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Charge localization phenomena in correlated oxides Comin, Riccardo 2013

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Charge localization phenomena incorrelated oxidesbyRiccardo CominB.Sc., Universita? degli Studi di Trieste, 2007M.Sc., Universita? degli Studi di Trieste, 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)December, 2013c? Riccardo Comin 2013AbstractCharge segregation is very common in correlated oxides, spanning from theextreme limit of the Mott insulator, characterized by strong charge localiza-tion and suppressed charge dynamics, towards more mildly correlated statesof matter, characterized by partial charge reorganization (charge-density-wave). In this thesis work I have investigated how the spatial organizationof valence charge evolves with electrostatics (carrier doping) and chemistry(going down in the periodic table). For this reason, the focus is on theexperimental study of the strongly-correlated 3d -oxides and the spin-orbit-coupled 5d -oxides.These investigations have been performed with a bundle of state-of-the-artspectroscopic techniques in the field of quantum materials, namely: angle-resolved photoemission (ARPES), low-energy electron diffraction (LEED),resonant elastic X-ray scattering (REXS), X-ray diffraction (XRD), scan-ning tunnelling microscopy (STM) and optical spectroscopy. To support ex-perimental data, we have used theoretical tools such as conventional densityfunctional theory (DFT) for the 5d -oxides and developed ad-hoc approachesfor the more complex 3d -materials.The all-around study of underdoped high-Tc Bi-based cuprates allowed usto shed new light on the universality and origin of charge-ordering instabili-ties in these materials and understand their interplay with superconductivityand pseudogap. These phenomena have been investigated in detail in under-doped samples of Bi2201, using a various experimental techniques (ARPES,REXS, XRD, LEED, STM) and various tailored theoretical approaches. Theresults of these studies are presented in Chapters 2 and 3.In the 5d -based iridates (in particular, Na2IrO3) we have revealed and char-acterized a novel form of Mott-Hubbard physics. This has been possibleiiAbstractthanks to the combination of ARPES and optics for probing the electronicstructure near the Fermi energy, and DFT for providing the theoreticalframework to understand the electronic ground state in these materials. Ul-timately, this approach helped demonstrate the crucial role of spin-orbitinteraction in driving a novel Mott phase in materials where the Mott cri-terion might be violated (Chapter 4).Altogether, the resulting phenomena discovered in copper- and iridium-based oxides have revealed novel unconventional aspects of the physics ofcorrelated materials, thus paving the way for future explorations of the com-plex but fascinating jungle of transition metal oxides.iiiPrefaceThis thesis work is the outcome of the various scientific projects with whichI have been involved. The theme of my research is the study of correlationeffects and many-body physics in transition metal oxides, and more specif-ically the investigation of charge-localized electronic states in this class ofmaterials. Out of the multiple studies I have performed as main investiga-tor, I selected only those which culminated in scientific publications, withmyself being first author. These are not presented in chronological order,but follow a rationale based on the idea of tracking the evolution of thisphenomenology from 3d - down to 5d -based oxides. Various other relatedongoing projects, belonging to the same field of research, and which are stillin progress or under completion, will not be incorporated nor discussed inthis thesis. Furthermore, besides the scientific aspect of my research, I havebeen engaged in various projects on the technical side. This componentof my activity as a doctoral student, albeit substantial, is not documentedhere, either.In the following sections, I will expand on my own contributions to eachspecific scientific project. In all cases, despite being involved as the primaryinvestigator, I have relied on the support of many coworkers toward success-ful completion of each investigation. For this reason, I will also acknowledgeall coauthors? role and contribution in detail.Chapter 2 ? Surface-bulk dichotomy and soft electronicphases in Bi2201This Chapter reports on the study of temperature- and doping-dependenceof structural supermodulations in Bi2201. This investigation has involvedivPrefacethe following experimental techniques: ARPES, LEED, REXS, XRD. Theresults and the related analysis and interpretation have been published in:Nature Communications 4 1977 (2013) (Ref. 1 in the List of Publications,Appendix). The manuscript was co-written by myself and J. Rosen, withinput and contributions from all coauthors, and finalized by A. Damascelli.This investigation was envisaged and designed by myself, J. Rosen, andA. Damascelli. Various facilities and groups have been involved: our ownARPES laboratory at UBC, the Italian Light Source (ELETTRA, Trieste)the Canadian Light Source (CLS, Saskatoon), the Berliner Synchrotron Fa-cility (BESSY, Berlin), the Max-Planck Institute for Solid-State Research(MPI-FKF, Stuttgart), the National Institute of Advanced Industrial Sci-ence and Technology (AIST, Japan), the Materials Science Centre (Univer-sity of Groningen, Netherlands), and Agilent Technologies UK.J. Rosen and I performed temperature-dependent ARPES at ELETTRA,with assistance from D. Fournier and L. Petaccia (L.P. is also responsiblefor beamline maintenance and local user support). J. Rosen carried outadditional ARPES and LEED experiments at UBC, with assistance fromG. Levy, on the in-house spectrometer maintained by myself, B. Ludbrook,Z.-H. Zhu, C. Veenstra, D. Wong, and P. Dosanjh. I performed REXSmeasurements at CLS with the assistance of A. Nicolaou, R. Sutarto andF. He (R.S and F.H. are in charge of beamline maintenance and local usersupport), and at BESSY with A. Frano and Y. Lu. G. Blake, F. White, andT. Palstra are responsible for the XRD measurements. Y. Yoshida and H.Eisaki grew and characterized the Bi2201 samples.I and J. Rosen were ultimately responsible for data analysis and interpreta-tion, together with G. Levy, B. Keimer, G.A. Sawatzky, and A. Damascelli(J. Rosen is in particular responsible for the Ginzburg-Landau model).Chapter 3 ? Charge order driven by Fermi-arc instability inBi2201This Chapter reports on the study of temperature- and doping-dependentcharge-ordering phenomena in Bi2201. This investigation has involved thevPrefacefollowing experimental techniques: REXS, STM, ARPES. The results andthe related analysis and interpretation have been recently accepted for pub-lication in Science (Ref. 8 in the List of Publications, Appendix). Themanuscript was written by myself, with input and contributions from allcoauthors, and finalized by A. Damascelli.This investigation was conceived and planned by myself and A. Damas-celli. Various facilities and groups have been involved: our own ARPESlaboratory at UBC, the Berliner Synchrotron Facility (BESSY), the Cana-dian Light Source (CLS), the STM laboratory at Harvard University, theNational Institute of Advanced Industrial Science and Technology (AIST,Japan), and the Max-Planck Institute for Solid-State Research (MPI-FKF,Stuttgart).I performed all of the REXS measurements: (i) at BESSY with the assistanceof A. Frano, E. Schierle and E. Weschke (E.S. and E.W. are responsible forbeamline maintenance and local user support); and (ii) at CLS with R.Sutarto and F. He. M. Yee, A. Soumyanarayanan and Y. He carried out theSTM measurements at Harvard University, in the group of J. Hoffman. Idesigned the phenomenological model and performed related susceptibilitycalculations. Y. Yoshida and H. Eisaki grew and characterized the samples.I was ultimately responsible for data analysis and interpretation, togetherwith A. Frano, M. Yee, J. Hoffman, M. Le Tacon, B. Keimer, I.S. Elfimov,G.A. Sawatzky, and A. Damascelli.Chapter 4 ? Na2IrO3 as a Novel Relativistic Mott InsulatorThis Chapter reports on the study of the low-energy electronic structure ofNa2IrO3. This investigation has involved the following experimental tech-niques: ARPES and optical spectroscopy. The results and the related anal-ysis and interpretation have been published in: Physical Review Letters109 266406 (2012) (Ref. 3 in the List of Publications, Appendix). Themanuscript was written by myself, with input and contributions from allcoauthors, and finalized by A. Damascelli.This investigation was conceived and planned by myself and A. Damas-viPrefacecelli. Various facilities and groups have been involved: our own ARPESlaboratory at UBC, the Optics group at the Universite? de Gene`ve, and thesample growth and preparation group and laboratory at the Georg-August-Universita?t Go?ttingen.I performed the ARPES measurements at UBC, on the in-house spectrom-eter maintained by myself, B. Ludbrook, Z.-H. Zhu, C. Veenstra, and J.Rosen. J. Hancock and D. Stricker carried out the optical absorption andellipsometry measurements at the Universite? de Gene`ve, in the group ofD. van der Marel. I performed the DFT calculations using the WIEN2Kcode, with the support of I. Elfimov (who manages and maintains the CPUclusters at UBC).I was ultimately responsible for data analysis and interpretation, togetherwith I. Elfimov, J. Hancock, D. van der Marel, and A. Damascelli.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Correlated oxides: a past, present, and future challenge . . . 11.2 Charge localization: a hallmark of correlations at work . . . 71.3 Experimental techniques . . . . . . . . . . . . . . . . . . . . 111.3.1 ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.2 REXS and XRD . . . . . . . . . . . . . . . . . . . . . 191.4 Physics of correlations . . . . . . . . . . . . . . . . . . . . . . 241.4.1 Origin of correlations in photoelectron spectroscopy . 251.4.2 Electron-phonon correlations in solids: the polaron . 291.4.3 Doping-controlled coherence: the cuprates . . . . . . 311.4.4 Temperature-controlled coherence: the manganites . . 341.4.5 Probing coherence with polarization: the cobaltates . 35viiiTable of Contents1.4.6 Correlated metals: the spin-orbit coupled 4d -TMOs . 381.4.7 Mott criterion and spin-orbit coupling: 5d TMOs . . 392 Surface-bulk dichotomy in Bi2201 . . . . . . . . . . . . . . . 412.1 Orthorhombic and modulated structure of the Bi-cuprates . 442.2 Probing the surface with ARPES and LEED . . . . . . . . . 452.3 Bulk-sensitivity with XRD and REXS . . . . . . . . . . . . . 472.4 Theoretical modeling and discussion . . . . . . . . . . . . . . 492.5 Chapter 2 ? Appendix . . . . . . . . . . . . . . . . . . . . . . 532.5.1 Materials and methods . . . . . . . . . . . . . . . . . 532.5.2 Experimental doping and temperature dependence . . 552.5.3 Fermi surface nesting . . . . . . . . . . . . . . . . . . 583 Charge order driven by Fermi-arc instability in Bi2201 . . 603.1 REXS results . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2 STM results . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.3 Fermiology of the pseudogap state . . . . . . . . . . . . . . . 663.4 Chapter 3 ? Appendix . . . . . . . . . . . . . . . . . . . . . . 703.4.1 REXS addendum . . . . . . . . . . . . . . . . . . . . 703.4.2 STM addendum . . . . . . . . . . . . . . . . . . . . . 773.4.3 Model Green?s function and particle hole-propagator 774 Na2IrO3 as a Novel Relativistic Mott Insulator . . . . . . . 904.1 ARPES results . . . . . . . . . . . . . . . . . . . . . . . . . . 924.2 Optical conductivity . . . . . . . . . . . . . . . . . . . . . . . 964.3 Density functional theory . . . . . . . . . . . . . . . . . . . . 984.4 Chapter 4 ? Appendix . . . . . . . . . . . . . . . . . . . . . . 1004.4.1 Materials and Methods . . . . . . . . . . . . . . . . . 1004.4.2 DFT methodology . . . . . . . . . . . . . . . . . . . . 1004.4.3 Temperature dependence and charging in PES . . . . 1014.4.4 Extraction of ?? from PES with K-doping . . . . . . 1045 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107ixTable of ContentsBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112AppendixList of publications . . . . . . . . . . . . . . . . . . . . . . . . . 134xList of Tables3.1 Summary of CDW parameters from REXS and STM . . . . . 66xiList of Figures1.1 Building blocks of correlated materials. . . . . . . . . . . . . . 21.2 Correlated states of matter in transition metal oxides. . . . . 51.3 Different types of electronic ground states. . . . . . . . . . . . 91.4 Phase diagram of YBCO. . . . . . . . . . . . . . . . . . . . . 101.5 Energetics of the photoemission process. . . . . . . . . . . . . 141.6 Geometry of an ARPES experiment. . . . . . . . . . . . . . . 151.7 Resonant vs. non-resonant scattering. . . . . . . . . . . . . . 211.8 Geometry of a REXS/XRD experiment. . . . . . . . . . . . . 231.9 Franck-Condon effect and correlations in spectroscopy. . . . . 251.10 Polaron physics in 1D. . . . . . . . . . . . . . . . . . . . . . . 301.11 Doping-dependent correlation effects in cuprates. . . . . . . . 331.12 Single-particle coherence in the manganites. . . . . . . . . . . 341.13 Correlations in multi-band systems: the example of cobaltates. 361.14 Weak correlations and spin-orbit physics in 4d -oxides. . . . . 371.15 Spin-orbit-driven correlated states of matter. . . . . . . . . . 402.1 Temperature-dependent electronic structure of Bi2201. . . . . 452.2 Temperature-dependent superstructure modulations in Bi2201. 472.3 X-ray measurements of the supermodulations in Bi2201. . . . 492.4 Mean-field theory and electronic susceptibility. . . . . . . . . 522.5 LEED doping and temperature dependence of Bi2201. . . . . 562.6 X-ray results on UD15K Bi2201. . . . . . . . . . . . . . . . . 572.7 Tight-binding fit of the experimental FS. . . . . . . . . . . . . 593.1 Momentum- and temperature-dependent REXS in Bi2201. . . 633.2 Signatures of charge ordering from STM maps. . . . . . . . . 65xiiList of Figures3.3 Modeling the ARPES fermiology in Bi2201. . . . . . . . . . . 673.4 Doping dependence of the charge order wavevector . . . . . . 683.5 Laue pattern in Bi2201 and axis definition. . . . . . . . . . . 703.6 Experimental geometry in REXS measurements. . . . . . . . 723.7 Fourfold symmetry of CDW in REXS. . . . . . . . . . . . . . 733.8 Signatures of charge modulations in Cu-O plane. . . . . . . . 743.9 Schematics of the REXS fitting procedure. . . . . . . . . . . . 753.10 Low- and high-temperature REXS scans. . . . . . . . . . . . . 753.11 Temperature dependence of REXS wavevector and FWHM. . 763.12 Bias voltage-dependent STM conductance map. . . . . . . . . 783.13 Model self-energy and Green?s function. . . . . . . . . . . . . 803.14 Fermi-arcs: experiment vs. theory, and modified DOS . . . . 823.15 Zero-frequency, zero-temperature electronic susceptibility . . 853.16 Doping-dependent spectral function and electronic response. . 863.17 Electronic response and related control parameters. . . . . . . 874.1 ARPES results on fresh-cleaved surfaces of Na2IrO3. . . . . . 934.2 ARPES results on K-covered surfaces of Na2IrO3. . . . . . . . 954.3 Optical absorption and experimental gap in Na2IrO3. . . . . . 974.4 DFT bandstructure and the role of correlations. . . . . . . . . 994.5 Charging effects: temperature-dependent UPS. . . . . . . . . 1034.6 Charging effects: flux-dependent UPS. . . . . . . . . . . . . . 1034.7 K-exposed UPS on Sample 2, and determination of ?? . . . . 1044.8 K-exposed UPS on Sample 1 and comparative profiles for ??. 105xiiiList of Abbreviations2D two-dimensional3D three-dimensionalAFM Anti-Ferro-MagneticAN AntiNodalARPES Angle-Resolved PhotoEmission Spectroscopya.u. arbitrary unitsBi2201 Bi2Sr2?xLaxCuO6+?BZ Brillouin ZoneCB Conduction BandCDW Charge Density WaveDFT Density Functional TheoryDOS Density Of StatesEDC Energy Distribution CurveEM ElectroMagneticEu-LSCO La2?x?ySrxEuyCuO4FL Fermi LiquidFS Fermi SurfacexivList of AbbreviationsFT Fourier TransformFWHM Full Width at Half Maximumh.c. hermitian conjugateHS Hot SpotHWHM Half Width at Half MaximumIPES Inverse PhotoEmission SpectroscopyLBCO La2?xBaxCuO4LDA Local-Density ApproximationLEED Low-Energy Electron DiffractionLSCO La2?x?ySrxCuO4MDC Momentum Distribution CurveNd-LSCO La2?x?ySrxNdyCuO4OD OverdopedOP Optimally-dopedPES PhotoEmission SpectroscopyPG PseudogapREXS Resonant Elastic X-ray ScatteringSC SuperconductivitySOC Spin-Orbit CouplingSPECS SPECS Surface Nano Analysis GmbHSTM Scanning Tunneling MicroscopyTB Tight BindingxvList of AbbreviationsTMO Transition Metal OxideUBC University of British ColumbiaUD UnderdopedUPS Ultraviolet Photoemission SpectroscopyVB Valence BandXPS X-ray PhotoEmission SpectroscopyXRD X-ray DiffractionYBCO YBa2Cu3O6+yxviList of Symbolsaq photon (of wavevector q) annihilation operatora?q photon (of wavevector q) creation operatorck electron (of momentum k) annihilation operatorc?k electron (of momentum k) creation operator?2 Chi-square likelihood estimate functionc speed of light? the origin in momentum space of the bulk Brillouin zone.e electronic chargeEF Fermi energy (equivalent to the chemical potential)I(k, E) photoemission intensity (energy and momentum dependent)~ Planck?s constantk generalized fermionic (electron) momentumk|| electron momentum parallel to the sample surfacek? electron momentum perpendicular to the sample surfacekx electron momentum projected along x?ky electron momentum projected along y?kz electron momentum projected along z?xviiList of SymbolsM(k) matrix element (momentum dependent)me electronic mass? chemical potential (equivalent to the Fermi Energy)? photon frequency? fermionic frequencyi?m imaginary fermionic frequency? bosonic frequencyi?n imaginary bosonic frequencyQ generalized bosonic (electron) momentum? electron spinT?? Time-ordering operator?SOC strength of atomic spin-orbit couplingxviiiAcknowledgmentsDuring these 4 (and counting) years of my PhD journey I have had the op-portunity of working on a series of very intriguing and challenging projects,some of which turned out to be also very rewarding, and eventually endedup constituting the bulk of my thesis.The person who deserves the first and most grateful acknowledgment formaking this possible is of course my supervisor, Andrea Damascelli. I trulyconsider his leadership, guidance, and vision to be the primary factors thatmade this learning experience an invaluable one in my academic career sofar. I came to UBC with little or no knowledge of UHV technology, lowtemperature physics, electronic spectroscopies, scientific writing, etc., and Ifound an ideal environment to learn all of these aspects and techniques, andmany more. I would have never imagined to amass so much experience insuch a short period of time.Looking back at when I decided to travel nearly 9000 kilometers from myhome town, I can now say it really has been the best possible choice I couldhave done at the time. Also, I owe a big thanks to Fulvio, for coming upwith the original idea of joining Andrea at UBC.I would like to dedicate my second acknowledgment to George, who I ef-fectively regard as my second supervisor over these last years. From himI have learned many of the trade secrets of X-ray spectroscopies ? whichturned out to be crucial for all my last publications ? and even though Ihave only been able to absorb a tiny percentage of his scientific wisdom,this was enough to stimulate my curiosity into many various directions inthe field of correlated oxides. I will miss those hour-long chats following my?I-only-have-a-very-quick-question-to-ask? incursions in his office.Of course, being part of a great research group at UBC has also been axixAcknowledgmentskey component of this quite unique career and life experience, and I wouldlike to acknowledge all the people I worked with, in the ARPES group:Giorgio, David, Ilya, Jason, Bart, Jeremy, Christian, Ryan, Jonathan, Elia,Simone, Alessandro, Ludivine, Kou.Besides the ARPES people, I would like to thank Art and David for involvingme in the joint effort to create a unique XUV-photoelectron setup in theworld. Even though this ambitious project has taken longer than we initiallythought, I am really happy to see us being so close to finally get the veryfirst data of this kind from our machine! I wish I had more time left now toplay with it now that it is becoming operational.Among the UBC people, Doug and Pinder deserve a special mention, fortheir patience in showing and teaching me the many technical aspects anddetails that often go unnoticed but that are really fundamental for most ofthe research we do. The various projects that we worked on together in thelab (Hydrogen lines, cryostat, etc.) unfortunately are not showcased in thisthesis, but I regard them as important and formative as the scientific ones.Finally, I also have many collaborators to acknowledge here. I would liketo start from the Max-Planck Institut people: Bernhard, Alex, Mathieu, Yi? the work that we have done together has produced in my opinion somequite spectacular results, both in terms of science and publications, waybeyond my most optimistic expectations.Following in this long list of coworkers I would like to mention: Jason,Damien and Dirk (University of Geneva) for joining efforts on the under-standing of Na2IrO3; Mike, Yang, Anjan and Jenny from Harvard for theexcellent STM work on Bi2201 and for playing a key role in the making ofour Science paper; Graeme and Thomas (University of Groningen) for pro-viding the very high-quality XRD data on Bi2201, which were instrumentalto understand such an incredibly complex material; Philip, Felix (St. An-drews) and Kyle for leading the great early work on Bi2201, which I stillconsider one of the brightest and most insightful results shining light on theall-but-easy phenomenology of these systems.A special acknowledgment goes to all the beamline scientists who helped meduring the (many) beamtime shifts, that ended up representing a prominentxxAcknowledgmentscomponent of my research experience: Ronny and Feizhou, for all the in-valuable help with measurements, discussions, planning and sharing of ideasthroughout these last 3 years and a dozen of beamtime experiments; Luca,for the continuing help and interaction on our many ARPES measurementson cuprates; Enrico and Eugen, for turning a single week of night shifts intoone of the most exciting and successful measurements of my PhD.A big ?thank you? also goes to all our samples suppliers ? Hiroshi, Yoshiyuki,Philipp and Yogesh ? without your beautiful single crystals to measure, myideas and efforts would have been aimless.Last but not least, I would like to acknowledge all the Canadian LightSource staff that I interacted with over these years. I have visited Saskatoonso many times that it almost became my second home, and I have to say thatas a user I was always granted the best working conditions to produce somequite spectacular scientific results ? many of which will extend even beyondthis thesis work. I could probably say that X-ray experiments have been asimportant as the ARPES ones at UBC for my formation and experience inthe study of quantum materials throughout these years.xxiAcknowledgmentsAd Alice,per essermi stata sempre vicina nel pensiero e nel cuore,e avermi dato il coraggio e la forza di affrontare questo lungo viaggioinsieme.xxiiChapter 1Introduction1.1 Correlated oxides: a past, present, andfuture challengeThe correlated materials discussed in this thesis work belong to the ample(and growing) class of transition metals oxides (TMOs). Despite giving riseto a rich variety of distinctive unconventional phenomena, these systems allshare the same basic structural elements: TM-O6 (near) octahedral units,where a central transition metal (TM) cation TMn+ is coordinated to 6neighboring O2? anions, sitting at positions ~?={(?ax,?ay,?az)}, ai beingthe TM-O nearest neighbor bond length (along the axis i?). Most TMOspossess either cubic (ax=ay=az) or tetragonal (ax=ay 6=az) TM-O6 units,although deviations from these two local symmetries are also present insame compounds. The remaining elements in the structure primarily servethe purpose of completing the stoichiometry and also, in most cases, to helpcontrolling certain material parameters (e.g. doping, bandwidth, structure,magnetism). The delicate interplay between the localized physics takingplace within these building blocks, and the delocalized behavior emergingwhen such local units are embedded in a crystalline matrix, is what makesthese systems so complex and fascinating.The emerging physics in this class of materials, as one goes from therow of 3d to that of 5d transition metals, is schematically introduced inFig. 1.1, where the relevant elements are highlighted (see caption). Thephenomenology of correlated oxides can be understood in terms of the com-petition between charge fluctuation (favored by the O-2p electrons) andcharge localization (driven by the TM-d electrons). The peculiarity of 3d11.1. Correlated oxides: a past, present, and future challengeControlparametersBandwidth  (U/W)Band fillingDimensionalityDegrees offreedomd - fopenshellsmaterialsU<<WCharge fluctuationsU>>WSpin fluctuations0.3 0.2 0.10100200300 SCAFTemperature  (K)Dopant Concentration x0.0 0.1 0.2 0.3SCAFPseudogap'Normal'MetalI II IIIb IVb Vb VIb VIIb VIIIb Ib IIb III IV V VI VII 0Lanthanides*Actinides*****Ru TcH HeLi Be B C N O F NeNa Mg Al Si P S Cl ArRb Sr Y Zr Nb Mo Rh Pd Ag Cd In Sn Sb Te I XeCs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At RnFr Ra Rf Db Sg Bh Hs MtTh Pa UK Ca Sc Ti V Cr Fe Co Cu Zn Ga Ge As Se Br KrCe Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb LuMnAcStrongly Correlated Electron SystemsNiNp Pu Am Cm Bk Cf Es Fm Md No LrNd2-xCexCuO4 La2-xSr xCuO4Charge / SpinOrbitalLattice? Kondo? Mott-Hubbard? Heavy Fermions? Unconventional SC? Spin-charge order ? Colossal MR0.111010010000 0.5 1 1.5 2(K)xTCa SrP-ICAF-IP-MM-M SCTPTM/NM   Ca2-xSrxRuO4Figure 1.1: Building blocks of correlated materials and related control pa-rameters. U refers to the on-site Coulomb repulsion, whereas W indicatesthe bandwidth of the valence band, proportional to the electron hopping am-plitude in the lattice. The yellow and blue circles pictorially represent thetwo extreme cases for an atomic wavefunction having UW and UW ,respectively. Blue boxes in the periodic table indicate those elements exhibit-ing strongly localized physics (U W ), due to the presence of localized 3d(transition metals) and 4f (rare earths) orbitals. The phase diagrams forthe special cases of Cu-based (from Ref. 1) and Ru-based (from Ref. 2) ox-ides, which exhibit unconventional superconductivity, are expanded in thebottom right and left panels, respectively.and 4f shells is that the radial part of the wave functions has an extensionwhich is small compared to typical interatomic distances, as opposed to theoxygen 2p orbitals which are more extended. This follows from the fact thatthe average squared radius ?r2?nl of the atomic wavefuctions decreases withincreasing angular quantum number l. As a results, the more localized 3dand 4f electrons are not well described within the independent particle pic-21.1. Correlated oxides: a past, present, and future challengeture, where electrons are assumed to interact with the average (electronic)charge density, which is hardly affected by the motion of a single electron.In reality, for the tightly confined 3d and 4f electrons, the addition of anextra electron in the same shell entails a large energy cost given by the extraCoulomb energy U (also commonly named Hubbard term, or parameter).This is at the heart of what is referred to as strongly-correlated electronbehavior, and it underlies most of the spectacular phenomena observed inthese materials. As a result, all the relevant degrees of freedom ? charge,spin, orbital and lattice ? are deeply entangled, and their mutual interplayis what governs the low-energy physics.Over time experimentalists have learned how to tune this delicate inter-play by means of selected control parameters ? bandwidth, band filling, anddimensionality. All of these parameters are primarily tuned chemically (e.g.via the choice of the specific TM ion, or by carrier doping) but they can alsobe controlled experimentally (e.g. by means of pressure, EM fields, or in-situ doping). The resulting novel phenomena and materials include Kondophysics [3] and heavy fermion systems [4] (found especially in ? but notlimited to ? 