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Correlated phenomena studied by ARPES : from 3d to 4f systems Zonno, Marta 2020

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Correlated phenomena studied by ARPES:from 3d to 4 f systemsbyMarta ZonnoBSc., University of Trieste, 2011MSc., University of Trieste, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)The University of British Columbia(Vancouver)June 2020c©Marta Zonno, 2020The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Correlated phenomena studied by ARPES:from 3d to 4 f systemssubmitted by Marta Zonno in partial fulfillment of the requirements for thedegree of Doctor of Philosophy in Physics.Examining Committee:Andrea Damascelli, University of British Columbia (Physics & Astronomy)SupervisorMona Berciu, University of British Columbia (Physics & Astronomy)Supervisory Committee MemberMarcel Franz, University of British Columbia (Physics & Astronomy)University ExaminerAlireza Nojeh, University of British Columbia (Electrical and ComputerEngineering)University ExaminerFelix Baumberger, University of Geneva (Quantum Matter Physics)External ExaminerAdditional Supervisory Committee Members:George A. Sawatzky, University of British Columbia (Physics & Astron-omy)Supervisory Committee MemberJeremy S. Heyl, University of British Columbia (Physics & Astronomy)Supervisory Committee MemberiiAbstractThe physics of strongly correlated materials is at the heart of current con-densed matter research. The inclusion of interactions in these materialsbetween electron themselves or with other excitations intertwines variousdegrees of freedom (orbital, spin, charge and lattice), leading to a num-ber of novel phenomena like Mott-Hubbard and charge-transfer insula-tors, high-temperature superconductivity and mixed-valence and Kondophysics. This thesis focuses on the study of two classes of correlated ma-terials: copper-oxide high-temperature superconductors, whose correlatedphysics is driven by the localized nature of the half-filled Cu 3d-orbitals,and the rare-earth hexaborides, which are characterized by the stronglycorrelated 4 f -shell.Recently, it has been shown that the interplay between different mech-anisms underlying the formation of the superconducting condensate in thehole-doped bi-layer Bi2Sr2CaCu2O8+δ can be addressed in the time do-main by means of time- and angle-resolved photoemission spectroscopy(TR-ARPES). Using this technique, the primary role of phase coherencehas been established. By exploiting the same dynamical experimental ap-proach, we show that such scenario also describes the ultrafast collapse ofsuperconductivity in the single-layer compound Bi2Sr2CuO6+δ. Moreover,by performing a comprehensive study on different doping levels of bothsingle- and bi-layer compounds, we provide new insights on the tempera-ture evolution of the nodal quasiparticle spectral weight.The second part of the thesis focuses on electron-doped cuprates, ad-dressing the putative relation between the spectroscopically observed pseu-iiidogap and the robust antiferromagnetic order. Employing TR-ARPES as atool to perform a detailed temperature dependent investigation allows usto explicitly link the momentum-resolved pseudogap spectral features tothe evolution of the short-range spin-fluctuations in the optimally-dopedNd2−xCe2CuO4.Lastly, we make use of chemical substitution to investigate the mixed-valent character of the rare-earth hexaboride SmxLa1−xB6 series. Our com-bined ARPES and x-ray absorption measurements reveal a departure froma monotonic evolution of the Sm valence as a function of x and the possibleemergence of a mixed-valent impurity regime.ivLay SummaryIn many materials, electrons behave independently from other particles.However, for a wide class of quantum materials each electron cannot beconsidered as an independent particle, but it is instead strongly linked tothe other electrons around and subjected to their interaction. This electronicconnection gives rise to new quantum phenomena, including supercon-ductivity where a material can conduct electricity without any loss. In thisthesis, we study how the interplay between electrons affects the physics oftwo classes of materials: high-temperature superconductors and rare-earthcompounds, by employing ultrafast laser excitations and chemical substi-tution, respectively. The results may contribute to a better understandingof the role of electron correlations in shaping the properties of quantummaterials, which may play a vital role in future technologies and applica-tions.vPrefaceThe work presented in this thesis is representative of the research activ-ity I conducted at UBC during my time as graduate student. It consists ofthree projects for which I have been the primary investigator. However,experimental physics is a highly collaborative endeavor and all the workpresented here was made possible thanks to involvement from numerousindividuals as of sample growth, technical support and complementary ex-perimental techniques. I will list here the specific contributions I and othershave made for each of the experimental chapters.Chapter 3. This chapter focuses on the analysis of the spectral func-tion of hole-doped Bi-based cuprates, both single- (Bi2201) and bi-layer(Bi2212) compounds, by means of time-resolved and equilibrium angle-resolved photoemission spectroscopy (TR-ARPES and ARPES). The exper-iments were conceived by F. Boschini, A. Damascelli and me. All the TR-ARPES measurements were performed at UBC by me, with the help of F.Boschini and E. Razzoli. F. Boschini, E. Razzoli, R. P. Day, M. Michiardi, B.Zwartsenberg, P. Nigge, A. Sheyerman, M. Schneider, S. Zhdanovich, A. K.Mills, G. Levy and me provided technical support and maintenance for theARPES setup at UBC. The equilibrium ARPES data were collected at theCanadian Light Source synchrotron facility in Saskatoon by me and F. Bos-chini, with assistance from S. Zhdanovich, T. Pedersen and S. Gorovikov.The Bi2201 and Bi2212 samples were provided by Y. Yoshida and G. D. Gu,respectively. Data analysis was done by me, with input from F. Boschiniand A. Damascelli. A. Damascelli supervised the project and was respon-sible for the overall project’s direction, planning and management. Theseviresults are currently being prepared for publication.Chapter 4. The results in this chapter are largely based on the pub-lication ”Emergence of pseudogap from short-range spin-correlations inelectron-doped cuprates” by F. Boschini*, M. Zonno* et al. (2020). F. Bos-chini and me performed the TR-ARPES experiments at UBC, with the assis-tance of E.Razzoli and M. Michiardi, and analyzed the data. F. Boschini, E.Razzoli, R. P. Day, M. Michiardi, B. Zwartsenberg, P. Nigge, A. Sheyerman,M. Schneider, S. Zhdanovich, A. K. Mills, G. Levy and me provided tech-nical support and maintenance for the ARPES setup. The interpretation ofthe work was done by me, F. Boschini and A. Damascelli, with preciousinputs from E. Razzoli, E. H. da Silva Neto, C. Giannetti and D. J. Jones.Single crystals of electron-doped NCCO were grown by A. Erb. A. Dama-scelli supervised the project. A. Damascelli was responsible for the overalldirection, planning and management of the project. The manuscript sum-marizing the main results was written by me, F. Boschini, R. P. Day and A.Damascelli with input from all authors.Chapter 5. This chapter investigates the evolution of the mixed-valencecharacter in the SmxLa1−xB6 series. The project was conceived by me, G. A.Sawatzky and A. Damascelli. I performed the ARPES experiments at UBCwith 21.2 eV photon energy, with assistance from M. Michiardi. AdditionalARPES data with 67 eV photon energy were collected by M. Michiardi incollaboration with the group of P. Hofmann in Aarhus at the ASTRID2 syn-chrotron facility, with support from K. Volkaert, D. Curcio and M. Bianchi.I analyzed all the ARPES data. X-ray absorption measurements were per-formed at the Canadian Light Source by R. Green and myself. R. Greenanalyzed the XAS data. M. Michiardi, F. Boschini, E. Razzoli, B. Zwart-senberg, P. Nigge, A. Sheyerman, R. P. Day, M. Schneider, G. Levy and meprovided technical support and maintenance for the ARPES setup at UBC.Single crystals of SmxLa1−xB6 were grown by P. F. S. Rosa and Z. Fisk. Theinterpretation of the work was done by me, R. Green, G. A. Sawatzky andA. Damascelli. G. A. Sawatzky and A. Damascelli supervised the overallproject. These results are currently being summarized into a manuscriptfor publication.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background: cuprate high-temperature superconductors . . 51.2 Background: samarium hexaboride . . . . . . . . . . . . . . . 102 Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . 162.1 Angle-resolved photoemission spectroscopy . . . . . . . . . 172.2 Time- and Angle-resolved photoemission spectroscopy . . . 272.2.1 TR-ARPES at UBC with 6.2-eV probe . . . . . . . . . 292.3 X-Ray Absorption Spectroscopy . . . . . . . . . . . . . . . . . 32viii3 Tracking the spectral function of Bi-based cuprates by TR-ARPES 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 TR-ARPES on Bi2201 . . . . . . . . . . . . . . . . . . . . . . . 453.3 Tracking the ultrafast enhancement of phase fluctuations inBi2201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4 Nodal coherent-spectral-weight meltdown in Bi-based cuprates 573.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 Emergence of pseudogap from short-range spin-correlations inelectron-doped cuprates . . . . . . . . . . . . . . . . . . . . . . . . 704.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2 Static 6.2-eV ARPES of Nd2−xCexCuO4 . . . . . . . . . . . . 744.3 Tracking the pseudogap spectral weight in an ultrafast fashion 804.4 Relation of the pseudogap to the spin-correlation length: the-oretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925 Exploring the mixed-valent state in SmxLa1−xB6 . . . . . . . . . 945.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 LaB6 and SmB6: ARPES comparison . . . . . . . . . . . . . . 985.3 Evolution of the mixed-valence character in the SmxLa1−xB6series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.4 Towards the limit of mixed-valent impurity regime . . . . . 1115.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121A Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145ixList of FiguresFigure 1.1 Electron localization in the periodic table . . . . . . . . . 2Figure 1.2 Different types of correlated quantum states of matter . . 4Figure 1.3 Crystal structure of cuprates . . . . . . . . . . . . . . . . 6Figure 1.4 Phase diagram of cuprates . . . . . . . . . . . . . . . . . . 8Figure 1.5 One-electron excitation spectrum of SmB6 . . . . . . . . . 12Figure 1.6 Energy level diagram of Sm2+ and Sm3+ . . . . . . . . . 14Figure 2.1 Energy scheme of the photoemission process . . . . . . . 18Figure 2.2 Electron mean free path in solids . . . . . . . . . . . . . . 20Figure 2.3 Geometry of an ARPES experiment . . . . . . . . . . . . 21Figure 2.4 Timescale of various carrier and lattice processes in asolid upon laser excitation . . . . . . . . . . . . . . . . . . 27Figure 2.5 Pump-Probe scheme for time-resolved ARPES . . . . . . 28Figure 2.6 Schematic of the 6.2-eV TR-ARPES setup at UBC . . . . . 30Figure 2.7 X-Ray Absorption technique . . . . . . . . . . . . . . . . 33Figure 2.8 Common detection approaches in XAS experiments . . . 36Figure 3.1 Electron pairing vs phase coherence and suppression ofquasiparticle spectral weight in Bi2212 . . . . . . . . . . . 41Figure 3.2 Static 6.2-eV ARPES of Bi2201 . . . . . . . . . . . . . . . . 44Figure 3.3 Energy-resolved TR-ARPES dynamics in Bi2201 OD24 . 46Figure 3.4 Energy-resolved decay times in Bi2201 OD24 . . . . . . . 47Figure 3.5 Transient electronic temperature Te . . . . . . . . . . . . 49Figure 3.6 Nodal single-particle scattering rate dynamics . . . . . . 53xFigure 3.7 Off-Node spectral function dynamics . . . . . . . . . . . 54Figure 3.8 Disentangling Γp and Te dynamics in Bi2201 . . . . . . . 56Figure 3.9 Suppression of coherent spectral weight in Bi2201 OD24 58Figure 3.10 Γs and nodal CSW vs. temperature in Bi2201 OD24 . . . 60Figure 3.11 Doping dependence of the temperature evolution of CSWand Γs in Bi2201 . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 3.12 Universal suppression of CSW in Bi-based cuprates . . . 63Figure 3.13 CSW from equilibrium temperature-dependent ARPES . 64Figure 3.14 Simulated ∆CSW(T) in a FL vs. MFL picture . . . . . . . 67Figure 4.1 Phase-diagram and AF order of Nd2−xCexCuO4 . . . . . 72Figure 4.2 Fermi surface of optimally-doped Nd2−xCexCuO4. . . . 75Figure 4.3 6.2-eV static ARPES spectra of optimally-doped NCCO . 76Figure 4.4 Pseudogap in static ARPES, comparison with previousstudies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 4.5 Pseudogap dispersion and amplitude . . . . . . . . . . . 79Figure 4.6 Experimental strategy for tracking transient filling of thePG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 4.7 Effective electronic temperature Te . . . . . . . . . . . . . 83Figure 4.8 Pseudogap spectral weight vs. Te in optimally-dopedNCCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 4.9 Energy- and momentum-resolved TR-ARPES dynamics 87Figure 4.10 Comparison of experimental and simulated TR-ARPESdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 4.11 Filling of the PG through a reduction of ξspin . . . . . . . 90Figure 4.12 Contribution of Γ and η to the pseudogap modeling . . . 91Figure 5.1 Crystal structure of SmxLa1−xB6 . . . . . . . . . . . . . . 96Figure 5.2 Electronic structure of LaB6 . . . . . . . . . . . . . . . . . 99Figure 5.3 ARPES dispersion of LaB6 and SmB6 . . . . . . . . . . . . 100Figure 5.4 ARPES iso-energy contours in LaB6 and SmB6 . . . . . . 101Figure 5.5 Photon energy dependence of LaB6 and SmB6 . . . . . . 102Figure 5.6 Dispersion of SmB6 probed with different photon energies 103xiFigure 5.7 ARPES spectra of SmxLa1−xB6 . . . . . . . . . . . . . . . 105Figure 5.8 X-pocket dispersion in SmxLa1−xB6 . . . . . . . . . . . . 106Figure 5.9 Evolution of the Sm valence from the ARPES dispersion 108Figure 5.10 XAS study of the SmxLa1−xB6 series . . . . . . . . . . . . 110Figure 5.11 Calculated Sm valence for three ideal cases . . . . . . . . 112Figure 5.12 Simplified model for the Sm valence in SmxLa1−xB6 . . . 114xiiGlossary2D two-dimensional3D three-dimensionalAFM AntiferromagneticARPES Angle-Resolved Photoemission SpectroscopyBCS Bardeen, Cooper, and Schrieffer theoryBi2201 Bi2Sr2CuO6+δBi2212 Bi2Sr2CaCu2O8+δBi2223 Bi2Sr2Ca2Cu3O10+δBZ Brillouin zoneCFPs Coefficients of Fractional ParentageCO Charge-OrderCSW Coherent Spectral WeightDFT Density Functional TheorydHvA de Haas van Alphen oscillationsDOS density of statesEDC Energy Distribution CurvexiiiEXAFS Extended X-ray Absorption Fine StructureFEL free electron laserFL Fermi-liquidFS Fermi SurfaceFWHM full width half maximumHHG high-order harmonic generationHTSC High Temperature SuperconductivityIPFY Inverse Partial Fluorescence YieldLHB Lower Hubbard BandLSCO La2−xSr2CuO4MDC Momentum Distribution CurveMFL marginal-Fermi-liquidMOs molecular orbitalsMV Mixed-Valent stateNCCO Nd2−xCe2CuO4NEXAFS Near Edge X-ray Absorption Fine StructureOD overdopedOP optimal-dopingPEY Partial Electron YieldPFY Partial Fluorescence YieldPG PseudogapSC SuperconductivityxivSDW Spin-Density WaveSEDC Symmetrized Energy Distribution CurveTEY Total Electron YieldTFY Total Fluorescence YieldTKI Topological Kondo InsulatorTMOs Transition Metal OxidesTM Transition MetalTR-ARPES Time- and Angle-Resolved Photoemission SpectroscopyUD underdopedUHB Upper Hubbard BandUHV Ultra High VacuumVB valence bandXAS X-Ray Absorption SpectroscopyxvAcknowledgmentsExperimental physics is by nature a collaborative environment and, al-though this thesis has a single author, it could have never been possiblewithout the help and support of numerous people around me.The first and bigger thank you goes to my supervisor Andrea Dama-scelli for offering me the unique opportunity to move thousands of kilo-meters from my home-town in this beautiful place I quickly learned tocall ”home”. Your deep understanding of physics, your enthusiasm andyour guidance have helped me to sail through the exciting (and at timesrough) waters of a PhD in condensed matter physics. I feel honored tohave worked with you and incredibly thankful for all the encouragementand support received, often times far beyond the pure scientific sphere.The uncountable hours spent in the lab and in the office would havefelt overwhelming without the emotional and intellectual support of all thepeople in the ARPES group: Giorgio Levy, Sergey Zhdanovic, Art Mills,Michael Schneider, Eduardo da Silva Neto, Berend Zwartsenberg, PascalNigge, Alex Sheyerman, Amy Qu, Ryan Day, Elia Razzoli, Ketty Na, SeanKung, Christopher Gutie´rrez, Danilo Kuhn and Sydney Dufresne. A partic-ular thank you to Matteo Michiardi and Fabio Boschini, who have played abig part in my research projects and encouraged me countless times. Eachof you have made UBC a special place to grow both scientifically and per-sonally.Thank you to George Sawatzky and Robert Green for their preciousinvolvement in the rare-earth project and to Philip Hofmann for offeringhis system in Aarhus to perform synchrotron measurements on these com-xvipounds. Thank you to Tor Pedersen and Sergey Gorovikov for their sup-port at the Canadian Light Source and to our collaborators Y. Yoshida, G.D. Gu, P. F. S. Rosa and Z. Fisk for providing the samples without whichnone of the projects here presented would have been possible.A big thank you to all my committee members (supervisory and exam-ining) for taking the time and effort to read and provide helpful feedbacksto improve my thesis.Life is not just work, and the six years I spent in Vancouver have beenenriched by new experiences and adventures with amazing people I feelincredibly lucky to call friends: Davide, Jose´, Mirtha, Fernando, Aris andIoanna, Hector and David, Daniel and Giulia, Tiffany and Edward in primis.Thank you to my ”Verona-ties” Francesco and Alessandro, who always re-mind me how the world can feel small when the friendship is strong. Anda deep thank you to Javiera for being there, showing me new horizons andmaking everything shine brighter when shared together. All the words inthe world cannot express how profoundly thankful I feel for the infinitelove and support I constantly receive from my parents, my brother and mysister: you all really are the pillars of my life. I do believe acknowledgmentsare better done in person, maybe in front of a nice meal and glass of goodwine, so I will cut this short and do the rest in real life!Marta ZonnoVancouver, June 9, 2020xviiDedicationTo my brother and my sisterand to my parentswho have never stopped believing in meand never will.A mio fratello e mia sorellae ai miei genitoriche non hanno mai smesso di credere in mee mai lo faranno.xviiiChapter 1IntroductionMany of our present-day technologies are based on materials such as sili-con, aluminum, copper and diamond, whose macroscopic properties (elec-trical, mechanical and thermal) are fairly well described in terms of non-interacting electrons. The combination of an electronic structure governedby delocalized states, stemming primarily from s and p orbitals, and thePauli exclusion principle leads to a ground state where the kinetic energyof the electrons dominates over the possible electron-electron interactions.The latter are thus often treated as afterthought in describing most famil-iar metals, band insulators and semiconductors. However, there is a wideclass of materials for which the interaction among electrons is strong andplays a vital role in shaping their electronic, magnetic and sometimes evenmechanical properties. These are the now-called strongly correlated electronsystems, whose physics cannot be described in terms of non-interactingsingle-particle Hamiltonian. The resulting entanglement of all the signif-icant degrees of freedom (orbital, spin, charge and lattice) gives arise toa plethora of intriguing exotic phenomena including Mott-Hubbard andcharge-transfer insulators [1–4], High Temperature Superconductivity (HTSC)[5–8], mixed-valence and Kondo physics [9–11], heavy fermions [12–14]and various forms of charge and orbital ordering [15–19].1   Lu     Hf       Ta      W      Re     Os      Ir       Pt      Au    Y      Zr      Nb    Mo      Tc     Ru     Rh      Pd      Ag   Sc      Ti        V      Cr      Mn    Fe      Co     Ni     Cu  