is barely noticeable at high magnetic fields, with most of the lines overlapping. At low magnetic fields, B ≈ 0.1 T, the inverse of the magnetic length λB is comparable with the distance in momentum space between the Dirac points present in the low-energy electronic dispersion due to the skew interlayer coupling γ3 . This leads to a “magnetic breakdown” [54, 55] and the n = 2 Landau levels merge with the zero-energy LL. As a result, at lower magnetic fields the zero-energy LL has a 16-fold degeneracy. In high quality samples, one might therefore expect filling factors ν = ±8 to be the most robust at low magnetic fields.

2.2.4

Effect of external strain on the electronic dispersion

As mentioned earlier, its high tunability makes bilayer graphene a unique material. External parameters are able to influence the low-energy dispersion of bilayer graphene. Strain for example has interesting effects on the above-mentioned four Dirac cones close to the charge neutrality point [56–58]. The influence of strain was not experimentally investigated during this thesis. However, since the Lifshitz transition is a major topic in this thesis, we will qualitatively describe the effects of strain on the low-energy band structure for completeness. In Ref. [56–58], the effect of homogeneous in-plane deformation and shear was considered. Homogeneous in-plane deformation implies that the vertical alignment between top and bottom layers is conserved (i.e. A2 still lies on top of B1 ), whereas shear allows the two layers to be misaligned. The combination of these two defor-

19

Chapter 2. Electronic properties of bilayer graphene

Figure 2.4: The low-energy band structures of differently strained bilayer graphene systems are shown. In the center (w = 0), the case with no strain is shown. For the band structures shown in the top row, the corresponding Landau level spectra are displayed underneath. The expected degeneracies are highlighted in colored circles for the lowest LLs. This figure is by courtesy of Marcin Mucha-Kruczy´ nski who performed the calculations.

mations is responsible for making the hopping parameters which have an in-plane component (γ0 and γ3 ) direction-dependent. Such deformations affect the low-energy dispersion of bilayer graphene as illustrated in Fig. 2.4 (w is a parameter describing how the hopping parameters are modified by the application of strain; the bigger |w|, the larger the strain). In Fig. 2.4, one can see how strain is responsible for the shifting and/or merging of the low energy Dirac cones. The Dirac cones are first shifted and can then collide and merge with increased strain, as illustrated with the most left and right dispersion relations. The LL spectrum of bilayer graphene is modified accordingly, as shown below each electronic spectrum. Both, for small and large strain, the highest range of magnetic field displayed here q shows that the LL spectrum is approximately described by the sequence  ≈ ±~ωc n(n − 1) of four-fold degenerate LLs at non-zero energy (n ≥ 2) and an eight-fold degenerate LL at  = 0 (n = 0, 1) [29], as in the pristine case. The low magnetic field behavior is however different according to the strength of the applied strain. The different expected degeneracies are highlighted in colored circles for the lowest LLs in Fig. 2.4. The quantum Hall effect can therefore be used as a tool to detect degeneracies and compare gap sizes, revealing the main characteristic features of the low-energy dispersion of strained bilayer graphene. Up to this day, experimental confirmation of this prediction by means of strain engineering in bilayer graphene has not been accomplished. Alternative options have to be found, to better control the low energy dispersion and the Lifshitz transition.

20

2.2. Bilayer graphene: a step by step introduction to the band structure

(c)

E [meV]

(b)-46

-45 E [meV]

(a) 80

py -52

py -80

px

py

px

px

-70

Figure 2.5: (a) Band structure obtained while applying an interlayer asymmetry u = 100 meV between the two graphene sheets and taking into account γ3 : close to the band gap, the trigonal distortion is still visible. (b) Magnifying the top of the valence band, one can clearly observe that there are three outer maxima and a minimum in the center. The energy separating them is around 3 meV for u = 100 meV. (c) Constant energy contours taken from (b) show that the continuous Fermi contour (dashed black lines) is broken as a function of energy, this time into three pockets (bright yellow, highlighted with solid black lines). This illustrates the Lifshitz transition in a gapped bilayer graphene system.

2.2.5

Inducing an asymmetry u with γ3

Accessing the Lifshitz transition of pristine bilayer graphene experimentally is challenging. However, by using the possibility of applying a vertical electric field, one can make this phenomenon more convenient to observe. In the Hamiltonian expressed in Eq. (2.1), one can now set the interlayer asymmetry to u = 100 meV and investigate the consequences on the band structure. As shown in Fig. 2.5(a), the band gap has opened in a similar way as in the case where no skew interlayer hopping was included. However, already on this scale of ±80 meV, one can see that the trigonal perturbation is still present. Zooming at the top of the valence band, as done in Fig. 2.5(b), we see that the opening of the band gap has a strong effect on our previous four Dirac cones: only three peaks are left and they do not exhibit a linear dispersion anymore. The central peak has been pushed down by the γ1 parameter, by the same process as for the “Mexican hat”. The energy contours however show the same kind of physics as a function of energy. Fig. 2.5(c) shows these contours as a function of energy for the valence band: again, at high negative energies, the band structure exhibits one unique and continuous energy contour (black dashed lines). While rising the Fermi level towards the gap, the contour breaks into three pockets when going across the Lifshitz transition [53] [in bright yellow in Fig. 2.5(c), solid black contours]. This time, however, the energy scale of this transition is increased: the band gap opening led to a stretching of the three outer peaks and the difference between the three maxima at the top of 0 the valence band and the minimum in the center is now around ELT = 3 meV, as illustrated in Fig. 2.5(b). One can therefore conclude from this estimate that the experimental observation of the Lifshitz transition is facilitated by the opening of a band gap and that the larger the gap, the more enhanced the experimental visibility

21

Chapter 2. Electronic properties of bilayer graphene

of this effect. This new topology affects the quantum Hall effect behavior. Instead of the extra fourfold symmetry arising from the four Dirac cones, the triplet at the top of the valence band gives rise to a threefold degeneracy [59]. Due to the opening of the band gap, the valley degeneracy is lifted. Thus the conductance plateaus that would remain quantized until the lowest range of magnetic fields are expected to correspond to ν = ±6 and, because of exchange interaction, ν = ±3. The full LL spectrum will be discussed in more detail in Chapter 6.

2.3

Conclusion

In this chapter, we described the role of each coupling parameter on the band structure of bilayer graphene and have revealed the importance of the skew interlayer coupling, which gives rise to a Lifshitz transition close to the charge neutrality point. We have discussed why this transition is challenging to observe experimentally, as it occurs at small energies compared to the energy scale on which disorder dominates transport in bilayer graphene samples. Using high quality samples is therefore a requirement to experimentally access this phenomenon. In the following chapter we will describe how such high quality samples can be obtained. Combining high sample quality and large displacement fields offers another way to facilitate the experimental access to the Lifshitz transition. Under the influence of a transverse electric field, the triplet at the top of the valence band is stretched in energy and the Lifshitz transition becomes therefore accessible on a more convenient range of energies. The technology allowing for this investigation will be presented in Chapter 4.

22

Chapter 3 Backgated bilayer graphene devices In this chapter, we describe how high quality graphene can be obtained. We will start by explaining the fabrication process that is used to transfer a (bilayer) graphene flake on a hexagonal boron nitride substrate. The etching and contacting steps will be mentioned as well. We will point out the fact that the whole process brings resist residues on the surface of graphene and we will consequently explain which techniques are available to obtain surfaces as clean as possible.

3.1

Introduction

In the early graphene-based transport experiments, graphene was deposited on Si/SiO2 substrates [3, 10, 11], which offer both good optical contrast to locate graphene and a way to electrostatically tune the Fermi energy using doped silicon as a backgate. However, it was rapidly found that this substrate, as convenient as it is, is a non-negligible source of disorder: the surface is rough, contains impurities, charge traps, as well as charged surface states, which all limit the electronic performance of the resulting devices [60–63]. To circumvent this issue, two main paths have been investigated. The first one consists in suspending graphene, i.e. by getting rid of the substrate. Until recently, this yielded the best quality graphene ever achieved [64, 65], with the observation of high mobilities and of clear broken symmetry states in the quantum Hall regime (for example in [37]). This technique, however, results in fragile devices with complicated architectures if additional electric gates are required. It also limits the strength of vertical electric fields that one can apply to graphene before the structure collapses. At the same time, this approach could potentially be used to induce controlled strain in the suspended graphene membrane [66]. As we have seen in the previous chapter, controlled strain can be interesting, as it is another

23

Chapter 3. Backgated bilayer graphene devices

parameter that allows tuning the low energy dispersion of bilayer graphene and therefore to influence the Lifshitz transition [56, 67]. In parallel, another option was developed by Dean et al. [68]: using hexagonal boron nitride (h-BN) as a substrate. This material is both a good substrate (low roughness and free of charge traps) and a good dielectric (it is very robust and allows for the application of high electric fields to the graphene). The first process developed by Dean et al., called the dry transfer technique, showed a clear enhancement of the mobilities [68] (values will be shown in detail later in the chapter). This technique, pioneered by the Kim group at Columbia University, was brought to our group by Dr. Susanne Dr¨oscher and Dr. Dominik Bischoff and perfected through the years, thanks to the combined efforts of the whole graphene team (especially Dr. Cl´ement Barraud, Tobias Kr¨ahenmann, Pauline Simonet and Michelle Gruner). In this chapter, we will describe the details of this transfer process. We will furthermore point out that, at the end of the fabrication process, even though h-BN brings non-negligible improvements, transport still suffers from the residues lying on the surface of the (bilayer) graphene flake. We will discuss the possible cleaning processes and illustrate their efficiency by transport measurements performed before and after the cleaning step.

3.2

Fabrication of backgated bilayer graphene devices

Our goal is to have a bilayer graphene flake transferred on top of a h-BN flake. Two preparations are done in parallel: on one hand, h-BN needs to be deposited on Si/SiO2 -chips and, on the other hand, graphene will be prepared on a transferable substrate. Once both flakes are found, the transfer can be performed and the graphene flake can be etched in the desired shape and contacted.

3.2.1

Preparation of h-BN flakes on Si/SiO2 : substrate

3.2.1.1

Si/SiO2 chips preparation

In a first step, Si/SiO2 -chips with markers are processed from commercial Si/SiO2 wafers. The thickness of the SiO2 layer is 285 nm. The thickness is an important parameter: it provides us with an optimized contrast to later identify the graphene flakes [69]. HDMS is first spin-coated on the wafer to help the photolithography resist to stick. Then ma-N 1405 is spin-coated and the wafer is baked at 100 ◦ C. Photolithography is then performed, using a mask which consists of a repetition of the pattern shown in Fig. 3.1(a). Putting the wafer into ma-D 533 allows to develop the pattern and the opened trenches are then further cleaned with a short oxygen plasma treatment done in a plasma asher. Once this is done, the metal can be deposited by evaporation (5 nm Ti and 50 nm Au) and the lift-off can be performed

24

3.2. Fabrication of backgated bilayer graphene devices

(a)

(b)

(d)

10µm

(c)

1mm Figure 3.1: (a) Schematic of the optical lithography mask used for the preparation of Si/SiO2 -chips. (b) Zoom of one cell. (c) Optical microscope picture of one cell, with some h-BN flakes (from dark blue to yellow) exfoliated on the surface. (d) Zoom of one of the h-BN flakes (light blue) present in (c).

in warm acetone (the use of the ultra-sound bath and of the pipette is necessary). The wafer is next cut in small chips of 0.7 × 0.7 cm2 size [each chip is patterned like in Fig. 3.1(a)]. First, a layer of protective resist is spin-coated on top and the wafer is diced in chips with the help of a dicing saw. In Fig. 3.1(b) and (c), zooms of one cell of Fig. 3.1(a) are shown. The purple color of the surface is due to the oxide layer thickness. The yellow pads are the photolithographically-defined Ti/Au-markers lying on the SiO2 surface. 3.2.1.2

Deposition of h-BN on Si/SiO2 -chips

Before starting the deposition, the protective resist from the previous step needs to be removed. The chips are cleaned in warm acetone for 5 minutes and put 5 more minutes in isopropanol, applying ultra-sound. The chips are then further cleaned in the UVOCS (UV Ozone Cleaning System). The chips are now ready to be used for deposition. The deposition is done in a “grey” room, in order to prevent contamination of the cleanroom. Small hexagonal boron nitride crystals are deposited on a blue tape piece, which is then folded many times until the full surface of the blue tape looks homogeneous. In the meantime, the Si/SiO2 chips can be heated to 50 ◦ C. The chips are then strongly pressed against the surface of the tape and the tape is carefully peeled off from the chip: the exfoliation is done. In order not to break the crystal surface, we believe the peeling-off has to be done as smoothly as possible and in one slow movement. In Fig. 3.1(d), a zoom-in of Fig. 3.1(c) is shown, showing a h-BN flake. Small markers are available to facilitate locating the flakes. To remove most of the glue and various contaminations from the surface of the different flakes, the chips can be annealed in an oven, in a Ar/H2 atmosphere at

25

Chapter 3. Backgated bilayer graphene devices

380 ◦ C for about 4 hours. However, when dirt is present somewhere on the surface, this annealing step can potentially lead to the spreading of the dirt. During the course of this thesis, this issue became more and more obvious and it was therefore decided not to proceed with any annealing step on bare h-BN flakes. 3.2.1.3

Locating the h-BN flakes

Once the deposition and the optional cleaning are done, the chips are examined under the optical microscope and the different flakes are located (using the preprocessed photolithography markers). Once a flake is located, it is checked under dark field illumination, which helps to visualize residues. If the flake appears clean enough under dark field illumination and if its color matches our height wishes, the flake is further examined with the Atomic Force Microscope (AFM). 3.2.1.4

Cleanliness check

For this, a large AFM scan is performed. This scan provides two pieces of information. First, the exact height of the flake is measured. This will be relevant for the measurements in order to evaluate the lever arm to the backgate. The second parameter is the surface roughness. The RMS surface roughness of SiO2 is 0.3 nm and therefore, if one wishes to get improvements from the h-BN substrate, the surface roughness needs to be strictly below this value. Typically, flakes with a roughness close to 0.1 nm, which is the lowest resolution of our AFM setup, are considered to be worth processing. At this stage of the process, h-BN flakes have been identified as potentially good substrates. The transferable graphene-match has therefore to be found.

3.2.2

Preparation of bilayer graphene flakes on transferable substrates

3.2.2.1

Si-chip preparation

The chips are processed with the same markers as in the previous section. However, instead of using Si/SiO2 wafers, silicon wafers are used. These wafers are also diced into square chips, this time slightly bigger (1.1 × 1.1 cm2 ). Two layers of resist are then spin-coated. First, a layer of PVA (Poly Vinyle Alcohol) is used. This resist is water-soluble. On top of the PVA, a layer of PMMA1 950K, which is a standard Ebeam resist, is spin-coated. This layer has to be spin-coated with a lot of attention: one needs to be able to find the speed parameters that generate a purple color of the surface, similar to the one obtained from the 285 nm of SiO2 . This way, the visibility of graphene will be optimized. From one day to another, the resist can change and therefore it is necessary to adjust this speed every time. 1

PMMA: Poly(methyl methacrylate)

26

3.2. Fabrication of backgated bilayer graphene devices

(a) Bilayer Trilayer

10µm 3.2.2.2

Single-layer

Graphite

(b)

10µm

Figure 3.2: (a) Optical microscope image of a flake deposited on a Si chip covered with the bilayer resist. The flake exhibits a single-layer, a bilayer and a trilayer part. (b) Example of another single-layer graphene flake.

Deposition of graphene on Si chips

Once all chips are spin-coated, the deposition of graphene can be done. It also takes place in the grey room. The same way as before, the chips are heated to 50 ◦ C: this allows to soften the resist and has been found to improve the adherence of the graphite pieces to the surface, providing larger graphene flakes2 . The warm chips are then deposited on top of a blue tape piece, homogeneously covered with graphite. They are pressed against the tape and removed in one slow movement. 3.2.2.3

Locating the graphene flakes

The same way as for boron nitride, the chips are then examined under an optical microscope. Pictures of the flakes are shown in Fig. 3.2. After checking under darkfield illumination, if a flake is considered as sufficiently clean, the location of this flake is written down. A Raman spectrum of the flake is next recorded and, from the full width at half-maximum of the 2D-peak, the single or bilayer nature of the flake is established. Typically, a FWHM between 27 cm−1 and 34 cm−1 is measured for single layer graphene and of about 45 cm−1 to 56 cm−1 in the case of bilayer graphene. In the special frame of this thesis, only bilayer flakes were studied. 3.2.2.4

Cleanliness check

Once the nature of the flake is confirmed, one needs to finally check the cleanliness of its surface. As with h-BN, an AFM analysis is performed and the RMS roughness is measured. At this stage of the process, h-BN and bilayer graphene flakes are ready to be matched for further processing.

3.2.3

Transfer of a graphene flake on a h-BN substrate

Once a good graphene flake is found that can match a h-BN flake, the transfer process can be started. This process is illustrated in Fig. 3.3 and is adapted from [68]. First, the location of the graphene flake on the PMMA film has to be made obvious. This way, once the PMMA layer is detached from the Si-substrate, one can still know where the flake lies on the (transparent) PMMA film. To do this, 2 to 2

Roman Gorbatchev and Pauline Simonet are to thank for this information.

27

Chapter 3. Backgated bilayer graphene devices

Figure 3.3: (a) Schematic of a transfer chip, with a graphene flake identified on its surface. (b) The chip is deposited in water, which slowly attacks the PVA layer. (c) The dissolution of the PVA layer is over: the Silicon chip falls to the bottom of the beaker and the PMMA layer remains floating at the surface of water, with the graphene on top. The PMMA layer can be fished with the volcano, in grey. (d) Schematic of the PMMA layer lying on top of the volcano. (e) The volcano is positioned under a microscope and handled with a micromanipulator to align the graphene flake on top of the h-BN flake. (f) The resulting graphene-on-h-BN stack is shown, after dissolving away the layer of transfer resist.

3 pieces of blue tape are deposited around the graphene flakes, delimiting an area of about 0.5 × 0.5 cm2 . On the edges of the chip, the resist is slightly scratched away to better expose the PVA layer. The whole chip [displayed in Fig. 3.3(a)] is then left on the surface of a water bath. The water will slowly dissolve the layer of water-soluble resist. This can be seen in Fig. 3.3(b). After a few minutes, the water-soluble resist is fully dissolved, leaving the Si chip at the bottom of the beaker and the PMMA film floating on the surface [see Fig. 3.3(c)]. Then, a macroscopic tool which is handled by hand, called “volcano”, is used to “fish” the PMMA film, placing the graphene flake in the center of the hole (using the blue tape markers as a guide). The volcano is then shortly put on a hotplate at 80 ◦ C (for roughly 5 minutes), just enough to remove the water from the film. In order to prevent any stress on the film due to potential evaporation of water, the volcano is tilted from time to time or put up or down. As represented in Fig. 3.3(e), the selected h-BN chip is then positioned in the center of the microscope field of the micromanipulator and the volcano is mounted on

28

3.2. Fabrication of backgated bilayer graphene devices

Graphite

10µm

(c)

Ohmic

(b)

BNb

Ohmic

(a)

Ohmic

BLG

Ohmic

10µm

10µm Ohmics

Figure 3.4: (a) Optical microscope picture of the result of a transfer: the bilayer graphene flake lies on a h-BN flake (BNb , for bottom). (b) AFM picture of the transferred graphene on top of h-BN. (c) Device after etching and contacting.

the arm of the micromanipulator. The graphene flake then lies in the middle of the volcano and can be aligned with the h-BN flake. The center of the volcano is brought in contact with the Si/SiO2 /h-BN-chip, and by heating the chip, the PMMA film will stick to the surface: the transfer is done. The resist has however to be removed: the Si/SiO2 /h-BN/graphene-chip/PMMA-stack is put in warm acetone for 2 hours. A schematic of the resulting stack, after resist removal, is shown in Fig. 3.3(f). An optical microscope image is also shown, in Fig. 3.4(a).

3.2.4

Patterning and contacting the graphene flake

Once the h-BN/graphene stack is done, the graphene flake can be contacted. The first thing to do is to perform a high resolution AFM scan of the device [see Fig. 3.4(b)]. This picture is then used to locate the cleanest area of the graphene (i.e. the regions worth processing). After transferring, some bubbles usually appear at the interface of the graphene and the h-BN. Performing an AFM scan allows as well to locate these areas and to avoid them while drawing the E-beam mask. Another important information which comes out of the AFM scan concerns the cleanliness of the flake. If the flake is not clean enough, the electronic transport will suffer from the disorder induced by the surface residues. Possibilities exist to improve the cleanliness of the graphene on h-BN: one can for example perform thermal annealing in an atmosphere of Ar/H2 or mechanical cleaning using an AFM in contact mode. These two processes are discussed in detail in the following section. However, since the rest of the process also brings residues on the surface, it is preferable to have the flake ‘naturally’ clean enough at this stage, in order to avoid the multiplication of the cleaning steps: cleaning can then be postponed to the very last moment. Once the graphene is judged clean enough, the processing can continue. The E-beam lithography for the contacts cannot be performed directly, as the surface of h-BN is so smooth that contacts do not stick properly to it. Hence, before starting to pattern the contacts, one needs to make the h-BN a bit rough along

29

Chapter 3. Backgated bilayer graphene devices

the path where the contacts will be deposited. This is done with a short exposure to an RIE (Reactive Ion Etching). In the same step, the graphene flake can be etched in the desired shape. For the experiments presented in this thesis, usually a ribbon 1 µm wide and a few µm long was patterned. Subsequently, the mask for the contacts is exposed and the evaporation of Cr/Au is performed. A resulting device is shown in Fig. 3.4(c).

3.2.5

Conclusion

The dry transfer technique, pioneered by the Kim group [68], allows to reach a better transport quality in graphene by enabling the change of substrate. h-BN is not only a smooth material, but also a better dielectric than SiO2 (less charge traps and surface states), reducing the bulk disorder. However, h-BN improves the transport properties of graphene only if one manages to get rid of the different process residues which appear during each fabrication step. In the following, we will present the different cleaning options that have been studied by us.

3.3

Improving the quality of bilayer graphene on h-BN

In order to observe the theoretically predicted graphene physics in the experiment, the process residues, accumulated during each step of the fabrication process, have to be reduced as much as possible. Different techniques exist and will be presented here. We will first explain how disorder can be studied and characterized through transport measurements and we will then present three of these cleaning techniques in detail: the thermal annealing technique, the current annealing and finally, the mechanical cleaning using contact mode AFM.

3.3.1

Quantifying disorder in bulk graphene

In order to measure quality improvements induced by the different cleaning processes that we will present in this section, we first need to introduce the parameters that allow to draw a conclusion on the quality and to quantify the amount of disorder present in the system. The application of an electric field to the graphene sheet allows to shift the Fermi level through the electronic dispersion. The density therefore depends linearly on the applied voltage. In principle, if no field is applied, the Fermi level lies at the charge neutrality point. However, in practice, doping can be present in the device, due to which the charge neutrality point is effectively shifted to a finite voltage 0 value, called VBG . The density is therefore defined as follows, as a function of the 30

G [S]

3.3. Improving the quality of bilayer graphene on h-BN

Figure 3.5: A log-log schematic plot of the conductance versus the carrier density at one side of the charge neutrality peak allows to distinguish between the regime where the density is tuned via a gate voltage and the regime where the density is fixed due to residual disorder. The saturation value of the disorder, extracted as depicted schematically here, allows to estimate the strength of the disorder in the device.

10-3

nsat

1011 -2 n [cm ] ~ VBG

1012

lever arm to the backgate β: 0 ) n = β(VBG − VBG

β =

0 BN SiO2 1 e BN dSiO2 +dbottom SiO2 BN

(3.1)

In diffusive systems (L, W > mf p, i.e. the dimensions of the graphene device are larger than the carrier mean free path), the conductivity is given by: σ=

1 L . RW

(3.2)

Due to the substrate on which the graphene lies or because of process residues, disorder is induced. This disorder plays a dominant role at low energies, where the induced spatially varying random potential is responsible for spatially shifting the charge neutrality point from the average Fermi energy. These spatial shifts can be seen as the formation of electron-hole puddles [70, 71]. The induced disorder can therefore be quantified from the density range within which the applied voltage is inefficient to cause significant changes in the conductivity, due to redistribution of charge carriers within these puddles. Below a certain voltage range, the density does not follow the predicted linear relation anymore. In principle, this information is contained in a standard resistance measurement as a function of the applied backgate voltage. In order to better visualize the effect of disorder, Du et al. proposed the use of a log-log plot [65], where the conductivity is plotted as a function of the density. From this, one can easily identify the threshold at which the effective carrier density is not changing anymore as a function of applied voltage. The saturation carrier density nsat can then easily be identified as shown in Fig. 3.5. In the diffusive regime, the conductivity depends linearly on the density via the relation: σ = µen. (3.3) Thus, the mobility µ as it is defined here is only meaningful for n > nsat . In the following, the mobility values will be always given at the saturation carrier density

31

Chapter 3. Backgated bilayer graphene devices

value. Note that the values are often different for electrons and for holes, as will be discussed later. To conclude, the parameters that are the most meaningful to characterize disorder are the saturation carrier density and the mobility (determined at the saturation density value). Another measurement that often reveals the quality of transport is the quantum Hall effect. As explained in Chapter 2, the Hall conductivity of bilayer graphene is quantized at values σxy = νe2 /h, with ν = ±4, ±8, ±12... In good quality samples, strong electron-electron interaction can lift the fourfold degeneracy and lead to the appearance of “broken symmetry” states, with conductance plateaus at integer multiples of e2 /h. Looking at the Landau level spectrum can therefore give qualitative information about the transport quality. In the next sections, we will present three different techniques of cleaning that have been investigated during this thesis.

3.3.2

Thermal annealing

Transferring a graphene flake onto a h-BN substrate usually leaves some residues on the surface of these two flakes. These residues mostly consist of PMMA, left from the transfer film which has been dissolved in acetone. In order to remove these residues, Dean et al. proposed a thermal annealing process [68]. Dr. Dominik Bischoff adapted this technique in our group. We use a quartz tube furnace, with the possibility to perform the annealing either in air or under a controlled argon/hydrogen atmosphere. The complete setup has been described in his thesis [72]. To get an idea of the ranges of mobilities and saturation carrier densities reached for graphene on different substrates, we show a summary of collected values in Table 3.1. The values shown for graphene on h-BN were measured on samples fabricated using the transfer technique presented earlier which were thermally annealed (the new stacking technique developed by Wang et al. in 2013 [73] is ignored at this stage and will be commented on in the next chapter). The values shown here are non-exhaustive and were taken from Refs. [10, 26, 39, 65, 68, 74–80], measured between 4.2 K and 30 mK, depending on the reference. Most of these mobility values were measured at the saturation carrier density. However, in some references, the density at which the mobility is extracted is not mentioned. For the samples investigated in this thesis (graphene on h-BN) however, the annealing step was rarely sufficient to completely remove the residues left by the transfer film. Furthermore, subsequent processing steps, like etching and contacting the graphene flake, also bring residues to the graphene surface. Hence we concluded that the cleaning step could be postponed to a later point. The graphene flake was therefore only processed if it was judged to be clean enough right after the transfer step. This flake was then etched and contacted and only at this point the cleaning was performed. Unfortunately it appeared to be even more difficult to clean the graphene surface

32

3.3. Improving the quality of bilayer graphene on h-BN

µ[cm2 /Vs] SLG nsat [cm−2 ] µ[cm2 /Vs] BLG nsat [cm−2 ]

on SiO2 2, 000 − 20, 000 5 × 1010 − 2 × 1011 1, 000 − 10, 000 1011 − 1012

on h-BN 15, 000 − 100, 000 109 − 1010 10, 000 − 80, 000 5 × 1010 − 1011

Suspended 100, 000 − 200, 000 109 − 5 × 109 15, 000 − 100, 000 1010 − 3 × 1011

Table 3.1: Comparison of mobilities and saturation carrier densities for devices built on SiO2 , on h-BN and for suspended graphene. These non-exhaustive values are taken from the Refs. [10, 26, 39, 65, 68, 74–80].

using thermal annealing once the contacts were surrounding the area to be cleaned. Thermal annealing as a cleaning technique was therefore not further investigated for the purpose of this thesis and other more local techniques were studied in greater detail.