4f -based materials), Mott-Hubbard/charge-transfer insulators[5] (e.g. CuO, NiO, CoO, MnO), unconventional superconductivity [6?8](cuprates, such as e.g. La2CuO4, and also Sr2RuO4), spin-charge orderingphenomena [9, 10] (e.g. La2CoO4, La2NiO4, La1?xCaxMnO3), and colos-sal magnetoresistance [11] (La2?xSrxMn2O7). Remarkably, some of thesephenomena can be found in the very same phase diagram, as is the case ofCu- and Ru-oxides1 (see bottom-right and bottom-left phase diagrams inFig. 1.1, respectively).To better elucidate the origin and nature of correlated behavior, I willfirst introduce one of the most fascinating manifestations of the novel, cor-related physics arising from the spatial extent of p- and d -orbitals: the Mottinsulator. In order to understand the origin of this concept, it is useful to1Note the different role of the variable x in these phase diagrams: for the Cu-oxidesx amounts to the effective number of holes that are injected in the CuO2 planes, due tothe imbalance in the valence of La (3+) and Sr (2+); for the Ru-oxides the substitutionis isovalent (Sr2+ for Ca2+) and is therefore not accompanied by carrier doping.31.1. Correlated oxides: a past, present, and future challengestart from classical band theory. One of the fundamental paradigms of bandtheory affirms that the nature of the electronic ground state in a single-bandmaterial is entirely determined by the band filling, which is directly relatedto the number of electrons NUC in the unit cell: if NUC is odd (even), thesystem must be metallic (insulating). For this reason, the discovery of in-sulating TMOs having odd NUC came as a surprise [12]. The breakdownof single-particle physics, and consequently of band theory (where electronsare assumed to interact only with the lattice ionic potential and the averageelectronic density), was originally proposed by Mott and Peierls in 1937 [13]:[Prof. Peierls] suggested that a rather drastic modification ofthe present electron theory of metals would be necessary in orderto take these facts [the low conductivity in NiO] into account.The solution of the problem would probably be as follows: if thetransparency of the potential barriers is low, it is quite possiblethat the electrostatic interaction between the electrons preventsthem from moving at all. At low temperatures the majority ofthe electrons are in their proper places in the ions. The minoritywhich have happened to cross the potential barrier find thereforeall the other atoms occupied, and in order to get through thelattice have to spend a long time in ions already occupied by otherelectrons. This needs a considerable addition of energy and so isextremely improbable at low temperatures.This new category of correlated insulators can be therefore seen as the man-ifestation of the dominant role of on-site interactions in 3d oxides: at half-filling (1 electron per site), the large on-site Coulomb repulsion U betweenthe strongly localized 3d electrons makes hopping processes unfavorable,thus leading to charge localization and subsequent insulating behavior, witha gap in the electronic spectrum opening up at the chemical potential. Whenthe lowest occupied and the first unoccupied bands both have mainly TM-dorbital character, as in Fig. 1.2(a), one uses the term Mott insulator. Thepresence of a Mott gap, with its characteristic scale of the order of U, istherefore a hallmark of correlated behavior in these systems. The lowest41.1. Correlated oxides: a past, present, and future challengeN+1N?1Jeff=1/2UO-2pMott-Hubbard Charge-Transfer Relativistic MottLHBTM-dLHBTM-dUHBTM-d UHBTM-dO-2pd-orbitals10 statesdxyegt2gCrystal fieldsplittingEnergyEFStrongly-correlated electronic states of matterSingle TM ionTM-O6unitsEnergyTM-t2gLeff=1LHBspin-orbitU?Jeff=3/2O-2pdxz dyzdz2 dx2-y2(a) (b) (c)?: charge transfer energy U: on-site Coulomb repulsionUHBU ?Figure 1.2: Different types of correlated quantum states of matter discoveredin transition metal oxides. The energy axis refers to spectrum of many-bodyexcitations, with the Fermi energy EF separating the electron addition part(N + 1, above EF ) from the electron removal part (N ? 1, below EF ). (a)when d -states are close to EF (i.e., they are the lowest-ionization states),the partially-filled d -band is split into a lower (LHB) and an upper Hubbardband (UHB) by the action of U. Whenever U >W a gap opens up at EF .(b) same as (a), but now the last occupied band has O-2p character, since Uis larger than the Cu-O charge-transfer energy ?; the corresponding gap isa charge-transfer gap. (c) the action of spin-orbit (SOC) interaction splitsthe t2g manifold into Jeff =3/2 and 1/2 submanifolds. The latter is higherin energy and lies close to EF . Again, the action of U can open a Mott-likegap, but this hinges on the previous SOC-induced splitting of the t2g band.electron removal and addition states (bands) are respectively termed lowerHubbard band (LHB) and upper Hubbard band (UHB). This is sketched inFig. 1.2(a), where a gap at the chemical potential is separating the LHB andUHB (both having mainly 3d -character).The stability of a Mott-Hubbard insulating ground state against a delocal-ized metallic behavior lies in the fulfillment of the Mott criterion, which es-tablishes the condition for the localizing energy scale (the Coulomb interac-tion U ) to overcome the delocalizing ones. The latter are usually condensed51.1. Correlated oxides: a past, present, and future challengein a single energy scale, W, or the bandwidth, which represents the energyspread of the valence band in the non-interacting limit. The bandwidthW is fully determined by: (i) the chemistry of the material, which controlswhich electronic orbitals are involved in the intersite hopping, besides theomnipresent O-2p states; (ii) the structure, which shapes the hopping pathsand bond orientations (systems with similar chemistry but slightly differentstructures may realize very different electronic states, like is the case forCa2RuO4 and Sr2RuO4 [2]). The Mott criterion, which is embodied by theinequality U >W , is based on the prerequisite that the correlated d -statesare the lowest-lying ones, i.e. those closest to EF . While this is in most casestrue, it fails to hold for the late 3d transition metals [14, 15], where 3d<2pinstead,  being the orbital on-site energy (or the band center-of-mass ina delocalized picture). In such cases one talks of charge-transfer insulators[14], the denomination following from the fact that the lowest-energy ex-citation involves the transfer of one electron from the last occupied band,of O-2p character, onto the first unoccupied band, of TM-3d character.The cost for such an excitation is the charge-transfer energy ?=U ? ?pd,given by the difference between the Hubbard parameter U and the 3d -2porbital energy separation (?pd). This is depicted in Fig. 1.2(b), where nowthe charge-transfer gap separates the 3d -derived UHB and the O 2p-derivedvalence band. A comprehensive classification can be found in Ref. 14.The bandwidth W and Coulomb repulsion U are not the only relevantenergy scales in the field of correlated materials. More recently, a newclass of materials has appeared on the stage, that are based on the late5d transition metals (osmium, iridium), and whose electronic states haveto be treated within a relativistic framework, due to the heavier nuclearmass. This results in a new energy scale making its way into the problem:spin-orbit coupling (SOC), whose strength will be indicated by ?SOC . Thisnew element in the Hamiltonian, despite being a single-particle term (com-ing from the expansion of the single-fermion Dirac Hamiltonian), stronglyaffects the balance governing the interplay between W and U, making thepreviously introduced Mott criterion not sufficient. This results in the emer-gence of a new class of correlated quantum states of matter, the relativistic61.2. Charge localization: a hallmark of correlations at workMott insulator, in which on-site Coulomb repulsion and spin-orbit interac-tion have to be treated on equal footing. The idea behind the existence ofsuch a state is sketched in Fig. 1.2(c). To summarize, three different classesof correlated TMOs have been introduced:1. Mott-Hubbard insulators ? Fig. 1.2(a).2. Charge-transfer insulators ? Fig. 1.2(b).3. Relativistic Mott insulators ? Fig. 1.2(c).Note that the 4d -based oxides have been left out of this overview, as thepresence of Mott physics in such systems is still debated, although they hosta variety of different many-body phenomena, including relativistic correlatedmetallic behavior and unconventional superconductivity.It is then clear how, as one goes down from 3d to 5d materials, U pro-gressively decreases whereas a new energy scale, spin-orbit coupling, gainsimportance and thus has to be accounted on equal footing. The evolution ofU/W and the interplay with SOC from 3d to 5d are central elements of thisthesis. The resulting charge-localized ground states have been here investi-gated at the experimental and theoretical level, and a brief introduction isprovided in the next section.1.2 Charge localization: a hallmark ofcorrelations at workIn the previous section, the Mott insulator was introduced as one of themost spectacular manifestations of correlated physics and of the concomitantbreakdown of band theory. Such an electronic state of matter relies on aparticular condition, namely having an integer (odd) number of electronsper site in order to ensure that no charge fluctuations are present. Whenthis requirement is attained, the lowest-energy electronic excitation involvesthe transfer of one electron between two neighboring sites, causing a doubleoccupation which for d-orbital systems always entails a large energy cost oforder U . The many-body ground state in this configuration ? with electrons71.2. Charge localization: a hallmark of correlations at worklocalized at the lattice site ? can be written as (A is the antisymmetrizingoperator): ?NMott=A?i ?i, where here the spin degrees of freedom are hereleft out for the sake of generality. Such state is, in a sense, the electronicequivalent of what a crystalline state of matter is for an ensemble of atoms.This situation is depicted in Fig. 1.3(a), which corresponds to an averagefilling of 1 electron/site, and in the specific case shows how the (energy-expensive) double occupancy states are completely absent.In the opposite limit, a metal is a material which is instead characterized byan electronic spectrum which now admits zero-energy excitations (no gapis present). In this case carrier hopping in the lattice, and consequentlyalso charge fluctuations, are energetically favored (and, in fact, are also re-sponsible for interatomic bonding). While the charge density of valenceelectrons is still concentrated around the parent ions (like in a Mott insula-tor), in a metallic ground state there is a nonzero probability to have empty,singly-, and doubly-occupied states (assuming that a single electronic or-bital forms the valence band, i.e. a maximum occupancy of two electrons)occurring in the lattice. This is related to the nature of a many-body stateformed from a Slater determinant of tight-binding-like single-electron states:?NTB=A?k<EF ?k, where the Bloch states ?k are derived from the atomic(Wannier) orbitals ?i as follows: ?k =?i exp (ik ? ri)?i. By construction,such state, which is the lowest-energy state whenever W>U , admits single-site electron occupations different from one, i.e. it promotes local chargefluctuations. This condition, illustrated in Fig. 1.3(b), occurs in all systemswhere the valence electrons are associated to s or p orbitals (or mixed), lead-ing to an itinerant behavior where hopping wins over Coulomb repulsion.The atomic/molecular counterpart here would be an homogeneous system,i.e. a gas or a liquid, where the single-particle kinetics are dominant overrepulsive interatomic interactions. Indeed, metallic states are analogouslysubdivided in two categories: (i) Fermi gases, or free electron-like electronicensembles where electron-electron interactions are instead negligible, andindependent-particle dynamics take place; (ii) Fermi liquids, where moder-ate Coulomb interactions are not large enough to completely suppress single-particle motion in the lattice, and charge dynamics still admit zero-energy81.2. Charge localization: a hallmark of correlations at workMott insulator Charge order - Stripes Charge order - CheckerboardMetaldcbaFigure 1.3: Different types of electronic ground states, and correspondingspatial diagrams of site occupation. (a) The Mott insulator, where only sin-gle occupancy is allowed. (b) The metallic state, where sites having double-or no-occupancy are possible due to charge fluctuations. Note that the trueground state of such a system is made of a linear combination of manysuch configurations. (c) Charge-ordered states admit deviations from strictsingle-occupancy, but the charge (occupancy) fluctuations crystallize in or-dered patterns: shown are the case for a stripe order (c) and checkerboardorder (d).?quasiparticle? excitations, yet with renormalized parameters (correspond-ing to a reduced electron velocity, mobility and lifetime); (iii) non-Fermiliquids, or strange metals, where the dynamics of electronic states vary be-tween Fermi liquid-like and insulating-like as a function of momentum k,as a consequence of, e.g., a momentum-dependent ?pseudogap? that wipesaway only a subset of the zero-energy quasiparticle excitations (like in thecase of underdoped cuprates).In-between these two extreme limits one can have many-body ground-stateswhich admit charge fluctuations that are purely static, i.e. frozen in anordered pattern. This phenomenon determines the formation of a charge-density wave (CDW), which represents a periodic rearrangement of the elec-tronic charge. Such a ground state, which now breaks the translationalsymmetry of the lattice, represents the compromise between the tendencyto segregate the charge (a reminiscence of the Mott physics) and the presenceof charge fluctuations in the lattice. Fig. 1.3(c,d) illustrates different possibleforms of charge ordering in a crystal, and specifically those that are relevantin the case of copper-based oxides (which will be the subject of Chap. 2 and91.2. Charge localization: a hallmark of correlations at work0.05 0.1 0.2 0.250.15SCT*THTNTSDWTemperature (K)Doping (holes/Cu)YBCOAFMPseudogap Fermi liquidCDWWeak correlationsMott Charge orderFigure 1.4: Phase diagram of YBCO as a function of doping and tempera-ture, readapted from [16]. Highlighted are the relevant phases: (i) the an-tiferromagnetic Mott-insulating state persisting up to 5% hole-doping (p);(ii) the superconducting (SC) dome which spans from 5 to 26%; (iii) thepseudogap regime bounded by the T ? line; (iv) the CDW regime between7-8 and 15% doping, with transition temperatures as determined from Halleffect (TH), nuclear magnetic resonance (TNMR), scattering (TCDW ); (v)the spin-density wave phase, occurring between the Mott and the CDWphase, with onset temperatures measured with neutron scattering (TSDW );and (vi) the Fermi liquid on the overdoped side, beyond p=16%. Note thatcorrelations decrease from left to right.3). In one case [Fig. 1.3(c)] electrons tend to form one-dimensional rivers ofcharge, also denominated stripes. An alternative form of ordering, which istwo-dimensional in nature, is instead the checkerboard [see Fig. 1.3(d)].An archetypical system where all these states of matter emerge as afunction of doping, or average electron/hole occupation per Cu site, is repre-sented by the cuprates. The corresponding phase diagram, shown in Fig. 1.4,is the ideal example where to pinpoint these different phenomena at work.The complexity in the landscape of electronic phases vs. doping and tem-perature is a recurring element in the phenomenology of copper-oxides. This101.3. Experimental techniquesphase diagram features a strongly correlated phase near zero doping, wherethe d-states are half-filled, and a Mott-Hubbard state sets in. As a functionof hole-doping p, the system gains coherence thanks to the doped carriersallowing access to many-body states characterized by larger charge fluctua-tions. At very high hole-doping, in the so-called overdoped region (p>16%),the system is well-described by Fermi liquid theory [17]. In the intermediateregion, between 5 and 16% hole-doping, many exotic have been found overthe years (see figure caption). Recently, a charge-density-wave has been de-tected in YBCO [18] at the crossover between the strongly correlated andFermi liquid regime, in the region where charge stripes were found earlyon in cuprates [19]. This phenomenon ultimately arises as the result ofthe competition between the strongly-localized d-states and the mobile p-orbitals where holes reside, a concept that links back to what introduced inthe previous section.In essence, charge localization can take up different forms, dependingon the strength of correlation effects. The study of these various types ofelectronic ground states in different classes of TMOs is the core subject ofthis thesis work.1.3 Experimental techniquesSince their original discovery, correlated oxides have been extensively studiedusing a variety of experimental techniques and theoretical methods, therebyattracting an ever-growing interest by the community. It was soon realizedthat the low-energy electronic degrees of freedom were playing a key role indetermining many of their unconventional properties, with the concepts of?correlations? and ?many-body physics? gradually becoming part of the ev-eryday dictionary of many condensed-matter physicists. It was only aroundthe mid 90?s that a series of considerable technological advancements, al-lowing for unprecedentedly high momentum- and energy-resolutions, madeARPES one of the prime techniques for the study of correlated materi-als. Shortly after, the advent of high-resolution/high-brilliance synchrotron-based photon sources allowed the use of REXS to probe the electronic struc-111.3. Experimental techniquesture in a selective way by performing diffraction experiments at resonance.In the last 15 years, these two techniques have progressively come closer to-gether, and are nowadays routinely used as complementary tools in the studyof the low-energy single-particle (ARPES) and particle-hole (REXS) excita-tions. While momentum resolution in ARPES is used to look at the chargedynamics of hole-excitations in a long-range ordered system ? and there-fore at the electronic excitation spectrum ? REXS is a valuable probe fordetecting the ground-state electronic density in Fourier space with extremesensitivity. In the following sections, these techniques will be introducedand their role with respect to our investigations on correlated materials willbe clarified.1.3.1 ARPESAngle-resolved photoemission spectroscopy (ARPES) is one of the most di-rect methods of studying the electronic structure of solids. By measuring thekinetic energy and angular distribution of the electrons photoemitted froma sample illuminated with sufficiently high-energy radiation, one can gaininformation on both the energy and momentum of the electrons propagatinginside a material. This is of vital importance in elucidating the connectionbetween electronic, magnetic, and chemical structure of solids, in particularfor those complex systems which cannot be appropriately described withinthe independent-particle picture.The basic experimental mechanism of ARPES hinges on the capabilityof exciting electrons bound to a material into free electron states above thevacuum level, which usually can be well described as plane waves. Withinsuch particle in ? particle out scheme, the peculiarity of ARPES is thatmonochromatic light (particle in) is used in the excitation process (here-after referred to as photoexcitation or photoemission), while removed elec-trons (particle out) are measured by a detector: ARPES thus measures theelectron-removal spectrum. Furthermore, due to very short mean free pathof electrons in the 10-50 eV range (the one that usually applies at the exper-imental level), ARPES is a very-sensitive technique, capable of measuring121.3. Experimental techniquesonly the very topmost crystallographic layers of any material, and requiringatomically-flat exposed (cleaved) surfaces, and a Ultra-High-Vacuum (UHV)environment.ARPES detectors are capable of measuring both the kinetic energy andthe emission angle of photoexcited electrons, which then allows to recon-struct the full free-electron momentum kout (see Fig. 1.5). The need forangular resolution follows from the electron momentum being a valid quan-tum number for electronic eigenstates in systems possessing long-range or-der. The crucial step in the probing scheme is then how to connect kout tothe energy and momentum eigenvalues of electronic states in the material.This follows from applying the energy and momentum (only the componentparallel to the surface) conservation laws to the closed system composed ofmaterial + interacting electromagnetic field:h? + Ein = Eout (1.1)Kph + kin? = kout? . (1.2)In Eqs. 1.1 and 1.2 the parallel components of electron momenta inside(outside) the material are called kin? (kout? ), while Eph = h? and Kph arethe photon energy and momentum, respectively. Note that the electronenergies Ein and Eout are conventionally referred to the vacuum level Ev,which implies that the energy of the bound electron can be decomposedinto a binding energy term EB and a material-specific work function ?,both positively defined (see Fig. 1.5): Ein =?(EB + ?). The knowledge ofthe electron takeoff angles ? and ? and kinetic energy Ekin allows then tofully reconstruct the electron momentum by using [the overall geometry isdisplayed in Fig. 1.6(a)]:kx =1~?2mEkin sin? cos? (1.3)ky =1~?2mEkin sin? sin? (1.4)kz =1~?2mEkin cos? . (1.5)131.3. Experimental techniquesWavefunction schemeEnergy schemeexcitationinto a dampedfinal statewave matchingat the surfaceE0Zh?h?h?EfEiEEvEFEBSampleN(E)N(Ekin)Valence BandCore LevelsSpectrumEFEkin?E0V0a bFigure 1.5: Energetics of the photoemission process (from Ref. 20). (a)The electron energy distribution produced by the incoming photons, andmeasured as a function of the kinetic energy Ekin of the photoelectrons(right), is more conveniently expressed in terms of the binding energy EB(left) when one refers to the density of states in the solid (EB =0 at EF ).(b) Pictorial representation of the single-particle-like initial and final statesinvolved in the photoemission process.The mechanism illustrated so far relies on the existence of single-particlestates in the material, and is therefore only exact for a non-interacting sys-tem. Beyond this approximation, one deals with a genuine many-body prob-lem, where N-particle fermionic quantum states cannot be approximatedwith inter-particle interaction (Coulomb repulsion). The formalism there-fore requires the use of many-body N-particle states, whereby Eqs. 1.1 and1.2 are rewritten as:h? + ENi = ENf (1.6)Kph + kNi = kNf , (1.7)141.3. Experimental techniqueskPhotoemission geometrye-Samplezyxh???aFermi liquid systemN-1 N+1 EEFcNon-interacting electron systemEEFkN-1 N+1 EEFkbanalyzerElectronFigure 1.6: (a) Geometry of an ARPES experiment, from [1]. The emissiondirection of the photoelectron is specified by the polar (?) and azimuthal (?)angles. Momentum resolved one-electron removal and addition spectra for:(b) a non-interacting electron system (with a single energy band dispersingacross the Fermi level); (c) an interacting Fermi liquid system.where now ENi is the total energy of the (initial) ground state |?NGS?, whilethe final state is one of the possible excited states with total energy ENf .Within this framework, a quantitative formula for the photoemission cross-section can be obtained starting from the light-matter interaction Hamilto-nian, which reads (in the Coulomb gauge):Hint =e2mc(A?p+ p?A) = emcA?p (1.8)In the perturbative regime Fermi?s golden rule can be used to express theprobability of photoexciting the system from its many-body ground state|?NGS? to one of the various possible excited states |?Nm?, with the prescrip-tion that the final state |?Nm? shall contain a plane wave state ?kf representingthe photoemitted particle:Wfi =2pi~|??Nm|Hint|?NGS?|2?(ENm?ENGS?h?) ? |??kf |Hint|?ki ???N?1m |?N?1GS ?|2(1.9)where the N-particle initial and final states have each been decomposed inan antisymmetrized product of a single-particle-like part (the electron thatis being photoexcited) and the (N-1)-particle-system left behind: ?NGS =A?ki ?N?1GS , ?Nm=A?kf?N?1m . Summing over all possible final states that can151.3. Experimental techniquesbe involved in the process one finally obtains:Wfi =?f,i|Mkf,i|2?m|cm,i|2?(Ekin+EN?1m ? ENi ?h?)? ?? ?A?(k,E)(1.10)whereMkf,i is the single-particle matrix element, while |cm,i|2= |??N?1m |?N?1i ?|2is the probability that the removal of an electron from state i, with momen-tum k, will leave the (N?1)-particle system in the excited state m. The sumof all this processes defines the spectral function (for electron removal in thiscase) A?(k, E), which can be calculated from exact diagonalization of themany-body Hamiltonian (for a more comprehensive treatment, see Ref. 1).From Eq. 1.10 it follows that, if ?N?1i =?N?1m0 for one particular m=m0,the corresponding |cm0,i|2 will be unity and all the others cm,i zero; in thiscase, if also Mkf,i 6= 0, the ARPES spectra will be given by a delta func-tion at the Hartree-Fock orbital energy EkB =??bk, as shown in Fig. 1.6(b)(i.e., non-interacting particle picture). In the strongly correlated systems,however, many of the |cm,i|2 terms will be different from zero because theremoval of the photoelectron results in a strong change of the system ef-fective potential and, in turn, ?N?1i will have an overlap with many of theeigenstates ?N?1m . Therefore, the ARPES spectra will not consist of singledelta functions but will show a main line and several satellites accordingto the number of excited states m created in the process [Fig. 1.6(c)]. Thisultimately shows how ARPES is a genuine probe of the single-particle exci-tation spectrum, or alternatively of the single-particle (hole) dynamics in asystem with long-range order.In mathematical terms, the propagation of a single electron in a many-body system is described by the time-ordered single-electron removal Green?sfunction G?(k, t? t?), which can be interpreted as the probability amplitudethat an electron removed from the system in a Bloch state with momentumk at a time zero will still be in the same state after a time |t?t?|. By takingthe Fourier transform, G(k, t? t?) can be expressed in energy-momentum161.3. Experimental techniquesrepresentation, yielding (at T =0):G?(k, E) =?m|??N?1m |ck|?Ni ?|2E ? EN?1m + ENi ? i?(1.11)where the operator c?k = ck? annihilates an electron with energy E, mo-mentum k, and spin ? in the N -particle initial state ?Ni ; the summationruns over all possible (N?1)-particle eigenstates ?N?1m with eigenvaluesEN?1m , and ? is a positive infinitesimal (note also that from here on wewill take ~ = 1). In the limit ? ? 0+ one can make use of the identity(x?i?)?1=P(1/x)?ipi?(x), where P denotes the principle value, to derivethe relationship between the single-particle spectral and Green?s function:A?(k, E)=? 1piImG?(k, E).The corrections to the Green?s function originating from electron-electroncorrelations can be conveniently expressed in terms of the electron properself energy ?(k, E) = ??(k, E)+ i???(k, E). Its real and imaginary partscontain all the information on the energy renormalization and lifetime, re-spectively, of an electron with band energy ?bk and momentum k propagatingin a many-body system. The Green?s and spectral functions expressed interms of the self energy are then given by:G(k, E) = 1E ? ?bk ? ?(k, E)(1.12)A(k, E) = ? 1pi???(k, E)[E ? ?bk ? ??(k, E)]2 + [???(k, E)]2. (1.13)Because G(t, t?) is a linear response function to an external perturbation,the real and imaginary parts of its Fourier transform G(k, E) have to satisfycausality and, therefore, also Kramers-Kronig relations. This implies that ifthe full A(k, E)=?(1/pi)ImG(k, E) is available from photoemission and in-verse photoemission, one can calculate ReG(k, E) and then obtain both thereal and imaginary parts of the self energy directly from Eq. 1.12. However,due to the lack of high-quality inverse photoemission data, this analysis isusually performed using only ARPES spectra by taking advantage of certain171.3. Experimental techniquesapproximations (such as, e.g., particle-hole symmetry near EF ; for a moredetailed discussion, see also Ref. 21, 22 and references therein).In general, the exact calculation of ?(k, E) and, in turn, of A(k, E) is anextremely difficult task. In the following, as an example we will briefly con-sider the interacting FL case [23?25]. Let us start from the trivial ?