Th  Pa   U   Np   Pu  Am  Cm  Bk  Cf   Es  Fm  Md  No  Ce  Pr  Nd  Pm  Sm  Eu   Gd   Tb  Dy  Ho  Er   Tm  YbIncreasing localizationIncreasing localization 5d4d3d5f4f3dcupratesrare-earths4fFigure 1.1: Electron localization in the periodic table. Illustration ofthe general evolution of the electron localization across the periodictable for the d- and f -compounds. Right inset: radial extend of the 3dand 4 f wave functions of the valence electrons of the cations.The investigation of this wide class of material dates back to the mid-dle of the 20th-century, motivated at first by the discrepancy between thevalue of the electronic band gap as measured experimentally (usually feweV) and as predicated by band-theory (often much smaller or even null) formany Transition Metal Oxides (TMOs). This inconsistency was resolvedby the inclusion of an on-site Coulomb repulsion energy U in the so-calledHubbard model [2, 20]. In fact, according to the conventional band-theorypicture, a half-filled band would result into a metallic ground state; how-ever, in the case of atomic-like electronic states, such as the 3d- and 4 f -shell, the addition of an extra electron on the same site occurs at a cost ofthe large extra energy U, thus preventing hopping of the valence electronin the crystal in the form of fluctuations dndn → dn−1dn+1. The resultingcharge localization drives the system into an insulating state (Mott-Hubbardinsulator): the broad metallic band is split into an occupied Lower HubbardBand (LHB) and an empty Upper Hubbard Band (UHB), the two separatedby a gap determined by the size of U (see Figure 1.2a, left panel). Thepresence of the large on-site Coulomb repulsion is then the keystone forstrongly-correlated physics. In the context of the periodic table, stronglyinteracting electrons occur in well localized orbitals that can be arranged in2order of increasing localization as follow:5d < 4d < 5 f < 3d < 4 f . (1.1)This trend is illustrated in Figure 1.1 and designates the compounds withpartially filled 3d orbitals, as the family of cuprates HTSC, and the 4 f rare-earth materials as prototypical examples of strongly-correlated electron sys-tems. Although Equation 1.1 gives an indication on the strength of corre-lations, it is important to compare U with another relevant energy scale inthe system, namely the bandwidth W. This parameter is related to the near-neighbour hopping and represents the extent in energy of the valence bandin the limit of non-interacting particles. The ratio U/W then encodes theinterplay/competition between atomic-like and delocalized charge fluctu-ations and it determines the stability of the Mott-insulating phase.While the Mott-Hubbard model marked a breakthrough for the inclu-sion of electron-electron interactions in the modeling of quantum materials,an additional step towards the understanding of correlated insulators wastaken in the 1980s with the development of the Zaanen-Sawatzky-Allentheory. This laid the foundation for the definition of charge-transfer insula-tors [3]. In these systems, the insulating gap is not determined by U, butrather by the charge-transfer energy ∆, which defines the energy cost formoving an electron between the anion and the cation within the same unitcell (see Figure 1.2a, middle panel). This picture describes the late 3d TMOsin which the lowest-energy excitations will occur between the occupied lig-and (e.g. O or S) 2p band and the transition-metal (TM) 3d orbitals (e.g. Nior Cu), in the form p6dn→ p5dn+1.By gradually reducing the charge-transfer energy ∆ up to the limit ∼Wlig+WTM2 (where Wlig and WTM are the bandwidths of the ligand and TMbands, respectively), the mixed-valent state emerges. Here, two different va-lence configurations of the TM (i.e. dn and dn+1) are simultaneously presentin the system and jointly characterize the ground state, with LHB and UHBnow separated by 2U (see Figure 1.2a, right panel). In this case, chargefluctuations can occur at very small energy scale involving both the TM3DOS DOSEnergyIndependent electronsCorrelated electronsEF UDOS DOSEnergyIndependent electronsCorrelated electronsEFU∆DOS DOSEnergyIndependent electronsCorrelated electrons2UEFMott-Hubbard Insulator Charge-transfer Insulator Mixed-valent systemd-bandsp-bandsdn-1dn+1 dn+1dn-1dn+2dn-1dn+1dndn dn dn-1 dn+1 p6 dn p5 dn+1 dn-1 dn dn dn-1Mott-Hubbard Insulator Charge-transfer Insulator Mixed-valent systemabLHBUHBFigure 1.2: Different types of correlated quantum states of matter.(a) Illustrative electron removal and addition spectra resulting fromstrong electron correlations in three different types of compounds:Mott-Hubbard insulator (left), charge-transfer insulator (middle) andmixed-valence system (right). A broad band is depicted in green,while the narrow correlated bands in orange (here associated to p-bands and d-bands, respectively, for illustrating purposes). Note thathere the hybridization between the broad and narrow bands is not in-cluded. U is the on-site Coulomb repulsion and ∆ the charge-transferenergy. (b) Schematic of the lowest energy charge fluctuations for thethree different classes of correlated compounds. Adapted from [4].and the ligand states. Various rare-earth compounds exhibit mixed-valentbehavior, where different valence configurations for the strongly correlated4 f -shell are observed.4The classes of materials introduced thus far represent the starting pointof the research presented in this thesis project on copper-oxide HTSC (knownas cuprates) and La-substituted SmB6. On one side, the Mott-insulatingphysics, driven by the strong correlations in the half-filled Cu 3d-states, de-fines the stochiometric cuprates, also known as parent compounds. Startingfrom this ground-state, many fascinating phases emerge upon hole- andelectron-doping, such as superconductivity, pseudogap regime and charge-order, to name a few. Moving down in the periodic table to the rare-earthcompounds, the strongly correlated 4 f -shell comes into play and, amongthe various emerging phenomena, there is the Mixed-Valent (MV) state.SmB6 is a prominent example where the description of the ground stategets particularly complicated by the physics of MV. Following, we will givea brief background on these compounds to set the stage for the specific in-vestigations carried out in this thesis project.1.1 Background: cuprate high-temperaturesuperconductorsSuperconductivity (SC) was first discovered in 1911 when co-workers inthe group of Heike Kamerlingh Onnes observed an abrupt vanishing ofthe electrical resistivity of mercury when cooling the sample below 4.2 K[21]. Following this first experimental result, an intense theoretical and ex-perimental effort focused in the developing of a theoretical description ofSC and in the search of compounds exhibiting a higher transition temper-ature, Tc. However it was only in the mid 1950s that the first theory ofsuperconductivity was developed by Bardeen, Cooper, and Schrieffer the-ory (BCS) [22, 23]. Twenty years later the maximum Tc achieved was∼ 23 Kin Nb3Ge. A major breakthrough was achieved in 1986 when Bednorzand Mu¨ller discovered superconductivity near 40 K in the copper-oxideLa2−xBaxCuO4 [5], marking the prelude of the field of HTSC. Since thatfirst discovery, cuprates became subject of an intensive investigation. Soonit was realized that the observed high range of Tc (as high as ∼ 138 K inHg0.8Tl0.2Ba2Ca2Cu3O8+δ [24]) was one of many indicators of the emer-5BiSrCaCuONd/CeCuOBi2201 Bi2212 NCCOa bFigure 1.3: Crystal structure of cuprates. (a) Crystal structure of thehole-doped Bi-based single-layer Bi2Sr2CuO6+δ (Bi2201) and bi-layerBi2Sr2CaCu2O8+δ (Bi2212) cuprate. The lattice constants are approxi-mately a = b = 3.86 A˚ and c = 24.7 A˚ (for Bi2201) and c = 30.7 A˚ (forBi2212). (b) Crystal structure of the electron-doped Nd2−xCe2CuO4(NCCO), with lattice constants of a = b = 3.9 A˚ and c = 12.1 A˚ .gence of a different kind of superconductivity, with respect to the con-ventional BCS theory. Various theories have been proposed, based on po-larons, quantum criticality, spin-fluctuations and resonating valence bands,to name some [25–29]. However, despite the uncountable attempts to ad-dress the origin of HTSC in cuprates (on both the experimental and the-oretical level) since Bednorz and Mu¨ller′s discovery, no consensus on thepairing mechanism has yet been achieved.The family of copper-oxides HTSC is characterized by a layered per-ovskite structure, in which the square planar CuO2 plane (where each cop-per ion is fourfold coordinated with oxygens) forms a single- or multi-layerconducting block separated by insulating layers, also called charge reser-voir layers. Depending on the number of copper-oxygen planes withinthe unit cell, the cuprates are classified into single-layer compounds, asLa2−xSr2CuO4 (LSCO), Bi2Sr2CuO6+δ (Bi2201) and Nd2−xCe2CuO4 (NCCO);6bilayer compounds, as Bi2Sr2CaCu2O8+δ (Bi2212); trilayer compounds, asBi2Sr2Ca2Cu3O10+δ (Bi2223) and beyond. Figure 1.3 shows the crystal struc-ture for Bi-based hole-doped cuprates (as it evolves from one to two CuO2layers) and for the electron-doped NCCO. Note that the anisotropy be-tween the c-axis and the in-plane axis makes these systems quasi-2D, asconfirmed by a recent Angle-Resolved Photoemission Spectroscopy (ARPES)experiment of monolayer Bi2212 [30]. Despite the difference in the crystalstructure, all cuprates share the central role of the Cu-O block in determin-ing the macroscopic electronic properties of the materials, including SC.In the parent compound, the Cu ions have a 3d9 electronic configuration,corresponding to having the four lowest energy orbitals (dxy, dxz, dyz andd3z2−r2) fully occupied, with the only highest energy dx2−y2 being half-filled.Due to the vicinity in energy, the latter strongly hybridizes with the planarO 2px and 2py orbitals, leading to the formation of an antibonding band atthe Fermi level. This same band is further split into a LHB and UHB bythe Coulomb repulsion among electrons on the same Cu site, making thesystem an insulator.Starting from this insulating state, novel phases of matter emerge ascharge carriers are introduced into the CuO2 plane, either via chemicalsubstitution of an element or change of oxygen content in the reservoirlayers. In accordance with the charge-transfer insulator picture, the intro-duced holes preferentially reside in the ligand band with primarily oxygencharacter, while doped electrons locate on the Cu site. Figure 1.4 illustratesthe simplified phase diagram for both electron- and hole-doped cuprates,here represented by NCCO and Bi2201, respectively. The two sides of thephase diagram show similarities and differences. First of all, in both casesa superconducting phase emerges at specific doping levels whose maxi-mum Tc is reached at optimal-doping (OP) – lower and higher dopinglevels are referrred to as underdoped (UD) and overdoped (OD), respec-tively –. The SC phase in cuprates is characterized by an order parameterwith d-wave symmetry: a node is observed along the (0, 0)− (pi,pi) direc-tion (called nodal direction), while the maximum amplitude occurs at theso-called antinode corresponding to the Fermi surface crossing along the7T*T (K)T*0100200300TNAFMSC0 0.1 0.2 PGNCCO Bi2201hole Figure 1.4: Phase diagram of cuprates. Simplified phase diagram forelectron- and hole-doped cuprates. An Antiferromagnetic (AFM) insu-lating state characterizes the parent compounds at zero doping. How-ever, new phases develop upon electron or hole doping, such as SC,Charge-Order (CO) and the enigmatic Pseudogap (PG) characterizedby the onset temperature T?.(pi, 0)− (pi,pi) direction [31]. Over the years, extensive studies on varioushole-doped cuprates disclosed fascinating phases as a function of dopingand temperature, therefore extending the phase diagram beyond the di-chotomy of antiferromagnetic (AFM) insulating regime and SC. States asspin-glass, Spin-Density Wave (SDW) and charge-order (CO) are now wellestablished phenomena. If, on one hand, the richness of their phase dia-gram is what renders cuprates so attractive for exploring the interplay be-tween various degrees of freedom in strongly correlated materials, on theother hand it makes the precise description of each separate phase challeng-ing. The investigation of the CO phase is a clear example of this challenge.This electronic phase is defined by a self-organization of the valence elec-tron density into periodic structures which break the native translational8symmetry of the lattice. Although CO has been reported as a ubiquitousproperty of cuprates experimentally observed in both hole- and electron-doped compounds [18, 19], a comprehensive description of its microscopicorigin and interplay with other phases has yet to be found. Another promi-nent example is the so-called pseudogap (PG) phenomenon observed inthe UD regime. Spectroscopically, it manifests as a suppression of spectralweight in proximity of the Fermi level persisting for temperature above Tcand eventually disappearing at the onset temperature T?. Up to now, theintrinsic nature of the PG is still an open question, as well as its link (orlack thereof) with SC. In particular, it is still debated whether the PG is di-rectly related to superconductivity and characterized by preformed Cooperpairs (therefore acting as a precursor of SC), or if it is a manifestation ofa completely different competing order [32–35]. This open question alsotranslates onto the expected structure of the phase diagram. While it is ex-perimentally established that the onset temperature T? decreases as dopingincreases, its precise doping dependence in the OD regime, particularly incoincidence with the SC dome, is still subject of discussion. The scenario ofa PG gradually merging the SC dome opposes to that of a T? line enteringthe SC dome at about optimal doping and defining a quantum critical point[32, 34].Numerous theoretical models have been proposed to describe the PGin hole-doped cuprates, ranging from different type of density waves (be-ing pair or charge) [36–38] to nematic order [39, 40]. In electron-dopedcuprates, the PG is believed to bear relation with the exceptionally robustAFM phase, that extends for a much wider doping range with respect to thep-type counterpart. The association of the PG with the buildup of AF cor-relations is based on various evidences, such as the development of long-range AFM order for a Ne´el temperature (TN) about half of T?, and a mo-mentum dependence of the PG from photoemission experiments consistentwith the picture of a 2D-AFM order characterized by a wave-vector (pi,pi).That said, a direct analogy of the PG phenomenon between the two sidesof the phase diagram is troublesome, mainly due to the large ambiguity inits definition for p-doped materials. Differences are observed in terms of9(but not limited to) [33, 35, 41, 42]: (I) gap anisotropy, for electron-dopedcuprates the PG is maximum at (pi/2,pi/2) in contrast to the (0,pi) observedin hole-doped cuprates ; (II) ground state of the UD regime where the PG isstrongest, in the electron-doped it is dominated by the AFM order while SCplays a major role in the hole-doped case; (III) optical conductivity data, asno signature of the PG is reported in the ab-plane for hole-doped cuprates.The study of the interplay between different phases of the cupratesphase diagram is a key aspect of this thesis project. In particular, in Chap-ter 3 we analyze the dynamical competition between pairing strength andphase coherence in the creation of the SC condensate, confirming that in thehole-doped single-layer Bi2201 the major contribution is given by the latter.Moreover, we explore the temperature evolution of the nodal quasiparticlespectral weight across different hole-doped cuprates (Bi2201 and Bi2212)and doping levels, revealing a universal T-dependence which is unrelatedto both Tc and T?. Chapter 4 focuses on the relation between PG and AFMin electron-doped cuprates. By exploiting an ultrafast approach on NCCO,we track the temporal and temperature evolution of the PG spectral fea-tures directly in momentum-space, and demonstrate its explicit relation tothe evolution of the spin-correlation length, even when only short-rangeAFM is present.1.2 Background: samarium hexaborideThe concept of mixed-valence (MV) was introduced more than 50 yearsago to describe the specific properties of the electronic state of a solid, inwhich a given element is present in the system in more than one oxida-tion state. After its discovery, this subject has attracted intense scientificinterest in the past few decades in relation to valence fluctuations [9, 43–46]. MV occurs in numerous rare-earth compounds, driven by the stronglyatomic-like and partially filled 4 f -shell (spatially located inside the 5d and6s outer shells). Within the wide class of MV compounds, it is important todiscriminate between inhomogeneously MV compounds and homogeneouslyMV compounds. In the former case, ions corresponding to different config-10urations occupy distinct crystallographic sites, and the MV state emergesat low temperature when the intersite Coulomb repulsion dominates thekinetic energy, thus suppressing the hopping of 4 f electrons between thetwo species of ions having different valences. Rare-earth materials in thiscategory include Sm3X4 (X=S, Se or Te) and Eu4As3. Contrarily, in the caseof homogeneous MV, all the rare-earth ions occupy equivalent sites in thecrystal structure, hence the MV state is primarily defined by a single-ionproperty where an exchange at the Fermi level between the atomic-likeelectrons and the conduction band arises via hybridization. Such scenariois expected to occur when the energies corresponding to the two differ-ent valence configurations are nearly degenerate. In the case of rare-earths,these are associated to different 4 f -shell occupation numbers, n and (n− 1):the ground state would then be a mixture of both 4 f n and 4 f n−1d1 configu-rations on each rare-earth ion. Stereotypical compounds exhibiting homo-geneous MV state are TmSe, SmS, SmB6 and YbB12.In this thesis we will focus on SmB6 in particular, whose homogenousMV was disclosed in the second half of the 20th century, when spectro-scopic studies reported the presence of two distinct valence configurationsfor the Sm ions: Sm2+ (associated to 4 f 6) and Sm3+ (representing 4 f 55d1)[47, 48]. Shortly after, additional peculiar properties were reported forSmB6, such as a resistivity plateau and a linear specific heat developingat low temperatures [49, 50]. These features, commonly associated to gap-less metallic states, were inconsistent with a description of SmB6 based onthe simplest hybridization picture, where an insulating state is driven bythe gap opening via hybridization between the localized 4 f states and thebroad conduction 5d-band. A possible solution to this discrepancy was pro-posed ten years ago, when SmB6 was indicated as the first possible realiza-tion of Topological Kondo Insulator (TKI) [51, 52]. Within this description,topologically protected surface states arise within the bulk band gap openthrough the hybridization of localized f electrons with the conduction elec-trons. These theoretical results triggered a renewed interest in SmB6, withmany experimental works trying to verify the TKI scenario. Although var-ious transport and ARPES measurements claim to have successfully dis-116H6D6G/ 6F6I6P8S5D 5G5G5H5I5 I5H5K5L67F5D3G/ 5G5F 5I65I55I46P/ 4P/ 4D6F6H f 6  f 7f 5  f 6f 6  f 5f 5  f 4Intensity (a.u.)0-2 2 4 6 8 10-4-6-8-10Energy (eV)Figure 1.5: One-electron excitation spectrum of SmB6. Calculated ex-citation spectrum for the mixed-valent SmB6 in the atomic limit, in-cluding all the multiplets from the Coulomb interaction of the 4 f -shell.Being a MV system, both Sm2+ (4 f 6) and Sm3+ (4 f 5) define the groundstate generating four sets of features in the excitation spectrum, eachcorresponding to the electron removal ( f 5→ f 4 in blue and f 6→ f 5 inred) and addition ( f 5→ f 6 in green and f 6→ f 7 in purple) from one ofthe Sm configurations. The standard 2S+1LJ notation is used to denotethe contribution from different multiplets. Adapted from [4].entangled the surface band dispersion from the bulk states, and to haveunraveled the topological nature of the in-gap states [53–58], a clear an-swer has yet to emerge, with more spectroscopy studies indicating insteada bulk origin of the Fermi surface in SmB6 [59–61]. Quantum oscillationsexperiments were also not conclusive, with two groups reporting de Haasvan Alphen oscillations (dHvA) in the