3.3.3

Current annealing

Another cleaning technique investigated in this thesis is the current annealing [81]: this method consists in applying high currents to a graphene sheet. While the graphene can survive this process, the residues and various contaminants lying on its surface are moved or removed. What really happens to these residues is not clear. The induced increase of temperature is suspected to either melt the residues, evaporate or sublimate them [81]. In this section we will present an example of current annealing performed on a bilayer graphene flake, deposited and contacted on a h-BN substrate. All measurements were performed at a temperature of T = 4.2 K. The following results were obtained in collaboration with Pauline Simonet (the sample processing as well as the measurements and data analysis). 3.3.3.1

Sample presentation

The sample studied in this subsection is shown in Fig. 3.6. It consists of a bilayer graphene flake transferred on top of a h-BN substrate. As the graphene flake was long and narrow, there was no need to etch the device and two pseudo Hall bars (graphene stripes contacted by two parallel Ohmic contacts at each extremity) were built out of it: device A and device B. Fig. 3.6(a) shows the AFM phase image of the whole device. Here, a lot of residues are visible on the surface of the h-BN flake (white dots). In the two zooms on the separate devices however, only very few residues can be seen directly on the graphene surface. 3.3.3.2

Device characteristics before current annealing

In the following, we will present the measurements performed on these two devices before the cleaning step. In order to judge the quality of the bilayer graphene,

33

Chapter 3. Backgated bilayer graphene devices

(a) BN

BLG

(b)50°(b)

(c)

Device A

Device A

Phase

Device B

BLG

Device B

5 µm

1 µm

BN

BLG 1 µm

BN

0

Figure 3.6: (a) AFM image of the whole device, displaying the phase signal (which usually provides a better visibility of the process residues). The bilayer graphene flake edges are highlighted with yellow dashed lines. This flake lies on top of h-BN (blue dashed lines) and is contacted with Cr/Au Ohmic contacts. Two devices are built out of this flake. (b) Zoom on the upper device, called device A. (c) Zoom on the lower device, called device B.

we will focus on the value of the mobility, the saturation carrier density and the quantum Hall effect characteristics. 3.3.3.2.1

Transport at B = 0

In Fig. 3.7(a), the 4-terminal resistance measured versus the backgate voltage on device A is shown. From this measurement, one can further extract the saturation carrier density for the hole and electron sides, as shown in Fig. 3.7(b). This value allows to estimate the energy range in which disorder is dominating the transport. The mobility is therefore only meaningful if taken at (or above) this value. The corresponding values, for the hole and electron sides, are displayed in Fig. 3.7(a). We find: µel = 11, 100 cm2 /Vs and µh = 19, 400 cm2 /Vs. The second device is as well measured in a 4-terminal configuration. The results are shown in Fig. 3.8(a-b). As done with sample A, from the backgate trace displayed in (a), the saturation carrier density is estimated [see Fig. 3.8(b)]. At these density values, the mobility is found to be: µel = 25, 100 cm2 /Vs and µh = 30, 700 cm2 /Vs. For both devices, it is worth mentioning that the measured mobility values are clearly above the ones previously obtained on similar bilayer devices deposited on SiO2 in our group. 3.3.3.2.2

Transport at B 6= 0

As an example of parameters illustrating a device’s quality, the quantum Hall effect

34

3.3. Improving the quality of bilayer graphene on h-BN

3

Device A

(b)

(b)

μh = 19 400 cm2/Vs μel = 11 100 cm2/Vs

10-3

hole conductance electron conductance nsat, h = 2.95 x1010 cm-2 nsat, el = 2.92 x1010 cm-2

G [S]

R [kΩ]

(a)

2

1µm 1

0

5 VBG [V]

10

1010

11 n [cm-2] 10

Figure 3.7: Device A: (a) The resistance measured in a 4-terminal configuration is shown as a function of the backgate voltage. The displayed mobility values are taken at the value of saturation carrier density. (b) As before, using a log-log scale, one can estimate the saturation carrier density in order to evaluate the disorder of the system.

5

Device B

(b)

μh = 30 700 cm2/Vs μel = 25 100 cm2/Vs

10-3

hole conductance electron conductance nsat, h = 1.62 x1010 cm-2 nsat, el = 1.93 x1010 cm-2

G [S]

R [kΩ]

(a)

3

1µm 1 0

5 VBG [V]

10

1010 n [cm-2]

1011

Figure 3.8: Device B: (a) The resistance versus backgate voltage, measured in a 4terminal configuration, is shown. (b) From this measurement, using a log-log scale, one can further access the saturation carrier density for the hole and electron sides.

measurements should be mentioned. For device B, the Landau fan (conductance versus the backgate voltage and the magnetic field) is shown in Fig. 3.9. The Landau fan is nicely developed already without any cleaning. The expected sequence of quantum Hall conductance plateaus is highlighted with the labeling of the filling factor ν. The lines locate the position of the plateaus expected from a plate capacitor model. As seen here, broken symmetry states below ν = ±4 are already visible, indicating a good device quality. We now continue to evaluate the effect of current annealing on these characteristics.

35

Chapter 3. Backgated bilayer graphene devices ν=4

ν = 8 ν = 12

3.3.3.3

1.5 dG/dVBG [x 10-4A/V2]

ν = -8 ν = -4

B [T]

Figure 3.9: (a) The derivative of (a) ν = -12 the conductance dG/dVBG of device B versus the backgate voltage and the 4 magnetic field is shown. The typical Landau fan structure with conductance plateaus corresponding to ν = ±4, ±8, ±12, ±16 is observed. The expected positions of the quantum Hall plateaus, based on the plate 2 capacitor model, are shown with lines and the corresponding filling factors are displayed at the top of the plot. In addition to the expected sequence of plateaus, broken symmetry states 0 -20 are also clearly visible: ν = ±2 are drawn and one can also see ν = −1.

0 VBG [V]

30

-1.5

Current annealing procedure

The current annealing procedure consists in sending a large current through the graphene flake. We expect this high current to generate a temperature gradient, which helps moving/removing the dirt from the graphene surface. To do so, we apply a DC voltage between the two innermost contacts of a device, while the others are left floating. The schematic circuit diagram is shown in Fig. 3.10 (adapted from Ref. [82]). Using a 100 Ω resistor in series with our graphene device enabled the simultaneous measurement of the current going through the device. The current density is then defined as follows: VHP I = , W 100Ω × W

(3.4)

where W is the width of the device, as shown in Fig. 3.10(a), and VHP is the voltage measured by the HP multimeter. To start the process, we need to slowly apply a voltage which leads to a known current density. Using existing literature, as well as Ref. [82], it appears that current densities up to 0.5 µA/nm do not lead to any change of the transport properties. This current density can therefore be considered as a safe starting point and we begin the process by ramping up the voltage until this current density is reached. In the case of the two presented devices, no changes were observed up to 1 µA/nm. Once the voltage is set to the desired value, the device is left in this state for a period of time varying between 30 minutes and 1 hour. During that time, due to the presence of an Allen-Bradley resistor on our sample holder, one can notice a non-negligible increase of temperature (up to 10K). Once the annealing time is over, the voltage is slowly brought back to zero and the sample is ready to be measured. After the current annealing procedure, our

36

3.3. Improving the quality of bilayer graphene on h-BN

(a)

L

1

(b) 2

W Au contact

Au contact

Graphene

Yokogawa DC source

+ -

HP Multimeter

+ -

100 Ω

1 Sample 2

Voltage A Amplifier B (gain 1)

Figure 3.10: (a) Sample schematic highlighting how the width W and length L are defined. (b) Electronic setup to perform current annealing: the voltage is applied to a Yokogawa voltage source and the resulting current through the graphene flake can be deduced from the voltage read of the HP multimeter, VHP .

samples did not reach immediately their steady state. Measuring the resistance as a function of backgate voltage after ramping down the voltage, one could observe that the charge neutrality peak was changed, but not improved. Very long waiting times were necessary for the samples to finally exhibit the expected improved transport characteristics (longer than a night). Warming up and cooling down the samples has the same effect as waiting for long times. In the following, we will present the results of our current annealing process, measured after warming up to room temperature and cooling down again the device. 3.3.3.4

Devices characteristics after current annealing

In Fig. 3.11(a), the resistance of device A measured as a function of the backgate voltage is shown. Many steps of annealing have been performed, with increasing current densities from 0.5 µA/nm up to 1.6 µA/nm. As before, the saturation carrier density for the hole and electron sides were extracted, as shown on the (b) panel: one can see a slight improvement for both sides. On the other hand, the improvement of the mobility at these values is quite noticeable: for the electron side, it is almost enhanced by a factor of three. The values are given in Fig. 3.11(a) and are summarized in Table 3.2. The second device was also current annealed in many steps. The resistance versus backgate voltage recorded after the final step is shown in Fig. 3.12(a). The saturation carrier density is extracted from a log-log plot of Fig. 3.12(a), which can be seen in Fig. 3.12(b). In this case the current density sent through the graphene device was ramped from 1 µA/nm up to 1.7 µA/nm. From the log-log plot shown in Fig. 3.12(b), it appears that the disorder increased slightly during the process. However, the mobilities measured at these values are still improved by a factor of two in both cases. The evolution of the saturation carrier density is currently still not understood. The evolution of the disorder-related values measured on both devices are summarized in Table 3.2. Another check for the quality evolution consists in recording magneto-transport measurements. However, since the quality of the quantum Hall effect did not change after current annealing, we are not able to comment any further. After warming up the sample, another comparison can be made using AFM

37

Chapter 3. Backgated bilayer graphene devices (a) 4

Device A

(b)

μh = 30 133 cm2/Vs μel = 29 569 cm2/Vs

G [S]

R [kΩ]

3

hole conductance electron conductance nsat, h = 2.49 x1010 cm-2 nsat, el = 2.89 x1010 cm-2

10-3

2 1µm 1 0

5 VBG [V]

10

1011n [cm-2]

1012

Figure 3.11: Device A, after current annealing: (a) The resistance measured in a 4terminal configuration versus backgate voltage is shown. The values of mobility recorded at the saturation carrier density are given: both show an improvement. The strongest improvement is seen on the electron side, where the mobility is enhanced by almost a factor of 3. (b) From the measurement shown in (a), using a log-log scale, the saturation carrier densities are estimated. Both are slightly reduced compared to the values obtained before cleaning.

4

Device B

μh = 61 200 cm2/Vs μel = 42 100 cm2/Vs

(b)

hole conductance electron conductance nsat, h = 1.71 x1010 cm-2 nsat, el = 2.52 x1010 cm-2

3 G [S]

R [kΩ]

(a)

10-3

2

1µm

1 00

5 VBG [V]

10

1011 -2 n [cm ]

1012

Figure 3.12: Device B, after current annealing: (a) The resistance measured in a 4terminal configuration versus backgate voltage is shown. It has been recorded after the final annealing step and after warming up and cooling down again the sample. The mobilities measured at the value of saturation carrier density from (b) are almost doubled compared to the first measurement. (b) Plotting the data from (a) in a log-log scale, the saturation carrier densities are estimated: even though the mobility has improved at these points, the disorder seems to be slightly enhanced.

imaging: one can now have a look at the surface residues and how they evolved with current annealing. The result is displayed in Fig. 3.13. In this case the dirt did not fully disappear, but one can see that the three “blobs” were disintegrated in many small pieces.

38

3.3. Improving the quality of bilayer graphene on h-BN

Parameter nsath [cm−2 ] nsatel [cm−2 ] µh [cm2 /Vs] µel [cm2 /Vs]

Device A Before After 10 2.95 × 10 2.49 × 1010 2.92 × 1010 2.89 × 1010 19, 400 30, 133 11, 100 29, 569

Device B Before After 10 1.62 × 10 1.71 × 1010 1.93 × 1010 2.52 × 1010 30, 700 61, 200 25, 100 42, 100

Table 3.2: Summary of the evolution of the quality parameters, before and after current annealing.

(b)

Before

After

BLG 1 µm

3.3.3.5

BN

Figure 3.13: (a) The phase of the AFM scan is shown before the current annealing: one can notice three “blobs” of PMMA lying within the area of the device. (b) After current annealing, the AFM image is changed: the three PMMA “blobs” have been disintegrated into smaller pieces.

50 signal

BLG

Phase [°]

(a)

BN

1 µm

0

Conclusion

Current annealing is a powerful technique to improve the mobility of the graphene flake. Residues lying on the surface are efficiently moved or removed by this process. However, compared to the annealing of suspended devices, the graphene-on-h-BN geometry has the drawback that the bottom surface cannot be improved as much as the top surface during this process, due to van-der-Waals interactions with the h-BN surface. Another drawback is that high currents are sent through the gold contacts and electromigration can occur, leading to the appearance of gaps in the gold circuits. Current annealing is usually appreciated for the fact that this is an in-situ technique and therefore it can be delayed until the very last moment, before starting other kinds of measurements. However, since the main topic of this thesis is to perform measurements on dual-gated bilayer graphene, it would be preferable to have a method to clean the graphene just before the encapsulation, without having to glue, bond and cool-down the device.

3.3.4

Mechanical cleaning

In 2012, Goossens et al. came up with an alternative idea which uses an AFM tip in contact mode to move the process residues away from the graphene surface [83]. This technique is called ‘mechanical cleaning’.

39

Chapter 3. Backgated bilayer graphene devices

Figure 3.14: (a) AFM image of the device before cleaning. The bilayer graphene (a) flake is highlighted by dashed black lines. (b) Image after cleaning: the device has been destroyed. The flake has been ripped away and a part of it can be seen next to its former position (black dashed line). Scratches are also visible on the h-BN sur- 1µm face.

(b)

1µm

This technique is an efficient way to push away the process residues. However, if not performed with full control (of speed, force and position), the graphene flake can easily be ripped away and the contacts can be moved, scratched or even broken by the tip. In order to adapt this delicate technique to our AFM setup (a Dimension V AFM from Veeco-Bruker), Ekaterina Kuznetsova focused her master’s thesis on the problem [84]. The following results have been obtained in collaboration with her and with the help of Pauline Simonet. 3.3.4.1

Method

In order to perform safe and efficient cleaning, the AFM probe has to be carefully chosen. Companies usually provide a description of the probe they sell, mentioning the expected parameters of the tips. However, discrepancies between the probes from the same batch can exist. It is therefore necessary to characterize a tip before starting the cleaning process. The key parameter to measure is the spring constant k. This is done using the thermal tune method. Performing a power spectral density analysis of the cantilever oscillations, the spring constant can be estimated from integrating and fitting the area under the peak of the fundamental mode. We found that ideal values for safe cleaning are spring constants between 0.1 − 0.4 N/m. Larger values are likely to rip the graphene flake away and even to scratch trenches in the h-BN, while at lower values, the tip does not sufficiently move the residues and therefore this leads to inefficient cleaning. An example of a violent cleaning is shown in Fig 3.14: in Fig. 3.14(a), one can see the shape of the graphene in between the Ohmic contacts, while Fig. 3.14(b) shows how the graphene has been destroyed by the tip and a part of what is left from the flake is lying, rolled up, next to its previous position. Once a tip with a suitable spring constant is found, the calibration has to be done. To do so, a force curve is recorded. This reveals how much force acts on the tip while it moves perpendicularly to the sample. In the case of our setup, it records the value of the deflection setpoint (DSP) as a function of the movement of the Z-motor. In order not to damage the area one wishes to clean, it is better to do this next to it (but still on graphene on h-BN to have the same surface properties). A typical force curve is shown in Fig. 3.15. First, the tip approaches the surface:

40

3.3. Improving the quality of bilayer graphene on h-BN

Figure 3.15: A typical force curve: in the case of our setup, the force is not directly recorded. Instead, the deflection setpoint (DSP) is measured as a function of the tip z-position (here in volts; the conversion to nm is done later). The DSP can then be converted into a force by a simple calculation (see text).

-1

-2

STOP START

DSP [V]

0

Approaching

Lost contact VCSmin

10

20

30

Z [V]

40

50

no force is felt, the DSP is constant. Once the tip is close enough, it gets attracted to the surface: contact is reached. Reaching the maximum of the Z scan size (the cantilever is as close as allowed by the AFM), the cantilever moves away from the sample again and more attractive forces are experienced. After the point labeled VCSmin , the tip looses the contact with the surface (this is therefore the last contact point, where no force is applied to the surface of the sample and which is very unstable). Once the flat part of the curve is reached, the tip feels no more force. Important to notice here is that we are talking about the force experienced by the tip and not about the force applied to the surface. The conversion DSP/force applied to the surface is done by locating the minimum VCSmin , which allows to define the point where no force is applied to the surface. Next one needs to know the spring constant (which is measured at the very beginning of the process) and the deflection sensitivity s (this is a measure of the reaction of the probe on the movement of the tip in Z-direction and allows the conversion from the signal from the photodiode (V) to a deflection of the cantilever (nm) - this can be calculated by the AFM software). The force applied can then be calculated from the relation: F = sk(VCSmin − DSP ).

(3.5)

The full description of the cleaning process is given in the Appendix A. 3.3.4.2

Results

In order to evaluate our cleaning process, different test samples were used: some dirty h-BN and graphene flakes on SiO2 and three processed samples. The processed samples consist of bilayer graphene flakes, which were etched into rectangular stripes of ∼ 1 µm width and a few microns length. 3.3.4.2.1

Improvement of the surface cleanliness

A straightforward check of the efficiency of the cleaning process consists in the evaluation of the surface cleanliness. This can be done by recording a tapping mode

41

Chapter 3. Backgated bilayer graphene devices

(a)

1µm (b)

(c)

(d)

(e)

Figure 3.16: (a) Tapping mode image of sample A: the contacted bilayer graphene flake (black dashed lines) on h-BN is shown before any cleaning. Same device after one (b), two (c), three (d) and four cleaning steps (e). The white dashed boxes illustrate the position of the remaining dirt, which has been pushed away from the surface of interest. The direction followed by the tip during the cleaning is shown with arrows.

image of the area after the contact mode cleaning. The conclusion is positive: if the parameters do not drift dramatically with time (small drifts are not a problem), the cleaning can be performed safely and easily using our process. Images of a controlled cleaning are shown in Fig. 3.16. In this figure, one can observe how the surface of the bilayer graphene (highlighted with the black dashed lines) gets cleaner after each repetition of the cleaning procedure. For every cleaning step, the force applied has been slightly increased: from 15 nN at the first try to 45 nN. We therefore conclude that our process can in principle be performed safely, efficiently and in a controllable way. 3.3.4.2.2

Consequences of cleaning on the electronic properties

Our goal is however not only to produce clean surfaces: we wish to see an effect on the electronic properties of the graphene flake. Two questions have to be addressed: - can the cleaning improve the transport properties of the (bilayer) graphene? - is there a threshold force after which the cleaning is either not efficient anymore or even becomes damaging? To answer these questions, the processed samples, called devices A, B and C, were used. They consist of a bilayer graphene flake transferred on a h-BN substrate, etched and contacted. The chip was then glued into a chip carrier and bonded. The bond wires were located such that it is convenient to access the graphene surface with an AFM tip (i.e. not too high and located on only one or two sides of the chip). In order to avoid electrostatic discharges, a grounding sample holder was used to keep the chip properly grounded in between measurements and during the cleaning steps. A picture of the chip in the chip carrier lying in the grounding sample holder is shown in Fig. 3.17. Two parameters were evaluated, before and after each cleaning step: the saturation carrier density, which describes the amount of disorder present in the sample and the value of the mobility taken at this density value. The quantum Hall effect was also measured and qualitatively evaluated (judging for example the ‘strength’ of the quantum Hall plateaus, their number, the value of magnetic field at which

42

3.3. Improving the quality of bilayer graphene on h-BN

Figure 3.17: The grounding sample holder is shown here: a processed chip glued in a chip carrier (yellow) using silver epoxy is bonded to this chip carrier. In order to avoid electrostatic discharges during the whole process, this chip carrier is kept in this grounding sample holder.

1cm

the Landau levels are formed, etc.). Out of the three fabricated samples, only two exhibited ‘proper’ transport properties from the beginning on. The third one (called device A) was very resistive, probably due to a bad contact between the Ohmic contacts and the surface of the flake. In the case of this resistive sample, where no quantum Hall plateau was visible up to 5 T, the improvement of the transport after cleaning was very limited, even though the surface cleanliness did improve dramatically (the measured value can be seen in Table 3.3). This points out the main limitation of the technique: if performed after the contact deposition, the area under the contacts is not accessible for cleaning anymore. However, since we wished to understand the effect of cleaning on the electronic properties, we needed to compare electronic measurements before and after cleaning which requires the presence of the contacts. In the end, only two reasonable quality samples were left, which is of course a very limited number to draw any strict conclusion. What we can however say is that, in the case of both samples (samples B and C), one cleaning step improved the quality: the saturation carrier density, nsat , was reduced by a factor 2 and the mobility µ was increased (up to a factor of 3 for one of the samples). The exact values are shown in Table 3.3. To illustrate the improvements, we show in the following the measurements performed on device C. An example of resistance versus backgate voltage measurement, before and after cleaning, is shown in Fig. 3.18. In Fig. 3.19, the quantum Hall effect measurements are also shown for the same sample, before (a) and after cleaning (b). We can see that the hole side is clearly improved, with more Landau levels being visible after the cleaning process, and that the magnetic field threshold for the formation of Landau quantization is lowered. Furthermore, broken symmetry states become more pronounced. However, further cleaning did not improve the parameters we chose as our quality benchmark: for one sample, the second cleaning lead to an enhancement of the quality, with the mobility increasing by a factor of 2, while for the other one, the mobility was reduced. The full summary of the evolution of the values as a function of the number of cleaning steps performed is shown in Table. 3.3.

43

Chapter 3. Backgated bilayer graphene devices

60

ν = -12 ν = -8

(a) ν = 4

ν = -4

ν=8

(b)

ν = 12

5

4

4

3

3

2

-20

ν = -12 ν = -8

(b)

-10

ν = -4

0

VBG [V]

(a) ν = 4

ν=8

10

20

ν = 12

-20

-10

0

VBG [V]

10

20

30

2

0 -30

30

0.2 (c)

1

1 0 -30

20

d²G/dBdVBG [e²/(h.V.T)]

5

40

0 -30

B [T]

B [T]

(a)

ρ [kΩ]

Figure 3.18: Example of resistance versus backgate voltage measured before and after AFM cleaning (device C). In this case, the cleaning was successful and the mobility increased while the saturation carrier density was reduced.

-20

-10

0

VBG [V]

10

20

30

-0.2

Figure 3.19: Quantum Hall effect measurements recorded on device C, before (a) and after (b) AFM cleaning. The conductance was measured as a function of backgate voltage and magnetic field. The conductance, differentiated along both axes, is displayed here. The black lines indicate the expected position of the quantum Hall plateaus from a plate capacitor model. The corresponding filling factors are indicated at the top of the measurements. After cleaning, the measured LL spectrum is not drastically changed, but one can still notice that more Landau levels are visible on the hole side (negative VBG values) and that the Landau levels are formed at smaller B-field values (this is highlighted with the black dashed line), which is a sign of improved electronic quality. Broken symmetry states, below filling factor −4 are also more strongly pronounced on the hole side.

44

3.4. Conclusion

Sample A

B

C

Cleaning effect Number of cleaning nsat [cm−2 ] µ[cm2 /Vs] None 14 × 1010 1, 850 10 1 2.4 × 10 5, 600 10 None 2.5 × 10 20, 000 1 1.45 × 1010 69, 150 10 2 13.6 × 10 9, 360 3 1.1 × 1010 22, 500 None 3.4 × 1010 15, 710 10 1 2.1 × 10 21, 600 2 3 × 1010 34, 360

Conclusion Ø Improvement Ø Improvement Degradation Improvement Ø Improvement Improvement

Table 3.3: Summary of the evolution of the quality parameters, before and after cleaning.

3.3.4.3

Conclusion

The conclusions one can draw from our experiments on the AFM cleaning efficiency are limited. The established cleaning procedure appears to be controlled enough to lead to clear improvement of the surface quality. However, the correlation between surface cleanliness and quality of transport is not straightforward. For some of the cleaning processes, improvements were observed whereas for others, a degradation took place. Another conclusion is that AFM cleaning seems to be facilitated by the addition of an air annealing step: the annealing seems to make the surface residues more ‘movable’. Already during a tapping mode scan, it was noticed that some dirt was moved by the tip. The amount of force needed to clean the surface should then be reduced. One can however point out that, in the frame of the dual-gated technology, it is probably preferable to have a homogeneous surface in order to induce homogeneous band gaps in bilayer graphene. For the work presented in this thesis, AFM cleaning has been performed on different dual-gated devices (before encapsulation). Despite the fact that no comparison can be done between the quality of such a device before/after cleaning (because the devices were not bonded and measured until the very last step), the concerned samples still exhibited the best quality achieved within the frame of this thesis, as will be shown in the next chapters.

3.4

Conclusion

While depositing graphene on h-BN substrate should provide better transport quality due to reduced bulk-induced disorder, getting to this point is not as easy as one might think. The process still brings a lot of process residues on the graphene surface, which limit the transport properties. Getting rid of these residues is the key for being able to access the true physics of graphene. Different techniques have

45

Chapter 3. Backgated bilayer graphene devices

been studied throughout this thesis. It appears that AFM cleaning is the most convenient, as it allows to “broom” away the residues from the graphene surface. This can be done at the very last moment, before encapsulating the device under another h-BN layer. We should however highlight at this stage that the substrate removal/replacement and the surface cleanliness do not solve every problem pointed out by transport experiments: besides the clear improvement of the bulk transport quality, the problems concerning nanostructures remain unchanged [26], as it seems that the performance of these devices is mostly dominated by the edges [27].

46

Chapter 4 Dual-gated bilayer graphene One of the strengths of bilayer graphene as a material lies in its high electronic tunability. In this chapter, we focus on explaining how a band gap can be induced using dual-gated technology. Two fabrication techniques will be discussed, which allow the encapsulation of a bilayer graphene flake between two hexagonal boron nitride layers and the definition of top gates. The electrostatic properties of such devices will then be explained, as well as their basic electronic characterization. Finally, we will study the band gap induced by the transverse electric field.

4.1

Introduction

The band structure of bilayer graphene can be influenced by external parameters such as strain or electric fields. In particular, transverse electric fields are able to generate a band gap close to the charge neutrality point, as it has been pointed out in Chapter 2. To do so, one needs to induce a potential asymmetry between the two layers of a bilayer graphene flake. To induce this asymmetry, chemical doping can be used [33] or external gates [34] can be patterned. In this chapter, we will focus on the latter, which was facilitated by the development of the dry transfer technique presented in the previous chapter.

4.2

Fabrication

As mentioned in the introductory part, bilayer graphene is an interesting material because its band structure can locally be modified under the influence of a transverse electric field. If the flake is under the influence of a back- and a top gate, a band gap can be induced, the size of which depends on the strength of the applied electric field. Hence, extending the fabrication process described in the previous chapter in order to pattern top gates is a way to not only probe but also tune the electronic spectrum of bilayer graphene.

47

Chapter 4. Dual-gated bilayer graphene (a)

BNt

(b)

(c)

60nm

500nm

10µm

10µm

500nm

0nm

Figure 4.1: (a) Optical microscope picture of the device presented in Chapter 3 (Fig. 3.4) after the transfer of the top h-BN layer (BNt ; this flake edges are highlighted with a dashed purple contour): the Ohmic contacts, covered by this flake, appear now slightly darker. (b) Optical microscope picture of the device after the deposition of the top gates: one quantum dot is designed (defined by the three upper top gates) as well as a quantum point contact (the two top gates at the bottom). (c) AFM pictures of the top gates defining the two devices: three top gates (red, blue and green dashed contours) are expected to form a QD (top), schematically shown with a white dot and two top gates (yellow and orange dashed lines) form a QPC (bottom; the expected channel is shown in white). Some resist residues are visible in between the gates.

4.2.1

Encapsulating graphene using the dry transfer technique

Since h-BN is both a smooth substrate and a good dielectric, it can be transferred on top of a BLG/h-BN stack as the ones presented in Chapter 3: this is the encapsulation of the graphene. This step is done as shown in Fig. 3.3, the only difference being that the graphene lying on the PMMA resist is now replaced with a h-BN flake. The resulting h-BN/BLG/h-BN stack is displayed in Fig. 4.1(a).

4.2.2

Top gate definition

Depending on what kind of device has to be produced (barrier, quantum point contact or quantum dot), the corresponding mask is designed. As explained before, the device is first shortly exposed to RIE to promote the adhesion of the gold on the h-BN surface. Only after this step, the top gates are patterned with E-beam lithography and Cr/Au is deposited. Fig. 4.1(b) shows an optical microscope image of the resulting device: the bright gold contacts are the top gates (they are located on top of the h-BN and therefore appear brighter). The sample shown now includes two different devices: at the top, the top gates define a quantum dot [see the top AFM image of Fig. 4.1(c)] and at the bottom, a quantum point contact is patterned (lower AFM scan).

