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Electron-phonon mediated superconductivity probed by ARPES : from MgB2 to lithium-decorated graphene Ludbrook, Bartholomew Mears 2014

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Electron-Phonon MediatedSuperconductivity Probed byARPES: from MgB2 toLithium-Decorated GraphenebyBartholomew Mears LudbrookB.Sc. Hons., Victoria University of Wellington, 2007B.A., Victoria University of Wellington, 2007M.Sc., Victoria University of Wellington, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)November 2014c© Bartholomew Mears Ludbrook 2014AbstractThis thesis traces a path from conventional superconductivity in a bulk ma-terial to the introduction of superconductivity and other novel phenomenain graphene.Magnesium diboride is a conventional superconductor, where the pairingis mediated by the electron-phonon coupling. ARPES (angle resolved pho-toemission spectroscopy) is shown to be an excellent probe to quantitativelystudy the momentum dependence of the electron-phonon coupling, demon-strating the origin of the distribution of superconducting gap sizes previouslyobserved with other experimental techniques.Next, we exploit our understanding of the electron-phonon coupling tostudy how it can be tuned in a low dimensional system. It is shown thatthe electron-phonon coupling in graphene can be strongly enhanced by thedeposition of alkali adatoms. High resolution, low temperature ARPES mea-surements provide the first experimental evidence of superconductivity inthis two-dimensional system, showing a temperature dependent pairing gap,and an estimated Tc of ∼ 6 K.Finally, we present a study of another graphene-adatom system expectedto show novel physics. Thallium on graphene has been predicted to enhancethe spin-orbit coupling, leading to a robust topological insulator state. FromARPES measurements characterizing this system, we disentangle the long-range and short-range scattering contributions and show that thallium atomsact as surprisingly strong short-range scatterers. Our results are consistentwith theoretical predictions for this system, indicating it is a good place tosearch for a two-dimensional topological insulator.iiPrefaceExperimental physics is a highly collaborative endeavour and the work pre-sented here could not have been done without the help and input of numerousindividuals. Collaborations including sample growth, theoretical support,and complementary experimental techniques, have all benefited this projectimmensely. This thesis consists of three projects where I have been the pri-mary researcher. Each will produce a publication on which I am lead author.Here I will detail the various contributions from myself, and others, to eachof the projects.In all cases, the contributions of the UBC-ARPES group must be recog-nized for the work done to maintain a world-class system that made the ex-perimental work possible. Under the supervision of A. Damascelli, I have con-tributed significantly, as have G. Levy, C.N. Veenstra, D. Wong, P. Dosanjh,and M. Schneider. The code used for visualizing and analyzing data has beenwritten primarily by G. Levy and C.N. Veenstra.For the work on MgB2 in Chapter 3, samples were grown by N. Zhigadloat ETH Zurich. The time-resolved spectroscopy and subsequent analysiswas done by S. dal Conte, C. Gianetti and G. Cerullo at the UniversitaCattolica del Sacro Cuore in Italy. I conducted the ARPES measurements,along with all of the analysis and interpretation of data. The code usedfor the self-energy analysis was written be C.N. Veenstra. The code forthe extraction of the Eliashberg function was a mix of code written by J.Shi (and made available online) and myself. I performed density functionaltheory and other electronic structure calculations with help from Z. Zhu. Iwrote the manuscript [1] with input from C. Giannetti, S. dal Conte, and A.Damascelli. A. Damascelli supervised and provided guidance on all aspectsiiiPrefaceof the measurements and analysis.For the project of lithium on graphene in Chapter 4, graphene sampleswere provided by A. Stohr, S. Forti, and U. Starke at MPI-Stuttgart. Thecode used for the self-energy analysis was written be C.N. Veenstra. Thecode for the extraction of the Eliashberg function was a mix of code writtenby J. Shi (and made available online) and myself. I designed and assembledthe UHV chamber used to prepare the graphene samples. I performed theARPES and analysis on Li-graphene with assistance from G. Levy. Measure-ments on niobium were done by D. Dvorak and S. Zdhanovich. A. Damascellisupervised and provided guidance on all aspects of the measurements andanalysis, and I wrote the manuscript [2] resulting from this work with inputand editing from A. Damascelli and G. Levy.The thallium on graphene project presented in Chapter 5 involved signifi-cant collaboration. The project was initiated by C. Ast and C. Strasser, andsamples were provided by U. Starke and S. Forti from MPI-Stuttgart. C.Strasser and I carried out preliminary measurements. I conducted the bulkof the experimental work presented here and in the resulting manuscript.Scanning tunnelling microscopy was done by S. Burke and A. MacDonaldwith assistance from a summer student under my supervision, K. Chow, andinput from me. The scattering theory work was done by T. Wehling. Themanuscript resulting from this work [3] was written by C. Strasser and C.Ast, with input from A. Damascelli and me.Over the course of my PhD, I have contributed to numerous other projectsin ways ranging from technical assistance to experimental assistance. Thesecontributions are not described in this thesis, but the resulting publicationsare listed in Appendix A.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Symbols and Acronyms . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 MgB2: a multi-gap superconductor . . . . . . . . . . . . . . . 21.2 Superconductivity in lithium-decorated graphene . . . . . . . 41.3 Graphene: a robust topological insulator? . . . . . . . . . . . 51.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ARPES: Experiment & Analysis . . . . . . . . . . . . . . . . 82.1 Introduction to ARPES . . . . . . . . . . . . . . . . . . . . . 82.1.1 The UBC system . . . . . . . . . . . . . . . . . . . . . 122.2 The spectral function . . . . . . . . . . . . . . . . . . . . . . 162.3 Self-energy analysis . . . . . . . . . . . . . . . . . . . . . . . 16vTable of Contents2.4 Signatures of bosons in ARPES . . . . . . . . . . . . . . . . . 192.4.1 Electron-phonon coupling . . . . . . . . . . . . . . . . 202.4.2 Extracting α2F (ω) from ARPES data . . . . . . . . . 212.5 From electron-phonon coupling to superconductivity . . . . . 233 Anisotropic Electron-Phonon Coupling in the Multi-Gap Su-perconductor MgB2 . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 ARPES on MgB2 . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Time-resolved optics . . . . . . . . . . . . . . . . . . . . . . . 403.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Superconductivity in Lithium-Decorated Graphene . . . . . 444.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Lithium adatoms on graphene and the effects of sample tem-perature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3 Enhancement of the electron-phonon coupling . . . . . . . . . 534.4 Spectroscopic gap: evidence of superconductivity . . . . . . . 614.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 Thallium on Graphene . . . . . . . . . . . . . . . . . . . . . . . 685.1 Preparation of Tl adatoms on graphene . . . . . . . . . . . . 695.2 Long-range vs. short-range scattering . . . . . . . . . . . . . 785.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Appendix A Additional Publications . . . . . . . . . . . . . . . 103viList of Tables3.1 Summary of λk for MgB2 . . . . . . . . . . . . . . . . . . . . . 37viiList of Figures2.1 Geometry of a typical ARPES measurement . . . . . . . . . . 102.2 ARPES 3D dataset . . . . . . . . . . . . . . . . . . . . . . . . 112.3 The ‘universal’ inelastic mean-free-path curve . . . . . . . . . 122.4 Diagram of an ARPES electron analyzer . . . . . . . . . . . . 132.5 Measuring the experimental energy resolution . . . . . . . . . 152.6 Illustration of the spectral function and self-energy . . . . . . 172.7 Example of the Dynes function . . . . . . . . . . . . . . . . . 243.1 MgB2 atomic and electronic structure . . . . . . . . . . . . . . 283.2 Fermi surface and band dispersion of MgB2 . . . . . . . . . . . 313.3 Determination of kz . . . . . . . . . . . . . . . . . . . . . . . . 333.4 KKBF procedure on MgB2 . . . . . . . . . . . . . . . . . . . . 343.5 Extraction of α2Fk(ω) in MgB2 . . . . . . . . . . . . . . . . . 363.6 Momentum resolved λk . . . . . . . . . . . . . . . . . . . . . . 393.7 Time-resolved optics on MgB2 . . . . . . . . . . . . . . . . . . 414.1 Graphene structure and band structure . . . . . . . . . . . . . 464.2 ARPES measurements of Li on graphene . . . . . . . . . . . . 494.3 Temperature dependence of the band dispersion in Li-Graphene 514.4 Temperature dependence of the doping in Li-G . . . . . . . . . 524.5 Li-G self-energy analysis . . . . . . . . . . . . . . . . . . . . . 554.6 Analysis of electron-phonon coupling in Li-decorated graphene 564.7 Reversible enhancement of the electron-phonon coupling . . . 594.8 Comparison of two methods for determining α2Fk(ω) and λk . 60viiiList of Figures4.9 Spectroscopic observation of a pairing gap in Li-decoratedgraphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.10 Reference pairing gap in polycrystalline niobium . . . . . . . . 644.11 Absence of a gap along KM . . . . . . . . . . . . . . . . . . . 655.1 Calibration of thallium deposition . . . . . . . . . . . . . . . . 705.2 Dispersion of the graphene pi bands with increasing Tl coverage 725.3 Charge transfer doping from Tl atoms . . . . . . . . . . . . . 745.4 Temperature dependence of thallium on graphene . . . . . . . 755.5 Annealing Tl on graphene, data . . . . . . . . . . . . . . . . . 765.6 Annealing Tl on graphene . . . . . . . . . . . . . . . . . . . . 775.7 Analysis of the ARPES linewidth . . . . . . . . . . . . . . . . 815.8 Calculations of the short range scattering . . . . . . . . . . . . 83ixList of Symbols and AcronymsΣ′′ imaginary part of the electronic self-energyΣ′ real part of the electronic self-energyα2F (ω) Eliashberg functionδ Short-range scattering potential dielectric constantbkm electronic bare-band dispersionλ electron-phonon coupling parameterMBE molecular beam epitaxyk electron momentumARPES angle resolved photoemission spectroscopyBCS Bardeen-Cooper-SchreifferCP cold phononsED Dirac pointEF Fermi energyEPC electron-phonon couplingxList of Symbols and AcronymsFS Fermi surfaceFWHM full-width at half-maximumGIC graphite intercalated compoundHWHM half-width at half-maximumKK Kramers-KronigKKBF Kramers-Kronig bare-band fittingMDC momentum distribution curveMEM maximum-entropy methodMgB2 magnesium diborideNb niobiumSCP strongly-coupled phononsSTM scanning tunnelling microscopyTc superconducting transition temperatureTl thalliumUBC University of British ColumbiaxiAcknowledgementsI would like to acknowledge and thank all of those who have contributedto this project over the years, beginning with the UBC ARPES lab, whosemembers work tirelessly to maintain a cantankerous, but ultimately state-of-the-art system. Giorgio Levy leads this effort, with assistance from Eduardoda Silva Neto, Michael Schneider, and past members Riccardo Comin, Chris-tian Veenstra, Jason Zhu, and Ryan Wicks. Thanks also to the technical staffPinder Dosanjh and Doug Wong, who build, fix, and advise, when we are introuble.This project benefited enormously from all of the collaborators who con-tributed directly, or helped frame this work in a wider context. Thankyou, Carola Strasser and Christian Ast, for a valuable collaboration on thegraphene projects. Thank you, Stefano dal Conte and Claudio Giannetti, forthe time-resolved measurements on MgB2. Thank you, Roberta Relabora,Jun Fuji, Piero Torelli, Ivana Vobornik, and Giancarlo Panaccione, the APEbeamline scientists in Trieste, for the assistance with a project that doesnot appear here, but was immensely enjoyable to work on. Thank you, An-drew MacDonald and Sarah Burke, for the STM measurements on graphene.Thank you, Ali Khademi and Josh Folk, for the frequent discussions andongoing collaboration on transport in graphene.Thanks also to Mona Berciu, Doug Bonn, Ilya Elfimov, Marcel Franz,Steve Plotkin, and George Sawatzky, for the discussions, advice, and feedbackover the course of my PhD, and on this thesis.Finally, thank you to my supervisor, Andrea Damascelli, for the guidanceand encouragement. Thank you for the hard work that you put in, thatmotivates us to work hard.xiiFor Fiona Jean Hodge,thank you for the distractions.xiiiChapter 1IntroductionOne of the major goals of condensed matter physics is the description ofelectrons in a solid. In the simplest case, the properties of the electronare determined by the periodic potential of the crystal lattice. In reality,electrons interact with each other, with the vibrations of the lattice, andwith a host of other possible excitations. The description of this systemof interacting particles defines the many-body problem; understanding thisbehaviour in complex systems is arguably the most formidable challenge incondensed matter physics. But as great as the challenge is, the rewards aregreater. Emergent from the soup of particles and interactions, phenomenasuch as superconductivity demonstrate the wealth of new physics, and newtechnology, waiting to be unlocked.Improvements in experimental probes have allowed ever more insight intothe complex electronic structure of correlated materials. ARPES, the pri-mary technique used in this thesis, can access the low energy electronic ex-citations resolved in both energy and momentum. As a result, ARPES canmeasure signatures of interactions in complex systems that are not accessi-ble to other techniques. However, the interpretation of ARPES data remainschallenging in such systems, partly because general descriptions of the elec-tronic phenomena arising from many-body physics are still very limited.A useful strategy for tackling problems in physics is to start from thesimplest solvable description, and add in the interesting complexity little bylittle. A similar approach works in the experimental investigation of complexinteracting systems. By studying systems that are dominated by one typeof interaction, theory can make accurate predictions and experiments can bereliably interpreted. An example of this is MgB2 where superconductivity at11.1. MgB2: a multi-gap superconductor39 K is induced by the interaction between electrons and the vibrations ofthe lattice; other interactions can be ignored. In Chapter 3 we present newmeasurements on this conventional superconductor, providing new insightsinto its phonon-mediated superconductivity. As a next step, in Chapter 4we show how the EPC (electron-phonon coupling) can in fact be tuned inmonolayer graphene through the deposition of lithium adatoms. We presentevidence that suggests, for the first time, the presence of superconductivityin this two-dimensional carbon crystal. Finally, in Chapter 5 we exploreanother way of tuning the properties of graphene based on a prediction of atopological insulator phase stabilized through heavy adatom deposition. Wepresent a thorough characterization of Tl adatoms on graphene, concludingthat this is a promising system to observe the predicted effect.1.1 MgB2: a multi-gap superconductorThe mechanism that allows electrons to overcome their Coulomb repulsionand form superconducting pairs in conventional superconductivity is the cou-pling between electrons and phonons. This interaction is described by the socalled ‘glue function’ or Eliashberg function which is akin to the spectrum ofphonon excitations weighted by the strength of the coupling with the elec-tronic states. The strength of this interaction is given by the dimensionlesselectron-phonon coupling constant, λ, which enters into the determination ofthe superconducting transition temperature, Tc.The experimental determination of the Eliashberg function is of paramountimportance, because it identifies the phonon modes that are coupling to theelectronic states, and are therefore directly involved in the superconductingpairing mechanism. There are a number of experimental probes that canaccess the Eliashberg function, including electron-tunnelling which gave thefirst example of this in lead (Pb) [4]. However, most techniques probe themomentum-integrated Eliashberg function, related to the average EPC overthe entire Fermi surface. Sophisticated calculations, including details of both21.1. MgB2: a multi-gap superconductorthe electronic and phononic structure, can now provide a fully momentum-resolved view of the electron-phonon coupling, showing that in many cases,certain parts of the Fermi surface are more strongly coupled than others.This calls for an experimental probe that can do the same. ARPES providesthe momentum resolved electronic structure, and in Chapter 2 it is shownthat this leads to the experimental determination of the momentum-resolvedEliashberg functions and electron-phonon coupling strength. The Eliashbergfunction is the link between experiment and theory for the electron-phononcoupling in conventional superconductors.Superconductivity in MgB2 at 39 K was discovered in 2001 [5] and wasquickly understood in terms of electron-phonon mediated conventional super-conductivity, albeit with a remarkably high Tc. When experiments startedto show some unexpected results relating to the size of the superconductinggap, it was proposed that this material might be the first known example ofa multi-gap superconductor, whereby the Fermi surfaces of different orbitalcharacter couple to different phonon modes with different strength, leadingto Fermi surface dependent superconducting gaps. This was confirmed byARPES [6], an ideal probe for disentangling the superconductivity of distinctFermi surfaces.With better samples, and better experiments, the picture became evenmore complex, but also more interesting. Electron tunnelling spectroscopymeasurements of the superconducting gap showed that not only are there twodistinct gaps, but that there is a distribution of gaps on each Fermi surface[7]. In Chapter 3 we present the highest quality ARPES measurements everperformed on this material, where the electron-phonon coupling is clearlymanifest in the kinks in the dispersion. It is shown that the anisotropy of theelectron-phonon coupling around the Fermi surface explains the distributionof gaps seen in recent experiments [7], and theory [8–12].31.2. Superconductivity in lithium-decorated graphene1.2 Superconductivity in