magnetization, yet giving oppositeinterpretations on whether the signal originates from a 2D or 3D Fermi sur-face [62, 63]. As a result, after years of intensive investigation, the groundstate of SmB6 still remains an open questions.The difficulties in describing the physics in SmB6 arise from the smallenergy scales involved. The hybridization between the partially filled stronglycorrelated 4 f states and the conduction electrons opens a gap of about∼ 20meV [56–58, 60, 64, 65] (note that this value is associated to the indirectbandgap, not excluding a direct gap as small as few meV). The Fermi levellies within this gap leading to a bulk insulating state and a mixture of theSm2+ (4 f 6) and Sm3+ (4 f 5). At low temperature, the reported Sm valence12is ∼ 2.505 [66–69], reflecting an almost equal presence of the two Sm con-figurations. Consequently, both of them need to be accounted for in thedescription of the electronic structure, as well as the 4 f -shell strong mul-tiplet effects. This gives rise to a particularly rich one-particle excitationspectrum for the 4 f -shell in SmB6, where the electron addition and removalcomponents from both Sm configurations are present, as illustrated in Fig-ure 1.5 for the atomic limit. In accordance to the schematic shown in Fig-ure 1.2a (right panel), the LHB and UHB – here constituted of f 5→ f 4 andf 6→ f 7 features, respectively – are separated by an energy of the order ofseveral eV corresponding to 2U. Therefore, despite the overall extensionof the excitations spectrum (as wide as ∼ 20 eV), the low-energy scale isdominated by the one-electron removal from 4 f 6-states reaching the lowenergy 4 f 5 (Sm2+ electron removal spectrum) and the one-electron addi-tion to 4 f 5 reaching the very low energy 4 f 6-levels (Sm3+ electron additionspectrum). The possible hole/particle excitations are narrowed down evenfurther by the inclusion of the Coefficients of Fractional Parentage (CFPs),which describe the transition amplitudes between the many-body atomicconfigurations ln−1 and ln via angular-momentum coupled electron addi-tion (a comprehensive list of coefficients can be found in [70]). In the par-ticular case of SmB6, the spatially contracted 4 f radial wavefunctions leadsto very weak one-electron overlapping integrals complimented with theCFPs to reach the lowest energy states. Thus, in the first approximation,the relevant interactions are solely involving the Hund’s ground states ofthe two Sm electronic configurations, namely the 7F0 singlet for Sm2+ (4 f 6)and the 6H5/2 state for Sm3+ (4 f 5) (see the schematic of Figure 1.6 with theground state and first excited states for the two configurations). The CFPsassociated to these two ground states reveal a dramatic reduction in thedirect f - f transition amplitudes between f 5 and f 6: the bandwidth of thef -shell gets reduced by almost 30 times with respect to the value predictedby Density Functional Theory (DFT) (which does not account for the mul-tiplets and the associated CFPs) [4, 64, 65]. These observations suggest ascenario in which the direct f - f hopping can be at zeroth order neglected,leaving the mediated f -d hopping (which is significantly less affected by13≈ 150 meV≈ 40 meVJ=1 (n=3)J=0 (n=1)J=7/2 (n=8)J=5/2 (n=6)Г6Г8Г7Г7Г8Sm2+ (4f 6) Sm3+ (4f 5)Figure 1.6: Energy level diagram of Sm2+ and Sm3+. Ground stateand first excited state energy diagram of the two Sm configurations.The ground state of Sm2+ is a singlet state, while for Sm3+ an addi-tional splitting into a quartet and a doublet is driven by the cubic crys-tal field. The degeneracy of the states is indicated by the n index. Thecharge density for 5 and 6 electrons are also shown. Adapted from[71].the CFPs effect) as the primary particle/hole excitation channel. In lightof the above discussion, it can be argued that SmB6 differs from the con-ventional Kondo system, where the unoccupied and occupied f -states arewithin range of the Fermi level; instead, it may be thought of as a highlyasymmetric Anderson-Hamiltonian impurity system, where the relevantlow-energy particle/hole excitation involving the f -shell are made possi-ble only by the hybridization between f - and d-states.In order to get further insights on the physics of SmB6, such as its MVcharacter and the possible realization of TKI, we exploit chemical substi-tution to compare the low energy electronic states in SmB6 with the analo-gous in LaB6, a compound characterized by the same crystal structure butexhibiting a metallic ground state due to the lack of 4 f electrons. Hereof,in Chapter 5 we report the study of the SmxLa(1−x)B6 series, exploringthe evolution of the electronic structure and the Sm valence by gradually14changing the stochiomety of the compounds.15Chapter 2Experimental techniquesIn this chapter we will briefly describe the experimental techniques em-ployed during this research project, namely Angle-Resolved Photoemis-sion Spectroscopy (ARPES), Time- and Angle-Resolved Photoemission Spec-troscopy (TR-ARPES) and X-Ray Absorption Spectroscopy (XAS).Since the mid ’90s, enduring technological advancements have fueledmajor improvements of both the momentum and energy resolutions, mak-ing ARPES the primary technique to study the electronic structure of stronglycorrelated materials. Based on the photoelectric effect, ARPES relies onthe information carried by a photoelectron emitted from a crystal about itsbinding energy (related to the kinetic energy) and momentum (related tothe emission angle) upon a monochromatic light excitation. By collectingand analyzing both, ARPES provides direct access to the material’s spec-tral function, which encodes information about the interactions betweenelectrons and other particles.Over the years, extensive efforts have been directed to include addi-tional degrees of freedom, such as spin and time, to the conventional ARPESframework. In particular, the extension in the time-domain is achievedvia pump-probe scheme in the so-called TR-ARPES. Here, an ultrashortlow-energy laser pulse (pump) is used to excite the system and a secondhigher energy pulse (probe) to photoemit the electrons. The relaxation backto equilibrium can be tracked by changing the time delay between these16two pulses, thus obtaining insights on the related dynamics of the spectralfunction.XAS is a core-level spectroscopy technique, in which monochromaticx-ray photons are employed to promote core electrons to the valence shellabove the Fermi level. At some specific photon energies, the absorptionprobability suddenly increases as resonances corresponding to specific ad-sorption processes are reached. Although in a momentum-integrated fash-ion, XAS represents therefore a complimentary technique to ARPES to probefor example the Upper Hubbard Band.2.1 Angle-resolved photoemission spectroscopyAngle-resolved photoemission spectroscopy (ARPES) is one of the mostpowerful techniques to study the electronic structure of solids. It is a “photon-in” −→ “electron-out” technique based on the photoelectric effect, a phe-nomenon that was first experimentally observed in 1887 by H. Hertz andW. Hallwachs [73, 74], but only in 1905 Einstein was able to theoreticallyexplain it [75]. Most of the information covered in this section follows fromexcellent review articles [76–78].When monochromatic light of energy hν illuminates the surface of asample, photons are absorbed by the electrons in the materials and, if theenergy is sufficiently high, a photoelectron can be emitted. From the energyconservation law, the kinetic energy of the emitted electron, Ekin, is relatedto the energy of the incoming photon hν by the expression:Ekin = hν−Φ− |EB| (2.1)where EB is the binding energy of the emitted electron in the material andΦis the work function of the material. The latter quantity, typically ∼4-5 eV,represents the energy barrier between the Fermi level (EF) and the vacuumlevel (Evac), i.e. the minimum energy required to remove one electron fromthe material, as illustrated in the schematic diagram of Figure 2.1. Differentincident photon energies are commonly employed in ARPES experiments,ranging from laboratory-based ultraviolet sources – such as helium gas dis-17EEVacEFEBSampleN(E)N(Ekin)SpectrumE0hνCorelevelsValence bandV0φhνEkinEFFigure 2.1: Energy scheme of the photoemission process. Schematicsof the electron energy distribution as a function of the photoelectronkinetic energy as a result of an incoming photon of energy hν. Φ andV0 are the material’s work function and inner potential, respectively.Adapted from [72].charge lamps (characterized by two emission lines, He I: 21.2eV and He II:40.8eV) or lasers – to synchrotron facilities with tunable photon energiesup to the x-ray regime. The relation of Equation 2.1 also dictates the highsurface sensitivity of ARPES via definition of the electrons mean free pathλ. This parameter represents the average length traveled by an electron in-side a solid with a probability 1/e of not undergoing any scattering events;hence, λ dictates the escape depth, i.e. the thickness of the sample whichcontributes to the primary signal. Interestingly, in spite of λ being propor-tional to the dielectric function of the material [72], a ”universal” curve is18found experimentally to describe the electron mean free path as a functionof the kinetic energy (Figure 2.2). In the particular case of ARPES, wherecommonly used photon sources are in the ultraviolet range (10-100eV), thisimplies that the resulting kinetic energies correspond to short mean freepaths (very few topmost atomic layers of the surface), making ARPES oneof the most powerful techniques for exploring the electronic properties atthe surface of quantum materials. The main drawback of ARPES surfacesensitivity is the requisite to perform the measurements in Ultra High Vac-uum (UHV) – base pressure better than 10−10 torr – to ensure the sampleremains clean on an atomic level throughout the experiment. This aspect ofthe ARPES technique also limits the investigation of those samples whichcan be appropriately prepared in-situ either by cleaving of bulk crystals,cleaning of the exposed surface via sputtering and annealing or by in-situgrowth.The key feature of ARPES is the possibility to analyze the emission an-gles of the photoelectrons in addition to their kinetic energy, allowing afull reconstruction of the electronic dispersion as a function of both energyand momentum. In this regard, a major advancement in the developmentof ARPES derived from the advent of two-dimensional detectors able tosimultaneously analyze both the kinetic energy of the photoelectrons andthe angles of emission. Such measurements are commonly achieved withhemispherical analyzers: the emitted electrons are collected through an en-trance slit, subjected to an electric field in the lens column to match a presetpass energy and then dispersed in the hemisphere accordingly to their ki-netic energy and emission angle, before reaching a phosphor screen andbeing subsequently detected by a camera. As in the case of Equation 2.1,conservation laws can be used to link the crystal momentum }k charac-terizing the electron inside the solid with the momentum }K of the photo-electron outside the material. However, since the translational symmetry isbroken at the surface of the solid, only the parallel component of the mo-mentum can be conserved:kx = Kx =1}√2mEkin sinθ cosϕ (2.2)190.1110100Electron mean free path λ (nm)1 10 100 1000Electron kinetic energy (eV)Figure 2.2: Electron mean free path in solids. ”Universal” curve forthe electron escape depth as a function of the electron kinetic energy invarious solids. Common energies involved in ARPES experiments (∼10-100 eV) correspond to few atomic layers, making it a high surface-sensitive technique. Adapted from [79].ky = Ky =1}√2mEkin sinθ sinϕ (2.3)where θ and ϕ are the polar and azimuthal angle, respectively, as definedin Figure 2.3. The determination of the perpendicular component of themomentum is not that straightforward and it requires the inclusion of theinner potential V0, describing the bottom of the valence band. The thirdcomponent of the momentum is then given by:kz =1}√2m(Ekin cos2 ϕ+V0) . (2.4)Different strategies are used in order to determine V0 [76, 80]; experimen-tally, the most commonly adopted one is by tuning the energy of the ex-citing photons and analyzing the resulting periodicity in kz. Note that theprecise evaluation of kz becomes relevant only in the investigation of three-20e-xyzϕθSamplehνFigure 2.3: Geometry of an ARPES experiment. General geometry ofan ARPES experiment where upon an incoming photon of energy hνan electron is emitted at polar angle θ and azimuthal angle ϕ . Adaptedfrom [76].dimensional materials, while for (quasi-) two-dimensional compounds theelectron dispersion is (mostly) fully confined in a plane and kz is irrelevant.In both cases, single crystals are needed to guarantee the translational sym-metry required to measure electron momentum.The creation and emission of a photoelectron is a more complex pro-cess than the one shown in Figure 2.1, where the electron is removed fromits initial state in the material and directly collected onto the detector. For-mally, it results easier to describe the photoemission process in terms of athree-steps model, in which each photoelectron undergoes three indepen-dent events:1. Photoexcitation into a final bulk state due to the absorption of an in-coming photon;2. Transport to the surface, during which several inelastic scatteringevents might happen giving rise to the so-called secondary electron sig-nal (a broad background with increasing intensity at low kinetic en-ergies), to be distinguished from the primary electrons, which did notexperience inelastic collisions and hence respond to the conservation21laws shown in Equation 2.1, Equation 2.2 and Equation 2.3;3. Escape from the surface by overcoming the potential barrier and con-sequent collection into the detector.In this framework, Fermi’s golden rule becomes instrumental for a formaldescription of the photoemission process. It describes the transition prob-ability wi f between an initial state, ψi, and an excited final state, ψ f , uponabsorption of a photon with energy hν:wi f =2pih¯∣∣∣∣ 〈ψ f |Hint|ψi〉 ∣∣∣∣2δ(E f − Ei − hν) , (2.5)where Ei and E f are the energies of the initial and final state, respectively.The term Hint describes the light-matter interaction between an electron ofmass m and the electromagnetic field A and, by using first order perturba-tion theory, it can be expressed by:Hint = − emc (A · p) . (2.6)So far the photoemission process has been addressed as a single-particlemechanism; however, in the realistic case of interacting systems, a N-bodydescription must to be adopted. In this regard, a commonly used assump-tion is the sudden approximation, which considers the photoemission processinstantaneous and therefore makes it possible to neglect any interactionsbetween the photoelectrons and the core hole left behind in the photoe-mission process (i.e. the effective potential of the system changes discon-tinuously at the instant the photoelectron is instantaneously removed) 1.Note that such approximation may be inappropriate when probing the sys-tem with low photon energies since the resulting low kinetic energy of thephotoelectrons may imply a longer time to escape in the vacuum than thesystem response time. In the following, the discussion to address in more1In particular, this assumption implies that the (N − 1)-particle wavefunction remainsunchanged when the interaction Hint is switched on and off, thus allowing the use of theinstantaneous transition probabilities wi f via Fermi’s golden rule [81].22details the analysis of the photoemission process will be restricted to thecontext of the three-step model and the sudden approximation.Under these assumptions, both the N-particle initial state, ψNi , and theexcited final state, ψNf , can be described in terms of single Slater-determinantproduct between the one-electron function Φk and a (N − 1)-particle term:ψNi =AΦki ψN−1i (2.7)ψNf =AΦkf ψN−1f (2.8)whereA is the antisymmetric operator ensuring the fulfillment of the Pauli’sprinciple. The matrix element of Equation 2.5 can then be written as:〈ψNf |Hint|ψNi 〉 = 〈Φkf |Hint|Φki 〉 〈ψN−1f |ψN−1i 〉 , (2.9)where Mki f ≡ 〈Φkf |Hint|Φki 〉 is the one-particle matrix element, while thesecond term represents the (N − 1)-particle overlap integral. By choos-ing the eigenstates of the (N − 1)-particle Hamiltonian (labelled by m) asconvenient basis, we can sum over all the possible final states that can beinvolved in the excitation process to obtain:Wi f =∑i, f|Mki f |2∑m|〈ψN−1m |ψN−1i 〉|2δ(Ekin + EN−1m − ENi − hν)︸ ︷︷ ︸A−(k,E). (2.10)The summation over m describes the probability of all possible electron-removal processes from an initial state i and momentum k to a (N − 1)-particle excited final state m, and it defines the spectral function (in this casefor electron removal), A−(k, E). As can be seen from Equation 2.10, thisquantity encodes fundamental information about the correlations presentin the system. In fact, in the non-interacting picture ψN−1i = ψN−1m∗ for aparticular m = m∗ and thus only the term corresponding to m∗ in the sum-mation is not zero. However, in the case of strongly correlated materials,the photoelectron removal leads to a strong overlap of ψN−1i with multi-ple ψN−1m , so A−(k, E) would deviate from a single series of delta functions23and show instead additional peaks or tails corresponding to all the excitedstates m created in the process.In mathematical terms, the most common approach to discuss the pho-toemission on solids is based on the Green’s function formalism, in whichthe time-ordered one-electron removal Green’s function (or propagator) de-scribes the time evolution of a state where an electron with momentum kis removed from the system at time zero. By taking the Fourier transform,the real space propagator can be expressed in the energy-momentum rep-resentation:G−(k, E) =∑m|〈ψN−1m |ck|ψNi 〉|2(E− EN−1m − ENi + iη), (2.11)where the operator ck annihilates an electron with energy E and momen-tum k in the N-particle initial state ψNi and η is a positive infinitesimal. Inthe limit η→ 0+, the identity (x+ iη)−1 = P(1/x)− ipiδ(x) is valid (whereP represents the principal value; furthermore, from here on we will takeh¯ = 1) and allows to rewrite the Green’s function as:G−(k, E) =∑m(|〈ψN−1m |ck|ψNi 〉|2)(P(1E− EN−1m − ENi)− ipiδ(E− EN−1m − ENi ))(2.12)By comparing Equation 2.12 with Equation 2.10, a simple relation can bederived connecting the Green’s function to the one-electron spectral func-tion:A−(k, E) = − 1piImG−(k, E) . (2.13)The main advantage of adopting the Green’s function formalism is thepossibility to extend the photoemission description to the case of many-body interactions via inclusion of the complex electron self-energy Σ(k,ω) =Σ′(k,ω) + iΣ′′(k,ω). Its real and imaginary parts describe the renormal-ization of the energy and lifetime of an electron with bare-energy εbk andmomentum k due to many-body interactions. The self-energy can be usedto expressed both the one-electron Green’s function and spectral functionas follow:G(k,ω) =1ω− εbk − Σ(k,ω)(2.14)24andA(k,ω) = − 1piΣ′′(k,ω)[ω− εbk − Σ′(k,ω)]2 + [Σ′′(k,ω)]2. (2.15)Following from Equation 2.10 and Equation 2.15, one can calculate the totalphotoemission intensity as a function of energy ω and momentum k:I(k,ω) = I0(k,ω) f (ω)A(k,ω)= − 1piI0(k,ω) f (ω)Σ′′(k,ω)[ω− εbk − Σ′(k,ω)]2 + [Σ′′(k,ω)]2,(2.16)where I0(k,ω) is proportional to the square of the dipole matrix element|Mki f |2 = |〈Φkf |Hint|Φki 〉|2 and consequently depends on the momentum ofthe electron, the incident photon energy and polarization and the particu-lar experimental geometry [76, 80]. Equation 2.16 also includes the Fermi-Dirac distribution term f (ω) = [e(ω−µ)/kBT + 1]−1 to properly describe thephotoemission process as a probe of only the occupied energy states (elec-trons are removed from the system, µ represents the Fermi energy). Thisterm also explains why, in order to limit the broadening arising from kBT,most ARPES experiments are carried out at low temperature. Althougheffects due to resolution broadening in both energy and momentum, aswell as possible background signal, may affect the photoemission intensity,Equation 2.16 shows the capability of ARPES to access the electron-removalspectral function and the encoded information about correlations via self-energy.Experimentally, the aforementioned connection is achieved through theanalysis of the spectral lineshape and two main