48

4.2. Fabrication

4.2.3

Alternative process: the stacking technique

As mentioned in the previous chapter, the drawback of the dry transfer technique is the accumulation of residues on the surface of the graphene flake due to multiple steps during which this surface is in contact with various polymers. This leads to the necessity of finding efficient and reliable cleaning techniques, as investigated in Chapter 3. This cleaning should be performed before the encapsulation step. The dry transfer technique developed by Dean et al. did bring improvements in the quality of the electronic transport, with clear enhancement of the mobilities. However, these numbers could still not compete with the values measured on suspended devices (see Table 3.1 in the previous chapter), mostly due to the difficulty of getting rid of process-induced residues. But very recently, the same group developed a new technique [73], simultaneously with other groups [85], called the “pick-up” technique, allowing for one-dimensional side contacts. This new process allows to create h-BN/Graphene/h-BN stacks that can be contacted at the edge, leaving the graphene free of any polymer residue. This process, which will be presented in detail in the following subsections and has been developed in our group under the guidance of Dr. Srijit Goswami from TU Delft, allows the graphene-onboron nitride technology to now compete with the quality of suspended devices. Further advantages of encapsulated bilayer graphene devices compared to suspended bilayer graphene devices are the almost non-existing strain and the possibility of applying large gate voltages through the h-BN flakes. 4.2.3.1

Preparation of the pick-up stamp

The fabrication process starts with the preparation of the PDMS stamp 1 . To handle the stamp, we use one of the glass slides generally used for optical microscopy as support. The glass slide has the advantage of being relatively thin and transparent (such that one can look through during the process). Such a glass slide has first to be cleaned properly (5 min in warm acetone followed by 5 min in warm IPA). A 1 × 1 cm2 piece is cut out from the PDMS Gel-Pak sheet shown in Fig. 4.2(a). The actual PDMS is only one layer of this sheet, as schematically shown in (b). Using clean tweezers, the top protective layer is removed. The PDMS can next be detached from its polymer substrate. This removed layer can then be slowly brought into contact with the glass slide: the idea is to go as slowly as necessary to avoid the formation of bubbles at the interface glass/PDMS. The PDMS-on-glass slide is depicted in Fig. 4.2(c). The next step would then be to deposit h-BN on this stamp. However, in order to improve the coverage and the quality of the later deposited h-BN flakes, a layer of copolymer resist has first to be spin-coated on top of the PDMS. The first trials were made using MAA(8.5)MMA EL 12. However, the process is now continued with MAA(17)MMA EL 11, as it better survives the contact with the blue tape. Once 1

PDMS: Polydimethylsiloxane

49

Chapter 4. Dual-gated bilayer graphene

2 cm

(a)

(b)

(c)

Protective film

(d)

PDMS polymer

Figure 4.2: (a) A commercial PDMS Gel-Pak sheet. (b) Cross-section of the sheet shown in (a): the PDMS is encapsulated between a thick polymer substrate and a protective layer. Removing the protective film, we can then carefully peel the PDMS layer off the substrate using clean tweezers (top part). (c) The PDMS layer from (b) now deposited on a glass slide. (d) A resist layer is spin-coated on top of the PDMS.

the PDMS is covered with the copolymer, the deposition is done, as for standard Si/SiO2 chips. The glass slide is brought into contact with the tape, with no force applied on the surface, and the blue tape is peeled off very slowly. The stamp is then examined with an optical microscope and flakes are located. Subsequently, the cleanliness is checked with an AFM. If the flake is clean enough, it will constitute the top layer of the future stack. An example of h-BN flake on such a stamp is shown on Fig. 4.3(a). 4.2.3.2

Picking-up graphene

The graphene is prepared and identified on Si/SiO2 chips. Once a good flake is chosen, it is placed on the stage of the micromanipulator. On the other hand, the glass slide with the chosen h-BN flake is positioned in the arm of the micromanipulator. The careful alignment of the glass slide and the chip seems to be crucial for an efficient pick-up: this is checked using a bubble level (also known as spirit level meter). For this purpose, the arm of the micromanipulator has been modified in order to be able to not only control the x, y and z motion of the arm and the angle between the z-axis of the arm and the z-axis of the stage, but as well the angle between the xy-plane of the arm and the table. This way, the PDMS stamp surface can be well aligned with the chip. In order to proceed to the pick-up, the arm of the micromanipulator is brought down towards the surface of the chip, making sure that the two flakes are aligned. The stage on which the chip sits is next heated to 100 ◦ C and the contact is established between the two surfaces. One can then observe through the microscope the growth and propagation of the area of the copolymer which is in contact with the SiO2 surface. This propagating area should move as slowly as possible and in a homogeneous way (not jerkily). Once this area encloses the two flakes, now lying on top of each other, the arm of the micromanipulator is brought up such that the area in contact quickly moves back and the contact is lifted: the graphene flake has

50

4.2. Fabrication (a)

BNtop

(b)

BNtop

(c)

Bi-Gr

BNtop

BNbottom

Bi-Gr

10 µm

10 µm

10 µm

Figure 4.3: (a) Optical microscope picture of a h-BN flake deposited on the PDMS stamp, previously covered with the copolymer. (b) The half-stack is shown: a bilayer graphene flake has been picked up by the h-BN flake lying on the copolymer. The full stack is shown in (c) (BNbottom /bilayer/BNtop ). The top layer has been cut compared to its original shape shown in (b), following the clear blue lines in (b).

been picked-up by the h-BN lying on the stamp. Such a situation is depicted in Fig. 4.3(b). However, if the copolymer around the two flakes stays too long in contact with the surface of the chip, it might be that the h-BN flake will get transferred on top of the graphene (instead of the graphene being picked-up by the h-BN flake). Another important trick to know is that it is most likely the larger flake which picks up the smaller one. Thus, it is convenient to start with a graphene flake smaller than the h-BN of the glass slide. Once the graphene is picked up, it should be checked with an AFM, such that every detail (wrinkles, bubbles etc.) can be identified and located to better decide where the future sample should be patterned.

4.2.3.3

Preparation of the bottom boron nitride

The next step consists in depositing this half-stack on the bottom h-BN. This bottom flake is prepared on a Si/SiO2 substrate in the standard way, as presented in the previous chapter. As mentioned in the previous paragraph, the biggest flake tends to attract the other flake(s). Thus if we want to transfer the half-stack on the bottom h-BN piece, it is preferable to choose a flake which is larger than the halfstack. In order to deposit the graphene-on-h-BN on the bottom h-BN, the glass slide is brought down towards the h-BN-on-SiO2 . The contact is made such that the contact area propagates slowly and controllably. Once the area surrounding the flakes is in contact, the arm is retracted such that this interface moves, this time, very slowly back (contrarily to the pick-up step, where the retractation was quick): the deposition is done. The result of this final step is shown in Fig. 4.3(c). In this picture, one can notice that the top layer of h-BN has been cut, unintentionally, during the retractation of the glass slide according to the pattern shown in blue dashed lines in Fig. 4.3(b). The cut followed existing cracks or defects, emphasizing the importance of selecting homogeneous flakes, free of any visible defects.

51

Chapter 4. Dual-gated bilayer graphene (a)

SiO2 Si BNtop Bi-Gr

(b)

BN BN

Au

SiO2 Si

(c)

BN BN

Au (c)

BNbottom

Au

Etched Stack

Au

10 µm

10 µm

Au

1 µm

Figure 4.4: (a) Schematic of a cut through the stack after etching (top part) and corresponding optical microscope picture of the real stack after etching (bottom part): only the blue rectangle is left. The position of the full flakes, etched away, is still highlighted and can be seen. (b) Schematic of a cut through the stack after contacting (top part) and corresponding optical microscope picture of the real stack after the lift-off (bottom part). (c) AFM image corresponding to (b) where the stack after contacting can be seen. The overlap between the gold contacts and the edges of the stack is clearly visible.

4.2.3.4

Patterning and contacting the stack

Once the three flakes are stacked, the patterning can be done. Since the graphene layer is buried under the top h-BN flake, we need to etch the stack such that the graphene can later be contacted on the edges. The first step is therefore to etch the stack in the desired shape (Hall bar, ring etc.). In the following, we design rectangular samples. In order to etch such a shape, we use a hard mask of hydrogen silsesquioxane (HSQ). First, a layer of PMMA 495 A4 is spun onto the surface and baked at 175 ◦ C for 15 min. This layer is necessary to enable the “lift-off” of the hard mask. The HSQ is then spin-coated and baked at 90 ◦ C for 5 min. HSQ is a very sensitive resist and is usually stored in a fridge. Thus, it should be taken out of the fridge in advance (a few hours before use) to be able to warm up to room temperature before spinning. The chip is now ready for E-beam lithography. The desired mask is exposed2 and developed, first in a solution TMAH3 , then in water and finally in IPA. At this stage, the developed stack consists of the h-BN/bilayer/h-BN stack covered by PMMA and HSQ. Everywhere around this rectangle, the HSQ has been dissolved by TMAH and only the PMMA is left (PMMA is not sensitive to TMAH). We 2

Note that HSQ is a negative resist, to the contrary of the resists used in the other processes presented here 3 TMAH: Tetramethylammonium hydroxide. This is usually used as a developer for photolithography resists and needs therefore to be diluted in water for our purpose.

52

4.2. Fabrication

therefore need to get rid of the layer of PMMA before starting the etching process. This is done with a short exposure to an oxygen plasma in the RIE setup. After this, only the E-beam-exposed resist stack is left and the rest of the chip is uncovered. The etching is done, in the RIE setup, using a plasma of CHF3 and oxygen. The concentration of gases is chosen to etch h-BN at a rate of 30 − 32 nm/min. We however do not need to etch through the whole stack. Since we want to pattern the graphene, we only need to etch until the graphene is reached (typically 1 min of etching is enough). The HSQ is then removed by putting the chip in warm acetone. The acetone dissolves the PMMA layer and therefore lifts off the hard HSQ mask. A schematic of the etched stack is shown in Fig. 4.4(a) and the corresponding optical microscope picture is shown below. The stack is now patterned and has to be contacted. This step is done following a similar recipe as in the dry transfer technique. It provides 1d side-contacts, as shown schematically in Fig. 4.4(b). However, as we can see here, and more clearly in Fig. 4.4(c), due to alignment uncertainty with the E-beam, one needs to plan a small overlap between the gold contacts and the top surface of the stack in order to be sure to make a contact at the side. This is the drawback of this technique: the gold overlap will act as a top gate and will be able to dope the graphene lying under it. It can result in the formation of barriers, influencing the transmission of carriers. This might be even more dramatic in the case of bilayer graphene where band gaps can be induced. This is why another technique has been developed by Dr. Srijit Goswami from TU Delft, which avoids these overlaps and provides perfect side-contacts. The idea is to use, instead of a hard mask, a sufficiently high PMMA layer. If the layer is high enough, only a small part will be etched away by the CHF3 /O2 plasma and the stack underneath will remain protected. This way, the first step consists now in exposing the contact paths, etching and depositing the metal immediately afterwards. The shape of the device is then defined in a second step, where the rectangular shape is now created. Top gates can be patterned on top of such a stack as well, though it is not as straightforward as it was in the dry transfer technique (the graphene is now reachable from every side of the rectangle - we would need either to avoid etching one edge or to deposit another dielectric to allow the metal to reach the top h-BN without touching the graphene edge). Since the results presented in this thesis do not make use of this technique, we won’t describe any further detail here. All top-gated devices presented in the following were patterned using the dry transfer technique presented earlier.

4.2.4

Conclusion

To conclude, in this thesis two techniques were studied which enable the encapsulation of a bilayer graphene flake between two h-BN flakes. Top gates can be patterned on top of the top layer of h-BN and displacement fields can be generated, allowing

53

Chapter 4. Dual-gated bilayer graphene

(a)

(b)

εTG

dTG

εBN

dBNtop

εGr

c0

εGr

c0

εBN

dBNbottom

εBG

dBG

εSiO2

dSiO2

Figure 4.5: (a) Schematic of the three plate capacitors. (b) Corresponding real stack, as produced and investigated in this thesis.

to study the physics of gapped bilayer graphene.

4.3

Electrostatics of dual-gated bilayer graphene

As demonstrated in Chapter 2, the layer symmetry of bilayer graphene can be broken if the two sublattices experience different potentials. Thus a way to electrostatically generate a band gap in its electronic structure is to control the two different layers with different gates. Sandwiching bilayer graphene between a top and a bottom gate, with dielectric spacers, provide a way to create an electric field perpendicular to the flake. Such a structure is depicted schematically in Fig. 4.5.

4.3.1

Basic electrostatic model

The structure shown in Fig. 4.5 can be seen as a series of three plate capacitors. The first one consists of the backgate and the bottom layer of graphene, the second one of the two layers of graphene and finally, the top layer of graphene facing the top gate. To understand what happens in the bilayer graphene sheet, we will now treat these capacitances as infinitely extended plate capacitors. This assumption is only valid if the dielectric thicknesses are much smaller than the width and length of the metallic plates, which is the case for a realistic sample. The voltage applied to each gate is responsible for a charge surface density σGr on each of the graphene layers. The relation between the charge surface density and the voltage is given by Gauss’s law and can be expressed as follows: BN dSiO2 + dbottom SiO2 BN SiO2 BN dtop top BN − 10 BN σGr

VBG = − 10 VTG =

54

bottom σGr

(4.1)

4.3. Electrostatics of dual-gated bilayer graphene

The total carrier density is then: top bottom σGr = σGr + σGr = −e˜ n 1 n ˜ = − σGr e 1 0 BN 1 0 BN SiO2 VBG + = VTG bottom e BN dSiO2 + dBN SiO2 e dtop BN

= β VBG + α VTG

(4.2)

(4.3)

where β and α are the relative lever-arms of the back- and top gates. ˜ resulting from the gating, as We next define the applied displacement field D, follows: top bottom ˜ = σGr − σGr D 0 e = (β VBG − α VTG ) 0

4.3.2

(4.4)

Environment

The above analysis does not take the environment into account. To better describe a real sample, environmental doping should be included in the model. This doping generates an intrinsic displacement field, which leads to an offset of the charge (0) (0) neutrality points VBG and VTG (0 stands for the charge neutrality point). This field can be expressed as a function of these two measurable quantities:  e  (0) (0) (4.5) Dintr = β VBG − α VTG 0 The intrinsic displacement field Dintr is added to the displacement field induced ˜ The effective density n of the graphene and the effective displaceby the gates D. ment field D are therefore expressed as follows: n

= β

D =

βe 0



(0)



(0)





(0)



VTG − VTG

VBG − VBG + α VTG − VTG



VBG − VBG −

αe 0



(0)



(4.6)

Without any voltage applied to the gates, the graphene is already under the influence of the doping. Thus the effective displacement field seen by the flake is non-zero. The zero-displacement field is reached when the intrinsic field is overcome, (0) (0) thus when the applied voltages are given by VBG = VBG and VTG = VTG .

4.3.3

Opening a band gap

Controlling the gate voltages is a way to control the density and the displacement field. Thus the application of gate voltages, generating asymmetry between the

55

Chapter 4. Dual-gated bilayer graphene

layers, can be directly related to the opening of a band gap of size Egap . However, estimating the asymmetry and the gap size requires more accuracy than the simple model previously described. Indeed, in the presence of an external displacement field D, as defined in the previous subsection, electrons in BLG tend to minimize their potential energy by rearranging themselves between the layers [30, 32]. As a result, the external field is screened and an electric field of smaller magnitude E exists between the two graphene layers. This electric field generates the true layer asymmetry u which is defined as the the on-site energy difference between the top and the bottom layer of the bilayer sheet. From the externally applied displacement field, one can self-consistently calculate the layer asymmetry u and further relate it to the band gap size Egap , yielding: | u | γ1 . Egap = √ 2 u + γ1 2

(4.7)

The complete description of the self-consistent calculation of u has been described in literature [42].

4.4

Transport in dual-gated bilayer devices

Now that the fabrication and the basic electrostatic model of dual-gated bilayer graphene have been presented, we discuss how such a device can be characterized.

4.4.1

Device presentation

For all the devices that will be presented in the following pages of Part II and in Part III, the basic stacking is identical and follows what has been presented earlier, using the dry transfer technique. The stack is shown in Fig. 4.6(a). Note that, before encapsulation, the surface of the bilayer graphene flake used in the device that will be investigated in this section has been cleaned using the mechanical cleaning technique [83], as presented in Chapter 3.

4.4.2

Inducing a band gap in bilayer graphene

To assess the possibility of opening a band gap with such a device, a uniform top gate is used [we call this kind of top gate geometry a “barrier” - see Fig. 4.6(a-b)]. The gate covers the whole width of the graphene (WBLG = 1.3 µm), such that, if the gate-induced displacement field is able to open a band gap, the transport between source and drain will be suppressed. In such a situation, a large increase of the resistance should be observed. However, the gate does not cover the whole length between the Ohmic contacts. Instead, Lgate = 1.1 µm, whereas the length of the bilayer graphene flake, i.e. the distance between Ohmic contacts, is LBLG = 3 µm. Such a device defines three

56

4.4. Transport in dual-gated bilayer devices

Bilayer

(b)

(c)

ngate

D

30

20

nn'n

np'n 0

G [e²/h]

SiO2 h-BN SiO2Si Si-BG

Au contacts

VBG [V]

(a)

-20

pn'p pp'p

-40 -6

D = -0,9 V.nm-1 -4

-2

0

VTG [V]

2

4

6

0

Figure 4.6: (a) Schematic of a dual-gated BLG device. The top gate (labeled TG) is a uniform Cr/Au barrier, covering the full width of the bilayer graphene flake lying under the top h-BN flake. (b) Optical microscope picture of the same device. (c) Measured 2d map of the previous device. The conductance G is plotted as a function of VBG and VTG .

regions in series. The two outer ones, called the leads and located close to the Ohmic contacts, are bilayer graphene regions which can only be tuned using the doped silicon backgate. Their length is Llead = 0.95 µm. In between these two regions, the central area, under the top gate, can be tuned by both the top- and backgate voltages [the carrier types – n or p – belonging to this area are labeled with a prime in Fig. 4.6(c)]. The use of these two gates also allows for the application of an interlayer asymmetry u and therefore allows to open a band gap in this region.

4.4.2.1

Influence of the gates - How to read a 2d map

A typical measurement result for such a dual-gated device is shown in Fig. 4.6(c). It shows the conductance G, measured between the two Ohmic contacts as shown in Fig. 4.6(a), as a function of the top- and backgate voltages. Two horizontal lines of low conductance (blue) are seen. They correspond to the charge neutrality in the two lead regions. Above these lines, the two lead regions of the device are both n-doped, while on the negative backgate voltage side, they are p-doped. Along the diagonal, another low conductance line is observed: this line, which defines the displacement field axis D (e.g. the line along which the asymmetry between top- and bottom-layer is increased), illustrates the situation where the Fermi energy lies in the band gap of the central region of the device. On the left side of the D-axis, the dual-gated area of the bilayer graphene flake is p-doped while on the right it is n-doped. As demonstrated in the previous section,

57

Chapter 4. Dual-gated bilayer graphene

Figure 4.7: Conductance along the D-axis, as a function of VBG . At very negative or very positive backgate voltages, i.e. at high displacement fields, the conductance is suppressed.

G [e²/h]

6 4 2 0 -60

-40

-20

0

VBG [V]

20

40

the applied displacement field is tuned according to: D=

  βe  αe  (0) (0) VBG − VBG − VTG − VBG . 0 0

(4.8)

The density of the graphene sheet under the top gate is then tuned as: ngate = β



(0)





(0)



VBG − VBG + α VTG − VBG .

(4.9) (0)

The intersection of these two axes (D = 0 and n = 0) lies at VTG = VTG and (0) VBG = VBG . The conductance along the displacement field axis is shown in Fig. 4.7. As we can see, the conductance is fully suppressed for VBG = −50 V, which corresponds to D = −0.9 V/nm. According to tight-binding calculations, such a displacement field would lead to a band gap of Egap = 80 meV. The gap size should nevertheless be accessible from transport experiments, by performing temperature dependent measurements. Here, the transport through a region with an opened band gap should obey an Arrhenius law. However, as demonstrated in [34, 36–38], transport through the gap is actually dominated by hopping processes, which obscure the true band gap size. This will be discussed in the following. 4.4.2.2

Characterization of the induced band gap

To get a better insight in our device, a good characterization of the band gap is needed. Measuring the resistance at a high constant displacement field should provide more information. In the following, we apply a constant backgate voltage VBG (VBG = 25.5 V) and record the current through the bilayer device as a function of the top gate voltage, for a constant DC bias Vbias = 100 µV. The current is next converted into a resistance value. The same trace is recorded for the whole range of temperature accessible in the measurement setup in use (T = 1.4 K ... 16.2 K). The outcome is shown in Fig. 4.8(a). We can observe a decrease of the resistance peak height, from 4.5 MΩ at T = 1.4 K to 236 kΩ at T = 16.2 K: the thermal energy facilitates the transport of charge carriers through the gap.

58

4.4. Transport in dual-gated bilayer devices

R [MΩ]

3

(c) 16

15

15

14

2 1 0

(b)16

T = 16.2 K -6

-5.5 VTG [V]

-5

14

13

12

0

Raw data Fit with ∆0 = 33.5 meV

ln(Rmax) [ln (Ω)]

T = 1.4 K

4

ln(Rmax) [ln (Ω)]

(a)

Raw data Fit with Egap = 1.6 meV 0.2

0.4 0.6 1/T [1/K]

0.8

13 12 0.2

0.4

0.6 0.8 1/T1/3 [1/K]

1

Figure 4.8: (a) Measured resistance versus the top gate voltage at VBG = 25.5 V, for different temperatures, ranging from T = 1.4 K ... 16.2 K. (b) From (a), the maximum of resistance at a constant temperature is extracted and plotted versus 1/T . A linear fit in the high temperature range is used to extract a lower bound of Egap . (c) Same data as in (b) plotted versus 1/T 1/3 : more data points can be linearly fitted, from which the characteristic energy scale ∆0 can be estimated.

To understand the involved thermal processes, we plot the maximum of the resistance for each trace as a function of 1/T . This is shown in a natural logarithm y-scale in Fig. 4.8(b). We observe two different behaviors. For the highest range of temperatures, the evolution of the maximum of resistance is exponential. However, it quickly becomes sub-exponential as the temperature is further decreased. The high temperature range is well described by an activated behavior. In the presence of an ideal band gap, the resistance is expected to follow an Arrhenius law such that Rmax ∝ exp(Egap /2kB T ), since a carrier needs to be provided with an energy of half of the gap size to reach the valence or the conduction band. From Fig. 4.8(b), an estimate of Egap can be extracted: Egap = 1.6 meV. However, since our range of temperatures is limited, one can consider this estimation to be a lower bound. Compared to this, the band gap expected from tight-binding calculations for the applied displacement field (D = 0.92 V/nm) is close to 80 meV, which is much larger than our range-limited estimation. The sub-exponential part reveals the important role played by disorder: transport is facilitated by the presence of mid-gap states where charge carriers are localized. Hopping transport occurs via these states and becomes the dominating transport mechanism at low temperatures. Similar observations were made on other dual-gated bilayer graphene devices [34, 36, 38, 86] and it was found that transport is better described by a 1/T 1/3 scaling law, characteristic for two-dimensional variable range hopping [87]. This is illustrated in Fig. 4.8(c). As done in Ref. [86], we use the Mott theory [87] for two dimensional variable range hopping to extract the energy scale ∆0 , on which this phenomenon takes place. Using the linear fit illustrated in Fig. 4.8(c), we find ∆0 = 33.5 meV. This allows to estimate an optimum hopping

59

Chapter 4. Dual-gated bilayer graphene

distance, 

doptimum = 

q

~vF 27/γ1 ∆0 πDOS(EF )kB T

1/3 

,

(4.10)

where γ1 is the interlayer hopping parameter, vF the Fermi velocity and DOS(EF ) the density of states at the Fermi energy of bilayer graphene [87]. We find doptimum = 65 nm at the lowest temperature, confirming that a non-negligible number of localized states must be present in order to allow traversing the gapped region of the device, which is of the size of the top gate, i.e. approximately 1 µm. 4.4.2.3

Influence of the oxide layer on the temperature dependence

We would like to highlight at this stage that the above-mentioned results did not match our expectations. The literature mentioned in the last subsection [34, 36, 38, 86] in comparison with our results reported measurements performed on dual-gated bilayer graphene. Nevertheless none of these devices was fabricated using h-BN as dielectric for both the top- and the backgate. The use of h-BN was expected to reduce the number of localized states and to increase the range of temperatures along which the transport is activated. This did not happen: the values estimated from our measurements are of the exact same order of magnitude as all the previously mentioned measurements. This indicates that localization might not depend on the choice of the oxide.

4.5

Conclusion

To summarize, we presented how dual-gated bilayer graphene devices can be fabricated and characterized. We furthermore pointed out that transport through the gapped region is dominated by hopping processes, which obscure the true gap size. Hence, there is no way to directly confirm the predicted gap size in the device using transport experiments. However, ARPES experiments [88] demonstrated that the gap matches well with theoretical predictions and can reliably be estimated using the self-consistent procedure, as presented in Refs. [30, 32]. We will see later that our quantum Hall effect measurements also bring an indirect confirmation of the gap size estimation (see Chapter 6).

60

Chapter 5 Magneto-transport in dual-gated bilayer graphene In this chapter, we present the behavior of a dual-gated bilayer graphene device exposed to a perpendicular magnetic field. We limit our investigation to a standard quality bilayer graphene device, where no broken symmetry states are observed. This allows us to use a description based on the LandauerB¨ uttiker approach, which is compared to the presented measurements. The model will be extended to high quality graphene in the next chapter, where electron-electron interaction plays an important role.

5.1

Introduction

The fabrication of locally dual-gated bilayer graphene was presented in the previous chapter. As demonstrated in Chapter 2, investigating the different degeneracies observed in the quantum Hall regime should enable the identification of the different low-energy features of the band structure. To do so, one first needs to understand the behavior of our devices under the influence of a magnetic field. For the geometry discussed in Chapter 4, three regions in series are created. We present an approach, based on the Landauer-B¨ uttiker formalism, which describes the behavior of the conductance through such a device.

5.2

Device presentation

The device studied in this chapter is similar to the one presented in Chapter 4. It consists of a bilayer graphene flake, sandwiched between two h-BN layers. The bilayer flake was cleaned only using thermal annealing before encapsulation. This cleaning step did not remove all the residues present on the surface and the resulting device is not one of the cleanest devices investigated by us. Two Ohmic contacts allow us to perform two-terminal measurements and a barrier top gate

61

Chapter 5. Magneto-transport in dual-gated bilayer graphene

40

12

np'n

10

0

8

-40 -4

6

pn'p

-20

4

pp'p -2

R [kΩ]

Figure 5.1: Two-terminal resistance measured as a function of VBG and VTG at T = 4.2 K. Along the displacement field axis, the resistance shows a moderate increase, from 2 kΩ up to 16 kΩ.

VBG [V]

20

14

nn'n

2 0 VTG [V]

2

4

has been patterned using E-beam lithography. The distance between Ohmic contacts is LBLG = 4.7 µm and the top gate is centered in-between, with a length Lgate = 1.5 µm. The device therefore defines three regions in series: a first BLG lead which is 1.2 µm-long, the dual-gated region with Lgate = 1.5 µm, and the second lead which has the same length as the first one. The two outer leads are tuned simultaneously by only the backgate and therefore always possess the same charge carrier density, whereas the dual-gated region is tuned by both the top- and the backgate. All the measurements shown in this chapter were recorded at a temperature T = 4.2 K. The standard electronic characteristics of the device is shown in Fig. 5.1. Here, the two-terminal resistance measured between the Ohmic contacts is shown as a function of the top- and backgate voltages. As explained in the previous chapter, the two charge neutrality lines define four regions of different polarity combinations, which are labeled in the figure (the prime refers to the dual-gated region). Along the diagonal line, which defines the displacement field D axis, the resistance increases, from R = 2 kΩ up to 16 kΩ. This moderate increase indicates that the barrier is not very efficient. Good devices with such a geometry usually show an increase of the two-terminal resistance up to more than 1 MΩ. This first measurement therefore points towards a poor quality device. To get more insight into the quality of the device, we perform magneto-transport measurements. Figure 5.2(a) shows the derivative of the conductance as a function of the backgate voltage and the magnetic field B. This measurement has been recorded at a constant top gate voltage value (VTG = 0 V). We observe quantum Hall conductance plateaus starting from B = 1.2 T. Solid and dashed lines indicate the position of the quantum Hall conductance plateaus expected from a plate capacitor model, taking into account the lever arm to the backgate β and the expected sequence of quantum Hall conductance plateaus of bilayer graphene.

62

5.2. Device presentation

4

3

3

B [T]

4

B [T]

(b) 5

2

-20

0

VBG [V]

(c)

0 -4

20

-2

(d) G [e²/h]

G [e²/h]

40

0 VTG [V]

ν=±4 ν=±8 ν=±12 ν=±16 ν=±20 ν=±24 -1.5 2 4

0T

30

60

20

0T

20 ν = -12

2 1

1 0

1.5 dG/dVTG(BG) [x 10-4A/V2]

(a) 5

10 ν = -8

ν = -4

0 -30 -20

-10

0 10 VBG [V]

5T 20

30

0 -4

5T -2

0 VTG [V]

2

4

Figure 5.2: (a-b) Landau fan showing the derivative of the conductance along the x-axis (which is the back or top gate voltage axis) measured as a function of the magnetic field B and as a function of VBG at VTG = 0 V (a) and of VTG at VBG = 9 V (b). The solid and dashed lines indicate the expected position of quantum Hall plateaus [legend indicated in (b)], according to a plate capacitor model. In (c-d), the corresponding conductance traces at constant B are shown for B = 0 ... 5 T. No contact resistance was subtracted.