(k, E)=0non-interacting case. The N -particle eigenfunction ?N is a single Slater de-terminant and we always end up in a single eigenstate when removing oradding an electron with momentum k. Therefore, G(k, E)=1/(???bk?i?)has only one pole for each k, and A(k, E) = ?(E??bk) consists of a singleline at the band energy ?bk [as shown in Fig. 1.6(b)]. On the other hand, inan interacting Fermi liquid (FL), we can describe the correlated Fermi seain terms of well defined quasiparticles, i.e. electrons dressed with a mani-fold of excited states, which are characterized by a pole structure similarto the one of the non-interacting system but with renormalized energy ?qk,mass m?, and a finite lifetime ?k = 1/?k. In other words, the propertiesof a FL are similar to those of a free electron gas with damped quasipar-ticles. As the bare-electron character of the quasiparticle or pole strength(also called coherence factor) is Zk<1 and the total (i.e., energy-integrated)spectral weight must be conserved, we can separate G(k, E) and A(k, E)into a coherent pole part and an incoherent smooth part without poles [26]:G(k, E) = ZkE ? ?qk + i?k+Gincoh (1.14)A(k, E) = Zk?k/pi(E ? ?qk)2 + ?2k+Aincoh (1.15)where Zk = (1? ????E )?1, ?qk =Zk(?bk+??), ?k =Zk|???|, and the self energyand its derivatives are evaluated at E = ?qk. It should be emphasized thatthe FL description is valid only in proximity to the Fermi surface and restson the condition ?qk??  |???| for small (E??) and (k?kF ). Further-more, ?k? [(pikBT )2+(?qk??)2] for a FL system in two or more dimensions[26, 27], although additional logarithmic corrections should be included inthe two-dimensional case [28]. By comparing the electron removal and ad-181.3. Experimental techniquesdition spectra for a FL of quasiparticles with those of a non-interactingelectron system (in the lattice periodic potential), the effect of the self-energy correction becomes evident [see Fig. 1.6(c) and 1.6(b), respectively].The quasiparticle peak has now a finite lifetime and width (due to ???), butsharpens rapidly as it emerges from the broad incoherent component andapproaches the Fermi level, where the lifetime is infinite corresponding toa well defined quasiparticle. Furthermore, the peak position is shifted withrespect to the bare band energy ?bk (due to ??): as the quasiparticle massis larger than the band mass because of the dressing (m? >m), the totaldispersion (or bandwidth) will be smaller (|?qk| < |?bk|). We note here, aslater discussed in more detail in relation to Fig. 1.10, that the continuumof excitations described by the incoherent part of A(k, E) in general doesstill retain a k and E-dependent structure with spectral weight distributedpredominately along the non-interacting bare band; this, however, is usuallycharacterized by remarkably broad lineshapes [see e.g. Fig. 1.9(c) and 1.12]and should not be mistaken for a quasiparticle dispersion.1.3.2 REXS and XRDResonant Elastic X-ray Scattering (REXS) and X-ray Diffraction (XRD)also belong to the category of particle in ? particle out spectroscopies. Inthis case the probing particles are in both cases photons, which get scatteredfrom a material due to the interaction with the electronic clouds. In the caseof PES/IPES, photons are absorbed and therefore the interaction operator(in second-quantized form) can be expressed as[aqc?k+qck + a??qck+qc?k],where the first (second) term represents electron removal (addition). ForREXS and XRD the interaction Hamiltonian has to instead contain operatorcombinations of the kind[aqa?q?Qb?Q + h.c.], where b?Q (bQ) is a generic bo-son creation (annihilation) operator (here bosons can be phonons, magnons,or electron-hole excitations). This form for the effective interaction Hamilto-nian can be derived from the full electron-matter (non-relativistic) minimal191.3. Experimental techniquescoupling Hamiltonian, which reads:Htot =?i[12me(pi ?ecA(ri, t))2+ V (ri, t)]+?i 6=je2|ri ? rj |2+HEM= Hel +HEM +emec?iA(ri, t) ? pi? ?? ?H(1)int+ e22mec2?iA(ri, t)2? ?? ?H(2)int, (1.16)where pi and ri represent the momentum and position coordinates of thei -th electron respectively, while V (r, t) and e2/|r? r?|2 are the lattice po-tential and the Coulomb interaction terms, respectively. A(r, t) representsthe vector potential and HEM is the Hamiltonian of the EM field alone.The interaction operators H(1)int and H(2)int are linear and quadratic in thevector potential, respectively. Expressing the vector potential in second-quantized notation as A(r, t)??q[exp(iq ? r? i?t)a?q + h.c.]immediatelyshows that combinations of the kind a?a ? which govern photon-in photon-out spectroscopies ? originate out of first-order perturbation theory for H(2)int ,and second-order forH(1)int . The respective transition probabilities in first andsecond-order Fermi?s golden rule, assuming the initial and final state to bothcoincide with the system?s ground state ?GS , then read:W (1)fi ???????GS |e22mec2?iA(ri, t)2|?GS?????2(1.17)W (2)fi ??????(emec)2?m??GS |?iA(ri, t) ? pi|?m???m|?iA(ri, t) ? pi|?GS?EGS ?Em ? h? + i?m?????2.(1.18)In Eq. 1.18, ?m represents a generic many-body excited state of the system,with corresponding energy Em and lifetime ~/?m. In the X-ray regime, thesetwo transition channels both involve excitation of a high-energy many-bodystate with a core hole. In a more intuitive perspective, this is equivalent tothe excitation of a core electron into an intermediate state through absorp-tion of the first photon, followed by re-emission of a (scattered) photon oncethe core hole is filled back. The first-order term in Eq. 1.17 corresponds201.3. Experimental techniques?GS?GS?m?GS?GSh?inEFnon-resonant resonant Photon energy h??E=EGS-EmWfi    - XRD(1)Wfi    - REXS(2)Cross - sectiona bcqinh?outqouth?inqinh?outqout?m0Figure 1.7: Resonant vs. non-resonant scattering. (a) Non-resonant elasticscattering or X-ray diffraction (XRD) proceeds through promotion of a coreelectron into a virtual state, followed by immediate decay back into theground state and consequent emission of a scattered photon. (b) Resonantelastic scattering (REXS) occurs whenever the photon energy is tuned toan electronic transition from the ground state ?GS to a real excited state?m; again, radiative recombination of the excited electron with the core holecreates the scattered photon. (c) The different photon energy dependenceof XRD and REXS, showing the resonant enhancement effect.to transition into a virtual intermediate state, whereas Eq. 1.18 describesprocesses involving real intermediate states ?m. As a consequence, thefirst-order mechanism is non-resonant, and is associated to Thomson scat-tering (which dominates the signal in XRD), which is proportional to theFourier transform of the squared electronic density; on the other hand, thesecond-order process is resonant, and therefore associated to REXS.This concept is depicted in Fig. 1.7(a,b), which schematize the mechanismfor non-resonant (XRD) and resonant (REXS) processes, respectively. Themechanism corresponding to XRD involves a single step (grey arrow), invirtue of its first-order nature; conversely REXS, being a second-order tran-sition, proceeds in two stages (black arrows) involving an intermediate state.This clearly reflects in the very different photon energy (h?) dependence ofthe two channels [see Fig. 1.7(c)]: whereas XRD is nearly energy-independent(red dashed curve), the cross-section for REXS is strongly peaked aroundthe energy of the electronic transition (blue curve), where the experimentalsignal undergoes a strong enhancement (while decaying to zero away from211.3. Experimental techniquesthe resonance). This usually occurs in correspondence of an absorptionedge, i.e. when electronic transition from a deeply bound core state into thevalence band (and beyond into the continuum) take place.In a real scattering or diffraction experiment, a monochromatic X-raybeam with wavevector qin, photon energy h?in= c ? qin and polarization inimpinges on a sample, and the EM part of the light-matter quantum stateas |?i?EM = |0?...|n?qin,in |0?..., meaning all other photon modes are empty, ornq,=0 for q 6=qin. An operator term of the kind a?qoutaqin from either of thetransition probabilities in Eqs. 1.17 and 1.18 is then responsible for the tran-sition from |?i?EM to the final state |?f ?EM?|0?...|n? 1?qin,in |1?qout,out |0?....A scattered photon will then be detected along the direction of the wavevec-tor qout using a photomultiplier (PM), such as a simple photodiode (PD), ora multichannel-plate (MCP) in combination with a phosphor-based photo-cathode. At the end of the process, a net momentum and energy have beentransferred to the sample, which can be derived from the correspondingconservation laws:h?in = h?out +?Esample (1.19)qin = qout +Qsample. (1.20)In the case of elastic scattering ? which is the relevant one for this work ?h?in=h?out, which implies that there is no net energy transfer (?Esample=0)and that the magnitude of the exchanged momentum can be expressed asQsample=2 qin sin (?sc/2) (?sc is the scattering angle, see also Fig. 1.8). Pro-jecting Qsample into the plane defining the sample surface then yields thein-plane (Q?) and out-of-plane (Q?) components of the transferred momen-tum, which will be often referenced in the rest of this work.This geometry is illustrated in Fig. 1.8. In general the PM moves on asingle-circle, that is, a single angular goniometer (the corresponding variableis often denominated 2?). The sample stage usually involves translationalmotion (xyz) and various rotations, whose number defines the type of diffrac-tometer. 2-circle diffractometers, which are the most common choice for softX-ray experiments, feature only a single angular motion for the sample (?),221.3. Experimental techniques(qout,?out??in)(qin,?in)PMSample?scDiffractometer?sc?out?inqinqoutQ||Q?Q sampleCounts-0.4 -0.2 0.0 0.2 0.4Q|| (r.l.u.) 300 K 20 Kh?=931.5 eVa b cFigure 1.8: Geometry of a REXS/XRD experiment. (a) Experimental scanof the in-plane exchanged wavevector on Bi2201, showing the appearanceof a charge-ordering peak near Q? = 0.25 (more on this in Ch. 3). (b)Schematics of a conventional diffractometer, with kinematics for the scat-tering/diffraction process outlined in (c).whose axis is perpendicular to the diffraction plane (the one spanned by thevectors qin and qout). Diffractometers with more circles are also routinelyused, especially at higher photon energies (hard X-rays, h?>2?3 keV) wheremore reciprocal space can be accessed.In general, the experimental signal comprises both resonant and non-resonant contributions, and in the two possible regimesW (XRD)fi W(REXS)fior W (REXS)fi  W(XRD)fi one ends up probing different phenomena. In thefirst case, where non-resonant processes are dominant, all electrons con-tribute equally to the measured signal, which is therefore simply propor-tional to the atomic number Z. As a consequence, the diffraction signal willbe dominated by the core electrons, which usually outnumber the valenceones, or ncore  nvalence (with the exception of lighter elements). Sincecore states are very tightly bound to their parent nucleus, in a diffractionexperiment one mainly probes the ionic lattice in reciprocal space. For thisreason, XRD is widely used in structural studies.In the second case, the scattering process has a strong enhancement in cor-respondence of a very specific electronic transition. As a result the signalbears the signature of the electronic distribution (in reciprocal space) of thefinal state of such transition. This characteristic of resonant scattering al-231.4. Physics of correlationslows it to be not only element-specific (whenever the absorption edges ofdifferent chemical species are enough spaced apart in photon energy), butalso orbital-selective. This unique capability of REXS has been establishedand employed in many different systems. Charge-ordering in cuprates [29]and orbital-ordering in the manganites [30] are among the most spectacularcase studies.1.4 Physics of correlationsThe sensitivity of ARPES to correlation effects is deeply connected to itscorresponding observable, which is the one-particle spectral function previ-ously introduced in Sec. 1.3.1. This physical quantity conveys informationnot only on the single-particle excitations, but also on the many-body finalstates which can be reached in the photoemission process. However, thedistinction between single-particle and many-body features in A(k, ?) atthe experimental level is often subtle. In order to disentangle the nature ofthe underlying excitations it is common practice to decompose the spectralfunction into a coherent, Acoh(k, ?), and an incoherent part, Aincoh(k, ?)(see Ref.1).Whereas in a purely non-interacting system all single-electron excita-tions are coherent, since they are insensitive to the behavior of the otherparticles, things can be quite different when electron-electron interactions,and therefore quantum-mechanical correlation effects, are turned on. Forthese reasons, the redistribution of spectral weight between the coherentand incoherent part in A(k, ?) is commonly regarded as a distinct signatureof correlations at work. In particular, the integrated spectral intensity ofthe coherent part, the quasiparticle strength Zk that can be extracted fromARPES, is a relatively direct measure of the correlated behavior of a givensystem. In general, Zk can vary from 1 (non-interacting case) to 0 (stronglycorrelated case, no coherent states can be excited).In the following subsections we will explain how the concept of cor-relation already emerges in simple molecular-like systems (i.e., few-body)and evolves into the complex structures found in solid-state materials (i.e.,241.4. Physics of correlationsH2+H20-00-60-2(a) (b)BA-1.0k =(pi/2, pi/2)H218 160-00.0Energy (eV)-0.5(c)Kinetic energy (eV) coordinatesEnergy ??=0125346?=0125346q01Figure 1.9: Franck-Condon effect and correlations in spectroscopy.(a) Franck-Condon effect and its relation to the single-particle spectral func-tion in atomic physics. (b) Photoionization spectrum of gaseous H2 (fromRef. 31), featuring a comb of lines corresponding to the various accessibleexcited (vibrational) final states of the H+2 system left behind; the red lineis an abstraction to what would happen in solid-state H2 (from Ref. 32).(c) A(k = (pi/2, pi/2), ?) from Ca2CuO2Cl2, showing the broad incoherentZhang-Rice peak, with the sharp Sr2RuO4 lineshape superimposed for com-parison (from Ref. 33).many-body). Different types of correlation effects will be reviewed, withparticular emphasis on those stemming from electron-phonon and electron-electron interactions. We will then discuss different aspects of correlatedelectron behavior in a few selected transition metal oxides, and show howcorrelations evolve with ? and to some degree can be controlled by ? theexternal control parameters introduced in Sec. 1.1 and Fig. Origin of correlations in photoelectron spectroscopyIn general, the connection between the one-particle spectral function andcorrelations is not immediately obvious and might look mysterious to thereader. It is useful and instructive to clarify what the photoelectron spec-trum for a correlated system looks like, beginning with an example frommolecular physics. In Fig. 1.9 we show the photoionization spectrum of themolecular gas H2 which, at variance with a simpler atomic gas (e.g., He251.4. Physics of correlationsor Ne), exhibits a fine structure made of a series of peaks almost evenlyseparated in energy. The underlying physical explanation for these spec-tral features relates to the Franck-Condon principle, which is explained inFig. 1.9(a). This is best understood if we write down the equation for thephotoionization cross section, which will involve: (i) an initial state wave-function ?i, assumed to be the ground state for the neutral molecule; and(ii) a final state wavefunction ?f , which can be a linear combination ofeigenstates for the ionized, positively charged molecule H+2 . The possibleeigenstates we consider here can be separated into an electronic part (thehydrogen-like 1s orbital ?1s) and a nuclear part, which in a diatomic moleculelike H2 can be vibrationally excited. The latter is given, to a good approx-imation, by one of the eigenfunctions of the harmonic oscillator, ?n(Req),which depend on the interatomic equilibrium distance Req. Combining to-gether electronic and nuclear (i.e., vibrational) components we obtain a basisset for the molecular Hamiltonian in the form ?n=?1s?n(Req). We can thenuse this set of functions in the matrix element governing the photoionizationprocess:IH2?H+2 ??m??H+2m |p ?A|?H2n=0? ??m??k,m|p ?A|?1s???m(RH+2eq )|?0(RH2eq )? .(1.21)Here p?A is the dipole interaction operator and the initial state is the groundstate for H2, or ?H2GS=?n=0. The term ??k,m|p?A|?1s?=Mkm is the matrix el-ement previously introduced in Sec. 1.3.1, representing the overlap betweenthe initial-state electronic wavefunction ?1s and the final state plane-wave?k,m. It is readily seen that, if RH+2eq =RH2eq , then IH2?H+2 ??m ?m,0 and thephotoionization spectrum would be composed of a single peak, correspond-ing to the ?0-0? transition between the initial and final ground states. Inreality, the neutral and ionized molecule will see a different charge distribu-tion (thus leading to a different electrostatic potential), due to the missingCoulomb interaction term for the 1s electrons in the Hamiltonian for H+2 .As a consequence, the molecule before and after photoexcitation will have a261.4. Physics of correlationsdifferent interatomic equilibrium distance, and many of the terms in Eq. 1.21will be different from zero resulting in multiple transitions in the experimen-tal spectrum [corresponding to the vertical excitations in Fig. 1.9(a) and tothe various peaks in Fig. 1.9(b)]. The lowest energy peak [labeled ?0-0?in Fig. 1.9(b)] still corresponds to a transition into the ground state of theionized molecule, but it only contains a fraction of the total photoemis-sion intensity, or spectral weight. At this point it is useful to introduce analternative definition of coherent and incoherent spectral weight:? The coherent spectral weight is a measure of the probability to reachthe ground state of the final-state Hamiltonian (HH+2 ) in the photoex-citation process. In experimental terms, it is represented by the totalarea of the 0-0 transition shown in Fig. 1.9(b).? The incoherent spectral weight is a measure the probability to leavethe ionized system in any of its excited states. It can be thereforecalculated from the integrated intensity of all the 0-m (m 6= 0) peaksin the ionization spectrum, as shown in Fig. 1.9(b).In solid-state, many-body systems, both molecular vibrations and elec-tronic levels are no longer discrete but have an energy dispersion (turninginto phonons and electronic bands, respectively). This is what gives a con-tinuum of excitations when many body interactions are at play, as opposedto the sharp excitation lines of the H2 case. However these concepts remainvalid, although we shall now restate them within a many-body framework:? The coherent spectral weight corresponds to the probability of reach-ing, via the electron addition/removal process (?N =?1, where N isthe initial number of electrons), the many-body ground-state for the(N ? 1)-particle Hamiltonian (H?N?1).? The incoherent spectral weight gives the cumulative probability thatthe (N ? 1)-particle system is instead left in an excited state.A photoelectron spectrum for a solid-state many-body system will looklike the dashed curve in Fig. 1.9(b), due to the multitude of final states that271.4. Physics of correlationscan be reached as a result of the photoemission process. The well-definedfeatures characterizing A(k, ?) in the molecular case will then broaden outinto a continuum of excitations. This was experimentally found to occur inthe strongly-coupled cuprate material Ca2CuO2Cl2 [see Fig. 1.9(c)], whichwill be discussed in more detail in the next section. We also note that forthe incoherent part of the spectral function two cases are possible in a solid:1. Aincoh(~k, ?) is composed of gapless many-body excitations, e.g. cre-ation of electron-hole pairs in a metal; this typically produces an asym-metric lineshape, as in the case of the Doniach-Sunjic model [34].2. Aincoh(~k, ?) originates from gapped excitations, e.g. coupling betweenelectrons and optical phonons; in this case the coherent part is wellseparated from the incoherent tail, and a quasiparticle peak can bemore properly identified.What we have just seen for the H2 molecule stems from the interactionbetween the electronic and the nuclear degrees of freedom. In the absence ofsuch interplay there would be no fine structure in the corresponding spectralfunction. This is a very important concept, which is deeply connected to theidea of correlations. The Hamiltonian of a given (few-body or many-body)system, in the absence of interaction terms, can be decomposed into a sumof single particle terms (non-interacting case). Correspondingly, the systemis unperturbed by the addition or removal of a particle during the photoex-citation process; due to the orthonormality of the involved eigenstates, the(N ? 1)-system left behind will not be found in a superposition of excitedstates but rather left unperturbed in its ground state (at zero temperature).Single-particle spectroscopy would then detect a single transition (e.g., the0-0 peak in Fig. 1.9) and the spectral weight is fully coherent. Conversely,when the electron-nucleus and/or electron-lattice interaction are switchedon, addition/removal of a single electron perturbs the molecular/lattice po-tential to some degree and this can trigger creation or annihilation of oneor multiple vibrational modes in the process. As we will see in the fol-lowing section for electron-phonon coupling in solids, the effect at the level281.4. Physics of correlationsof spectral function can be very different according to the strength of theinteraction.1.4.2 Electron-phonon correlations in solids: the polaronThe interaction of the mobile charges with the static ionic lattice is whatunderlies the formation of electronic bands in all crystalline materials. How-ever, the lattice is never really static and its low-energy excitations, thephonons, are present even at very low temperatures. As they hop aroundin the lattice, electrons can interact (through the ionic Coulomb potential)with ? or become ?dressed? by ? phonons, thereby slowing their quantummotion. These new composite entities, known as polarons, represent thetrue quasiparticles of the coupled electron-lattice system: the properties ofthe ?bare? electrons, in primis bandwidth and mass, are now renormalizedin a fashion which directly depends on the strength of the electron-phononcoupling. Here we show two examples of polaronic physics, one experimentaland the other theoretical, which exhibit different features but relate to thesame underlying interactions.The first case is that of Ca2CuO2Cl2 (CCOC). This compound is the un-doped parent compound of the high-Tc superconducting cuprates, wherethe low-energy physics originates from the hybridized Cu 3d ?O 2p statesof the CuO2 planes. In Fig. 1.9(c), the ARPES spectrum of CCOC at~k=(pi/2, pi/2) is shown [33]. This value of electron momentum correspondsto the lowest ionization state of the Zhang-Rice singlet (ZRS) band [35]. Thelatter is a 2-particle state, made of a combination of one O 2p and one Cu3d hole in a total spin-zero state (singlet). Hence the nature of such a stateis intrinsically correlated, and cannot be described within a single-particleframework. While this feature was originally recognized as the quasiparticlepole of the same Cu?O band that is also found in the doped compounds[36], the Gaussian lineshape, together with the broad linewidth (??0.5 eV)suggest that this feature might instead be identified as the incoherent partof the spectral function. It follows that the spectral function has no actualquasiparticle weight Zk, because the intensity of the lowest energy excita-291.4. Physics of correlations?=0.1(c)0 1 2?=0.5 ?=1.0 ?=10(b) (d) (e)E B  (meV)200EF?Zk~0Momentum   (1/?)0 1 2 0 1 2 0 1 2 3?=0bkqk0 1 2(a)??A(k,?)Figure 1.10: Polaron physics in 1D.A(k, ?) for the Holstein model, showing the quasiparticle band and the 0-, 1-and n-phonon dispersing poles in the spectral function for increasing valuesof the electron-phonon coupling ?=0, 0.1, 0.5, 1.0 and 10 (panels a, b, c, dand e, respectively). The dashed black, and the orange lines represent thebare band ?bk, and the renormalized quasiparticle band ?qk, respectively. TheFermi energy EF has been set at the top of the quasiparticle band (fromRef. 22).tion [the ?0-0? line marked by the ?B? arrow in Fig. 1.9(c)] approaches zero.While the previous example illustrates a case where the strong electron-boson interaction entirely washes away the coherent spectral weight, in dif-ferent physical systems it is possible to have sizeable weight in the quasipar-ticle pole. This is illustrated with the second example discussed here, theone-dimensional (1D) Holstein model [22], shown in Fig. 1.10. This modelis represented by the following Hamiltonian:H1DHolstein =?k?bkc?kck +??Qb?QbQ +g?n?k,Qc?k?Qck(b?Q + b?Q) . (1.22)The evolution of the associated spectral function as a function of the di-mensionless electron-phonon coupling parameter ?= g2/2t? (in this calcu-lation ? = 50meV) is shown in Fig. 1.10. The case ? = 0 (no coupling),301.4. Physics of correlationsshown in Fig. 1.10(a), yields a spectral function which exactly follows thebare electronic band ?bk (indicated by the black dashed line in Fig. 1.10), i.e.A(k, ?)=?(???bk), where the ?-function is here broadened into a Lorentzianfor numerical purposes. Already in the small coupling limit ?=0.1 a quasi-particle band branches off the original band, with a k-dependent spectralweight [see Fig. 1.10(b)]. The latter is substantially redistributed, with thespectral weight at binding energies higher than the quasiparticle band be-longing to the incoherent part of the spectral function, which forms a con-tinuum of many-body excitations for EB>?. As the electron-phonon cou-pling is further increased, it can be noted from Fig. 1.10(c,d) how: (i) thereis a progressive renormalization of the quasiparticle band (?qk), which im-plies a reduction in the total bandwidth and a change in the slope ??qk/?k(quasiparticle velocity); (ii) the spectral weight is redistributed between thequasiparticle band ?qk and the incoherent features at higher binding energy,corresponding to a photohole copropagating with, or ?dressed? by, one ormultiple phonons. In the strong-coupling regime ?= 10, the quasiparticleband and its n-phonon replicas are nondispersive (i.e., corresponding to a di-verging quasiparticle mass), and the coherent spectral weight Zk has almostcompletely vanished [Fig. 1.10(d)].After having discussed quasiparticle renormalization due to electron-phonon coupling, in the following we will turn our focus onto electron-electron interaction effects, which are particularly pronounced in 3d -TMOs,and dominate the low-energy electrodynamics in these systems.1.4.3 Doping-controlled coherence: the cupratesAs anticipated, copper-based oxide superconductors exhibit a rich phase di-agram, encompassing a variety of unconventional phases (Fig. 1.1), which in-clude: high-temperature superconductivity, Mott insulating behavior, pseu-dogap regime, strange metal (non-Fermi liquid), and possibly electronic liq-uid crystal (nematic phase), to name a few. In particular, their remarkablepeculiarity lies in the possibility of realizing these different phases simply bycontrolling the charge carriers doped into the CuO2 planes.311.4. Physics of correlationsA manifestation of the underlying correlated nature of these materi-als can be found in the doping-dependent evolution of coherent behaviorin the low-energy electrodynamics. This is the case of the ARPES resultson YBa2Cu3O6+x (YBCO), one of the most studied within the family ofcuprates owing to oxygen-ordering and subsequent low degree of disorder. Inthis and similar materials, hole-doping is usually controlled at the chemicallevel, by tuning the stoichiometric ratio between the O and Cu content. Asfor the study of the low-energy electronic structure by ARPES, this has beenhampered by the lack of a natural cleavage plan and especially the polarityof the material, which leads to the the self doping of the cleaved surfaces[37?39]. As a result, while the bulk of YBCO cannot be doped beyond 20%by varying the oxygen content, the surfaces appear to be overdoped up toalmost 40% (the highest overdoping value reached on any cuprate [39]). Anapproach devised to resolve this problem involves the control of the carrierconcentration at the surface [38, 39] by in-situ potassium deposition on thecleaved crystals, which enables the investigation of the surface electronicstructure all the way from the overdoped (p?0.37) to the very underdopedregion of the phase diagram (p?0.02).Concurrent with a modification of the Fermi surface, which evolves fromlarge hole-like cylinders to Fermi arcs [38, 39], there is also a pronouncedchange in the ARPES spectral lineshape [see Fig. 1.11(c), where the corre-sponding energy distribution curves have been extracted from the ARPESmaps in panels (a,b), for k = kF ]. In particular, two major effects are ob-served going from the over- to the under-doped surface: (i) the progressiveloss of the nodal coherent weight with no quasiparticle peak being detectedat p=0.02 [Fig. 1.11(c)], accompanied by an increase in the incoherent tail,and consistent with conservation of the total spectral weight; (ii) the sup-pression of the nodal bilayer splitting ?B,ABN shown in the ARPES intensitymaps of Fig. 1.11(a,b) [an even more pronounced suppression can be ob-served at the antinodes], which goes hand-in-hand with the redistribution ofspectral weight from the coherent to the incoherent part of the spectral func-tion. Since correlation effects suppress hopping within and between planesin a similar fashion, the renormalization of the measured bilayer splitting321.4. Physics of correlationsBinding energy (eV)0.00.220 40 60MinMaxk  (% of ?S)0.00.2p=0.24p=0.06(a)(b) (c)d0.0 0.3Hole doping (p)0.0Z N0.20.40.6Bilayer SplittingRescaledSWR @ EF(d)O6.34 =0.24=0.06 = energy (eV)Intensity (a.u.)Figure 1.11: Doping-dependent correlation effects in cuprates.