strategies are often used, fo-cusing either on the energy- or the momentum-dependence of the photoe-mission signal. The first approach consists in computing the photoemissionintensity at a fixed moment k∗ (where now k refers to the two-dimensionalmomentum conserved during the photoemission process) and thus ana-lyzing the energy-dependence of the so-called Energy Distribution Curve(EDC). The second focuses instead on the momentum-dependence at afixed binding energy ω via the so-called Momentum Distribution Curve25(MDC). Although the precise determination of Σ(k,ω), and consequentlyof A(k,ω), often turns out to be very challenging, the analysis of the spec-tral lineshape via EDC and MDC is a fundamental tool to this goal. In thisregard, it is instructive to illustrate the simple interacting Fermi liquid case.Close to the Fermi level (i.e. EF − h¯ω| Σ′′ |), the system can be modeledin terms of quasiparticles, describing electrons ”dressed” with a collection ofexcited states moving coherently with them. Within this model, the quasi-particle spectral function can be divided into a coherent part and incoherentpart (Aincoh), with only the former associated to the low energy excitations:AFL(k,ω) = 2piZk 1pi12τk(ω− ε˜k)2 +(12τk)2+ Aincoh . (2.17)Here, the term Zk = (1 − ∂Σ′/∂ω)−1 is a renormalization constant, alsocalled coherent factor and it gives a measure of the quasiparticle strength(generally speaking, Zk equal to 1 corresponds to the non-interacting case,while Zk equal to 0 to the strongly correlated case where no coherent statecan be excited). The other two terms involved in Equation 2.17, τk and ε˜k,are the quasiparticle lifetime and its renormalized energy, respectively. Tofirst order, they are related to the real and imaginary part of the self energyby the following expressions: ε˜k = Zk(εbk + Σ′) and τk = (−2Σ′′)−1. Notethat in the formalism of Equation 2.17, the self-energy and its derivativesare evaluated at ω = ε˜k. EDC and MDC provide thus access to different as-pects of the quasiparticle spectral function: while the former relates to theenergy-dependence of Σ(k,ω), the latter assumes a Lorentzian lineshape(assuming that εbk can be linearized about the MDC maximum) whose fullwidth half maximum (FWHM) can be used to estimate the lifetime of thequasiparticle τk.2610fs ps ns s0.1 11.0 0111.00111.0011CARRIEREXCITATIONTHERMALIZATIONCARRIERREMOVALTHERMAL ANDSTRUCTURAL EFFETSabsorption of photonsimpact ionizationcarrier-carrier scatteringcarrier-phonon scatteringAuger recombinationradiative recombinationcarrier diffusionablation & evaporationthermal diffusionresolidificationFigure 2.4: Timescale of various carrier and lattice processes in asolid upon laser excitation. Overview of relevant timescale of differ-ent relaxation processes in solids following ultrafast optical excitation.Adapted from [82].2.2 Time- and Angle-resolved photoemissionspectroscopyARPES provides the capability to directly probe the electronic structure ofquantum materials by simultaneously resolving energy and momentum.However, being based on the photoelectron effect, its application is limitedto the investigation of occupied electronic states in equilibrium conditions.In order to gain access to out-of-equilibrium phenomena and associated dy-namical properties, a time resolved approach is needed. Non-equilibriumspectroscopies have attracted increasing interest in the context of the studyof strongly correlated materials. They offer the unique opportunity to ex-plore new phases of matter otherwise unattainable at equilibrium or to dis-entangle in the time domain energetically intertwined orders based on thedistinctive timescale that describes the recovery of the equilibrium initialstate. Figure 2.4 outlines the main relaxation processes occurring in a solidfrom a femto-second to micro-second timescale as a result of an optical ex-citation, where four regimes can be distinguished: carrier excitation, ther-malization, carrier removal and thermal and structural effects [83]. In par-27   xyz  xyz    xyz δtδtInfrared pumpUV probee-SampleFigure 2.5: Pump-Probe scheme for time-resolved ARPES. Illustra-tion of the pump-probe approach adopted in a time-resolved ARPESexperiment: pump and probe pulses, delayed by δt, are sent to thesample; at t=0 the infrared pump pulse hits the sample inducing anonequilibrium state; the probe pulse ejects photoelectrons at a time δtafter the excitation, while the system is relaxing back to equilibrium.ticular, in this thesis work we will focus on the fast dynamics, 0.1 ps - 10 ps(yellow shadow in Figure 2.4). The advent of ultrafast laser sources able todeliver ultrashort and coherent light pulses (of few to several hundreds offs duration) triggered the development of time-resolved optical techniquesbased on the pump-probe scheme. The same approach is adopted to extendthe conventional ARPES technique into the time domain, as shown in Fig-ure 2.5. While in equilibrium ARPES only one ultraviolet light beam isemployed, time-resolved ARPES (TR-ARPES) makes use of two ultrashortlaser pulses of different frequency hitting the sample in close succession.The first pulse, called pump and typically in the infrared regime, excites thesample into a non-equilibrium state. Shortly after such excitation a secondpulse, called probe and with sufficiently high photon energy, is sent to thesample to induce the photoemission process. By varying the time delaybetween the two pulses, snapshots of the electronic structure are collectedas the system relaxes back to equilibrium, providing insights on the relatedelectron dynamics and recovery mechanisms of the material under consid-eration.At present, various TR-ARPES systems are available worldwide. Themost common class of instruments relies on solid-state laser sources wherethe ultraviolet probe pulse is generated by sum-frequency generation in28nonlinear crystal [84–87]. This approach retains an upper limit on the max-imum achievable probe energy of 6-7 eV, which provides remarkable mo-mentum resolution but at the same time strongly constrains the accessi-ble momentum-space via Equation 2.2 and Equation 2.3. Therefore, thesesystems are well suited for the investigation of those electronic states inthe vicinity of the first Brillouin zone’s Γ point, such as the Dirac statesin Topological Insulators or the nodal region in high-temperature cupratesuperconductors. In order to overcome the low-photon energy limitationand get access to larger momenta, high-order harmonic generation (HHG)techniques have been developed by exploiting the nonlinear processes oc-curring when a strong laser pulse interacts with an atomic gas (e.g Kr orAr). Although the necessity of having large electric field to induce nonlin-ear processes in atomic gases negatively affects the energy resolution of theoutcoming harmonics, recent developments of new HHG schemes yield toa resolution comparable to the standard 6 eV-setup but on a probe photonenergy of the order of several tens of eV [88–91]. Finally, at the largest scale,TR-ARPES is recently being addressed in the context of newly built (or un-der construction) x-ray free electron laser (FEL) sources [92, 93]. The possi-bility of tunable photon energy, variable pulse duration and wide range ofaccessible momenta makes TR-ARPES experiments based on FELs sourcesan exciting prospect for future investigations of quantum materials.2.2.1 TR-ARPES at UBC with 6.2-eV probeFigure 2.6 shows the experimental apparatus employed for the TR-ARPESmeasurement presented in this thesis work. It is based on a commerciallypurchased Ti:sapphire laser consisting of the VitesseDuo oscillator (pumpedby a 10W Nd:YVO4 continuous-wave Coherent Verdi laser) and the regen-erative amplifier RegA 9000 both by Coherent Inc. This combined systemprovides trains of light pulses in the near-infrared range (800 nm, corre-sponding to 1.55 eV) with a pulse duration of ∼ 180 f s and at a repetitionrate of 250 kHz. The output beam is split in two: the reflected portion isused as pump while the transmitted one is driven through a cascade of non-29pumpprobeSHGTHGFHGpump-probedelay stageVitesseDuo + RegA 9000output beam:800nm - 180fs - 250kHzsampleprobe 200nmpump 800nmseparatorbeamsplittermirror BBOcrystallensFigure 2.6: Schematic of the 6 eV TR-ARPES setup at UBC. A simpli-fied sketch of the experimental TR-ARPES setup in Damascelli’s groupat UBC.linear processes to generate the fourth-harmonics probe at 200 nm, equiva-lent to 6.2 eV. In particular, the latter is achieved by three subsequent sum-frequency generations in phase-matched BBO (β-Barium Borate) crystals.Once undertaken the nonlinear generation, the probe is recombined to thepump beam in a collinear geometry (to guarantee spacial overlap) and fo-cused on the sample in the UHV experimental chamber at a 45◦ angle ofincidence. The spot size of the two beams have been estimated by using abeamprofiler to 250µm and 180µm for pump and probe, respectively. Fi-nally, the time delay between the pump and the probe beam is controlledby a motorized translation stage placed in the pump path. Once the probepulse hits the sample, the photoemitted electrons are collected and mea-sured by means of their energy and momentum using an hemisphericalanalyzer conforming to the conventional ARPES technique.The characterization of the overall temporal and energy resolution isimperative in any TR-ARPES experiment. One approach to estimate thesystem’s temporal resolution relies on tracking the photoemission inten-30sity as a function of the pump-probe delay for states well above the Fermilevel (>2 eV) in polycrastilline gold. At this energies, the time resolutionis expected to exceed the intrinsic electronic states lifetime so that in goodapproximation the temporal response of the photoemission signal is solelydetermined by that of the pump and probe themselves. The resulting tem-poral profile can be fit by a Gaussian whose FWHM returns a duration forthe probe pulse of 250 fs (assuming a 180 fs pump pulse). As for the total en-ergy resolution, an estimate can be achieved by measuring at low temper-ature the Fermi edge width of any samples with electronic bands crossingthe Fermi level. In the case of TR-ARPES measurements on cuprate super-conductors presented in this thesis, the equilibrium (i.e. acquired withoutthe pump beam) band dispersion along the nodal direction can serve to thisgoal and sets the system’s energy resolution to 17 meV.Beside the overall temporal and energy resolution, other aspects of theprobe and pump pulses need to be examined in performing a TR-ARPESexperiment. As the counting rate scales linearly with the incident photonflux, one might imagine that the statistics and the integration time could bedrastically improved by simply increasing the intensity of the probe pulse.However, we must consider the space charge effect originating from re-pulsive Coulomb forces that photoelectrons exert on each other when theyare emitted nearby in time and space. This effectively leads to a broad-ening and distortion of the ARPES spectrum, both in energy and momen-tum [94–97]. Although potentially relevant in any photoemission exper-iment, space charge becomes crucial in TR-ARPES measurements due tothe combination of ultrashort laser pulses, small spotsize and typically lowkinetic energy of the photoelectrons. In this work, space charge effects wereminimized by decreasing the intensity of the probe beam until the relatedenergy-resolution broadening effects at the Fermi level became comparableto the background resolution (17 meV). Note that in this setup the maxi-mum pulse energies are typically< 50nJ, thus excluding any field emissioneffects from a large (mm size) flat metal surface (comparable pulse energiesare in fact used to induce field emission in nanometer tips) [98–100].31Lastly, the pump fluence represents another fundamental quantity in aTR-ARPES experiment as it directly relates to the initial density of excitedcarriers. By changing the pump fluence is thus possible to explore differentexcitation regimes of the system under study. Similarly to the probe pulsecase, very high pump fluences may lead to distortions in the ARPES spec-trum: as the excitation density increases, multi-photon photoemission fromthe pump pulses might take place creating additional photoelectrons whichmight even dominate the direct photoemission signal [94]. In our work, thevalue of the pump fluence was varied by directly adjusting the power of thepump beam using a variable filter placed along the beam path. For each se-lected pump fluence the absence of multi-photon effects was verified bythe lack of photoemission signal with the only pump beam illuminatingthe sample. The inaccuracy in determining the average power of the pumpbeam (usually measured with a powermeter) and the spotsize on the sam-ple, alongside with losses from the viewport, reflect in the uncertainty ofthe fluence values given for each TR-ARPES experiments presented in thisthesis.2.3 X-Ray Absorption SpectroscopyX-ray spectroscopy techniques involve the production and monitoring ofradiative transitions between ground and excited states. After the discov-ery of x-rays in 1895 by Rontgen [101], it took two decades before the firstx-ray absorption spectrum was observed by De Broglie [102]. Since then,core-level x-ray spectroscopies have come a long way promoted by the re-markable advancements made in both the experimental and theoretical ca-pabilities. As an example of groundbreaking theoretical progress, the ap-plication of Fourier analysis to the theory of x-ray absorption (XAS) madein 1971 by Sayers, Stern, and Lytle paved the way to a quantitatively esti-mation of structural parameters (such as bond distance and coordinationnumber) from the experimental spectrum [103]. On the experimental side,the advent of broadband synchrotron radiation sources starting from the1970s argubly represent the main advancement which fueled the develop of32Attenuation(m-1)Energy (eV)NEXAFSEXAFSBackgrounda bFigure 2.7: X-Ray Absorption technique. (a) Schematic of the x-rayabsorption process and related decay processes involving the emissionof Auger electrons or fluorescent photons. Filled and open spheresrepresent electrons and holes, respectively. From [104]. (b) Illustrationof the three main regions characterizing a XAS spectrum. Adaptedfrom [105].x-ray spectroscopy techniques into reliable and routine experimental tools.The possibility to select a specific incoming photon energy and polariza-tion, along with a high flux are just some of the significant benefits offeredby modern synchrotron radiation facilities.In a simple one-electron picture, XAS involves the interaction betweenan incoming photon and a core-level electron, as shown in Figure 2.7a. Theabsorption of the photon by the electron can either excite the latter to abound unoccupied state or eject it into the continuum. A typical XAS ex-periment involves scanning the incoming photons energy while detectingthe absorption of such photons: the resulting spectrum would then pro-vide the x-ray absorption coefficient as a function of the incident x-ray en-ergy (an example is displayed in Figure 2.7b). When such energy hits par-ticular values, mainly determined by the atomic number of the absorbingatom, the absorption probability results strongly enhanced marking whatis called an ”edge”. Each edge represents a distinct core-level binding en-ergy and by convention is labelled accordingly to the principal quantumnumber of the excited electron: K associated to n = 1, L to n = 2, M to n = 333and so on. The absorption below an edge is dominated by background pro-cesses which include elastic and inelastic scattering processes, lower energyedges absorption, etc. In the immediate vicinity of an edge, in the regionoften called the Near Edge X-ray Absorption Fine Structure (NEXAFS), theabsorption is instead characterized by the presence of fine structure associ-ated with resonant transition between core electrons and excited states ofthe material (Figure 2.7b). It is this fine structure that encodes informationabout the electron and magnetic structure of the absorbing atom. In thepresence of a strong Coulomb interaction between the core-hole and thevalence electron (as often the case in TMOs and rare earth compounds), thefine structure observed in NEXAFS (combined with the dipole selectionrules discussed lateron in this section) provides direct information aboutthe ground state occupation of the valence state [106]. The investigation ofthe part of the spectrum well above the edge is instead called Extended X-ray Absorption Fine Structure (EXAFS), where the excited core electron hassufficiently high energy to reach the continuum (Figure 2.7b). This regionshows a gradual decrease in the absorption rate along with superimposedoscillating features reflecting the scattering processes undergone by the ex-cited electrons with neighbouring atoms.Any XAS experiments begin with the interaction between the incidentphoton and the bound core level electron. Although such interaction be-tween x-rays and matter can be much more complicated, a single-particlepicture can be used to provide a simplified approach which grasps the un-derlying physics and the fundamental derivation of the key dipole selectionrules. As for the case of the ARPES technique presented in Section 2.1, thestarting point for a formal description is the expression via Fermi’s goldenrule of the transition rate of an initial state ψi into a particular final state ψ fas a result of the absorption of a photon with energy hν, as in Equation 2.5.By applying a first order dipole approximation for the perturbing potential,the absorption rate can be written as:Ri→ f ∝ ∑i→ f2pih¯∣∣∣∣ 〈ψ f |eˆ ·~p|ψi〉 ∣∣∣∣2δ(E f − Ei − hν) (2.18)34where eˆ is the unit vector describing the polarization of the incident electro-magnetic wave and ~p the electron linear momentum operator. This prob-ability is essentially what is being measured in XAS spectroscopy. By ex-panding the term〈ψ f |eˆ ·~p|ψi〉making use of the spherical harmonics andthe Clebsch-Gordan coefficients, Equation 2.18 can be used to derive thedipole selection rules. These rules describe the restrictions in the transitionprocess in terms of the quantum numbers and play a key role in the theoryof core-level x-ray spectroscopy techniques. Equation 2.19 lists the moregeneral selection rules for multi-electron atoms. In the presence of signifi-cant spin-orbit interactions, the restrictions on the total angular momentumJ = L+ S and its projection along the z axis, MJ , have to be used rather thenthose for independent L and S.