Figure 5.2(c) shows the conductance cuts corresponding to the Landau fan where each trace corresponds to a constant magnetic field value, between B = 0 ... 5 T. On the hole side (negative backgate voltages), we observe the formation of nicely defined conductance plateaus at G = 4, 8, 12, 16...×e2 /h (indicated in the figure). A conductance plateau is also visible at G = 2 e2 /h, as seen in Fig. 5.2(a). Contrarily to the previous measurement, shown in Fig. 5.1, these observations indicate this time a good quality device. On the electron side, conductance plateaus are seen as well, but are slightly less pronounced. Figure 5.2(b) shows another quantum Hall measurement. This time the dualgated region is investigated by measuring the conductance as a function of the top gate voltage and of the magnetic field B. Here, the numerical derivative along the top gate voltage axis is shown. This measurement has been performed at a constant backgate voltage value (VBG = 9 V) and by sweeping VTG , i.e. the density under

63

Chapter 5. Magneto-transport in dual-gated bilayer graphene

the top gate. The plate capacitor model indicates that the conductance plateaus correspond to the ν = ±4, 8, 12, 16 quantum Hall states, seen from the overlap with the solid and dashed lines on top of the measurement. No broken symmetry state is visible. In this measurement, we can also see that the magnetic field threshold for the formation of plateaus is higher than in the case where the density in the bilayer graphene leads was the one being tuned, with plateaus appearing now at B ≈ 3 T. In a naive picture, one could interpret this observation as an indication that the quality under the top gate is reduced compared to the quality of the graphene leads. Another striking observation can be made in (d). Here, the conductance is shown at constant magnetic fields values, ranging from 0 to 5 T. It exhibits conductance plateaus at unexpected values, some of them not fitting any integer multiple of e2 /h. To summarize, we observed signatures of good quality transport in the graphene leads, where the quantum Hall plateaus were formed at smaller B-field values and where broken symmetry states are observed. Under the top gate, however, no strong resistance increase was seen as the displacement field was increased and the quantum Hall plateaus were formed at higher B-field values. Additionally, no broken symmetry state was observed. The Landau fan measured as a function of the top gate voltage in Fig. 5.2(b-d) however indicates an unusual quantization. This will be further investigated in the next section.

5.3

Conductance in strong magnetic fields

In 2007, two groups studied a similar device geometry as the one presented here, but with encapsulated single-layer graphene [89, 90]. The observations were similar, with an unusual conductance quantization observed in magnetic field. This geometry was also studied in Refs. [79, 91, 92]. To understand the behavior of such a system, we need to study how edge modes behave in our device. Three cases have to be distinguished, as shown in Fig. 5.3.

5.3.1

Unipolar regime

We first consider the regime where the carrier type is identical in the three regions of the device: this is the unipolar regime. Two cases exist. They are shown schematically in Fig. 5.3(a-b): in (a), the situation where the density is larger in the BLG leads than in the dual-gated region is shown. This means that more edge modes are present in the two outer regions (|ν| > |ν 0 |). We emphasize here that due to the disorder commonly observed in our devices, we do not consider any fractional quantum Hall state nor any edge reconstruction. Thus only the edge modes that exist through the whole device will be fully transmitted, while the others will be reflected at the interface, as shown in Fig. 5.3(a). The conductance will therefore be determined by 2 the filling factor sequence of the dual-gated area and follow: G = eh |ν 0 |. 64

5.3. Conductance in strong magnetic fields

(a)

(c)

(b) p

p

p

p’

𝝂𝝂𝝂𝝂′ > 𝟎𝟎 𝒂𝒂𝒂𝒂𝒂𝒂 |𝝂𝝂′ | < |𝝂𝝂| G=

𝒆𝒆𝟐𝟐 𝒉𝒉

p

p’

𝝂𝝂𝝂𝝂′ > 𝟎𝟎 𝒂𝒂𝒂𝒂𝒂𝒂|𝝂𝝂′ | > |𝝂𝝂|

|𝝂𝝂′ |

G=

𝒆𝒆𝟐𝟐

|𝝂𝝂′ |

𝝂𝝂 𝒉𝒉 𝟐𝟐 𝝂𝝂′ −|𝝂𝝂|

p

n’

p

𝝂𝝂𝝂𝝂′ 0) are illustrated: either the density, and therefore the number of edge modes, in the dual-gated region is smaller (a) or larger (b). In (c), the bipolar case is shown (νν 0 < 0).

The second possibility arising in the unipolar situation is when the density in the dual-gated region is larger than in the leads: |ν 0 | > |ν|. This is depicted in Fig. 5.3(b). As seen here, additional edge modes exist in the dual-gated region. Since they are at different potentials, partial equilibration between co-propagating edge states can occur [89]. We therefore follow a Landauer-B¨ uttiker approach and consider the different currents through the device, as shown in Fig. 5.4.

I

I1 = I + I4 It = (1-r) I1

|ν|

|ν|' |ν|

|ν| Ir = (1-r) I3

3

1

2

|ν’| – |ν|

4

|ν|

Figure 5.4: Sketch illustrating the labeling of the different currents propagating along the edges used in the detailed case presented in the main I2 = r I1 text (νν 0 > 0 and |ν 0 | > |ν|). We assume here that the current is only injected from the left to the right Ohmic contact.

We call I the current of the scattering state that is incident from the left Ohmic contact alone. At the top edge of the dual-gated region, the current will then be given by I1 = I + I4 [where I4 is the current along the edge labeled with a 4 in Fig. 5.4]. In the following, we assume that no current is injected from the right Ohmic contact. We furthermore assume that edge channels equilibrate on the upper and lower edge of the dual-gated region [89–92], i.e we assume that the gate width is larger than the equilibration length. This gives I2 = I3 = rI1 , with r being the reflection coefficient (r = (|ν 0 | − |ν|)/|ν 0 |). We obtain the following relations:

65

Chapter 5. Magneto-transport in dual-gated bilayer graphene

   I1          

= I + I4 = I + r2 I1 =

I2 =

            I4

=

I , 1 − r2

r I = I3 , 1 − r2

(5.1)

r2 I. 1 − r2

Next, we express the reflected and transmitted currents, Ir and It : r I, 1+r

  Ir    

= (1 − r)I3 =

    It

1 = (1 − r)I1 = I. 1+r

(5.2)

Hence, the reflection and transmission probabilities can be expressed as:     R          T 

= |ν|

r |ν|(|ν 0 | − |ν|) = , 1+r 2|ν 0 | − |ν| (5.3) 0

= N − R = |ν| −

0

|ν|(|ν | − |ν|) |ν ||ν| = , 0 2|ν | − |ν| 2|ν 0 | − |ν|

Thus, the conductance is given by: G=

e2 |ν 0 ||ν| . h 2|ν 0 | − |ν|

(5.4)

Such a configuration with integer filling factors can therefore be responsible for the appearance of conductance plateaus at fractional multiples of e2 /h. For example, in the case where |ν 0 | = 8 and |ν| = 4, the conductance would be G = (8/3) × e2 /h.

5.3.2

Bipolar regime

The last configuration to be investigated is depicted in Fig. 5.3(c): the bipolar case. In this case, following a similar Landauer-B¨ uttiker approach, the conductance is found to be [89]: e2 |ν 0 ||ν| . (5.5) G= h 2|ν 0 | + |ν| Hence, this case also gives rise to a conductance which is a fractional multiple of e2 /h.

66

(a) 50

(b) 50

32

VBG [V]

VBG [V]

5.3. Conductance in strong magnetic fields

G [e²/h]

0

-50 -4

-2

2

-50 -4

4

VBG [V]

(d) 50

0

-50 -4

-2

0 VTG [V]

-2

2

4

0 VTG [V]

2

4

10

0

-50 -4

0

dG/dVTG [e²/h/V]

VBG [V]

(c) 50

0 VTG [V]

0

-2

0 VTG [V]

2

4

-20

Figure 5.5: (a,c) The conductance and its derivative along the top gate axis are measured as a function of the backgate voltage and the top gate voltage at B = 5 T. (b,d) Calculation of the conductance and its derivative along VTG , based on the LandauerB¨ uttiker approach and the plate capacitor model which are described in the text.

5.3.3

Comparison between model and experiment

Now that the influence of the different filling factor combinations has been investigated, one can try to visualize the effect of the conductance quantization at a constant magnetic field. Fig. 5.5(a) shows the conductance measured at B = 5 T as a function of top and backgate voltages (T = 4.2 K). As expected, a rhombus pattern is observed. To make it more visible, we display the derivative of the conductance along the top gate voltage axis in Fig. 5.5(c). Using the previously presented Landauer-B¨ uttiker formalism, the calculated lever (0) (0) arms and the extracted doping offsets (VTG and VBG ), we next compute the predicted conductance as a function of the gate voltages. For this, we calculate the filling factors using the relations:

67

Chapter 5. Magneto-transport in dual-gated bilayer graphene

hnlead , eB hβ (0) = (VBG − VBG ) eB

ν =

hngate , eB hα (0) = ν+ (VTG − VTG ) eB

(5.6)

ν0 =

(5.7)

with α and β the relative lever arms of the top- and backgate (as defined in Chapter 4). The Landau level filling factors are then rounded to the closest multiple of 4, to mimic the expected bilayer graphene sequence (ν = 0, ±4, ±8...). This sequence is expected to occur in the dual-gated region (due to the limited quality, disorder is dominant and prevents the observation of the valley splitting which should arise due to the band gap). In the leads however, ν = 0 is not expected, but our observation is best reproduced if this filling factor is implemented in the model. The result is shown in Fig. 5.5(b). Both experimental and calculated results are displayed using the same color scale to facilitate the comparison. Since our simple model only describes a step function, the calculated conductance function has been smoothed using a moving average filter. The derivative of the conductance is also shown in Fig. 5.5(d). The overall agreement is satisfactory: qualitatively, the pattern is reproduced. However, details vary. Comparing the conductance data of calculation and experiment, we observe some discrepancy in the quantization. This is especially visible at high densities, where the measured conductance does not match the calculated one. This mismatch is likely to be due to the limited quality of the device, or to an uncomplete equilibration, as assumed in the model.

5.4

Conclusion

In this chapter, we have gained insight into magneto-transport properties of the dualgated bilayer graphene devices studied in this thesis. The device used to illustrate the Landauer-B¨ uttiker model of the three regions in series, with different filling factors, did not show a particularly good quality. Still, a qualitative agreement between theory and experiment has been observed. Better quality graphene, exhibiting a full lifting of the degeneracies, will be investigated in the next chapter.

68

Chapter 6 Probing the Lifshitz transition of high quality BLG using large displacement fields In this chapter, we present transport experiments which allow to identify the Lifshitz transition in a gapped bilayer graphene system. The quantum Hall effect is used as a probe to relate the different degeneracies of the Landau levels to different parts of the electronic dispersion. Investigating the very large displacement field regime, we present and analyze a kink in the measured conductance, close to the displacement field axis, which could be related to the van Hove singularity. The following results have been partially published in:

Anomalous Sequence of Quantum Hall Liquids Revealing a Tunable Lifshitz Transition in Bilayer Graphene A. Varlet, D. Bischoff, P. Simonet, K. Watanabe, T. Taniguchi, T. Ihn, K. Ensslin, M. Mucha-Kruczy´ nski and V. Fal’ko Physical Review Letter 113, 116602 (2014)

6.1

Introduction

As mentioned in Chapter 2, a way to tune and observe the Lifshitz transition in experiments is to apply an interlayer asymmetry to the bilayer graphene flake. To do so, one needs to apply a vertical displacement field D, generated using a combination of top- and backgates. Encapsulating the bilayer graphene flake is therefore required. As the results presented in this chapter have been obtained with a device fabricated using the dry transfer technique [68], we will first shortly describe this fabrication process. Nevertheless, the reader should keep in mind that recent improvements of fabrication techniques may lead to even better quality and therefore clearer observations [73, 85]. The electrostatics of the device will be explained as well

69

Chapter 6. Probing the Lifshitz transition of high quality BLG using large displacement fields 0 Amplitude error (mV) 30 1µm Ohmic

BLG h-BN

1µm Ohmic

(b)

Ohmic

BLG

(a)

h-BN Ohmic

1µm

(c)

h-BN BLG

Figure 6.1: (a) AFM image of the device after the first cleaning attempt using thermal annealing. A 1.3 µm-wide bilayer graphene flake (edges highlighted with dashed green lines), deposited on h-BN (entire picture frame) is contacted by four Ohmic contacts (their contours are drawn with solid black lines). A lot of residues are visible as bright yellow spots. (b) AFM image of the same device after an annealing step in air: the surface did not get cleaner but the residues are now movable, as they are moved by the tip during the scan. (c) AFM scan of the device after the last cleaning step, done performing an AFM scan in contact mode. The surface of the bilayer graphene stripe in between the innermost Ohmic contacts is now free of visible residues.

as the band gap properties. In the second part of the chapter, we will reveal how, using the quantum Hall effect as a tool, one can identify the presence of the Lifshitz transition via the study of the different Landau level (LL) degeneracies. Finally, we will present results from a second cool-down during which a higher displacement field regime was investigated, which revealed the presence of an extra feature, a dip in the conductance, which could be related to a van Hove singularity. The following results were obtained in collaboration with Prof. Vladimir Fal’ko and Dr. Marcin Mucha-Kruczy´ nski, who performed the self-consistent calculations for the gap estimation, the Landau level spectrum and the van Hove singularity.

6.2

Device presentation

The device fabrication starts with the exfoliation of h-BN on Si/SiO2 . Once a good candidate flake is identified and its cleanliness is confirmed by AFM, a bilayer graphene flake can be transferred on top. The transfer is done, as presented in Chapter 3, using the dry transfer technique: bilayer graphene is exfoliated on top of a Si-chip covered with a double-layer resist (in our case PVA + PMMA 1 ), the first layer being water-soluble while the second one is a standard polymer; once the bilayer nature of the flake is confirmed by Raman spectroscopy and its cleanliness checked with the AFM, the chip can be put in water: the sacrificial layer is dissolved and we are left with the graphene on PMMA floating on the surface. This floating polymer piece can then be “fished”, heated up and brought in contact with the h-BN using a micromanipulator. The flake is then etched and contacted using electron beam lithography. At the end of this step, the graphene surface is usually covered with resist residues. 1

PVA: Polyvinyl alcohol; PMMA: Poly(methyl methacrylate)

70

6.2. Device presentation

(a)

Bilayer Graphene S

BNbottom SiO2 Si-Backgate

BNtop

TG

D

(b)

5µm

BNtop

Bilayer Graphene

SiO2

Figure 6.2: (a) Sketch of the device. A contacted bilayer graphene flake is sandwiched between two h-BN flakes. The whole device can be tuned using the Si backgate. The central region can also be independently controlled using the central top gate. (b) Optical microscope image of the real stack. Two devices are shown and the measurements presented here were recorded on the lower one.

BNbottom

Different cleaning techniques have been established: thermal annealing [68, 93], current annealing [81] or mechanical cleaning [83], as presented in the second section of Chapter 3. The presented device has been cleaned using two of these techniques. This is illustrated in Fig. 6.1. First, an attempt was made to clean the device with a standard thermal annealing step in an Ar/H2 atmosphere. The surface did not get clean, as shown in Fig. 6.1(a). In a second attempt, we tried to remove more residues by performing an annealing in step air. As shown in Fig. 6.1(b), the surface cleanliness did not improve, but it helped to make the residues more movable [we can see them being dragged by the tip during the scan in Fig. 6.1(b)]. Finally, mechanical cleaning was performed, which appeared to be very efficient, as illustrated in Fig. 6.1(c). Since no resist residues can be observed on the bilayer graphene surface anymore, the device is ready for encapsulation. For this, a second h-BN flake is transferred on top on the stack in the same way as the graphene flake, and a top gate is patterned with electron beam lithography in the middle. The resulting device is sketched in Fig. 6.2(a) and shown in Fig. 6.2(b). All measurements that will be shown in this chapter were recorded in a twoterminal configuration. The two inner Ohmic contacts shown in Fig. 6.1(c) were used. The linear conductance G = I/U was obtained by applying a DC bias U = 100 µV and measuring the current I. A small AC top gate voltage superimposed on the DC voltage VTG was used to measure the normalized transconductance dG/dVTG . The measurements have been performed at a temperature of 1.6 K. The electrical characteristics of the device is shown in Fig. 6.3. Here, G has been been measured as a function of back- and top gate voltages, VBG and VTG . The two horizontal black stripes correspond to the charge neutrality in the two only backgated regions (the leads), which differ slightly in carrier density. The diagonal charge neutrality line corresponds to the dual-gated region and defines the axis of the displacement field, D. Along the direction of the ngate -axis, the density in the (0) (0) dual-gated region changes at constant D. These two axes cross at (VBG , VTG ) and define four quadrants, corresponding to different combinations of carrier polarities

71

Chapter 6. Probing the Lifshitz transition of high quality BLG using large displacement fields 30 0

20

32

nn'n

10

np'n

0

VBG [V]

Figure 6.3: Conductance measured as a function of the voltages applied to the backgate, VBG , and the one applied to the top gate, VTG . The star shows the displacement field value D = −0.9 V/nm, which is commented on in the main text.

G [e2/h]

-10 -20

pn'p

-30

pp’p

-40 -50 -6

-4

-2

0

VTG [V]

2

4

6

in the three areas. These configurations are indicated in Fig. 6.3 as nn0 n, pn0 p, np0 n and pp0 p, where the prime refers to the polarity of the central region. Increasing D (0) (0) from (VBG , VTG ) decreases the conductance by increasing the asymmetry u between the layers, which results in the opening of a band gap. As we can see, larger u widens the insulating (black) region. The largest displacement field values reachable at the two extremities of the displayed axis (one is highlighted with a star in Fig. 6.3) are D ≈ ±0.9 V/nm, which would lead to an asymmetry u close to 80 meV. The estimation of the gap size is done using a self-consistent procedure, as presented in Refs. [30, 32]. This allows to take the charge redistribution between the layers as a function of changing gap size into account, which leads to a different screening of the applied D-field. The good agreement between theoretical predictions and the induced gap size in the asymmetrically doped bilayer graphene devices was confirmed in the past by ARPES experiments [88]. The band gap size should nevertheless be accessible by performing temperature dependent transport measurements (transport through the gap should obey an Arrhenius law). However, as discussed in Chapter 4 and demonstrated in [34, 36–38], transport through the gap is actually dominated by hopping processes, which obscure the true gap size in the analysis of the transport data. Hence, there is no experimental way to directly confirm the estimated gap size in the device. We can nevertheless confirm that our estimation of u = 80 meV provides a realistic estimate, as confirmed by the quantum Hall effect measurements which will be shown in the next section. The strength of such an encapsulated geometry lies in the following arrangement: the h-BN material is so robust against electric fields that it is experimentally possible to open large band gaps, by applying large voltages. This is for example not possible with suspended devices, which would collapse due to electrostatic forces. In our case,

72

6.3. Probing the Lifshitz transition using the quantum Hall effect

VBG [V]

20

(a)

0

G [e2/h]

28

(b)

Figure 6.4: (a) Conductance as

dG/dVTG [e2/(h.V)] a function of the voltages applied -15 13

0

-20 -40 -5

0

VTG [V]

5

-5

0

VTG [V]

5

to the backgate, VBG , and the one applied to the top gate, VTG , measured at B = 6 T. (b) Normalized transconductance map measured in the same voltage ranges: a number of lines are revealed, running parallel to the D-axis. They correspond to the transition between quantum Hall conductance plateaus.

the flake remains unaltered and we are able to investigate this high displacement field regime. Robustness against electric fields, together with quality, is an important requirement to access the Lifshitz transition experimentally.

6.3

Probing the Lifshitz transition using the quantum Hall effect

As mentioned in Chapter 2, inducing a large asymmetry between the two layers of a high quality bilayer flake leads to a situation amenable for the observation of a Lifshitz transition, which in this case is a transition from one unique and continuous Fermi contour at positive or negative energy to a broken contour with three separate pockets close to the edges of the bands. This has consequences on the conductance quantization observed in the quantum Hall regime. At low magnetic fields a sixfold degeneracy should be observed (three without spin degeneracy). Thus a way to confirm the presence of the Lifshitz transition is to perform quantum Hall measurements on the previously presented device and to analyze the number of degeneracies. The results discussed in this section are described in more detail in Ref. [59]. Fig. 6.4(a) shows the conductance as a function of top- and backgate voltages taken at B = 6 T. This map is not qualitatively different from the measurement shown in Fig. 6.3. However, the conductance does get quantized, which becomes more visible when recording the normalized transconductance signal dG/dVTG . Two triangular regions are observed in Fig. 6.4(b), exhibiting a series of pronounced lines running parallel to the displacement field axis. These lines delimit a regime in which the density in the two outer regions of the device is larger than the density in the central one, e.g. more edge modes are present in the two outer regions (ν > ν 0 ). Hence, the measured conductance, which reflects the number of edge modes transmitted from source to drain, will be given by G = e2 ν 0 /h [89, 90, 92, 94]. In this regime, the measured conductance, and therefore the transconductance, reflects the quantum Hall effect in the dual-gated region. This regime was explained in detail in

73

Chapter 6. Probing the Lifshitz transition of high quality BLG using large displacement fields Chapter 4. Each of the lines running parallel to the D-axis can be interpreted as the transition between plateaus of quantized conductance in the gapped central area (a plateau gives rise to a zero transconductance and a transition between plateau to a maximum, in black, or to a minimum, in yellow). To make this more obvious, we investigate a conductance cut at high displacement fields (VBG = −61 V). Fig. 6.5(a) shows the corresponding conductance measured close to B = 6 T in red. It exhibits conductance plateaus at multiples of e2 /h, indicating that all the degeneracies are lifted at large magnetic fields. This reveals the presence of strong electron-electron interaction and therefore indicates a good quality of the sample. The same curve, measured on a wider density range, is shown in Appendix B. In Fig. 6.5(b), the Landau level spectrum is shown, where we measured the normalized transconductance between B = 0 and 6 T, allowing one to follow the evolution of the broken-symmetry states as a function of the magnetic field (the same measurement is shown, on a wider density range in Appendix B). Here, the conductance plateaus correspond to the black zero-transconductance regions. Going from high magnetic fields to lower values, an unexpected behavior is observed: following filling factors ν 0 = −4 and ν 0 = −5, it appears that, close to B = 5 T, these two quantum Hall states merge. Going to the lowest magnetic fields (B ≈ 2.5 T) where plateaus are still present, two strong conductance plateaus are observed. Cuts of the conductance around this magnetic field value (yellow dashed line) are shown in Fig. 6.5(a) (orange curves). This reveals that these two conductance plateaus correspond to the quantum Hall states ν 0 = −3 and ν 0 = −6. The threefold degeneracy indicates the presence of three spin-split orbits at the same energy, which could be related, as mentioned earlier, to the presence of the Lifshitz transition in the gapped bilayer graphene system. To get a better insight into these observations, we want to calculate the expected Landau level spectrum of our system. To do so, one first needs to evaluate the asymmetry parameter u. This is done taking the influence of the skew interlayer hopping γ3 into account and using a self-consistent procedure. The result is shown in Fig 6.6 as a function of D for ν 0 = 0 and ν 0 = −6 at B = 6 T. This demonstrates that the asymmetry values at the two filling factors differ by less than 10% from each other for a sufficiently large displacement field. Fig. 6.5(c) shows the calculated Landau level spectrum corresponding to gapped bilayer graphene. This calculation assumes spin-degeneracy, meaning that each branch in the spectrum would count twice if compared to the above measurement. In this case, the asymmetry value was estimated from the calculation shown in Fig 6.6 and adjusted within the allowed interval, defined by the boundaries set by ν 0 = 0 and ν 0 = −6, to better fit the our observation, providing the value u = 82 meV. Starting from low magnetic fields, this spectrum exhibits a threefold orbital degeneracy corresponding to the gap at ν 0 = −6 in the experiment. As mentioned earlier, this is due to the presence of the triplet at the top of the valence band, which gives rise, in magnetic field, to three equivalent orbits, separately confined

74

6.3. Probing the Lifshitz transition using the quantum Hall effect 8

(a)

𝜈𝜈 = −6

G [e²/h]

6

6T

2T

5.8 T

4

-8

dG/dVTG [e2/(hV)] -15 13

ngate [x1011 cm-2]

(b)

2.5 T

𝜈𝜈 = −3

2 0 VBG = -61 V -12 -10

𝜈𝜈 = −6

-6

ngate [1011 cm-2]

-1 -2 -2

-2

-3

-3

-4 -6

-4 -5

-6

-6

-2

(d)

-1

-5

-4

𝜈𝜈 = −3 0

i) ii) iii)

-10

E [meV]

(c)

B=0T

VBG = -61 V

-30

i)

valley K

u = 82 meV

ii)

valley K’

-40 iii)

-50

0

2

B [T]

4

6

Figure 6.5: (a) Conductance cuts taken along VBG = −61 V, measured sweeping VTG ∼ ngate , at two different magnetic field ranges: the measurements reveal a lifting of all the degeneracies at high magnetic fields and a threefold degeneracy at lower fields. (b) Measured Landau level spectrum, showing the evolution of the red conductance cuts from (a) as a function of magnetic field. (c) Corresponding calculated Landau level spectrum, obtained for a gapped trigonally-warped bilayer graphene system, with an interlayer asymmetry set to u = 82 meV. (d) Band structure of bilayer graphene at B = 0 T, close to the top of the valence. Some representative energy cuts are taken from it to highlight the possible orbital degeneracies of the system which will become relevant in a magnetic field.

75

Chapter 6. Probing the Lifshitz transition of high quality BLG using large displacement fields

120

B=6T

100

u [meV]

Figure 6.6: The interlayer asymmetry u is calculated selfconsistently as a function of the applied displacement field at B = 6 T for ν 0 = 0 and ν 0 = −6. The inset shows a schematic of bilayer graphene. The picture is by courtesy of Marcin MuchaKruczy´ nski who performed the calculations.

ν' = 0 ν' = -6

80 60

-u/2

40

A2 B 2

20 0

0.2

0.4

u/2

A1 B

1

0.6 0.8 D [V/nm]

1

1.2

within the three pockets defined by this triplet [see Fig. 6.5(d)i]. We tentatively ascribe the observed gap at ν 0 = −3 to the lifted spin degeneracy due to exchange interaction. While increasing the magnetic field, the spin-degenerate orbits grow larger in momentum space and start to mix, leading to a splitting of the triplet, until this degeneracy is finally fully lifted (already at B ≈ 4.6 T in the experiment). At even higher fields (B = 6 T) only one unique orbit is left [to be compared with Fig. 6.5(d)iii]. In-between these two fields, the Lifshitz transition takes place. In the band structure at B = 0 T, as shown in Fig. 6.5(d)ii, two contours coexist. The Fermi sea possesses, in its center, an electron-like island. This gives rise, in a magnetic field, to two separated counter-propagating orbits which do not mix and hence lead to an occasional orbital degeneracy [point encircled in Fig. 6.5(c)]: in the calculated spectrum, the gap of ν 0 = −4 closes and correspondingly the quantized plateau in the experiment disappears (dotted arrow). The magnetic field position of this crossing corresponds fairly well to the magnetic field position of the crossing in the measurement (see yellow arrow), indicating that the size of the band gap in the system must be quite close to the estimated one. The investigation of the Landau level degeneracies is therefore an appropriate tool to locate the presence of the Lifshitz transition in bilayer graphene. Beyond that, it is observed that odd filling factors ν 0 = −3 and ν 0 = −5 remain quantized with the expected conductance quantization as the Landau levels cross. Also, ν 0 = −3 vanishes at magnetic fields slightly below the field of the Lifshitz transition. This behavior of the odd filling factors may be due to interaction effects and remains to be further investigated on a theoretical level. To conclude this first experimental section, clear signatures of trigonal warping in a gapped bilayer graphene system were observed. The careful study of the Landau level degeneracies revealed the different topologies of the Fermi contours. However, more information is contained in the measurement. In between the two extreme regimes (the three equivalent orbits at both low field and low energy, and the lifting of all the degeneracies at high magnetic fields), a rich variety of interaction-driven transitions between quantum Hall ferromagnets was observed.

76

6.4. Evolution of the position of the Landau level crossing as a function of the displacement field

Figure 6.7: (a) At a constant magnetic field, one can convert VBG and VTG into ngate and D. This way, one can identify the position of the crossing (black arrow). We consider two points of constant transconductance surrounding the crossing region (encircled) and extract their position in D. (b) The result of the self-consistent calculation is shown for ν 0 = 0 (black) and ν 0 = −6 (red). The measured values are shown as dots for the two features encircled in (a) (red and blue dots).

6.4

Evolution of the position of the Landau level crossing as a function of the displacement field

Next, we analyze the displacement field dependence of the magnetic field value B∗ where the 2nd and 3rd Landau levels cross near the top of the valence band. The crossing observed in the measurements between ν 0 = −4 and ν 0 = −5 [dotted arrow in Fig. 6.5(c)] can be seen for a wide range of displacement fields and magnetic fields. It corresponds to the closing of a gap between Landau levels. To track the evolution of this crossing, we measure the transconductance at a constant magnetic field as a function of VBG and VTG , in a limited region of the pp0 p-quadrant and locate the position of the crossing in displacement field [see Fig. 6.7(a)]. Since the shape of the crossing evolves with magnetic field, we focus on two points of constant transconductance and track their positions [these two points are highlighted with black circles in Fig. 6.7(a)]. The results are shown in Fig. 6.7(b) as red and blue dots (representing the two points of constant transconductance). We clearly see a linear dependence of the crossing position in magnetic field on the displacement field. Calculations performed by Marcin Mucha-Kruczy´ nski allow to draw a comparison between theoretically predicted values for B∗ and our measurements. For this, a computation scheme was developed, in which the asymmetry parameter u is calculated self-consistently according to the calculated value of D (which is obtained from the gate voltages). Next, the Landau level spectrum at the given asymmetry value is calculated and the position of the crossing is observed. The value of D

77

Chapter 6. Probing the Lifshitz transition of high quality BLG using large displacement fields

80 60 40

np'n

60

nn'n

VBG [V]

20

G [e²/h]

Figure 6.8: New 2d map of the device during a second cool-down. The conductance G is measured as a function of VBG and VTG at T = 1.5 K. No contact resistance was subtracted. Compared to Fig. 6.3, the maximum displacement field that is reached is twice as big as in the previous sections: D = −2.1 V/nm. This is due to a change in the position of the charge neutrality points and to the higher voltages applied.