lithium-decoratedgrapheneBased on a solid understanding of electron-phonon mediated superconduc-tivity, one can begin to explore how these properties can be manipulated orintroduced in a system. Tailoring new bulk materials by combining partic-ular elements into a specific crystal structure is one approach, but it facesmany (often insurmountable) challenges from the restrictions of chemistry.Working with low dimensional systems, on the other hand, the properties ofthe entire system can be tuned by modifications to the surface. Graphene, aperfect two dimensional crystal of carbon atoms, is an ideal place to start.Graphene is host to a wide array of spectacular phenomena [13, 14], mak-ing it of interest for both fundamental and applied research. Superconduc-tivity is arguably the most important property that has not been observedin monolayer graphene. This is not for lack of theoretical and experimentaleffort [15–21], and a recent theoretical work has suggested a way to makegraphene superconducting through the adsorption of lithium adatoms [17].The proposed superconductivity is electron-phonon mediated, as in MgB2.In fact, the phenomenology of superconductivity in Li-graphene should bevery similar to that of MgB2, barring its two-gap nature.While not observed in pure bulk graphite, superconductivity occurs incertain GICs (graphite intercalated compounds), with Tc of up to 11.5 K inthe case of CaC6 [22, 23]. The origin of superconductivity in these materialshas been identified in the enhancement of electron-phonon coupling inducedby the intercalant layers [24, 25]. The observation of a superconducting gapon the graphitic pi∗ bands in bulk CaC6 [26] suggests that realizing super-conductivity in monolayer graphene might be a real possibility.Although the Li-based GIC – bulk LiC6 – is not known to be supercon-ducting, Li decorated graphene emerges as a particularly interesting casewith a predicted superconducting Tc of up to 8.1 K [17]. The proposed mech-anism for this enhancement of Tc is the removal of the confining potential of41.3. Graphene: a robust topological insulator?the graphite C6 layers, which changes both the occupancy of the Li 2s band(or the ionization of the Li) and its position with respect to the graphenelayer, ultimately leading to an increase of the electron-phonon coupling fromλ = 0.33 to 0.61 in going from bulk to monolayer LiC6. Based on similarconsiderations, it has also been argued that the LiC6 monolayer should ex-hibit the largest value for both λ and thus Tc, among all alkali-metal-C6superlattices [17].In Chapter 4 we demonstrate how the electron-phonon coupling in a sin-gle layer of graphene can be strongly enhanced by the presence of lithiumdeposited at low temperature. We provide the first evidence for the presenceof a temperature-dependent pairing gap on part of the graphene-derived pi∗Fermi surface. The detailed evolution of the density of states at the gap edge,as well as the phenomenology analogous to the one of known superconductorssuch as niobium, indicate that the pairing gap observed at 3.5 K in grapheneis associated with superconductivity. Based on the BCS (Bardeen-Cooper-Schreiffer) gap equation, 2∆ = 3.5 kb Tc, the measured gap of ∆ = 0.9meVsuggests that Li-decorated graphene is superconducting with Tc' 5.9 K, re-markably close to the value of 8.1 K found in density-functional theory cal-culations [17].1.3 Graphene: a robust topologicalinsulator?Tailoring the properties of low-dimensional systems in general, and graphenein particular, opens immense possibilities for both fundamental and appliedresearch. A natural progression from our study of Li-graphene is to use sim-ilar methods and techniques to explore what other novel physics graphenecan host. In the relatively new field of topological insulators, the topology ofthe electronic wavefunction can be tailored by controlling which symmetriesare respected in a sample, leading to topologically protected surface or edgestates. Thallium adatoms on graphene have been predicted to open a signif-51.3. Graphene: a robust topological insulator?icant spin-orbit gap at the Dirac point, transforming graphene into a robusttopological insulator [27].Graphene was the first material predicted to be a topological insulator,where it was claimed that a gap would open at the Dirac point of the elec-tronic structure, while gapless, spin-polarized edge modes persist at the sam-ple boundaries [28]. The origin of this gap is the spin-orbit coupling, which isgenerally proportional to the atomic mass, and hence is very weak for carbon:calculations have estimated the size of the gap to be on the order of µeV, or∼ 0.01K [29, 30]. Proposals to enhance the size of the gap focus on enhancingthe spin-orbit coupling experienced by the low-energy electronic excitationsin graphene. A particularly interesting proposal, from an experimental pointof view, aims to achieve this through the adsorption of heavy adatoms [27].There are several criteria that a heavy adatom must meet in order to beconsidered a candidate. First, the adsorption site should be the hollow site,at the centre of the graphene hexagon. This allows the adatom to mediatethe second-neighbor hopping without breaking the sublattice symmetry. Theadatoms must be non-magnetic so as not to break time-reversal symmetry,which implies the complete charge transfer of any unpaired electrons. Andfinally, the adatom should interact strongly with the graphene pi electrons.Extensive density functional theory and tight binding calculations identifiedindium and thallium as the best options, and they were shown to open spin-orbit gaps of up to 6 and 20 meV respectively, many orders of magnitudelarger than the intrinsic gap of graphene.The theoretical prediction of a large spin-orbit gap the Dirac point ofthallium decorated graphene motivated our experiments. In Chapter 5 wepresent a comprehensive phenomenology of thallium adatoms on the mono-layer graphene surface, highlighting where our results agree with theoreticalpredictions for this system. In particular, a detailed study of the scatteringmechanisms in this system reveals the important contribution of short-rangescattering, indicating the graphene electrons do ‘feel’ the Tl atoms. Whetherthis leads to the predicted gap remains to be seen.61.4. Outline1.4 OutlineChapter 2 begins with a description of ARPES, the experimental techniqueused in all of the projects presented in this thesis. This is followed by adiscussion of what ARPES can tell us, beyond the simple band structure ofa material. In particular, it is shown how the electron-phonon interaction,important for conventional superconductivity, can be quantitatively analyzedfrom ARPES spectra. These techniques are applied to MgB2 in Chapter 3,where we gain new insight into this multi-gap superconductor. In Chapter 4we provide the first ever evidence for superconductivity in graphene, showinghow lithium deposited at low temperature stabilizes this state via the samemechanism as in MgB2. In Chapter 5 we explore another way to modify theproperties of graphene, with the aim of creating a 2D topological insulator.We describe how thallium adatoms interact with the graphene monolayers,and show that this system is a good place to look for the quantum spin hallstate.This thesis demonstrates the power of the ARPES technique for under-standing the origins of superconductivity in both 3D and 2D systems, andfor characterizing new low-dimensional systems. This is also a thesis aboutmaterials. It is about how novel materials can be made and measured, all inultra-high vacuum, and at temperatures a few degrees above absolute zero.And while these conditions are not conducive to the imminent developmentof ground-breaking new technology, hopefully the work presented in this the-sis will contribute in some way to bringing future materials and technologiesto fruition.7Chapter 2ARPES: Experiment &AnalysisARPES measures the energy and momentum of the occupied electronic statesin a solid. The highest occupied states form a surface in momentum space atconstant energy, known as the Fermi surface, which defines many of the elec-tronic, thermal, and optical properties of the solid. ARPES is one of very fewprobes that can directly measure the Fermi surface, and hence it is a power-ful tool in determining the electronic structure of materials. ARPES is alsosensitive to the signatures of an interacting system via the kinks and otheranomalies that appear in the electronic dispersion, making it an importanttechnique for understanding the origin of complex many-body phenomena.This chapter introduces the experimental technique, and outlines what itcan tell us about interactions. This is followed by a sketch of the theoreti-cal framework for performing a quantitative analysis of the electron-phononcoupling, and an explanation of how this relates to the superconducting prop-erties of a material.2.1 Introduction to ARPESARPES traces its origins to the photoelectric effect first explained by Al-bert Einstein in 1905, the year now referred to as the annus mirabilis. Therealization that light was quantized explained why light incident on a con-ducting material leads to the emission of electrons only when the energy ofthe light, hν, is above a certain threshold, regardless of the intensity. The82.1. Introduction to ARPESkinetic energy (Ekin) of the outgoing electron is given byEkin = hν − φ− |Eb| , (2.1)and hence from the measured kinetic energy, the binding energy of the elec-tron in the solid Eb can be determined. The work function φ is a materialdependent quantity that defines the minimum energy required for an electronto escape the solid.The photoemitted electron has wavevector or momentum K = p/~ withmagnitude K =√2mEkin/~. The angle at which the electron is photoemit-ted is related to its momentum by:Kx =1~√2mEkin sin(θ) cos(ϕ), (2.2)Ky =1~√2mEkin sin(θ) sin(ϕ), (2.3)Kz =1~√2mEkin cos(θ), (2.4)where the polar (θ) and azimuthal (ϕ) emission angles are defined by theexperimental geometry shown in Fig. 2.1.Given the components of the electrons momentum in vacuum, we wouldlike to find the crystal momentum k of the electron in the solid. Due to thetranslational symmetry in the x - y plane, the components of K parallel tothe sample surface are conserved across the sample-vacuum interface, givingK‖ = k‖ =1~√2mEkin sin(θ) , (2.5)where k‖ is the crystal momentum of the electron. The momentum of theincoming photon is much smaller than the electron momentum, and can beignored.There is no such symmetry in the direction perpendicular to the surface,and so the perpendicular component of the momentum is not conserved.Rather,k⊥ =1~√2m(Ekincos2θ + V0) , (2.6)92.1. Introduction to ARPESFigure 2.1: Geometry of a typical ARPES measurement. The absorp-tion of photons with energy hv leads to the photoemission of electrons withenergy Ekin. The positions of the photon source and the analyzer are fixed.Rotating the sample about the x and y axes defines which emission angleswill be detected. From Ref. [31]102.1. Introduction to ARPESFigure 2.2: The 3D dataset produced by the UBC ARPES system.ARPES measurement of Sr2RuO4 showing how the spectral function can becut at a constant energy or momentum. The Fermi surface is the constantenergy cut at E=0, while the band dispersion along several high symmetrydirections in the Brillouin zone are seen in the side panels. From Ref. [32].where V0 depends on details of the band structure and the work function.While these details can be determined experimentally or from calculations,the more common approach is to compare the experimental Fermi surfacewith calculated ones for different values of k⊥.The typical ARPES experiment measures the kinetic energy and the emis-sion angles of a photoelectron relative to the sample plane. This is all theinformation necessary to build up a picture of the 3-dimensional kx, ky, Espace, and the occupied electronic states within, as shown for the perovskitematerial Sr2RuO4 in Fig. 2.2.112.1. Introduction to ARPESFigure 2.3: The ‘universal’ inelastic mean-free-path curve. The in-elastic mean-free-path, or escape depth, of photoelectrons shows a minimafor kinetic energies between 20 and 50 eV. ARPES using photon sources inthis range are particularly sensitive to the electronic structure from samplesurface, which is not always the same as in the bulk. From Ref. [31]2.1.1 The UBC systemMany systems studied with ARPES are layered or 2D materials and as suchpossess a natural cleavage plane. Measuring involves cleaving a single crys-tal sample in vacuum, exposing a clean, well ordered face. For the manysystems that cannot simply be cleaved in vacuum, it is necessary to be ableto prepare clean and atomically flat surfaces. To this end, I designed andbuilt a preparation chamber where samples could undergo cycles of sputter-ing and annealing, and where deposition of adatoms or thin films could bedone at temperatures between 300 K and 1200 K, and in ultra-high vacuum.The addition of these capabilities opened up new avenues of research includ-ing the preparation of atomically flat metal single crystal samples, graphene,MBE-grown topological insulator samples, and niobium thin-film growth.In the case of metallic single crystals, sputtering involves the bombard-ment of the sample with energized (100-3000 eV) ionized argon particles.122.1. Introduction to ARPESFigure 2.4: Diagram of an ARPES electron analyzer. A voltageapplied between the hemispheres selects the kinetic energy that will hit thecentre of the detector, and the range of kinetic energies that will be detectedFrom Ref. [33]The Ar+ ions hit the sample surface, imparting enough energy to removeadsorbed atoms and sample material. Sputtering leads to significant surfaceroughness. Annealing involves heating the sample, typically to the pointwhere the mobility of the surface atoms is significantly increased, allowingthe surface to reconstruct. In the case of graphene samples, annealing simplyremoves any surface contaminants due to the transportation of the sample inair, leaving a pristine single crystal surface. Samples are transferred directlyfrom the preparation chamber to the ARPES chamber without breaking vac-uum, and thus preserving the clean surface for measurements.The other key components of a state-of-the-art ARPES system are thephoton source, the electron analyzer, and the cryogenic sample manipula-tor. The UBC ARPES chamber uses a gas discharge lamp1 to provide amonochromatized source of photons. Helium gas is typically used, giving aprimary excitation line at HeI=21.2 eV, and a less intense line at 41.8 eV,which lie in the ultra violet spectrum. At these photon energies, the kineticenergy of the photoelectrons corresponds to the minimum of the inelastic1SPECS UVS300132.1. Introduction to ARPESmean free path for excited electrons in solids. Put another way, ARPESat these photon energies is extremely surface sensitive, as shown in Fig. 2.3.This can be a blessing or a curse, depending on the experimental aims, andshould be kept in mind when interpreting results.The electron analyzer consists of two concentric hemispheres where aradial voltage may be applied. Photoelectrons with kinetic energy Ekin ± δare directed onto a detector where δ can be set between several hundredmeV, and several eV. Photoelectrons with energy Ek hit the centre of thedetector, and different kinetic energies are resolved along one axis of the 2Ddetector [Fig. 2.4]. The other axis of the detector resolves the angle of thephotoelectron (θ), and in this way, the detector collects a 2D image of energyvs. momentum.The energy and momentum resolution are in principle determined by theparameters of the analyzer, but in reality are also affected by stray magneticfields and electronic noise at the experiment chamber. The latter two arereduced as much as possible by shielding the photoemitted electron pathfrom magnetic fields, and by eliminating any loops to the electronic ground,which can pick up noise by induction from stray magnetic fields. In the bestconditions, the angular resolution has been determined to be 0.3◦. The energyresolution is determined by fitting the Fermi edge of a polycrystalline goldsample with a Fermi-Dirac distribution function convoluted with a Gaussianwhose width corresponds to the resolution:I∫dk(ω)=[f(ω, T ) (a+b ω) ]⊗Rω . (2.7)This is shown in Fig. 2.5 for two temperatures. Resolution values of ∼ 5 meVare typical for the high resolution measurements presented in this thesis.Samples are mounted on a 6-axis cryogenic manipulator that allows thesample temperature to be controlled between 300 and 3.5 K, with an accuracyof ±0.1 K. The analyzer and lamp are fixed in position, and so by changingthe angle of the sample with respect to the analyzer, different emission anglesare sampled, and the 2D Fermi surface may be mapped out, as shown in142.1. Introduction to ARPESGoldT=15 KT=3.5 K−10 −5 0 5 1000.51Energy (meV)Intensity(a.u.)Figure 2.5: Measuring the experimental energy resolution. Fittingthe Fermi edge of a polycrystalline gold sample at a known temperaturewith Eq. 2.7 allows one to determine the energy resolution, as shown here fortwo temperatures. The thermal smearing of the Fermi-Dirac distribution isshown between 3.5 K and 15 K. The resolution is found to be 5± 0.5 meV atboth temperatures.152.2. The spectral functionFig. 