(a,b): ARPES dispersion in YBCO, along the nodal cut [? ? (pi, pi)] forp=0.24 and 0.06, respectively, showing the lack of bonding-antibonding (B-AB) bilayer splitting and the spectral function being mostly incoherent forp=0.06. (c) A(k=kF,N , ?) as a function of doping for the bonding Cu-Oband, showing the progressive suppression of the quasiparticle peak. (d) ZNas determined from the B-AB splitting and the spectral-weight ratio SWR(see text). Also shown are guides-to-the-eye and the 2p/(p + 1) Gutzwillerprojection relation (from Ref. 39).with respect to the prediction of density functional theory can be used asan equivalent measure of the coherent weight Zk=?B,ABN /2tLDA? (N), withtLDA? (N)'120meV. This is a more quantitative and more accurate methodthan estimating the spectral weight ratio between quasiparticle and many-body continuum, SWR =? ??EF I(kF,N , ?)d?/? ??0.8eV I(kF,N , ?)d?, since inthis case the coherent and incoherent parts of A(k, ?) are not well separated.Using both methods, it is possible to observe a suppression of the coherentweight as one goes underdoped, with Zk vanishing around p = 0.1 ? 0.15[Fig. 1.11(d)], consistent with the observation that the underdoped (p<0.1)ARPES spectra are mostly incoherent, A(k, ?) ? Aincoh(k, ?). The prox-imity of the Mott phase (p = 0), with its strongly correlated behavior, isbelieved to be the reason underlying the loss of coherent behavior as holedoping is progressively reduced, and forces a departure from the Fermi liquid331.4. Physics of correlations(b) (c)Wavevector, k (?-1)(a)T=Figure 1.12: Single-particle coherence in the manganites.(a) ARPES image plot of the Mn-eg valence band along the (0, 0)-(pi, pi)direction in La1.2Sr1.8Mn2O7 (T = 20K); note the quasiparticle band ?qkbranching off the bare band near EF , in analogy to the case of the Holsteinmodel [Fig. 1.10(b)]. (b,c) Stack of low-energy EDCs for T =15 and 120K,respectively, emphasizing the emergence of the quasiparticle peak below TC(from Ref. 40).description much more rapid than predicted by the mean field Gutzwillerprojection Z=2p/(p+ 1).1.4.4 Temperature-controlled coherence: the manganitesAnother family of 3d -based oxides characterized by a rich phase diagramis that of the manganites. These materials, which exhibit the fascinatingphenomenon known as colossal magnetoresistance, have been extensivelystudied by ARPES [40?46]. In one of these compounds, La1.2Sr1.8Mn2O7,the high temperature spectra (T > 120K) do not qualitatively differ fromthose seen for undoped cuprates, previously presented in Fig. 1.9(c). Asshown in Fig. 1.12(d), there is no spectral weight at the Fermi energy, and341.4. Physics of correlationsthe lowest-energy excitation is a broad peak dispersing between -1 and -0.5eV [40]. This finding suggests we are again looking at a strongly correlatedsystem, where all the spectral weight is pushed into a broad and incoherentstructure away from EF . Surprisingly, when the temperature is loweredthrough the Curie value TC?120K, a sharp feature emerges at the chemicalpotential, with an intensity progressively increasing as the sample is cooleddown to 15K [see Fig. 1.12(a,b,c) and related insets]. This is a remarkableexample of how temperature can lead to a transfer of spectral weight fromAincoh to Acoh, in this case associated with the ferromagnetic transitionoccurring at TC . Note the large ratio Aincoh/Acoh, i.e. a incoherent-to-coherent transition, and the subsequently small Zk: this is an indicationof the fact that we are still in a regime where electronic correlations arevery strong, similar to the case of undoped and underdoped cuprates and,as we will see in the following section, also of cobaltates [47]. In addition, itis important to note how the coherent spectral weight does not necessarilyappear throughout the entire Brillouin zone, but might instead be limitedto a reduced momentum range, where electronic excitations can propagatein a coherent manner.1.4.5 Probing coherence with polarization: the cobaltatesAs discussed in the previous sections, the distinction between coherent andincoherent parts of A(k, ?) ? and thus the determination of the quasiparticlestrength Zk ? although conceptually well defined, is often not easy to esti-mate from ARPES experiments. An additional complication is encounteredwhenever more overlapping bands contribute to the low energy electronicstructure in the same region of momentum. In such instances, there is onecharacteristic of the ARPES technique which can be exploited, namely theexplicit dependence on light polarization of the photoemission intensity froma band of specific symmetry, as a result of matrix-element effects. For a sin-gle band system, changing any of the experimental parameters would changethe ARPES intensity as a whole, thus preserving the shape of the spectralfunction and in particular the ratio between Acoh(k, ?) and Aincoh(k, ?),351.4. Physics of correlations(a)0.0-0.5-1.0-1.5 0.0-0.5-1.0Intensity (a.u.)kF = 0.52 ?-1k = 0.78 ?-1k = 1.10 ?-1k = 1.34 ?- (a.u.)-1.2 -0.8 -0.4 0.0 LH LV Difference LH-LV??Intensity (a.u.)(c)  T = 5 K T = 160 K T = 220 KBinding Energy (eV) Binding Energy (eV)Binding Energy (eV)(b)Figure 1.13: Correlations in multi-band systems: the example of cobaltates.(a) top panel: A(k = kF , ?) from the misfit cobaltate [Bi2Ba2O4][CoO2]for two different polarizations - red curve is linear horizontal (LH), bluecurve is linear vertical (LV); (a) bottom panel: linear dichroism ALD =ALH ?ALV . (b) momentum-dependence of ALD near k=kF (from Ref. 48).(c) temperature-dependence of ALD [49].since these terms are weighted by an identical matrix element. The situa-tion is very different in a multiband system since, whenever the quasiparticlepeaks and the many-body continua originate from different single-particlebands, they will be characterized by a different overall symmetry. In thiscase, one may use the polarization dependence of the single-particle matrixelements to disentangle the different spectral functions contributing to thetotal ARPES intensity. This approach, shown in Fig. 1.13 and discussed inmore detail in Ref. 48, has been used in the study of misfit cobaltates, a fam-ily of layered compounds, where the low-energy electronic states reside inthe CoO2 planes. These compounds all have 3 bands crossing the chemicalpotential and, while detecting Acoh is relatively simple due to its sharpnessin proximity to EF , evaluating the ratio between Acoh and Aincoh is a muchmore complicated task due the the overlap of contributions stemming fromdifferent orbitals.Two close-lying bands, of respectively a1g and eg ? orbital character, havedifferent symmetries and can thus be selected using polarization [see toppanel in Fig. 1.13(a)]. Taking the difference between spectra measured with361.4. Physics of correlationsSr 2RhO 4(b)Sr 2RuO 4MX M XM0-0.20-0.25E-EF (eV) LDA LDA+SOM MM XX (c)(d) (e) (f)????SB?XX(a)Figure 1.14: (a-c): Stack of experimental ARPES EDCs in Sr2RuO4, alongthe high-symmetry direction ? ? X (a), with LDA (b) and LDA+SO (c)predictions for the Fermi surface (from Ref. 50, 51). (d-f): same as (a-c), forthe case of Sr2RhO4 (from Ref. 51?53).different polarization (linear dichroism), one can isolate the full spectralfunction for the a1g band [Fig. 1.13(a), bottom panel]. The momentum- andtemperature-dependence are then displayed in Fig. 1.13(b) and (c), respec-tively, evidencing a very similar behavior to the one found in the manganites[49]. With this approach is thus possible to track the quasiparticle weightZk as a function of temperature and doping, even in multiband systems andin those regions of momentum space where bands overlap.371.4. Physics of correlations1.4.6 Correlated metals: the spin-orbit coupled 4d-TMOsStepping down one row in the periodic table we find the 4d transition metaloxides. Based on simple arguments, one would expect correlations to playa less important role in these materials. This is due to the larger spatialextent of the 4d orbitals, as compared to the 3d case, which at the sametime favors delocalization (larger W ) and reduces on-site electron-electroninteractions (smaller U ), thus positioning these systems away from the Mottcriterion. Following such intuitive expectations, one indeed finds an evidentsuppression of correlation effects, which is accompanied by the emergence ofcoherent charge dynamics even in undoped (i.e., stoichiometric) compounds.However, marking the difference from 3d oxides, a new important term hasto be considered for 4d materials: the spin-orbit (SO) interaction. The asso-ciated energy scale ?SO becomes increasingly important for heavier elements(with an approximate ?SO?Z4 dependence on the atomic number Z ), whichthen have to be treated within a relativistic framework. Whereas these ef-fects are largely neglected in cuprates, where ?SO(Cu2+)?20?30meV, theyare important in ruthenates and rhodates (and even more in 5d materials,as we will see later), where ?SO(Ru4+)=161meV and ?SO(Rh4+)=191meV[54]. Furthermore, in 4d systems correlation effects continue to play a role,hence these systems are commonly classified as correlated relativistic metals.ARPES results on two of the most studied 4d -based oxides, namelySr2RuO4 and Sr2RhO4 [50?53] are shown in Fig. 1.14, together with predic-tions for the Fermi surface from density functional theory in the local densityapproximation (LDA). While one indeed finds intense and sharp quasiparti-cle peaks in the energy distribution curves (EDCs) ? and consequently largevalues of Zk supporting the strongly reduced relevance of many-body corre-lations ? the matching between experimental and predicted Fermi surfacesis not perfect for Sr2RuO4 and is actually poor for the even more cova-lent Sr2RhO4. Experiments and theory are almost fully reconciled whenSO coupling is included in the single-particle methods used to describe thelow-energy electronic structure of 4d -oxides; on the other hand, the experi-mental bands still appear renormalized with respect to the calculations, by381.4. Physics of correlationsapproximately a factor of 2 similar to overdoped cuprates [39, 55], whichindicates that electronic correlations cannot be completely neglected. Thisultimately qualifies Sr2RuO4 and Sr2RhO4 as correlated relativistic metals.1.4.7 Mott criterion and spin-orbit coupling: 5d TMOsBased on the reduced correlation effects observed in 4d -oxides, a progres-sive evolution into an even less correlated physics in 5d materials would beexpected. For this reason, the discovery of an insulating state in Sr2IrO4, acompound isostructural and chemically similar to cuprates and ruthenates,came as a big surprise. The first resistivity profiles to be measured in thisiridate [56] showed an insulating behavior, as also later confirmed by opticalspectroscopy [57]. ARPES data on this material, showing the low-energydispersions of the Ir 5d -t2g states, consistently found no spectral weight atEF [see Fig. 1.15(a)]. Furthermore, and most importantly, there is a signifi-cant disagreement between experimental data and LDA(+SO) calculationsthat, as displayed in Fig. 1.15(b,c), would predict the system to be metallic,with a Fermi surface corresponding to a large Luttinger counting. This is asituation reminiscent of the 3d -oxides, where fulfillment of the Mott crite-rion would yield a correlated S=1/2 insulating state at variance with bandtheory. This novel underlying physics emerges because of the prominentrole of the SO interaction, whose strength is ?SO ? 500meV for Ir4+, andwhich now acts in concert with the other relevant energy scales (W andU ). In the atomic limit, the action of SO would splits the otherwise degen-erate t2g orbitals into two submanifolds with the total angular momentumJ2eff =L2eff +S2, and its projection Jzeff , as new quantum numbers. Withinsuch a framework, local correlations would then split the Jeff =1/2 mani-fold into lower and upper Hubbard bands thus opening a Mott gap, providedU >WJeff=1/2. This mechanism, sketched in Fig. 1.15(f,g), yields a noveltype of correlated ground state, the so-called relativistic Mott-insulator .One should note that the validity of such pseudospin-1/2 approximationis still debated, and alternative mechanisms are being discussed. Also, re-cent works have questioned the Mott-like nature of the electronic ground391.4. Physics of correlations(a) (b) (d) (e)(f) (g)t2g bandJeff = 1/2 band??SOJeff = 3/2 bandJeff = 1/2 Mott ground statewide t2g?band MetalUA BUHB ULHB S = 1/2 Mott ground state Jeff = 1/2 UHBJeff = 1/2 LHBJeff band split due to SOJeff = 3/2 band?SO?MXLDAM?X ??M ?XLDA+SO2 1 0 Binding energy (eV)?XM?(c)Figure 1.15: Spin-orbit-driven correlated states of matter.(a) ARPES data along high-symmetry directions within the first Brillouinzone (EDCs at high-symmetry points are marked in red), showing no spec-tral weight at EF (blue dashed line). (b,c): density-functional calculationswithin the LDA and LDA+SO approximations, respectively. (d,e) Possiblelow-energy scenarios in a 5d5 system: (d) U = 0, ?SO = 0, yielding an un-correlated metallic ground state; (e) U > W , ?SO = 0, yielding a S = 1/2Mott-insulating state; (f) U=0, ?SO?W , giving a spin-orbit coupled metal;(g) ?SO?W , U?WJeff=1/2, producing a Jeff =1/2 Mott-insulating groundstate (from Ref. 57).state in Sr2IrO4 and rather suggested that this system could be closer to aSlater-type (thus, non-correlated) insulator [58, 59]. In this latter case, theinsulating gap would result from the onset of long-range magnetic orderingand not strong electron correlations of the Mott type, i.e. the metal-insulatortransition would coincide with the magnetic ordering transition. In the nextand last section, we will present an unambiguous experimental realizationof relativistic Mott physics in iridates, and will highlight similarities anddifferences with respect to Sr2IrO4.40Chapter 2Surface-bulk dichotomy inBi2201Neutron and X-ray scattering experiments have provided mounting evidencefor spin and charge ordering phenomena in underdoped cuprates. Theserange from early work on stripe correlations in Nd-LSCO to the latestdiscovery of a CDW in YBCO. Both phenomena are characterized by apronounced dependence on doping, temperature, and an externally appliedmagnetic field. In this chapter, it will be shown that these electron-latticeinstabilities exhibit also a previously unrecognized bulk-surface dichotomy.Surface-sensitive electronic and structural probes uncovered a temperature-dependent evolution of the CuO2 plane band dispersion and apparent Fermipockets in underdoped Bi2201, which is directly associated with an hitherto-undetected strong temperature dependence of the incommensurate super-structure periodicity below 130K. In stark contrast, the structural modu-lation revealed by bulk-sensitive probes is temperature independent. Thesefindings point to a surface-enhanced incipient CDW instability, driven bynesting of the FS. This discovery is of critical importance in the interpre-tation of single-particle spectroscopy data and establishes the surface ofcuprates and other complex oxides as a rich field for the study of electroni-cally soft phases.41Chapter 2. Surface-bulk dichotomy in Bi2201The underdoped cuprates, with their pseudogap phenomenology [60?62]and marked departure from Fermi liquid behavior [63], have led to propos-als of a wide variety of possible phases ranging from conventional chargeand magnetic order to nematic and unconventional density wave instabili-ties [19, 64?85]. Despite the extensive theoretical and experimental effort,the generic phase behavior of the underdoped cuprates is still a matter ofheated debate, primarily because of the lack of an order parameter thatcould be universally associated with the underdoped regime of the copper-oxide high-Tc superconductors. For instance, early on evidence was ob-tained for long-range spin and charge order in the form of uniaxial stripes[19]. This phenomenology has been detected in compounds belonging tothe La2?x?y(Sr,Ba)x(Nd,Eu)yCuO4 family [19, 86?88], namely Eu-LSCO,Nd-LSCO, LBCO, and recently also pristine LSCO (where stripe order ap-pears as a near-surface effect [89]), and it is historically associated withthe family-specific reduction of superconducting Tc near 12% doping, theso-called ?1/8-anomaly?.More recently, high-field quantum oscillations [75], Hall resistance [90]and thermoelectric transport [91] results on underdoped YBCO were in-terpreted as a signature of a magnetic-field induced reconstruction of thenormal-state FS, suggesting that stripe order and/or a CDW phase might bemore general features of HTSCs? underdoped regime. Interest in this direc-tion has been burgeoning with the latest nuclear-magnetic resonance (NMR)[92], resonant X-ray scattering (REXS) and X-ray diffraction (XRD) results[16, 18, 93], providing direct evidence for a long-range incommensurate CDWin YBCO around 10-12% hole doping, which further shows a suppression forT <Tc and an enhancement with increasing magnetic field. Although thisphenomenon bears some differences with respect to charge stripes, a com-mon intriguing aspect is that they both are electronically-driven forms ofordering and appear to compete with superconductivity.If a CDW phase in underdoped cuprates is universal, it should be observ-able in compounds with similar doping levels regardless of their structuraldetails. In addition, it is of fundamental importance to connect structuralobservations (XRD, REXS) to those of electronic probes, such as ARPES42Chapter 2. Surface-bulk dichotomy in Bi2201and STM. However, for YBCO this might be prevented altogether by thepolar instability and self-doping of the (001) surface; in fact, ARPES stud-ies have not yet directly detected a folding of the electronic band structure[63, 94] carrying the signature of a symmetry-broken CDW state as other-wise seen in either quantum oscillation [75] or X-ray diffraction experiments[16, 18, 93]. To broaden the search and attempt this connection, the most in-teresting family is the one of Bi-cuprates which, owing to their extreme two-dimensionality and natural cleavage planes, have been extensively studied bysingle-particle spectroscopies [17, 95]. ARPES and STM have provided richinsight into the electronic properties of the CuO2 plane, including signaturesof broken symmetries [68, 71, 72, 74, 82, 84, 85, 96] and hints of a ?pseudo-gap phase-transition? [84], although the identification of a bona fide orderparameter has remained elusive. More specifically in regards to a poten-tially underlying CDW instability, pristine Bi-cuprates have been shown toexhibit multiple superstructures, and while some of these modulations origi-nate from the structural mismatch between BiO and CuO2 lattice planes andhence are non-electronic in origin [17, 97?100], others have been recognizedby STM to evolve strongly with doping and magnetic field [69, 70, 78, 83];however, their relationship to the ?structural? superstructures and the FShas remained unclear.In this work the structural and electronic properties of Bi2201 havebeen studied, whose crystal structure exhibits a stacking of well-spaced, sin-gle CuO2 layers in the unit cell and a highly-ordered superstructure [100],by means of surface-sensitive ARPES and LEED probes, as well as bulk-sensitive resonant (REXS) and non-resonant (XRD) X-ray diffraction. Thefocus is on the temperature dependence of the electronic structure fromunder (p ' 0.12, Tc = 15K, UD15K) to nearly optimal doping (p ' 0.16,Tc = 30K, OP30K). This allowed the discovery of a temperature depen-dent evolution of the CuO2 plane band dispersion and apparent FS pockets,which is directly associated with the evolution of the incommensurate super-structure. Surprisingly, this effect is limited to the surface (ARPES-LEED),with no corresponding temperature evolution in the bulk (XRD-REXS).The quasilinear, continuous variation of the surface modulation wavelength432.1. Orthorhombic and modulated structure of the Bi-cuprates2pi/Q2 from ? 66 to 43A?, below a characteristic TQ2 ' 130K, provides ev-idence for a surface-enhanced CDW instability, driven by the interplay ofnodal and antinodal FS nesting.2.1 Orthorhombic and modulated structure ofthe Bi-cupratesAn important aspect to consider for the study of Bi-cuprates is that thesematerials are not structurally tetragonal, but instead orthorhombic, with2 inequivalent Cu atoms per CuO2 plane [17, 97?100]. This leads to a45? degree rotated and?2??2? 1 larger unit cell, as compared to thetetragonal one, with lattice parameters a? ?= b? ?=?2?3.86A?, where 3.86A?is the planar Cu-O-Cu distance (c?= 24.9A? for both structures). Here theorthorhombic unit cell is used, with momentum axes expressed using thereciprocal lattice units (r.l.u.) 2pi/a?, 2pi/b? and 2pi/c. The orthorhombicityand consequent band backfolding have been shown to be responsible for theobservation of the so-called ?shadow bands? [99], a replica of the hole-likeCuO2 FS centered at the ? point, thus settling a longstanding debate ontheir possible antiferromagnetic origin [101]. In addition, the presence ofincommensurate superstructure modulations, arising from a slight latticemismatch between the BiO layers and the CuO2 perovskite blocks [102],further adds to the complexity of the FS. As for the single-layered Bi2201specifically, while a single Q1 superstructure vector is known to give rise toadditional folded replicas along the orthorhombic b? axis at optimal doping(OP) [17], two distinct structural modulations with Q1 and Q2 wavevectorsarise with underdoping (UD). If these complications are not fully taken intoaccount in analyzing ARPES data, the resulting highly complex FS appearsto be composed of a small set of closed pockets [100].442.2. Probing the surface with ARPES and LEEDMomentum  (2pi/b?) M8 eV10K Q210K0.50.3 0.402040fBinding energy  (meV)M8 eV100K Q2100K02040  e10 K100 KM?Q2Q2100K Q210KdIntensity  (a.u.) 10   b (pi,pi)? MQ2Q1Q1+Q2-Q2T = 10-Q1KaMQ2Q1Q1+ Q2-Q1 +Q2   T = 100K-Q1?(pi,pi)0.1 0.3 0.5 0.7- energy  (eV)cMomentum  (2pi/b?)-Q 1 M-Q 1Q 2Q 1 Q 1Q 2  T=100KUD15Kh?= 19eV++MinMaxQ 2Figure 2.1: Temperature-dependent electronic structure of Bi2201. (a,b)Sketch of one quadrant of the tetragonal BZ for T =100 and 10K, respec-tively; indicated are the expected FS belonging to the main band (M) andits replicas due to different Q1 and Q2 superstructure vector combinations(solid lines), as well as all the corresponding backfolded features due to theorthorhombicity of the crystal (dashed lines, so-called ?shadow-bands?). Thenodal strip in (a,b) highlights the region measured by ARPES with variousphoton energies and temperatures in (c,e,f), and the six bands detected forthis experimental geometry and polarization (the photon polarization is setin the plane of detection to suppress all but the main band and its replicas).MDCs at EF for 10 and 100K are directly compared in (d).2.2 Probing the surface with ARPES and LEEDThe UD15K Bi2201 ARPES data from along the nodal direction are firstpresented in Fig. 2.1. As demonstrated in previous work [100], and heresketched in Fig. 2.1(a,b) for a simpler identification of the various bands,the high crystallinity of these samples allows resolving the FS of Bi-cupratesto an unprecedented level of detail: the main (M) CuO2-plane band (blacksolid line), its Q1 and Q2 superstructure replicas stemming from the BiO-layer-induced incommensurate superstructure (red and blue solid lines), andall the corresponding backfolded bands due to the orthorhombicity of thecrystal (dashed lines). Furthermore, as shown in Fig. 2.1(c) for UD15K at452.2. Probing the surface with ARPES and LEEDT = 100K, and emphasized in the highlighted nodal strip in Fig. 2.1(a),by taking advantage of the polarization-dependent selection rules [100] onecan selectively suppress the redundant backfolded bands to highlight morecleanly the behavior of main (M) and Q1-Q2 bands. The ability to simul-taneously detect all superstructure replicas allows us to uncover ? in thetemperature dependence ? a new and unprecedented aspect of the data:while the position of the main CuO2 band is completely temperature inde-pendent, between 100 and 10K there is a significant shift in momentum ofonly (and all) the Q2-related bands [see Fig. 2.1(e) and (f), and Fig. 2.1(d)for the direct comparison between 10-100K momentum distribution curves(MDCs) at EF]. This is summarized in the 10K FS sketch of Fig. 2.1(b),which illustrates that a critical consequence of this effect is a seeming vol-ume change of all ostensible FS pockets defined by the various backfoldedbands, despite the fact that the actual number of carriers is not changing atall. The ARPES results are complemented by a detailed analysis of the su-perstructure diffraction vectors from LEED. On UD15K at 6K, rather thanindividual Bragg peaks [Fig. 2.2(a)], the experiment gives lines of Q1 and Q2fractional spots along the orthorhombic b? axis. From the fit of the LEEDdata [Fig. 2.2(c)], the magnitude of the superstructure wavevectors are foundto be Q6K1 =0.285?0.015A??1 and Q6K2 =0.142?0.015A??1, corresponding to?1/4 and 1/8 in r.l.u., respectively. Also LEED, on this highly-resolved su-perstructure, reveals a remarkable temperature dependence [Fig. 2.2(b,c)].Consistent between LEED and ARPES-MDC analysis [Fig. 2.2(d)], whileQ1 is virtually temperature independent from 5 to 300K, Q2 increases withrespect to its high-temperature value Q300K2 = 0.095?0.015A??1 (? 1/12 inr.l.u.) below a TQ2 ' 130K. The evolution of Q2 ? as seen by both elec-tronic and structural probes ? implies an inter-unit-cell structural and/orelectronic modulation, with a wavelength 2pi/Q2 evolving from 66 to 44 A?(i.e. from 12 to 8? b?) upon cooling from 130 down to 5K.462.3. Bulk-sensitivity with XRD and REXSQ2 ARPESQ2 LEEDQ1/2 LEED0 100 15050 200Temperature  (K)  legnth (?-1 ) d-1-0.500.51a-2.5 -2 -1 -0.5 0 0.5Q b ( pi/) b? 2Qa  ( pi/ )a?2Intensity (a.u.)b0.2 0.3 0.4 0.5 0.6 0.7?Q2 ?Q2CIntensity (a.u.)Qb  ( pi/ )b?2UD15KT = 6 K T = 150 KQ21230Q1T = 6K41230Q1Q2Exp. T-windowFigure 2.2: Temperature-dependent superstructure modulations in Bi2201.