D

ngate

0

-20

pn'p

-40 -60

pp'p

-80 -10

-5

D = -2.1V/nm 0 5 VTG [V]

10

0

(and therefore of u) is then adjusted to make the position of the crossing fit our observation. This enables us to the display in Fig. 6.7(b) the relation between D and the magnetic field value B∗ for the two filling factors ν 0 = −6 and ν 0 = 0. These two values set margins for the theoretical expectation of B∗ . As we can see, the experimental and the theoretical dependencies of B∗ on D are qualitatively similar but differ in the details. The offset can be related to the error in the estimation of D, but the different slopes are at the moment still not understood.

6.5

Possible observation of the van Hove singularity at higher displacement fields

In this section, we present additional observations that we believe could be related to the Lifshitz transition. During a second cool-down of the above presented device, the environmental doping around the bilayer device changed a lot. This can be seen in Fig. 6.8: the position of the charge neutrality points of two untop-gated regions (blue horizontal lines) moved both towards positive voltages. This observation is unexpected, as the encapsulation should make the device robust against a changing environment. This shift is also responsible for a shift of the position of the origin of the displacement (0) (0) field axis, (VBG , VTG ), allowing us to reach larger displacement fields at comparable negative voltages. This, combined with the fact that larger gate voltages were applied during this cool-down (no leakage current was observed up to VBG = ±80 V), enabled the investigation of the very high displacement field regime. The largest reachable value is now D = −2.1 V/nm, as highlighted by the arrow in Fig. 6.8. The high displacement field regime has been investigated in the measurement

78

6.5. Possible observation of the van Hove singularity at higher displacement fields

(a) -30 -40 Figure 6.9: (a) Normalized transconductance dG/dVTG measured at high negative displacement fields [lower right region of the 2d map shown in Fig. 6.8]. We can observe an additional feature developing to the left of the D-axis, evolving as an orange line. (b) Conductance cuts corresponding to the lowest part of the range displayed in (a), taken between VBG = −70 V and VBG = −80 V: in conductance, the additional feature takes the shape of a plateau-like kink, evolving parallel to the gap (area where the conductance is suppressed). Here, a contact resistance Rcontact = 300 Ω was subtracted (estimated from quantum Hall effect measurements).

VBG [V]

-50 -60 -70 -80

60 dG/dVTG [e²/h/V] -120

4

6

(b) 140

VTG [V]

8

10

120 G [e²/h]

100 80

VBG = -80 V

60 VBG = -70 V 40 20 0

7

8

VTG9[V]

11

10

shown in Fig. 6.9. Fig. 6.9(a) shows the normalized transconductance dG/dVTG , which has been measured using a modulation of the top gate (no numerical derivation). The measurement shows the region close to the D-field axis which evolves diagonally here. On the lower left of the D-axis, we can see an orange line, developing at around VBG = −50 V which is running parallel to the axis. To better understand this feature, we show in Fig. 6.9(b) the corresponding conductance cuts, taken between VBG = −70 V and VBG = −80 V. Here, the feature is the strongest. We can clearly see that the previously observed orange line, parallel to the D-axis, is related to a plateau-like kink in the conductance, located at around G = 20 e2 /h. Since this kink is observed at very high displacement fields and close to the gap, we suspect that this feature is another consequence of the Lifshitz transition. We highlight here that the effect of the Lifshitz transition is still present in quantum Hall measurements and therefore survived the second cool-down and the changes of

79

Chapter 6. Probing the Lifshitz transition of high quality BLG using large displacement fields doping previously mentioned. The saddle point present in the band structure of gapped bilayer graphene at the Lifshitz transition gives rise to a peak in the density of states. This peak is called a van Hove singularity [95]. In the following we would like to argue that our observation, the appearance of a conductance dip at high displacement fields, is related to this singularity. We furthermore study the position of the conductance kink in density as a function of the applied displacement field. The result is shown in Fig. 6.10(a), where the transconductance from Fig. 6.9(a) is shown as a function of D and of the density of the dual-gated region, ngate . As seen here, for this range of displacement fields, the position of the observed singularity, which we emphasize with black dashed lines, does not have a linear evolution and occurs at densities between 1 × 1011 cm−2 and 3 × 1011 cm−2 . However, theoretical predictions do not match our experimental observations, as described in the following. Fig. 6.10(b) (black curve) shows the calculated position of the van Hove singularity of bilayer graphene in density as a function of the applied displacement field. Even though there is a qualitative agreement between theory and experiment for the overall evolution [comparing the black dashed lines in Fig. 6.10(a) and the black curve in Fig. 6.10(b)], there is a mismatch in the expected position of the singularity in density by a factor of 10. Further efforts have been made in order to match the theoretical behavior to our observation. First, as shown with the red and the two blue curves in Fig. 6.10(b), the input parameters of the Hamiltonian were slightly tuned (v, v3 and γ1 ). This did help providing smaller densities, but changes did not correct the order of magnitude mismatch between experiments and calculation. Next, refinements concerning the initial Hamiltonian and the integration limits in the momentum space were made and the screening model was improved. None of this helped to get closer to the density range observed in the measurement. The next reason that could explain this mismatch could be the physics happening at the interface of the barrier. The momentum matching at this interface could be further studied. However, since the crystallographic orientation of the sample is not known (the device was not etched along any “natural” edge), every possibility would have to be studied and computing blindly such a device does not seem appropriate. The position of the van Hove singularity is still under investigation at this stage.

6.6

Conclusion

Bilayer graphene is a highly tunable material. Small topological changes, such as the Lifshitz transition, can be largely influenced by external parameters, such as displacement fields and magnetic fields. This enabled us to reach a regime where the transition is brought to a more convenient range of density and where the quantum Hall effect can be used to probe the different degeneracies of the band structure. This resulted in the observation of the Lifshitz transition under the influence of the

80

6.6. Conclusion

-1.2

60 dG/dVTG [e²/h/V] -120

-0.6

v = 1.0 x 106 m/s, v3 = 0.1 x 106 m/s, γ1 = 0,38eV v = 1.0 x 106 m/s, v3 = 0.05 x 106 m/s, γ1 = 0,38eV v = 1.25 x 106 m/s,v3 = 0.0675 x 106 m/s,γ1 = 0,38eV v = 1.5 x 106 m/s, v3 = 0.05 x 106 m/s, γ1 = 0,38eV

D [V/nm]

-1.4 D [V/nm]

-1.6 -1.8

-1.0 -1.4

-2 -2.2 -1

-1.8 0 -0.5 ngate [1012cm-2]

0.5

1

-2.5

-2

-1.5 -1 -0.5 ngate [1012cm-2]

0

Figure 6.10: (a) Normalized transconductance dG/dVTG from Fig. 6.8(a) now plotted as a function of density and displacement field. We see that that the position of the conductance kink, highlighted with a black dashed line, has a non-linear evolution with increasing displacement field. (b) Calculation of the theoretically expected position of the van Hove singularity performed by Marcin Mucha-Kruczy´ nski. This calculation was done for the standard parameter values that were used in the previous sections (black curve), but was also extended to slightly varying parameters (red and blue curves). This figure is by courtesy of Marcin Mucha-Kruczy´ nski.

magnetic and displacement fields. Reaching even higher displacement field values, we could identify what could be another signature of the Lifshitz transition, which appeared as a conductance kink. This observation could be related to the van Hove singularity occurring at the transition. Bilayer graphene represents therefore a unique system, in which the topology of the band structure can be externally influenced and chosen.

81

Part III Band gap and chirality

© M-H Liu

Chapter 7 Introduction to Klein tunneling In this chapter, we review the basics of the Klein paradox, a relativistic phenomenon, and how it takes place in single-layer graphene. This phenomenon leads to high transparency of potential barriers and should therefore prevent the efficient confinement of an electron wave. Engineering smooth potential landscapes helps to circumvent this problem by generating what is called “Klein collimation”. As we will show it here, this effect enhances the confinement strength leading, in the case of a Fabry-P´erot interferometer, to a better visibility of the interference pattern.

7.1

Motivation

Interference of particles is a manifestation of the wave nature of matter. A wellknown realization is the double-slit experiment, which cannot be described by the laws of Newtonian mechanics, but requires a full quantum description. This experiment has been performed with photons [96, 97], electrons [98] and even molecules [99]. Another system widely studied in optics is the Fabry-P´erot (FP) interferometer, where a photon bounces back and forth between two coplanar semitransparent mirrors. Partial waves transmitted after a distinct number of reflections within this cavity interfere and give rise to an oscillatory intensity of the transmitted beam as the mirror separation or the particle energy is varied. In solid state physics, graphene has proven to be a suitable material for probing electron interference at cryogenic temperatures [100, 101]. However, in single-layer graphene (SLG), the realization of FP interferometers is challenging. The absence of a band gap and the Klein tunneling hamper the efficiency of sharp potential steps between n- and p-type regions, which play the role of the interferometer mirrors [12, 102, 103]. Theory suggests that smooth barriers enhance the visibility of interference effects [13, 91] due to Klein collimation [104]. Recently, ultraclean suspended SLG devices have shown FP interference with stunning contrast, using cavity sizes of more than 1 µm [78, 105, 106]. In this chapter, we will address different questions. We will first give a basic

85

Chapter 7. Introduction to Klein tunneling

Figure 7.1: Schematic of the potential barrier problem: the transmission of a particle with energy E, incident from region I on a potential barrier of height V0 , is studied. Classically, the particle cannot be transmitted when E < V0 . However, quantum mechanically, the particle has a certain probability to be transmitted in region (III), even if its own energy is less than the potential height V0 . If the particle is relativistic, the barrier can even become highly transmissive under certain circumstances.

L = 2a

E V0 I 0

II -a

III a

x

introduction to the Klein paradox, in order to later introduce how this phenomenon takes place in SLG. We will then further explain the effect of a smooth potential barrier and introduce the concept of Klein collimation. Finally, the concept of the Berry phase will be introduced and we will show how, in a Fabry-P´erot interferometer, its appearance under a magnetic field can be seen as a signature of Klein physics in graphene.

7.2

Particles scattering on a potential step: introduction to the Klein paradox

The behavior of a particle incident on a potential step has been widely studied since the beginnings of quantum mechanics. Whereas a classical particle is forbidden to overcome a potential barrier higher than the energy of the particle itself, a quantum particle would still have a certain probability to ‘tunnel’ through, as a consequence of its wave nature. In relativistic quantum mechanics, another odd situation has been pointed out by Oskar Klein in 1929 and remains known as the ‘Klein paradox’ [107]: under certain conditions, a potential barrier can become nearly transparent for relativistic particles. In non-relativistic quantum mechanics, a particle of energy E incident on a barrier of width L and height V0 , with V0 > 0, as the one depicted in Fig. 7.1, will be transmitted as an evanescent wave with a transmission probability T ∝ e−2κL √ 2m(V −E)

0 ). However, the wider or the higher the barrier is, the more rapidly (κ = ~ this probability vanishes. This celebrated problem is fully based on the Schr¨odinger equation. In 1928, Paul Dirac established the corresponding version of this equation for relativistic particles, now known as the Dirac equation [108]. The way was then open to solve the problem for relativistic particles. One year later, the Swedish

86

7.3. Klein tunneling in graphene

physicist Oskar Klein came up with a paradoxical result, which differs strongly from the non-relativistic one in the case of a sufficiently high barrier [107]. To illustrate this paradox, one can consider the most dramatic case where V0 → ∞. In this case, the relativistic transmission probability is given by E 2 − m2 c4 T = 2 1 2 4 E − 2m c

(7.1)

and therefore the barrier becomes transparent when E  mc2 . The key difference between the behavior of relativistic and non-relativistic particles is the fact that the latter have the possibility to tunnel through the barrier as evanescent waves, whereas the former will be able to propagate as their own antiparticles. This is a direct consequence of charge-conjugation symmetry. However, one has to highlight here that this ‘textbook’ situation, with a sharp barrier, is hardly achievable in experiments. The result only holds for barriers having a potential h ). If this condition is drop sharp compared to the Compton wavelength (λc = mc not fulfilled, an exponential decay of the transmission would as well take place, as demonstrated by Sauter [109] in 1932. Because of the difficulty of obtaining a such a sharp potential interface, the Klein paradox is still lacking an experimental confirmation for free relativistic particles. However, there exists a solid state physics system where the Klein paradox takes place as well: graphene.

7.3

Klein tunneling in graphene

Near the K-point, charge carriers in graphene follow a linear dispersion relation, with E = ~vF k. This was demonstrated in Chapter 1. They therefore behave as massless relativistic particles. Hence, graphene represents a solid state physics equivalent to a relativistic system, where the Klein paradox could be observed. In the following we will address the question of the transmission through a potential barrier taking into account the shape of the barrier and we will present how this translates in the experiments.

7.3.1

Sharp potential barriers

The problem of Klein tunneling in SLG has been addressed by Katsnelson et al. in 2006 [12]. In this chapter, as well as in the next one, we will follow a similar approach. We consider an electron wave incident on the barrier displayed in Fig. 7.1 (V0 > 0), where we assume invariance along the y-direction. When hole states are populating the barrier (V0 > E + ~vF |ky |), we find that the wavefunction in the three

87

Chapter 7. Introduction to Klein tunneling

L = 2a

y Figure 7.2: Schematic of the potential barrier in the xy-plane. Charge carriers are incident from region I on the barrier (region II) and are transmitted in region III. The incident and refractive angles, φ and θ are indicated.

I

II

III ϕ

θ

ϕ

-a

a

x

different regions can be written as follows: !

!

1 1 i(kx x+ky y) i(−kx x+ky y) ψI (x, y) = +r , iφ e i(π−φ) e se se !

(7.2a)

!

1 1 ψII (x, y) = a 0 iθ ei(qx x+ky y) + b 0 i(π−θ) ei(−qx x+ky y) , se se

(7.2b)

!

1 ei(kx x+ky y) , ψIII (x, y) = t seiφ

(7.2c)

where s and s0 the are the sign functions [s =sgn(E) and s0 =sgn(E − V0 )], φ and θ are the incident and refractive angles defined in Fig. 7.2, q is the transmitted wavevector and a, b, r, t are the amplitudes of the respective partial waves. The angles and wavevectors are related to each other in the following way: φ = arctan (ky /kx ), kx = kF cos φ, ky = kF sin φ, θ = arctan (ky /qx ), From the continuity of the wavefunction at the two interfaces, the coefficients a, b, r and t can be determined. In particular, we have: sin φ − ss0 sin θ , ss0 [e−2iqx a cos (φ + θ) + e2iqx a cos (φ − θ)] − 2i sin 2qx a (7.4) 2 from which the transmission probability can be calculated (T = 1 − |r| ). From this expression, one can first notice that a normal incidence always leads to full transparency of the barrier: this is the Klein tunneling in graphene. Two examples of T (φ), calculated using Eq. (7.4), are shown in Fig. 7.3, considering the incident energy E = 80 meV, the barrier height V0 = 300 meV, and the width L = 2a = r = 2ei(φ−2kx a) sin (2qx a)

88

7.3. Klein tunneling in graphene

(a) T=1 0,8

𝜋𝜋 22

5 12

0,6



4

𝜋𝜋  6 6

0,2 0.2

0.4

2

T=1

0,4

T = 0 0.

𝜋𝜋

(b)

𝜋𝜋 33 

0.6

0.8

0,8

2

5

𝜋𝜋 33

12



0,6 

12 0 0 1.



4

𝜋𝜋  66

0,4 0,2 T = 0 0.

0.2

0.6

0.4

00 1.

0.8

23  12

3 𝜋𝜋  −2 2

19  12

5 𝜋𝜋 −3

3

7 4



12

23  12

11𝜋𝜋  −6

6

7 5

3 𝜋𝜋 −2 2

19  12

𝜋𝜋 −3 3

4

11 𝜋𝜋

−6

6

Figure 7.3: Calculated transmission probability T as a function of the incident angle φ for barrier of width (a) L = 50 nm and (b) L = 200 nm, and of height V0 = 300 meV. The incident wave is set to an energy E = 80 meV.

50 nm in (a) and L = 200 nm in (b). In this plot, we can see that the transmission probability is very high for small incident angles and then exhibits additional fulltransmission resonances at finite angles which are (sometimes) called ‘magic angles’. One should mention at this stage that this result, based on the Dirac equation, can be reproduced by the tight-binding model-based Green’s function approach, which can easily handle arbitrary shaped barriers and also allow more general band structures [110]. This will be used for example in Chapter 9 to implement a band gap in bilayer graphene. While the fact that graphene allows for the observation of Klein tunneling is a great achievement, it brings however major drawbacks. The consequences of this effect on possible applications of graphene in electronics are far-reaching. Graphene is not only a gapless material and therefore possesses no ‘OFF’ state, but one also cannot use a potential barrier to confine the charge carriers. However, the smoothness of the barrier brings a partial solution to this problem.

7.3.2

Smooth potential steps

In the above problem, we have considered a perfectly sharp barrier, where the Fermi wavelength λF is large compared to the width of the smooth edge of the cavity d (i.e. kF d  1; d is the length over which the potential rises from 0 to V0 ). However, for electrostatically defined barriers or steps, the potential change is likely to be smooth compared to the Fermi wavelength. This problem has been addressed by Cheianov and Fal’ko [104] for a single pn interface and it was found that if the incident angle

89

Chapter 7. Introduction to Klein tunneling

(a) T=1

Figure 7.4: Calculated transmission probability T as a function of the incident angle φ for a smooth pn interface. The results are diplayed for two different widths of the smooth region: (a) d = 10 nm and (b) d = 50 nm. The incident energy has been set to E = 80 meV.

0,8

𝜋𝜋

22

π 2

𝜋𝜋  33

0,6

(b) T=1 

4

𝜋𝜋

66

0,4 0,2 T = 0 0.

0.2

0.4

0.6

0.8

0,8

𝜋𝜋

22

5 12

𝜋𝜋 33 

0,6 



4

𝜋𝜋  66

0,4

12

0,2

00 1.

T = 0 0.

0.2

0.4

0.6

0.8

23  12

3 𝜋𝜋 −2 2

is not too close to

5 12

19  12

5 𝜋𝜋 −3

7 4

00 1. 23  12

11 𝜋𝜋  −6 6 3𝜋𝜋  −2 2

3



12

19  12

5 𝜋𝜋 −3

7 4

11𝜋𝜋  −6 6

3

, the transmission probability is given by: 2

T = |t|2 = e−πkF d sin φ ,

(7.5)

where d is the width of the interface (in the limit of kF d  1). An example of such an angle dependence is shown in Fig. 7.4 for a width of (a) 10 nm and (b) 50 nm. The resulting function still exhibits a perfect transmission at normal incidence, but the trajectories incident at finite angles are exponentially suppressed. This result is known as “Klein collimation”. Charge carriers incident at finite angles therefore contribute less to the overall conductance. In the case of interference experiments, such as Fabry-P´erot (FP) interferometers, this results in an enhancement of the visibility of the interference pattern [13]. Recently, the combination of very high quality graphene with smooth potential profiles yielded the observation of FP interference over distances exceeding 1 µm [78, 105].

7.3.3

The true hallmark of Klein physics: the Berry phase

As explained before, collimated transmission is a signature of Klein scattering. However, graphene systems usually contain disorder, which reduces the visibility of the interference pattern. In addition to this, there is (so far) no direct way to access the angle-dependent transmission T (φ), since transport experiments only measure the total conductance G. This is why a true hallmark of Klein scattering has been sought. In 2008, Shytov et al. [49] realized that a signature of Klein tunneling in ballistic pnp FP cavities should appear in the magnetic field dependence of the interference pattern.

90

7.3. Klein tunneling in graphene

(a)

L

y x C

(b)

R

ky kx

(a)

L

y x C

(b)

R

ky kx

Figure 7.5: (a) Trajectory of an incident wave in real space through an npn junction (L for left cavity, C for center and R for right), in the case of a system having an imperfect transmission at zero magnetic field (not graphene). The double arrow indicates the reflected path. (b) In momentum space, such a trajectory would go through the origin, where the Berry phase would contribute to the phase. Figure 7.6: (a) In the case of SLG, a finite magnetic field is needed to obtain enough backscattering. (b) In momentum space, such a trajectory as the one shown in (a) would enclose the origin. At this magnetic field value, the Berry phase of π would be picked up, leading to a π-shift in the interference pattern.

Under a magnetic field influence, semiclassical electron trajectories are bent due to the Lorentz force. The different paths are elongated, leading to an additional kinetic phase ΦWKB . As soon as the trajectories form closed loops, areas are enclosed and therefore an Aharanov-Bohm phase ΦAB is added. Thus, the first effect of the magnetic field is to lead to a change of the resonance condition which will therefore be met at a higher density: this leads to a slow drift of the oscillations as a function of B, with a parabolic trend. Furthermore, the cyclotron bending contributes to the phase in a different way. At B = 0, the trajectory propagating at normal incidence will be fully transmitted. Thus there exists no such a trajectory as the one depicted in Fig. 7.5 which, in momentum space, would directly enclose the origin [see Fig. 7.5(b)]. However, as soon as B is large enough to lead to enough backscattering to form closed loops, the corresponding momentum-space trajectories will now enclose the origin. This is shown in Fig. 7.6: at this precise moment, due to the existence of a singularity at this point, the π-Berry phase is picked up. In the measurements, this will therefore lead to a shift of the oscillation pattern by half a period. Hence this π-shift constitutes a direct proof of Klein tunneling, through the analysis of backscattering. This observation has now been made for graphene cavities of various qualities and widths [13, 78, 105, 111].

91

Chapter 7. Introduction to Klein tunneling

7.4

Conclusion: Fabry-P´ erot interference in singlelayer graphene

In this first chapter, we have shown that single-layer graphene constitutes a solid state system where the Klein paradox can be probed. However, besides the fact that this allows to experimentally access this phenomenon, this also has a dramatic consequence: barriers in graphene are highly transmissive, preventing any efficient confinement. Engineering smooth barriers helps to attenuate this problem by collimating the transmission. In contrast to single-layer graphene, bilayer graphene is supposed to be a good candidate to build efficient potential barriers, as we will discuss it in the next chapter.

92

Chapter 8 Anti-Klein tunneling in bilayer graphene We now address the problem of the transmission of an electron wave through a potential barrier in the case of bilayer graphene. In this chapter, we will therefore demonstrate how, in bilayer graphene, anti-Klein tunneling takes place. We will then mention the fabrication of Fabry-P´erot interferometers.

8.1

Sharp barriers

In this chapter, we will focus on the transmission properties of potential barriers in bilayer graphene. We will demonstrate the presence of ‘anti-Klein tunneling’, meaning zero transmission at normal incidence. To do so, one investigates the same system as the one depicted in Fig. 7.2 of Chapter 7, but with bilayer graphene as a material. The Hamiltonian to be considered is the following: H = H0 + V0 ,

(8.1a) !

2

H0 = where m∗ =

γ1 2v 2

−~ 0 (kx − iky )2 , 2 0 2m∗ (kx + iky )

(8.1b)

= 0.033m0 . The Schr¨odinger equation can be written as:

0 (kx − iky )2 2 (kx + iky ) 0

!

2m∗ (E − V (x)) ψ1 ψ1 , =− ψ2 ψ2 ~2 !

!

(8.2)

yielding the following equation: d2 − ky2 dx2

!2

ψi =

2m∗ (E − V (x)) ~2

!2

ψi , i = 1, 2.

(8.3)

In order to illustrate the case where hole states are present within the barrier, ~2 k2 we will assume that we are in the situation where V0 > E + 2m∗y . We therefore define 93

Chapter 8. Anti-Klein tunneling in bilayer graphene

the wavevectors in each region as follows: √ 2m∗ E , region I : k = q ~ 2m∗ (V0 − E) region II : q = , ~ √ 2m∗ E region III : k = . ~

(8.4a) (8.4b) (8.4c)

The solutions in the three different regions are given by: !

ψI (x, y) = a1

!

!

ψII (x, y) = a2

!

1 1 ei(qx x+ky y) + b2 0 −2iθ ei(−qx x+ky y) s0 e2iθ se !

+ c2

!

1 1 eλx x+iky y + d2 e−λx x+iky y , 0 s h2 −s0 /h2 !

ψIII (x, y) = a3

!

1 1 1 i(kx x+ky y) i(−kx x+ky y) + b1 + c1 eκx x+iky y , 2iφ e −2iφ e se se −sh1 (8.5a)

(8.5b)

!

1 1 ei(kx x+ky y) + d3 e−κx x+iky y , se2iφ −s/h1

(8.5c)

where a1 is the amplitude of the incoming wave, b1 the amplitude of the reflected wave and a3 is the amplitude of the transmitted wave. The other parameters are defined as follows: kx = k cos φ ky = k sin φ

qx = q cos φ qy = q sin φ = ky √ λx = q 1 + sin θ2

q

κx = k 1 + sin φ2

s0 = sgn(E − V0 )

s = sgn(E) q

q

h1 = ( (1 + sin φ2 ) − sin φ)2 h2 = ( (1 + sin θ2 ) − sin θ)2 . As before the different amplitudes (ai , bi , ci ) are derived from the continuity condition at the two interfaces. This yields 8 equations with 9 unknown parameters and therefore only numerical solutions are possible. We can however express the analytical solution of the simple case arising at normal incidence (φ = θ = 0), when: 2

T = |t| =

2 a 3 a1

=

(q

2

4ikqe2ika . 2 −2qa 2 2qa + ik) e − (q − ik) e

(8.6)

From this expression, we see that when a → ∞, T approaches zero, in sharp contrast to the SLG case. Solving the general expressions numerically, one obtains the angle dependence of the transmission displayed in Fig. 8.1. Unlike for SLG, where the massless Dirac

94

8.2. Observing Fabry-P´erot interference in an ideal ballistic BLG np0 n junction (a) T = 11.0 0,8

𝜋𝜋 2

(b)

𝜋𝜋 3

0,6

T = 11.0 𝜋𝜋 6

0.5

0,4 0,2

T=0

0.2

0.4

0.6

0.5

1.0

𝜋𝜋 − 2

0.8

1.0

− −

𝜋𝜋 3

0,8

𝜋𝜋 6

𝜋𝜋 2

𝜋𝜋 3

0,6

𝜋𝜋 6

0.5

0,4 0,2 0

T=0

0.2

0.4

0.6

0.8

0.5

1.0

𝜋𝜋 − 2

1.0

− −

𝜋𝜋 3

𝜋𝜋 6

0

Figure 8.1: Calculated transmission probability T as a function of the incident angle φ for a np0 n barrier in bilayer graphene. The incident electron wave has an energy E = 30 meV and the barrier height is set to V0 = 100 meV. The result is displayed for two different barrier widths: (a) L = 2a = 50 nm and (b) L = 100 nm. Unlike the SLG case, there is always perfect reflection at normal incidence.

fermions are always perfectly transmitted at normal incidence, a perfect reflection is observed. For non-zero incidence angles, some ‘magic angles’ appear, where the transmission increases sharply to one. The total transmission, which is obtained by integrating the transmission over all incident angles, is therefore much smaller than in the SLG case. This is an interesting property, in the sense that electrostatic barriers in BLG are then highly efficient to confine carriers, making the material suitable for electronic applications. For very wide barriers, the amount of resonances can be quite large, increasing the computation time dramatically and therefore making it challenging to simulate real size devices.

8.2

Observing Fabry-P´ erot interference in an ideal ballistic BLG np0n junction

As previously mentioned, in transport experiments, the total conductance G is the only accessible parameter. In the case of a ballistic barrier, the total conductance 1 R relates to the transmission probability via G ∝ π T (φ) cos φdφ. Hence, the transmission function has to be integrated over the whole range of angles in order to be able to draw a comparison with experimental results. However the sharp resonances can easily pose a problem if one tries to calculate the conductance: the wider the barrier, the larger the amount of sharp resonances in the transmission function (see Fig. 8.2). In experiments, different groups have been reporting ballistic interference through 1 µm-wide SLG barriers [78, 105]. Thus a challenge for the BLG community would be to reproduce these results on such large scales. To the best of our knowledge, there exists no publication reporting the observation of ballistic interference in BLG. One reason might be that, since the transmission function exhibits such sharp reso-

95

Chapter 8. Anti-Klein tunneling in bilayer graphene (a)

Figure 8.2: Calculated transmission probability T as a function of the incident angle φ for a np0 n barrier in bilayer graphene (the energy and the barrier height are the same as in Fig. 8.1). The result is displayed for two different barrier widths: (a) L = 100 nm and (b) L = 400 nm. A larger number of resonances is observed when increasing the size of the ballistic barrier.

𝜋𝜋 2 T = 11.0

(b)

𝜋𝜋 3

0,8 0,6

T = 11.0

0.5

0,4 0,2

T=0

0.2

0.4

0.6

0.5

0.8

1.0

−

1.0

𝜋𝜋 − 2

−

𝜋𝜋 3

0,8

𝜋𝜋 6

𝜋𝜋 6

𝜋𝜋 2

𝜋𝜋 3

0,6

𝜋𝜋 6

0.5

0,4 0,2 0

T=0

0.2

0.4

0.6

0.5

1.0

𝜋𝜋 − 2

0.8

1.0

− −

𝜋𝜋 3

0

𝜋𝜋 6

nances, the overall amplitude of the conductance oscillations would be rather small and therefore hard to identify. Another reason comes from the way the FP interferometers are usually designed: the cavity is commonly defined using a top gate. However, such a top gate has another important effect on BLG: it induces a vertical displacement field which opens a band gap. The system is therefore not exactly the same as the one described in this chapter. However, how the band gap influences the transmission function is an interesting question. In the following chapter, we will therefore focus on the effect of the band gap opening on the observation of this well-known interference phenomenon.