∆L = 0,±1, (L = 09 L = 0)∆l = ±1∆S = 0∆J = 0,±1∆MJ = 0,±1(2.19)Note that the selection rules listed above are valid only within the elec-tron dipole approximation, which holds only in the soft x-ray absorptionprocess where the wavelengths are relatively long compared to the radialextend of the core level wave functions.Different detection techniques can be employed in a XAS experiment.The simplest way to measure the absorption rate is by placing the samplebetween the source and the detector, in the transmission detection scheme(Figure 2.8a): the attenuation of the x-rays would be quantified by howmuch the signal drops on the detector. This process is described by Beer-Lambert law:I(ω) = I0(ω)eµ(ω)D ⇒ µ(ω) = − 1D ln(I(ω)I0(ω))(2.20)where I and I0 are the transmitted and incident beam intensities respec-35Transmissione-IDTEY, PEYTEYTFY, PFY, IPFYx-raydetectorsamplea b cFigure 2.8: Common detection approaches in XAS experiments. (a)Illustration of the transmission technique, where the XAS is recorderby monitoring the fraction of photons passing through the sample. (b)Scheme of electron yield techniques (TEY or PEY), where the emit-ted electrons during the relaxation of the x-ray excited states are de-tected (either by direct collection of surface-emitted photoelectrons orby monitoring the replenishing current through the sample). (c) De-piction of the fluorescence yield technique (TFY, PFY and IPFY), whereemitted fluorescence photons are detected. The blue shadow qualita-tively represents the probing depth of each detection technique. From[105].tively and h¯ω is the energy of the x-rays. The other two terms are thesample thickness D and the absorption coefficient of the sample µ (alsocalled the attenuation coefficient). This parameter is related to the absorp-tion cross-section and depends on the material’s properties [106]. The stan-dard XAS spectrum would then be given by plotting µ(ω).In many cases this transmission approach is not the most practical. First,for each sample it requires an energy-dependent optimization to matchthickness and density. Second, if µ(ω)D  1, the photons cannot pene-trate through the sample in order to reach the detector, so no signal wouldbe detected. Even in the case when the attenuation is not big enough toprevent any signal to be detected, it can still lead to artificial suppressionof the absorption peaks and thus affect the analysis [105]. As an alterna-36tive approach, one can resort to yield techniques. After the absorption ofan x-ray the system is left in an excited state which quickly decays by re-leasing energy in the form of Auger electrons or photons. In practice, itis often found that the number of transmitted photons (i.e. the absorptionsignal) can be mapped quite consistently into the number of these emittedAuger electrons or photons. When the emitted electrons are detected as afunction of incident photon energy to reconstruct the XAS signal, the tech-nique is known as Total Electron Yield (TEY) [see Figure 2.8b; if instead ofdetecting all the electrons, only the ones with certain energies are collectedis called Partial Electron Yield (PEY)]. Note that, instead of directly detect-ing the electrons as they are emitted from the surface, a common procedureis to measure the magnitude of the positive current induced in the sampleupon x-ray irradiation. In either case, a fundamental aspect of TEY/PEY isthe small probing depth limited by the mean free path of the electrons thatare freed in the absorption process (similar argument that applies for thesurface-sensitivity of ARPES).Total Fluorescence Yield (TFY) [or Partial Fluorescence Yield (PFY) ifonly a particular set of fluorescence transition is monitored] relies on thedetection of the photons emitted in the relaxation process following the x-ray excitation (Figure 2.8c). The primary strength of this approach is thehigher bulk sensitivity respect to electron yield techniques: the probingdepth of TFY and PFY is dictated by the photons penetration depths whichis of the order of ∼ 100 nm for soft x-rays. This comes as a main advantagein the study of reactive materials where surface oxidation and contamina-tion may affect the intrinsic signal probed by a surface-sensitive technique.Note that the fluorescence yield spectra are susceptible to some distortionssuch as saturation and self-absorption effects; these have then to be takeninto account while analyzing the XAS signal [107–110].Lastly, a new yield technique termed Inverse Partial Fluorescence Yield(IPFY) has been recently developed, in which fluorescence photons froma non-resonant lower energy edge are detected. As more incident pho-tons are absorbed at the resonant edge, the non-resonant fluorescence back-ground decreases giving an inverted signal directly proportional to the true37XAS spectrum [109, 110]. Although very recent and not always attainable(as it requires a lower energy edge available), when applied IPFY may pro-vide the most reliable spectra among all the yield techniques. A combina-tion of PFY and IPFY have been used to collect the XAS spectra presentedin Chapter 5.38Chapter 3Tracking the spectral functionof Bi-based cuprates byTR-ARPESCopper-oxide high-temperature superconductors are the prototypical ex-ample of 3d strongly correlated systems and a multitude of competing or-dering phenomena and phases characterize their phase diagram. Here, weemploy TR-ARPES to directly access the temporal and temperature dynam-ics of the one-electron spectral function in hole-doped Bi-based cuprates.By optically perturbing the system, we report the ultrafast enhancement ofthe phase fluctuations in the single-layer Bi2Sr2CuO6+δ, while only mildlyaffecting the pairing strength, consistent to what was previously reportedfor the bi-layer compounds. Furthermore, this transient approach allows usto also address the temperature-dependent meltdown of the nodal quasi-particle peak with unprecedented sensitivity. The observed suppression ofspectral weight is shown to be ubiquitous of the Bi-based cuprates familyand proposed to stem from the temperature dependence of the imaginarypart of the self-energy within the Fermi-liquid description.393.1 IntroductionHigh-temperature superconductivity (HTSC) is one of the most fascinat-ing phenomena and yet unsolved puzzle of the last 30 years in condensedmatter. Among the different families of materials exhibiting HTSC, theBi-based hole-doped cuprates are the most intensively studied [111, 112].Over the years, extensive theoretical and experimental efforts (by meansof different techniques ranging from optics, transport, photoemission spec-troscopy and magnetic probes) gradually disclosed the richness of the cuprates’phase diagram. Besides the Mott-insulating state characterizing the parentcompounds and the superconducting dome, numerous intertwined phasessuch as charge-order, spin-density waves and pseudogap emerge upondoping, making the cuprates an exceptional platform to investigate the in-terplay and competition among different correlated states of matter. Alongwith this exciting opportunity comes the challenge to precisely describeeach separate phase, with thermal excitations often obfuscating the appear-ance of different phases and potential thermodynamic phase transitions. Inthis context, the use of experimental tools capable to disentangle degreesof freedom associated to different competing orders become instrumental.Time- and angle-resolved photoemission spectroscopy (TR-ARPES) takesthe capability of directly visualizing the electronic dispersion in momen-tum space typical of ARPES into the time domain, providing access to itsdynamics and disclosing transient regimes otherwise unattainable for in-vestigation within the equilibrium framework.The interplay among different phenomena characterizing the SC phasetransition in cuprates is a remarkable example where the possibility to dis-entangle in the time domain underlying mechanisms occurring on the sameenergy scale plays a crucial role in establishing the hierarchy in the forma-tion of the condensate. In superconducting materials the transition tem-perature Tc has been proposed to be controlled by two distinct phenom-ena, namely the creation of electron pairs (characterized by the pairing en-ergy Ep) and the persistence of macroscopic phase coherence (whose en-ergy scale h¯ΩΘ is related to the zero-temperature superfluid density) [113,40TcT (K)TpairTΘDoping 0 0.5T/Tceq1.51Tel TΘΓpΔTΘ Tel0 0.5T/Tceq1.51ΓpΔBefore Aftera b(p,p)(p,0)(0,0)(0,p)05101500.51ΔIpeak (%)lab 2(0)/lab 2(T)Bi2212OD84FWHM (meV)30200 0.5T/Tc1Bi2212YBCOYBCO, µSRMicrowave0-0.4Energy (eV)Intensity (a.u.)c dFigure 3.1: Electron pairing vs phase coherence and suppression ofquasiparticle spectral weight in Bi2212. (a) Schematic phase diagramof cuprates illustrating the interplay between electron pairing (withonset Tpair) and the stability of phase coherence (with ordering temper-ature TΘ) in defining Tc. Adapted from [113]. (b) Cartoon of the pos-sible suppression of SC driven by the loss of phase coherence: uponan optical excitation, while the gap size ∆, which relates to the elec-tronic temperature, gets only mildly increased (red sphere), the on-set of phase fluctuations TΘ gets reduced below the equilibrium Tc(green sphere and dashed line), driving the system out of the SC state.Adapted from [114]. (c) Temperature dependence of the momentum-integrated spectra collected at the antinode (grey circle in the inset)for overdoped Bi2212 (Tc ∼84 K). (d) Temporal evolution of the ampli-tude of the quasiparticle peak ∆Ipeak and FWHM (top panel) and thecomparison with the penetration depth ratio λ2ab(0)/λ2ab(T) – propor-tional to the superfluid density – (bottom panel), suggesting a directrelation between the suppression of quasiparticle spectral weight andsuperconductivity. Panels (c)-(d) adapted from [115].41116, 117]. While for conventional BCS superconductors the pairing energyis significantly smaller than the onset of phase fluctuations, hence unques-tionably defining the SC phase transition, the low superfluid density withinthe Cu-O planes in cuprates leads the two phenomena to occur on the sameenergy scale (∼ tens of meV in Bi-based compounds [113, 118]). The criti-cal temperature Tc may then be defined by the onset of phase fluctuationsTΘ rather than the temperature Tpair below which the pairing mechanismbecomes locally relevant (see schematic in Figure 3.1a) [113]. This proposalfinds supports in various equilibrium measurements on UD cuprates re-porting a non-zero pairing gap up to ∼ 1.5Tc, while pair-breaking scatter-ing processes emerge sharply at Tc [119–122]. These findings suggest thepossibility to induce a collapse of SC via manipulation of the density ofphase fluctuations alone, independently of the number of across-gap ex-citations, as schematize in Figure 3.1b. The viability of this scenario hasbeen validated in the UD bi-layer compound Bi2212 (Tc ∼ 82 K) by a re-cent TR-ARPES experiment by Boschini et al. [114]. Upon an infrared ul-trashort pulse excitation, while the amplitude of the SC gap is only slightlyaffected, the pair scattering rate Γp (associated to the presence of phase fluc-tuations) exhibits a significant change and a faster dynamics with respectto that of charge excitations, designating the phase coherence as the pri-mary mechanism controlling the formation of the macroscopic condensate(i.e. determining Tc). Furthermore, as an additional result of the ultrafast IRexcitation, a clear suppression of the quasiparticle spectral weight has beenobserved, consistent to what was reported as a function of temperature inother ARPES studies on Bi2212 [114, 115, 123, 124]. Such a suppressionof quasiparticle spectral weight, manifesting near the antinodal region (pi,0), as well as along the nodal direction where the SC gap is null, seemsto exhibit an abrupt change around the superconducting phase transitiontemperature fading off for T>Tc, and thus suggesting a direct relation tothe condensate density (see Figure 3.1c-d).In order to further investigate the stability of the phase coherence inBi-based cuprates as a function of the number of Cu-O planes, as well asthe viability of the putative relation between the nodal quasiparticle spec-42tral weight and the superconducting state, we employed TR-ARPES onthe single-layer Bi2Sr2CuO6+δ (Bi2201). By adopting the same ultrafast ap-proach proposed by Boschini et al. [114], we perturb the system with a IRultrashort pulse and follow the relaxation dynamics of the spectral func-tion. The comparison between our results and those reported for Bi2212points towards the key role of phase coherence in the stability of the SCcondensate in HTSC cuprates, and suggests the enhancing of phase fluc-tuations within the Cu-O plane as the major result triggered by the opticalpumping. Moreover, by extending the equilibrium and ultrafast ARPESstudy to various compounds of both Bi2201 and Bi2212, we track the tem-perature evolution of the nodal quasiparticle spectral weight as a functionof hole doping and materials, disclosing a universal dependence whichlinks the suppression of the quasiparticle amplitude to the electronic tem-perature. The observed behaviour, which persists for temperatures muchlarger than Tc, rules out an explicit relation to the superconducting state.Single-crystals of Bi2201 and Bi2212 were grown using the floating-zone method and hole-doped by oxygen annealing. All the samples werealigned along the Γ-Y direction and cleaved in situ. TR-ARPES measure-ments were performed at UBC using the 1.55 eV-pump and 6.2 eV-probesetup in Damascelli Laboratory (refer to Section 2.2.1 for details). A vari-able filter was used to adjust the incident pump fluence in different datasets.Both the pump and the probe beams were s-polarized. Measurements werecarried out at pressures lower than <5·10−11 torr and at a base temperaturewithout the pump beam of 10 K. The angle and energy of the photoelec-trons were resolved using a SPECS Phoibos 150 electron analyser. The en-ergy and temporal resolutions of the system were estimated 17 meV and250 fs, respectively. Complementary equilibrium ARPES measurementswere carried out on Bi2212 at the Quantum Materials Spectroscopy Centrebeamline at the Canadian Light Source in Saskatoon using a photon energyof hν = 27eV at base pressure <10−10 Torr with energy resolution betterthan 5.3 meV.43(0,p) (p,p)(p,0)(0,0)(0,p) (p,p) (0,p) (p,p)(p,0)(0,0) (p,0)(0,0)f0.0-0.1-0.2Energy (eV)0.0-0.1-0.2Energy (eV)0.0-0.1Energy (eV)0.0-0.1-0.2Energy (eV)0.0-0.05 0.05-0.1k-kF (Å-1)0.0-0.05 0.05-0.1k-kF (Å-1)0.0-0.05 0.05-0.1k-kF (Å-1)0.0-0.05 0.05-0.1k-kF (Å-1)Nodef= 45ºNodef= 45ºNodef= 45ºOff-Nodef= 32ºUD15 OD30 OD24 OD241500 0.3T (K)pAFMSCUD15OD30OD24a bdeUD15 OD30 OD24cFigure 3.2: Static 6.2-eV ARPES of Bi2201. Equilibrium Fermi Surface mapping (FS, top) and ARPES spec-trum along the nodal direction ( (0,pi)-(pi,pi), red cut in the FS) of Bi2201 for (a) an underdoped sample,Tc ∼ 15K (UD15), (b) slightly overdoped, Tc ∼ 30K (OD30) and (c) overdoped sample, Tc ∼ 24K (OD24).The FS are obtained by an energy integration of 20meV and the solid green lines are tight-binding constantenergy contours at ω=0 [125, 126]. (d) ARPES dispersion collected along Off-Node directions (blue cut in FS)defined by ϕ '32o (ϕ angle between (0,pi)-(pi,pi) and the nodal direction) of OD24. (e) Schematic of the p-Tdiagram illustrating the position of the three explored dopings. All data acquired with 6.2-eV probe light at10K base temperature.443.2 TR-ARPES on Bi2201Three different doping levels of the single-layer Bi2Sr2CuO6+δ were stud-ied in this investigation: underdoped sample with Tc ∼ 15K (Bi2201 UD15),slightly overdoped with Tc ∼ 30K (Bi2201 OD30) and overdoped with Tc ∼24K (Bi2201 OD24) – the schematics in Figure 3.2e shows the correspond-ing position in the p-T diagram –. Figure 3.2a-c illustrate the equilibriumFermi surfaces (FS) and the ARPES spectra along the nodal direction (redsolid cut in the FS) for the three dopings acquired with 6.2-eV probe pulsed-light at 10 K. The solid green lines are the tight-binding constant energycontours at ω=0 [125, 126] showing the evolution of the arc dispersion asa function of the doping level. The low photon energy of the probe pulserestricts the accessible k-space to the nodal direction and its vicinity, pre-cluding the exploration of the antinodal region where the d-wave SC gapexhibits its maximum value. As a result, to capture non-zero contributionsfrom the SC gap in the ARPES spectrum we instead focus on the momen-tum direction defined by ϕ'32o (Off-Node cut), ϕ being the angle between(0,pi)-(pi,pi) and the nodal direction. By visual inspection of the Off-Nodecut collected on OD24 and presented in Figure 3.2d, a modest bending ofthe electronic dispersion can be distinguished approaching the Fermi levelas a manifestation of the (small) SC gap.Given the high-quality of the observed spectral features, we first focusour experimental efforts on performing a comprehensive TR-ARPES studyof Bi2201 OD24. The overall modification of the ARPES spectrum occur-ring upon the 1.55-eV pump excitation (mainly restricted to those stateswithin range of the Fermi level) can be visualized by computing the dy-namics of the phototemission intensity as a function of energy and mo-mentum. Figure 3.3a shows the transient evolution of the photoemissionintensity about the Fermi momentum kF as a function of binding energyand momentum direction (Node, upper panel and Off-Node, lower panel;directions as defined in Figure 3.2) acquired with a incident pump fluenceof (12± 3)µJ/cm2, hereon referred to as LF regime. We note that for bothmomentum directions the depletion (i.e. a suppression of spectral weight)45010-10-20010-10-20010-1005-50 2 4 6 80 2 4 6 8NodeOff-NodeNodeNodeLF MFHFIntensity (a.u.)Intensity (a.u.)Intensity (a.u.)Intensity (a.u.)Delay (ps)Delay (ps)-10 meV-15 meV-20 meV-25 meV-30 meV10 meV15 meV20 meV25 meV30 meVa bLFFigure 3.3: Energy-resolved TR-ARPES dynamics in Bi2201 OD24.(a) Temporal evolution of the photoemission intensity about theFermi momentum kF at different binding energies (5meV integra-tion window) along the nodal direction (top) and Off-nodal (bottom)of Bi2201 OD24. Data collected with an incident pump fluence of(12 ± 3)µJ/cm2, referred to as Low Fluence (LF) regime. The solidlines are fits to an exponential function convoluted to a Gaussian ac-counting for the temporal resolution (for the ω > 0 along the node,bi-exponential curves given by the sum of two exponential decays areused). (b) Dynamics of the nodal photoemission intensity in OD24 forincident pump fluence of (42± 6)µJ/cm2 (Medium Fluence MF, top)and (90± 10)µJ/cm2 (High Fluence HF, bottom). The solid lines arebi-exponential decay fits.46 -20 0 20Energy (meV)05101520τ (ps)NodeOff-NodeBi2212LF510151.50τ (ps)τ fast (ps)-20 0 20Energy (meV)40-40Node LFMFHFa bFigure 3.4: Energy-resolved decay times in Bi2201 OD24. (a) Relax-ation decay times as a function of binding energies, describing theexponential decay of momentum-resolved photoemission intensitycurves in the LF regime (as the ones presented in Figure 3.3a), alongthe nodal and off-nodal direction. Open red circles are decay timesextracted with the same procedure for optimally-doped Bi2212 em-ploying an incident pump fluence of (15± 4)µJ/cm2. (b) Comparisonof the decay times obtained in Bi2201 OD24 along the nodal directionfor the three different fluences regimes explored in this work. The bot-tom inset shows the values characterizing the fast recovery componentemerging at high fluences.observed for ω< 0 appears almost twice as strong as the population (i.e. in-crease of spectral weight) occurring at positive energies. The dynamics canbe described by a single exponential decay convoluted by a Gaussian whichaccounts for the experimental temporal resolution and the intrinsic quasi-particle build-up time (solid lines in Figure 3.3a). In the case of the nodallow-energy population, a bi-exponential curve (given by the sum of two ex-ponentials with different decay times) is instead used to fit the experimen-tal data upon convolution to a Gaussian. The extracted decay times τ forthe two momentum cuts are presented in Figure 3.4a, along with the valuesobtained along the node in optimally-doped Bi2212 (Tc ∼ 91 K, OP91) withan incident pump fluence of (15± 4)µJ/cm2. Despite the smaller SC gapcharacterizing Bi2201 and the consequent reduced scattering restrictionsfollowing phasespace considerations, the observed dynamics appear over-all slower than in Bi2212 (especially clear for ω< 0). In addition to the slow47few picoseconds recovery feature, proposed to be closely connected to SC,a sharp femtosecond relaxation dynamics has been reported in Bi2212 forpump fluences above the critical fluence (Fc, defined as the fluence at whichthe near-nodal gap is suppressed) [127–130]. In the case of Bi2201, the re-ported critical fluence FBi2201c ∼ 8µJ/cm2 is comparable to the LF regime re-ported here, and indeed a faster dynamics can be seen starting to emerge inthe population traces along the nodal direction (see Figure 3.3a). In order toverify this scenario, we performed additional TR-ARPES measurements onBi2201 OD24 by increasing the incident pump fluence to (42± 6)µJ/cm2,MF regime and (90± 10)µJ/cm2, HF regime. The resulting nodal dynam-ics for selected binding energies are presented in Figure 3.3b and consis-tently exhibit two contributions to the relaxation recovery. While the decaytime of the slow component decreases with increasing the incident pumpfluence, the time constant of the fast component τf ast < 1.5ps does not ex-hibit a significant variation within the explored pump fluence range (seeFigure 3.4b). This observation is in agreement with what was reported foroptimally doped Bi2212 [128], although in the present case a much higherfluence (∼ 10 Fc) has been employed.By increasing the incident pump fluence, more energy gets transferredinto the sample. In order to provide an estimate of the overall thermal ef-fect induced by the presence of the pump beam on the sample before thetemporal incidence with the probe (i.e. at negative delays) and on howthe electronic bath responds to the optical excitation on a thermal level,we compute the evolution of the transient electronic temperature for eachemployed pump fluence. In virtue of the zero SC gap at the node, theanalysis of the momentum-integrated energy distribution curves (EDCs)along the nodal direction can serve to this purpose as it maps into theFermi-Dirac