(a) Typical LEED pattern measured at T =6K. The white box in (a) high-lights the region shown in detail for T = 150 and 6K in (b) and (c), re-spectively. In (b,c) symbols represent the data from a vertical cut along thecenter of the box in (a), while blue and red curves are a Voigt fit of the Q1and Q2 superstructure peaks. (d) Magnitude of Q1/2 Q2 (in A??1) vs. tem-perature, as inferred from LEED and ARPES-MDC analysis at 21 eV photonenergy (and in agreement with ARPES from 7 to 41 eV, see e.g. Fig. 2.1).The yellow and red/blue boxes indicate the temperature integration windowof each data point, for ARPES (5K) and LEED (3K), respectively, whilethe error bars show the goodness of fit for the Voigt profiles determined froma chi-square test. Note that, for the almost temperature independent Q1,half of the actual value is plotted for a more direct comparison with Q2..2.3 Bulk-sensitivity with XRD and REXSThe surface sensitivity of ARPES and LEED calls for an investigation ofthe same phenomenology by means of light scattering techniques, which areknown to probe materials deeper in the bulk. In the following discussion,reciprocal space coordinates are labeled H?, K?, L (representing the re-ciprocal axes of respectively a?, b?, c), and reciprocal lattice units will beused. At all photon energies it is possible to clearly identify the supermod-ulation associated with Q1. In particular, REXS maps taken on UD15KBi2201 at the Cu, La, and O soft X-ray edges all exhibit a clear enhance-ment at this wavevector (see Sec. 2.5.2 for details). This confirms that thecorresponding modulation is present throughout the unit cell, and thereforealso in the CuO2 plane, explaining the strong folded replicas observed in472.3. Bulk-sensitivity with XRD and REXSARPES. However, modulations with longer periods are not detected. Thesecan be probed by XRD maps measured at 17 keV photon energy, thus reveal-ing a much larger portion of reciprocal space, as shown in Fig. 2.3(a-c) forT =300K. The H?-K? section in Fig. 2.3(a), which can be compared to theLEED map in Fig. 2.2(a), also features a multitude of superstructure satellitepeaks along K? (the Bragg peaks being the most intense ones). Fig. 2.3(b)displays the K?-L sections for H? =1, 2, which reveal the presence of newfeatures exhibiting a peculiar elongation along L and period-8 modulationalong K? with positions given by K?=(2n+ 1)/8. The latter are thereforeincompatible with the near period-4 modulation associated with Q1 or anyof its harmonics (also note that no similar features are found for H? = 0,thus explaining the lack of period-8 rods in soft X-ray REXS, which due tokinematic constraints can only probe a reduced portion of reciprocal space).Different orders of this period-8 modulation can be seen when zooming into the H?=2 slice in Fig. 2.3(c), with their assignment given more schemat-ically in Fig. 2.3(d). These are located at positions Qij2 =nG? iQ1 ? jQ2,where G is a reciprocal lattice vector, Q1 = 1/4 u?K? , and Q2 = 1/8 u?K?(corresponding to the ARPES and LEED low-temperature Q6K2 value). No-tably, the same features can be seen in resonant scattering at Cu and Bideeper edges (i.e., in the hard X-ray regime). Fig. 2.3(e,f) show correspond-ing K?-L sections (H?=1), taken at the Cu-K edge at low (6K) and high(120K) temperature, respectively. Additional data for the Bi-L3 edge andthe temperature-dependent XRD maps are shown in Sec. 2.5.2 . The inten-sity of the Qij2 rods is approximately 1 order of magnitude smaller than themost intense Q1 peak, within the same K?-L sections. Considering the largeprobing depth of hard X-rays, this intensity ratio is too large to identify theseas crystal truncation rods, or ascribe them to surface modulations. Theseperiod-8 spots therefore originate from an additional supermodulation whichmust be present in the bulk of the material, and characterized by poor c-axiscoherence, as the elongated structure suggests. On the other hand, thesefeatures exhibit long-range order in the a?-b? plane, as evidenced by theirwell-defined shape in H?-K? sections, with correlation lengths ? > 100? b?.To summarize the findings from XRD and REXS on UD15K, no signifi-482.4. Theoretical modeling and discussionT=300 KEph=17 keVK*- L planesH*=2H*=1K* (r.l.u.)L (r.l.u.) 261 2 34 c58 78 98 278258218 = 2n+1  8Q1BraggK* (r.l.u.)L (r.l.u.) 26 1 2 34 d16128L (r.l.u.)0.40.20 K* (r.l.u.)Cu K edgeT=6 Ke H*=116128L (r.l.u.) 0.40.20 K* (r.l.u.)T=120 Kf H*=1Cu K edge18 380612180 2 424 H* K*LX-RAY DIFFRACTION RESONANT X-RAY SCATTERING14a bQ2bulkFigure 2.3: X-ray measurements of the superstructure modulations inUD15K Bi2201. (a) 3D view of the basal planar sections of XRD mapsat Eph ? 17 keV (only positive axes are shown). (b) H?=1, 2 slices, showingthe appearance of period-8 diffraction rods, while no similar features arefound for H?=0 [see bottom plane in panel (a)], thus explaining the lack ofperiod-8 features in soft X-ray REXS. (c) Enlarged view of correspondingregion of interest in (b) for the H?=2 slice. (d) Schematic cartoon explain-ing the multiple features that are visible in (c): blue circles correspond toBragg peaks for integer K? and L orders; black crosses are the 1/4-orderQ1 peaks; red ellipses are the 1/8-order Q2 peaks. (e) REXS map acquiredat the Cu-K edge (Eph ? 8.9 keV) at 6 K, representing a K?-L plane atH?=1. (f) Same as (e), but acquired at 120 K.cant temperature dependence is observed between 300 and 6K in all scans,neither in the peak positions nor in the relative intensities. Altogether, theseresults suggest a scenario involving the presence of an additional bulk super-modulation with a well-defined periodicity along b? (? 8 lattice periods),stable over a broad range of temperatures, and characterized by large cor-relation lengths within the (001) planes but poor coherence perpendicularto them.2.4 Theoretical modeling and discussionThe combination of surface (ARPES, LEED) and bulk (XRD, REXS) sen-sitive probes has enabled us to establish that Q1 and Q2 superstructuremodulations are present both in the bulk and at the surface of underdoped492.4. Theoretical modeling and discussionBi2201, close to 1/8 doping (UD15K). In addition, an unprecedented bulk-surface dichotomy was uncovered in the temperature dependence of the su-perstructure modulations and corresponding electronic structure. While nodependence is observed for the Q1 and Q2 superstructure in the bulk andalso for Q1 at the surface, a pronounced temperature evolution is present,that is associated with the surface Qsurf2 . As for the doping dependence ofthis phenomenon, while the Q1 modulation survives all the way to optimaldoping (Q1'0.280 and 0.273A??1 for UD23K and OP30K, respectively), theQ2 modulation is substantially weakened and temperature independent forUD23K (p=0.14, Q2' 0.135A??1), and can no longer be detected in eitherLEED or ARPES on OP30K (p> 0.16). This is discussed in the Sec. 2.5.2based on the doping and temperature dependent LEED, ARPES, and X-raydata.The dependence of the Q1/Qsurf2 ratio versus temperature for UD15Kis summarized in Fig. 2.4(a) and allows some important phenomenologicalobservations: (i) the temperature dependence ofQsurf2 below TQ2 shows com-mensurability with the static Q1 modulation, as evidenced by the Q1/Qsurf2ratio varying from 3 to 2 over a range of 130K. (ii) The evolution ofQ1/Qsurf2 exhibits a possible transient lock-in behavior when the wavelengthof the Qsurf2 modulation is commensurate with the orthorhombic lattice:2pi/Qsurf2 = n ? b?, with n ranging from 12 to 8, as marked by red arrowsin Fig. 2.4(a) (see also the Supplementary Material of Ref. 103 for a moreextended discussion). A similar albeit more pronounced behavior has beenobserved for charge-stripe order in La2NiO4+? from neutron scattering [104].(iii) In analogy to what was reported for manganites [105], the continuousevolution of incommensurate wavevectors over a wide temperature rangehints at competing instabilities, which can lead to a soft electronic phase.(iv) Finally, the fact that at low-temperature (LT) also Qsurf,LT2 ' Qbulk2indicates a direct connection between the bulk and surface modulations.The details of the observed CDW instability and its temperature evolu-tion can be reproduced using a twofold analysis involving: (i) the evaluationof the electronic susceptibility [through the zero-temperature, zero-frequencyLindhard function ?(Q,?=0)] (see also Sec. 2.5.3); and (ii) a mean-field502.4. Theoretical modeling and discussionGinzburg-Landau model based on an ad-hoc phenomenological free-energyfunctional F [?], typical of that applied to CDW systems (more details canbe found in the Supplementary Material of Ref. 103. The electronic suscepti-bility ?(Q,?=0) has been calculated for various doping levels starting froman electronic structure comprised of main, shadow, and Q1-folded bands[see Fig. 2.4(b)] and is shown for p=0.12, 0.14, and 0.16 in the right-handside panel of Fig. 2.4(a). Two peaks occur in the susceptibility along theK? direction in reciprocal space at QK? =0.095 and 0.140 A??1 for p=0.12,closely matching the Q2 supermodulation vectors for UD15K. This allowsassociating Qsurf,HT2 'Q1/3 and Qsurf,LT2 'Q1/2 with nodal and antinodalFS nesting, i.e. Qsurf,HT2 = QNK? = 0.095 and Qsurf,LT2 = QANK? = 0.140A??1[these denominations designate the region in k-space where bands overlapmaximally, as pictorially shown in Fig. 2.4(c,d)]. These nesting instabilitiesare very sensitive to the hole doping, especially for the steeper nodal dis-persion; for p= 0.14 and 0.16 the nodal peak abruptly vanishes, while theantinodal is split ? and therefore ceases to be commensurate to Q1 ? andgradually reduced and broadened towards optimal doping, yielding a pro-gressively less pronounced instability. These findings qualitatively explainthe experimentally-observed progressive weakening of the features associatedwith Q2 as hole doping is increased (discussed in Sec. 2.5.2). Ultimately, thisestablishes the specific high- and low-temperature values observed for thesurface CDW modulation on UD15K,Qsurf,HT2 andQsurf,LT2 , to be associatedwith competing FS nesting instabilities of the Q1-modulated orthorhombiccrystal structure. Most importantly, this identifies the temperature depen-dent Q2 surface CDW as a phenomenon limited to the underdoped regime,near 1/8 doping, consistent with experimental observations.As for the origin of the observed temperature dependence, the evolutionof Q1/Qsurf2 (ratio of wavevector magnitude) is well captured by a phe-nomenological Ginzburg-Landau description based on the minimization ofthe surface free energy functional F [?], and is thus consistent with an in-cipient CDW instability at the surface. This is shown by the comparisonof LEED and theoretical results (red trace) for the evolution of Q1/Qsurf2in UD15K, shown in Fig. 2.4(a). More details on the underlying theoreti-512.4. Theoretical modeling and discussion?Q1p=0.12?pi-pi 0pi-pi0bpi0?M-pi 0MM+Q1daS-Q1+Q2      sur,HTS-Q1M+Q1+Q2      sur,LTc0piNodal NestingAntinodal NestingUnderlying FS320 50 100 150Temperature  (K)Q 1/Q2n = 8n = 9n = 10n = 11n = 12VB=0VB=2.5 0.511.52?(QK*,?=0)p= x103)13 14Figure 2.4: Mean-field theory and nesting effects in the charge susceptibility.(a, main panel) Temperature evolution of the Q1/Qsurf2 ratio (black dots),as inferred from the LEED data in Fig. 2.2, compared to the evolution ofthe mean-field predicted wavevector; red arrows mark those Q values atwhich the modulation associated with Q2 becomes commensurate with theunderlying orthorhombic lattice Q2 = (2pi/b?)/n, for various values of n.The colored curves illustrate the effect of increasing VB, showing how thebulk structure can pin the CDW and suppress the temperature dependenceof Q2. (a, side panel) Calculated electronic susceptibility ?(Q), cut alongthe K? direction in reciprocal space, for p=0.12 (red), 0.14 (blue) and 0.16(green). (b) Cartoon of the FS modeling used in the calculation of ?(Q);orange traces mark the FSs involved in the nesting mechanism (M+Q1 andS?Q1). (c,d) Schematics of the antinodal and nodal nesting instabilities,which connect the main (M, black trace) band with M+ Q1 and S?Q1(orange traces), respectively. The resulting nesting vectors Qsurf,LT2 andQsurf,HT2 are represented by the thick red connectors, while the correspondingQ2-derived FS are shown in dashed red.cal framework and method are provided in the Supplementary Material ofRef. 103. Commensurability to the susceptibility peaks (Q1/Qsurf2 =2 and522.5. Chapter 2 ? Appendix3) underpins the low- and high-temperature limits, while the free energyF [?], in absence of a bulk potential, provides a modeling of the surface,and accounts for the temperature dependence of the Q2 wavevector. InGinzburg-Landau mean-field theory, this can be understood as a conse-quence of the temperature dependent harmonic content of a non-sinusoidalCDW (see again Ref. 103 for more on this point), which here coincides withthe Q1/Q2 commensurability effects.In conclusion, the temperature dependent evolution of the CuO2 planeband dispersion and Q2 superstructure on the highly-ordered Bi2201 surfacecan be understood to arise from the competition between nodal and antin-odal FS nesting instabilities, which give rise to a dynamic, continuouslyevolving wavevector. This also indicates that such a remarkable electron-lattice coupling is directly related to the ordinary, static Q1 superstructure? as a necessary precursor to FS nesting at the low- and high-temperatureQsurf2 ? and giving rise to commensurability effects. Since the nodal nesting-response is very sensitive to the hole-doping, this also explains why thesurface temperature-dependence disappears towards optimal doping. Thisestablishes the importance of surface-enhanced CDW nesting instabilitiesin underdoped Bi-cuprates, and reveals a so-far undetected bulk-surface di-chotomy. The latter is responsible for many important implications, suchas the temperature-dependent volume change of all apparent FS pocketsin ARPES, and could play a hidden role in other temperature-dependentstudies.2.5 Chapter 2 ? Appendix2.5.1 Materials and methodsFor this study two underdoped (x = 0.8, p ' 0.12, UD15K and x = 0.6,p ' 0.14, UD23K) and one optimally doped (x = 0.5, p ' 0.16, OP30K)Bi2Sr2?xLaxCuO6+? single crystals were used (p is the hole doping per pla-nar copper away from half-filling). The superconducting Tc = 15, 23, and30K, respectively, were determined from in-plane resistivity and magnetic532.5. Chapter 2 ? Appendixsusceptibility measurements. For UD15K T ? ' 190K was found, based onthe onset of the deviation of the resistivity-versus-temperature curve fromthe purely linear behavior observed at high temperatures.ARPES measurements were performed at UBC with 21.2 eV photon en-ergy (HeI), and at the Elettra synchrotron BaDElPh beamline with photonenergy ranging from 7 to 41 eV. In both cases the photons were linearlypolarized and the polarization direction ? horizontal (p) or vertical (s) ?could be varied with respect to the electron emission plane. Both ARPESspectrometers are equipped with a SPECS Phoibos 150 hemispherical an-alyzer; energy and angular resolution were set to 6-10meV and 0.1?. Thesamples were aligned by conventional Laue diffraction prior to the experi-ments and then mounted with the in-plane Cu-O bonds either parallel or at45? with respect to the electron emission plane. LEED measurements wereperformed at UBC with a SPECS ErLEED 100; momentum resolution wasset to 0.01 A??1 by using a low electron energy of 37 eV, at which value thesignal intensity reaches a maximum. During the LEED measurements, thesamples were oriented with the orthorhombic b?-axis vertical in reference tothe camera, and rotated by 7 ? in the horizontal plane to detect more spots.For both LEED and ARPES, the samples were cleaved in situ at pressuresbetter than 5?10?11 torr. The detailed temperature-dependent experimentswere performed on the UBC ARPES spectrometer, which is equipped with a5-axis helium-flow cryogenic manipulator operating between 2.7 and 300K.The ARPES (LEED) data were acquired at 0.5 frame/sec (30 frame/sec),while the sample was cooled at a continuous rate of 0.1K/min (1K/min).The ARPES (LEED) data were averaged over 1,500 (5,400) images, result-ing in ARPES spectra (LEED curves) with a temperature precision of 5K(3K). The higher temperature accuracy achieved in LEED stems from itstenfold signal-to-noise ratio as compared to ARPES.Resonant elastic soft X-ray measurements were taken using a 4-circlediffractometer at the REIXS beamline at the Canadian Light Source, work-ing at the O-K (Eph ? 530 eV), La-M4,5 (Eph ? 836 eV) and Cu-L2,3(Eph ? 930 eV) absorption edges. Hard X-ray scans were performed usinga psi-8 diffractometer (8-circle) at the Mag-S beamline at BESSY, work-542.5. Chapter 2 ? Appendixing at the Cu-K (Eph ? 8.9 keV) and Bi-L3 (Eph ? 13.2 keV) deep edges.Both soft and hard X-ray scattering measurements were performed in thetemperature range 15-300K. X-ray diffraction reciprocal space maps wereacquired using an Agilent Technologies SuperNova A diffractometer. Thedata were collected at 300K and 100K using Mo-K? and Cu-K? radiation,respectively. The excitation energies used for these experiments correspondin turn to approximate attenuation lengths ? of ? 150 nm (Cu-L2,3), 6?m(Cu-K ) and 12?m (Mo-K?). In all cases samples were pre-oriented usingLaue diffraction and mounted b? and c axes in the scattering plane. In orderto expose an atomically flat (001) surface, in- and ex- situ cleaving proce-dures were adopted for soft and hard X-ray measurements, respectively.2.5.2 Experimental doping and temperature dependenceThe doping and temperature dependence of the Q2 superstructural modu-lations is shown in Fig. 2.5 for UD23K and OP30K samples. The appear-ance of the Q2 modulation in UD23K and UD15K samples is in contrastto OP30K, where only the Q1 modulation is observed in both LEED andARPES. In addition, the observation of the Q2 modulation in UD23K isalso at variance with the temperature dependence discussed in the maintext for UD15K, as the LEED and ARPES diffraction features associatedwith the Q2 wavevector remain temperature-independent in UD23K at thevalue of the bulk/low-T surface Q2 in UD15K. X-ray diffraction (XRD)and resonant scattering (REXS) data on UD15K as a function of temper-ature are displayed in Fig. 2.6. XRD maps show no qualitative differencebetween low- [T = 100K, Fig. 2.6(a,b)] and high-temperature [T = 300K,Fig. 2.6(c,d)]. Although the different acquisition parameters yield a slightlydifferent contrast and signal-to-noise ratio in the intensities, the rod-likeperiod-8 features exhibit the same pattern, showing no evolution from 300to 100K. REXS data at the Cu-K edge (8.98 keV) show no signs of temper-ature dependence, as well [see Fig. 2.6(e,f)]. The scattering map acquiredon the Bi-L3 edge (13.4 keV), plotted in Fig. 2.6(g), is qualitatively indis-tinguishable from the Cu-K edge, a signature that the associated period-8552.5. Chapter 2 ? AppendixT=120KT=15KUD24K, p= (a.u.)M+Q2MQ2kN (pi/a) kN (pi/a)OP30K, p=0.160.0IntensitykN (?2pi/a)0.40.2-0.2T=14K T=30KQ2 Q1T=224KUD23K, p=0.140.0IntensitykN (?2pi/a)0.40.2-0.2dcbaQ10.50.4 (a.u.)MOP30K, p=0.16Figure 2.5: LEED doping and temperature dependence of Bi2201, for (a)UD23K (p=0.14) and (b) OP30K (p=0.16) samples. A period-8 Q2=Q1/2modulation is present in UD23K, but temperature-independent, while no Q2modulation is observed in OP30K. (c) ARPES MDCs along the nodal cut??Y around the Fermi crossing of the main band, shown for UD23K, in thesame configuration as for the UD15K ARPES data [Fig. 2.1(d)]; ARPES,in agreement with the LEED results, provide evidence for a Q2 surfacemodulation in UD23K that is temperature independent and fixed to thebulk value. (d) Similar MDCs for OP30K show no indication of Q2-derivedbands.modulation is ubiquitous in the unit cell. The Qbulk2 rods are absent in thesoft X-ray scattering scans [Fig. 2.6(h)], consistently with the XRD resultsfor the basal plane H? = 0 [see Fig. 2.3(a) in the main text], whereas theQ1-related peak (at H? ? 0.25) exhibit a clear enhancement at all the in-vestigated absorption edges, thus reflecting the presence of a well-definednear-period-4 modulation throughout the entire unit cell.562.5. Chapter 2 ? AppendixT=300 KH*=2H*=1L (r.l.u.)L (r.l.u.) 161280.40.20 K* (r.l.u.)T=100 K0612180 2 40612180 2 4UD15K - XRD - (K*L) slicesc da bT=120 KT=6 KUD15K - REXS - (K*L) slicesCu-K edge - 8.98 keVK* (r.l.u.) Bi-L3 edge - 13.4 keV16128T=6 KefgSoft X-ray edgesH*=00.05 0.15 0.25T=18 KT=200 KLa-M5Cu-L3O-KhQ1Q2bulkQ1Q2bulkQ1Q2bulkH*=1 H*=1H*=1Figure 2.6: X-ray results on UD15K Bi2201. Left panels: K?-L sectionsof the X-ray diffraction maps, taken at T =100K, H? =1 (a), and H? =2(b) ? and T = 300K, H? = 1 (c), and H? = 2 (d). Right panels: resonanthard X-ray scattering K?-L maps (H?=1) with photon energy tuned to theCu-K edge, T =6K (e) and T =120K (f) ? and Bi-L3 edge, T =6K (g). (h)Scattering scans along K? for the soft X-ray edges: La-M5 (red, L=2.05),Cu-L3 (yellow, L= 2.05) and O-K (blue, L= 1.1), acquired at 18K (top)and 200K (bottom). Note: in (h) the intensity is plotted on a logarithmicscale, and the 18K profiles have been offset for clarity.572.5. Chapter 2 ? Appendix2.5.3 Fermi surface nestingIn addition the electronic susceptibility, or Lindhard function,?0 =?knF(k+q)? nF(k)k ? k+q, (2.1)has been calculated for p=0.12 to p=0.16, from the tight-binding fit of theFS with hopping parameters extracted from the experimental ARPES FS[100] to accurately reproduce nesting for this material. The tight-bindingmodel used for the calculation is depicted in Fig. 2.7(a) for p = 0.12, andincludes the main band and shadow band, plus their Q1 replicas. Thecalculation is performed as described in Ref. 55, and the result is shown inFig. 2.7(b). Two important nesting susceptibility peaks occur at q = 0.140and q=0.095 A??1, which closely match the low and high-temperature valuesof the Q2-wavevector measured in the experiment. The effect of doping wasincorporated by a shift of the chemical potential in the tight-binding model,thus changing the size of the FS, and resulting in the suppression of theQ1/3-peak and a gradual splitting, weakening and broadening of the Q1/2-peak [Fig. 2.7(b)]. This suggests that the Q1/3 instability only exists in anarrow range of dopings, and the Q1/2 peak in a comparatively larger rangeof dopings, near p=1/8. In particular, the Q1/2-peak comes from antinodalnesting between the main band (M) and its own Q1 replica [M+Q1, seealso Fig. 2.7(c)], while the Q1/3-peak arises from nodal nesting betweenthe main band (M) and the Q1 replica of the shadow band (S?Q1), asshown in Fig. 2.7(d). For the antinodal nesting related to the Q1/2-peak,the reduced coherence at the antinode (due to the opening of the pseudogap)does not substantially affect the Lindhard function, for two reasons: (i) themain contribution to the antinodal peak in ?(QK?) comes from the flattopology of the FS at the antinode (making the nesting quasi-1D); (ii) thelargest antinodal nesting overlap [see Fig. 2.7(c)] occurs at a portion of theFS around ?=10?, where spectral weight is still substantially coherent [here? represents the polar angle in k-space, measured from k=(pi, pi), with ?=0?corresponding to the antinode, ?=45? to the node]. This ?=10? value lies582.5. Chapter 2 ? Appendix?Q1Underlying FS Susceptibility?pi?pipipi?S-Q1+Q2sur,HT M?pi?pipipi?pi?pipipi00? (Q K*,?=0)0.06 0.10 0.14 0.18a bc dM?M+Q1+Q2sur,LTQK* (?-1)p= NestingAntinodal NestingFigure 2.7: Tight-binding fit of the experimental FS (a) for underdopedBi2201 (x= 0.8, p= 0.12). The main band and its Q1 replicas are shownin black and blue, respectively, with folded ortho-derived features (shadowbands) dashed. Nesting-susceptibility calculated from the tight-binding FS(b) is shown for p=0.12 (red), 0.14 (blue) and 0.16 (green). For p=0.12,there are two peaks near q = 0.095 and q = 0.14 A??1, corresponding tonodal and antinodal nesting, (c,d) respectively. The weaker nodal peakdisappears with increased hole-doping, while the antinodal peak splits intotwo progressively smaller and broader peaks.in the proximity of the gapless portion of the FS, the so-called ?Fermi arcs?,whose tips are located around ?=15 ? 20?.59Chapter 3Charge order driven byFermi-arc instability inBi2201A common theoretical description of the remarkably high Tc in cuprateshas been hindered by the apparent diversity of intertwining electronic or-ders out of which superconductivity emerges. In the underdoped regime,the link between these competing orders and the pseudogap regime withits ?Fermi-arc? phenomenology represents an unresolved enigma and a keyto understand these materials. Here a combined investigation of a singlecuprate family by REXS, STM, and ARPES is reported. By bringing to-gether these real and momentum space, bulk and surface probes ? and withthe aid of electronic response calculations ? charge order is quantitativelydemonstrated to emerge as a many-body effect triggered by an instabilityof the Fermi arcs, rather than from simple antinodal nesting in the single-particle limit. These converging findings suggest the existence of a universalcharge-ordered state in underdoped cuprates and demonstrate the generalrole of the fermiology in controlling the phases with which superconductivitycompetes.60Chapter 3. Charge order driven by Fermi-arc instability in Bi2201Since the discovery of copper-oxide high-temperature superconductors25 years ago, several unconventional ordering phenomena have been foundat the borderline between the strongly localized Mott insulator at zero dop-ing and the itinerant Fermi-liquid state emerging beyond optimal doping[63, 106]. In this ?underdoped? regime, the competition between varioussymmetry-broken electronic and magnetic phases ? charge and spin-densitywave order [68?71, 82, 84, 96, 107], in addition to superconductivity ? mightprevent these materials from developing a true long-range-ordered state.The origin of these incipient instabilities, as well as their connection to theunderlying pseudogap (PG) phenomenon [62], remains one of the grand chal-lenges in the field of cuprates and ? more generally ? of modern condensedmatter physics.Charge ordering in the PG state has been long known to occur at thesurface of various cuprate compounds [69?71, 96], but the recent discovery ofan incommensurate CDW in the bulk of YBCO [16, 18, 92] has revitalizedthe scientific interest in the universality of a charge-modulated phase incuprates. At first glance, the observed wavevectors might appear suggestiveof a CDW driven by the nesting of the antinodal (AN) segments of thenoninteracting FS, as previously proposed [16, 72, 78, 96]. However, in such aconventional Peierls-like CDW mechanism the onset of structural distortions(TCDW) coincides with the emergence of gaps in the electronic structure (T ?),in contrast to experimental reports that TCDW<T ? in cuprates [16, 18, 83,92].Here this puzzle is resolved by revealing that the CDW wavevector in un-derdoped cuprates corresponds to the Fermi arc tips rather than the antin-odal nesting vector - a novel mechanism that requires the opening of apseudogap, in line with experiments. For this investigation, a particularcompound ? Bi2201 [see also Table 3.1 for the investigated doping levels]? was identified as the ideal system owing to its two-dimensionality andhigh degree of crystallinity, which make it suitable for a combined studybased on surface (ARPES and STM) and bulk (REXS) probes. In partic-ular, we studied three underdoped (x=0.8, p' 0.115, UD15K and x=0.6,p' 0.13, UD22K and x=0.5, p' 0.145, UD30K) Bi2Sr2?xLaxCuO6+? sin-613.1. REXS resultsgle crystals. The superconducting Tc =15, 22 and 30K, respectively, weredetermined from magnetic susceptibility measurements. The Tc-to-dopingcorrespondence is taken from Ref. 108. While previous chapter dealt withstructural effects and found multiple supermodulations running along theorthorhombic axis b? [100, 103], here the electronic effects which may bemore intimately related to superconductivity are discussed. Here the goalis to search for charge modulations along the Cu-O bond directions usingREXS and STM, whose direct sensitivity to the behavior of the valencecharge in real- (STM) and reciprocal-space (REXS) makes them a uniqueand complementary combination.3.1 REXS resultsREXS is an X-ray scattering technique, where photons are used to exchangemomentum with the electrons and the ionic lattice, in order to gain infor-mation on the electronic charge distribution. As opposed to conventionalX-ray diffraction which is widely used for structural studies, in REXS thephoton energy is tuned to resonance with one of the element-specific ab-sorption lines. This results in a strong enhancement of the sensitivity tospecific chemical species, and allows the detection of very small variationsin the electronic density profile within the CuO2 planes [87], otherwise inac-cessible using nonresonant methods. REXS scans were performed along thetetragonal crystallographic a and b axes, with the corresponding reciprocalaxes labeled QH and QK . Due to their near-equivalence, the common no-tation Q? is used hereafter, together with reciprocal lattice units (r.l.u.) formomentum axes: 2pi/a0=2pi/b0=1, with a0'b0'3.86A? (see also Sec. 3.4.1and Fig. 3.5 and 3.6 for more details). Fig. 3.1(a) shows a series of scansat high (300K) and low temperature (20K) acquired on a UD15K samplenear the Cu-L3 absorption peak at a photon energy h? =931.5 eV. An en-hancement of scattering intensity, in the form of a broad peak, is clearlyvisible at 20K at |Q?