2.2 for Sr2RuO4.2.2 The spectral functionARPES probes the imaginary part of the single-particle Green’s functionpropagator describing an interacting system, a quantity known as the spectralfunction:A(k, ω) = − 1piΣ′′(k, ω)[ω − bk − Σ′(k, ω)]2 + [Σ′′(k, ω)]2. (2.8)Here, bk is the bare-band of the non-interacting system, and Σ(k, ω) =Σ′(k, ω) + iΣ′′(k, ω) is the complex electron self-energy describing the inter-actions. In a non-interacting system, Σ(k, ω) = 0, and the spectral functionconsists of a single pole at each k along the bare-band bk. In an interact-ing system, the form of the self-energy determines how the spectral functionis modified: the real part relates to the renormalization of the electron en-ergy, affecting the observed dispersion of the band, while the imaginary partrelates to the lifetime, and the broadening of the spectral function.2.3 Self-energy analysisAccessing the self-energy is an increasingly important goal of ARPES mea-surements, as it provides the link between experiment and theory. In thecase where the self-energy is weakly momentum dependent, it is convenientto analyze the spectral function in terms of constant energy cuts, or MDCsat ω = ω˜ [see Fig. 2.6]. As long as bk can be linearized about the MDC peakmaximum at k = km, the MDC will have a Lorentzian lineshape [34]. Usingthe Taylor expansion of the bare-band about the MDC peak maximum k=km(i.e. bk = bkm + vbkm(k − km) + ...) ignoring higher order terms, and realizingthat ω˜ = Σ′ω˜ + bkm , Eq. 2.8 can be rewritten asAω˜(k) 'A0pi∆km(k − km)2 + (∆km)2. (2.9)162.3. Self-energy analysisFigure 2.6: Illustration of the spectral function and self-energy. (a)Strong electron-phonon coupling modifies the spectral function, and the ex-tracted MDC dispersion (orange) differs from the known bare-band by thereal part of the self-energy (green). (b) The Lorentzian lineshape of the MDCat ω˜ with peak maximum at km and HWHM ∆km. From Ref. [33]172.3. Self-energy analysisThis is recognizable as the form of a Lorentzian with HWHM ∆km andweight A0 given by∆km = −Σ′′ω˜/vbkm ,A0 = 1/vbkm =∫Aω˜(k)dk. (2.10)The Lorentzian fits at ω = ω˜, having peak maximum km, determine allthe quantities required to calculate the self-energy, provided the bare-bandis known:Σ′ω˜ = ω˜ − bkm ,Σ′′ω˜ = −∆kmvbkm . (2.11)Of course, the bare-band is never really known a priori. It can, how-ever, be found from experimental data through a self consistent, iterativeprocedure that simultaneously determines the real and imaginary parts ofthe self-energy, and the bare-band [34]. This routine relies on the fact thatΣ′(ω) and Σ′′(ω) are determined independently (although both dependingon the bare-band), but each must be consistent with the KK (Kramers-Kronig) transform of the other2 (this method is known as Kramers-Kronigbare-band fitting, or KKBF). Starting with a guess for the parameters of thesecond order polynomial bare-band, the self-energies are determined fromΣ′MDC = ω˜ − bkm and Σ′′MDC = −∆kmvbkm , and compared with their KKtransforms:Σ′KK(ω) =1piPV∫ ∞−∞Σ′′MDC(x)x− ω dx,Σ′′KK(ω) = −1piPV∫ ∞−∞Σ′MDC(x)x− ω dx, (2.12)2This is simply a result of the general theorem that the real and imaginary parts of ananalytic complex function are KK related.182.4. Signatures of bosons in ARPESwhere PV denotes the Cauchy principle value. The bare-band is varied untilΣ′MDC(ω) (Σ′′MDC(ω)) is self-consistent with Σ′KK(ω) (Σ′′KK(ω)).The methods presented here are valid for a weakly momentum-dependentself-energy, and will fail when the self-energy is strongly momentum depen-dent. It is difficult to know, in a real system, what amounts to ‘strong’momentum dependence. One clue is a non-Lorentzian MDC lineshape, how-ever, a Lorentzian lineshape does not guarantee weak momentum dependence[34, 35]. The failure of the KKBF (Kramers-Kronig bare-band fitting) rou-tine is a fairly good indicator, but again, simulations have shown that incertain cases it is possible to have self-consistent ΣMDC and ΣKK that donot give the true self-energy [36]. There is an alternative method of calculat-ing Σ′′ without any knowledge of the bare-band, and that has been shown tobe immune to strong local momentum dependence of the self-energy. FromEqns. 2.10 and 2.11 we see thatΣ′′Ratio = −∆km/A0 , (2.13)which can be calculated directly from the parameters of the MDC fits. Thisquantity can act as a consistency check on the self-energy analysis as anyspurious k-dependent effects appearing in the self-energy calculated in theKKBF routine would lead to deviations between Σ′′Ratio and Σ′′MDC . Theagreement of these two quantities can be used as an indicator of the reliabilityof the self-energy determined from real materials, which often show somemomentum dependence.2.4 Signatures of bosons in ARPESThe self-energy describes how a bare particle is ‘dressed’ by its interactionswith electrons and bosons. The composite of the bare particle and its rele-vant interactions is known as a quasiparticle, and both its energy and lifetimeare renormalized with respect to those of the bare particle. These renormal-izations introduce features in the spectral function that are measured with192.4. Signatures of bosons in ARPESARPES. For the systems studied in this thesis, the relevant bosons are thequantized lattice vibrations known as phonons. In the following, we discussthe electron-phonon coupling, and demonstrate how ARPES measurementsaccess details of this interaction that is so important for conventional super-conductivity.2.4.1 Electron-phonon couplingCertain interactions in a solid introduce an attractive potential between elec-trons, allowing them to overcome their Coulomb repulsion and form compos-ite bosons known as Cooper pairs. The condensate of these new bosons is thesuperconducting state. An early successful description of superconductivity,BCS theory, treated the pairing interaction of two electrons at the Fermienergy with opposite spin and momentum by a constant potential, withoutreference to its origin.In conventional superconductors, it is the exchange of phonons that me-diates the pairing interaction. The link between BCS theory and the theoryof electron-phonon coupling in metals is credited to Eliashberg, and the the-ory bears his name [37, 38]. The central quantity is the bosonic spectrum,Eliashberg function, or simply α2F (ω), which is described below. This func-tion defines the properties of this class of superconductors.The electron-phonon interaction involves the scattering of an electron ofmomentum k to a state with momentum k′ via a phonon of momentumq = k′ − k, with probability given by the electron-phonon coupling matrixelements g(k,k′, v) (v is the phonon mode index). The Eliashberg functionis found by summing over the ways to scatter an electron from k to k′weighted by their probability:α2Fk(ω) =∑k′,v|g(k,k′, v)|2δ(ω − ωv,q)δ(k − k′) . (2.14)Here the first delta function relates to the phonon energy, while the secondis the electron energy within the quasi-elastic approximation [56]. This func-202.4. Signatures of bosons in ARPEStion is akin to the phonon density of states weighted by the strength of thecoupling to electrons, although is it not simply a product of the two as thename misleadingly suggests. This is a very important quantity that definesthe electron-phonon coupling in a system. The dimensionless momentum-resolved electron-phonon coupling constant is given by:λk = 2∫ ω0dω′α2Fk(ω′)ω′, (2.15)These quantities are given here in their momentum-resolved form to pro-vide grounds for comparison with ARPES, which measures the momentum-resolved electronic structure. The momentum-integrated quantities are re-trieved by summing over all pairs of k,k′ in Eq. 2.14, or by averaging λk overthe Fermi surface.The Eliashberg function is related to the real and imaginary parts of theself energy via the integral relationsΣ′k(ω) =∫ ∞−∞dv∫ ∞0dω′ α2Fk (ω′)2ω′v2 − ω′2 f(v + ω) , (2.16)|Σ′′k(ω)| = pi~∫ ∞0dω′α2Fk(ω′)[1− f(ω − ω′) + 2n(ω′) + f(ω + ω′)] , (2.17)where f(ω) and n(ω′) are the Fermi-Dirac and Bose-Einstein distributionfunctions for the electrons and phonons respectively.2.4.2 Extracting α2F (ω) from ARPES dataGiven the self-energy determined from ARPES measurements, the challengeis to pull out the momentum resolved α2Fk(ω) function. The simplest ap-proach is to perform an integral inversion by minimisingχ2 =∑i[Di − Σ′k(i)]2σ2i(2.18)against α2Fk(ω), where Di are the data, σi is the error, and Σ′k is defined byEq. 2.16. However, this tends to fail due to the noise in the data. There need212.4. Signatures of bosons in ARPESto be some constraints imposed on the form of α2Fk(ω). Next, we discusstwo ways to achieve this.Maximum entropy methodThe MEM (maximum-entropy method) is an integral inversion procedure toextract the Eliashberg function from ARPES data originally presented inRef. [39], and used in the analysis of numerous systems since [40]. Instead ofthe direct inversion of Eq. 2.18, this method minimizes the functionalL =χ22− aS , (2.19)where χ2 is defined as above, and S is the generalized Shannon-Jaynes en-tropy:S =∫ ∞0dω[α2F (ω)−m(ω)− α2F (ω)lnα2F (ω)m(ω)] . (2.20)Here, m(ω) is a ‘best guess’ at the form of the Eliashberg function, and theentropy term will be maximized for m(ω) = α2Fk(ω). The parameter a inEq. 2.19 is a multiplier that determines how much importance to place on theentropy term, and hence the constraint function on the Eliashberg function.Self consistent methodAn alternative method uses the experimentally determined imaginary partof the self-energy as a constraint on the inversion procedure. This is achievedby fitting the real and imaginary parts of the self-energy simultaneously, withthe same Eliashberg function. This amounts to minimisingχ2 =∑i[Di − Σ′k(i)]2σ2i+∑j[Dj − Σ′′k(j)]2σ2j, (2.21)where Di and Dj are the data for the real and imaginary parts of the selfenergy respectively, and Σ′k and Σ′′k are defined in Eqs. 2.16 and 2.17. A222.5. From electron-phonon coupling to superconductivityfurther constraint is involved by taking α2Fk(ω) to be a sum of Lorentzianswhose peak position, width, and area can vary.The merits of the two methods are discussed where they are applied inchapters 3 and 4, but the main point is that they both work, in the sensethat they both provide reasonable α2Fk(ω) functions with spectral weightat the same frequencies. Values of λk agree, although the values from theself-consistent method are consistently 10-20% larger, probably reflecting theapproximation of the Eliashberg function.2.5 From electron-phonon coupling tosuperconductivityStrong-coupling Eliashberg theory is very successful at describing the super-conducting properties of conventional superconductors based on the details ofthe electron-phonon coupling. Some of the relationships relevant to ARPESmeasurements are outlined below.The critical temperature, below which the superconducting state can per-sist, is related to λ through the Allen-Dynes formula:kBTc =~ωln1.2exp[− 1.04(1 + λ)λ− µ∗(1 + 0.62λ)], (2.22)where ωln is the logarithmic average phonon frequency defined as,ωln = exp[2λ∫ ∞0dωωα2F (ω)lnω)], (2.23)and µ∗ is the dimensionless Coulomb potential, typically on the order of 0.1.The other key parameter of the superconducting state of relevance toARPES is the superconducting gap, which opens below Tc. This gap appearsas a suppression of spectral weight in the single-particle density of statesat the Fermi level. Its physical origin is the energy required to break aCooper pair. BCS theory gives a universal relationship between the transition232.5. From electron-phonon coupling to superconductivity∆ = 1.4meV−0.005 0 0.005Energy (eV)Γ = 0Γ = 0.1∆Γ = 0.2∆Figure 2.7: Example of the Dynes function. The Dynes function(Eq. 2.25) describes a linear density of states with a BCS-like gapped Fermisurface, and includes a phenomenological pair-breaking term Γ that broadensthe features. Simulations are plotted here for Γ =0,0.1, and 0.2 times thegap, ∆ = 1.4 meV.temperature and the magnitude of the gap,2∆0kBTc= 3.53 , (2.24)which serves as a useful guide, but in reality the proportionality constant ismaterial specific. Corrections to this equation based on the material specificEliashberg function can achieve good agreement with experiment [38].The gap in the single particle density of states as measured by a spectro-scopic probe such as photoemission or electron tunnelling is described by theDynes function [41],D(ω) = Re[ω − iΓ[(ω − iΓ)2 −∆2]1/2], (2.25)where Γ is a phenomenological pair-breaking term that broadens the edge ofthe gap. This function is plotted with a gap value ∆ = 1.4 meV and variousvalues of Γ in Fig. 2.7.To summarize, the ability of ARPES to measure both the k-resolved su-perconducting gap, and the signatures of the phonon (or more generally, bo-242.5. From electron-phonon coupling to superconductivityson) coupling is one of the reasons this technique is so important for the studyof the high-Tc cuprate superconductors, where the magnitude (and phase) ofthe gap varies strongly over the Fermi surface. In those materials, the originof the coupling, and hence the gap, is poorly understood. In the next twochapters of this thesis we use ARPES to understand the anisotropic electron-phonon coupling in the conventional multi-gap superconductor MgB2, andthen to demonstrate how superconductivity can be created in graphene.25Chapter 3Anisotropic Electron-PhononCoupling in the Multi-GapSuperconductor MgB2Certain interactions in a solid introduce an attractive potential between elec-trons, allowing them to overcome their Coulomb repulsion and form Cooperpairs. The condensate of these new bosons is the superconducting state. Theorigin of this attractive potential is the question at the heart of research intohigh-temperature superconductivity. Conventional superconductors, on theother hand, are well understood, and it is the electron-phonon coupling thatmediates the pairing interaction.As discussed in Chapter 2, the electron-phonon interaction involves thescattering of electrons from momentum k to k’ by a phonon with momentumq = k’−k. This is described by the Eliashberg function, α2F (ω). Momentumresolved techniques such as ARPES access a momentum resolved quantityα2Fk(ω) which includes the phonon mediated interactions between one pointk on a Fermi surface and all other points k’ on all Fermi surfaces. Onthe other hand, optical measurements provide the function α2F (ω) whichinvolves a momentum average over the entire Brillouin zone.We study the conventional multi-gap superconductor MgB2 with twocomplementary techniques: ARPES and time-resolved optical spectroscopy.With ARPES, we study the strength of the electron-phonon coupling to dif-ferent parts of the Fermi surface, demonstrating a strong anisotropy. Next,we use a novel time-resolved optical spectroscopy technique that discrimi-nates between strongly and weakly coupled phonons based on the relaxation263.1. Introductiondynamics of the transient reflectivity. This shows that a small population ofphonons is responsible for the majority of the coupling strength. In short, wedemonstrate an anisotropic coupling between electronic states and a subsetof strongly coupled phonon modes.3.1 IntroductionMgB2 consists of graphene like boron planes separated by a magnesium layer[Fig. 3.1(a)]. The electronic structure near the Fermi level is entirely deter-mined by the boron orbitals. In the honeycomb structure, the boron 2s,2px and 2py orbitals form sp2 hybrids. The direct overlap of these alongthe B-B bond direction forms three covalent σ bands, two of which crossthe Fermi level. The 2pz orbitals form two pi bands, both of which crossthe Fermi level. The Mg donates two electrons to the boron planes, andthe Mg2+ ion interacts strongly with the B pz orbitals, pulling the pi bandsdown in energy [43]. The σ bands, being formed from in-plane orbitals, areessentially two-dimensional. They have very little kz dispersion, and formtwo concentric tube-like Fermi surfaces about ΓA in the 3D Brillouin zonein Fig. 3.1(b). The pi bands on the other hand, are very strongly dispersingalong kz, forming the network of Fermi surfaces near the zone boundary. Theelectronic band structure, calculated by density functional theory, is shownin Figs. 3.1(d-f) for the high symmetry cuts through the Brillouin zone shownin Fig. 3.1(c). The character of the orbital is represented by the width of thebands, showing that the σ bands consist of the B px and py orbitals, whilethe pi bands are of pz character.The superconducting properties of MgB2 were discovered in 2001 [5] andwith its relatively high Tc of 39 K, there was immense interest in the mech-anism behind its superconductivity. MgB2 was quickly established as a con-ventional superconductor, where the pairing mechanism is mediated by theelectron-phonon interaction [44], but with a twist. Early experiments in-dicated that MgB2 might have two distinct superconducting energy gaps273.1. IntroductionMgBa bcpxd KM  A LH A10010Energy(eV) pye KM  A LH Apzf KM  A LH AFigure 3.1: MgB2 atomic and electronic structure. (a) The structureof MgB2 consists of graphene-like hexagonal boron planes separated by Mglayers. (b) The Fermi surface is defined by the B orbitals, leading to twotubular σ sheets around the Γ point, and two pi surfaces out near the zoneboundary [from Ref. [42] ]. A guide to the hexagonal Brillouin zone and thelabels of the high symmetry points is shown in (c). Density functional theorycalculated band structure shows the dispersion of the boron derived σ and pibands which cross EF . The width of the bands is proportional to the boronpx, py, pz orbital character in (d), (e), (f) respectively.283.2. ARPES on MgB2[45–49]. Calculations based on strong-coupling Eliashberg theory supportedthis [8, 9] and ARPES measurements shortly thereafter confirmed it withmeasurements of a 3 meV gap on the pi band, and a 7 meV gap on the σband [6].It was recently shown experimentally that even the two gap picture isnot quite the full story. Tunnelling data shows a distribution of gaps aboutthe values expected for each Fermi surface [7]. This was in fact predicted bycalculations using a fully anisotropic Eliashberg model [9, 11, 12], where theanisotropy originates in the electron-phonon coupling. However, this momen-tum dependence of the electron-phonon coupling has never been observed.In the following section, we use ARPES to examine the momentum-resolvedelectron-phonon coupling. We show that the coupling at the σ band Fermisurfaces is significantly larger than the Fermi surface averaged value mea-sured by other techniques, which explains the large superconducting gap.Furthermore, the electron-phonon coupling is stronger on the inner band,and anisotropic in momentum on the outer band, in agreement with theory,and explaining the observed distribution of superconducting gaps.High-quality single crystals of MgB2 were grown using a cubic anvil tech-nique. The details of the crystal growth and extensive characterization aregiven in Ref. [50]. Optical measurements were done on polished samples at300 K. For ARPES, samples were oriented with Laue diffraction, then cleavedin vacuum to expose a clean surface. Measurements were done at 10 K, andin vacuum better than 4×10−11 Torr. Energy and angular resolution were setto 15 meV and 0.01 A˚−1, respectively.3.2 ARPES on MgB2There are few reports of ARPES measurements on MgB2 [6, 49, 51, 52], andthe quality of the data has been limited by the quality of the cleaved samplesurface. MgB2 single crystals are typically on the order of 0.5 - 1 mm2 with athickness less than 0.3 mm, which provides a technical challenge to preparing293.2. ARPES on MgB2the samples for measurement. A further problem is that, despite the layeredboron planes, there is strong electronic bonding along the c-axis, and samplesdo not cleave easily. The cleaved sample surface is typically multi-facetedand rough. The disorder scatters the outgoing photoelectrons, which leadsto a poorly defined k‖, and hence poorly momentum resolved bands. Themeasurements presented here were obtained by careful sample preparation,and by cleaving many samples in order to increase the statistical likelihoodof obtaining a clean, flat surface.The high quality ARPES data reported here reveal a number of featuresnot observed in previous studies [6, 51]. The Fermi surface measured withp-polarised light in Fig. 3.2(a) shows the two hole-like σ sheets around theΓ point, and one of the pi sheets out towards the zone boundary. There is astrong intensity modulation of the σ bands around the Fermi surface due tothe interplay between the px, py orbital symmetries and the polarization ofthe incoming light [53]. Switching between s and p polarized light selects theinner or outer σ band, as shown in Figs. 3.2(c) and (d) for the dispersion alongΓ-K (the red dashed line in Fig. 3.2(a)). The two σ band Fermi surfaces havebeen colourized artificially based on these polarization dependent dispersionswhich clearly show two bands.303.2.ARPESonMgB2KM(a)1 0.5 0 0.5 110.500.51kx (A 1)k y( A 1)Expt. Calc.