96

Chapter 9 Inducing a cavity in a gapped BLG system: observation of Fabry-P´ erot interference In bilayer graphene, potential steps between n- and p-type regions are responsible for evanescent interface states, leading to a zero-transmission at normal incidence, known as anti-Klein tunneling. Furthermore, in bilayer graphene a band gap can be induced by a transverse electric field. The Berry phase of π in single-layer graphene has been predicted and observed to cause a phase jump of π in the Fabry-P´erot fringes at a weak magnetic field. In gapless bilayer graphene the Berry phase is known to be 2π, but it is yet to be understood how the opening of the band gap changes this value and how this further influences the interference effect. The present chapter is based on the experimental observation of ballistic interference in a dual-gated bilayer graphene device, which we identify as a signature of Fabry-P´erot interference in a gapped area of the device. This will enable the study of the effect of the gap on the transmission function and the Berry phase. The following results have been partially published in:

Fabry-P´ erot interference in gapped bilayer graphene with broken anti-Klein tunneling A. Varlet, M.H. Liu, V. Kr¨ uckl, D. Bischoff, P. Simonet, K. Watanabe, T. Taniguchi, K. Richter, K. Ensslin and T. Ihn Physical Review Letter 113, 116601 (2014)

9.1

Introduction

In this experimental chapter, we investigate Fabry-P´erot (FP) interference in a gapped bilayer graphene (BLG) system. The high quality of the BLG flake, combined with the device’s electrical robustness provided by the encapsulation between

97

Chapter 9. Inducing a cavity in a gapped BLG system: observation of Fabry-P´erot interference two hexagonal boron nitride layers, allows us to observe ballistic phase-coherent transport through a 1 µm-long cavity. In order to confirm the phase coherent origin of our experimental observation, a collaboration with the ‘Complex Quantum Systems’ group of Prof. Klaus Richter in Regensburg was established. Two postdocs from the Richter group, Dr. Ming-Hao Liu and Dr. Viktor Kr¨ uckl, performed quantum transport simulations and evaluated the Berry phase for gapped BLG, respectively, to confirm the origin of the observed interference patterns. Comparing experiment and theory allowed us to reveal the roles of the anti-Klein tunneling, the gap and the Berry phase. Most importantly, the gap was shown to destroy the perfect reflection for electrons traversing the barrier at normal incidence (anti-Klein tunneling). The dependence of the interference pattern on the carrier density, the magnetic field, the applied bias, and the temperature was further investigated, all consistently pointing towards the conclusion of FP interference. The theory data which is presented below is used with the permission from their authors, Dr. Ming-Hao Liu and Dr. Viktor Kr¨ uckl.

9.2

Motivation: Tunable transmission function

We first point out a simple but striking theoretical fact regarding a BLG np0 n junction (the prime is used in the following to refer to the central region). The perfect reflection across a bipolar potential step in BLG was predicted in Ref. [12], in the absence of the gap. Little attention has been paid however to the fact that the presence of the gap destroys the perfect reflection at normal incidence [112, 113]. By calculating the angle-resolved transmission T (φ) (using a Green’s function method based on a tight-binding model as in Ref. [110]), we find that a non-zero normal incidence transmission can be recovered by introducing a gap. As illustrated in Fig. 9.1(a–c), T (φ = 0) increases with the so-called asymmetry parameter u defined as the on-site energy difference between the two graphene layers [42]. Here we consider transport through an ideal, symmetric, sharp np0 n junction with the potential step height fixed at 40 meV (the Fermi level is kept in the center of the potential step) and u increasing from zero to 30 meV. Due to the two-fold valley degeneracy incorporated in the tight-binding model, the transmission function T has a maximum possible value of 2 within the single-band regime (|E| ≤ γ1 , where γ1 ≈ 0.39 eV is the nearest-neighbor interlayer hopping). As seen in Fig. 9.1, for such an energy/potential configuration, T (0) = 0 is only observed when u = 0 [Fig. 9.1(a)], which is the celebrated anti-Klein tunneling. It acquires finite values when u 6= 0 [Fig. 9.1(b–c)]. Recovering a finite T (0) with increasing band gap thus reveals the counter-intuitive role played by u, which increases the transmission by suppressing anti-Klein tunneling.

98

9.3. Device characterization

T

1.6 1.2 0.8 0.4 0 -90 -60 -30

EF VR VL

VC

L = 150 nm

1.6 1.2 0.8 0.4 0 60 90 -90 -60 -30

0

Φ

30

L = 150 nm

EF VR VL

VC VR

1.6 1.2 0.8

T

EF VL

VC

(c) Large gap (u = 30 meV)

(b) Small gap (u = 10 meV)

L = 150 nm

T

(a) Zero gap

0

Φ

30 60 90

0.4 0 -90 -60 -30

0

Φ

30

60 90

Figure 9.1: Effect of the band gap opening on the transmission function T as a function of the incident angle φ for a np0 n barrier in BLG. The step height is 40 meV and the Fermi level is kept in the center of this step. Three different gap sizes have been used: (a) u = 0 meV, (b) u = 10 meV and (c) u = 30 meV. We observe a strong modification of the transmission function, which is changed from anti-Klein tunneling to a ‘broken’ anti-Klein tunneling situation. The pictures are by courtesy of Ming-Hao Liu who performed the calculation.

9.3

Device characterization

The device under investigation is sketched in Fig. 9.2(a) and (b) (right device). It consists of a h-BN/BLG/h-BN stack which is deposited on a Si/SiO2 -substrate, prepared as described in Ref. [59] and in Chapter 4, using the transfer technique presented in Ref. [68]. The measurements were performed at a temperature of 1.6 K in a two-terminal configuration (using the inner Ohmic contacts). Unless stated otherwise, a constant bias voltage was applied symmetrically between the Ohmic contacts [S and D in Fig. 9.2(a)]. The current was measured, giving access to the conductance G. Modulating the top gate voltage with an AC component enabled the measurement of the normalized transconductance dG/dVTG . The device consists of three areas, as shown in Fig. 9.2(a): two outer areas (denoted as L and R, of length `L = `R = 0.95 µm) which are simultaneously tuned by the voltage VBG , and the central area (denoted as C, `C = 1.1 µm) which is tuned by both top- and backgate (VTG ,VBG ). The two-dimensional map presented in Fig. 9.2(c) shows the conductance of the device measured as a function of both VTG and VBG . Depending on the voltages applied, four different polarity combinations are possible: two where the three areas in series have the same polarity (pp0 p and nn0 n the prime referring to the central region C) and two where the outer regions and the central one have a different polarity (np0 n and pn0 p). The charge neutrality point ~ axis (diagonal of the dual-gated area spans along the so-called displacement field D ~ line of low conductance, labeled D). While increasing |D|, the conductance at the ~ charge neutrality point decreases as a gap is opened. This D-axis is defined as

99

Chapter 9. Inducing a cavity in a gapped BLG system: observation of Fabry-P´erot interference

5µm SiO2

g [e²/h]

2

Theory nn'n

V BG [V]

VBG [V]

Figure 9.2: (a) Schematic of the de- (a) vice: a BLG flake is sandwiched between two h-BN layers. The whole flake can be tuned by the Si backgate and the central area can additionally be independently Bilayer tuned using the central top gate. S and Graphene D S D are the Ohmic contacts enabling the BNbottom measurement of the conductance G. The SiO 2 orange plane at the top is a schematic of Si-Backgate the three regions defined by the gates: L (b) and R are the two leads and C is the dual-gated region. (b) Optical microscope image of two devices fabricated out of the same BLG flake. The dark yelBilayer Graphene low areas are Ohmic contacts (buried unBNtop BNbottom der BNtop ) and the clear yellow ones are G [e²/h] 0 0 30 the top gates. The measurements were (c) (d) Experiment carried out on the right device, using 20 → 𝐷𝐷 20 ngate the inner Ohmic contacts. (c) Conducnn'n np'n 0 np'n 0 tance versus top and backgate voltages measured at 1.6 K. The box indicates -20 -20 the range of the measurement shown in pn’p Fig. 9.3. (d) Calculated normalized con- -40 pp’p -40 ductance versus top and backgate volt-5 -5 0 5 V [V] ages. TG

pn’p pp’p V

0

TG

[V]

5

the zero-density line, meaning that it corresponds to the situation where the Fermi energy within this dual-gated region lies exactly in the middle of the gap. This band gap has been characterized with temperature-dependent measurements, which yielded an activated gap size of Egap = 2 meV at VBG = 28 V (corresponding to ~ = 1.1 V/nm). As previously reported [34, 36–38], transport in the gapped region |D| is dominated by hopping processes, preventing the experimental determination of the real gap size. Our calculations for the gate-dependent conductance use a Green’s function formalism [78, 114], based on the nearest-neighbor tight-binding model for BLG [43, 44] with the on-site energy profile extracted from the gate-modulated carrier density. To relate the carrier density to the gate voltages in the respective areas, we adopt a parallel-plate capacitance model. In area X (X = L, C, R), the carrier density nX = nX (VTG , VBG ) is composed of a top gate contribution ntX (only for X = C), a backgate contribution nbX , and an intrinsic doping contribution n0X . Following the tight-binding theory of the gap [42], the inputs of ntX , nbX , n0X allow us to compute the asymmetry parameters uX (VTG , VBG ). Finally, with nX and uX , the band offset for region X, which is to be added to the diagonal matrix elements (on-site energy) of the tight-binding Hamiltonian, VX , is calculated. This fixes the global Fermi level

100

9.4. Observation of an oscillatory behavior of the transconductance signal in the pn0 p regime at energy E = 0 where linear-response transport occurs. To keep the model as simple as possible, we neglect stray fields of the top gate and assume the potential profile to be ideally flat and sharp. We focus on the transport properties of the dual-gated region, modelling L and R as semi-infinite leads and regarding C as the scattering region [with the only exception in Fig. 9.5(e); see below]. Such a simplified model, which takes into account an ideal band gap [42], turns out to work surprisingly well: the main features of the measured conductance map [Fig. 9.2(c)] are well-captured by our calculation, as shown in Fig. 9.2(d). Note that different from Ref. [78], throughout this work we will not worry about mode counting and concentrate only on the normalized conductance g (with a maximum of 2 due to valley degeneracy).

9.4

Observation of an oscillatory behavior of the transconductance signal in the pn0p regime

We now focus on the pn0 p regime: Fig. 9.3(a) depicts the conductance signal as a function of the backgate and top gate voltages, showing a clear oscillatory behavior. To increase the visibility of the oscillations, one can look at the corresponding transconductance map shown in Fig. 9.3(b), which is measured within the same voltage ranges. Cuts within these two maps are shown in Fig. 9.3(c–d). They allow to better see the strength of the oscillating pattern. In the following, we will focus on the transconductance signal to analyze the oscillatory pattern in more detail. However, the same study could be carried out utilizing the conductance. (a)

G [e²/h]

8,8

(b)

10

-24

-15

dG/dVTG [e²/h/V] 15

D||

VBG [V]

-25

-27 -28 4.5

5

(c)

6 4.5

5.5 VTG [V]

(d) 15

dG/dVTG [e²/h/V]

VBG = -26 V G [e²/h]

9.6

5.5 VTG [V]

6

VBG = -26 V

5

-5

9.2 4.5

5

5

5.5 VTG [V]

6

-15 4.5

5

5.5 VTG [V]

6

101

Figure 9.3: (a) Conductance measured as a function of top- and backgate voltages in the pn0 p regime [this region is highlighted with a purple rectangle in Fig. 9.2(c)]. Conductance oscillations are clearly visible. (b) Normalized transconductance map, measured as a function of the same voltages as in (a): the oscillations appear more clearly. (c)/(d) Cuts along the top gate voltage axis taken from (a)/(b) (black lines) at VBG = −26 V.

Chapter 9. Inducing a cavity in a gapped BLG system: observation of Fabry-P´erot interference 2

2

e −1 −10 dG/dVTG [ h V ] 10

0.3

−24

−24

(d)

D||

−25

−26

−26

−27

−27 −28

−28 4.5

5

5.5

6

4.5

5

6

(b)

(e) 0.3

2

2

dG/ dVTG [ eh V −1]

5.5

V TG [V]

VTG [V] 10

VBG [V]

−25

g [ eh ]

VBG [V]

(a)

0

0.2

−10 2

2.2

2.4

2.6

2

nC [1012 cm − 2]

2.2

2.4

2.6

nC [1012 cm − 2]

2

(c)

(f)

0.02 0.01

1 0

2.2

4.4

6.6

ω [µm]

8.8

2.2

4.4

6.6

8.8

|F (ω)|

3

|F (ω)|

Figure 9.4: (a) Measured transconductance and (d) calculated normalized conductance in the pn0 p region as a function of the top- and backgate voltages. The grey arrows indicate faint horizontal patterns which originate from the graphene leads and ~ the slope of the D-axis is shown as a black solid line. (b)/(e) Cut along the dashed line in (a)/(d). By projecting the density nC onto the wavevector k axis, the discrete Fourier transforms of (b) and (e) are shown in (c) and (f), respectively, both of which exhibit a sharp peak at frequency ω = 2.2 µm, which is precisely twice the cavity length 2`C .

g [ eh ]

0.2

0

ω [µm]

~ As highlighted in Fig. 9.4(a), these oscillations evolve parallel to the D-axis (the ~ ~ slope of the D-axis is displayed in Fig. 9.4(a) as a black line labeled Dk). This indicates that they arise from a mechanism taking place in the dual-gated part of the device. In Fig. 9.4(b), an example of a transconductance trace taken at VBG = −26.7 V is shown. To confirm the origin of the oscillatory signal, we analyze the oscillation frequency of Fig. 9.4(b) by q first projecting the displayed density axis, nC , onto the wavevector axis, k = π|nC |, and then performing a discrete Fourier transform from k- to frequency space. Since the main contribution to the FP interference is given by phase difference ∆Φ = k · 2`C , the oscillation frequency is expected to be ω = 2`C = 2.2 µm. This is in agreement with our observations in Fig. 9.4(c), where we show the result of the discrete Fourier transform. A similar behavior has been observed for different density ranges in the pn0 p regime, indicating the robustness of the phenomenon (see Appendix C). Figures 9.4(d-f) are the theory counterparts of experimental Figs. 9.4(a-c), except that the theoretical analysis is performed using the normalized conductance (the calculation is ideally ballistic and therefore leads to a clear-enough visibility). We emphasize that the calculation was done based on the same simplified model as the one used in Fig. 9.1, free of any tuning parameters. In Fig. 9.4(f), in addition to the main peak, higher harmonics are visible at integer multiples of the fundamental period. The agreement between theory and experiment is satisfactory, confirming the ballistic origin of the fringes observed in Fig. 9.4(a) as FP interference in the

102

9.5. Magnetic field dependence

dual-gated area C.

9.5

Magnetic field dependence

To further understand the FP interference in BLG, we performed magneto-transport measurements and calculations in the low magnetic field regime, where the behavior of the conductance is still dominated by FP interference, without entering the quantum Hall regime. Figure 9.5 shows the measurement of dG/dVTG (a) as a function of the top gate voltage and the magnetic field and the corresponding calculation for g (b). The overall agreement between experiment and calculation is satisfactory: the number of oscillation periods, which can be directly counted (both 17 stripes), and the general field dependence (oscillations shifting towards higher densities with increasing magnetic field) are consistent. We can however distinguish two different regimes. In the higher range of magnetic fields, i.e. 20 mT . |Bz | . 60 mT, the agreement between measurement and calculation is good. This is depicted in Figs. 9.5(c) and 9.5(d) which are a zoom-in of Figs. 9.5(a) and 9.5(b) (dashed box). In these plots, one can easily see that the expected parabolic shifting trend of the oscillations is indeed observed in the experiment. However, some non-negligible discrepancies can be seen in the low-field behavior, in the range of |Bz | . 20 mT: starting from zero magnetic field and going towards higher values, it first seems that the oscillations are not affected by the increasing magnetic field. Then a sudden backwards shift happens while approaching Bz = 20 mT. To elucidate the peculiar field behavior observed only in the experimental map of Fig. 9.5(a) at |Bz | . 20 mT, we first investigate the accumulated phases for a closed loop in the dual-gated cavity C: it is given by a classical trajectory, which arises from backscattering at the interfaces, and encloses the area δA, as sketched in the inset of Fig. 9.5(b). The total phase can be decomposed into a kinetic part ΦWKB , the AharonovBohm phase ΦAB = eBz δA/~ and the Berry phase ΦBerry , as explained in Chapter 7 for the SLG case. Based on the phase difference between a transmitted and a twice reflected electron wave, we obtain the FP resonance condition: ΦWKB + ΦAB + ΦBerry = 2πj,

j ∈ Z.

(9.1)

As explained earlier, ΦWKB and ΦAB are magnetic field dependent and are the origin of the parabolic trend of the oscillations evolution as a function of Bz . What strongly differs from the previously studied cases of SLG is the Berry phase. In SLG, due to perfect transmission at normal incidence, the system requires finite magnetic field to build up trajectories which, in momentum space, enclose the origin and therefore pick up the Berry phase. In the present case, since even at Bz = 0 the transmission is finite (in the studied energy range), there already exists trajectories which go through k = 0 and therefore the Berry phase is already picked up. The Berry phase is therefore not a magnetic field dependent parameter here.

103

20 60

V BG = − 26.2 V

(d)

40

0.3

40 20 0

0.2 2

g [ eh ]

(b) 60

L

C

20 5.2

5.4

VTG [V] 40

(e)

20 0

R

δA

−20

−60 4.5

B z [mT]

−40 −60

−20 −40

40

2

20 0 −20

60

(c)

B z [mT]

5 0 −5

(a)

dG/dVTG [ eh V −1]

B z [mT]

60 40

B z [mT]

Figure 9.5: (a) Measured and (b) calculated magnetic field dependence of the FP oscillations at VBG = −26.2 V. The inset of (b) shows the closed loop, which is considered in Eq. (9.1). The dashed curves (sketched only for Bz ≤ 0 for clarity) are numerical solutions to the resonance condition (9.1) with (black) and without (red) the Berry phase. (c) and (d) are close-ups from (a) and (b), respectively, indicated by the black dashed boxes. (e) Calculated dg/dVTG map for a model with the scattering region simulating the full L, C, R areas.

B z [mT]

Chapter 9. Inducing a cavity in a gapped BLG system: observation of Fabry-P´erot interference

5

5.5

VTG [V]

6

−40 5.05 5.1 5.15 5.2

VTG[V]

It is nevertheless a non-constant parameter. As shown in more detail in Fig. 9.6(a), our computation of the Berry phase of BLG shows that it is only 2π for a vanishing asymmetry uC and can generally take values between 0 and 2π, depending on the carrier density nC and layer asymmetry uC . For a gapped configuration, the Berry phase will be zero close to the band edges (i.e. close to the gap) and then goes asymptotically towards 2π for increasing energy: the broken chirality is restored at higher energies. For the bipolar region presented in Fig. 9.4(a), ΦBerry is nearly constant, ranging between 1.22π and 1.46π [see Fig. 9.6(b)]. Thus the Berry phase only adds a phase offset which varies as a function of density. In order to better visualize this effect, one then numerically solves Eq. (9.1). We find a set of resonance contours well matching the periodicity of the FP fringes and located quite closely to the conductance maxima [see black dashed contours in Fig. 9.5(b)]. Furthermore, neglecting ΦBerry in Eq. (9.1) leads to the red dashed contours, which do not coincide with the positions of constructive interference. We therefore conclude that an effect of the Berry phase exists in the transport calculation, and is always involved independent of Bz . This is in sharp contrast to the case of SLG, where the Berry phase effect requires a weak magnetic field in order to overcome the perfect transmission of Klein tunneling [13, 49]. From the above discussion, the Berry phase can be ruled out as a possible reason

104

9.6. Bias dependence

ΦBerry / 2π

0

0.5

1

ΦBerry / 2π

(a)

20

VBG[V]

0.62

(b)

0 −20 −40 −6

−3

0

3

6 4.5

VTG [V]

0.72

5

5.5

−24

Figure 9.6: (a) Berry phase −25 as a function of top- and backgate voltages within the dual−26 gated area C of the device. (b) Zoom-in of the bipolar −27 block indicated by the white box in (a).

−28 6

VTG [V]

for the peculiar magnetic field behavior observed for |Bz | . 20 mT in the experiment in Fig. 9.5(a). We next suspect that the effect originates from the outer areas L and R. While the fringes reported in Fig. 9.4(a) have been confirmed to arise from FP interference in the C area, the L and R areas that are independent of VTG and slightly shorter than C may exhibit FP interference as well. Indeed, by taking a closer look at Fig. 9.4(a), one can identify a few horizontal patterns (see grey arrows) that are independent of VTG and may be attributed to FP interference within the outer regions L and R. Taking the cavity length of `L = `R = 0.95 µm, we estimate the expected voltage spacing to be around 0.6 V, consistent with those horizontal patterns observable in Fig. 9.4(a). By performing a ballistic calculation, taking into account the total 3 µm length of the L, C, R areas, we show in Fig. 9.5(e) a gatefield map focusing on a smaller VTG range. The peculiar behavior at low field close to zero is indeed recovered, supporting the idea that the outer areas are the main cause. The interplay between the different field dependencies of the FP interference in the L, C, R cavities is, however, beyond the scope of this discussion.

9.6

Bias dependence

We now focus on the bias dependence of the FP-modulated signal, shown in Fig. 9.7(a). The oscillations undergo a linear shift in top gate voltage as a function of the source-drain bias. Furthermore, their amplitude is modulated. To highlight this modulation, we calculate the standard deviation of the signal at each bias value [Fig. 9.7(b)]. A modulation can be observed, with two minima at Vbias ≈ ±0.6 mV. This dependence reflects the energy averaging effect of the bias voltage. We may understand the situation by considering the oscillating transmission as a function of energy within the bias window eVbias : each time the window encloses an integer number of oscillation periods, a minimum is reached in the signal. This bias characterization confirms once more that we interpret the data in the correct framework.

105

Chapter 9. Inducing a cavity in a gapped BLG system: observation of Fabry-P´erot interference

(a)

9.7

-10

dG/dVTG [e²/h/V]

10

(b)

1 Vbias [mV]

Figure 9.7: (a) Bias dependence of the FP oscillations. The oscillations undergo a linear shift and their amplitude appears to be modulated. (b) To make the amplitude modulation of the oscillations obvious, the standard deviation of the data measured in (b) is calculated along the same bias interval. Three maxima are visible.

2

0

-1 -2

VBG = -28 V 4 4.5

V

5 TG

[V]

5.5

6 3

4 5 σ [a.u]

Temperature dependence

Finally, we analyze the experimental effect of temperature T on the FP interference signal. Fig. 9.8(a) shows the FP interference for a temperature range of 1.37 K ≤ T ≤ 17.5 K. We can distinguish two regimes: at low temperature, the effect of the temperature on the oscillation amplitude is strong. However, for higher value of T , the effect saturates with some persistent oscillations. These almost temperature-independent features are not of coherent origin and we therefore subtract their contribution from the oscillations for the following analysis. 17,5 5 T [K]

0 T [K]

dG/dVTG [e²/h/V]

(a)

-5

-10

VBG = -28 V 1.8

2

ngate

(b)

Thermal damping

Tfit [K]

6

2.2 [x 1012 cm-2] (c) 1

4 2 0

0

1

2

3 4 Tmeasured

5

6

7

1,37

2.4

0.8

Data Thermal damping term

0.6 0.4 0.2 0

0

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐴𝐴

20

40 ∆E/kBT

2 𝜋𝜋 2 𝑘𝑘𝐵𝐵 𝑇𝑇 Δ𝐸𝐸

1 2 𝜋𝜋 2 𝑘𝑘𝐵𝐵 𝑇𝑇 sinh( Δ𝐸𝐸 )

60

80

Figure 9.8: (a) Temperature dependence of the oscillations. (b) Result of the temperature fit compared to the measured temperature. (c) Using the fitted temperature value, we compare the behavior of our signal with the thermal damping (black dashed line).

106

9.8. Conclusion

To get a better insight in the observed temperature dependence, we wish to find out how the effect of temperature can be modeled. From the raw data it is however impossible to find a matching model to describe our observation, which implies that one of the parameters used must contain errors. This can be seen from considering the derivative of the Fermi-Dirac distribution: 1 ∂f = ∂EF 4kB T

cosh2



1 , E − EF 2kB T

(9.2)

where T is the measured temperature (obtained from an Allen-Bradley resistance value) and E = ~2 πnC /2m∗ . Convoluting the lowest temperature curve with the derivative of the Fermi-Dirac distribution at T = Tmeasured , while varying the density on the same range as the one covered by the experiment, we indeed notice that we do not recover the full amplitude of the signal measured at T = Tmeasured : this confirms that one of our parameters is poorly estimated. This might be due to an unprecise evaluation of the energy (possibly due to screening or deformation of the band structure by trigonal warping, which are not taken into account in our simple calculation). In order to correct the discrepancy, we next fit the convoluted curve, using the temperature as a free parameter. The best fit temperature, once used as input parameter in the derivative of the Fermi-Dirac distribution, allows to recover the full amplitude of the measured signal. The result showing the correction on the temperature is shown in Fig. 9.8(b). Using the corrected temperatures, we now compare the standard deviation of each curve to the thermal damping term: 2π 2 kB T dG ∼A dVTG ∆E

1 !, 2π 2 kB T sinh ∆E

(9.3)

where A is a scaling parameter and ∆E is the averaged period of the oscillations in the studied interval. The result is displayed in Fig. 9.8(c), with A = 0.2 and ∆E = 1.2 meV. We find good agreement between the data points and the model and therefore attribute the damping of the oscillations to thermal averaging. This is in agreement with what is expected from FP interference.

9.8

Conclusion

In conclusion, we have observed Fabry-P´erot oscillations in a 1 µm-long gapped BLG cavity. We characterized the origin of these oscillations by studying their density-, magnetic field-, bias- and temperature-dependencies. Our calculations were able to reproduce our experimental observations and therefore demonstrate the importance of the tunable band gap in the system, which leads to a lifting of anti-Klein tunneling. This allowed us to confirm the ballistic phase-coherent nature of transport through

107

Chapter 9. Inducing a cavity in a gapped BLG system: observation of Fabry-P´erot interference the dual-gated region. Our work, combined with the recent advances in the quality of sandwiched structures [73] is a step towards future electron optics experiments in gapped BLG. The mechanism responsible for the lifting of the anti-Klein tunneling and the 2π Berry phase, however, remains to be explained. This will be addressed in the following chapter.

108

Chapter 10 Band gap and broken chirality In this chapter, we present the effect of the band gap opening on the pseudospin. We interpret the results presented in the previous chapter in terms of a breaking of the chirality in the energy range close to the gap. We show how the opening of a band gap, both in single-layer and in bilayer graphene, leads to the appearance of an out-of plane component for the pseudospin, which has consequences on the Berry phase.

10.1

Motivation

In Chapter 9, we described the observation of Fabry-P´erot interference in a dualgated bilayer graphene device. We demonstrated that the band gap which is opened in the bilayer graphene system plays a role in our observations and we could furthermore identify some consequences on transport measurements. We especially pointed out that, as a result, the anti-Klein tunneling gets broken and the Berry phase is not fixed to 2π anymore. However, the above-mentioned properties are only “symptoms” and one needs to understand further what basic property is so strongly perturbed by the gap that consequences on transport become measurable. To do so, we will start by introducing in more detail the concept of the Berry phase and explain what its quantization at integer multiples of π represents.

10.2

Berry phase

In this section, we will introduce the concept of Berry phase [115]. The Berry phase, also known as a “geometrical phase” or “Pancharatnam phase”, is the phase acquired by a system during an adiabatic evolution. The origin of such a phase can easily be understood with an illustrative example. Consider a vector moving on a sphere, as shown in Fig. 10.1. This vector is, at the starting point (point 1), tangent to the sphere and is, in this case, pointing

109

Chapter 10. Band gap and broken chirality

3 θ

2

Figure 10.1: Vector moving on a sphere, from the position 1 to the north pole and back to its original position by going along a different meridian. We see that the vector does not recover its original alignment: it has acquired a phase.

1

4 5

upwards. Moving the vector along a meridian towards the north pole while keeping its direction locked, brings the vector to the position labeled 3. Going down towards the equator, along another meridian, brings us to position 4. Finally, moving back along the equator to the initial position, we clearly realize that the vector is not aligned with its original direction: it has acquired a phase. It has turned by an angle θ which is nothing else than the angle separating the two meridians. This phase is called a geometrical phase. The appearance of such a phase is a direct consequence of the evolution of the system, and therefore it is a direct property of the Hamiltonian. We will start by presenting the Berry phase of a simple two-level system and subsequently turn to graphene and its bilayer version.

10.2.1

Berry phase in a two-level system

To be able to visualize what a Berry phase quantized in multiples of π indicates, we first introduce the concept of the Berry phase for a standard two level system (like the spin for example, which can be described by a superposition of spin up and down states). The generic Hamiltonian for such a system can be written as: H =h·σ

(10.1)

with h a real vector and σ the vector of Pauli matrices, defined as follows: 0 1 σx = 1 0

!