|=0.265?0.01, while at 300K it disappears into thefeatureless background. The disappearance of the peak is most evident inthe scan for negative-Q? (Fig. 3.1(a), left profiles), while in general the signal623.1. REXS results1008060Counts/I 0-0.4 -0.2 0.0 0.2 0.4Q|| (r.l.u.) 300 K 20 KUD15Kh?=931.5 eVaCu-L3Peak area934932930Photon energy (eV) REXS  XAS 01bAbsorption (arb. units)Peak intensity (arb. units)300250200150100500Temperature (K)UD22KUD15K S1 (cool.) S1 (warm.) S5 (Q | |<0) S5 (Q | |>0) S4 (Q | |<0) S4 (Q | |>0)cUD30KT*=185KT*=240KT*=205KFigure 3.1: Momentum- and temperature-dependent REXS scans in Bi2201.(a) Low- and high-temperature scans of the in-plane momentum Q?, show-ing the emergence of two CDW peaks around QCDW'?0.27. (b) Resonanceprofile at QCDW after background subtraction (red markers), superimposedonto the Cu-L3 absorption edge (XAS). (c) Temperature dependence of theCDW-peak area for the various samples and geometries utilized (see leg-end), with shaded linear guide-to-the-eye. The pseudogap temperature T ?is from [109] (red boxes cover an associated phenomenological indeterminacyof ?10K).modulation with angle (or Q) observed in the high-temperature scans canbe most likely attributed to simple fluorescent emission (since no ostensi-ble temperature evolution is observed for T >T ?). By subtracting out thebackground fluorescence, it is possible study the dependence of such featureon photon energy, which reveals its resonant behavior at the Cu-L3 edge[Fig. 3.1(b)]. In addition, the weak dependence on the out-of-plane com-633.2. STM resultsponent of the wavevector (Q?), and the absence of features at the La-M5absorption edge (see Fig. 3.8), confirm that this feature originates from aCDW occurring in the CuO2 planes. In Fig. 3.1(c) one can follow its tem-perature evolution to find out that the onset is around TCDW ? 200K and175K, respectively, for UD15K and UD22K (see Sec. 3.4.1 and Figs. 3.9 and3.11 for a more detailed analysis). The proximity between TCDW and T ? forboth dopings suggests an intimate relationship between the CDW and thePG phase. At the same time, while it is apparent that the charge modu-lation breaks translational symmetry in some sense, the system lacks long-range order, with short correlation lengths ?CDW? 20 ? 30A? evolving onlyweakly with doping and temperature (see Fig. 3.11). Such short correlationlengths suggest either strong disorder or substantial fluctuations persistingdown to low temperatures [110]. Note how the error bars associated to thetemperature-dependent CDW intensity points in Fig. 3.1(c) are determinedfrom a chi-square fitting analysis of the REXS scans. The underlying scat-ter in the data points is instead most certainly due to an instability in thesample position with respect to the X-ray beam, which causes the latter toilluminate portions of the sample with a variable strength of the CDW state.3.2 STM resultsThe magnitude of the wavevector observed in resonant scattering is withinclose range of previous STM results on the checkerboard in Bi2201 [78],although data have been lacking in the very underdoped region. This workfeatures new scanning tunnelling microscopy and spectroscopy data on theUD15K compound. Fig. 3.2(a) shows the topographic map T (r), acquiredat 4.5K on a field of view of approximately 29?29 nm, which images thetopmost BiO layer (the natural cleavage plane in these materials). Thewell-known structural supermodulation Q1 can be already seen in the formof corrugated ripples, but is best visualized in the Fourier-transformed mapT? (q) as strong satellite spots [Fig. 3.2(b)] forming a line oriented at 45? withrespect to the a and b axes. Notably, the second supermodulation Q2 is alsoseen in T? (q), consistent with the results presented in the previous chapter.643.2. STM resultsIn the conductance map dI/dV (r, V = 24mV) [Fig. 3.2(c)], a checker-board modulation is clearly detected along a and b, over an ample rangeof bias voltages (-56 to 104mV, see also Fig. 3.12). Again, this is bestseen after taking the FT of dI/dV (r, V ) [Fig. 3.2(d)], which shows that thecheckerboard-related peaks are found at |Q?| = 0.248? 0.015 (see inset).This is in close proximity to the QCDW observed in scattering ? with aslight discrepancy that might arise from surface relaxation effects ? and alsoto the value recently reported in the context of phonon anomalies in Bi22012 nmHighLowHighLow5 nm5 nmTopography - FTdI/dV(24mV) dI/dV(24mV) - FTBragg Q1 Q2 CDWacbdQ|| (r.l.u.)0.17 0.25 0.33TopographyUD15KQK*a bb*a bb*(0,1)(1,0)Figure 3.2: Signatures of charge ordering from STM maps. (a) STM topog-raphy with atomic resolution over a 29 nm region (T =4.5K, -200mV and20 pA). The inset shows the supermodulation in higher resolution (8 nm, -1.2V, 100 pA) (b) FT of (a). Bragg, Q1 and Q2 peaks are indicated by blackcircles, blue dots, and green triangles, respectively. (c) dI/dV map at 24mVbias over a 29 nm region (9K, -200mV and 250 pA). A checkerboard mod-ulation with period ?4a0 is clearly seen in real space. (d) FT of (c). Insetshows a linecut of (d) in the range indicated by the blue rectangle, with thepeak at 0.248?0.015 corresponding to the checkerboard modulation seen in(c).653.3. Fermiology of the pseudogap state[111]. Furthermore, the features found in STM possess a correlation length?CDW?28 A?, again in agreement with REXS. A summary of the analysis onREXS and STM results is presented in Table 3.1, and shows a good agree-ment for both QCDW and ?CDW, also with previous reports for the samematerial [78].Technique Sample ParametersQCDW (r.l.u.) TCDW (K) ?CDW (A?)REXS UD15K (p'0.115) 0.265? 0.01 200? 20 26? 3UD22K (p'0.130) 0.257? 0.01 175? 5 23? 3UD30K (p'0.145) 0.243? 0.01 n/a 21? 3STM UD15K (p'0.115) 0.248? 0.015 n/a 28? 2Table 3.1: Comparative summary for the CDW peak parameters, as seenwith REXS and STM, for the various doping levels.3.3 Fermiology of the pseudogap stateThe next step is to link the universal surface and bulk charge order to thefermiology. We quantify and clarify this connection by using ARPES to mapthe Fermi surface on the same UD15K Bi2201 sample studied by REXS andSTM [100, 103]. In a similar context, the ARPES-derived ?octet model? inthe interpretation of quasiparticle scattering as detected by STM [112], is asuccessful example of such a connection, and demonstrates the importanceof low-energy particle-hole scattering processes across the ?pseudogapped?Fermi surface.From the raw ARPES data (Fig. 3.3(c) [100]), we deduce that the chargeordering wavevector connects the Fermi arc tips, not the antinodal Fermi sur-face sections as it had been previously assumed [16, 72, 78]. To better under-stand the empirical link between charge order and fermiology, we first derivethe non-interacting band structure by fitting the ARPES-measured spectralfunction Aexp(k, ?) to a tight-binding model [100, 103, 113]. The corre-sponding Fermi surface is shown in Fig. 3.3(a) for the case of p=0.12, equiv-663.3. Fermiology of the pseudogap statebMaxMinQHS=0.255ARPES - UD15Kkx (units of 1/a0)?kHSQHS=0.255APG(k,?=0) - p=0.12 kx (units of 1/a0)AN?(0,pi) (pi,pi)(0,-pi) (pi,-pi)k y (units of 1/a 0)kx (units of 1/a0)A0(k,?=0) - p=0.12QAN=0.139a b c?(k,?)Figure 3.3: Modeling the ARPES fermiology in Bi2201. Evolution of theFermi surface for hole-doping p= 0.12 from (a) the non-interacting to (b)the interacting case, via the inclusion of the self-energy ?PG(k, ?). A fur-ther Gaussian smearing (c), with ?kx = ?ky = 0.03 pi/a representing theeffective experimental resolution, allows comparison between the calculatedand measured Fermi surface from UD15K Bi2201 [100, 103]. Also shown isthe progression from antinodal (AN) nesting at QAN ? highlighted by thewhite arrow ? to the QHS-vector associated with the tips of the Fermi arcs(hot-spots, HS) ? marked by the gold connector.alent to UD15K [108]. The AN nesting, marked by the white arrow, yieldsan ordering wavevector QAN ?0.139, in disagreement with the REXS/STMaverage value QCO ? 0.256. To account for the suppression of antinodalzero-energy quasiparticle excitations ? a hallmark of the pseudogap (PG)fermiology ? we construct a model spectral function APG(k, ?) with an ap-propriate self-energy ?PG(k, ?), which combines the features found from ex-act diagonalization of the Hubbard model [114] with the doping-dependentparameters introduced in Ref. 115 (see Supplementary Note 3 for more de-tails). Figure 3.3(b) shows how the non-interacting Fermi surface is trans-formed by the action of our ?PG(k, ?), and also highlights the concurrentshift in the smallest-Q zero-energy particle-hole excitation (gold connectors).The interacting spectral function APG(k, ?) used here is tuned to optimizethe match with the corresponding ARPES data [100, 103]; after accountingfor instrumental resolution ?k, the agreement with the experimental data673.3. Fermiology of the pseudogap state0. wavevector (r.l.u.) dopingREXS - QCO Bi2201 - This workSTM - QCO Pb-Bi2201 - Ref. 7 Bi2201 - This work QAN from A0(k,?) QHS from APG(k,?) Q?el  Hubbard ModelFigure 3.4: Doping dependence of the charge order wavevector QCO as de-termined by REXS and STM on Bi2201 (this work and [78]); note thatbars represent peak widths, and not errors. Also shown are evolution ofthe Fermi surface-derived wavevectors QAN (antinodal nesting) and QHS(arc tips) measured from the ARPES spectral function A(k, ?), as well asthe doping dependent wavevector Q?el from the Hubbard-model-based elec-tronic susceptibility [114].is excellent, as shown in Fig. 3.3(c). The vector connecting the tips of theFermi arcs, called hot-spots (HS), is found to be QHS?0.255, closely match-ing the experimental values of QCO found for the UD15K sample (see alsoTable 3.1).In Fig. 3.4 we report the doping dependence of the charge-order wavevec-tor QCO as seen experimentally, as well as QAN and QHS as obtained fromthe spectral function A0(k, ?) and APG(k, ?) for the non-interacting andinteracting cases, respectively. The Tc-to-doping conversion for the experi-mental points is taken from previous studies on La-substituted Bi2201 [108];for Pb-substituted Bi2201 [78] this correspondence might be altered becausePb may contribute holes as well. The mechanism based on electron-holescattering between AN excitations, with wavevector QAN , proves to be in-683.3. Fermiology of the pseudogap stateadequate throughout the whole doping range. On the other hand, both thewavevector magnitude QHS and doping dependent-slope dQHS/dp agreewith the Bi2201 experimental data, thereby establishing a direct connectionbetween charge order and HS scattering. To gain further phenomenologicalinsights into a possible link between the ordering of the electronic densityand the available charge dynamics, we evaluate the momentum-dependentelectronic response (susceptiblity) near the Fermi surface, or ?el(Q,?) (seeSupplementary Note 3 for more details). We approximate ?el(Q,?) as aself-convolution of the single-particle Green?s function G(k, ?), in line witha similar approach successfully used in the study of magnetic excitationsin cuprates [116?118]. Despite the simplicity of our model, the results forRe{?el} along the direction of the experimental charge ordering confirmthat there is an enhancement of particle-hole scattering at a wavevectorQ?el closely following QHS (dashed red line in Fig. 3.4). This convergencesupports the idea that accounting for the empirical role played by the hotspots is of critical importance for future, more quantitative studies of theelectronic instability.The convergence between the real- and reciprocal-space techniques inour study indicates a well-defined length scale and coherence associated withthe electronically-ordered ground state. These findings on Bi2201 suggestthat the short-ranged charge correlations in Bi-based cuprates [70, 71, 78],and the longer-ranged modulations seen in Y-based [16, 18, 119, 120] andLa-based compounds [19, 87, 121], are simply different manifestations of ageneric charge-ordered state (see Ref. 122 for related findings on Bi2212).That the experimental ordering wavevectors can be reproduced through thecorrelation-induced Fermi arcs in the PG state demonstrates a quantitativelink between the single-particle fermiology and the collective response of theelectron density in the underdoped cuprates.693.4. Chapter 3 ? AppendixQHQKK* QK*Figure 3.5: Laue diffraction pattern of sample UD15K-S4. The arrows defineour convention for the reciprocal axes H and K we will refer to hereafter.3.4 Chapter 3 ? Appendix3.4.1 REXS addendumResonant elastic soft X-ray measurements (REXS) were performed: (i) atBESSY ? beamline UE46-PGM-1, using a XUV-diffractometer; and (ii) atthe Canadian Light Source ? beamline REIXS, using a 4-circle diffractome-ter. In REXS, three different UD15K crystals were studied, hereafter labeledS1, S2 and S3, and one sample each for UD22K and UD30K. The photonenergy was tuned to the La-M5 (h?=834.7 eV) and Cu-L3 (h?=931.5 eV)absorption edges. The probing scheme hinges on control of incoming polar-ization (we measured both ? and pi channels), while no outgoing polarizationanalysis was performed. Reciprocal-space scans were acquired by rocking thesample angle at fixed detector position, in the temperature range 10-300K.In all cases samples were pre-oriented using Laue diffraction and mountedwith either a or b axis in the scattering plane. Both in- and ex-situ cleavingprocedures were used, yielding consistent results.The structural symmetry of La-substituted Bi2201 crystals is character-ized by an orthorhombic distortion of the tetragonal unit cell, whose newaxes are at 45? with respect to the nearest-neighbour Cu-O bond direc-703.4. Chapter 3 ? Appendixtions (which define the a and b axes) and by the presence of long-rangeordered supermodulations along a single axis (b*). From the point of viewof symmetry, the a and b axes are equivalent, hence we had to establishan arbitrary convention to distinguish between them in the actual measure-ment. Supplementary Figure S3.5 shows a representative Laue pattern ofa Bi2201 single-crystal. Besides the lowest-order Bragg reflections, one cansee a streak with a high-density of diffracted spots, which corresponds tothe direction of the superstructural modulations Q1 (and Q2), commonlyassociated to the orthorhombic QK? reciprocal axis. The QH and QK axeshave been defined as indicated by the arrows.The scattering measurements have been performed in two different ge-ometries, which are conventionally associated to positive and negative wavevec-tors. This situation is elucidated by Fig. 3.6(a,b), for the case of Q?<0 andQ?>0, respectively. The former case corresponds to ?in>?out, and vice versafor the latter (angles are measured from the surface normal). An additionalexperimental parameter is light polarization. In the soft X-ray regime con-trol over incoming polarization is straightforward, and two geometries canbe used - ? or pi - the former referring to having the polarization perpen-dicular to the scattering plane [see Fig. 3.6(c)], while for the latter the lightpolarization vector lies in such plane [Fig. 3.6(d)]. In the same energy rangeit is much more difficult and less efficient to select the outgoing polarization,therefore for this study no polarization analysis of the scattered light wasperformed in any of the measurements. The reciprocal axes QH and QK areexperimentally found to be equivalent, in agreement with expectations basedon symmetry arguments. Along these axes, the same features are found inUD15K, i.e. a broad peak around QH =QK =0.265 (see Fig. 3.7), with verysimilar temperature dependence. Fig. 3.8 elucidates the various aspects thatmotivate the assignment of the detected peaks in REXS to a modulation ofthe electronic charge within the CuO2 planes. Panel (a) shows a series ofrocking curves (scans of the sample angle ?), projected onto the planar com-ponent of the wavevector (Q?), for different values of the detector angle 2?.A broader Q?-dependence (i.e., probing periodicities along the c axis) ofthe scattering signal near the CO wavevector (Q??0.265) is plotted in the713.4. Chapter 3 ? AppendixQ//<0QkinkoutQ//Q//>0QkinkoutQ//kinkout?kinkoutpiaba bc dFigure 3.6: Schematics of the experimental geometry in REXS measure-ments, which illustrate how the negative (a) and positive (b) sign of Q? aredefined based on the position of the exchanged wavevector Q with respectto the surface normal. (c,d) delineate the two incoming polarization geome-tries, vertical (?) and horizontal (pi), respectively. No polarization analysison the scattered beam is performed in the current study.inset panel. The presence of a peak structure at different detector angles,and the associated weak modulation along Q? implies the two-dimensionalnature of the underlying CO. Fig. 3.8(b) shows two Q?-scans, taken at theCu-L3 and La-M5 edges (photon energies were defined based on the maxi-mum of the absorption signal) at 10K. The absence of any features at theLa-edge reveals that the charge modulation is confined to the CuO2 layers.Fig. 3.8(a) shows the temperature and light polarization dependence of themomentum scans, which indicate that the CO signal is maximum when theincoming light is ?-polarized. This implies that the charge modulation orig-inates primarily from the intermediate states that can be reached in thisgeometry, i.e. Cu-dx2?y2 . Also, throughout the rest of the discussion, unlessotherwise specified, it is assumed that all reciprocal space scans have been723.4. Chapter 3 ? Appendix0. (r.l.u.)20 K300 K100500. (r.l.u.)UD15K - S5UD15K - S110 K300 KCounts/I0a bFigure 3.7: Fourfold symmetry of CDW in REXS. (a) REXS scans alongQH at low- (10K) and high-temperature (300K). (b) REXS scans along QKat 20 and 300K.performed at a fixed detector angle of 2? = 167?, and at the Cu-L3 edgeresonance. The analysis of the Q?-scans, in presence of weak CDW featureslike is the case of Bi2201 samples, is complicated by the issue of a reliablebackground subtraction. One simple method consists in subtracting out thesmoothed high-temperature profiles, in order to highlight the residual peakwhich grows as temperature is lowered through the transition temperatureTCDW. Also, based on the contrast between the low- and high-temperatureprofiles [see, e.g., Fig. 3.1(a) or Fig. 3.8(c)] the region of the CDW peak canbe identified, and therefore define a masking window. This window, indi-cated by the gray boxes in Fig. 3.9 is then used as a mask within a ?2-basednon-linear regression method used to fit the highest-temperature spectrawith a quartic (quadratic) polynomial for Q? > 0 (Q? < 0). This profileis then used as the background for all other (lower) temperatures (dashedblack curves in Fig. 3.9), and the full scattering profile (full yellow curvesin Fig. 3.9) is fitted using an unconstrained normalized Gaussian peak ontop of it: I(Q)= (1/?2piw) ? A ? exp[?1/2((Q?QCDW)/w)2]. Using this733.4. Chapter 3 ? Appendix908682780.|| (r.l.u.)Counts/I02?: UD15KCu-L320K180140100600. La-M5 (Eph=834.7 eV) Cu-L3 (Eph=931.5 eV)Q|| (r.l.u.)UD15K - S420K1401201008060-0.4 -0.2 0.0 0.2 0.4Q|| (r.l.u.)20 K300 K? piLight polarizationUD15K - S5Cu-L390858075 Q? (r.l.u.)Q||=0.27a b cFigure 3.8: Signatures of charge modulations in Cu-O plane. (a) Projectionalong Q? of REXS ?-scans (rocking curves), taken at the Cu-L3 edge fordifferent values of detector angle 2? (profiles are vertically offset for clarity).The inset shows the Q? dependence of the scattering signal at Q? = 0.27(no fluorescence subtracted, the black circle denotes the Q? value at whichall Q?-scans were performed). (b) REXS scans at the La-M5 (834.7 eV) andCu-L3 (931.5 eV) edges. (c) Dependence of the scattered intensity on theincoming light polarization (? and pi) at low (20K) and high temperature(300K), at the Cu-L3 edge. In panels (a) and (c), the gray stripes mark therange where the charge-ordering (CO) peak was observed.method one can extract from the experimental data the following parame-ters: peak area (A), position (QCDW) and linewidth (w). From the latter thefull-width-at-half-maximum can then be calculated as FWHM=2?2 ln 2 wFig. 3.10 displays the high- and low-temperature REXS scans for the differ-ent doping levels investigated, while Figs. S3.11(a,b) show the peak position(left) and FWHM (right) for the various samples and doping levels inves-tigated. The temperature axis is restricted to the range T < 100K, wherethese parameters can be more reliably extracted in virtue of a better signal-to-noise ratio. For the UD15K, note that red (gray) markers are used forQ? < 0 (Q? > 0). The bottom panels [Figs. S3.11(c,d)] report the resultsof a statistical analysis applied to the data shown in the top panels, withaveraged parameter values and error bars extracted for each dataset (graymarkers for UD15K and red markers for UD22K), and further averaged forthe UD15K samples to provide a single-valued final estimate for this dopinglevel (blue markers). Overall, FWHMs range between 0.08 to 0.1 (in recip-743.4. Chapter 3 ? Appendix1008060-0.35 -0.30 -0.25 -0.20Q|| (r.l.u.)Counts/I 012010080600. Q|| (r.l.u.)300 K Fit20 K Bckgr.UD15K - S5abFigure 3.9: Schematics of the fitting procedure applied to the REXS scans,shown on two representative datasets, for Q? < 0 (a) and Q? > 0 (b). Thevertical bars represent square-root errors used for the normalized counts.The dashed black and full yellow profiles represent the background andthe full peak+background fit result. The overlaid gray window marks themasked region, used in the estimation of the background.1008060Counts/I 0-0.4 -0.2 0.0 0.2 0.4Q|| (r.l.u.) 300 K 20 KUD15Kh?=931.5 eVa0.2 0.4 200 K 10 KOP30Kc0.2 0.4 200 K 10 KUD22KbQ|| (r.l.u.)Figure 3.10: Low- and high-temperature scattering scans for the dopinglevels investigated with REXS: (a) UD15K; (b) UD22K; (c) UD30K. Bluestars indicate the location of the CDW wavevector QCDW.rocal lattice units), which correspond in turn to correlation lengths of 6 to8 unit cells, or approximately 20-30 A?.753.4. Chapter 3 ? Appendix0. wavevector (r.l.u.)1008060402000. (r.l.u.)100806040200Temperature (K) | |<0Q | |>0UD15K - S4UD15K - S5UD15K - S10. wavevector (r.l.u.)S1 (warm)S1 (cool)S4 (Q //<0)S4 (Q //>0)S5 (Q //<0)S5 (Q //>0)UD15KUD22KOP30KAverageS1 (warm)S1 (cool)S4 (Q //<0)S4 (Q //>0)S5 (Q //<0)S5 (Q //>0)UD15KUD22KOP30KAverage FWHM (r.l.u.) 3.11: Temperature dependence of REXS wavevector and FWHM.(a) Summary of the temperature-dependent results for the CDW wavevec-tor and full-width-at-half-maximum (FWHM) for the UD22K sample, in therange T <100K. (b) Same as (a), for the various UD15K samples measured.(c) Statistical analysis of the CDW wavevector: the gray markers (bars)represent the temperature-averaged values (errors) within each dataset inUD15K, with the blue marker (and dashed line) being the overall averageover different samples and geometries; the red and green markers are thesingle points for UD22K (averaged over temperatures) and UD30K, respec-tively. (d) Same as (c), for the FWHM.763.4. Chapter 3 ? Appendix3.4.2 STM addendumScanning Tunnelling Microscopy experiments were performed on a home-built cryogenic UHV STM at 9 and 40K. STM tips were cut from Pt/Irwire and cleaned by field emission on polycrystalline Au foil prior to theexperiments. Bulk crystals were cleaved in-situ in cryogenic UHV and im-mediately inserted into the STM. Topographic images were measured inconstant current mode at a fixed sample bias of -200mV, and a tunnellingcurrent of 200-250 pA. Differential conductance spectra were measured outof tunnelling feedback at a fixed tip-sample separation using a lock-in tech-nique with an excitation voltage of 5 to 20mV and a frequency of 1.115 kHz.Spectroscopic maps were acquired over a 24-48 hour time frame.Fig. 3.12(a) shows the topographic map T (r), acquired at 4.5K on a fieldof view of approximately 29?29 nm, which images the topmost BiO layer(the natural cleavage plane in these materials). The well-known structuralsupermodulation Q1 can be already seen in the form of corrugated ripples,but is best visualized in the Fourier-transformed map T? (q) as strong satel-lite spots [see blue dots in Fig. 3.12(b)] forming a line oriented at 45? withrespect to the a and b axes. Notably, the second supermodulation Q2 isalso seen in T? (q) (red triangles), consistent with our previous findings byARPES and low-energy electron diffraction [100, 103]. The conductancemap dI/dV (r, V =24mV) and its Fourier transform [Fig. 3.12(c,d)] show acharge-modulated pattern. The latter is shown in Fig. 3.12(e) to be presentfor an ample range of bias voltages V ? 8 to 104mV ? and to be nondis-persive with V, ruling out the possibility that it arises from quasiparticlescattering, which instead would possess a distinctive dependence on V [123].3.4.3 Model Green?s function and particle hole-propagatorThe basis for the self-energy ?(k, ?) is derived from the parametrized ver-sion provided in [114], which was here extended to multiple doping val-ues than its original formulation (which was specific to the cases p = 0,0.12 and 0.24). Using this self-energy, one can then proceed to evalu-ate the single-particle propagator, or the retarded Green?s function, as:773.4. Chapter 3 ? AppendixHighLowHighLowBragg Q1 Q2dI/dV(24mV)5 nm a bb*c2 nm5 nmTopographyaa bb*dI/dV(24mV) - FTdHighLowdI/dV(-24mV)5 nm a bb*e dI/dV(-24mV) - FTfbQK*Topography - FT0.15 0.25Fourier Transform Intensity (a.u.)0.35104 mV40 mV0 mVQ|| (r.l.u)-8 mV-24 mV-40 mV-56 mV8 mV24 mV56 mV72 mV88 mVgFigure 3.12: (a) STM topography with atomic resolution over a 29 nm re-gion (T =4.5K, -200mV and 20 pA). The inset shows the supermodulationin higher resolution (8 nm, -1.2V, 100 pA) (b) Fourier transform (FT) of (a)[hollow circles indicate Bragg vectors (?1, 0) and (0,?1)]. The inset showsa linecut through the Bragg, Q1 and Q2 peaks as indicated by black circles,blue dots, and red triangles, respectively. Note that Q1 and Q2 are thestructural supermodulation peaks along the orthorhombic reciprocal axisQK? as defined in Ref. 103. (c,e) dI/dV map at respectively 24 and -24mVbias over a 29 nm region (9K, -200mV and 250 pA). A checkerboard mod-ulation with period ? 4a0 is clearly seen in real space. (d,f) SymmetrizedFT of (c,e), respectively. (g) Stack of linecuts of (d), with vertical offset forclarity, in the range indicated by the orange rectangle in (d,f), as a functionof bias voltage V and wavevector Q?, expressed in reciprocal lattice units[red traces correspond to the bias values used in (d,f)]. A charge-order peakcan be seen around Q? ? 0.25 in the full range of bias voltages spanningfrom -56 to 104mV.783.4. Chapter 3 ? AppendixG(k, ?)=(? ? barek ? ?(k, ?))?1. This particular form for ?(k, ?), obtainedby extrapolating to all momenta the exact diagonalization results on a single-band t?t??U Hubbard model, incorporates sharp features ? in the form ofa dispersing pole [see red peak in Fig. 3.13(b)] - and broader continua [seedashed green profile in Fig. 3.13(b)]: ?(k, ?)=?pole(k, ?)+?cont(k, ?). Thedispersing pole is defined as ?pole(k, ?)= |?PGk |2(? + polek )?1, where ?PGk =?PG0 (cos(kx) ? cos(ky)) and polek =2tpole(p) ? (cos(kx) + cos(ky)) + ?pole isthe reversed nearest-neighbor hopping (plus a constant offset). The disper-sion parameter of the self-energy pole tpole is taken from [115]. In general,the functional form of the pole and the continua was retained [with theonly addition of a slope component, see Fig. 3.13(b)], and slightly adjustedthe underlying parameters in order to guarantee optimal matching with theexperimental spectral functions. The underlying bare band parameters forthe dispersion in the (kx, ky)-plane are as follows: t = 0.4 eV, t?/t = ?0.2,t??/t=0.05, t???=0. The doping-dependent chemical potential ? has been de-fined using Luttinger sum rule. The overall momentum-energy map (alonghigh-symmetry directions) is shown in Fig. 3.13(a), while a stack of constant-momentum slices at high-symmetry points is plotted in Fig. 3.13(b). Notethe vanishing of the pole residue ?PGk along the nodal direction ? ? (pi, pi).In the case where ?pole=0, which is assumed in this model, the additionalline of zeros for Re G(k, ? = 0) [which arises from a diverging ?(k, ? = 0)]coincides with the AFM zone-boundary [see Fig. 3.13(c)], and is de factoresponsible for the formation of the Fermi arcs within this framework. Theso-called ?hot-spots? are also here defined as the points where the remnant(non-interacting) FS intersect the AFM zone-boundary, or equivalently asthe loci in momentum space where barek =polek =0.Such mechanism is highlighted in the inset of Fig. 3.13(d), which showsa zoom-in of the spectral function A(k, ?)