(b)0.4 0.2 0 0.2 0.40.40.200.20.4kx (A 1)k y( A 1) P(c)0 0.25 0.50.60.40.20E ne r gy (e V) S(d)0 0.25 0.5k (A 1)Figure 3.2: ARPES measurements showing the Fermi surface and band dispersion of MgB2(a)The Fermi surface of MgB2 showing the inner (red) and outer (blue) σ bands, and the pi band (black)crossing near the zone boundary. Panel (b) shows the σ bands in more detail on the left, and the kzintegrated calculated Fermi surface on the right. The band dispersions measured along ΓK (red dashed linein panel (a)) are shown in panel (c), along with the calculated dispersions (blue) which have been rigidlyshifted by -140 meV to get agreement at EF . (d) Outer σ band dispersion measured with s polarized light,which suppresses the intensity of the inner band.313.2. ARPES on MgB2While the perpendicular component of the electron momentum is notconserved, it is well defined, as discussed in Chapter 2. kz is determinedby the photon energy, as well as unknown, material specific quantities suchas the inner potential V0. A comparison of the experimental Fermi surfacewith calculated ones for different values of kz shown in Fig.3.3 indicatesthat our measurements are cutting the 3D Fermi surface [see Fig. 3.1(b)] atapproximately kz=0.The electronic dispersion of the σ bands is calculated with density func-tional theory at kz=0, and is compared to the experimental dispersions inFigs. 3.2(c) and (d). The calculations are rigidly shifted in energy by -140meV to get the correct Fermi level crossing (i.e. agreement at E=0) butotherwise the agreement is good. We note that a similar shift was neededto match calculated band structure to bulk sensitive de Haas-van Alphenmeasurements [54], which measures the Fermi contour areas. This impliesthere remains some tweaking of the band structure parameters to be done,but also that the discrepancy observed is not due to a surface effect. TheFermi contour areas (which are proportional to the number of electrons inthe band) for the light and heavy holes are found to be 0.06 and 0.125 ±0.01 A−2, or approximately 3.5 and 7.5 % of the Brillouin zone, again, inagreement with the bulk sensitive de Haas-van Alphen measurements forkz=0 [54]. This shows that despite its surface sensitivity, ARPES is in factmeasuring the bulk electronic structure of this material. The good agreementbetween the shifted band structure and the ARPES dispersions also indicatesthat electron-electron interactions are not likely to be important in MgB2,as these often cause strong deviations between density functional theory andmeasured dispersions [51].The σ band dispersions measured along ΓK (red line in Fig. 3.2(a)) shownin Figs. 3.2(c) and (d) show striking ‘kinks’ at around 70 meV binding energy.These renormalizations are signatures of the strong electron-phonon coupling.Next, we show how a detailed analysis of the ARPES spectra yields uniqueinsight on the EPC, and how it relates to superconductivity in MgB2.323.2. ARPES on MgB2Data kz = 0 kz = 0.05 kz = 0.15 kz = 0.25 kz = 0.5 kz = 0.75Figure 3.3: Determination of kz. The Fermi surface measured by ARPEScan be compared with calculated Fermi surfaces for different values of kz.The central σ bands show little kz dispersion due to their 2D nature. Thepi bands, on the other hand, vary strongly with kz throughout the Brillouinzone. Comparison between the data and calculation shows that for thisphoton energy (21.2 eV) we are measuring at approximately kz=0. Here, kzis given in units of 2pi/c.Band dispersions were measured along the two high symmetry directionsΓK and ΓM, providing four datasets for analysis corresponding to the innerband (labelled σ1) along ΓK, and the outer band (σ2) along ΓK, ΓM, andΓM′ (where the prime indicates a different M point). Measurements of the piband dispersion did not show any signs of strong renormalization, but this islikely due to the strong kz dispersion which broadens features in the ARPESspectral function [55].The intensity profiles at constant energy, known as momentum distribu-tion curves or MDCs, are fitted with a Lorentzian and a constant background,as described in Chapter 2. This generates the peak maximum (km) and peakwidth (∆km) at energy ωm. These parameters are related to the real andimaginary parts of the self-energy as follows,Σ′ = ωm − bkm ,Σ′′ = −∆kmvbkm , (3.1)where bkm is the energy of the bare band at km and vbkm is its velocity. We usea self-consistent fitting procedure to extract the self energy and bare-bandtogether without any a priori knowledge of the bare-band. The Kramers-Kronig relation between the experimentally determined real and imaginary333.2. ARPES on MgB2a0 0.1 0.2 0.3 0.4 0.50.40.30.20.10k (A1)E-EF(eV)b10100200⌃0b20 0.1 0.2 0.30100200⌃00 (meV)c10100200⌃0c20 0.1 0.2 0.30100200Energy (eV)⌃00 (meV)d0.40.30.20.1 0 0.1 0.2 0.3 0.40.40.30.20.10k (A1)E-EF(eV)e10100200⌃0e20 0.1 0.2 0.30100200⌃00 (meV)f10100200⌃0f20 0.1 0.2 0.30100200Energy (eV)⌃00 (meV)Figure 3.4: Kramers-Kronig bare-band fitting to determine the self-energy. MDC dispersions are plotted on top of the raw data in (a) and(d). The momentum space cut in (a) is along ΓK, while in (d) it is alongMΓM. The real and imaginary parts of the self-energy are plotted in black in(b,c,e,f), while the KK transforms are shown in red. The imaginary part ofthe self energy calculated from the spectral weight, Σ′′ratio is shown in dashedgreen, and its good agreement with the MDC width Σ′′(ω) is a confirmationthat the KKBF analysis can be applied (see text).343.2. ARPES on MgB2parts of the self-energy is the self-consistency criteria [34, 36]. Figures 3.4(a) and (d) show the MDC dispersion overlayed on the raw data, where the‘kinks’ are clear signatures of coupling with phonon modes.The real and imaginary parts of the self-energy (black) and their Kramers-Kronig transforms (red) are determined from Eq. 3.1 and the KKBF proce-dure (described in further detail in Chapter 2), and plotted in Figs. 3.4(b,c,e,f).In each case there is a one-to-one correspondence between the peak in Σ′ andthe step in Σ′′ at an energy of ∼70 meV. While Σ′ describes the renormaliza-tion of the band, Σ′′ is related to the inverse lifetime. The step in Σ′′ impliesan additional scattering channel opens at that energy, reducing the lifetimeof the quasiparticles, causing the increase in the measured MDC linewidth.The self energy is related to the bosonic spectrum by [56]Σ′k(ω) =∫ ∞−∞dv∫ ∞0dω′ α2Fk (ω′)2ω′v2 − ω′2 f(v + ω) , (3.2)where f(ω) is the Fermi distribution function. The momentum resolvedα2Fk(ω) function is extracted from the self-energy by an integral inversionprocedure described in detail in Chapter 2. Briefly, the real part of the self en-ergy is calculated from a trial α2Fk(ω), and compared to the experimentallydetermined one. α2Fk(ω) is varied, subject to a constraint function, untilthe generated self energy fits the experimental one, and hence the bare-bandplus self-energy agrees with the MDC dispersion. We use the self-energyextracted from the self consistent fitting procedure and a histogram modelbased on the equilibrium optics (Fig.3.7(a)) for the constraint function.The key results of the electron-phonon coupling analysis are summarizedin Fig. 3.5. We observe a strong anisotropy of the electron-phonon couplingstrength between the two σ bands, and along different momentum directions.The MDC dispersions, all showing the strong kink at ∼70 meV, are plottedin orange in Fig. 3.5(a-c) along with the bare-band (gray) determined by theKKBF procedure described above. The MEM fitting to the real part of theself-energy [data in black, fit in red in Figs. 3.5(d,e,f)] determines the α2F (ω)shown in the lower panels. The success of the fit is also seen in the MDC353.2. ARPES on MgB2aσ1ΓK0 5 · 10−2 0.1−0.4−0.3−0.2−0.10k (A−1)E-EF(eV)d0100200Σ′g0 100 2000100200E (meV)Σ′′(meV)λk1.6j0 50 100 150012E (meV)α2F,λ(ω)σ2ΓKb0 5 · 10−2 0.1−0.4−0.3−0.2−0.10k (A−1)e0100200h0 100 2000100200E (meV)λk1.0k0 50 100 150012E (meV)σ2ΓMc0 5 · 10−2 0.1−0.4−0.3−0.2−0.10k (A−1)f0100200i0 100 2000100200E (meV)λk1.2l0 50 100 150012E (meV)Figure 3.5: Extraction of α2Fk(ω) in MgB2. (a-c) MDC dispersions (or-ange) for the three unique momentum cuts measured with ARPES. Thebare-band determined from the KKBF is shown in gray. The real part of theself energy (black) is fitted with Eq. 3.2 and the MEM procedure describedin the text. The successful fits (red) correspond to the Eliashberg functionsin panels (j,k,l). Further confirmation of the fitting can be seen in the MDCdispersion calculated from the MEM fit in panels (a,b,c). The imaginarypart of the self-energy determined from the KK transform of the fit to thereal part is shown in (g,h,i). The Eliashberg functions, plotted in colour in(j,k,l), are used to calculate λ, shown as the black line in (j,k,l).363.2. ARPES on MgB2Band Direction λσ1 ΓK 1.6 ± 0.1σ2 ΓK 1.0 ± 0.1σ2 ΓM 1.25 ± 0.1σ2 ΓM′ 1.15 ± 0.1σav All 1.25 ± 0.2Table 3.1: Summary of λk for MgB2.dispersion calculated from the α2Fk(ω) [i.e. k = bk + Σ′, black line in panels(a,b,c)]. The imaginary part of the self-energy is calculated from the KKtransform of the fit to Σ′, and in each case agrees with the experimental onein Fig. 3.4.The momentum resolved electron-phonon coupling strength, sometimesreferred to as the mass-enhancement parameter, in the Eliashberg formalismis given by:λk = 2∫ ω0dω′α2Fk(ω′)ω′, (3.3)and is plotted in Figs.3.5(j,k,l), showing how λk is calculated from the energyweighted integral of the α2Fk(ω) function (black lines). The inner σ band hasa larger mass-enhancement parameter along ΓK (1.6 ± 0.1) than the outerband does along either of the high symmetry directions (1.0 ± 0.1 along ΓK,and 1.2 ± 0.1 along ΓM ). These results are summarized in Table 3.1, andvisually in Fig. 3.6.The agreement with fully anisotropic Eliashberg calculations for the val-ues of the k-resolved electron-phonon coupling on the σ bands is very good.In particular we make the comparison with the results of Ref. [12] in Fig. 3.6,who used a denser k-mesh than earlier calculations, and is therefore betterresolved. However, other theoretical studies find a similarly anisotropic result[9, 11].Taking the values of λk presented here, we can weight them by the Fermicontour areas (where σ2 is over twice the size of σ1) to get an approximate σ373.2. ARPES on MgB2band average coupling of λσ = 1.25±0.2. This number can be compared withvalues of 1.0 to 1.2 which are often reported from calculations and experiment[54, 57].As an interesting aside, a common approach to extracting the electron-phonon coupling (or the mass-enhancement factor) from ARPES data is tolook at the ratio of the bare-band slope to the renormalized band velocityand equate this to (1 + λ). However, exceedingly large values are oftenreported, and the validity of this approach has recently been questioned[34, 36]. Indeed, from the measurements presented here, we would find valuesof λk between 1.7 and 2.7, almost a factor of two larger than predicted.This provides some experimental evidence supporting the conclusions of Refs.[34, 36], showing the need for a complete self-energy analysis in order to accessinformation on the electron-phonon coupling from ARPES data.383.2. ARPES on MgB2Figure 3.6: Momentum resolved λk. The summary of experimental datain (a) compares very well with fully anisotropic Eliashberg calculations in (b).(from Ref. [12]). Note that in the experimental data, the points along ΓKwere measured at one point, and are repeated here based on the symmetryof the Fermi surface. The point along ΓM was measured along two directionsand found to be the same.393.3. Time-resolved optics3.3 Time-resolved opticsIn order to further understand the strong coupling measured on the elec-tronic σ bands, we use a novel time-resoved optical spectrocopy to study theelectron-phonon coupling. This technique is bulk sensitive, and momentum-integrated, and therefore provides complimentary information to the momentum-resolved, surface sensitive ARPES measurements.We begin by determining the momentum integrated, or Fermi surfaceaveraged, Eliashberg function from equilibrium (as opposed to time-resolved)optical reflectivity. Reflectivity data shown in Fig. 3.7(a) from Ref. [59] is fitwithin the extended Drude model, which describes the infra-red responseof metals with strong electron-phonon coupling [60]. While the details ofthe model will not be discussed here, the Drude-like response depends onthe electron-phonon coupling, and the details of the latter are recoveredfrom fitting the optical reflectivity. The extracted α2F (ω) is shown insetin Fig. 3.7(a) and is dominated by a peak at 70 meV. The electron-phononcoupling strength is determined to be λ = 1.1±0.1, which agrees with severalother bulk sensitive probes [54], but is larger than the average value foundin theory [57].The great advantage of a time-resolved optical technique lies in its abilityto single out different bosonic modes involved in the coupling with quasi-particles on the basis of the decay time of their characteristic relaxationdynamics [61]. In a pump-probe experiment, the system is ‘pumped’ intoan excited state by an initial laser pulse, and then ‘probed’ after some delaytime. The earliest dynamics after photoexcitation is phenomenologically de-scribed by treating electrons and phonons as two interacting systems bothcharacterized at each delay time τ by their own temperature [62]. In layeredmaterials like graphite and the cuprates the excited electrons exchange theirexcess energy through two decay channels: they thermalize with a fractionof SCP (strongly-coupled phonons) on a fast time scale (< 1 ps) and, on aslower time scale, with the complementary fraction of less interacting lattice403.3. Time-resolved opticsHot electronsStrongly coupled phonons Cold phonons⌧ < 100fs⌧ < 1psb0 0.4 0.8012·103Delay (ps)R/Ra0 0.5 1 1.5 2 2.5 3 3.50.20.40.60.81Energy (eV)R DOSph0 20 40 60 80 10000.51Energy (meV)↵2FFigure 3.7: Time-resolved optics measurements on MgB2. (a) Equi-librium reflectivity measured with spectroscopic ellipsometry (red line) andextended Drude model fit to the data (black line). The fit determines themomentum-integrated Eliashberg function, shown as the histogram inset.The calculated phonon density of states from Ref. [58] is shown in red forcomparison in arbitrary units. (b) Sketch of a pump-probe experiment: afterlaser excitation the hot electrons thermalize with a fraction of SCP on a timescale of 100 fs and after 1 ps with the rest of the lattice (cold phonons). Thetrace is the temporal trace at probe energy of 1 eV. The black line is the3TM fit to the data (explained in text).413.3. Time-resolved opticsexcitations CP (cold phonons) [61].Here, the transient variation of reflectivity δR/R is measured with un-precedented temporal resolution (∼15 fs) over a broad spectral region be-tween 1100 and 1600 nm. The MgB2 transient response is reported inFig. 3.7(b) measured with a photon energy of 1 eV. It has previously beenshown that the relative variation of the reflectivity in the infra-red regionfar from the plasma frequency of 6 eV can be explained as a broadening ofthe Drude peak due to the change in the electronic temperature through theincrease of the temperature of the bosonic modes strongly coupled to thehot electrons [61]. For that reason we fit the δR/R(t) trace within the threetemperature model used to interpret time resolved ARPES and reflectivitydata [61, 62]. The key elements of this model are that the rate of energytransfer from the hot electrons to the strongly coupled phonons depends onthe number of such phonon modes (or the phonon density of states), andthe strength of the coupling. These two quantities are related to parametersdetermined in the fitting procedure: the fraction of total phonon modes thatare strongly coupled, and the SCP contribution to the total α2F (ω) (andhence also λ). According to the fit, plotted in black in Fig. 3.7(b) the SCPmodes correspond to a small fraction f=0.15-0.22 of the total lattice modes,but contributes a coupling strength λSCP = 0.53− 0.75.By comparing α2F (ω) and the phonon density of states in Fig. 3.7(a), itis seen that the SCP modes must relate to the peak around 70 meV, as itis the only part of the spectrum where such a small fraction of the phononmodes can contribute such a large coupling strength. This energy correspondsto the boron-boron bond stretching modes with E2g symmetry [58]. Thesemodes change the orbital overlap of the px, py electronic states which formthe electronic bands at the Fermi level. This leads to the very clear, strongsignature of the electron-phonon coupling in the electronic spectral functionmeasured in ARPES.423.4. Conclusion3.4 ConclusionWe have presented a direct experimental evidence of the anisotropy of theelectron-phonon coupling in MgB2. Time resolved reflectivity directly iden-tifies the strongly coupled phonons responsible for the large Tc in MgB2,while ARPES measures the momentum-resolved coupling via the electronicexcitations. Not only does the coupling vary between the pi and σ bands, butthere is a strong anisotropy between the σ bands, and even along differenthigh-symmetry directions on same sheet. These results are the first directevidence of the detailed anisotropy of the electron-phonon coupling in MgB2.This anisotropy is the root cause of the distribution of superconducting gapsmeasured by electron tunnelling [7]. It is also an essential ingredient in theaccurate calculation of the superconducting properties of this material [8, 9](i.e. calculations that do not include the anisotropy fail to determine thecorrect Tc).MgB2 is often considered to be at the limit of what the electron-phononcoupling can achieve in terms of superconductivity. Despite much effort, Tchas not been increased. So while MgB2 might be considered a dead-end forapplications, it certainly serves as a test-bed for our understanding of theelectron-phonon coupling, and phonon-mediated superconductivity. Many ofthe features observed in this system, including multi-gap superconductivity,and the phonon induced band renormalizations, are observed in the noveliron-based superconductors [63]. For the methods used here to be success-fully applied to more complex systems, it is essential to