0 −i σy = i 0

!

!

1 0 σz = 0 −1

(10.2)

An arbitrary state can be represented by a superposition of the two basis states |−i and |+i (|↓i and |↑i for the spin) as: |ψi = α |−i + β |+i θ θ = cos |−i + sin eiφ |+i , 2 2

110

(10.3) (10.4)

10.2. Berry phase

|+⟩ 𝑧𝑧

𝑥𝑥

|𝜓𝜓⟩

𝑃𝑃

𝜃𝜃

𝜙𝜙

Figure 10.2: Representation of a quantum state |ψi on the Bloch sphere. This quantum state can 𝑦𝑦 be fully described by the polarization vector P, defined by its two angles θ and φ.

|−⟩

with |α|2 + |β|2 = 1, θ the polar angle (θ = 0 . . . π) and φ the azimuthal angle (φ = 0 . . . 2π) [116, 117]. The evolution of this state can be fully described by the polarization vector P. It can be obtained by taking the expectation values of the Pauli matrices as follows: 







sin θ cos φ Tr[σx |ψi hψ|]     P = Tr[σy |ψi hψ|] =  sin θ sin φ  . cos θ Tr[σz |ψi hψ|]

(10.5)

This vector describes a unit sphere, called Bloch sphere, or sometimes Poincar´e sphere. The quantum state |ψi and its polarization vector are shown on the Bloch sphere in Fig. 10.2. The two eigenstates of the Hamiltonian H, with energies E± = ±h, are given by: !

!

sin 2θ e−iφ |u− i = , − cos 2θ

cos 2θ e−iφ |u+ i = . sin 2θ

(10.6)

Using these eigenstates, the Berry connection A can be expressed. For a quantum state, it is defined, in its most general form, as follows [115, 117, 118]: A(k) = i hu(k)| ∇k |u(k)i .

(10.7)

In the spherical coordinate system, we find, for the lower energy level |u− i: Aθ = hu− | i

∂ |u− i = 0, ∂θ

Aφ = hu− | i

∂ θ |u− i = sin2 . ∂φ 2

(10.8)

To calculate the Berry phase Φ, which is the loop integral of the Berry connection on a closed path for an adiabatic evolution of the quantum state, we consider the simplest situation where the polarization vector evolves along a circle of constant latitude θ. This situation is depicted in Fig. 10.3. Here, the Berry phase is given by: Z 2π

Φ=

0

Aφ dφ = π(1 − cos θ).

111

(10.9)

Chapter 10. Band gap and broken chirality

|+⟩ 𝑧𝑧

Figure 10.3: We display here the polarization vector which has an adiabatic evolution at constant polar angle θ (angle between the vector and the z-axis) on the Bloch sphere. It encloses the solid angle Ω, which describes a portion of the sphere that we emphasize in yellow.

𝑥𝑥

𝛀𝛀

𝑦𝑦

|−⟩

The Berry phase is gauge invariant modulo 2π. Another way to visualize what the Berry phase really represents is shown in Fig. 10.3. Indeed, while precessing, the polarization vector describes a portion of sphere which is described by a solid angle Ω. Since the solid angle is given by: Ω=

Z θ Z 2π 0

sin θdθdφ = 2π

Z θ

sin θdθ = 2π(1 − cos θ),

(10.10)

0

0

we see that the expressed Berry phase is nothing else than half the value of this solid angle: Ω Φ= . (10.11) 2 The Berry phase is therefore half the area of the portion of sphere (yellow shaded region). In the case of an evolution in the equatorial plane (θ = π/2), the Berry phase would be equal to π (half the area of a half-sphere).

10.2.2

Berry phase in pristine single- and bilayer graphene

In single-layer and bilayer graphene, charge carriers are chiral. This means that their pseudospin is locked to their direction of motion. This gives rise to interesting properties that we will present here. In the vicinity of the K-point, single- and bilayer graphene are described by the Hamiltonians: !

HSLG = vF

0 π† , π 0

!

HBLG

1 0 (π † )2 =− ∗ , 0 2m π 2

(10.12)

with π the momentum operator as defined in Chapter 2, π = px + ipy . In the case of bilayer graphene, this Hamiltonian is an approximation of the 4×4 Hamiltonian presented in Chapter 2, which is appropriate to describe the system when the skew interlayer hopping is neglected. These Hamiltonians act on the

112

10.3. Berry phase in gapped single-layer and bilayer graphene

spinors (ψA , ψB )T and (ψB1 , ψA2 )T respectively. The associated eigenstates follow: E ± ψSLG

!

1 ±e−iφ , =√ 1 2

E ± ψBLG

!

1 ±e−2iφ =√ , 1 2

(10.13)

where φ = arg (kx + iky ). As mentioned in the previous section, we use the polarization vector P which is defined as the expectation value of the pseudospin operator σ, in order to visualize the motion of the pseudospin when the momentum rotates by 2π. We find: 

PSLG





cos φ   =  sin φ  , 0

PBLG



cos 2φ   =  sin 2φ  . 0

(10.14)

P has no z-component, due to the absence of diagonal terms in the Hamiltonians. This therefore depicts a pseudospin which rotates in the equatorial plane of the Bloch sphere (θ = π/2). This is shown in Fig. 10.4. In the case of SLG, the pseudospin rotates as fast as the wavevector and is therefore normal to the circle of constant energy [see Fig. 10.4(a)], whereas for BLG, the pseudospin winds twice as fast as for SLG [see Fig. 10.4(b)]. In both cases, the process is only momentum-dependent and completely energy-independent. This is highlighted by the different constant energy cuts in each dispersion. To summarize, in the case of pristine single- or bilayer graphene, the pseudospin is free of any z-component. Hence, the Berry phase represents the number of rotations of the pseudospin vector when the wavevector completes one rotation around the charge neutrality point. This is why the Berry phase is often called the pseudospin winding number. This integer number represents the degree of chirality [42]. Interestingly, this description is valid as well for multi-layer graphene. A J-layer graphene system can be described by the Dirac-like Hamiltonian: † J

HJ = gJ

0 (π ) πJ 0

!

  

g1 = vF g2 = 2m1 ∗ ,   g = v ... 3 γ2

(10.15)

1

The Berry phase is accordingly Jπ [42].

10.3

Berry phase in gapped single-layer and bilayer graphene

The pseudospin in single- and bilayer graphene is constrained to the xy-plane as a consequence of the lattice inversion symmetry. If this symmetry is broken, the situation is different. By applying a different potential to the two sublattices, the inversion symmetry can be lifted. In single-layer graphene, this can be done by aligning the flake on a

113

Chapter 10. Band gap and broken chirality

(b)

(a)

SLG

BLG

E ky kx

|𝜓𝜓𝐴𝐴 ⟩

|𝜓𝜓𝐵𝐵𝐵 ⟩

|𝜓𝜓𝐵𝐵 ⟩

|𝜓𝜓𝐴𝐴𝐴 ⟩

Figure 10.4: Pseudospin projection (blue arrows) along constant energy cuts (gray circles) in the dispersion of (a) single-layer and (b) bilayer graphene, around the K-point. The pseudospin motion is energy-independent in both cases. The lower parts show the pseudospin motion on the Bloch sphere. For single-layer graphene, it evolves in the equatorial plane and its projection encloses an area equal to half the sphere (yellow shading), leading to a Berry phase of π. In the case of bilayer graphene the same happens, except that the pseudospin spans twice the equatorial plane (orange shading), giving rise to a Berry phase of 2π.

114

10.4. Conclusion

hexagonal boron nitride substrate [119–122]. The closely similar lattice structure of graphene and h-BN results in locally aligning the atomic site A of graphene with a boron atom and its B site with a nitrogen atom (or vice-versa). The two carbon sites therefore experience different potentials. The same can be done in bilayer graphene, by setting the two layers to different potentials, using for example external gates. The resulting Hamiltonians are changed in the following ways: !

!

HSLG

u/2 vF π † , = vF π −u/2

HBLG

u/2 − 2m1 ∗ (π † )2 , = − 2m1 ∗ π 2 −u/2

(10.16)

with vF the Fermi velocity, u the asymmetry parameter and m∗ the effective mass of bilayer graphene (m∗ = γ1 /2v 2 ). What we can directly see here is that adding an asymmetry is related to adding diagonal terms to the Hamiltonians, i.e. a σz component. This therefore gives rise to a z-component of the polarization vector itself. In this situation, the pseudospin is not bound to the momentum plane anymore. This is shown schematically in Fig. 10.5(a) for the case of the conduction band of single-layer graphene, in the valley K. Here, we see that for small momenta (i.e. small energies, close to the band gap), the pseudospin points completely out of plane (red arrows). On the Bloch sphere, this means that no area is enclosed by the motion of the polarization vector and the Berry phase is 0. This is shown in Fig. 10.5(b) with the red arrow. The chirality is broken. Moving towards higher energies, the pseudospin tends to asymptotically recover its in-plane motion (blue arrows) and therefore a π Berry phase. The chirality is slowly restored [122]. In between the two extreme cases, the pseudospin is partially z-polarized, leading to a Berry phase varying between 0 and π, as shown with the green arrows and the green sphere segment in Fig. 10.5(b). Exactly the same thing happens with bilayer graphene when a band gap is induced. Close to the edges of the valence or conduction band, the pseudospin is fully z-polarized, leading to a Berry phase of 0. For increasing energies, the chirality is slowly restored, until the pseudospin returns to the xy-plane and the Berry phase is set back to 2π. This therefore explains the tunable Berry phase result presented in the Chapter 9.

10.4

Conclusion

In this chapter, we illustrated the concept of the Berry phase in a standard two level system. By analogy, we extended our description to the cases of single-layer and bilayer graphene. We further described how this well-defined value evolves as a band gap is being opened. We showed that the breaking of the inversion symmetry at the origin of the gap opening is responsible for the pseudospin acquiring a z-component at small energies. This has consequences on the value of the Berry phase. This chapter brings a tangible explanation for our experimental and theoretical findings presented in Chapter 9.

115

Chapter 10. Band gap and broken chirality

Figure 10.5: (a) Pseudospin projection (a) along constant energy contours of the conduction band of gapped single-layer graphene (gray circles), in the valley K. The process is now energy-dependent. ky (b) The pseudospin motion on the Bloch sphere is shown for three cases. The red kx arrows shows the orientation of the pseu- (b) dospin close to the band gap. No area is enclosed and the Berry phase is therefore 0. When going away from the gap, the zcomponent is decreased and the enclosed area grows (green arrows and green portion of sphere), until the chirality is restored at higher energies and the pseudospin returns to the equatorial plane (blue arrows), leading to a Berry phase of π.

116

|𝜓𝜓𝐴𝐴 ⟩

|𝜓𝜓𝐵𝐵 ⟩

Part IV Towards dual-gated bilayer graphene nanostructures

Chapter 11 Defining nanostructures by combining local top gates and the Si backgate In this last chapter, we present an attempt of inducing a quasi one-dimensional conduction channel in bilayer graphene, using a split gate geometry for the top gates of a dual-gated device. We demonstrate that a band gap is opened under the gates, but we point out that pinch-off cannot easily be achieved. We discuss the electrostatics of the device and the competing processes involved in the formation of a channel. We conclude that the standard split gate design does not provide us with enough control of the channel. We finally present an improved gate geometry, which might allow us to solve the problem.

11.1

Motivation

In Chapter 1, we explained the difficulties of engineering controllable nanostructures in graphene, using standard etching techniques [25–28]. We pointed out that even when bulk disorder is reduced, nanoribbons [26] and quantum dots [123] still strongly suffer from edge disorder. Alternative ways of patterning the edges are therefore needed, in order to control quantum confinement without being dominated by edge disorder. Bilayer graphene is a potential candidate to solve this issue. In bilayer graphene, combining the action of a top- and a backgate generates an asymmetry u between the two layers, which results in the opening of a band gap. This concept was theoretically introduced in Chapter 2 and experimentally investigated in the Chapters 4, 6 and 9. Hence, similar to the case of a sufficiently negative voltage applied on a top gate on GaAs leading to the depletion of the underlying 2DEG [124, 125], a top- and a backgate will induce a band gap in the sandwiched bilayer graphene region. Either way, this should enable the electrostatic confinement of charge carriers.

119

Chapter 11. Defining nanostructures by combining local top gates and the Si backgate (a)

(b)

Au TGs

200 nm

h-BN Au TG2

Amplitude error [mV]

h-BN SiO2 Si

40

Au TG1

0

Figure 11.1: (a) Schematic of the QPC device. It is similar to the devices presented in the previous chapters of this thesis, with the only exception of the top gate, which is now split into two separate gates. (b) AFM image of the two top gates. The white dashed lines highlight the outlines of the gates, which we expect to define a channel in between. The minimum distance between the two gates is Wchannel = 70 nm.

In this chapter, we investigate a split gate geometry in a dual-gated bilayer graphene device. We discuss the competing electrostatic processes that occur and emphasize how they affect our ability of controlling the potentially induced channel. We finally propose an improved design, enabling the full control of the channel.

11.2

Split gate geometry

Following the fabrication process described in Chapter 4, different top gate geometries can be defined during the very last E-beam lithography step. To electrostatically confine charge carriers in one-dimension, quantum point contacts (QPCs) are formed in GaAs heterostructures by using split gates [124, 125]. Similar top gate geometries can be designed on top of the top h-BN layer of a h-BN/bilayer graphene/hBN stack. An example of such a device is shown schematically in Fig. 11.1(a). This figure shows a standard dual-gated bilayer graphene device, as presented and investigated all along this thesis, with the only difference that instead of a “barrier” top gate, covering the full width of the graphene, there are now two top gates converging at the center of the device. A small gap exists in between these gates, in order to create a tunable bilayer graphene channel. Figure 11.1(b) shows an AFM image of a real device, showing this central region. The top gate outlines are highlighted with white dashed lines, which also indicate the shape of the potentially induced channel. The minimum distance between the two gates is Wchannel = 70 nm. The gates have a width of Wgate = 1.2 µm at their widest point. In the following, we present transport measurements where the current is recorded in a two-terminal geometry at a constant bias voltage Vbias = 100 µV and at a temperature of T = 4.2 K. The same voltage VTG is applied to the two top gates simultaneously. The two-terminal conductance, measured as a function of the top- and backgate voltages, is shown in Fig. 11.2(a). Here, we first identify a (dark blue) low conductance line evolving horizontally, close to VBG = 0 V. This line corresponds to the

120

11.2. Split gate geometry (a) 50

27

6

(b)

22 VBG [V]

D0

20 18

-50 -1

0

1

VTG [V]

2

3

1

0

200

VTG [V]

400

600

0.6

Figure 11.2: (a) Two-terminal conductance measured as a function of the top- and backgate voltages. The two top gates were swept simultaneously. A minimum in the conductance close to G = e2 /h is observed. (b) Conductance measured as a function of the same two voltages, but on a reduced range, centered around the minimum of conductance (which changed value). In the two measurements, no contact resistance was subtracted.

charge neutrality line of the regions which are not covered by the top gates (bilayer graphene leads): below it, the carriers in these regions are holes whereas they are electrons above. In addition, we can distinguish a diagonal line, which corresponds to the displacement field axis. In order to provide the reader with a good visibility, we do not mark this axis in the figure, but simply indicate the slope and direction of the displacement field with grey arrows. As for barriers, the displacement field axis indicates the charge neutrality under the top gates. This means that along this axis, the Fermi energy lies in the center of the gap which is induced under the split gates. In order to define a channel, we should therefore ideally keep the voltages along this axis, to keep the Fermi level in the center of the gap. Deviating from the displacement field axis would make the top gated regions conductive again, by shifting the Fermi energy in the conduction or valence band. A first striking observation can be made by examining the evolution of the conductance along the displacement field axis. Contrarily to the barrier case, by increasing the displacement field, the conductance does not exhibit a continuous decrease. A minimum is even clearly visible for positive values of displacement fields in Fig. 11.2(a). At this point, the conductance is close to G = e2 /h. During a second measurement shown in Fig. 11.2(b), the conductance was recorded as a function of the top- and backgate voltages in the region close to this minimum. For the same reasons as before, we do not draw the full displacement field axis, but simply highlight its slope with white dashed arrows. This time, the conductance was below e2 /h. In this figure, we can see how the conductance along the displacement field axis first decreases with increasing the displacement field, then reaches a minimum (dark blue region) and finally increases again. This behavior is unexpected. In a ‘naive’ picture, one would rather expect the

121

Chapter 11. Defining nanostructures by combining local top gates and the Si backgate (a)

BLG

(b)

DG-BLG

DG-BLG

(c)

BLG

Figure 11.3: (a) Schematic of the bilayer graphene (BLG) flake. The green color indicates the areas which are only under the influence of the backgate and the yellow color indicates the areas of the flake which are non-conducting due to the band gap which is opened by the gates. In this case, the gapped regions are exactly the dual-gated bilayer graphene (DG-BLG). Due to the gap, a channel is induced in the center and the electrons flow through this constriction. (b) Pinch-off situation: the stray field originating from the top gates opens a band gap in the constriction as well, resulting in barrier-like gapped region. (c) Schematic of a device in which the stray field is not strong enough to compete with the increasing density in the channel: the channel is then very conductive, preventing the pinch-off.

following scenario. By increasing the displacement field, a gap is slowly opened under the two gates. When the induced gap is large enough, the situation should be as depicted in Fig. 11.3(a). Here, the yellow areas represent the bilayer graphene regions located directly below the top gates. This part of graphene is therefore nonconducting (because the Fermi energy lies in the center of the gap and Egap  kB T ) and the charge carriers are forced to move through the constriction (red arrows). When the displacement is further increased, the influence of the stray field should be enhanced, leading to the gapped regions extending inside the constriction, until finally reaching the “pinch-off” at sufficiently high displacement fields. This is depicted in Fig. 11.3(b). Here, the gapped region has extended to the whole yellow region, similar to a barrier case. No charge carrier can be transmitted through this region anymore and transport is suppressed (G = 0). Increasing the displacement field further should enlarge the gap, but not affect the conductance, which should stay zero. However, as seen in the measurements shown in Fig. 11.2, this ‘naive’ scenario is not correct: the pinch-off never occurs and after reaching a minimum, the conductance increases again. This indicates that the stray field is never strong enough to close the constriction.

11.3

Electrostatic limitations

In this simple description, we didn’t take into account that a competing process takes place in the channel. When increasing the displacement field, the backgate voltage increases. However, the entire device is under the influence of this gate, and in particular the regions which are not covered by the top gates. Therefore

122

11.3. Electrostatic limitations 1

22 dG/dD [a.u.]

18

-1

VBG [V]

20

0

200

VTG [V]

400

Figure 11.4: Same data set as the one displayed in Fig. 11.2(b), now showing the derivative of the conductance along the displacement field axis (highlighted with the white dashed arrows) as a function of top- and backgate voltages. Conductance resonances become visible (black dashed line). These resonances span nearly parallel to the top gate axis.

600

the density in these regions keeps increasing. If our device is unable to reach the pinch-off situation, we can conclude that the backgate has a stronger effect on the channel than the stray field originating from the top gates. Instead of the situation depicted in Fig. 11.3(b), our device is in the situation shown in Fig. 11.3(c). Here, a band gap is opened under the top gates. However, upon further increasing the displacement field, the density in the channel increases more and more and the stray field is not strong enough to compete and lower the density of the channel. A confirmation of this scenario is shown in Fig. 11.4. Here, we show the same measurement as in Fig. 11.2(b), but we now display the derivative of the conductance along the displacement field axis. A series of conductance resonances appears, with slopes that are almost horizontal (indicated by the black dashed line). These conductance resonances correspond to localized states in the channel area [126]. These resonances evolve with a relative lever arm γchannel = αT G /βBG which is much smaller than the one of the dual-gated bilayer graphene regions (DG-BLG). The dual-gated bilayer graphene is tuned with a relative lever arm which is given by the slope of the displacement field axis, γDG−BLG = α/β (α and β were defined in Chapter 4). We find here γchannel = −0.5 while γDG−BLG = −31. It is therefore clear that the channel is mostly influenced by the backgate and that the stray field induced by the top gates is therefore rather inefficient. In the case of the presented device, one can blame the top layer of h-BN as the reason for the small stray field. Indeed, this layer is, in this particular device, only 7 nm-thick, whereas the distance between the two split gates is Wchannel = 70 nm. Thicker h-BN top layers might be a solution to induce more stray fields. The weak stray field is a clear disadvantage of the presented device. However, the conductance is still noticeably reduced and quantized conductance might occur. Many plateau-like conductance features are observed, adjacent to the minimum conductance region. However, none of these conductance plateaus is located at the expected conductance value. Fig. 11.5(a) shows the conductance along the displacement field axis. Along this line, the Fermi energy lies in the middle of the gap, opened under the top gates. As

123

Chapter 11. Defining nanostructures by combining local top gates and the Si backgate

30

6

G [e²/h]

15 10

20

5 0 -50

(c) 8

(b) VBG [V]

G [e²/h]

(a) 20

0

VBG [V]

50

4

dG/dVTG [x10-3 e2/(h.V)] 50 10 -30

-1

-0.5 0 VTG [V]

0.5

2 10

20 30 VBG [V]

Figure 11.5: (a) Conductance along the displacement field axis [blue line in (b)] plotted versus backgate voltage (to avoid inaccuracy in calculating D). Several “kinks” are observed in the conductance. A contact resistance of 200 Ω was subtracted, in order to make two conductance plateaus fit to G = 2 e2 /h and G = 4 e2 /h (black arrows). (b) Derivative of the conductance along the top gate voltage axis in the voltage range of the minimum of conductance. The direction of the displacement field is indicated with a blue solid line. Many conductance resonances are seen, at various positions. (c) Cut taken in (b), at VTG = 470 mV. The previously observed resonances give rise to plateau-like features in the conductance, which cannot be distinguished from a possible conductance quantization.

we can see, several plateau-like features are present below G = 5 e2 /h. They are however more numerous than expected and removing a contact resistance does not help to restore a sequence of conductance plateaus at integer multiples of e2 /h. In Fig. 11.5(a), we have subtracted Rcontact = 200 Ω, in order to fit some conductance plateaus at G = 2 e2 /h and G = 4 e2 /h (black arrows). However, due to spin and valley degeneracies, we expect a conductance quantization in integer multiples of G = 4 e2 /h. To emphasize the presence of a large number of conductance kinks in the signal, we display the derivative of the conductance along the top gate voltage axis, close to the minimum of conductance, in Fig. 11.5(b). Many parallel lines are observed, indicating conductance resonances. In this figure, no dominant resonance is observed, making it impossible to distinguish between quantized conductance plateaus and mid-gap states resonances (which are usually observed close to the displacement field axis [126]). Depending on the voltage configuration, some conductance plateaus can nevertheless appear close to integer multiples of e2 /h, as shown in Fig. 11.5(c). However, as no finite bias spectroscopy was performed on the device, we cannot conclude on their nature. Further measurements would have been required to better characterize the conductance evolution. Our observations reveal a major limitation of the split gate geometry. Due to the competing effects of the top- and the backgate on the channel, the channel is difficult to tune and the pinch-off might be unreachable. It would be preferable to be able to tune the displacement field (and therefore the gap below the top gates) independently from the density in the channel. We would therefore need an

124

11.4. Improved design

Au TTG

hBNtop-top

SiO2 Si

Figure 11.6: Improved geometry allowing for a better control over the channel. Starting from the device shown in Fig. 11.2(a), an extra h-BN layer has to be transferred on the top surface. A toptop gate (TTG) can then be patterned at the top, allowing to independently tune the region in between the split gates.

additional parameter, which would enable the independent tuning of the constriction without increasing its density, as will be discussed in the next section.

11.4

Improved design

To have more control over the channel, an additional top gate is needed. This can be done in two different ways. The first one would consist in adding a third top gate, called a finger gate, in between the two split gates. This idea was used in Ref. [35] 1 . However, this would require the three gates to be located very close to each other, in order to avoid creating two channels (one in between each split gate and the finger gate). As the realization of split gates separated by 50 nm up to 70 nm is already challenging (some gold residues are often left between the gates, leading to shorts), this option is too risky. As an alternative, one can position this additional gate in a higher level. The idea is to have one layer for the channel definition (where the top- and backgates are only used to open a band gap under the split gates) and an additional layer for the tuning of the channel. This is shown in Fig. 11.6. Starting from the structure shown in Fig. 11.2(a), one can add a layer of h-BN (called h-BNtop−top ). We have to highlight here that the use of the dry transfer technique is probably necessary for this step (presented in the Chapters 3 and 4). With the pick-up technique, presented in Chapter 4, it is, first of all, very challenging to deposit h-BN on such a structure and moreover, it is also probably too violent to come in contact with the glass slide on the electrodes. Once this transfer is done, the top-top gate can be patterned using E-beam lithography. As depicted in Fig. 11.6, this additional gate can follow the path of one of the split gates, in order to avoid tuning the rest of the bilayer graphene flake. An image of a real device, fabricated following the above-mentioned scheme, is shown in Fig. 11.7. We first present in Fig. 11.7(a) a device, similar to the previously studied one, which exhibits split gates at the top (the two leftmost yellow gates, labeled with the red letters “TG”). A zoom on the center of the two gates is shown in Fig. 11.7(b) with an AFM image. The same device is shown after the last 1

In this paper however, the finger gate tuned much more than the channel only: it covered the whole graphene width and should therefore have acted as a barrier.

125

Chapter 11. Defining nanostructures by combining local top gates and the Si backgate (a)

(b) BNt

(c) TG1

BLG

BLG

10µm

TG2

10µm

Figure 11.7: (a) Optical microscope image of a stack. A bilayer graphene flake is sandwiched between a bottom layer of h-BN (BNb in dark blue) and a top layer of h-BN (BNt in clear blue). The flake is contacted with 6 Ohmic contacts (which appear dark yellow as they are buried under BNt ). Three top gates are also visible, in bright yellow (they are labeled “TG”, in red). The two leftmost top gates form a split gate geometry, as shown with an AFM image in (b). (c) The same device after the transfer of the top-top h-BN (edges highlighted with pink dashed lines) and the definition of the top-top gate (edges highlighted with solid black lines). This gate covers the channel between the two split gates underneath.

encapsulation in Fig. 11.7(c). The third h-BN layer has been transferred on top of the previous stack (pink dashed lines) and a final top gate (solid black lines) has been patterned, covering the gap between the two split gates. Unfortunately, this part of the device didn’t work and no measurement can confirm our expectation of better tunability of the channel.

11.5

Conclusion

In summary, we have demonstrated that dual-gated bilayer graphene QPCs are challenging to implement in encapsulated devices. Due to a limited resolution, patterning very narrow split gates is challenging and thick layers of h-BN are required as upper dielectric, if one wants to obtain large enough stray field to fully control the channel. This however didn’t prevent Goossens et al. to report on the first dual-gated bilayer graphene quantum dot using h-BN encapsulation [39]. We should however point out that the pinch-off of the constrictions was also not achieved (G ≥ e2 /h), even though the upper h-BN layer was 50 nm high. To tackle this issue, we proposed an improved design for a quantum point contact geometry, which allows to separate an electrostatic definition layer, which is responsible for the opening of a band gap under the top gates and therefore for the creation of the channel, from the tuning layer, which is located higher up and allows to fully control the density in the channel. This idea is based on a work from Allen et al. [35] on dual-gated suspended bilayer graphene, where gate-defined

126

11.5. Conclusion (a) BNb

BNt

Source BLG

(b)

TG Drain

PG TG TG

(c)

TG

Source

Drain TG TG

T PG G

Source BNtop-top

Drain TG TG

Figure 11.8: (a) Schematic top view of a dual-gated bilayer quantum dot (QD). The dot is electrostatically defined by using the backgate and the top gates labeled TG, and the plunger gate is used to tune the dot levels. This geometry requires good coupling between the center of the dot and the plunger gate, i.e. the stray field should be strong enough to tune the central region. If the stray field is a problem, an alternative geometry can be used: (b) In this figure, we show three gates which allows to define a QD. In (c), the same device is encapsulated under BNtop−top and the plunger gate can directly come on top of the dot, for a better coupling.

BLG nanostructures were for the first (and only) time successfully demonstrated. Experiments on QPCs, designed with a finger gate in their center, were shown in the supplementary material. Again, even though quantized conductance was observed, no pinch-off was demonstrated (G ≥ e2 /h). This is possibly due to the gaps between the finger gates and the split gates. Positioning this tuning gate on a higher level would enable to cover the full distance between the split gates and might solve this issue. Similar geometries can be adopted to realize quantum dots. This is illustrated in Fig. 11.8. Instead of having a plunger gate on the same level as the dot-defining gates, as illustrated in Fig. 11.8(a), the plunger gate could be positioned on top of the structure. To do so, only three top gates are required to define the two constrictions and the dot. This is shown in Fig. 11.8(b). Then the top-top h-BN layer is transferred on the existing structure and the plunger gate can be patterned as depicted in Fig. 11.8(c), allowing to tune the quantum dot. For an even more complete control, one could also think of having top-top gates on the two constrictions. The proposed geometries could be implemented for various electronic nanostructures and therefore open the way to experiments on better controlled electrostaticallyinduced nanostructures.