=(?1/pi)ImG(k, ?) around (pi, 0).The Green?s function and the self-energy are related in such way that thequasiparticle band QPk and the pole band polek cannot cross each other.This avoided crossing drives the ?back-bending? of QPk as it approaches thecrossing between barek and polek (see white and blue curves, respectively)and causes the opening of a ?pseudogap? near the antinode. The latter also793.4. Chapter 3 ? Appendix?(pi,pi)Im? (k,?=0)840-4 Energy (t)?(pi,pi)(pi/2,pi/2)(pi,pi)(pi,0)(pi,pi/2)(pi/2,0)?=-0.01Im? (kHS,?)Momentum840-4 Energy (t)Pole Continuum840-4?kbare?kpole?(pi,pi) (pi,0) (pi,pi)MomentumEnergy (t)A(kHS,?)p=0.12-101(pi,0)Energy gaps (in units of t=0.4 eV)Hole dopingTemperature (K)?SC Tc0.20.10*?PG-4-3-2-10? (pi,pi)QPIncoherentspectral weightA(kHS,?)ad e fb cFigure 3.13: Model self-energy and Green?s function. (a) Momentum andenergy dependence of the imaginary part of the self-energy ?(k, ?), for thefull energy range used in this model. (b) Stack of constant-momentum cutsof Im ?(k, ?) at the high symmetry-points specified in panel (a). (c) Im?(k, ?=0), showing that the line of zeros for Re ?(k, ?) coincides with theantiferromagnetic zone boundary (dashed light blue line). (d) Momentumand energy dependence of the spectral function A(k, ?), with overlaid thepole in the self-energy (light blue curve) and the bare band (white curve); theinset is an enlarged view near EF and the antinode (pi, 0). (e) Nodal cut ofA(k, ?), which highlights the crossover from coherent excitations (quasipar-ticles) near the Fermi energy, onto incoherent, broad excitations at higherenergies. (f) Diagram of the energy- and temperature-scales used in thepresent model; full symbols mark the doping levels considered in the numer-ical calculations.inherits its unconventional momentum-dependence from the pole residue?PGk . Note that, when only the sharp dispersing pole is considered, thismodel coincides with the RVB-derived self-energy originally proposed in[115], in which case the derived spectral function is purely coherent, andtherefore not normalized due to the absence of the incoherent features, which803.4. Chapter 3 ? Appendixare instead observed experimentally. Such features are better seen in themain panel of Fig. 3.13(d), and appear particularly prominent around the? point, in agreement with experiments. This apparent band of incoherentexcitations along the nodal direction [shown in more detail in Fig. 3.13(e)],often termed the ?waterfall?, in this model bottoms down around 1-1.3 eV,in good agreement with experimental data on different compounds [63, 124].Fig. 3.13(f) shows the energy- and temperature-scales used in the subsequentanalysis. Most of the results only use the doping dependence of ?PG0 , whichwas calibrated by comparing the pseudogap-induced humps in the DOS tothe expected trend for the pseudogap magnitude [62]. The flexibility of thismodel allows to achieve an excellent agreement with the experimentally de-termined spectral functions, as is shown in Fig. 3.14(a,b) for the comparisonbetween a calculated FS and the corresponding experimental map in un-derdoped YBCO [63] and La-substituted Bi2201 [100], respectively. Withinthe framework of Fermi-arcs, the hot-spots are defined as those points in thefirst BZ where zero-energy excitations become gapped out; they are markedby the black circles in Fig. 3.14(c), with the underlying intensity map repre-senting a calculated FS for p=0.12. Fig. 3.14(d) schematizes the mechanism:zero-energy electronic excitations lying on the non-interacting FS (dashedred line) are pushed away from the Fermi level between the antinode (AN)and the HS (the full blue line is the pseudogap profile ?PGk ). As a conse-quence, the band topology is altered due to the PG-induced band-bending[see also Fig. 3.13(d)] and the pile-up of the local (in momentum-space) DOSmigrates from the AN to the HS. This behavior can be quantitatively cap-tured by computing the cumulative DOS ??(kx), defined as follows:??(kx)=? ?PG??PGd?? 2pi/a?2pi/adkyA(kx, ky, ?) (3.1)which is plotted in Fig. 3.14(e) for the case with (full blue line) and without(dashed red line) the PG. It is clear that when the latter is turned on, asingle peak (at the AN) in ??(kx) evolves into a profile with two smaller peaksnear the HS, thus setting the ground for enhanced electron-hole scattering813.4. Chapter 3 ? AppendixBi2201 - p~0.12YBCO - p=0.06?(0,pi)?ExperimentModel(pi,0)(pi,pi)(0,pi)(pi,0)(pi,pi)MaxMin?(-pi,0) (pi,0)?PGA(k,?=0)A(k,EF)HSANabcdeFigure 3.14: Fermi-arcs: experiment vs. theory, and modified DOS. (a)Comparison between experimental (YBCO, p=0.06, from [63]) and calcu-lated FS; a Gaussian convolution has been applied to the calculated mapto mimic momentum-resolution effects. (b) Same as (A), for the case ofUD La-substituted Bi2201 (p ? 0.12, from [100]). (c) calculated FS forp=0.12, with black circles marking the location of the hot-spots (HS). (d)Energetics of lowest-energy excitations near the FS in presence of the pseu-dogap; the blue curve is the k-dependent PG, the red dashed profile is thenon-interacting FS. (e) Cumulative densities of states ??(kx) for the non-interacting (red dashed) and interacting case (full blue); the kx-positions ofenhanced cumulative DOS are marked with circles.between these momentum points.With this phenomenological self-energy in hand, the next step is to calculatethe full Green?s function as a function of doping. The corresponding FSsand momentum-energy maps (for p=0.05, 0.08, 0.12 and 0.2) are shown inFig. 3.16(a1-a4,b1-b4). The panels c1-c4 are the same as b1-b4, but zoomedin energy around the Fermi energy. The associated doping-dependent den-sities of states (DOS) are obtained by tracing out the spectral functionover momenta: ?(?) =?dk A(k, ?). The resulting profiles are plottedin Fig. 3.17(a), showing the progressive opening of the pseudogap as hole-823.4. Chapter 3 ? Appendixdoping is decreased, in the form of two side-humps around EF .We made use of our phenomenological Green?s function to calculate the low-energy electronic response in Fourier space, as encoded in the particle-holepropagator G2(Q, i?n) (?n denotes a bosonic frequency). This quantity,which is generally referred to as the electronic susceptibility ?el, and re-duces to the Lindhard function for a non-interacting system, is defined asthe retarded charge-charge correlation function:?el(Q, ?) ?? ?0d?ei?n? ?T? (?el(Q, ?)?el(?Q, 0))? (3.2)=?k,k?,?,???T? (c?k,?(?) ck+Q,?(?) c?k,??ck?Q,??)? (3.3)where T? is the imaginary-time ordering operator, and the Fourier-transformedelectronic density operator ?el(Q, ?) has been expanded in terms of chargecreation and annihilation operators c and c?. We now adopt a generalizationof a Lindhard function (independent particle) approach to the interactingproblem, which consists of rewriting Eq. 3.3 as a self-convolution of the in-teracting Green?s function G(k, ?). This approach neglects the correlationbetween the particular electron-(k, ?) and hole-(k + Q, ?) state, but re-tains the effects of correlations with all other states, which enter throughthe single-particle self-energy ?PG(k, ?) [in analogy to similar approachesin optical spectroscopy [125, 126]]. This procedure has been previously usedfor the study of magnetic excitations in cuprates [116?118]. In more generalcontexts, the real part Re{?el} is in some cases indicative of the electroniccontribution to instabilities of the ionic lattice or of the charge density [127].In reality, a complete assessment of the system?s tendency toward orderingphenomena requires the detailed evaluation of all the coupling terms betweenthe different interconnected degrees of freedom, in primis electron-phononcoupling [111, 128, 129] and exchange interactions [117, 130].Our approach is therefore based on expressing the susceptibility as fol-833.4. Chapter 3 ? Appendixlows:?el(Q, ?) ??k,?G(k+Q, ?, ?)G(k, ?, ?) (3.4)with G(k, ?, ?) = ?T? (ck,?(?) c?k,?)?.This is now equivalent to approximating the electronic susceptiblity withthe particle-hole bubble diagram of the full Green?s function G. By Fourier-transforming Eq. 3.4 into the frequency-domain, we ultimately arrive to aform of ?el(Q, i?n) which is conveniently expressed as a frequency-momentumautocorrelation of G:?el(Q, i?n) ?1??k,i?m,?G(k+Q, i?m + i?n, ?) G(k, i?m, ?) (3.5)where we have here defined ? = (kT )?1, with ?n = 2npi/? and ?m =(2m+1)pi/? representing the bosonic and fermionic Matsubara frequencies,respectively. The summation in Eq. 3.5 is then converted to an integrationon the real-frequency axis (analytic continuation) using the substitution:i?m ? ? + i? (hereafter ?=0.01, unless otherwise specified).Analytic continuation onto the real-frequency axis was performed, andan FFT-based algorithm was utilized for fast-computation of the correlationfunctions; unless otherwise specified, grids of 256 ? 256 k- and Q-pointshave been used. The (kx,ky)-momentum ranges were set between ?pi/a andpi/a, and periodical boundary conditions (PBCs) have been imposed in thecorrelation to avoid boundary effects. Values of ?=?0.01 and ??=0.005(both in units of t=0.4 eV) have been used to guarantee a proper sampling ofthe energy axis, which is crucial especially near EF where excitations becomemore coherent, and the related features in G(k, ?) get sharper. The validityand applicability of the numerical code has been benchmark-tested on thetwo-dimensional electron gas (2DEG), whose analytic solution is known.The zero-temperature, zero-frequency electronic susceptibility ?T=0el (Q,?=0) for p=0.12 is plotted in Fig. 3.15(a), with overlaid the cut along QH (lightblue trace). The shape of this profile is determined by two contributions: (i)a peak at Q?el corresponding to enhanced particle-hole scattering between843.4. Chapter 3 ? Appendix454035300.40.20 0.18 0.20 0.05 0.06 0.08 0.10 0.11 0.12 0.14 0.16? (Q||=0) (eV-1 x103 )p:Q|| (r.l.u.)b Q*?el(Q,?=0)p=0.12QH (r.l.u.) 0.500.5QK (r..l.u.)Q*50454035aeV -1 x103Figure 3.15: (a) Zero-frequency, zero-temperature electronic susceptibilityfor p = 0.12; overlaid is the cut along QH (full blue line), and the localmaximum at Q?el?0.26 highlighted. (b) Stack of doping-dependent ?el(Q?)profiles, with red and gray guides-to-the-eye tracking the contributions fromscattering between HS and AN, respectively.the hot-spots; and (ii) particle-hole excitations across the PG from the ANregions, which populate the ?hump? at lower Q values. This profile showsa local maximum at Q?el = 0.252, which closely matches the experimentalQCO observed in Bi2201 (UD15K) and the QHS vector introduced before.In addition, two-dimensional maps of the electronic susceptibility have beencalculated for a series of different doping levels, and the corresponding cutsalong Q? are plotted in Fig. 3.15(b). The doping evolution of the HS and ANcomponents is indicated by, respectively, the red and gray guides overlaidon the plot.Figs. S3.16(d1-d4) display the two-dimensional maps of the zero-frequency,zero-temperature electronic susceptibility as a function of momentum, fordifferent doping levels. Overlaid on top of the color maps are the profilesalong QH calculated in different configurations: (i) ?(Q?,? = 0) from thefull Green?s function (full light blue profile); (ii) ?coh(Q?,?=0) from the co-853.4. Chapter 3 ? Appendix?=-0.05?kbare0.080.120.20?(pi,pi)0.05?=-0.05?k=0.05 .2pi/ap A(k,?=0) A(kHS,?) A(kHS,?) - near EF86420-2-43210-1Energy (eV)Energy (t)?Momentum(pi,pi) (pi,pi)(pi,0)-101-0.400.4Momentum?(pi,pi) (pi,pi)(pi,0)? (Q,?=0)?=-0.01T=0FullCoherentNoninteractingQ K (r.l.u.)0.500.5QH (r.l.u.)MaxMinMaxMin50454035eV-1x103 a1a2a3a4b1b2b3b4c1c2c3c4d1d2d3d4Figure 3.16: Doping-dependent spectral function and electronic response.(a1-a4) FSs for different hole-doping levels (p=0.05, 0.08, 0.12 and 0.2), ob-tained as a constant-energy slice of the spectra function at the Fermi energyA(k, ? = 0), followed by convolution to an isotropic momentum-resolutionGaussian function with ?kx =?ky =0.05 pi/a. (b1-b4) Momentum-energymaps of A(k, ?) along high-symmetry directions within the first BZ (thewhite curves represent the bare band dispersion). (c1-c4) Same as (b1-b4),but zoomed in around EF . (d1-d4) Resulting zero-frequency (? = 0) sus-ceptibility profiles, calculated as the particle-hole bubble of the full Green?sfunction G(k, ?), at T =0. Overlaid are Q? cuts of ?el(Q,?=0) for differentunderlying Green?s functions: (i) full (light blue full line); (ii) coherent partonly (dashed white); (iii) noninteracting (dotted-dashed red).863.4. Chapter 3 ? Appendix10.5DOS (states/eV)-0.4 0 0.4Energy (t)-0.2 0 0.2Energy (eV) 0.05 0.06 0.08 0.10 0.11 0.12 0.13 0.14 0.16 0.18 0.20p:?=-0.054037. 0.1 0.09 0.08 0.07 0.06 0.045 0.03 0?SC:p=0.12?(Q|| ) (eV -1 x10 3)0.25Q|| (r.l.u.) 0.30.2 0 K 40 K  80 K  120 K 160 KT:40p=0.12?=-0.05Q|| (r.l.u.)40350. Q|| (r.l.u.) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09  0.1  0.125 0.15  0.175 0.2?:p=0.12 ?(Q|| ) (eV -1 x10 3)a b c dFigure 3.17: Electronic response and related control parameters. (a) Doping-dependent DOS. (b,c,d) Dependence of ?el on superconducting gap ?SC (b),temperature T (c), and scattering rate ? (d), for p=0.12.herent part of Green?s function only (dashed white); (iii) ?nonint(Q?,?=0)from the noninteracting Green?s function (dash-dotted red). For illustra-tive convenience, the various profiles are normalized to their minimum andmaximum; also note that a proper normalization of ?coh(Q?,?= 0) is notpossible, due to the missing incoherent contributions. These susceptibilityprofiles reflect important aspects of the underlying fermiology. The peaksin ?coh(Q?,? = 0) are mainly controlled by scattering between the hot-spots, while the non-interacting susceptibility has a quasi-1D divergence atlower momenta, which is driven by nesting at the antinodes. As it can beseen at all doping levels, the full susceptibility bears both tendencies, withthe antinodal scattering being progressively suppressed as hole-doping is re-duced, and correspondingly the pseudogap increased. The presence of someremnant contribution from the antinodes is not surprising, as all electronicstates contribute, and not only the zero-energy single-particle excitations atthe FS. Nonetheless, the contribution from lower-lying states decreases as1/E and is therefore less prominent, in agreement with the observed dopingdependence of the (pseudo)gapped antinodal component in ?(Q?,? = 0).Also, the incoherent spectral weight in G(k, ?) contributes to the slowly-varying (with Q?) background in ?, in contrast with the sharper profiles of?coh and ?nonint.This framework and machinery allows to calculate the dependence on var-ious other control parameters. In principle it is possible to extend the cal-873.4. Chapter 3 ? Appendixculated Green?s function to the case where a superconducting ground stateis present. In the simplest approximation, this requires adding the follow-ing term to the self-energy: ?SC(k, ?) = |?SCk |2(? + barek +?PG)?1, where?SCk is the d-wave superconducting order parameter, and barek is the fullbare band dispersion. The total self-energy then reads ?=?PG + ?SC. Atthe level of susceptibility, this introduces an additional diagram besides theparticle-hole bubble, which arises from the self-correlation of the anomalouspropagator F (k, ?) (see [115] and [131] for more details on the subject).The resulting profiles for p=0.12, as a function of the superconducting gap?SC0 , are shown in Fig. 3.17(b). As it can be seen, near the wavevector Q?that maximizes ?(Q?,? = 0), there is hardly any dependence on ?SC0 ? amaximum reduction of ?(Q?,?=0) of 0.5% between the profiles for ?SC0 =0and 0.1 is seen to occur.Temperature-dependent suceptibility profiles are shown in Fig. 3.17(g). Astemperature is increased, the peak at Q?el is progressively reduced andbroadened, whereas the antinodal component at lower Q rises due to thermally-enhanced access to excitations around the antinode. The weak temperaturedependence of Q?el vs. temperature is a direct consequence of our model,which requires the underlying self-energy ?PG ? and therefore the spectralfunction ? to be temperature-independent. This assumption implies, in par-ticular, that the poles of the spectral function, and hence also the hot-spots,will not change location as a function of temperature. This is not incon-sistent with the empirical observation of temperature-dependent Fermi-arclength, which was recently explained to arise from the variation in temper-ature of a single parameter (the scattering rate ?), without needing the restof the bandstructure parameters (t, t?, t??, tpole, and ?PG) to change withtemperature [132].Lastly, the dependence of ?(Q?,? = 0) on the single-particle inverse life-time, or scattering rate ? (which is introduced through the imaginary partof the complex frequency using the substitution ? ? ? + i?), shown inFig. 3.17(h) reveals that the ?-induced broadening of the Green?s functiondrives a similar broadening in the main features of the susceptibility, in afashion which resembles the one observed with temperature. This might883.4. Chapter 3 ? Appendixsuggest that susceptibility-driven electronic instabilities are potentially sup-pressed by incipient disorder and/or impurity scattering.89Chapter 4Na2IrO3 as a NovelRelativistic Mott InsulatorThe study of the low-energy electronic structure of Na2IrO3 by ARPES,angle-integrated PES with in-situ potassium doping, optics, and DFT-LDAcalculations, is here reported. The narrow bandwidth of the Ir-5d mani-fold observed in ARPES highlights the importance of SOC coupling andstructural distortions in driving the system towards a Mott transition. Inaddition, at variance with a Slater-type description, the gap is already openat 300K and does not show significant temperature dependence even acrossTN '15K. From the potassium-induced chemical potential shift and comple-mentary optical conductivity measurements, the insulating gap is estimatedto be ?gap'340meV. While LDA+SOC already returns a depletion of theDOS at EF , this only corresponds to a ?zero-gap?. The observed 340meVgap value can be reproduced only in LDA+SOC+U calculations, i.e. withthe inclusion of both SOC and U (with U ' 3 eV). This demonstrates that,while the prodromes of an underlying insulating state are already found withSOC alone, on-site Coulomb interaction is necessary to explain the chargegap and the local spin, thus establishing Na2IrO3 as a novel relativistic Mottinsulator, in which Coulomb and relativistic effects have to be treated on anequal footing.90Chapter 4. Na2IrO3 as a Novel Relativistic Mott InsulatorThe proposal of an effective Jeff =1/2 Mott-Hubbard state in Sr2IrO4[133] came as a surprise since this case departs from the established phe-nomenology of Mott-insulating behavior in the canonical early 3d transition-metal oxides. There, the localized nature of the 3d valence electrons is respon-sible for the small bandwidth W , large Coulomb repulsion U , and suppres-sion of charge fluctuations [5, 134]. In particular, Sr2IrO4 appears to violatethe U >W Mott criterion, which for the very delocalized 5d Ir electrons isnot fulfilled. It was proposed that the strong spin-orbit (SOC) interactionin 5d systems (?SOC'485meV for Ir [135]) might lead to instability againstweak electron-electron correlation effects, and to the subsequent emergenceof a many-body insulating ground state [133]. However, the strong-SOC limitJeff =1/2 ground-state scenario has recently been put into question [136],and theoretical [137] and time-resolved optical studies [138] suggest that theinsulating state of Sr2IrO4 might be closer to a Slater than a Mott type: aband-like insulating state induced by the onset of AFM ordering and con-sequent band folding at TN ' 240K (Slater), as opposed to be driven byelectron-correlations with an insulating gap already open at temperatureswell above TN (Mott).Despite intense experimental and theoretical effort, the nature of the in-sulating state in the 5d iridates remains highly controversial. This is reminis-cent of the situation in 3d oxides, for which the Mott versus band-insulatordebate has lasted over four decades [134, 139?141]. For instance, in thecase of the prototypical AFM insulator NiO, this debate was conclusivelyresolved only after the correlated nature of the insulating state was estab-lished based on: (i ) the magnitude of the gap as measured by direct andinverse photoelectron spectroscopy (PES/IPES) [142], much larger than ex-pected from DFT [139]; (ii ) its persistence well above the Ne?el temperatureTN [143]; and (iii ) the detailed comparison between DMFT results [144] andmomentum-resolved electronic structure as measured by ARPES [143, 145].To address the nature of the insulating state in iridates, including therole of many-body electron correlations for their extended 5d orbitals andthe delicate interplay between W , U and SOC energy scales, a particularlyinteresting system is the newly discovered AFM insulator Na2IrO3 [146].914.1. ARPES resultsStarting from a Jeff = 1/2 model in analogy with Sr2IrO4, this systemwas predicted to exhibit quantum spin Hall behavior, and was considered apotential candidate for a topologically insulating state [147]. Further theo-retical [148, 149] and experimental [150] work emphasized the relevance ofstructural distortions, which lower the local symmetry at the Ir site fromoctahedral (Oh) to trigonal (D3h). Together with the structure comprisedof edge-sharing IrO6 octahedra, this leads to an effective bandwidth for theIr 5d-t2g manifold of ?1 eV. This potentially puts Na2IrO3 closer than otheriridates to the U ?W Mott criterion borderline ? and thus to a Mott in-sulating phase [148]. Most importantly, its lower TN ' 15K provides theopportunity of studying the electronic structure well above the long-rangeAFM ordering temperature and ? with the aid of novel DFT calculations ?establishing the nature of its insulating behavior.4.1 ARPES resultsThe 130K angle-integrated PES spectrum (see Sec. 4.4.1 for more details)in Fig. 4.1(a) shows two broad spectral features belonging to the Ir 5d-t2gbands (0-3 eV binding energy), and to the O 2p manifold (beyond 3 eV).The insulating character is evidenced by the lack of spectral weight at thechemical potential, which appears to be pinned to the top of the valenceband (no temperature dependence is observed in the 130-250K range, seeSec. 4.4.3). EDCs measured by ARPES alongM???K for the Ir 5d-t2g bandsare shown in Fig. 4.1(b). The detected features are only weakly dispersingin energy, with the most obvious momentum-dependence being limited totheir relative intensity. The electronic dispersion can be estimated from thenegative second derivative map ??2I(k, E)/?E2, and shown in Fig. 4.1(c).For a more quantitative analysis of the experimental data, EDCs were fittedusing 4 Gaussian components [Fig. 4.1(d)], plus a Shirley-type backgroundin order to account for secondary electrons (dashed black line). At leastfour peaks are found to be necessary to fit the dataset over the full mo-mentum range, and that also matches the number of DOS features fromDFT [Fig. 4.4(e,f)]. The direct comparison of fit (black diamonds) and sec-924.1. ARPES resultsIr 5d - t2gBinding energy (eV)123 0KM?MK?(b)2 134Binding energy (eV)123 0?2 134(d)KM?(c)246 0Ir 5d - t2gO 2pNa2IrO3T=130 Kh?=21.2 eV(a)MaxMinO 2pFigure 4.1: ARPES results on fresh-cleaved surfaces of Na2IrO3. (a) Angle-integrated O and Ir valence-band photoemission spectrum of Na2IrO3; thegrey portion is shown in detail in (b-d). (b) ARPES EDCs for the Ir 5d -t2gbands from along M???K. (c) Negative second derivative of ARPES maphighlighting the experimental dispersion; superimposed (black diamonds)are the fit analysis results from (d). (d) Model fit of the ?-point EDC with4 Gaussian peaks for the Ir valence bands (VBs), and a Shirley background[151]: 4 peaks are necessary to fit the dataset over the full momentumrange [matching the number of DOS features from DFT in Fig. 4.4(e,f)],with Gaussian lineshapes yielding better agreement than Lorentzians. Theaverage over M ???K gives (EV B,?V B,?EV B) in eV, for peak 1 to 4:(0.50,0.30,0.06); (0.94,0.39,0.08); (1.39,0.44,0.09); (1.89,0.43,0.07).ond derivative results in Fig. 4.1(c) yields a good overall agreement in thedispersion of the 4 features (small deviations stem from the peaks? relativeintensity variation, which is differently captured by the two methods). TheIr 5d-t2g valence band (VB) dispersions do not exceed ?EV B ? 100meV934.1. ARPES resultsin bandwidth ? at variance with the generally expected larger hopping am-plitude for 5d-t2g states. Further remarkable aspects of the Ir t2g bandsare: (i) their linewidth, with values ?V B =?2?V B ? 300?450meV; (ii)the Gaussian lineshape, which overall yields a better agreement than simpleLorentzian. A possible origin might be many-body electron correlation ef-fects as discussed for NiO [144], and strong electron-phonon coupling leadingto polaronic behavior in the spectral function [152].The results in Fig. 4.1 already provide one very important clue: thegap is open well above TN ' 15K, which directly excludes a Slater-type,magnetic-order-driven nature for the insulating state. As for the size of thegap, this cannot be readily identified by ARPES since photoemission canlocate the valence band, as the first electron-removal state, but not the CBwhich belongs to the electron-addition part of the spectral function [151].Alternatively, one can measure the gap in an optical experiment; however,one needs to discriminate between in-gap states of bosonic character (e.g.,phonons, magnons, excitons) and those particle-hole excitations which in-stead determine the real charge gap. This complication can often hinder theprecise identification of the gap edge [153]. Such complexity underlies thepast controversy on NiO: while the 0.3 eV gap obtained by DFT [139] wasdeemed consistent with optical experiments [154], the combination of directand inverse PES revealed the actual gap value to be 4.3 eV [142]. This iswell beyond the DFT estimate and establishes NiO as a correlated insulator.Here, for the most conclusive determination of the insulating gap mag-nitude, angle-integrated PES is used in combination with in-situ doping bypotassium deposition, as well as optics. A quantitative agreement betweenthe two probes would provide validation against possible artifacts stem-ming, e.g., from excitonic contamination in optics and/or surface sensitivityin PES, including effects specific to the substrate-adsorbate system in PESwith in-situ doping.To estimate the energy of the first electron-addition states and the DOSgap between the VB and CB, electrons are doped across the gap by in situpotassium deposition on the cleaved surfaces, and then follow the shift in944.1. ARPES resultsBinding energy (eV)123 0(a)K-evap:0 sec15 sec90 sec30 sec3 Rescaled energy (eV)012(b)K evaporation (sec)0 100 200?? (eV) (eV)0 0.1 0.2 0.3Sample 1Sample 2Sample 2 Sample 2(c) ?PES  ~ 340 meV50 secFigure 4.2: ARPES results on K-covered surfaces of Na2IrO3. (a)Background-subtracted angle-integrated EDCs for selected values of K-exposure [see colored markers in (c)]; the chemical potential shift ?? isrevealed by the motion of the high binding-energy trailing edge. (b) Sameas in (a), but shifted by the corresponding ??. (c) ?? vs. K-depositiontime for two different samples; in the inset, K-induced low-energy spectralweight ?SW as a function of ??, for Sample 2 only. In panel (c) data anderror bars are estimated from the comprehensive analysis of both O and Irtrailing edges and peak positions (see Sec. 4.4.4); also note that blue and redcurves in (c) and its inset are both a guide-to-the-eye.chemical potential by angle-integrated PES. The results are summarized inFig. 4.2 for K-evaporation performed at 130K on two different freshly cleavedsurfaces (see Sec. 4.4.4). The most evident effect is the shift towards higherbinding energy of both Ir and O valence bands, as shown in Fig. 4.2(a) forthe Ir 5d-t2g manifold, as shown in Sec. 4.4.4. This arises from the (equaland opposite) shift of the chemical potential ?? when electrons donatedby potassium are doped into the system; after an initial rapid increase,954.2. Optical conductivity?? saturates at ? 340meV [Fig. 4.2(c)]. When the K-deposited spectraare shifted in energy by the corresponding ?? so that their high binding-energy trailing edges match the one of the fresh surface [Fig. 4.2(b)], onecan observe the emergence of additional spectral weight (SW ) in the regionclose to and above EF . This low-energy K-induced spectral weight, ?SW ,can be computed as:?SW =?dk? E+F?1eVdE [I(k,E, xK+)? I(k,E, 0)] , (4.1)where I(k, ?) is the PES intensity, xK+ represents the K-induced surfacedoping, and E+F is the Fermi energy of the K-doped surface, which movesprogressively beyond the undoped-surface EF . The evolution of ?SW plot-ted versus ?? in the inset of Fig. 4.2(c) evidences an approximately linearSW increase up to ??'200? 