demonstrate theirreliability on a well understood system such as MgB2.43Chapter 4Superconductivity inLithium-Decorated GrapheneMonolayer graphene exhibits many spectacular electronic properties [13, 14],with superconductivity being arguably the most notable exception despiteintense theoretical and experimental efforts [15–21]. To overcome this lim-itation, it was theoretically proposed that superconductivity might be in-duced by enhancing the electron-phonon coupling through the decoration ofgraphene with an alkali adatom superlattice [17]. While experiments have in-deed demonstrated an adatom-induced enhancement of the electron-phononcoupling [16, 20, 64], superconductivity has never been observed. Usingangle-resolved photoemission spectroscopy (ARPES) we show that lithiumdeposited on graphene at low temperature strongly modifies the phonon den-sity of states, leading to an enhancement of the electron-phonon coupling bya factor of 3, or λ'0.58. On part of the graphene-derived pi∗-band Fermi sur-face, we then observe the opening of a ∆' 0.9 meV temperature-dependentpairing gap. This suggests, for the first time, that Li-decorated monolayergraphene is superconducting at 3.5 K.4.1 IntroductionGraphene is a two dimensional crystal of carbon in a honeycomb structure[Fig. 4.1(a)]. The lattice is held together by strong, fully occupied σ bonds,while the electronic structure is completely determined by the carbon pz or-bitals, which form the conical pi bands at the K points of the Brillouin zone,shown in Fig. 4.1(b) and (c). In freestanding graphene, the pi bands are half444.1. Introductionfilled, putting the crossing of the bands at the Fermi level. The relationshipbetween the energy of the band and the momentum (the dispersion) aroundthe crossing point is linear, and formally equivalent to the solution of the 2Dmassless Dirac equation [65]. The crossing point is therefore known as theDirac point, ED. Many of the unique and fascinating properties of graphenefollow from this unusual electronic structure [13, 14, 66], but it is the 2D na-ture of this material that makes it so tunable. Adatoms on the surface havebeen used to control the carrier density [67–71], but also different interactions(electron-electron or electron-phonon) [16, 72, 73], and metal-insulator tran-sitions [74, 75] to name a few examples. In the following we are interested inhow adatoms can induce superconductivity in monolayer graphene.Superconductivity occurs in certain bulk graphite intercalated compounds(GICs) with Tc of up to 11.5 K in the case of CaC6, and 6.5 K for YbC6[22, 23]. KC8 and RbC8 both show superconductivity below 2 K [76]. Super-conductivity in these materials is well described by DFT calculations, andis understood to be conventional, driven by the electron-phonon coupling(EPC) [24, 25, 77]. The intercalant atoms sit in the centre of the hexagons(the hollow sites) between graphite layers, forming an ordered√3×√3R30◦(CaC6, YbC6) or 2 × 2 (KC8, RbC8) structure. There is a charge transferfrom the intercalant atoms to the graphite, leading to electron doping of thepi∗ bands.There is some evidence to suggest that the electron doping alone is theorigin of superconductivity in the graphite planes, and that the intercalantatom’s sole role is to provide extra charge carriers [78–80]. The prevailingview, however, is that the intercalant atoms alter both the electronic andthe phononic structure of the material, leading to an enhanced EPC, andsuperconductivity [24, 25, 76]. There is a correlation between the GICs thatare superconducting, and the presence of a partially occupied intercalantderived band at the Brillouin zone centre [76], arising due to the incompleteionization of the intercalants outer shell electron(s). DFT calculations showthat the intercalant layers introduce low-energy, in-plane phonon modes that454.1. IntroductionFigure 4.1: Graphene structure and band structure. (a) The honey-comb structure of graphene with the primitive unit cell and lattice vectorsshown in cyan. (c) The electronic structure consists of cone-like bands ateach corner of the hexagonal Brillouin zone in (b). Lithium on grapheneis expected to form a√3 ×√3R30◦ supperlattice, where the new unit cell(shown in green) is three times larger than that of pristine graphene, androtated by 30◦.can couple strongly to the intercalant electronic states at the Fermi level[24, 25]. While theory is clear on the importance of the intercalant electronicstates, the recent observation of the superconducting gap on the graphitic pi∗band in bulk CaC6 shows that the carbon derived bands are also involved[26]. This suggests that superconductivity in 2D monolayer graphene is areal possibility.Recent DFT based calculations have suggested that superconductivitycan be induced in graphene through the ordered adsorption of certain alkalimetals [see Fig. 4.1(d)] [17]. Although the lithium GIC, LiC6, is not knownto be superconducting, Li decorated graphene emerges as a particularly in-teresting case, with a superconducting transition temperature calculated tobe ∼8 K. The proposed mechanism for this enhancement of Tc is the removalof the confining potential of the graphite layers, which shifts the Li banddown to the Fermi level where it becomes partially occupied. The presence464.1. Introductionof these states at EF enables a strong coupling with new lithium in-planephonon modes (Lix,y), and with out of plane carbon modes (Cz) that waspreviously forbidden by symmetry. To understand this last point, note thatthe electron-phonon perturbation operator for the Cz modes is odd with re-spect to the z axis [81], and can only couple electronic states with oppositeparity. The graphene pi∗ bands have universally odd z-parity, and thereforeintraband coupling via the Cz mode is forbidden. On the other hand, the Li2s band has even z-parity, enabling interband coupling with the pi∗ bands.This picture was partially validated by ARPES measurements of Cs, Rb,K, Na, Li, and Ca deposited on graphene, with the observation of an adatom-dependent vibrational mode across the series [20]. The presence of the alkaliadatoms introduces a low-energy (< 100meV) phonon mode to the system,and/or enhances the coupling to the out-of-plane carbon vibrations. How-ever, the strength of the EPC was still too low to lead to superconductiv-ity. In this chapter, we present an ARPES study of lithium deposited onmonolayer graphene at low temperature. We observe evidence of the Li 2sband, and a strong enhancement of the EPC, in line with recent theoreticalpredictions. High-resolution, low temperature ARPES measurements showevidence of a partially gapped Fermi surface, indicative of superconductivity.Epitaxial graphene monolayers with a carbon buffer layer were grownunder argon atmosphere on hydrogen-annealed 6H-SiC(0001) substrates, asdescribed in Ref. [82]. The carbon buffer layer passivates the dangling bondsat the SiC surface, enabling the formation of the decoupled graphene mono-layer. The samples were annealed at 500 ◦C and 8×10−10 Torr for 1 hour,immediately prior to the ARPES measurements. Lithium adatoms were de-posited from a commercial SAES alkali metal source, with the graphenesamples held at 8 K. Bulk Nb polycrystalline samples, with Tc= 9.2 K, werefractured in the ARPES chamber to expose a clean surface prior to the exper-iments. The ARPES measurements were performed with s-polarized 21.2 eVphotons on an ARPES spectrometer equipped with a SPECS Phoibos 150hemispherical analyzer, a SPECS UVS300 monochromatized gas discharge474.2. Lithium adatoms on graphene and the effects of sample temperaturelamp, and a 6-axes cryogenic manipulator that allows controlling the sam-ple temperature between 300 and 3.5 K, with an accuracy ±0.1 K. Band andFermi surface mapping, as well as the study of electron-phonon coupling,were performed at 8 K with energy and angular resolution set to 15 meV and0.01 A˚−1, respectively. For the measurements of the superconducting gaps,energy and angular resolution were set to 6 meV and 0.01 A˚−1, while thesample temperature was varied between 3.5 and 15 K. During the ARPESmeasurement the chamber pressure was better than 4×10−11 Torr.4.2 Lithium adatoms on graphene and theeffects of sample temperatureARPES measurements of the electronic structure of pristine and Li-decoratedgraphene at 8 K, characterized by the distinctive Dirac cones at the corners ofthe hexagonal Brillouin zone, are shown in Fig. 4.2(a-d): Li adatoms electron-dope the graphene sheet via charge-transfer doping, leading to a shift of theDirac point to higher binding energies. As evidenced by the evolution of thegraphene sheet carrier density in Fig. 4.2(h), this trend begins to saturateafter several minutes of Li deposition; concomitantly, we observe the emer-gence of new spectral weight at the Brillouin zone centre [see the comparisonof the Γ-point ARPES dispersion for pristine and 10 minute Li-decoratedgraphene in Fig. 4.2(e,f)].The spectral weight at Γ is likely to be a superposition of a superstructure-induced folding, and the Li 2s band, but these are difficult to disentangleas they occur in the same energy and momentum region. The lithium ispredicted to form an ordered√3×√3R 30◦ superlattice [shown in Fig. 4.1(d)].The new periodic potential would create a new Brillouin zone, backfoldingthe band structure along the zone boundaries. The expected reconstructedBrillouin zone is shown as the dashed white hexagon in Fig. 4.2(g), and thegraphene pi bands would be backfolded to Γ. In support of this, we notethat the shape of the new feature shown in Fig. 4.2(f) is reminiscent of the484.2. Lithium adatoms on graphene and the effects of sample temperatureK0 min Lia0.1 0.110.50KRelative Momentum, k-K (A˚1)Energy(eV)3 min Lib0.1 0.1K6 min Lic0.1 0.2K10 min Lid0.1 0.2K0 min Lie0.2 0.21.510.50Energy(eV)10 min Lif0.2 0.2Momentum (A˚1)Bandsat h0 10 20 30024681012Li deposition time (min)n 2D(⇥1013cm2 )10 min LigFigure 4.2: Charge-transfer doping of graphene by lithium adatoms.(a) Dirac-cone dispersion measured by ARPES at 8 K on pristine grapheneand (b,c,d) after 3, 6 and 10-minute Li evaporation respectively. The mea-surements were taken along the K-point momentum cut indicated by thethick white line in the Fermi surface plot in (g). The Dirac point, already lo-cated below EF on pristine graphene due to the charge-transfer from the SiCsubstrate (a), further shifts to higher energies with Li evaporation (b-d); thepresence of a single well-defined Dirac cone indicates a macroscopically uni-form Li-induced doping. While no bands are present at the Γ-point on pris-tine graphene (e), new spectral weight is detected on 10-minute Li-decoratedgraphene in (f) and (g), indicating a possible√3×√3R 30◦ backfolding ofthe electronic structure induced by Li ordering, and the Li 2s band. As illus-trated in the 8 K sheet carrier density plot versus Li deposition time in (h),which accounts for the filling of the pi∗ unfolded Fermi surface, the spectralweight at Γ is achieved for charge densities n2D&9×1013cm−2 (but completelydisappears if the sample temperature is raised above ∼40 K; see Fig. 4.3).494.2. Lithium adatoms on graphene and the effects of sample temperatureDirac cone, and that this form of reconstruction has been observed for bulkLiC6 as well as intercalated bilayer C6LiC6 [83, 84]. On the other hand, theLi 2s band is expected to form as a parabolic, free electron-like band at Γ.Here, we note that the symmetry of the Fermi surface shown in Fig. 4.2(g)is circular, and not hexagonally warped as one would expect for the foldedbands. Additionally, the area of the Γ Fermi contour is larger than that ofthe pi band Fermi contour, as predicted for the parabolic band [17].At the saturation of doping, we find a sheet charge-carrier density n2D'9×1013 cm−2. Assuming Li forms an ordered LiC6 structure [Fig. 4.1(d)] after4 minutes evaporation (i.e. one Li per three unit cells), this corresponds toa charge transfer to the graphene pi∗ bands of 0.14±0.02 electrons per Liadatom. This is significantly lower than what is reported in Li-intercalatedcompounds, where the Li is found to be completely ionized [85]. The latteris an important point, since the incomplete ionization of the Li 2s electronsis necessary in order to form a Li band at the Γ point, which in turn hasbeen identified as a key element in the enhancement of the electron-phononcoupling [17, 25, 76].After lithium deposition, for temperatures between 3.5 K and 20 K, the Liadatoms appear stable on the graphene surface. Fermi contours measured at7 K, 13 K, and 17 K shown in Figs. 4.4(a1-a3) exhibit no change in area withinthe bounds of uncertainty, as seen in the ky-integrated MDCs in Fig. 4.4(b).Increasing the temperature above 20 K leads to a progressive reduction ofthe charge transfer doping, seen in the shift of the Dirac point in the rawdata in Fig. 4.3(a) and in the extracted carrier density in Fig. 4.4(c). Asignificant broadening of the pi∗ bands with increasing temperature is alsoapparent, signifying increasing surface disorder. At 50 K there is a suddendisappearance of the Γ spectral weight [Fig. 4.3(b)], and it does not reappearupon subsequent cooling.The strong, irreversible temperature dependence of the doping, the tem-perature induced disorder, and the disappearance of the spectral weight areall in stark contrast with behaviour seen in Li-intercalated bulk graphite and504.2. Lithium adatoms on graphene and the effects of sample temperature10 K 20 K 40 K 60 K 100 Ka10.2 0 0.21.510.50Energy(eV)a20.2 0 0.2a30.2 0 0.2Momentum, k-K (A˚1)a40.2 0 0.2a50.2 0 0.2b10.2 0 0.21.510.50Energy(eV)b20.2 0 0.2b30.2 0 0.2Momentum (A˚1)b40.2 0 0.2b50.2 0 0.2Figure 4.3: Temperature dependence of the band dispersion in Li-Graphene. (a1-a5) ARPES measurements through the Dirac point at Kshowing a reduction in the n-type doping as the temperature is increased. Asignificant broadening of the bands can also be seen, particularly above 50 K,which is an indication of surface disorder. (b1-b5) ARPES measurements atΓ showing the complete disappearance of the backfolded bands at 50 K.514.2. Lithium adatoms on graphene and the effects of sample temperature7Ka10.6 0.8 11.61.4k y(A˚1)13Ka20.6 0.8 1Momentum kx (A˚1)17Ka30.6 0.8 1c0 25 50 75 100024681012Temperature (K)n(⇥1013cm2)b7 K13 K17 K0.6 0.8 1051015kx (A˚1)Intensity(a.u.)Figure 4.4: Evolution of the charge density with temperature. Mea-suring the Fermi surface of a Li-doped sample as a function of temperature(a1-a3) while warming up shows that the carrier density remains constant upto ∼ 20K. The ky-integrated MDCs in (b) emphasize this point, showing nochange in the area of the Fermi contour in this temperature range. Combin-ing this data with the temperature dependence in Fig. 4.3(a), we show thetemperature dependence of the graphene pi∗ band carrier density between7 K and 100 K. At temperatures below ∼20 K the doping is stable. Above∼20 K, however, there is a reduction of n with increasing temperature, withthe sheet carrier density returning to the starting value of clean graphene(1× 1013 cm−2) at around 100 K.bilayer graphene [85, 86] where the Li intercalates easily at room tempera-ture, and the doping is stable between 30 K and 300 K. This suggests that thelithium resides on the graphene surface, as opposed to intercalating betweenthe graphene and substrate.The evidence so far indicates the lithium forms an ordered layer on the524.3. Enhancement of the electron-phonon couplinggraphene surface, and the Li 2s band is partially occupied, only for lithiumdeposited on graphene at low temperature. The ordering disappears at 50 K.This is an important point that distinguishes this work from previous studies,and explains why we succeed in observing an enhancement of the electron-phonon coupling. While the doping-saturation regime was already reachedin previous in-situ alkali-deposition studies [20], the spectral weight at Γ wasnot observed presumably because the sample temperatures of ∼ 50 K weretoo high.4.3 Enhancement of the electron-phononcouplingHere, we perform a detailed analysis of the electron-phonon coupling in thegraphene pi∗ band as a function of lithium coverage. The lithium layer is pre-dicted to modify the phononic structure, which in turn modifies the electronicstructure via the electron-phonon coupling (EPC). These renormalizations ofthe electron energy and lifetime are described by the real and imaginary partsof the self-energy respectively. In an actual ARPES measurement, as pre-sented below, these features would appear as ‘kinks’ in the dispersion, wherethe band velocity changes abruptly, or as steps in the MDC linewidth at theenergy of the strongly coupled phonon.Measurements were performed along the cut shown by the white line inFig. 4.6(e) for three different lithium deposition times: 3 minutes, 6 min-utes and 10 minutes. The electron-phonon coupling analysis discussed belowcannot be performed on the clean, undoped sample due to the proximityof the Dirac point to the Fermi level, which makes it difficult to follow theMDC dispersion. At 3 minutes, the doping is not saturated, and the spec-tral weight at Γ is not yet apparent. Depositions of 6 and 10 minutes areboth in the saturated doping regime, and the spectral weight at Γ is present.This cut was chosen because it intersects the K point, giving the full disper-sion from the Fermi energy to the Dirac point. The MDCs are fitted with534.3. Enhancement of the electron-phonon couplinga Lorentzian plus linear background to determine the peak maximum kmand ∆km, and the KKBF routine was used to self-consistently determine thebare-band and self-energies [see Chapter 2 for more details]. The results ofthis process are summarized in Fig. 4.3. The MEM was used to fit the realpart of the self-energy, and extract the Eliashberg function for the differentLi coverages, and finally the k-resolved electron-phonon coupling parameterwas calculated from λk = 2∫dωα2Fk(ω)/ω. These results are summarizedin Fig. 4.6.Graphene doped with alkali adatoms always shows a strong kink in the pi∗band dispersion at a binding energy of about 160 meV [20, 64]. This is truefor the Li-decorated samples at all coverages studied here, as seen in the rawdata in Fig. 4.2(a-d). This high-energy feature is even better visualized inthe extracted momentum-distribution curve (MDC) dispersions [Fig. 4.3(a-c)and Fig. 4.6(b-d)]. This structure stems from the coupling to carbon in-plane(Cxy) phonons [17, 25] of E2g symmetry which, despite the apparent strength,contribute little to the overall coupling parameter due to their high energy(note that ω appears as a weighting factor in the integral calculation ofλ). The contribution to λk from these high-energy modes between 100 and200 meV can be isolated by integrating over α2Fk(ω) in that energy range.We find λk,HE = 0.14±0.05, and it remains approximately constant for allLi coverages studied here, as illustrated by the white symbols in Fig. 4.6(i).544.3. Enhancement of the electron-phonon coupling3 min 6 min 10 min6×1013 cm−2 1×1014 cm−2 1×1014 cm−2c0 0.02 0.04 0.06k − kf (A˚−1)b0 0.02 0.04 0.06k − kf (A˚−1)a0 0.02 0.04 0.06−400−300−200−1000k − kf (A˚−1)E-E F(meV)f1050Σ′f20 100 200050Energy (meV)Σ′′ (meV)e1050Σ′e20 100 200050Energy (meV)Σ′′ (meV)d1050Σ′d20 100 200050Energy (meV)Σ′′ (meV)Figure 4.5: Self-consistent self-energy analysis. MDC dipersions, de-termined by fitting the ARPES data with Eqn. 2.9, are plotted in orange in(a-c) and are used to extract the self-energy using the self-consistent Kramers-Kronig procedure described. The bare-band [gray in (a-c)] corresponds to thereal and imaginary parts of the self-energy calculated from Eqn. 2.10 plottedin orange in panels (d-f). The correct bare-band is identified by the goodagreement between the real and imaginary parts of the self-energy, and theirKK transforms (blue).554.3.Enhancementoftheelectron-phononcoupling3 min Lia0 0.15004003002001000k  kF (A˚1)E n er g y( m eV )3 min Lib0 0.053002001000k  kF (A˚1)E n er g y( m eV )⌃025 50(meV)3 min LiExpt Theory↵2F (!)