distribution,∫Inode(ω,k)dk ∝∫Anode(ω,k) f (ω)dk ∝ f (ω), asshown in Figure 3.5a. Photoexcited quasiparticles in cuprates, and in gen-eral in strongly correlated electron systems, release the energy depositedby the pump pulse through electron-electron and electron-boson scatter-ing, thermalizing and reaching a state of quasi-equilibrium on an ultrafasttime scale of about 100 fs [131–136]. Thus, the analysis of the nodal Fermi-481008060402010080604020-0.5 ps0.33 ps95 K25 K-40 0 40Energy (meV)Intensity (a.u.)HF0 2 4Delay (ps)6 8T e (K)0 2 4Delay (ps)6 8 0 2 4Delay (ps)6 8T e (K)LFMF HFτ=10 psτ=1.2 psτ=0.9 psa bcTcFigure 3.5: Transient eletronic temperature Te. (a) Momentum-integrated EDCs along the nodal direction before (-0.5ps) and after(0.33 ps) the pump excitation, HF regime. The black lines are Fermi-Dirac distribution fits. (b) Transient electronic temperature Te(t) forLF. The solid line is a phenomenological exponential decay fit. (c)Same as in (b) but for MF and HF regimes. The temporal evolutioncan be described by double exponential-decay fits (solid lines), wherea fast component (filled green and brown curves) grows on top of thesame slow recovery which characterizes the LF dynamics (filled bluecurve). Error bars in (b)-(c) reflect the systematic errors associated withthe experiment and the number of averaging cycles acquired for eachfluence.49edge width – approach already adopted in previous TR-ARPES studieson cuprates [124, 131] – allows for extracting an effective electronic tem-perature as a function of the pump-probe delay t. The resulting Te(t)for the three different fluences are displayed in Figure 3.5b-c, along withexponential-decay fits (solid lines) and show how, although only in the LFregime the base temperature can be safely placed below Tc (dashed blackline), the system does not relaxes back to equilibrium within the probedtemporal window in any of the fluence regimes explored. Interestingly, thesame long relaxation dynamics of∼ 10ps (filled blue curve) well character-izes all the observed Te(t), on top of which a second fast dynamics emergesat high fluences on the timescale of 1 ps after the pump excitation (filledgreen curve, MF ; filled brown curve, HF). The knowledge of the temporalevolution of the effective electronic temperature upon the photo-excitationwill be crucial in establishing the extension of pure thermal effects to thedynamics of the one-electron spectral function as discussed in the follow-ing section.3.3 Tracking the ultrafast enhancement of phasefluctuations in Bi2201As an extension into the time domain of the standard ARPES technique,TR-ARPES provides access to more information beyond the overall dynam-ics of the photoemission intensity as a function of energy and momentum.In particular, the dynamical response of the one-electron spectral functioncan be studied as a result of an ultrafast optical excitation, providing uswith a tool to study the dynamical interplay between the electron pairingmechanism and the stability of phase coherence in the formation of the con-densate in HTSC cuprates. As proposed by Norman et al. [137], these twophenomena may be accounted for in the approximated expression of theelectron self-energy at the Fermi momentum (k = kF) for a superconductor.It can be expressed as:Σ(ω) = −iΓs + ∆2(ω+ iΓp). (3.1)50Here, ∆ represents the amplitude of the SC gap which describes the across-gap excitations and whose value is momentum-dependent according to theestablished d-symmetry of the SC order parameter in cuprates. Γs is thesingle-particle scattering rate while Γp is the inverse pair lifetime proposedto reflect the presence of phase fluctuations and thus expected to vanish forTTc [137, 138]. TR-ARPES potentially provides access to the individualdynamics of these three parameters by tracking the temporal evolution ofthe spectral-function expressed by:A(k,ω) = − 1piΣ′′(ω)[ω− ek − Σ′(ω)]2 + [Σ′′(ω)]2 , (3.2)where ek is the bare energy dispersion. Here we remark that while thepresence of phase fluctuations has been shown to lead to a pair-breakingscattering rate as the one proposed in Equation 3.1 [138], a non-vanishingΓp does not uniquely identify phase fluctuations as the driver for the ob-served sepctral function phenomenology. Therefore, only in the UD andclose to optimally-doped regime a direct correspondence Γp-phase fluctua-tions may be legitimate whitin the context of Emery and Kivelson’s phasediagram [113] (see Figure 3.1a), whereas the validity of such univocal rela-tion remains questionable in the strongly OD regime.Disentangling the contributions of ∆, Γs and Γp to the ARPES spectrumcan be challenging, especially in the presence of small SC gaps as in Bi2201(∆max ∼ 15meV at the antinode [111]). However, along the nodal directionthe SC gap is null, making the second term in Equation 3.1 vanish, andthus allowing the investigation of the effect of Γs alone. Moreover, the es-timate of the overall thermal effect due to the pump beam on the samplepresented in Figure 3.5 safely identifies the system in a SC state (before thepump excitation) only in the LF regime. We then start our analysis focusingon the temporal evolution of the nodal ARPES spectrum of Bi2201 OD24in the LF regime. Figure 3.6a displays EDCs at k = kF acquired along thenode for different pump-probe delays showing an evolution of the quasi-particle peak. In order to get quantitative information about the associatedspectral function dynamics in terms of Equation 3.1 and Equation 3.2, we51compute the corresponding symmetrized EDCs (SEDCs) of the curves inFigure 3.6a. This commonly used procedure consists in adding to an EDCat kF the symmetrized about E = EF counterpart to rule out the contribu-tion of the Fermi-Dirac distribution term f (ω) to the ARPES spectra [139].Note that this approach relies on the particle-hole symmetry property of thespectral function at k = kF, experimentally verified in the near-nodal regionof Bi2212 [114, 140]. Assuming constant matrix elements in the exploredmomentum range, SEDC(ω) ∝ A(kF,ω). Figure 3.6b shows nodal SEDCsat different delays alongside fits to Equation 3.1 and Equation 3.2, obtainedwith ∆ = 0. The photoinduced modifications to the spectral function areaccounted for by a change of the single-particle scattering rate Γs. The fulltemporal evolution Γs(t) resulting from the fitting procedure is plotted inFigure 3.6c: starting from a value of (9.7± 0.3) meV at negative delays, itexhibits an increase of almost 25% upon the pump excitation. This rangeof values, as well as the characteristic several-ps long relaxation dynamics(exponential decay fit represented by the solid line in Figure 3.6c), are con-sistent with what is reported for optimally-doped and overdoped Bi2212[114, 121].Once we established the transient evolution of Γs along the nodal direc-tion, we now move to the analysis of the spectral function along the Off-Node direction (ϕ '32o as defined in Figure 3.2). In this case, the SC gap∆ has a non-zero value and needs to be included in the fitting of SEDCs,as well as the possible spectral broadening induced by Γp. In order to esti-mate the unperturbed value of ∆, we first fit the SEDC at negative delays(blue circles in Figure 3.7a) by assuming Γp = 0 (as expected in the caseof fully coherent condensate). The best fit returns Γeqs = (10.5± 0.5)meVand ∆eq = (5 ± 1)meV, which are in agreement with what is reported inprevious equilibrium ARPES measurements [111, 121], further validatingthe base line of the fitting procedure. Next comes the analysis of the dy-namics of the spectral function along the Off-Node cut as a function of thepump-probe delay when all the three parameters in Equation 3.1 possiblychange. Having three unknown variables combined to the small value of∆eq challenges the stability of the fitting procedure via Equation 3.1 and52-40 0 40Energy (meV)-80 -40 0 40Energy (meV)-80 8001Intensity (a.u.) EDC SEDC-0.5 ps+0.4 ps+6.6 ps-0.5 ps+0.4 ps+6.6 ps0 2Delay (ps)4 6 8131211109Γ s (meV)a b cFigure 3.6: Nodal single-particle scattering rate dynamics. (a) EDCsat the Fermi momentum k = kF for selected pump-probe delays, ac-quired along the nodal direction in Bi2201 OD24, LF regime. (b) Sym-metrized EDCs (SEDCs) of the curves presented in (a). The solidblack lines are fits obtained using Equation 3.1 and Equation 3.2 with∆ = 0. (c) Temporal evolution of the single-particle scattering rate Γsextracted by fitting SEDCs. The solid line represents a phenomenolog-ical exponential decay fit convoluted with a Gaussian. Error bars in (c)represent the confidence interval in the fit correcponding to 3σ.Equation 3.2, urging us in setting some constraints. To this purpose, wenote that the equilibrium values of Γs obtained along the nodal and off-nodal directions differ by less than 1 meV, a finding in agreement with thenearly momentum-independent behaviour reported by Kondo et al. [121]in the near-nodal region for Bi2212. This observation motivates our choiceto lock the temporal evolution of Γs along the Off-Nodal direction to the dy-namics extracted at the node (solid line in Figure 3.6c), setting a toleranceof ±1meV on the resulting fitting value. With this constraint in place, wefit the full temporal range of Off-Node SEDCs. Figure 3.7a presents SEDCsand corresponding fits for three selected pump-probe delays and, althoughthe small gap size precludes even at negative delays the precise identifica-tion of a defined ”double-peak” feature typical of the SC state, we observe amodification of the spectra occurring mainly about ω= 0. Such a modifica-tion encodes the temporal evolution of Γs, ∆ and Γp through Equation 3.1.In Figure 3.7b-d we present the resulting dynamics of these three param-eters obtained by SEDCs fitting which reveals a transient modification ofΓp more than five times larger than what observed for the gap. Moreover,53-0.5 ps+0.4 ps+6.6 psOff-Nodef= 32ºIntensity (a.u.)-40 0 40Energy (meV)-80 800 2Delay (ps)4 6 813121110Γ s (meV)6420Δ (meV)201510Γ p (meV)05abcdBi2201Bi22120200 8Figure 3.7: Off-Node spectral function dynamics. (a) SEDCs at k = kFalong the Off-Node direction (ϕ '32o as defined in Figure 3.2), LFregime. The solid black lines are fits obtained using Equation 3.1 andEquation 3.2. (b)-(d) Ultrafast dynamics of the single-particle scatter-ing rate Γs (b), the gap size ∆ (c) and the pair scattering rate Γp (d)obtained from the SEDCs fitting. The solid lines represents an ex-ponential decay fit convoluted with a Gaussian. The dashed line in(b) is the temporal evolution of Γs extracted along the nodal direc-tion and used to constraint the fit of SEDCs shown in (a), as describedin the main text. Inset in (d): Comparison between the Γp dynam-ics obtained for Bi2201 OD24 (F = (12± 3)µJ/cm2) and Bi2212 UD82(F = (30± 4)µJ/cm2, from [114]). Error bars in (b)-(d) represent theconfidence interval in the fit corresponding to 3σ.54while both Γs and ∆ do not recover the equilibrium values within the tem-poral range explored, the dynamics of Γp is completely decoupled exhibit-ing a relaxation time of ≈ 1ps, about ten times faster than what extractedfor the former two parameters. In order to offer a more quantitative discus-sion of our results in terms of previous studies, the inset in Figure 3.7d of-fers a direct comparison between the temporal evolution of Γp as extractedin this work for the single-layer Bi2201 OD24 to what reported for the bi-layer Bi2212 UD82 [114]. Although the two datasets have been collectedemploying substantially different incident pump fluence – (12± 3)µJ/cm2and (30± 4)µJ/cm2 for Bi2201 and Bi2212, respectively –, note that in bothcases this value corresponds to F∼ 2Fxc (x=Bi2201, Bi2212) substantiating adirect comparison between the two results. A remarkable agreement is ev-idenced from the inset in Figure 3.7d not only in terms of the characteristicrelaxation time on the 1ps timescale but also of the absolute magnitude ofthe observed pump-induced increase. In fact, in both studies the transientpair-breaking scattering rate Γp has its maximum at∼ 15 meV which corre-sponds to the energy scale at which phase fluctuations become relevant inBi-based cuprates [113, 118]. This observation points towards the interpre-tation of the temporal evolution of Γp in terms of an ultrafast enhancementof phase fluctuations upon IR excitation, as reported for the case of UDBi2212 [114]. The analogy between the two studies also extends to the char-acteristic build-up time of the enhancement of Γp. The ultrafast responseto the pump excitation is delayed with respect to that of Γs and ∆ with themaximum value reached about 600fs, as shown by the normalized differ-ential dynamics (defined as |x(t) − x(0)|/max[x(t)]) in Figure 3.8a. Thisobservation seems consistent with the proposal of a pump-induced non-thermal bosonic population as the key player in the increase of the phasefluctuations speculated for UD Bi2212 [114].To conclude our discussion on the dynamics of Γp, we address the ques-tion on whether or not the observed ultrafast increase stems from a merethermal effect induced by the presence of the pump on the sample. Infact, as shown in Figure 3.5b, the optical excitation generates a transientevolution of the electronic temperature casting doubt on what extend such550 2Delay (ps)4 6 80 1Delay (ps)01Norm. Intensity (a.u.)ΓpΓsΔ≈600 fs01Norm. Intensity (a.u.)1020304050Te  (K)ΔΓsΓpTea bFigure 3.8: Disentangling Γp and Te dynamics in Bi2201. (a) Normal-ized differential dynamics computed as |x(t)− x(0)|/max[x(t)] for Γs(red line), ∆ (ochre) and Γp (blue) from data in Figure 3.7b-d. (b) Com-parison between the Γs, ∆ and Γp normalized differential dynamics(left axis) and the temporal evolution of the electronic temperature ex-tracted along the nodal direction (filled black circles, right axis; fromdata in Figure 3.5b).temperature change reflects onto the dynamics of the spectral function andconsequently of the three parameters in Equation 3.1. To this goal, we plotin Figure 3.8b the normalized transient evolution of Γs, ∆ and Γp (left axis)alongside the electronic temperature as extracted along the nodal direc-tion (filled black circles, right axis; refer to Figure 3.5b). By direct visualinspection is evident that both the single-particle scattering rate and thegap size exhibit a dynamics closely locked to the temporal evolution of theelectronic temperature with a recovery time of ∼ 11ps. Thermally inducedacross-gap charge excitations solely determine both the pairing strength ∆and the increase of single-particle linewidth Γs (in agreement with what isreported from equilibrium ARPES [115, 137]). Contrarily, the completelyindependent dynamics exhibited by Γp with respect to Te testifies to thenon-thermal origin of the observed transient enhancement of phase fluctu-ations.All together the results reported here for Bi2201 OD24, along with thestrong analogy to what was reported for Bi2212 UD82, suggest the loss56of phase coherence as the key mechanism in the ultrafast collapse of su-perconductivity (and the recovery thereof), even in the slightly overdopedregime and independently on the number of Cu-O layers within the unitcell. Future experiments will be instrumental to address whether such sce-nario holds in the very OD regime.3.4 Nodal coherent-spectral-weight meltdown inBi-based cupratesWhile off-nodal quasiparticle spectral features observed below Tc are stronglylinked to the pairing strength and the Cooper pairs phase coherence, low-energy excitations along the nodal direction are usually regarded as min-imally affected by SC. Therefore, while a meltdown of coherent spectralweight was reported by equilibrium ARPES studies at the antinode of thebi-layer Bi2212 and related to SC [115, 123], such an effect was not ex-pected along the nodal direction where the SC gap is null. However, thisnodal-antinodal dichotomy seems softened on the basis of a suppressionof quasiparticle spectral weight along the node reported in a recent TR-ARPES study on optimally-doped Bi2212 and primarily observed in theSC state, thus suggesting a direct link between the nodal spectrum and SCitself [124]. As evinced from data reported in Figure 3.6a-b, a clear suppres-sion of spectral weight is evident also in the present case of Bi2201 OD24:the ultrafast broadening of the quasiparticle peak alone (reflected in theincrease of the broadening term Γs) does not compensate for the loss ofCoherent Spectral Weight (CSW) in the vicinity of EF. The SEDCs fittingprocedure by Equation 3.1 and Equation 3.2 described in Section 3.3 pro-vides us with access to the transient modification of the CSW amplitude interms of an overall normalization constant in the fitting function. Note that,since each pump-probe delay is acquired in the same experimental config-urations, this approach naturally sets the same background baseline for allSEDCs allowing for a direct measure of the relative variation of spectralweight, ∆CSW, upon the optical pumping. Figure 3.9a-c present ∆CSW(t)extracted via fitting of SEDCs along the nodal direction for the three dif-57 SEDCs fitting MDCs fitting0 2 4Delay (ps)6 80 2 4Delay (ps)6 81.000.950.900.850.80∆CSW (a.u.)1.000.950.900.850.80∆CSW (a.u.)LF MFHF MFa bc dτ=10 psτ=1.2 psτ=0.8 ps SEDCs areaFigure 3.9: Ultrafast suppression of coherent spectral weight inBi2201 OD24. (a)-(c) Relative variation of the nodal quasiparticle co-herent spectral weight, ∆CSW, as a function of the pump-probe de-lay extracted by fitting SEDCs in Bi2201 OD24 for LF (a), MF (b) andHF (c) regime. The solid lines are phenomenological single (LF) ordouble (MF and HF) exponential decay fits convoluted to a Gaus-sian. Error bars reflect the systematic errors associated with the exper-iment and the number of averaging cycles acquired for each fluence.