250meV, followed by a steeper rise once thesaturation value ??' 340meV is being approached. This behavior can beunderstood as due to the initial filling in of in-gap defect states ? either pre-existing or induced by K deposition ? which makes the jump of the chemicalpotential not as sudden as for a clean insulating DOS. Only when electronicstates belonging to the Ir 5d-t2g CB are reached one observes the saturationof ?? and the more pronounced increase in ?SW . This combined evolu-tion of chemical potential shift and spectral weight increase points to a DOSinsulating gap ?PES'340meV.4.2 Optical conductivityThe optical conductivity data (measured as outlined in Sec. 4.4.1), shown inFig. 4.3, reveal an insulating behavior with an absorption edge starting at300-400meV at 300K, with negligible temperature dependence down to 8Kand thus also across TN ' 15K (see inset). The experimental data are fitusing a joint DOS with Gaussian peaks for the CB and each of the 4 VBs964.2. Optical conductivity(see caption of Fig. 4.1):J(E) ?4?i=1?dE?AiGCB(E? + E)GVBi(E?) . (4.2)Here the prefactors Ai represent the optical transition strengths, with theband index i running over the 4 VB features extracted from the ARPES datain Fig. 4.1, and are left free. J(E) provides an excellent fit to the opticaldata in Fig. 4.3, and a least-squares analysis returns ECB'680meV for thelocation of the CB above EF , with a width ?CB=?2?CB'160meV (a valuein agreement with DFT, as shown later). The consistency of the combinedPES-optical conductivity analysis is confirmed by the optical gap obtainedfrom the onset of the simulated CB, ?OPT = EonsetCB . Following [152], thelatter is estimated as EonsetCB = ECB?3?CB, leading to ?OPT ' 340meV.Photon energy (eV)0 0.5 1.0 1.5 2.0 2.5? 1(?) (??1 cm?1)0500100015002O 2p134T=300 K0 0.5 eVphononT=250 KT=8 K?OPT ~ 340 meVFigure 4.3: Optical absorption in Na2IrO3. Optical conductivity data (redline), together with the simulated Ir 5d-t2g joint particle-hole DOS (blackline), and its individual components from the simultaneous fit of ARPESand optical data [colors and labels are consistent with the valence bandfeatures in Fig. 4.1(c), which represent the initial states of the lowest-energyoptical transitions]. Inset: temperature dependence of the gap edge.974.3. Density functional theoryThis matches ?PES from the K-induced ?? saturation in PES, providing adefinitive estimate for the insulating gap, ?gap'340meV.4.3 Density functional theoryWith a gap much smaller than in typical Mott insulators, discriminatingbetween correlated and band-like insulating behavior in Na2IrO3 requires adetailed comparative DFT analysis. The corresponding framework is out-lined in more detail in Sec. 4.4.2. Unlike the case of Sr2IrO4, the t2g degener-acy and bandwidth in Na2IrO3 are affected by structural distortions and thepresence of Na in the Ir plane. This is revealed by calculations performedfor a distortion- and Na-free hypothetical IrO2 parent compound: while inIrO2 the individual t2g-dispersions are as wide as ?1.6 eV, the Na-inducedband folding in Na2IrO3 opens large band gaps, leading to much narrowert2g-subbands (? 100meV). Even though this accounts well for the narrowbandwidth observed in ARPES, the material is still metallic in LDA with ahigh DOS at the Fermi level [Fig. 4.4(a,d)], at variance with the experimentalfindings. Remarkably, when SOC is switched on in LDA+SOC a clear gapopens up at EF in the kz=0 band dispersion [Fig. 4.4(b)], although not in theDOS where only a zero-gap can be observed [Fig. 4.4(e)]. A closer inspectionof the LDA+SOC dispersion for first occupied and unoccupied bands ? ver-sus both k? and kz in Fig. 4.4(g) ? reveals that the lack of a DOS gap stemsfrom the overlap of VB and CB at ? and K points for different kz values.In other words, while the direct gap (?k=0) is non-zero and ranges from aminimum of 54meV to a maximum of 220meV over the full BZ [Fig. 4.4(h)],the indirect DOS gap (?k 6= 0) is vanishing. This is still in contrast withthe experimentally determined band-gap magnitude ?gap ' 340meV (notethat PES with K-doping provides a measure of the indirect DOS gap, whileoptics probes the direct optical gap averaged over the whole 3-dimensionalBZ). The disconnection between insulating behavior and onset of AFM or-dering, together with the quantitative disagreement between observed andcalculated gap even in LDA+SOC, reveal that Na2IrO3 cannot be regardedas either a Slater or a band insulator. Also, given the narrow t2g bandwidths984.3. Density functional theorykz=pi/ckz=0BE (eV)-0.200.2 (g)Direct gap (eV) M? Kkz=00.1 pi/c0.2 pi/c 0.4 pi/c0.5 pi/c0.3 pi/c(h)Momentum?(kz=pi/c)LDA+SO?(kz=0)LDA LDA+SO LDA+SO+U0-1-2Binding energy (eV)K M? K M? K M?(b)(a) (c)MomentumLDA+SOLDA0.5-1.5 -1.0 -0.5 0Density of statesLDA+SO+U(d)(e)(f )Binding energy (eV)?SO~ 0.5 eVU=3 eV?DOS 365 meV Figure 4.4: DFT bandstructure and the role of correlations. (a-c) Ir 5d -t2gband structure (kz=0), and (d-f) corresponding DOS, obtained with LDA,LDA+SOC, and LDA+SOC+U (U =3 eV, JH =0.6 eV). (g) kz dispersion,for the last occupied and first unoccupied Ir 5d -t2g bands from LDA+SOC,as indicated by the filled region between the kz =0 and pi/c extreme lines.While the filled areas overlap, resulting in a vanishing indirect gap (i.e.?k 6=0), the direct gap (i.e. ?k=0) between the VB and CB is finite forall kz and k?. (h) LDA+SOC direct gap distribution along M ? ??K, fordifferent kz in the 3-dimensional BZ.(?100meV), one might expect the system to be even more unstable againstlocal correlations than anticipated. Indeed, a good overall agreement withthe data is found in LDA+SOC+U, for U = 3 eV and JH = 0.6 eV, thischoice corresponding to a Ueff =U?JH =2.4 eV, consistent with the valueUeff = 2 eV used in [148]. This returns a gap value ?U=3eVDOS ' 365meV[Fig. 4.4(f)] close to the experimental ?gap, and a 2 eV energy range for theIr 5d -t2g manifold [Fig. 4.4(c)] matching the spectral weight distribution inFig. 4.1. Note that a doubling of bands is seen in LDA+SOC+U due to theimposed AFM ordering (see also Sec. 4.4.2), but this is of no relevance here.At a first glance, U=3 eV might seem a large value for 5d orbitals; however,in the solid, the effective reduction of the atomic value of U strongly dependson the polarizability of the surrounding medium, which is the result of manyfactors, in primis the anion-cation bond length [155]. In this perspective,the value derived here is not unreasonable, and is also consistent with theexistence of local moments above TN revealed by the Curie-Weiss magneticsusceptibility behavior with ?'?120K [146].These findings point to a Mott-like insulating state driven by the delicate994.4. Chapter 4 ? Appendixinterplay between W , U , and SOC energy scales, in which co-participatingstructural distortions also play a crucial role. This establishes Na2IrO3, andpossibly other members of the iridate family, as a novel type of correlatedinsulator in which Coulomb (many-body) and relativistic (spin-orbit) effectscannot be decoupled, but must be treated on an equal footing.4.4 Chapter 4 ? Appendix4.4.1 Materials and MethodsNa2IrO3 single crystals were grown by a self flux method [146] and their highdegree of cristallinity was confirmed using LEED. Samples were preorientedby Laue diffraction in the two studied configurations, i.e. with ??M and??K in the electron emission plane. Crystals were then cleaved in situ ata base pressure of 5?10?11 mbar, exposing the (001) surface (parallel to theIr layers).ARPES measurements were performed at UBC with 21.2 eV linearlypolarized photons (He-I? line from a SPECS UVS300 monochromatizedlamp) and a SPECS Phoibos 150 hemispherical analyzer. Energy and an-gular resolutions were set to 30meV and 0.2?. During all measurements thetemperature was kept at 130K to guarantee stable conditions for as-cleavedand K-deposited surfaces (lower temperatures were prevented by the onsetof charging observed below 120K).The complex optical conductivity data were acquired in the 8-300K tem-perature range on freshly-cleaved (001) surfaces of Na2IrO3 using specularreflectivity measurements below 750meV and ellipsometry at higher fre-quencies.4.4.2 DFT methodologyBand-structure calculations were performed using a basis set of linearizedaugmented plane waves (LAPW), as implemented in the WIEN2K codepackage [156]. Unit cell parameters and atomic positions were taken fromthe most recently refined monoclinic C2/m crystal structure with 2 formula1004.4. Chapter 4 ? Appendixunits per unit cell [157]. Given the availability of high-resolution neutronscattering data, no further optimization of the structural parameters wasperformed. We used a PBE (Perdew...) LDA-functional, where exchangeand correlation effects are treated within the generalized gradient approxi-mation [158]. SO interaction was included as a second variational step usingeigenfunctions from scalar relativistic calculation [159], in which the SO cou-pling strength is calculated from the gradient of the potential, and the fulloperator reads: HSO= 12m2c21rdVdr l ?s (m being the electron mass, and l and sare the orbital and spin angular momentum operators, respectively) [ref toWien2K documentation]. The LDA+U method is implemented as an effec-tive spin and orbital dependent potential, thus also requiring the impositionof magnetic order (for an antiferromagnetic state, like it is the case here,this also implies a doubling of the unit cell). LDA+U was applied to theIr 5d states by varying U between 1 and 5 eV [160] (using a Hund?s cou-pling term JH =0.6 eV [161]) in a spin-polarized calculation, and assuminga zigzag-like antiferromagnetic spin arrangement with moments along the aaxis (zigzag-a) [162]. All calculations show were performed using Muffin-Tinradii of: RNa = 1.43 A?, RIr = 1.22 A?, RO = 0.79 A?and a mesh of 10?8?10k-points in the first Brillouin zone (resized to 227 inequivalent points in theirreducible portion of the Brillouin zone).4.4.3 Temperature dependence and charging in PESCharging effects are a well-known hindrance in photoelectron spectroscopy,as they can produce extrinsic artifacts in the spectra. They arise wheneverthe (positive) charge build-up at the surface, due to the photoelectric effect,is not compensated by neutralizing carriers supplied through the ohmic con-tact between sample and spectrometer. Charging is triggered not only bytemperature (making insulating samples exponentially more resistive) butalso by photon flux (more electrons are being pulled out as light intensity isincreased).Concerning ARPES and PES measurements, two approaches have beenused to ensure that no charging effects are taking place. (i) Temperature1014.4. Chapter 4 ? Appendixdependence was investigated from 300K down to 70K on a series of differentsurfaces in order to establish a safe temperature range for subsequent mea-surements (see main panel in Fig. 4.5); subsequently, T =130K was selectedas the optimal lowest temperature for the present study. (ii) For all mea-surements performed at 130K, before starting with extended acquisitionsangle-integrated photoemission spectra were also measured as a function ofphoton flux (see Fig. 4.6), as an additional check against charging (prior toand during evaporation the sample temperature was kept extremely stable,charging could not be verified via cooling).In particular, from Fig. 4.5 it can be seen that: (i) down to 125K nocharging is taking place (the curves at 150 and 125 K overlap almost per-fectly); (ii) at 95K (blue curve) the spectrum gets broader and shifts down-wards in binding energy. These are two distinctive signatures (broaden-ing+ shift) of spectral weight being inhomogeneously moved to higher bind-ing energies due to charging. In addition, a profile taken at 70 K (greycurve) is also shown, where charging effects are so severe that all the va-lence band features are completely washed away. At 130K, there is nosignature of charging; this is demonstrated by the lack of any variation inangle-integrated PES spectra taken upon reducing the photon flux on thesample, once the spectra have been normalized to compensate for the corre-sponding drop in intensity (Fig. 4.6). This allowed us to verify that chargingeffects could be ruled out for all measurements performed at 130K ? on bothas-cleaved as well as K-deposited surfaces.One last important insight obtained from these temperature dependentmeasurements is that the PES spectra exhibit no intrinsic temperature de-pendence in the range 250-130K. An enlarged view of the leading edge ofthe angle-integrated PES spectra between 250 and 150K is shown in theinset of Fig. 4.5. This highlights the lack of any temperature dependence inPES, consistent with what is observed also in optics all the way down to8K (see Fig. 4.3), in stark contrast with what is expected for a Slater-typeinsulating state.1024.4. Chapter 4 ? AppendixIntensity (arb. units)-8 -6 -4 -2 0Binding energy (eV) 150 K 125 K 95 KIr 5dO 2p-1 0 250 K 70 K 150 KFigure 4.5: Angle-integrated photoemission (PES) spectra from around ?-point and up to 9 eV in binding energy plotted for various temperatures:from 150 to 70K (main panel) to show the onset of charging below 120K, andfor 150 and 250K (inset) to highlight the lack of temperature dependencein the low-energy leading edge.Binding energy (eV)0-2-4Intensity (arb. units)Raw dataNormalized dataIr 5dO 2p T=130 K1 AI!ux1.5 A2 AFigure 4.6: PES spectra at T=130K, as a function of photon flux(parametrized using the He lamp anode current: raw spectra (bottom), andnormalized to the total area (top).1034.4. Chapter 4 ? AppendixBinding energy (eV)Intensity (arb. units)0-2-4Sample 2K-evap:0 sec15 sec90 sec30 secIr 5dO 2pEvaporation time (sec)?? (eV)0.300. 806040200 1: ?? from Ir peak 2: ?? from O peak 3: ?? from Ir-TEM 4: ?? from  O-LEM 5: ?? from  O-TEMIr-TEMO-LEMO-TEM(a) (b)Figure 4.7: K-exposed UPS on Sample 2, and determination of ??. Mainpanel: background-subtracted angle-integrated PES spectra showing bothIr-5d and O-2p manifolds, for selected values of potassium deposition, takenat T=130K on Sample 2 (same sample as in Fig. 4.2). Top-left panel: plotof ?? as extrapolated from the 5 different procedures discussed in the text.4.4.4 Extraction of ?? from PES with K-dopingPotassium was evaporated from a commercial SAES alkali-metal dispenser,at a constant rate and in steps of equal exposure, with the following evap-oration current/time: Ievap = 4.2A/5 sec for sample 1; Ievap = 4.5A/30 secfor sample 2. Note that no K desorption between consecutive steps wasobserved, a sign of the stability of the evaporated surfaces at these temper-atures; and also no detectable change in angle-to-momentum relations, andcorrespondingly of work-function.The data points corresponding to the chemical potential shift ?? (as inFig. 4.2) have been extracted in a variety of ways. Fig. 4.7 showcases thedifferent methods employed here, which are based on the analysis of theangle-integrated ARPES curves as a function of potassium evaporation (seepanel (b) in Fig. 4.7), for the case of Sample 2 (the same analysis was ap-plied to Sample 1). Based on the idea that a chemical potential shift drives1044.4. Chapter 4 ? AppendixBinding energy (eV)123 0(a)K-evap:0 sec30 sec60 sec180 sec270 sec3 Rescaled energy (eV)012(b)Sample 1 Sample 1K evaporation (sec)0 100 200?? (eV) 1Sample 2(c) ?PES~340 meVFigure 4.8: K-exposed UPS on Sample 1 and comparative profiles for ??.(a) Background-subtracted angle-integrated PES spectra from Sample 1, forselected values of K-deposition [see colored markers in (c)]; the chemical po-tential shift ?? is evidenced by the shift of the high binding-energy trailingedges. (b) Same as in (a), but shifted by the corresponding ??. (c) ?? vs.K-deposition time for two different samples.a rigid downwards energy shift of all bands, quantitative estimates for ??have been extracted from the shift of the intensity maximum (peak) of theIr-5d (1) and O-2p (2) manifold, the trailing-edge-midpoint (TEM) of theIr-5d (3) and O-2p (4) manifold, and the leading-edge-midpoint (LEM) ofthe O-2p manifold (5). Note that the Ir-5d LEM cannot be used for thisanalysis as it lies in an energy region where new states are populated withincreasing potassium coverage. Panel (a) in Fig. 4.7 plots the values cor-responding to the above-listed quantities, as a function of K-evaporationtime. It is apparent how the data have a bit of a scatter, which was used tostatistically define ?? and its error ??? in Fig. 4.2(c) in the main text, as1054.4. Chapter 4 ? Appendixthe average and standard deviation over the different methods employed.The last element discussed here is the full dataset for K-evaporation onSample 1. These results are presented in Fig. 4.8, which is the exact analogueof Fig. 4.2 and allows following the saturation of ?? upon approaching thehighest K-deposition.106Chapter 5ConclusionI will here summarize the conclusions that emerged from each investigation,and outline how they come together in the more general context of chargelocalization in correlated oxides.Surface-bulk dichotomy and soft electronic phases in Bi2201Structural supermodulations are a well-known complication in the phe-nomenology of single- and double-layered Bi-based cuprate compounds. Itis also common practice to neglect such effects when it comes to chargedynamics in these materials. We have studied under-to-optimally dopedBi2Sr2?xLaxCuO6 (x=0.8, 0.6, 0.5) compounds by means of surface probes:angle-resolved photoemission (ARPES), low-energy electron diffraction (LEED)- as well as bulk probes: resonant and non-resonant x-ray scattering anddiffraction (REXS/XRD). The complementarity between single-particle elec-tron spectroscopy and structural probes allowed us to unveil an unprece-dented doping- and temperature-dependence of the superstructural wavevec-tor. In addition to this, we find a striking dichotomy between the structuralresponse in the bulk, where no temperature evolution in observed, and atthe surface, where a pronounced, quasi-linear drift of the wavevector is de-tected. This phenomenology is suggestive of the existence of a near-surfacesoft electronic phase in these materials, as confirmed by a Ginzburg-Landaumodeling which shows that the experimental data are reproducible if reduceddimensionality effects at the surface are suitably accounted for.Furthermore, we find that the extremal wavevectors bounding the range forthe new detected superstructural modulation are intimately tied to the un-derlying electronic structure. In particular, the abrupt emergence of new107Chapter 5. Conclusionstructural features with doping is shown to be driven by an electronic in-stability near the complex Fermi surface of Bi2201. This explains both thepeculiar doping-dependence of the supermodulation Q2 (which is not evenpresent for x>0.6) and its specificity to Bi2201, since it requires the presenceof a Q1-modulated Fermi surface.Our results have profound implications on the understanding and inter-pretation of low-energy physics in underdoped cuprates, since they revealan unconventional mechanism of electron-lattice coupling in single-layeredcuprates. The resulting surface enhanced charge density wave is here foundto emerge out of the unique interplay between the charge and lattice degreesof freedom.This paves the way for future studies in which the information on electronicexcitations coming from ARPES can be matched with complementary struc-tural probes (REXS, XRD, LEED) and uncover the origin and nature ofcoupling phenomena in these and other materials. Such an approach hasa tremendous potential towards the understanding of spin/charge orderingphenomena in other materials, including many transition metal oxides. Fur-thermore, the observed surface-bulk dichotomy stands out as an importantprecedent in the field, showing that phenomena detected at the surface ofthese materials might not be fully representative of the bulk properties,which are studied by most conventional meso- and macro-scopic methods(such as transport, susceptibility, calorimetry). This ultimately urges fu-ture surface studies to rely on support from bulk probes.Charge order driven by Fermi-arc instability inBi2Sr2?xLaxCuO6+?Over the years underdoped cuprates have progressively established them-selves as a one-of-a-kind territory for unconventional physics at all levels.Despite multiple evidences for different forms of electronic instabilities, theexistence and universality of an underlying phase has been strongly debated,in part owing to unresolved discrepancies between electronic vs. structural,and surface vs. bulk probes. In this work we provide direct evidence for108Chapter 5. Conclusiona charge-ordered state below the pseudogap temperature T ? in underdopedBi-based cuprates, using resonant soft X-ray scattering. The experimen-tal information available from REXS reveals how this reorganization of theelectronic charge is idiosyncratic to the CuO2 planes, emerging only whenthe measurements are performed at resonance with the electronic transitiononto empty states at the Cu sites.Remarkably, the same feature is observed in real space, by means of scan-ning tunnelling microscopy, in the form of the well-known checkerboard pat-tern, which is thereby proven to be a genuine form of static ordering of theelectronic charge. Again, the charge ordering features are uncovered onlyby performing selective tunnelling from the STM probe onto the near-EFstates which live in the CuO2 planes.The quadrature and reconciliation between experiments and theory is fi-nally achieved by performing complementary calculations of the electronicresponse from an ARPES-derived phenomenological model. These numeri-cal results indeed show that scattering between the Fermi-arc endpoints ?the hot-spots ? very closely reproduces the observed wavevectors. This ul-timately reveals the long-sought connection between the low-energy fermi-ology, experimentally represented by the Fermi surface, and the charge-ordered ground state which emerges from the instability of the electronicsystem.All in all, these converging findings suggest the existence of a univer-sal charge-ordered state in underdoped cuprates and clarify its deep con-nection to the pseudogap state and the Fermi arcs which characterize thecorresponding spectrum of electronic excitations. Our results successfullyresolved a long-standing controversy, which was possible only thanks to thecombination of multiple techniques, and highlight a new framework for theunderstanding of cuprates in the underdoped side of the phase diagram.This methodology was proven very powerful, and has a wide range of appli-cability in similar classes of materials. In particular, this work has demon-strated how state-of-the-art REXS has reached a maturity level which makesit competitive with advanced real-space techniques in detecting minusculevariations in the charge density of selective electronic states. This approach109Chapter 5. Conclusionis ready to be extended to systems showing similar emerging phenomenain the presence of a comparative (with respect to Bi2201) degree of chargeinhomogeneity and disorder, such as cobaltates, nickelates, manganites, andpotentially new members from the growing classes of 4d - and 5d -based ox-ides.Na2IrO3 as a Novel Relativistic Mott Insulator5d -based transition metal oxides represent a novel class of materials whichattracted a growing interest in the last decade for giving rise to unconven-tional states of matter. In particular, a few members from the family of iri-dates, so far the most investigated, have been proposed to be the realizationof an unconventional spin-orbit assisted Mott-Hubbard physics. To test thisproposal, we have studied the newest addition to the iridate family, sodiumiridate (Na2IrO3), which is the first material where the insulating state, per-sisting up to room temperature, is completely uncorrelated to the magneticordering (which only kicks in at very low temperatures) ? and thereforethe best system to look into a pure Mott-like mechanism. We based ourinvestigation of the low-energy electronic structure in this compound usingARPES, optics, and band structure calculations in the local-density approx-imation (LDA).The experimental band structure as revealed by performing ARPES onfreshly cleaved surfaces is found in good agreement with theoretical cal-culations, especially for what concerns the bandwidth and overall extent ofthe Ir 5d -t2g states. Also, no spectral weight was found at the Fermi energyup to room temperature, consistent with the transport results which showan insulating-like resistivity all the way up to high temperatures.ARPES on potassium-doped surfaces and optics were subsequently per-formed in order to quantify the charge gap. Photoemission and optics turnedout to be in remarkable agreement, returning an insulating gap ??340meV.On the theoretical side, the LDA analysis revealed an unprecedented find-ing: without spin-orbit coupling (SOC) the system would be metallic, butwith SOC density functional theory predicts an insulating behavior. Further110Chapter 5. Conclusionanalysis then showed how the correct gap magnitude could only be repro-duced with the addition of an effective on-site Coulomb term of Ueff =2.4 eV.Ultimately, we found that Na2IrO3 realizes a novel type of Mott-likecorrelated physics, in which Coulomb and relativistic effects (SOC) bothcooperate towards driving the system insulating, therefore establishing thismaterial as a relativistic Mott insulator.This paves the way for the realization of new states of matter in 5d -basedmaterials, whose phase diagram is still to a very large degree unexplored.Controlling electronic phases through carrier doping has proven extremelysuccessful in 3d -based oxides, where some of the many unconventional phe-nomena have had a profound impact on the technological side: the revolutionin the data storage industry brought about by the discovery of colossal mag-netoresistance in the manganites is one of the most famous examples. 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Rosen, R. Comin , S. Kittaka, Y. Maeno, I. S. Elfimov,and A. Damascelli, ?Determining the Surface-To-Bulk Progression inthe Normal-State Electronic Structure of Sr2RuO4 by ARPES andDFT?, Physical Review Letters 110 097004 (2013).3. R. Comin , G. Levy, B. Ludbrook, Z.-H. Zhu, C.N. Veenstra, J.A.Rosen, Yogesh Singh, P. Gegenwart, D. Stricker, J.N. Hancock, D.van der Marel, I.S. Elfimov, and A. Damascelli, ?Na2IrO3 as a NovelRelativistic Mott Insulator with a 340-meV Gap?, Physical ReviewLetters 109 266406 (2012).4. Z.-H. Zhu, G. Levy, B. 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?, Physical ReviewLetters 107 186405 (2011).5. P.D.C. King, J.A. Rosen, W. Meevasana, A. Tamai, E. Rozbicki, R.Comin , G. Levy, D. Fournier, Y. Yoshida, H. Eisaki, K.M. Shen,1joint first-authorship134Appendix . List of publicationsN.J.C. Ingle, A.Damascelli, and F. Baumberger, ?Structural Origin ofApparent Fermi Surface Pockets in ARPES of La-Bi2201?, PhysicalReview Letters 106 127005 (2011).6. M. Bianchi, D. Cassese, A. Cavallin, R. Comin , F. Orlando, L.Postregna, E. Golfetto, S. Lizzit, and A. Baraldi, ?Surface core levelshifts of clean and oxygen covered Ir(111)?, New Journal of Physics11 063002 (2009).Reviews7. R. Comin , and A. Damascelli, ?ARPES: A probe of electronic cor-relations?, Chapter to appear in the book ?Strongly Correlated Sys-tems: Experimental Techniques?, edited by A. Avella and F. Mancini,Springer Series in Solid-State Sciences (2013).Papers in press8. R. Comin , A. Frano, M. M. Yee, Y. Yoshida, H. Eisaki, E. Schierle,E. Weschke, R. Sutarto, F. He, A. Soumyanarayanan, Yang He, M.Le Tacon, J. E. Hoffman, B. Keimer, G.A. Sawatzky, and A. Damas-celli, ?Charge ordering driven by Fermi-arc instability in underdopedcuprates?, Accepted in Science (2013).9. E. H. da Silva Neto, P. Aynajian, A. Frano, R. Comin , E. Schierle,E. Weschke, A. Gye?nis, J. Wen, J. Schneeloch, Z. Xu, S. Ono, G. Gu,M. Le Tacon, B. Keimer, A. Damascelli, and A. Yazdani, ?UbiquitousInterplay between Charge Ordering and High-Temperature Supercon-ductivity in Cuprates?, Accepted in Science (2013).Papers in preprint10. F. Cilento, S. Dal Conte, G. Coslovich, S. Peli, N. Nembrini, S. Mor,135Appendix . List of publicationsF. Banfi, G. Ferrini, H. Eisaki, M. Greven, D. van der Marel, R.Comin , A. Damascelli, L. Rettig, U. Bovensiepen, M. Capone, C.Giannetti, and F. Parmigiani, ?Non-equilibrium spectroscopies revealthe short-range correlations governing the phase diagram of cuprates?,Submitted (2013).136


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