(!)f0 50 100 150 20000.20.40.6Energy (meV)↵2 F( ! ),  ( !)6 min Lic0 0.053002001000k  kF (A˚1)⌃025 50(meV)6 min Lig0 50 100 150 20000.20.40.6Energy (meV)10 min Lid0 0.053002001000k  kF (A˚1)⌃025 50(meV)10 min Lih0 50 100 150 20000.20.40.6Energy (meV)10 min Liei Experiment3 6 1000.20.40.6Li deposition time (min)Figure 4.6: Analysis of electron-phonon coupling in Li-decorated graphene. (a) Dirac dispersionfrom 3-minute Li-decorated graphene, along the k-space cut indicated in the Fermi surface plot in (e),exhibiting kink anomalies due to electron-phonon coupling (white line: MDC dispersion). (b-d) MDC dis-persion and bare-band obtained from the self-consistent Kramers-Kronig bare-band fitting (KKBF) routine[34, 36], for several Li coverages (see Chapter 2); the real part of the self-energy Σ′ is shown in the side pan-els (orange: Σ′ from KKBF routine analysis; black: Σ′ corresponding to the Eliashberg function presentedbelow). (f-h) Eliashberg function α2F (ω) from the integral inversion of Σ′(ω) [39], and electron-phononcoupling constant λk = 2∫dω α2F (ω)/ω, where the colour shading represents the strength of the totalEPC. In (h) the theoretical result from Ref. [17] for a LiC6 monolayer are also shown (gray shading). (i)Experimentally-determined contribution to the total (black symbols) electron-phonon coupling from phononmodes in the energy range 100 - 250meV (blue shading, white symbols) and 0-100meV (orange shading);the coupling of low-energy modes strongly increases with Li coverage.564.3. Enhancement of the electron-phonon couplingThe effects of lithium become more significant in the modifications tothe low-energy part of the dispersion, below ∼ 100 meV. With 10 minutesof Li deposition [Fig. 4.6(d)], an additional kink is clearly visible at around30 meV binding energy, along with the associated peak in the real part of theself-energy Σ′. These changes occur with increasing Li coverage, which canbe seen in the progressive enhancement of both MDC-dispersion kink andΣ′ peaks in Fig. 4.6(b-d). The lack of any appreciable variation in carrierdensity between the 6 and 10 minute Li depositions [Fig. 4.2(h)] suggests itis really the increase in the number of Li atoms – and thus perhaps in thedegree of order within the Li layer – that is driving these changes.The extracted Eliashberg functions in Fig. 4.6(f-h) tell the same story:at high Li coverage, spectral weight in the Eliashberg functions appears atenergies below 60 meV, indicating the presence of phonon modes couplingstrongly to the graphene electronic excitations. It is these low energy con-tributions to α2Fk(ω) that lead to the large values of λk, as seen in theenergy-resolved λk(ω) plotted as the solid line in Figs. 4.6(f-h).In order to understand the origin of these new modes, we can look toexisting calculations for this system and similar ones such as CaC6 [17, 25].There is a strong consensus that the alkali in-plane (Lixy) vibrational modeslie between 10 and 40 meV, while the carbon out-of-plane (Cz) modes arearound 50-80 meV. Comparison with the experimental Eliashberg functionsin Figs. 4.6(f-h) leads to the conclusion that the Lixy modes are involved,while it is less clear if the Cz modes are also important.As for the total electron-phonon coupling λk for each coverage, repre-sented by the black symbols in Fig. 4.6(i), we note that our values mea-sured on the pi∗-band Fermi surface at an intermediate location betweenΓK and KM directions [Fig. 4.6(e)] provide an effective estimate for themomentum-averaged coupling strength. We state this based on observationsof anisotropic electron-phonon coupling around the pi∗-band Fermi surface inboth decorated graphene [20] and intercalated graphite [79]. In those cases,the maximum and minimum values of λ were measured along ΓK and KM,574.3. Enhancement of the electron-phonon couplingand the value measured at the intermediate Fermi crossing corresponds ap-proximately to the momentum-averaged coupling strength along the pi∗-bandFermi surface.Remarkably, the value λk = 0.58 ± 0.05 observed at the highest Li cov-erage [Fig. 4.6(i)] is approaching λ= 0.61 predicted for monolayer LiC6 [17];close agreement is also found for experimental and calculated energy-resolvedλk(ω) in Fig. 4.6(h), with most of the electron-phonon coupling enhancementcoming from the the low-energy phonon modes. Finally, λk' 0.58 achievedhere on 10 minute Li-decorated graphene is much larger than the momentum-averaged results previously reported for both Li and Ca deposition (λ'0.22and 0.28 respectively [20]), and is even comparable to λ' 0.58 observed forbulk CaC6 [79].We note that this enhancement is critically dependent on the presenceof the spectral weight at Γ, and disappears when the latter is disorderedby increasing temperature. The enhancement of the EPC with increasingLi coverage can be reversed by increasing the temperature above 50 K, thepoint at which the spectral weight at Γ vanishes. The EPC was studied ona Li-decorated sample (10 min) annealed at 60 K, and is compared to theanalysis pre-annealing in Fig. 4.7. We see the near total disappearance of thelow-energy renormalizations of the band. The self-energy is dominated bya peak around -180 meV, and the coupling is determined to be 0.13± 0.05,close to the value of the low-coverage sample. While we assign this effectto the disappearance of the Γ bands, it must also be noted that the chargecarrier density is reduced from 1 × 1014 to 5 × 1013 cm−2 after annealingat 60 K. It is possible that the doping plays a role in the modification of theelectron-phonon coupling, but it is not possible to disentangle the two effectsas the saturation of doping is associated with the appearance of the spectralweight at Γ.As a consistency check on the analysis and extraction of the EPC strength,we have done a parallel analysis using a simplified model of α2Fk(ω) consist-ing of four Lorentzians whose position, width, and area can be varied in584.3. Enhancement of the electron-phonon coupling10 min Lia0 0.05−300−200−1000k − kF (A˚−1)Energy(meV)Σ′25 50(meV)10 min Li+ annealb0 0.05−300−200−1000k − kF (A˚−1)Σ′25 50(meV)c0 50 100 150 20000.20.40.6Energy (meV)α2F(ω),λ(ω)Figure 4.7: Reversible enhancement of the electron-phonon cou-pling. The MDC dispersion (orange) and real part of the self energy (orange,right-hand panel) for the 10-minute Li sample in (a) show the strong kinkat around 30 meV. (b) After annealing to 60 K for several minutes, the lowenergy feature is no longer apparent. (c) The Eliashberg functions corre-sponding to the fits to the self energy (black) in (a) and (b) are shown inlight orange and light blue respectively. The absence of the low energy featureresults in a severely diminished λk, shown as the solid lines in (c).order to simultaneously fit the real and imaginary parts of the self-energy.The α2Fk(ω) found with this method is in good agreement with that foundby the MEM, with the main peaks at the same energies. The resultingλ = 0.55± 0.05 is slightly larger than the value from the MEM (0.46± 0.05),an effect that has been previously observed when comparing these analysismethods in other materials.[40] While the MEM is sensitive to the noise ofthe data, the agreement of these two methods implies they both capture theunderlying structure of α2Fk(ω).For an electron-phonon coupling parameter on the order of λk ∼ 0.6,the expected superconducting transition temperature can be calculated withthe Allen-Dynes formula3 to be ∼ 8 K. With this in mind, the next sectiondescribes the direct measurement of a signature of superconductivity in Li-graphene.3The Coulomb parameter is set to µ∗=0.115, [Eq. 2.22] based on what is known ofCaC6, and following Ref. [17]594.3. Enhancement of the electron-phonon couplingMax Entropy Method Lorentzian Methodb1 6 min Li050Σ′b20 100 200050Energy (meV)Σ′′(meV)a1 6 min Li050Σ′a20 100 200050Energy (meV)Σ′′(meV)α2F (ω)λ(ω)c0 100 20000.20.40.6Energy (meV)α2F(ω),λ(ω)d0 100 20000.20.40.6Energy (meV)α2F(ω),λ(ω)Figure 4.8: Comparison of two methods for determining α2Fk(ω) andλk. The real and imaginary parts of the self-energy for the 6-minute Li-decorated sample are shown in orange in (a) and (b). The maximum-entropymethod is used to fit Σ′ in black in (a1), and its Kramers-Kronig transform isshown in blue in (a2). The Eliashberg function and electron-phonon couplingconstant λk are shown in (c). An alternate method for extracting α2Fk(ω)by simultaneously fitting the real and imaginary parts of the self-energy withequal weighting is shown in (b). The Eliashberg function, which consists offour Lorentzians whose position and size is varied to achieve a good fit, isshown in (d), along with λk.604.4. Spectroscopic gap: evidence of superconductivity4.4 Spectroscopic gap: evidence ofsuperconductivityNext we use high-resolution, low-temperature ARPES to search for the open-ing of a temperature-dependent pairing gap along the pi∗-band Fermi surface,as a direct spectroscopic signature of the realization of a superconductingstate in monolayer LiC6. To increase our experimental sensitivity, as illus-trated in Fig. 4.9(a) and following the approach introduced for FeAs [87]and cuprate [88] superconductors, we perform an analysis of ARPES en-ergy distribution curves (EDC) integrated in dk along a one-dimensionalmomentum-space cut perpendicular to the Fermi surface. This provides theadded benefit that the integrated EDCs can be modelled in terms of a sim-ple Dynes gap function [41] multiplied by a linear density of states and theFermi-Dirac distribution function, all convolved with a Gaussian resolutionfunction:I∫dk(ω)=[f(ω, T ) (a+b ω)∣∣∣∣∣Reω − iΓ√(ω − iΓ)2 −∆2∣∣∣∣∣]⊗Rω , (4.1)where the parameters a and b describe the linear density of states, ∆ is thesize of the superconducting gap, and Γ is a phenomenological scattering term.As shown in Fig. 4.9(a) and especially 4.9(b) for data from the k-spacelocation indicated by the white circle in Fig. 4.6(e), a temperature depen-dence characteristic of the opening of a gap can be observed near EF : atvariance with the case of Au spectra crossing precisely at EF according tothe Fermi-Dirac distribution function [Fig. 4.10(a)], the leading edge mid-point of the Li-graphene spectra moves away from EF in cooling from 15to 3.5 K. When fit to Eq. 4.1, this returns a gap value ∆ = 0.9±0.2 meV at3.5 K (with Γ'0.09 meV). Given its small value compared to the experimen-tal resolution, the gap opening is best visualized in the symmetrized datain Fig. 4.9(c), which minimizes the effects of the Fermi function, and in thedensity of states in Fig. 4.9(d) when the effect of the 6 meV energy resolution614.4. Spectroscopic gap: evidence of superconductivityon the fitting function is simply removed (owing to the integration of theARPES intensity in dk, this analysis is unaffected by momentum resolution[87]).624.4.Spectroscopicgap:evidenceofsuperconductivityLi-GrapheneT=15 KT=3.5 K∆=0.9 meVb−10 −5 0 5 1000.51Energy (meV)I n te ns i ty (a . u. )Li-GrapheneT=15 KT=3.5 K00.51I n te ns i ty (a . u. )a−30 −20 −10 0 10Energy (meV)k Li-G15 K3.5 Kc−10 −5 0 5 100.70.80.91Energy (meV)Li-GDynesDy⊗Rωd2∆ 1.8meV−6 −3 0 3 6012Energy (meV)Figure 4.9: Spectroscopic observation of a pairing gap in Li-decorated graphene. (a) Diracdispersion from 10-minute Li-decorated graphene measured at 15 and 3.5K, at the k-space location indicatedby the white circle in Fig. 4.6(e); the temperature dependence is here evaluated for EDCs integrated in the0.1 A˚−1 momentum region about kF shown by the white box (bottom), with the only changes occurring nearEF (top). A closer look at the near-EF region in (b) shows the crossing point of the Li-graphene spectra isshifted away from EF (cyan dashed line), due to the pull-back of the leading edge at 3.5K. A fit to the Dynesgap equation yields a gap ∆'0.9meV at 3.5K (and 0meV at 15K). The superconducting gap opening isbest visualized in the symmetrized data in (c), i.e. by taking I(ω)+I(−ω) which minimizes the effects ofthe Fermi function even in the case of finite energy and momentum resolutions [89, 90], and in the densityof state plot in (d) where the effect of the 6meV resolution on the fitting function is simply removed [blueand red symbols in (c) represent the smoothed data, while the light shading gives the root-mean-squaredeviation of the raw data].634.4.Spectroscopicgap:evidenceofsuperconductivityNiobiumT=12 KT=4.5 K∆=1.4 meVb−10 −5 0 5 1000.51Energy (meV)I n te ns i ty (a . u. )Nb12 K4.5 Kc−10 −5 0 5 100.70.80.91Energy (meV)NbDynesDy⊗Rω2∆ 2.8meVd−6 −3 0 3 6012Energy (meV)GoldT=15 KT=3.5 Ka−10 −5 0 5 1000.51Energy (meV)I n te ns i ty (a . u. )Figure 4.10: Reference pairing gap in polycrystalline niobium. (a) The Fermi edge measured onpolycrystalline gold shows the crossing point of the two temperature curves at E=0, and no change inthe position of the leading edge with temperature, as expected for a normal metal. This is in contrastto niobium, a known superconductor with Tc ∼ 9K, where the leading edge of the EDC in (b) shifts atlow temperature. A Dynes fit to the Fermi edge yields a gap ∆' 1.4meV at 3.5K (and 0meV at 15K).The symmetrized data are shown in (c) where the temperature dependent gap is more obvious. The samesymmetrized fit is shown in blue in (d), and is compared to a simulation with the 6 meV energy resolutionset to 0.644.4. Spectroscopic gap: evidence of superconductivityThe spectroscopic gap appears to be anisotropic: it is either absent orvanishingly small along the KM direction. High-resolution measurementsperformed at the high symmetry point along KM do not show a gap (Fig.4.11). The EDC of the Li-G at this point on the Fermi surface has its crossingpoint at 0, and the shape of the edge is the same as a reference measurementon polycrystalline gold. The slight difference in the slope of the edge crossingzero is attributed to a small difference in the experimental resolution betweenthe two measurements. The gap is either closed or too small to measure withthe resolution of this study. Either way, it demonstrates an anisotropy of thegap. This is perhaps not so surprising as bulk CaC6 is thought to have ananisotropic superconducting gap.[91–93]KMLi-G, 3.5 KAu, 3.5 K∆=0 meV−10 −5 0 5 1000.51Energy (meV)Intensity(a.u.)Figure 4.11: Absence of a gap along KM. EDC at EF at the cornerof the triangular Fermi surface (shown inset) for Li-Graphene (blue) andpolycrystalline gold (yellow). Both measurements were done at 3.5 K.The detection of a temperature-dependent anisotropic gap with a leading-edge profile described by the Dynes function – with its asymmetry aboutEF and associated transfer of spectral weight to just below the gap edge –suggests the gap is a superconducting pairing gap. The phenomenology wouldbe in fact very different in the case of a Coulomb gap, i.e. the depression in654.4. Spectroscopic gap: evidence of superconductivitydensity of states at EF due to the combination of disorder with long-rangeCoulomb interactions [94]. In addition to being isotropic in momentum, thiswould lead to a rigid shift of the spectra leading edge, as observed in theemergence of the small nodal gap – and thus of a fully gapped excitationspectrum – in deeply underdoped cuprates [95]. Similarly, the observed gapis inconsistent with a charge density wave origin, since the gap is tied tothe Fermi energy as opposed to a particular wavevector (the latter mightoccur at the M points, when graphene is doped all the way to the Van Hovesingularities resulting in a highly-nested hexagonal Fermi surface [18], or atthe K points in the case of a√3 ×√3R 30◦ reconstruction leading to aDirac-point gap).To further explore the nature of the gap observed on Li-decorated graphene(and also demonstrate our ability to resolve a gap of the order of 1 meV), inFig. 