(d) Comparison between ∆CSW extracted from SEDCs fitting (yellowcircles), momentum-distribution curves (MDCs) fitting (orange line)and SEDCs integrated area (dashed light blue line); MF regime. ALorentzian-like function is used to fit the MDCs at E = −20meV.ferent pump fluence regimes exploited in the investigation of Bi2201 OD24(LF, MF and HF, respectively). The magnitude of the observed variationincreases as the pump fluence increases going from ∼ 7% for LF to as largeas 20% at HF. Interestingly, the temporal evolution can be phenomenologi-cally described by a single (LF) or double (MF and HF) exponential decayreminiscent of the dynamics characterizing the transient electronic temper-ature shown in Figure 3.5, where a slow 10 ps-dynamics appears for allthe three pump fluences, on top of which a fast feature develops for MF58and HF. This observation suggests a close connection between the extractedsuppression of CSW along the node and the transient electronic tempera-ture. Before moving on to the investigation of such a relation, we test thereliability of our approach based on SEDCs fitting for determining ∆CSWas a function of the pump-probe delay. Momentum Distribution Curves(MDCs) can be addressed as an alternative procedure in which the ampli-tude and broadening of the Lorentzian-like lineshape map into ∆CSW(t)and Γs(t), respectively. Figure 3.9d compares the temporal dynamics of∆CSW as extracted from fitting of SEDCs (yellow circles) and MDCs (or-ange line), showing a solid agreement. To go beyond the specific modelused to fit the data, we overimpose in Figure 3.9d also the total integratedarea of SEDCs (dashed light blue line): the observed consistency across thethree methods validates our experimental estimate of ∆CSW.In order to explicitly expose the relation between CSW and electronictemperature, we plot ∆CSW directly as a function of the extracted Te inFigure 3.10b. Note that in order to account for the different base tempera-ture observed for each pump fluence regime, the curves have been rescaledto match at Te ∼ 30K. Joint together, the results collected for the three dif-ferent pump fluences show an almost-linear suppression of CSW as a func-tion of the increasing electronic temperature for the entire range explored.Such significant linear suppression (up to∼ 20%) appears unaffected by theSC phase transition and persists for TeTc (blue shadow in Figure 3.10b).Note that, contrarily to what was previously reported for Bi2212 [124], thisfinding points towards a minimal (if not null) contribution of supercon-ductivity itself to the here-reported nodal quasiparticle meltdown in Bi2201OD24. For completeness, we present in Figure 3.10a also the temperatureevolution of the single-particle scattering rate Γs as extracted from fittingof nodal SEDCs for the different fluence regimes. The three curves are de-scribed by the same linear T-dependence increase, further supporting themajor role played by thermally induced charge excitations in determiningthe ultrafast increase of Γs, as also reflected in the coupled Γs and Te dy-namics (see Figure 3.8b).59LFMFHFLFMFHF100Te (K)80604020100Te (K)8060402016141210Γ s (meV)1.000.950.900.850.80∆CSW (a.u.)0.75baTcFigure 3.10: Nodal CSW and Γs vs. temperature in Bi2201 OD24. (a)Temperature evolution of the single-particle scattering rate Γs as ob-tained from nodal SEDCs fitting with different pump fluences. (b)Relative variation of the nodal coherent spectral weight ∆CSW as afunction of the electronic temperature Te for the three employed pumpfluences. Curves have been rescaled to match at Te ∼ 30K to accountfor the different base temperature (i.e. at negative delays) observedin the three fluence regimes. The blue shadow area indicates Tc. Datapoints in (a)-(b) are displayed vs the electronic temperature derived bythe phenomenalogical exponential decay fits shown in Figure 3.5b-d.To establish the observed ∆CSW behaviour as a general property ofsingle-layer Bi-based cuprates, we extend our TR-ARPES study to differ-ent doping levels, namely Bi2201 UD15 and Bi2201 OD30, in the LF regime(for a general overview of the FS and ARPES dispersion of the three dif-ferent Bi2201 compounds studied in this project, as well as the correspond-ing position in the p-T diagram, see Figure 3.2). Following the same pro-cedure outlined above, we extract the temporal evolution of the effectiveelectronic temperature from momentum-integrated EDCs along the nodaldirection (Figure 3.11a) and then relate it to the transient dynamics of Γsand ∆CSW as shown in Figure 3.11b-c, respectively. As expected from vi-sual inspection of the nodal cuts presented in Figure 3.2, the single-particlebroadening term is higher in the UD15 and OD30 samples with respect towhat is detected for OD24; nevertheless, a similar slope describes the in-600 2 4Delay (ps)6 8 0 2 4Delay (ps)6 8 0 2 4Delay (ps)6 8204060T e (K)1.000.950.900.85∆CSW (a.u.)16141210Γ s (meV)18Te (K)605020 4030Te (K)605020 4030UD15 OD30 OD24UD15OD30OD24UD15OD30OD24acbTcFigure 3.11: Doping dependence of the temperature evolution ofCSW and Γs in Bi2201. (a) Transient electronic temperature Te(t) ex-tracted from momentum-integrated EDCs (as described in Section 3.2)for three doping levels of the single-layer Bi2201: UD15, OD30 andOD24, respectively. The solid lines are phenomenological exponen-tial decay fits. Data acquired employing the low fluence for the pumpbeam ((12 ± 3)µJ/cm2). (b)-(c) Temperature evolution of Γs (b) and∆CSW (c) obtained from nodal SEDCs fitting for Bi2201 UD15 (lightblue data points), OD30 (orange) and OD24 (blue); LF regime. Shad-owed areas in (c) indicate the SC critical temperature Tc for the threedopings. Data points in (b)-(c) are displayed vs the electronic temper-ature derived by the phenomenalogical exponential decay fits shownin (a).61crement of Γs as Te increases. Interestingly, despite being characterized bydifferent single-particle broadening, the three dopings present a consistentT-dependence of the relative suppression of the nodal CSW, which doesnot exhibit any abrupt modifications across the corresponding Tc (coloredshadows in Figure 3.11c).The findings presented thus far are in contrast to what was previouslyreported for bi-layer Bi2212 by ARPES and TR-ARPES studies [115, 123,124] on at least two aspects: (i) no strong enhancement of the quasiparti-cle peak suppression is observed to occur below Tc, as a comparable effectis here detected in Bi2201 UD15 and Bi2201 OD30 despite the fact that inthe former case the system is already out of the SC state before the opticalpumping, while in the latter the condensate is fully coherent at negativepump-probe delays; (ii) no direct link is evinced between the ∆CSW andthe collapse of SC driven by the ultrafast enhancement of phase fluctua-tions, arguable in terms of the completely distinct dynamics of Γp to thatof ∆CSW, which instead closely resembles the evolution of Te. This dis-crepancy motivates additional investigations by employing the same TR-ARPES experimental approach described thus far but now applied on thebi-layer Bi2212. Moreover, conventional equilibrium temperature-dependentARPES measurements are also imperative to address whether the observedeffect is limited to the transient regime or consistent with a mere increase ofthe sample temperature. The next paragraphs will focus on these aspects.First we discuss the modifications of the nodal CSW in the bi-layerBi2212 induced by an ultrafast optical excitation equivalent to that em-ployed for Bi2201. Figure 3.12a shows the extracted transient electronictemperature in optimally-doped Bi2212 OP91 (Tc ∼ 91K) as a result of in-creasing pump fluences (in this specific dataset (14± 5) µJ/cm2, (23± 7)µJ/cm2and (62± 10)µJ/cm2, respectively). The corresponding ∆CSW from nodalSEDCs are plotted as a function of Te in Figure 3.12b. Although in thepresent case the high Tc precludes any conclusive claim on the influence ofSC in the evolution of CSW, the observed suppression of spectral weightexhibits a dependence on the extracted electronic temperature which re-sembles that presented in Figure 3.10b and Figure 3.11c for Bi2201. This62LF204060T e (K)a801000 2 4Delay (ps)1.000.950.900.850.750.80Te (K)1008020 6040∆CSW (a.u.)MFHFb1.000.950.900.850.750.80∆CSW (a.u.)Te (K)1008020 6040cUD15OD30OD24Bi2201OP91OD80OD60Bi2212LFMFHFFigure 3.12: Universal suppression of CSW in Bi-based cuprates. (a)Transient electronic temperature Te(t) extracted from momentum-integrated EDCs for Bi2212 OP91 (Tc ∼ 91K) in three different fluencesregimes, namely (14± 5)µJ/cm2, (23± 7) µJ/cm2, (62± 10)µJ/cm2,here labelled for LF, MF and HF, respectively. (b) Temperature evolu-tion of ∆CSW for all three fluence regimes. Data points are plotted vsthe electronic temperature determined by the phenomenalogical expo-nential decay fits shown in (a). (c) Comparison between ∆CSW(T) ex-tracted for the single-layer Bi2201 (UD15, OD30 and OD24; full mark-ers) and the bi-layer Bi2212 (OP91, OD80, OD60; open circles). Allcurves follow an almost-linear behaviour as a function of the electonictemperature. Shadowed areas mark the range of Tc for the differentcompounds.observation testifies the capability of our experimental approach in track-ing the loss of quasiparticle spectral weight upon optical pumping in bothBi2201 and Bi2212.A comprehensive picture of the nodal CSW meltdown in Bi-based cupratesis offered in Figure 3.12c. Here, computed ∆CSW(Te) for different dop-ing levels of single- (full markers) and bi-layer (open circles) compoundsare combined and compared. Despite the large variation of Tc across the6317K27K35K50K65K85K100K120K130K145K-40 0 40Energy (meV)-80 -40 0 40Energy (meV)EDC SEDC01Intensity (a.u.)Te (K)8040 1201.∆CSW (a.u.)Bi2212 OP91Bi2212 OD60EquilibriumBi2212 OP91Bi2212 OD60TR-ARPESa bFigure 3.13: CSW from equilibrium temperature-dependent ARPES.(a) EDCs at the Fermi momentum k = kF (left) and correspondingSEDCs (right) as a function of sample temperature in optimally-dopedBi2212 OP91. Data acquired at the Canadian Light Source sync-thron facility with a photon energy of 27 eV. (b) Comparison between∆CSW(T) as extracted from TR-ARPES (full markers) and equilib-rium ARPES (open circles) measurements of Bi2212 OP91 (green) andBi2212 OD60 (purple).studied compounds (shadows in Figure 3.12c), a remarkable agreement isachieved between the tracked CSW vs. Te in Bi2201 and Bi2212 in termsof both functional form and magnitude, establishing the nearly linear tem-perature dependence of the nodal coherent spectral weight as a universalbehaviour of Bi-based cuprates and uncorrelated to SC.All the data presented thus far in this section (for both Bi2201 and Bi2212)have been acquired in an ultrafast fashion, where the effects induced bythe pump excitation are tracked as a function of the pump-probe delay andsubsequently discussed in terms of the evolution of the effective electronictemperature determined by the Fermi-edge width at each delay. Althoughthis ultrafast approach ensures equivalent experimental conditions for eachtemperature (e.g. the same spot on the sample’s surface and momentum-space position), it might rise doubts on whether the observed behaviour isan exclusive property of the transient state. In this regard, we performed64conventional temperature-dependent ARPES studies on Bi2212 (OP91 andOD60) at the Canadian Light Source synchrotron facility. Figure 3.13a dis-plays the temperature evolution of EDCs at k = kF (left) and correspondingSEDCs (right) acquired along the nodal direction of Bi2212 OP91 with aphoton energy of 27 eV. Also in this case, the major effect of the tempera-ture increase to up ∼ 140K is the reduction of the quasiparticle amplituderather than the broadening of the spectral features. To quantify this modifi-cation in the context of the data shown above, in Figure 3.13b we offer a di-rect comparison between ∆CSW(T) as obtained from equilibrium ARPES(open circles) and TR-ARPES (full markers) for Bi2212 OP91 and Bi2212OD60. While equilibrium ARPES measurements cover a much wider tem-perature range, the curves obtained from the two experimental approachesare in good agreement and clearly exclude a vanishing of the CSW sup-pression within range of Tc, as instead previously reported at the antinodeof Bi2212 [115, 123].Here, it is important to remark that the observed behaviour of∆CSW(T)is not caused by nonlinearity effects of the detector. Nonlinearity effects arecommon in electron spectometers, especially when strong lineshapes occuralong with a low level background, as often the case of low photon en-ergy ARPES measurements on cuprates. It has been shown that this canresult into an artificial enhancement of the quasiparticle peak, and post-acquisition procedures have been developed to correct the data for sucheffects [127, 141]. However, these procedures may mask a correct evalu-ation of small modifications in the quasiparticle peak of the order of fewpercentage units, like that reported in this section for ∆CSW(T) in Bi2201and Bi2212. To address the possible contributions of the detector’s nonlin-earity in the observed evolution of ∆CSW(T), we collected the TR-ARPESdata shown in Figure 3.12a-b and the equilibrium data in Figure 3.13 in”single pulse counting mode”, a detection regime of the electron analyzernot affected by nonlinearity effects. In spite of this, Figure 3.12c and Fig-ure 3.13c show full consistency across all the performed measurements,therefore ruling out a suppression of spectral weight artificially inducedby nonlinear effects in the detecting process.65As last note we recognize that, in virtue of the persistence of the herebyreported suppression of CSW for TTc for both Bi2201 and Bi2212, it couldbe tempting to speculate an onset temperature associated to a differentorder than SC, such as the pseudogap. However, while the pseudogapcharacteristic temperature T? ranges well above 100 K for underdoped andoptimally-doped compounds (and is thus beyond the probed temperaturerange of this work), no equilibrium pseudogap has been reported for thevery overdoped Bi2212 OD60 compound [142–144]. In spite of this dif-ference, a substantial modification of the nodal CSW appears in our TR-ARPES and equilibrium ARPES results on Bi2212 OD60, comparable inmagnitude to that observed for the other compounds over the entire ex-plored temperature range (including for Tc <T<T? for the Bi2212 OP91sample, see Figure 3.13b). This observation excludes a direct connection tothe physics of the pseudogap.We propose that the observed CSW meltdown stems from the tempera-ture evolution of the imaginary part of the electron self-energy included inthe spectral function. Here, we limit the discussion to the spectral functionat k = kF. In fact, if we evaluate the spectral function always at the Fermimomentum (as is the case for all the experimental SEDCs reported thus farin this section), in first approximation the term εbk − Σ′(k,ω) in the denomi-nator of Equation 2.11 (which describes the renormalization of the electrondispersion) can be set to zero, to explore the effect of the imaginary part.This allows us to express the change in the spectral function solely in termsof Σ′′, as follow:AkF(ω) = − 1piΣ′′kF(ω)ω2 + [Σ′′kF(ω)]2. (3.3)This approach suggests that the evolution of ∆CSW(T) may represent ametric tool to distinguish between the canonical Fermi-liquid (FL) and themarginal-Fermi-liquid (MFL) behaviour. The MFL model has been pro-posed as a phenomenological description of HTSC cuprates in particular toaccount for the observed strange metal behaviour, i.e. the linear electricalresistivity [145–147]. FL and MFL exhibit distinct temperature dependence661.000.950.900.850.750.80∆CSW (a.u.)Te (K)1008020 6040bUD15OD30OD24Bi2201OP91OD80OD60Bi2212FLMFLSimulations0. S (eV)1.00.0Ak F (a.u.)-40 0 40Energy (meV)-80 80aFLMFLFigure 3.14: Simulated ∆CSW(T) in a FL vs. MFL picture. (a) Imag-inary part of the electron self-energy (top) and associated spectralfunction (bottom) for the Fermi-liquid (FL, black lines) and marginal-Fermi-liquid (MFL, red lines) models simulated via Equation 3.3 andEquation 3.3, at temperature equal to 10 K. (b) Comparison betweenthe simulated suppression of CSW, as extracted from Equation 3.3and Equation 3.4, and the experimentally observed ∆CSW(T) (filledand open circles for Bi2201 and Bi2212, respectively; data from Fig-ure 3.12c). The best agreement is achieved within the FL picture (blackline). In all simulations β and λ coefficients were set to 19.5 and 0.9, re-spectively, to match the relative variation of ∼ 22% at 100 K exhibitedby the experimental data.of the imaginary part of the self-energy [76]:Σ′′FL(ω, T) = β[ω2 + (pikBT2)]Σ′′MFL(ω, T) = λpi2[max(| ω |, T)] ,(3.4)where β and λ are coupling constants. Figure 3.14a shows Σ′′(ω) (top) andthe corresponding spectral function (bottom) simulated at 10 K using Equa-tion 3.3 and Equation 3.4, for both FL and MFL. Following the same proce-67dure by varying the temperature parameter, the evolution of the coherentspectral weight can be computed as a function of T for the two differentmodels. Figure 3.14b displays the results of such computation overimposedto the experimental data already shown in Figure 3.12c. Here, the values ofthe two constants β and λ are set such that the magnitude of the simulatedCSW suppression matches the ∼ 0.78% at 100 K observed experimentally.The best agreement is achieved by a FL description (black line), while theMFL (red line) seems to fail to reproduce the experimentally observed T-dependence. This finding is consistent with a recent ARPES study report-ing true Femi-liquid quasiparticles only along the nodal direction in over-doped La1.77Sr0.23CuO4 cuprate, while proposing an angular breakdown ofsuch FL behaviour moving towards the antinode [148].3.5 ConclusionsTo summarize, we have performed a comprehensive TR-ARPES study ofboth the single- and bi-layer hole-doped Bi-based cuprates. Firstly, by per-forming a detailed analysis of the dynamics of the spectral function inBi2201-OD24 upon optical pumping, we have disentangled the temporalevolution of the pair scattering rate Γp (related to the presence of phase fluc-tuations) from that of the gap amplitude ∆. In addition, we have demon-strated that the ultrafast enhancement of phase fluctuations and subse-quent fast relaxation (within ∼ 2 ps) do not map into the dynamics of theeffective electronic temperature, and thus cannot be ascribed to a merethermal effect. These results complement a similar TR-ARPES study onthe bi-layer Bi2212 [114], thus establishing the persistence of macroscopicphase coherence as an essential ingredient in the formation of the SC con-densate for underdoped and close to optimally-doped Bi-based cuprates,independently on the number of CuO2 layers. Secondly, we have reporteda comprehensive study of the nodal CSW meltdown as a function of theincreasing temperature. By investigating various doping levels and com-pounds (Bi2201 and Bi2212 alike), we have revealed a universal tempera-ture dependence unrelated from the characteristic critical temperatures of68the SC and PG, namely Tc and T?. Instead, the observed suppression ofCSW can be simulated in terms of the quadratic temperature dependenceof the imaginary part of the self-energy in the Fermi-liquid description. Theresults reported here suggest that the analysis of the temperature evolutionof the CSW via TR-ARPES may be used as a sensitive investigation tool toexplore as a function of momenta the potential Fermi-liquid to marginal-Fermi-liquid transition of the normal-state of Bi-based cuprates.69Chapter 4Emergence of pseudogap fromshort-range spin-correlations inelectron-doped cupratesThe strong electron Coulomb repulsion is considered the key ingredientto describe the emergence of exotic phases of quantum matter from high-temperature superconductivity to charge- and magnetic-order. However, acomprehensive understanding is often complicated by the appearance of apartial suppression of low-energy electronic states, known as the pseudo-gap. To elucidate the degrees of freedom associated with the pseudogapphenomenology in electron-doped cuprates, we apply TR-ARPES to un-veil the temperature evolution of the low-energy density of states in theoptimally-doped Nd2−xCexCuO4, an emblematic system where the pseu-dogap intertwines with magnetic degrees of freedom. By photoexciting theelectronic system across the onset temperature T?, we report the direct re-lation between the momentum-resolved pseudogap spectral features andthe spin-correlation length. This transient approach, corroborated by meanfield model calculations, allows us to establish the pseudogap in electron-doped cuprates as a precursor to the incipient antiferromagnetic order evenwhen long-range antiferromagnetic correlations are not established, as inthe case of optimal doping.704.1 IntroductionExotic electronic phenomena are often masked by the interactions withinand among various degrees of freedom [149–152]. A prominent exampleis the puzzling pseudogap (PG) phenomenon, a mysterious state of corre-lated matter by now notorious from systems as diverse as unconventionalsuperconductors [153–155], dichalcogenides [156, 157], and ultracold atoms[158–160]. Generally speaking, the PG is associated with a partial suppres-sion of the electronic spectral weight in the vicinity of the Fermi level (ω=0),and evidence for the PG has been widely reported [153–161]. This behav-ior may be anticipated in the presence of long-range (or mesoscopic) order,e.g. spin- or charge-order, which breaks the translational symmetry of thecrystal: the loss of spectral weight in particular momentum-energy regionswould be a simple consequence of the avoided crossings in the symmetry-reduced bandstructure [156, 157, 162, 163]. However, this argument may beunsatisfactory in the presence of strong electronic correlations and short-range orders with correlation length of few unit cells. Copper-oxide high-temperature superconductors are a clear example where the origin of thePG (which presents different phenomenology for hole and electron dop-ing) and its relation with other phases are still debated, and a universalunderstanding has yet to emerge [41, 149, 153–155, 162–165].In the specific case of electron-doped cuprates, the PG is believed tobear a relation to the robust antiferromagnetic (AF) order [41, 42, 166–173].In fact, the family of electron-doped cuprates (much less studied than theirhole-doped counterpart) is characterized by an AF phase extending overa wide doping range up to the SC dome, as illustrated in Figure 4.1a forthe archetypal electron-doped cuprate Nd2−xCexCuO4 (NCCO). However,scattering experiments on electron-doped cuprates have shown that thelong-range AF order disappears when entering the narrow SC dome [169,174], as evinced by the evolution of the instantaneous spin-correlation length(ξspin) presented in Figure 4.1b. Moreover, the commonly reported charge-order in cuprates does not exhibit a clear connection to the AF order [175,176], although a coupling to dynamic magnetic correlations has been re-71AF SC[42][167][166]PCCO [168]T*0.11 0.13 0.15 0.170100200300T (K)Ce doping, xNd2-xCexCuO4x = 0.154x = 0.150x = 0.145x = 0.134x = 0.129x = 0.106x = 0.075x = 0.038xspin / a 5205010020050010T (K)0 100 200 300 400a bFigure 4.1: Phase diagram and AF order of Nd2−xCexCuO4.