127

Conclusion In this thesis, we have investigated experimental techniques which allow engineering a band gap in bilayer graphene. Using these techniques, we were able to further investigate the consequences of the presence of the band gap on the transport properties of dual-gated bilayer graphene devices.

We presented fabrication techniques, and in particular cleaning processes, that enabled the production of high quality dual-gated bilayer graphene devices. With such devices, the low energy part of the band structure of bilayer graphene could be studied experimentally with an unprecedented level of detail. The quantum Hall effect was found to be a very powerful tool to probe the characteristics of the band structure and enabled the identification of the triplet present at the top of the valence band, revealing the presence of a Lifshitz transition. This was accomplished by a systematic study of the Landau level degeneracies under a high displacement field. At even higher displacement fields, an additional kink in the conductance was observed. This kink occurred at relatively low energies and its position was found to depend on the displacement field strength. Calculations were performed in order to relate this observation to the van Hove singularity occurring at the Lifshitz transition, but a mismatch between the predicted and the measured densities has been observed. A higher energy range of the band structure was also studied, this time through the observation of Fabry-P´erot interference, which is a signature of ballistic transport. This helped us to address the question of the effect of a band gap on the anti-Klein tunneling. We demonstrated that the anti-Klein tunneling gets broken by the band gap. Indeed the tunable band gap was found to lead to a tunable transmission through a pn interface and therefore to a tunable Berry phase. One of our key findings is that the physics of bilayer graphene can mimic the behavior of single-layer graphene because the tunability of the transmission function allows to recover a perfect transmission at normal incidence for a certain energy range, together with a Berry phase of π, characteristic of the Klein tunneling in single-layer graphene. Our experiments allow for a better understanding of the physical effects which are caused by the opening of a band gap. This understanding is a prerequisite for the realization of efficient dual-gated nanostructures. Attempts to realize a quantum

129

Chapter 11. Defining nanostructures by combining local top gates and the Si backgate point contact were discussed in this thesis. Our results indicate that the geometry needs to be further improved for a better control of the induced conductance channel. The work presented in this thesis may serve as a foundation for the realization of better controlled dual-gated bilayer graphene nanostructures. Impact of this thesis The presented work addressed the question of the effect of a band gap opening on the transport properties. For the first time, we could experimentally access the Lifshitz transition of bilayer graphene. This observation was enabled by the astonishing quality of the bilayer graphene flake, together with the electrical robustness of the hexagonal boron nitride. The latter allowed for the application of high displacement fields to the system, which is responsible for stretching the energy range on which the Lifshitz transition occurs in the band structure. With this, we could reveal the rich topology of bilayer graphene by the mean of quantum Hall measurements. Through the investigation of Fabry-P´erot interference, we further addressed the question of the effect of the induced asymmetry on the chirality of the charge carriers. We found that, close to the band gap, the chirality of the charge carriers is destroyed, demonstrating how relevant the presence of the band gap is for such a system. In conclusion, this thesis demonstrates how an induced asymmetry between the two layers of a bilayer graphene flake is not only responsible for the opening of a band gap, but also strongly affects both the topology of the band structure as well as the chirality of the charge carriers. Such findings are key elements for future studies of dual-gated bilayer graphene nanostructures. Outlook The results presented in this thesis have revealed some questions which remain to be carefully addressed in the future. All the devices presented in this thesis were fabricated using the dry transfer technique, as presented in Chapter 3 and in Chapter 4. Another process was presented Chapter 4, which brings improvements to the quality of the transport. Engineering dual-gated devices using this technique should therefore generate better quality devices than the ones presented here. Since the stacking is facilitated, the fabrication time scale should be reduced and one could therefore think about slightly more complicated stacks. Adding a layer of graphite below the bottom h-BN layer would for example allow to replace the Si-backgate and therefore avoid using SiO2 as a dielectric. With better quality devices, further insights in similar measurements as presented here could be made and the following insights could be gained: The measured Landau level spectrum which revealed the presence of the Lifshitz transition contains for example more information than only the change of the topology of the Fermi contour. In between the three-fold degeneracy and the crossing of the Lifshitz transition, a rich variety of interaction-driven transitions between quantum Hall ferromagnets is observed. These transitions could be further investigated. A possible nematic phase transition could as well be studied in high quality devices.

130

11.5. Conclusion

Further experiments studying the Fabry-P´erot interference with a better quality device might allow for the observation of oscillations down to a lower energy. This would for example allow to reach the density regime where the Klein tunneling and the Berry phase of π are recovered. As demonstrated all along this thesis, one of the biggest experimental achievement consists in being able to apply such large displacement fields to the bilayer graphene flake. This is enabled by the robustness of the high quality dielectric, hexagonal boron nitride. Under such high displacement fields, a Rashba-type spinorbit interaction might be induced. This could be for example probed through weak localization measurements. If spin-orbit interaction would indeed be induced by the gates, this would mean that, in bilayer graphene, spin-orbit interaction can be turned on and off by the application of a displacement field. This interesting idea has been formulated by Prof. Dr. Daniel Loss and Prof. Dr. Jelena Klinovaja from Basel University. However, such a spin-orbit interaction might as well be a strong drawback for dual-gated nanostructures. The reason we are interested in such a technology is for the potential quantum information applications. Graphene is expected to be good candidate for spin qubits, as its spin coherence time is expected to be long (because of weak spin-orbit coupling and hyperfine interaction). Inducing spin-orbit interaction in the device would therefore definitely be an undesired consequence in this context.

131

Appendices A

A step by step process sheet to perform a safe and efficient AFM cleaning

To perform a safe cleaning, the AFM has to be properly calibrated. This calibration is sample-dependent, as the required force will depend on the sample specifics. Below, we present a step by step process sheet to perform AFM cleaning on grapheneon-h-BN samples. In order to be able to quickly locate the area of interest, we recommend to start the procedure with a large scan frame and a reduced amount of lines (frame of 10 µm and 32 lines within this frame). The tip should move with a relatively slow speed, in order to not damage the surroundings (scanning rate of 0.5 Hz). As suggested in the Bruker manual [127], we adopt standard gains for contact mode (integral gain = 15 and proportional gain= 39).

A.1

Setting up the workspace

• Set the AFM in contact mode. The opened windows should be “Thermal Tune”, “Ramp Parameter List”, “Ch1 Ramp Plot” and “PicoForce”. • Set the horizontal deflection (HD) and the vertical deflection (VD) at 0 V. • Set the deflection setpoint (DSP) at 2 V. • In the “Ramp Parameter List”, set the parameters in the Ramp menu and in the Channel 1 one. The “Data Center” should be set to zero and “Data Type” to “Deflection Error”.

A.2

Measuring the deflection sensitivity s

• Engage the tip. In order to obtain meaningful feedbacks, this should be done on a stiff surface. • In the “PicoForce” menu, choose “ramp continuously”: the force curve is recorded.

133

• Position the markers on the extremity of the steep slope and update the value of the sensitivity by clicking the “Def. Sens” button. • Withdraw the tip.

A.3

Estimating the spring constant k

• The value of the vertical and horizontal deflection can have drifted. Make sure they are at 0 V before continuing and set the DSP to 0 V. • In the “Thermal Tune” menu, set the value of deflection sensitivity correction (for rectangular cantilevers, we use 1.106) [127]. Set the frequency range in “Thermal Tune Range” to 0 kHz to 100 kHz. • Click on “Get data”. The spectrum is recorded and displayed in the same window. • Position the markers at the boundaries of the peak and click the “Fit data” and make sure the fit worked. • Click “Calc. Spring K”: the spring constant is measured.

A.4

Evaluating the necessary DSP and force

• Set the vertical deflection to −2 V and DSP to 0 V. • Position the tip in an area similar to the area to be cleaned (graphene on h-BN ideally) and engage the tip. • In the “PicoForce” menu, click on “Ramp continuous”. • In the “Channel Plot”, a green line is displayed. It corresponds to the DSP. The DSP value should be adjusted, such that the green line is at the minimum of the force curve (VCSmin , as introduced in Chapter 3), where no force is applied to the surface. • In the “Scan” window, a yellow or red error should appear. To recover a green signal, the DSP should be adjusted (likely reduced - not too much to not loose the contact). • Estimate the force using Eq. (3.5). If the value is in the desired range, the cleaning can be continued. In case the force is too small, the DSP can be set to a larger value. If it is too large, start over the calibration or use a new tip.

134

A.5

Cleaning the device

• The tip is still engaged on the surface: move in direction of the area to be cleaned using the X-Y offset. • Set the number of lines to 512 for a fine cleaning. • Zoom-in the area to be cleaned and make sure the tip velocity is low enough (' 1 µm/s): the cleaning is started.

A.6

End of the cleaning procedure

• When the frame is done, one can either clean another time (by letting the frame being scanned again), move in another area (if the area to be cleaned is large) or withdraw the tip. • The parameters usually slightly drift during the cleaning. Position the tip at a position close to the device and measure a force curve. • Measure VCSmin and estimate the force using Eq. (3.5). This value is an interesting feedback for the future cleaning steps. • Change the tip to a tapping mode one and record a tapping mode image of your device to appreciate the effect of the cleaning step. If the device is not clean enough, the cleaning procedure can be repeated with more force.

B

Full Landau level spectrum at high displacement fields

In Chapter 6, we showed the low Landau level part of the Landau level spectrum of our dual-gated bilayer graphene device. In this appendix, we would like to highlight the astonishing quality of the graphene by showing a larger energy range of the spectrum. This is shown in Fig. B.1. First, we observed the full conductance quantization of the Landau levels at integer multiples of e2 /h up to 20 e2 /h. To the best of our knowledge, this is the first time that broken symmetry states are observed up to such a high order. Next, we show the Landau level spectrum on a broader energy range. Here, we can nicely see how the higher Landau levels are grouped: the fourfold symmetry of bilayer graphene becomes visible by comparing the gaps between the different Landau levels. We observe that the larger gaps occur at ν = −8, −12, −16.

135

(a)

20

G [e²/h]

16 12 8 4 0 (b)

VBG = -61 V -20

6

5

6

7

VTG [V] 8

9

dG/dVTG [e²/h/V]

-16

-12

-8

10

0

-6

B [T]

4

2

0

VBG = -50 V 0

2 VTG [V]

4

Figure B.1: (a) Conductance cuts taken close to B = 6 T at VBG = −61 V. The conductance is quantized at each integer multiple of e2 /h up to 20 e2 /h: this observation is unprecedented and reveals the high quality of the bilayer graphene flake. (b) Landau level spectrum measured at VBG = −50 V. The filling factors of the Landau levels having the wider gaps are labeled. The fourfold symmetry can be seen, as the higher Landau levels are grouped.

136

C

C.1

Additional information on the observation of Fabry-P´ erot interference in dual-gated bilayer graphene Theory

In this part, the theoretical methods employed by Dr. Ming-Hao Liu and Dr. Viktor Kr¨ uckl in order to build the model used in Chapter 9 are presented in details. C.1.1

Electrostatic model for the device

We apply the parallel-plate capacitor model to deduce the top- and back-gate efficiencies for our bilayer graphene (BLG) device. The thicknesses of the top and bot(top) tom hexagonal boron nitride (h-BN) layers, determined by AFM, are dh-BN = 30 nm (bot) = 3.0 as their dielectric conand dh-BN = 23 nm, respectively, and we adopt h-BN r stant. The top gate capacitance for area C is then given by: CTG h-BN 0 = r (top) = 5.53 × 1011 cm−2 V−1 . e edh-BN

(12.1)

Together with the SiO2 substrate with thickness dSiO2 = 285 nm and dielectric con2 stant SiO = 3.9, the backgate capacitance is given by: r 

(bot)

−1

0 dh-BN dSiO2  CBG =  h-BN + SiO2 e e r r

= 6.53 × 1010 cm−2 V−1 .

(12.2)

Next, we deduce the intrinsic doping by inspecting the full conductance map. We assume that in region X (X = L, C, R), the residual carrier density is uniformly described by n0X . In Fig. 9.2 of the main text, the conductance dip at VTG = −2.1 V along the VBG = 0 V horizontal line cut suggests n0C =

CTG × 2.1 V = 1.16 × 1012 cm−2 . e

(12.3)

For the outer areas L and R, the residual density is deduced from the two top gate-independent horizontal Dirac lines, one at VBG = −7.6 V, suggesting n0L =

CBG × 7.6 V = 4.97 × 1011 cm−2 , e

(12.4)

and one at VBG = −14 V, suggesting n0R =

CBG × 14 V = 9.15 × 1011 cm−2 . e 137

(12.5)

Collecting Eqs. (12.1)–(12.5), we obtain the gate-dependent carrier density:  CTG    VTG    e

+

  CBG    VBG

+ n0X ,

CBG VBG + n0X , X = C e

nX (VTG , VBG ) = 

e

C.1.2

.

(12.6)

X = L, R

Asymmetry parameter

To calculate the gate-dependent asymmetry parameter u for our device, we follow the review by McCann and Koshino [42]. Let us temporarily suppress the area index X and consider the total carrier density n = nt + nb + n0 , where nt is the top gate contribution, nb is the backgate contribution, and the intrinsic doping is assumed to be equally distributed in the two graphene layers: nb0 = nt0 = n0 /2. This assumption allows us to rewrite Eq. (65) of Ref. [42] as uext =

γ1 Λ (nb − nt ) , n⊥

where

(12.7)

c0 e 2 n ⊥ 2γ1 εr ε0

(12.8)

γ12 n⊥ = π~2 vF2

(12.9)

Λ= is the screening parameter, and

is the characteristic carrier density. In Eqs. (12.7)–(12.9), γ1 = 0.39 eV is the nearestneighbor hopping for the interlayer coupling, c0 ≈ 0.335 nm is the interlayer spacing of the BLG, εr = 1 is the effective dielectric constant between the two layers of BLG, and vF is the Fermi velocity of graphene related to the tight-binding parameters through ~vF = (3/2)ta, t ≈ 3 eV being the nearest-neighbor intralayer hopping and a ≈ 0.142 nm being the carbon-carbon bond length. Using Eq. (12.7), Eq. (74) of Ref. [42] reads 



Λ 1 n⊥ u 0 |n| ≈ 1 − ln  + Λ(nb − nt ) 2 2n⊥ 2

s

n n⊥

2

u0 + 2 

2

−1 

,

(12.10)

where u0 = u/γ1 is defined. Finally, we rewrite Eq. (12.10) as 



Λ(nb − nt )  Λ |n| 1 u0 = 1 − ln  + n⊥ 2 2n⊥ 2 in order to avoid the divergence at nb = nt .

138

s

n n⊥

2

u0 + 2 

2

−1 

(12.11)

U L (meV) −60 −40 −20

0

VBG (V)

−100

0

100

(a)

20

U R (meV)

UC (meV) 20

−40 −20

0

20

(b)

(c)

0 −20 −40 −6 −3

0

3

6

−6 −3

VTG (V) U L (meV) −28

−26

VBG (V)

3

6

−6 −3

−95

−85

3

6

U R (meV)

−75

(d)

0

VTG (V)

U C (meV)

−24

−24

0

VTG (V) −26

−24

−22

(e)

(f)

Figure C.2: Asymmetry parameter uX in region X = L, C, R of the device as a function top- and back-gate voltages. Panels (a)– (c) in the upper row show the full range, where the white boxes mark zoom-in range for panels (d)–(f) in the lower row.

−26

−28 4.5

5

5.5

6 4.5

VTG (V)

5

5.5

6 4.5

VTG (V)

5

5.5

6

VTG (V)

The nonlinear Eq. (12.11) can be solved numerically to obtain the asymmetry parameter u = γ1 u0 , when the inputs nt , nb , and n0 are given. Using Eqs. (12.3)– (12.6) we obtain the asymmetry parameters uX for the respective areas X = L, C, R of our BLG device. Numerical results are shown in Fig. q C.2. The actual size of the band gap Egap is related with u through Egap = |u|γ1 / γ12 + u2 . C.1.3

Local energy band offset

From the calculated carrier density nX and asymmetry parameter uX based on the electrostatic model, the band offset VX for area X is given by v u u 2 uγ − sgn(nX )t 1

v u γ1 u tγ 2

u2X

!

u2X VX = + + ~2 vF2 π |nX | − , 1 + (2~vF ) π |nX | 1 + 2 4 2 γ12 q (12.12) which is obtained by replacing the two-dimensional wave vector k by π|nX | in the energy dispersion E(k) for gapped BLG [42] and by adding the minus sign. Application of Eq. (12.12) on the diagonal matrix elements of the model Hamiltonian for transport calculation therefore fixes the global Fermi level at energy E = 0, at which the transmission function is evaluated (linear-response transport). C.1.4

2

Berry phase in gapped bilayer graphene

In order to take into account the characteristic band structure of the BLG, we incorporate the Berry phase into the resonance condition of the Fabry-P´erot (FP) oscillations in Eq. 9.1 of the main text. Therefore, we describe the low energy

139

ΦBerry / 2π

ΦBerry / 2π

0.5

1

0.62

0.67

0.72

(a)

20

(b)

−24 −25

VBG (V)

Figure C.3: (a) Berry phase as a function of top- and backgate voltages for the state closest to the avoided crossing at the Dirac point within the dual-gated area C of the device. (b) Zoom-in of the bipolar block indicated by the white box in (a).

0

0 −26 −20 −27 −40 −6

−3

0

3

6 4.5

VTG (V)

5

5.5

−28 6

VTG (V)

excitations of BLG for a single valley by the Hamiltonian [42], u/2 ~vF k− 0 0 ~v k u/2 γ1 0    H= F + ,  0 γ1 −u/2 ~vF k−  0 0 ~vF k+ −u/2 



(12.13)

using the layer coupling γ1 , the asymmetry u and k± = kx ± iky . Without losing generality we focus on a single valley, as the results of the second one can be obtained by time reversal, leading to an inverted Berry phase. The four eigenstates ψσ (k) of H can be associated with the different bands, which we label as σ = n2, n1, p1, p2 from high to low energy. [For the present discussion, the relevant bands are the two inner bands n1, p1 as those sketched in Fig. 9.1 of Chapter 9.] Using these eigenstates, we can calculate the Berry connection [115], Aσ (k) = ihψσ (k)|∇k ψσ (k)i ,

(12.14)

and the corresponding Berry phase Φσ =

I

Aσ (k) · dk ,

(12.15)

k=const

which describes the additional phase the state ψσ (k) picks up upon traveling adiabatically one complete circle in momentum space. Since transport within the central area C in the pn0 p regime is carried only by the n1 band as sketched in Fig. 9.1 of the main text, FP oscillations pick up the Berry phase ΦBerry = Φσ=n1 . This additional phase changes upon varying the top- and back-gate voltages in the whole possible range from 0 to 2π, as shown in Fig. C.3(a). Also for the experimental transport data presented in Fig. 9.3(b) of the main text, the Berry phase ΦBerry is not constant but takes values between 1.22π and 1.46π, as presented in Fig. C.3(b). Consequently, the Berry phase has to be included in the resonance condition in order to achieve a precise prediction of the conductance maxima.

140

(a) 6

∆n (x 1010 cm-2)

5.5 5 4.5 4

VBG = -20V VBG = -25,9 V VBG = -28 V VBG = -29 V VBG = -30,6 V

3.5 3 3.8

4

4.2 2

C.2

4.4 π nc Lc

4.6

4.8

5

Figure C.4: Dependence of the peak spacing as a function of density for different backgate voltages (different colors). The expected behavior is displayed as a black dashed line.

5.2

(x 1010 cm-2)

Experiment: density dependence of the oscillations at other backgate voltages

As explained in [105], Fabry-P´erot interference should give rise to oscillations spaced √ in density by ∆n = 2 πnC /LC , where nC is the density in dual-gated area and LC the width of the cavity. In this case, LC = 1.1 µm. To confirm the origin of the oscillatory signal, we therefore study the peak spacing dependence, as done in the main text, but on a broader range of voltages. To highlight the square root dependence, we show in Fig. C.4(a) more data points: each color corresponds to a different backgate voltage value. We see that the observed behavior follows the expected behavior, represented with the black dashed line, reasonably well.

141

Publications Electronic triple-dot transport through a bilayer graphene island with ultrasmall constrictions Dominik Bischoff, Anastasia Varlet, Pauline Simonet, Thomas Ihn and Klaus Ensslin New J. Phys. 15, 083029 (2013) Ultrasmooth metallic foils for growth of high quality graphene by chemical vapor deposition P. Prochazka, J. Mach, D. Bischoff, Z. Liskova, P. Dvorak, M. Vanatka, P. Simonet, Anastasia Varlet, D. Hemzal, M. Petrenec, L. Kalina, M. Bartosik, K. Ensslin, P. Varga, J. Cechal and T. Sikola Nanotechnology 25, 185601 (2014) Anomalous sequence of quantum Hall liquids revealing a tunable Lifshitz transition in bilayer graphene Anastasia Varlet, Dominik Bischoff, Pauline Simonet, Kenji Watanabe, Takashi Taniguchi, Thomas Ihn, Klaus Ensslin, Marcin Mucha-Kruczy´ nski and Vladimir I. Fal’ko Phys. Rev. Lett. 113, 116602 (2014) Fabry-P´ erot interference in gapped bilayer graphene with broken antiKlein tunneling Anastasia Varlet, Ming-Hao Liu, Viktor Krueckl, Dominik Bischoff, Pauline Simonet, Kenji Watanabe, Takashi Taniguchi, Klaus Richter, Klaus Ensslin and Thomas Ihn Phys. Rev. Lett. 113, 116601 (2014) Measuring the local quantum capacitance of graphene using a strongly coupled graphene nanoribbon Dominik Bischoff, Marius Eich, Anastasia Varlet, Pauline Simonet, Thomas Ihn and Klaus Ensslin Phys. Rev. B 91, 115441 (2015)

143

Tunable Fermi surface topology and Lifshitz transition in bilayer graphene Anastasia Varlet, Marcin Mucha-Kruczy´ nski, Dominik Bischoff, Pauline Simonet, Kenji Watanabe, Takashi Taniguchi, Vladimir Fal’ko, Thomas Ihn and Klaus Ensslin Submitted to Synthetic Metals, Special issue on Graphene (2015) Hybrid Graphene-GaAs nanostructures Pauline Simonet, Clemens R¨ossler, Tobias Kr¨ahenmann, Anastasia Varlet, Thomas Ihn, Klaus Ensslin, Christian Reichl and Werner Wegscheider Submitted to Applied Physics Letter (2015) Localized Charge Carriers in Graphene Nanodevices Dominik Bischoff, Anastasia Varlet, Pauline Simonet, Marius Eich, Hiske Overweg, Thomas Ihn and Klaus Ensslin Submitted to Applied Physics Review (2015)

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Acknowledgements First of all, I would like to thank Prof. Klaus Ensslin for giving me the opportunity to join his group. From the very first minute of my interview at ETH, I have known how much of a privilege it is to work in such a great environment. I am very grateful for the past three and a half years that I could spend here. The unconditional trust and faith that Klaus has in his students is truly amazing. Together with his unbeatable optimism and good mood, his deep understanding of physics makes the Nanophysics group a very special place. Prof. Thomas Ihn is also to be thanked. Thomas and I spent a lot of time, sitting in his office, discussing all kinds of questions: from the most basic to the most complicated topics, Thomas always showed the same patience towards my questions and uncertainties. His unlimited knowledge and his kindness are undoubtedly part of the strengths of the group. I benefited from every single second I spent next to him. I would also like to acknowledge here the great advantage of having two professors in one group. Without Klaus’ and Thomas’ constant support, I wouldn’t have had the chance of getting to my results so quickly. I always found their door open, and inside the office someone willing to help me shed light on my experiments. I feel extremely lucky for the excellent supervision that I have worked under. I would also like to thank Prof. Sch¨onenberger for accepting to co-examining this thesis. I also would like to mention that the excellent work of his group was a motivation for searching for Fabry-P´erot interference in the first place. For financial support, I would like to thank the Marie Curie network S 3 N ano and its coordinator, Prof. Tartakovskii. I greatly enjoyed being part of this network and being able to meet specialists from other fields. I also thank all the fellows with whom I always had nice times during our various meetings and conferences, especially Stefan Schwarz with whom I did the secondment in Munich. During my PhD work, I also greatly benefited from a priceless theory support. Prof. Vladimir Fal’ko and his amazing understanding of experiments turned my prettiest Landau level spectrum into the signature of the Lifshitz transition in bilayer graphene. The collaboration with him marked a turn in my PhD and I am grateful for everything I learned from him. I acknowledge as well Dr. Marcin MuchaKruczy´ nski for our collaboration and for the various precious discussions that we had. The biggest motivation boost I got came from Dr. Ming-Hao Liu. After an intense

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discussion around my poster at the Graphene Week, Ming-Hao convinced me that simple barriers on bilayer graphene were worth spending some time on. I came back to ETH and could not think about anything else than searching for oscillations. And when I finally found them, I had only one thing in mind: getting to the bottom of this story with him! In Ming-Hao, I found the post-doc I was looking for. I am grateful for every email you answered, always with the same patience. I learned a lot from you Ming-Hao, from the Fabry-P´erot physics to how to produce nice figures for a paper. I am confident that you will soon be one of the best professors that a physics student can dream of. I am also very grateful to Prof. Klaus Richter and Dr. Viktor Krueckl for our fruitful collaboration. Another post-doc who played an important role in this thesis is Dr. Aleksey Kozikov. Aleksey provided me with great advice concerning my measurement setup. I learned something from every discussion we had together. The Graphene Team, through the years, is also to be acknowledged. I thank Dr. Susanne Dr¨oscher, Dr. Arnhild Jacobsen, Dr. Cl´ement Barraud, Dr. Theodore Choi, Dr. Yuan Tian, Pavel Prochazka, Georgia Tsoukleri, Ekaterina Kuznetsova, H¨ useyin Atci, Marius Eich and Hiske Overweg for the discussions and the fun moments in the office or in the cleanroom. I wish in particular all the best to Hiske and Marius who are taking over the dual-gated graphene business. Two members of the Graphene Team are missing on this list. Dr. Dominik Bischoff and Pauline Simonet. Dominik, I think that I could almost write that you taught me everything and that would not be too wrong... Having my desk next to yours helped me considerably going through my PhD and I am grateful for everything. Pauline, in you, I found more than a colleague but a friend. Working with you at the beginning of our PhDs was the best collaboration of my thesis: from Matlab questions until I-V converter details, you also provided me with great support. Some members of our big graphene family have also not yet been mentioned. Dr. Srijit Goswami, Dr. Romain Maurand, Dr. Peter Makk and Peter Rickhaus, you all contributed to make the conferences that I attended much more interesting places. I am grateful for all the conversations we had and for the inputs you gave me at different stage of my thesis. Among them, Dr. Srijit Goswami should be mentioned in particular: he helped us getting started with the new pick-up technique and his support, patience and kindness were very much appreciated. None of the results presented in this thesis would have been obtained without the great support of Cecil Barengo. Cecil was always willing to help me and took a great care of the previous VTI I was measuring in. Peter M¨arki and Paul Studerus are also acknowledged for their precious help in the lab. Another special thank is addressed to Claudia Vinzens. Without Claudia and her nice laughter, the E-floor would not be such a happy place! I have always considered ETH Z¨ urich as a paradise to do a PhD. It is a true luxury to have such working conditions, with a reliable helium supply and such a sophisticated cleanroom. I therefore thank Isabelle Altorfer and the FIRST staff for their reliable work.

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I would also like to thank the rest of the Nanophysics group. I had a great time here thanks to the excellent atmosphere. In particular, I would like to thank Dr. Fabrizio Nichele, Tobias Kr¨ahenmann and Richard Steinacher for all the fun moments and Friday beers. I would also like to thank the Solid Bodies for welcoming me in the soccer team and the extended board of AMP for the nice time I had being a member of this team. To survive in the wild physics world, I had huge external support from my two BFFs, V. and Michelle. You guys are my rock and it is easier to face life when I know that you two will always have my back. Furthermore, I also have to thank my family for the unconditional love and support. Your unwavering belief in me enabled everything that I have been doing and I will always be grateful for that. Last but not least, I would like to thank Stephan. Meeting you was the best thing that happened to me during this PhD and I am the luckiest person to have you in my life. Thank you for your encouragement and support, and for helping me staying sane in this crazy world.

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Curriculum Vitae Name: born:

Anastasia Varlet February 28, 1988 in Avignon Citizen of France

July 2006

Baccalaur´eat, Lyc´ee Ren´e Char (Avignon, France)

2006-2008

Classes pr´eparatoires aux Grandes ´ecoles, Lyc´ee Thiers (Marseille, France)

2008-2011

Engineering School PHELMA (PHysics, ELectronics, MAterials) (Grenoble, France) Semester Thesis: ‘Quantum transport in GaAs/AlGaAs heterostructures with parallel conduction’, Max Planck Institute for Solid State Research, Von Klitzing Department (Stuttgart, Germany) Master Thesis: ‘Transport in single molecule magnet-based transistors’, N´eel Institute, Molecular Transport and Nanospintronics group (Grenoble, France)

2010-2011

Master Nanosciences and Nanotechnologies (N2), Universit´e Joseph Fourier (Grenoble, France)

2011 2011

Master of Science in Engineering, specialization in Nanosciences Master of Science, specialization in Nanophysics

October 2011

Start of Ph. D. in the Laboratory of Solid State Physics at ETH Z¨ urich in the group of Prof. Klaus Ensslin

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