4.10(b-d) we show as a benchmark comparison analogous results from abulk, polycrystalline niobium sample – a known BCS superconductor withTc ' 9.2K and gap ∆ ' 1.4 meV. The edge shift [Fig. 4.10(b)] and the dipin the symmetrized spectra [Fig. 4.10(c)] are more pronounced than in theLi-graphene case, owing to the larger gap, but the behaviour is qualitativelyvery similar. The Dynes fit of the integrated EDCs Fermi edge determinesthe gap to be ∆=1.4±0.2 meV (with Γ'0.14 meV), in excellent agreementwith the reported values [38].It has been shown that in a 2 dimensional superconductor, there will bea thermodynamic instability relating to the unbinding of vortex-anti-vortexpairs above some critical temperature [96]. This is known as a Kosterlitz-Thouless transition, and if TKT ≤Tc then superconductivity will be sup-pressed. Estimates of TKT rely on parameters such as the superfluid densitywhich are not known for this system. However, a phenomenological relation-ship between the sheet resistance and TKT/Tc has been demonstrated [97],showing that quite large sheet resistances are needed to reduce TKT belowTc (on the order of kΩ). Li-graphene is heavily doped, with a typical sheetresistance below 1 kΩ [98], and so the possibility of a KT transition can664.5. Conclusionreasonably be set aside at this stage.4.5 ConclusionThe electron-phonon coupling and its relation to superconductivity is wellenough understood that plausible predictions about new superconductingmaterials can be made from theory. Such a prediction was made for Li-decorated graphene, and we present the experimental evidence to supportthis. It is only through low-temperature sample preparation in an ultra-clean,ultra-high vacuum environment, that lithium leads to a strong enhancementof the electron-phonon coupling, as measured by ARPES on the graphene pi∗band Fermi surface.Taken together, our ARPES study of Li-decorated monolayer grapheneprovides the first evidence for the presence of a temperature-dependent pair-ing gap on part of the graphene-derived pi∗ Fermi surface. The detailedevolution of the density of states at the gap edge, as well as the phenomenol-ogy analogous to the one of known superconductors such as Nb – as wellas CaC6 and NbSe2, which also show a similarly anisotropic gap around theK point [91–93, 99, 100] – indicate that the pairing gap observed at 3.5 Kin graphene is associated with superconductivity. Based on the BCS gapequation, ∆ = 3.5 kb Tc, this suggests that Li-decorated graphene might besuperconducting with Tc'5.9 K, remarkably close to the value of 8.1 K foundin density-functional theory calculations [17].67Chapter 5Thallium on GrapheneMany interesting properties of graphene can be tuned by the appropriateintroduction of impurities. This concerns not just charge-transfer dopingthrough adatoms or molecules to tune the position of the Dirac point withrespect to the Fermi level [67–71], but also different interactions (electron-electron or electron-phonon) [16, 72, 73], metal-insulator transitions [74, 75]and the emergence of superconductivity [2, 15, 17, 18, 21]. It has also beenproposed that graphene could be tuned to become a 2D topological insulator[27, 101, 102].Graphene was originally predicted to be a 2D topological insulator [28]however the spin-orbit coupling resulting from carbon’s small atomic number[103] is too weak to produce any experimentally observable, or technologicallyuseful, effects. The path to making graphene into a robust topological insu-lator is through opening a gap at the crossing of the pi bands at the corners ofthe hexagonal Brillouin zone, and in order to do this, the spin-orbit couplingmust be enhanced. Following an exhaustive search using electronic structurecalculations, thallium was identified as an element that when deposited onmonolayer graphene, could enhance the spin-orbit coupling, and open a gapat the Dirac point on the order of tens of meV, several orders of magnitudelarger than in intrinsic graphene [27]. In addition to being heavy, thalliummet several other criteria. Firstly, the atoms sit at the hollow sites, in themiddle of the carbon hexagons, where they can most effectively mediate thesecond-neighbour hopping that gives rise to the spin-orbit term in the Kane-Mele model [28]. Secondly, the adatom must have no magnetic moment soas not to break time-reversal symmetry. The calculations indicate that thesole unpaired 6p electron is fully ionized (ie. it is transferred to the graphene685.1. Preparation of Tl adatoms on graphenesheet), leading to electron doping of the pi bands.Using ARPES, we studied the system of thallium adatoms on graphene,characterizing the modifications to the electronic structure, and the role oftemperature. Many of the predictions from the theoretical work were foundto be correct, and this information will stimulate future studies of Tl ongraphene with ARPES and other experimental techniques, and is presentedin Section 5.1. Studying the evolution of the linewidth at the Fermi level withincreasing Tl coverage revealed details of the scattering mechanisms at playin this system. In Section 5.2 it is shown that short-range δ-like scatteringdue to the adatoms is significant in this system, despite the expectation thatCoulomb scattering would dominate. Interestingly, this too lends support toone of the predictions from theory that implies Tl is a very good candidatefor realizing a topological insulator from graphene.5.1 Preparation of Tl adatoms on grapheneEpitaxial graphene monolayers with a buffer layer were grown under argonatmosphere on hydrogen annealed 6H-SiC(0001) at MPI-FKF in Stuttgart[82]. ARPES and scanning tunneling microscopy (STM) measurements wereperformed at UBC. The ARPES experiments used linearly-polarized photonswith an energy of 40.8 eV and an ARPES spectrometer equipped with aSPECS Phoibos 150 hemispherical analyzer and UVS300 monochromatizedgas discharge lamp. Energy and angular resolution were set to 15 meV and0.3◦. The graphene samples were slightly post-annealed after insertion intoultra-high vacuum, and right before Tl deposition in the ARPES chamber.It has been calculated that thallium is one of the most mobile atoms onthe graphene surface [104]. Therefore, Tl atoms were evaporated from anelectron beam evaporator on a cold sample to avoid surface diffusion (at4.5K for STM (scanning tunnelling microscopy), and 8K for ARPES). Allsubsequent measurements were also done in the same temperature range.During measurements the samples were kept at a pressure better than 7 ×695.1. Preparation of Tl adatoms on graphene10−11 mbar.STM measurements demonstrate that, at very low temperatures and con-centrations, thallium atoms are present as immobile monomers on the surface[see Fig. 5.1]. The topographic image shows a thallium covered graphenemonolayer for a coverage of 0.2 %. Even at this low coverage, a small fractionof adsorbed thallium appears to be in dimer or trimer form, although the vastmajority appear to be thallium monomers. This has also been confirmed byTl deposition on a clean Au(111) surface [Fig 5.1(a)] using the same parame-ters and yielding the same number density. It is difficult to find regions whereboth the thallium adatom and the graphene lattice can be visualized simul-taneously. This is due to electronic inhomogeneities in the substrate andbuffer layer. Thallium on more uniform graphene patches appears to sit onthe hollow site, in agreement with predictions from density functional theory(DFT) calculations [27, 104], although the large apparent size of the thal-lium adatoms makes a more conclusive assignment difficult. and graphene’sinherent electronic inhomogeneity make an identification of the adsorptionFigure 5.1: Calibration of thallium deposition. STM images of Tladatoms deposited from and electron-beam evaporator on Au(111) [panel(a)], and graphene/SiC [(b) and (c)]. Tl atoms are imaged as the brightspots in (a), and as the orange/gold spots in (b) and (c). Samples wereheld at 4.5 K during deposition and measurement, and the adatoms showedno signs of diffusion, and were predominantly isolated, single atoms. Thecoverage shown here corresponds to 2 minutes evaporation and 0.2 % of amonolayer.705.1. Preparation of Tl adatoms on graphenesite difficult, thallium on more uniform graphene patches appears to sit onthe hollow site, in agreement with predictions from density functional theory(DFT) calculations [27, 104]. As the precise coverage is important for thesubsequent quantitative analysis of the ARPES data, the STM results werealso used for the precise flux calibration of the Tl evaporator.715.1.PreparationofTladatomsongraphene0 0.06 0.09 0.15 0.21 0.3 0.45 0.71 1.0 1.65 3.2 4.80.1 0 0.110.50E ne r gy (e V)0.1 0 0.1 0.1 0 0.1 0.1 0 0.1 0.1 0 0.1 0.1 0 0.1 0.1 0 0.1Momentum, kx (A˚1)0.1 0 0.1 0.1 0 0.1 0.1 0 0.1 0.1 0 0.1 0.1 0 0.1Figure 5.2: Dispersion of the graphene pi bands with increasing Tl coverage. Raw ARPES datameasured the the K point of the Brillouin zone shows how the electronic structure is modified with increasingTl coverage from left to right. The numbers along the top indicate the coverage of Tl as a percentage of amonolayer, where a monolayer would correspond to one Tl atom per carbon atom. There is some dopingof the pi bands due to charge transfer from the Tl adatoms, but only up to 1%. With increasing coveragebeyond this, the trend reverses, and the Dirac point can be seen to shift back towards the Fermi level. TheMDC width increases with increasing coverage.725.1. Preparation of Tl adatoms on grapheneFigure 5.2 shows the experimental band structure of an epitaxial graphenemonolayer near the K-point, measured in the geometry shown in Fig. 5.3(a).The concentration x of thallium atoms on the surface is given in percentof a graphene monolayer, i.e. the number of thallium atoms per carbonatom. The left-hand side data set is from the pristine graphene monolayer,exhibiting an initial n-type doping due to the intrinsic charge transfer fromthe SiC substrate [82]. Upon thallium deposition, there is a charge transferfrom the thallium atom to the graphene sheet: electron doping increases sothat the Dirac point shifts to higher binding energies, away from the Fermilevel. For concentrations higher than ∼ 1%, the trend is reversed, and theDirac point shifts back towards the Fermi level. We ascribe this to the onsetof pairing and clustering of atoms, which reduces the efficiency of chargetransfer doping. For this reason, in the scattering analysis in Section 5.2,we concentrate on the dilute limit below 1%, where clustering effects canbe neglected. The strong broadening of the ARPES line width at the highercoverages makes it very difficult to determine whether there is a spectroscopicgap appearing at the Dirac point. Theory predicts that there should be aspin-orbit coupling induced gap of 20 meV at 6 % coverage [27], but thiswould be unobservable with the level of broadening seen.The shift of the Dirac point to higher binding energies upon thalliumdeposition illustrated in Fig. 5.2 corresponds to an increase of the Fermi con-tour area. The corresponding evolution of Fermi wave vector kF , as well ascharge carrier density n and energy shift are shown in Fig. 5.3 (b), (c), and(d) respectively. Below a coverage of 1%, all three parameters are increasingmonotonically. The energy shift reflects the increase in occupation due tothe donation of electrons by the thallium atoms. A simple model employinggraphene’s linear band dispersion is used to estimate the charge transfer perthallium atom from the energy of the Dirac point (ED):ED =√pi~νF√N0 +Ne , (5.1)where νF is the band velocity near the Fermi level, N0 is the number ofelectrons in the clean graphene, and Ne is the number of electrons donated735.1. Preparation of Tl adatoms on graphenea b0 2 4 6−0.2−0.100.10.2Tl coverage (%)k F(1/A˚)c0 1 2 3 4 50123Tl coverage (%)n(e− /cm2 ×1013 ) d0 1 2 3 4 500.10.20.3Tl coverage (%)∆ED(eV)Figure 5.3: Charge transfer doping from Tl atoms. ARPES measure-ments were performed at the K point of the Brillouin zone, shown with theblue line in (a). MDCs at the Fermi level show the evolution of the Fermiwavevector in (b). The charge density in (c), calculated from the FS con-tour area, increases linearly with coverage until about 1 %, where the trendreverses. A similar trend is seen in (d), where the energy shift of the Diracpoint is shown. A simple model relating ∆ED, the coverage, and the numberof electrons donated per Tl is shown in green, indicating that the Tl donatesone electron per atom.by the adatoms (proportional to the number of adatoms times the number ofelectrons per adatom). The best fit is found for 1.0 ± 0.1 electrons donatedper thallium atom to the graphene layer, shown in green in Fig. 5.3(d). Thisconfirms one of the results from DFT calculations for thallium adatoms ongraphene [27], implying the Tl atoms donate their sole unpaired 6p electronto the graphene, and therefore have no magnetic moment.While the Tl atoms are immobile below 8 K in the topographic images[Fig. 5.1], we observe that the thallium atoms are extremely mobile on thegraphene surface at higher temperatures (cf. Ref. [104]). This is illustratedin the temperature dependence of the 2D carrier density n in Fig. 5.4 for aTl coverage of 0.6 %. Above 15 K, a rapid decrease in doping efficiency canbe observed, which we attribute to the clustering of the mobile Tl atoms,indicating that Tl is only very weakly bonded to the graphene. Comparingthe temperature dependence of the doping and linewidth with that of potas-sium on graphene is instructive. Both show a decrease in n back towards the745.1. Preparation of Tl adatoms on graphene1.5 % Potassiumb0 50 100012340.6 % Thalliuma0 50 10000.511.52n(e−/cm2×1013)d0 50 1000123·10−2Temperature (K)c0 50 100012345·10−2Temperature (K)MDClinewidth(A˚−1)Figure 5.4: Temperature dependence of thallium on graphene. Panels(a) and (b) compare the temperature dependence of the doping from thal-lium and potassium atoms between 8 K and 100 K. In both cases there is areduction in the carrier concentration as the temperature is increased. TheMDC linewidth measured at the Fermi level shows a clear difference for thetwo adatoms. The linewidth increases with temperature in the case of thal-lium (c), but decreases for potassium (d). In all figures, the black data pointrepresents the clean graphene, while the black arrow shows how the quantitychanges following the low temperature deposition of 0.6 % Tl or 1.5 % K.755.1. Preparation of Tl adatoms on grapheneinitial value of graphene/SiC, but the MDC linewidth of Tl/MLG increaseswith temperature, while that of K/MLG decreases. This is consistent with apicture where potassium is re-evaporated with increasing temperature, andthe graphene reverts to its pristine state. The opposite behaviour of Tl ongraphene indicates that it is not leaving the surface, but rather, forming clus-ters that make the surface more inhomogeneous and leading to the observedbroadening of the linewidth in Fig. 5.4(c).(a)1 0 110.50Energy(eV)(b)1 0 1Momentum, k-K (A˚1)(c)1 0 1(d)1 0 1Figure 5.5: Annealing Tl on graphene. (a) ARPES spectrum of cleanmonolayer graphene. (b) ARPES from a Tl-deposited and annealed sample,showing the Dirac point shifting towards the Fermi level, opposite to what isseen for low temperature deposition. Subsequent deposition and annealingcycles (c,d) continue to push the system towards charge-neutrality, where theDirac point is at the Fermi level, and the carrier density is zero.Annealing the Tl-decorated graphene to 700◦C for 30 minutes reduces thecarrier density to zero, bringing the Dirac point to the Fermi level, shownin Fig. 5.5. Fig. 5.6 shows the charge carrier density after annealing, as afunction of coverage. The reduction in doping observed with increasing cov-erage (blue data points) tends to approach the values of the clean graphene,shown as a dotted line in Fig. 5.6. The further reduction past this value forthe annealed samples implies the Tl is interfering with the charge transferfrom the SiC substrate, perhaps by intercalating between the graphene andsubstrate. This behaviour has been seen with gold atoms [105], and withoxygen [106] on a similar system and so is perhaps not surprising. It is an765.1. Preparation of Tl adatoms on graphene0 2 4 60123Tl coverage (%)n(e−/cm2×1013)AnnealedAs depositedFigure 5.6: Annealing Tl on graphene. The carrier concentration for lowtemperature deposited Tl is shown in blue. Depositing Tl at low tempera-ture followed by annealing at 700◦C leads to a large reduction in the carrierconcentration, plotted in gold.interesting prospect, however, due to the theoretical interest in studying howTl adatoms can modify the electronic structure of graphene.To summarize the characterization of Tl on graphene, we have shownthat isolated monomers exist only up to ∼ 1% concentration, and are onlystable as such below 15 K. There is evidence that Tl intercalates between thegraphene and substrate at high temperatures, bringing the Dirac point to theFermi level. The Tl adatoms are fully ionized, as predicted, and thereforeact as long-range charged impurity scatterers at the graphene surface. In thenext section, we examine the scattering more closely, and show that the Tlatoms also act as strong short-range scatterers775.2. Long-range vs. short-range scattering5.2 Long-range vs. short-range scatteringAdatoms on graphene tend to reduce the quasiparticle lifetime by introducinglong-range (Coulomb) scattering and/or short-range (δ-potential) scattering[107]. This depends on how the impurity modifies the electronic structurethrough its interaction with the graphene layer, i.e. donating charge to thegraphene layer and acting as a charged impurity, or being attached in someway to the graphene layer. This ranges from weakly-attached physisorption[108] to the formation of a covalent bond as in the example of hydrogen, whichsp3-hybridizes the carbon bond severely affecting the graphene electronicstructure [74]. While it is generally agreed upon that the charge carriermobility µ is inversely proportional to the density of charged impurities nimp,resulting in an increase of the scattering rate upon doping, other reports claiman increase of the quasiparticle lifetime upon doping with charged impurities[109]. Such behaviour critically depends on screening, which is determinedthrough the dielectric constant  of the system.Although Coulomb scattering is often assumed to be the dominant scat-tering mechanism in graphene, short-range scattering can become appreciableif dielectric screening is highly effective in reducing Coulomb scattering rela-tive to short range scattering, or if there are impurity states close to the Fermilevel giving rise to a particularly strong scattering potential. Therefore, inreal systems, it is not necessarily a priori clear which scattering mechanismis dominant. Here, we use the experimental spectral function measured byangle-resolved photoemission spectroscopy (ARPES) on thallium-doped epi-taxial graphene to observe the scattering behaviour induced by the thalliumadatoms, and disentangle Coulomb (long-range) from δ-like (short-range)scattering contributions.At a constant low temperature, linewidth broadening at the Fermi levelfrom interactions (i.e., electron-electron, electron-phonon, electron-plasmon)can be regarded as a small, constant contribution. In a Fermi liquid, linewidthbroadening at the Fermi level due to interactions vanishes as T 2 when thetemperature goes to zero. Consequently, the spectral linewidth at the Fermi785.2. Long-range vs. short-range scatteringlevel reflects predominantly defect and disorder contributions. Both longrange and short-range scattering mechanisms can be described by an ana-lytic expression for the self-energy in the dilute limit. Fitting the self-energyto the experimental data allows us to extract the fundamental parameters ofthe scattering mechanisms at play in the graphene layers.The line widths carrying the scattering information are taken from thefull-width at half maximum (FWHM) of momentum distribution curves (MDCs)at the Fermi level; these are shown for a selection of thallium coverages inFig. 5.7(a). The MDCs were fitted with a Voigt function – a convolution ofa Gaussian and Lorentzian – which represents the intrinsic Lorentzian line-shape of the spectral function convolved with a Gaussian function due to theexperimental resolution. The width of the Gaussian resolution function wasset to match the experimental momentum resolution 0.016 A˚−1, while theextracted Lorentzian FWHM (full-width at half-maximum) can be relatedto the scattering, or lifetime.The linewidth increases monotonically with increasing thallium coverageshown in Fig. 5.7(b) due to scattering from the randomly adsorbed thal-lium atoms. We analyze data up to x = 0.5 % in order to avoid effects ofclustering that we have shown to appear around 1 %. We assume that thelinewidth which we measure for the clean sample comes from initial disorder,defects and residual interactions, and does not change with Tl coverage. Inorder to unravel the contributions from the thallium-induced disorder to thelinewidth broadening and thus the corresponding contribution to graphene’selectrodynamics, we discuss two scattering mechanisms in more detail.The first mechanism is due to scattering from charged impurities, whichhas a long-range effect due to the screened Coulomb potential. Since thalliumdonates one electron per atom to the graphene sheet, it can be treated asa long-range Coulomb scatterer. The linewidth broadening, which can bedirectly modelled by the imaginary part of the self-energy, has the following795.2. Long-range vs. short-range scatteringsimplified expression:Σ′′long = α2nimpvFpiI(2α)/kF , (5.2)where α is the effective fine-structure constant, I(2α) is a function definedin Ref. [110], vF is the Fermi velocity, nimp is the impurity density (whichrelates to the thallium concentration x via nimp = 2x/A where A is the areaof the graphene unit cell), and kF =√pi(νnimp + ν0) is the Fermi momentumwith ν being the number of electrons donated per impurity atom, and ν0 thecharge density in graphene before thallium deposition.805.2. Long-range vs. short-range scatteringFigure 5.7: Analysis of the ARPES linewidth. (a) The MDC linewidth isfitted with a Voigt lineshape, which consists of a Lorentzian peak convolutedwith a Gaussian accounting for the experimental resolution. The linewidthas a function of Tl coverage is shown in blue in (b). Modelling using variousvalues of the parameters δ and  are also shown.815.2. Long-range vs. short-range scatteringThe only free parameter in Eq. 5.2 is the effective fine structure constant ofgraphene; all other parameters can be determined from the experiment. Theeffective fine structure constant depends on the dielectric constant  throughα = 2.2/. The dielectric constant in turn is determined by the underly-ing substrate; the smaller its value, the stronger the long-range scatteringcontribution to the linewidth broadening. From the analysis of plasmaronsignatures in epitaxial graphene on a buffer layer on SiC(0001) [111], thedielectric constant has been determined to be  = 22 ± 8. The long rangescattering calculated from Eq. 5.2 using this value of  is shown in Fig. 5.7(b)in red, and clearly does not fit the measured linewidth evolution with thal-lium coverage. Recent calculations, however, point towards an overestimateof the dielectric constant [112], suggesting that the actual value is likelysmaller. Indeed, by analyzing the band velocity renormalization in epitaxialgraphene, a reduced value  = 7.26 ± 0.02 was found [113]. The linewidthbroadening from long-range scattering for this value of the dielectric con-stant is shown as a yellow line in Fig. 5.7(b), and again, it only accounts forpart of the experimental linewidth broadening. One must go to values of far from those reported in the literature in order to adequately describe thedata in Fig. 5.7(b). Therefore, another mechanism is likely to contribute tothe experimental linewidth.A second prominent scattering mechanism that arises in the presence ofdisorder is short-range scattering from an effective δ-potential. This approachdoes not distinguish between inter- and intra-valley scattering in graphene,but it does capture the general strength of short-range scattering. Shortrange scattering is appreciable when adatoms have states close to the Fermilevel, thereby inducing a sizeable scattering potential [114, 115]. Its effecton the linewidth broadening can be modelled by the self-energy within theWolff-Clogston-model [116, 117]. We adopt a self-consistent implementationin the dilute limit within the coherent potential approximation, whereby thevarying potentials of a random distribution of adatoms is replaced by anordered lattice of effective potentials [118]:825.2. Long-range vs. short-range scatteringFigure 5.8: Calculations of the short range scattering. The real andimaginary parts of the lattice Green’s function are shown in (a) in blue andgreen respectively. (b) The imaginary part of the self-energy for the short-range scattering model described by Eq. 5.3 with parameters δ = −5 eV andx = 1%. The inset shows the dependence of ImΣshort on δ at an energyof 0.5 eV relative to the Dirac point, which corresponds to near EF in themeasurements.835.2. Long-range vs. short-range scatteringΣ′′short(ω) = x δ Im(1 +δG0(ω − Σ(ω))1− δG0(ω − Σ(ω)))(5.3)where G0(ω) is the lattice Green’s function of the unperturbed graphenelattice [119] and δ is the scattering potential. As the thallium concentrationx as well as the lattice Green’s function are known [the latter is here shownin Fig. 5.8(a)], the only free parameter in the short-range scattering model isthe scattering potential parameter δ.The overall behaviour of the imaginary part of the short-range scatter-ing self-energy for graphene is shown in Fig. 5.8(b). Its rather strong energydependence is due to the special energy dependence of the graphene latticeGreen’s function [Fig. 5.8(b)] in conjunction with that of the scattering po-tential itself. The dependence on the scattering potential δ at an energyω = 0.5 eV, thus in the vicinity of the experimentally relevant energies, isshown in the inset of Fig. 4(b). For δ → ±∞, which can be interpreted asa missing atom in the lattice, the imaginary part approaches the same finitevalue. For small values of δ, the imaginary part of the self-energy variesmore rapidly, although it remains bounded, and exhibits a maximum valuefor δ ≈ −5 eV.To evaluate the combined effect of both scattering processes, we calcu-late the linewidth from a theoretical spectral function generated by includingthe long and short-range scattering self-energies from Eqs. 5.2 and 5.3 in thesingle particle Green’s function of graphene. Using the most accurate ex-perimental determination of  from Ref. [113], δ remains as the only fittingparameter. The model agrees well with the experimental data for  = 7.26and δ = −3.2 ± 1eV shown as the green shading in Fig. 5.7(b). As it turnsout, the contributions from short-range and long-range scattering have com-parable magnitudes.When comparing the value here obtained for δ with available calcula-tions in the literature [27], we have to keep in mind that our short-rangescattering model does not distinguish between different angular momentumchannels, but rather attributes equal scattering amplitude to all scattering845.3. Conclusionchannels. Nevertheless, we can obtain a ballpark estimate for δ from theformula δ = t2ad/|∆E|, where tad is the hopping to the Tl adatom and ∆E isthe energy difference between the unperturbed (free) Tl adatom states andthe Fermi level [114, 115]. From the parameters reported in Ref. [27], wecan estimate a theoretical value δth ranging from -1 to -4 eV. Therefore ourresults suggest that short range scattering in this system lies on the largerside of the theoretically predicted range.Similar effects of enhanced scattering due to impurities have been ob-served in a number of transport experiments [107, 120, 121], in agreementwith our findings. There, the main scattering indicator is the carrier mobil-ity, which decreases linearly with increasing impurity density. By contrast,a decreasing photoemission linewidth upon potassium doping has been re-ported by Siegel et al. [73]. We attribute this variance to the different dopingregime, and correspondingly different intrinsic screening, that were probedin the experiments. Carrier scattering mechanisms in graphene (includingcharge impurities, resonant scatterers, and ripples) have been strongly de-bated [120, 122], and predominance of one or the other mechanism is largelysample dependent. For graphene subjected to charged impurities, e.g. alkaliadatoms, long-range scattering is typically expected to be much more effec-tive in charge carrier scattering than short range scattering [121, 123, 124].Thallium adatoms on graphene, having an impurity charge of +1, might beexpected to behave in a similar fashion. We showed, however, that this is asystem where both mechanisms have nearly equal contributions due to theproximity of thallium impurity states to the Fermi level. More generally, thisimplies that it cannot be a priori assumed what scattering mechanism willdominate in any given impurity-graphene system.5.3 ConclusionIn summary, we have quantitatively shown that Tl adatoms on a monolayer ofepitaxial graphene grown on a buffer layer on SiC(0001) have a large effect on855.3. Conclusionthe quasiparticle scattering rate. The Tl adatoms introduce disorder and acton the graphene electronic structure both as Coulomb long-range scatterersas well as short-range scatterers with a δ-like potential. By modelling theself-energy for both scattering mechanisms, we are able to extract a strongshort-range scattering potential δ = −3.2±1 eV. Thus, short-range scatteringcan contribute to a sizeable increase of the scattering rate, even in the case ofcharged impurities where long-range Coulomb scattering is usually expectedto dominate. These findings, and the ability to predict and/or account forconflating scattering mechanisms, will have important implications in thedevelopment of novel impurity-graphene-based electronics.Our study of this system was motivated by the prediction that Tl adatomscould tune the topology of the wavefunction in graphene. While our datais unable to confirm this, we verify that some of the necessary conditionsfor this are met in this system, including showing that isolated adatomsexist at low temperature and low coverage, and the complete ionization ofthe Tl 6p electron. The strong δ scattering is also an indication that thetheory work is on the right track, implying a significant hopping term betweenthe adatom and the graphene. This is an essential ingredient for the spin-orbit coupling of a heavy adatom to rub off on the graphene and create thepredicted topological insulator.86Chapter 6ConclusionsThis thesis brings together three projects, employing angle-resolved pho-toemission spectroscopy to quantitatively understand interactions in a wellcharacterised many-body system, and then to study similar interactions innew systems.Combining the ability of ARPES to measure the momentum-resolvedelectronic structure and advanced data analysis techniques based on theself-energy, we can access the momentum-resolved electron-phonon coupling,which is a crucial quantity for the description of superconductivity. Similarapproaches have been used on simple metals with great success, but its appli-cation to materials such as the high Tc cuprate superconductors is hinderedby the unknown, but complex nature of the bosonic excitations. We studyMgB2, a well characterized conventional multi-gap superconductor, and un-equivocally demonstrate the anisotropy of the electron-phonon coupling thatleads to the high Tc of 39 K.The electronic states formed from the in-plane boron orbitals couple verystrongly to phonon modes centred around 70 meV. One of these bands showsan electron-phonon coupling strength approximately twice as large as theexpected average value from calculations. The coupling measured on theother band is also larger than the expected average, and its magnitude variesaround the Fermi surface. These results are entirely consistent with cal-culations, and explain a number of recent measurements of the anisotropicsuperconducting gap.Next, we use sample preparation techniques in ultra-high vacuum, andat low temperature, to modify graphene in such a way that it demonstratesphonon mediated superconductivity. We do this by depositing lithium adatoms,87Chapter 6. Conclusionswhich form new electronic and phononic states that are available to partic-ipate in the electron-phonon coupling. Using the techniques established onMgB2, we demonstrate a large enhancement of the electron-phonon couplingup to λ ∼ 0.6.High-resolution, low-temperature ARPES measurements then provide di-rect evidence for the presence of a temperature-dependent pairing gap on partof the graphene-derived pi∗ Fermi surface. The detailed evolution of the den-sity of states at the gap edge, as well as the phenomenology analogous tothe one of known superconductors such as Nb, indicates that the pairing gapobserved at 3.5 K in graphene is associated with superconductivity. Based onthe BCS gap equation, ∆=3.5 kb Tc, this suggests that Li-decorated graphenemight be superconducting with Tc' 5.9 K, remarkably close to the value of8.1 K predicted from density-functional theory calculations.The third project also involves the modification of graphene with adatoms,and in that sense, is a natural progression from the second. Motivated by theprediction that thallium adatoms could transform graphene into a topologi-cal insulator, we performed a detailed characterization of this system. UsingARPES supported by STM measurements, we show that isolated adatomsexist at low temperature and low coverage, and that the Tl 6p electron iscompletely ionized. These are both considered necessary conditions for theappearance of the topological insulator state.From a detailed analysis and modelling of the ARPES linewidth at theFermi level, we quantitatively show that Tl has a large effect on the quasi-particle scattering rate. The adatoms introduce disorder and act on thegraphene electronic structure both as Coulomb long-range scatterers as wellas short-range scatterers with a δ-like potential. This strong δ scatteringis another indication that the theory work is on the right track, implying asignificant hopping term between the adatom and the graphene. This is anessential ingredient for the spin-orbit coupling of a heavy adatom to rub offon the graphene and create the predicted topological insulator.To reiterate from the introduction, this thesis demonstrates the power of88Chapter 6. Conclusionsthe ARPES technique for understanding the origins of superconductivity inboth 3D and 2D systems, and for characterizing new low-dimensional sys-tems. This is also a thesis about materials. It is about how novel materialscan be made and measured, all in ultra-high vacuum, and at temperaturesa few degrees above absolute zero. And while these conditions are not con-ducive to the imminent development of ground-breaking new technology,hopefully the work presented in this thesis will contribute in some way tobringing future materials and technologies to fruition.89Bibliography[1] Ludbrook, B. M. et al. Anisotropic electron-phonon coupling in themulti-gap superconductor MgB2. In preparation (2014).[2] Ludbrook, B. M. et al. Evidence for superconductivity in Li-decoratedgraphene. Submitted to Nat. Phys. (2014).[3] Strasser, C. et al. Long-range versus short-range scattering in dopedepitaxial graphene. Submitted to Phys. Rev. Lett. (2014).[4] McMillan, W. & Rowell, J. Lead phonon spectrum calculated fromsuperconducting density of states. Physical Review Letters 14, 108–112 (1965).[5] Nagamatsu, J., Nakagawa, N., Muranaka, T., Zenitani, Y. & Akimitsu,J. Superconductivity at 39 K in magnesium diboride. Nature 410, 63–4(2001).[6] Souma, S., Machida, Y., Sato, T. & Takahashi, T. 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Physical Review Letters 98, 076602 (2007).102Appendix AAdditional PublicationsOver the course of my PhD, I have contributed to numerous other projectsin ways ranging from technical assistance to experimental assistance. Thesecontributions are not described in this thesis, but the resulting publicationsare listed below.1. B.M. Ludbrook, G. Levy, M. Zonno, M. Schneider, D.J. Dvorak,C.N. Veenstra, S. Zhdanovich, D. Wong, P. Dosanjh, C. Strasser, S.Forti, C. Ast, U. Starke, A. Damascelli. Evidence for superconductivityin Li-decorated graphene. Nat. Phys. (under review)2. C. Strasser∗, B.M. Ludbrook∗, G. Levy, A.J. Macdonald, S.A. Burke,T.O. Wehling, K. Kern, A. Damascelli and C.R. Ast. Long-range versusshort-range scattering in doped epitaxial graphene. Phys. Rev. Lett.(under review)3. B.M. Ludbrook, S. dal Conte, G. Levy, C.N. Veenstra, N. Zhigadlo,C. Giannetti, A. Damascelli. Anisotropic electron-phonon coupling inthe multi-gap superconductor MgB2. In preparation4. S. Dal Conte, L. Vidmar, D. Golez, M. Mierzejewski, G. Soavi, S.Peli, F. Banfi, G. Ferrini, R. Comin, B.M. Ludbrook, N.D. Zhigadlo,H. Eisaki, M. Greven, S. Lupi, A. Damascelli, D. Brida, M. Capone,J. Bonca, G. Cerullo, and C. Giannetti. Snapshots of the retardedinteraction of charge carriers with ultrafast fluctuations in the cuprates.In preparation5. G. Levy, W. Nettke, B.M. Ludbrook, C.N. Veenstra and A. Dama-scelli. Deconstruction of Resolution Effects in Angle-Resolved Photoe-103Appendix A. Additional Publicationsmission. Phys. Rev. B 90, 045150 (2014).6. S. Chi, S. Johnston, G. Levy, S. Grothe, R. Szedlak, B.M. Ludbrook,Ruixing Liang,P. Dosanjh, S. A. Burke, A. Damascelli, D. A. Bonn, W.N. Hardy, and Y. Pennec. Sign inversion in the superconducting orderparameter of LiFeAs inferred from Bogoliubov quasiparticle interfer-ence. Phys. Rev. B 89, 104522 (2014).7. C.N.Veenstra, Z.-H.Zhu, M.Raichle, B.M. Ludbrook, A.Nicolaou,B.Slomski, G.Landolt, S.Kittaka, Y.Maeno, J.H.Dil, I.S.Elfimov, M.W.Haverkortand A. Damascelli. Observation of strong spin-orbital entanglement inSr2RuO4 by spin-resolved ARPES. Phys. Rev. Lett. 112, 127002(2014).8. J. A. Rosen*, R. Comin*, G. Levy, D. Fournier, Z.-H. Zhu, B. M. Lud-brook, A. Nicolaou, C. N. Veenstra, D. Wong, P. Dosanjh, Y. Yoshida,H. Eisaki, G.R. Blake, F. White, T. T. M. Palstra, R. Sutarto, F. He,A. Frano, Y. Lu, B. Keimer, G.A. Sawatzky, L. Petaccia, and A. Dam-ascelli. Surface-enhanced charge-density-wave instability in underdopedBi2201. Nature Communications 4, 1977 (2013).9. C. N. Veenstra, Z.-H. Zhu, B. M. Ludbrook, M. Capsoni, G. Levy,A. Nicolaou, J. A. Rosen, R. Comin, S. Kittaka, Y. Maeno, I. S. Elfi-mov, and A. Damascelli. Determining the Surface-To-Bulk Progressionin the Normal-State Electronic Structure of Sr2RuO4 by ARPES andDFT. Phys. Rev. Lett. 110, 097004 (2013).10. R. Comin, G. Levy, B. M. Ludbrook, Z.-H. Zhu, C. N. Veenstra,J. A. Rosen, Y. Singh, P. Gegenwart, D. Stricker, J.N. Hancock, D. van der Marel,I. S. Elfimov, A. Damascelli. Na2IrO3 as a Novel Relativistic Mott In-sulator with a 340-meV Gap. Phys. Rev. Lett. 109, 266406 (2012).104Appendix A. Additional Publications11. Z.-H. Zhu, G. Levy, B. M. Ludbrook, C. N. Veenstra, J. A. Rosen,R. Comin, D. Wong, P. Dosanjh, A. Ubaldini, P. Syers, N.P. Butch,J. Paglione, I. S. Elfimov, A. Damascelli. Rashba spin-splitting controlat the surface of the topological insulator Bi2Se3. Phys. Rev. Lett.107, 186405 (2011).105

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