(a) Phase-diagram of the archetypal electron-doped cuprateNd2−xCexCuO4 showing the onset temperature of the pseudo-gap T? as an orange shadow [42, 166–168]. The doping measured inthis study is highlighted by the yellow arrow. (b) Spin-correlationlength ξspin (here plotted in units of a, where a is the lattice parameter)as a function of temperature detected by inelastic neutron scatteringmeasurements [169]. Note that close to optimal doping, i.e. x ≥ 0.145,ξspin does not diverge a low T, indicating the lack of a long-range AForder.cently shown [177]. In electron-doped cuprates, a stable PG has been re-ported above the entire AF and SC domes by spectroscopy and transportprobes, with its onset temperature indicated by T? (orange shadow in Fig-ure 4.1a [41, 42, 166–168]). In the presence of long-range AF order, i.e.when the instantaneous spin-correlation length ξspin diverges at low tem-perature and a Ne´el temperature is defined, T? has been proposed to be atemperature crossover for which the quasiparticle de Broglie wavelength(λB ' vF/piT, where vF is the Fermi velocity) becomes comparable to ξspin[169, 178]. However, these considerations seem to fail at optimal dopingwhere only short-range spin-fluctuations (ξspin ' 20− 25 a, where a is theunit cell size) are detected by inelastic neutron scattering (Figure 4.1b) [169].Indeed, for dopings where the long-range AF order disappears, i.e. when72the short-range ξspin does not diverge at low temperature, an unambiguousidentification of a temperature crossover with λB is prevented. In addition,the underlying superconducting phase has been proposed to limit the de-velopment of ξspin [169].In order to tie together the momentum-resolved PG spectral featuresand short-range AF correlations in electron-doped cuprates, we performeda time- and angle-resolved photoemission (TR-ARPES) study of optimallydoped NCCO (Tc '24 K, yellow arrow in Figure 4.1a), which is charac-terized by ξspin ' 20a at low temperatures (T'Tc) [169]. TR-ARPES pro-vides an alternative, more effective and controlled experimental approachto measure a detailed temperature-dependence than the standard equilib-rium ARPES, which is often complicated by surface degradation as wellas coarse and uncorrelated sampling. As in standard pump-probe spec-troscopy, a near-infrared pump pulse is used to perturb the system, withits relaxation studied by varying the temporal delay of a subsequent UVprobe pulse. After the initial fast relaxation within 100 fs [131–136], an ef-fective electronic temperature Te may be defined at each point in time, al-lowing a temperature-dependent scan to be performed continuously andwith remarkable accuracy [124]. Since the acquisition of TR-ARPES data isperformed by cycling continuously the pump-probe delays, each time de-lay (and consequently each electronic temperature) is acquired in the sameexperimental conditions. By applying this transient approach, we demon-strate the direct relation between the subtle momentum-resolved spectro-scopic features of the PG and short-range ξspin(Te), as extracted from inelas-tic neutron scattering [169]. In particular, we identify T? as the crossovertemperature above which the spectral broadening due to the reduction ofξspin exceeds the PG amplitude, establishing the PG as a precursor of theunderlying AF order.The optimally doped Nd2−xCexCuO4+δ (x '0.15) single crystals mea-sured in this study were grown by the container-free traveling solvent float-ing zone technique and an additional post-growth annealing treatment wascarried out to remove the oxygen surplus exhibited directly after growth.Such annealed crystals exhibit a transition temperature Tc=23.5 K with tran-73sition widths of 1 K [179, 180]. All the TR-ARPES experiments were per-formed using the 6.2-eV probe setup in Damascelli Laboratory at UBC ex-ploiting the classic pump-probe scheme: the photon energy of the pumpbeam is 1.55 eV while the 6.2 eV probe is generated by fourth-harmonicgeneration of the fundamental wavelength (see Section 2.2.1 for details).The data presented here were acquired with two different incident pumpfluences: 28±5 µJ/cm2 hereon denoted as low fluence (LF) and 50±10 µJ/cm2denoted as high fluence (HF). The pump and probe beams were s-polarized.The samples were cleaved and measured at <5·10−11 torr base pressureand 10 K temperature and the photoemitted electrons were detected by aSpecs Phoibos 150 hemispherical analyzer. The overall energy and tempo-ral resolutions were estimated at 17 meV and 250 fs, respectively.4.2 Static 6.2-eV ARPES of Nd2−xCexCuO4The equilibrium Fermi Surface (FS) mapping of optimally-doped NCCOacquired with 6.2 eV probe pulsed-light is shown in Figure 4.2a. Note thatdespite the low photon energy of the probe beam, the experiment as de-signed allows for access to the intersection of the AF zone boundary (AFZB,red dashed line) and the tight-binding at ω=0, commonly referred to asthe hot-spot (HS, purple dotted circles in Figure 4.2). This region in k-space coincides with the location where an AF-driven PG is expected tobe particle-hole symmetric [41]. In a mean field description, the commen-surate q= (pi,pi) folding of the FS originates from a strong quasi-2D AForder in the copper-oxygen plane [41]. The Green’s function can then bewritten as [163, 168, 178]:G−1(k,ω) = ω− ek + iη − ∆2PGω− ek+q + iΓ , (4.1)where ek is the bare energy dispersion, ∆PG the AF-driven pseudogap spec-troscopic amplitude determined by the local Coulomb interaction and spinsusceptibility [178], η the single-particle scattering rate, and Γ a broad-ening term that leads to a filling of the pseudogap via the reduction of74a b(p,p)(p,0)(0,0)w=0 meV w=0 meVNodeAFZB(0,p) (p,p)(p,0)(0,0)(0,p)fHS HSFigure 4.2: Fermi surface of optimally-doped Nd2−xCexCuO4. (a) Ex-perimental FS of optimally-doped NCCO measured with 6.2-eV probepulse, 10 K base temperature. The integration window in energy is20 meV at the Fermi level. The solid blue line is a tight-binding con-stant energy contour at ω=0 [181], the red dashed line the AF zoneboundary (AFZB) and the purple dotted circle encloses the hot-spot(HS). The black dashed line represents the nodal direction, while thegreen and the black solid lines the two momentum directions mainlyexplored in this work, ϕ '39o and ϕ '26.5o respectively (ϕ angle be-tween (0,pi)-(pi,pi) and the nodal direction). (b) Simulated Fermi sur-face using Equation 4.1, ∆PG=η=Γ=85 meV.ξspin [173, 178]. Using Equation 4.1 we can calculate the spectral func-tion A(k,ω) = − 1pi Im[G(k,ω)] [182] and compute the Fermi surface, asshown in Figure 4.2b. It agrees well with our experimental data and pre-vious ARPES mapping studies [170, 171], clearly displaying a suppres-sion of spectral weight at the HS (purple dotted circle). Here, we used∆PG = η= Γ= 85meV for simulation purposes, as evinced from our experi-mental data (see Figure 4.5 and Section 4.3) and in agreement with previousoptical and ARPES studies [42, 166, 167, 170, 171, 183].In order to verify the capability of our 6.2-eV ARPES setup to probethe pseudogap spectral features in NCCO, we plot in Figure 4.3 the equi-librium ARPES spectra acquired at 10 K along the nodal direction and at75Momentum (Å-1)Energy (eV) 0.0Energy (eV)NodeHot-SpotEDC @ kF (a.u.)a b0.60.50.4-0.2-0.10.0NodeFigure 4.3: 6.2-eV static ARPES spectra of optimally-doped NCCO.(a) ARPES dispersion along the nodal direction (left) and at the HS(right, as defined in Figure 4.2), measured with 6.2-eV probe pho-ton energy, s-polarization, at the base temperature of 10 K. The blackdashed line in the left panel is the tight-binding nodal dispersion fromRef.[181]. (b) Energy distribution curves (EDCs) at the Fermi momen-tum (kF, arrows in panel (a) ) along the two momentum directions.the HS, together with the corresponding energy distribution curves (EDCs)at the Fermi momentum (arrows in panel (a) ). The black dashed linein Figure 4.3a, left panel, displays the nodal dispersion as predicted bythe tight-binding model from Ref.[181]. Note that no significant boson-mediated band-renormalization (i.e. kink) is observed along both the nodaldirection and the HS. Electron-doped cuprates present broad spectral fea-tures in comparison to their hole-doped counterpart making a detailed line-shape analysis particularly challenging [167, 170, 171, 183]. However, byvisual inspection of both ARPES spectra and EDCs, a clear suppressionof the spectral weight is observed in the vicinity of the Fermi level onlyat the HS, as contrary to the nodal direction. In addition, to substantiateour work in the context of previous ARPES studies on the electron-doped76-0.15Energy (eV)-0.15 -0.150.0 0.0 0.0IntensityMatsui et al. [21] 6.2 eV data 6.2 eV Matsui [21]aEnergy (eV)bc(0,0) (p,0)(p,p)NodeIntensityFigure 4.4: Pseudogap in static ARPES, comparison with previousstudies. (a) Comparison of the momentum-integrated energy distri-bution curves (EDCs) ranging from the nodal direction (black) to theHS (ochre): left panel, data extracted from Matsui et al. [167]; rightpanel, 6.2-eV data of this work. (b) Fermi surface mapping with 6.2-eV. The four solid lines indicate the momentum cuts displayed in (a),right panel. (c) Direct comparison of the momentum-integrated EDCsat the HS for the two works. EDCs have been rescaled to match atω '-0.15 eV.NCCO, we provide a direct comparison between our equilibrium 6.2 eVARPES mapping with that reported by Matsui et al. [167] for optimally-doped NCCO, acquired with 21 eV photons. This is done by computingmomentum-integrated EDCs along different momentum cuts of the FS rang-ing from the nodal direction to the HS (see the FS shown in Figure 4.4b). Asdisplayed in Figure 4.4a, the suppression of the spectral weight associatedto the PG distinctly emerges approaching the HS (ochre curves) both in theARPES spectra reported by Mastui et al. (left panel) and in the correspond-ing EDCs acquired with 6.2 eV light (right panel). A more unequivocalcomparison of the observed PG spectral features is offered in Figure 4.4c77in which momentum-integrated EDC at the HS acquired with 6.2 eV (darkgreen) is overimposed to that from Matsui’s work (orange) [167], show-ing a remarkable agreement (EDCs have been rescaled to match at ω '-0.15 eV) and thus validating our experimental approach. Note that the evo-lution of the EDCs as a function of the momentum cut displayed in Fig-ure 4.4a reflects the dispersion of the pseudogap central energy along theFS. In particular, the PG center is expected to disperse from above to be-low the Fermi level moving from the nodal direction to the antinode [41].This behaviour stems from the incipient 2D-(pi,pi) antiferromagnetic orderincluded in Equation 4.1: as illustrated in Figure 4.5a only at the HS thepseudogap is centered at the Fermi level. This observation designates theHS as the primary choice for a momentum-resolved study in order to havedirect access to the pseudogap spectral features and their evolution as afunction of temperature.To conclude the characterization of the 6.2-eV equilibrium ARPES map-ping of NCCO, we present a novel scheme to visualize the energy de-pendence of the PG and its amplitude ∆PG. The symmetrization of themomentum-integrated EDC [i.e. Symmetrized Energy Distribution Curve(SEDC)] removes any dependence of the photoemission signal on the Fermi-Dirac distribution function [184], giving access to underlying modificationsof the local density of states (DOS). Note that while the DOS is defined asthe integral of A(k,ω) over the full Brillouin zone, in this work we focus onindividual momentum cuts perpendicular to the FS, effectively analyzinga tomographic density of states [122]. Figure 4.4a shows how the extrin-sic background level for ω < -0.05 eV depends on the momentum directionand its intensity may be comparable to the one of the spectral features ap-proaching the HS. The definition of a baseline may then be difficult, makingthe interpretation of SEDCs acquired along different momenta directionschallenging. In this regard, here we define the quantity RcutDOS(ω) as:RcutDOS(ω) =SEDCcutNoPump(ω)SEDCcutPump(ω)∝DOScutNoPump(ω)DOScutPump(ω), (4.2)78RDOS (a.u.)f=45° f=38° f=32° f=26.5°-0.1 0.1 -0.1 0.1Energy (eV)k-kF (Å-1)a Near Node HSLF HFNearNode  HS-0.1 0.1 -0.1 HS-0.1 0.10.0 -0.1 0.10.0Energy (eV) Energy (eV) Near NodeEFFigure 4.5: Pseudogap dispersion and amplitude. (a) Momentum dis-persion of the pseudogap as predicted by the simple (pi,pi)-foldingmodel of Equation 4.1 (η=Γ=10 meV). ϕ represents the angle between(0,pi)-(pi,pi) and the nodal direction as illustrated in Figure 4.3a. (b)RDOS(ω) as defined in Equation 4.2 along the near-nodal direction(green markers) and at the HS (red markers) for the two pump fluencesexploited in this work, LF (left) and HF (right). Data computed at themaximum effect of the pump excitation, i.e. ∼0.6 ps. Blue dashed lineshighlight the pseudogap amplitude.where the label ”Pump” refers to the time delay corresponding to the max-imum effect of the pump excitation, i.e. ∼0.6 ps. This function intrinsi-cally mimics the energy dependence of the pseudogap along the specificmomentum direction kcut, independently of the underlying backgroundcharacterizing different momenta along the FS. Figure 4.5b shows the com-puted RDOS(ω) along the near-nodal direction (as defined in Figure 4.5a,second panel) and at the HS for the two pump fluence regimes used in79this work, LF and HF respectively. While we do not observe any rele-vant modifications of the RDOS along the near-nodal direction (green mark-ers), a clear PG feature appears at the HS (red markers) for both fluenceregimes. From Figure 4.5b, we can estimate the pseudogap amplitude byassessing the extent of the spectral weight suppression from the gap cen-ter, i.e. the minimum in RTDOS (blue dashed lines). The obtained value∆PG ' 85meV is consistent with what reported in previous optical andARPES studies [42, 166, 167, 170, 171, 183] . The remarkable qualitative(Figure 4.4) and quantitative (Figure 4.5) agreement in the identification ofthe PG features between our work and previous spectroscopic studies onthe electron-doped NCCO establishes a solid base for the definition of thepseudogap phenomenology and its characterization in our investigation.4.3 Tracking the pseudogap spectral weight in anultrafast fashionBefore moving to a detailed analysis of our data, we illustrate our exper-imental strategy for tracking the PG via TR-ARPES. Figure 4.6a comparessimulated and experimental EDCs integrated along the momentum direc-tion through HS for two (transient) electronic temperatures. The corre-sponding experimental SEDCs in Figure 4.6b, bottom panel, show the fill-ing of the PG at high temperature, which can be well modeled by increasingthe spin-fluctuation spectral broadening term Γ from 85 meV (50 K, blackcurves) to 160 meV (130 K, red curves), while ∆PG=η=85 meV are fixed (Fig-ure 4.6b, bottom panel). However, we recognize that any experimental es-timate of the temperature dependence of the PG spectral weight by fittingof SEDCs may be affected by intrinsic and uncorrelated noise, as well as ex-trinsic electron background (arising from irregular cleaves, secondary elec-trons, and electrons scattered in the detection process). We overcome thislimitation by computing the difference between the photoemission inten-sity for high temperature (130 K) and its counterpart for low temperature(50 K), as illustrated in Figure 4.6c. As discussed later in more detail inEquation 4.3, this quantity is proportional to the differential momentum-80EDCa1. -0.1 0.0SEDCdEDCEnergy (eV) Energy (eV) Energy (eV)Te=50 KTe=130 KSim. Sim. Sim.Exp. Exp. Exp.PumpNoPumpPump - NoPump130 K - 50 Kb c1.00.0-0.1 4.6: Experimental strategy for tracking transient filling of thePG. (a) Momentum-integrated EDCs at the HS for 50 K (black) and130 K (red). Top panel: EDCs simulated using Equation 4.1, andΓ=85 meV and 160 meV for low and high temperature conditions, re-spectively. Bottom panel: experimental background-subtracted EDCsin HF regime (dots, raw data; solid lines, smoothed data; the back-ground is estimated from the integrated ARPES intensity in regionswhere no dispersive spectral features are detected). (b) Simulated (top)and experimental (bottom) symmetrized EDCs (SEDCs). For Te=130 Kthe shortening of ξspin leads to a filling-up of the PG (red curves). (c)Simulated (top) and experimental (bottom) differential EDCs (dEDCs,as defined in Equation 4.3) demonstrating how a filling of the PG man-ifests as an increase of the photoemission intensity for ω '-50 meV(green arrows).integrated EDCs (dEDCs) and, by removing spurious contributions, high-lights the temperature evolution of the PG spectral features and Fermi-Dirac distribution. While the latter would lead to a symmetrical suppres-sion (increase) of the photoemission intensity for all ω < 0 (ω > 0), inde-pendently of the explored momentum region, it is evident that a filling ofthe PG may lead to an increase of the photoemission intensity at ω ' −50meV (green arrows in Figure 4.6c). Since thermal contributions are negligi-ble at ω'−50meV within the range of electronic temperatures explored inthis work (4kBT∼50 meV for T=150 K), this approach allows us to identifythe increase of photoemission intensity for ω ' −50meV as the signatureof the filling of the PG independently from the thermal broadening and81thereupon track its temperature evolution with high sensitivity. Note thata similar scheme has been recently used to track the electron-boson interac-tion in hole-doped cuprates via a transient analysis/modeling of the bandrenormalization (ie. kink) [185]. However, as shown in Figure 4.3a, in thespecific case of NCCO only a very modest kink feature is observed (see alsoRef. [183]), thus not affecting our analysis and conclusions.Having defined the framework for our investigation, we now trackthe temperature-dependent modification of the low-energy DOS at the HSin optimally-doped NCCO by introducing thermal excitations via opticalpumping. In order to establish a direct connection between the TR-ARPESphenomenology and the PG onset temperature T?, we must convert themeasured time dependence of our data to an effective temperature evolu-tion [124]. This is done by fitting the Fermi edge width of the momentum-integrated EDCs along the near-nodal direction (ϕ '38o, green solid linein Figure 4.2a), as depicted in Figure 4.7a for various pump-probe delays.Following this approach an effective electronic temperature Te can be de-termined as a function of time delay for both the pump fluences exploitedin this work (Figure 4.7b, red and black circles for LF and HF, respectively).The transient Te can be phenomenologically fit by a double exponentialfunction (solid lines in Figure 4.7b), reminiscent of the two-temperature-model framework [124, 131–134]. Note that the discrepancy between thevalues at negative pump-probe delays for the two fluences is the result ofdifferent overall thermal effects due to the pump beam. To further sup-port our estimation of the transient electronic temperature, we performeda similar fitting procedure along the HS by centering the Fermi-Dirac fit-ting range to ω > 0 for tracking the modification of the slope of the dis-tribution. As shown in Figure 4.7c for the HF regime, the extracted Te(although naturally more scattered) appears in good agreement with thenear-nodal counterpart. Another important aspect to consider while es-timating the transient electronic temperature relies on the identification ofpossible non-thermal contributions to the electronic distribution, especiallywhen pump and probe are overlapped in time. Figure 4.7d shows dEDCsfor three different pump-probe delays in HF regime at the HS (we employ820.010.1-30 30 T e (K)160120804086420 LF HFb Near Node HS0.08 ps / 190 K0.8 ps / 120 K0.3 ps / 160 K8642016012080400 50 100Delay (ps)Delay (ps)T e (K)Energy (meV)dEDC (a.u.)HS0 50-50Energy (meV)LF 45 K-0.5 psHF 58 K-0.5 psHF 157 K+0.33 psHF 105 K+1 psHF 90 K+2 psHF 80 K+7 pscdIntensity (a.u.)aFigure 4.7: Effective electronic temperature Te. (a) Momentum-integrated EDCs along the near-nodal direction (green line in Fig-ure 4.2a) for selected pump-probe delays. The solid black lines areFermi-Dirac distribution fits. The inset displays three delays (-0.5 psLF, +0.33 ps HF, +2 ps HF) in logarithmic scale. (b) Transient electronictemperature for both LF and HF. The solid lines are phenomenolog-ical double exponential-decay fits (decay times are: (0.6±0.1) ps and(7.5±1.3) ps for LF, (0.45±0.15) ps and (8.3±2) ps for HF). Error barsrepresent the confidence interval in the fitting procedure correspond-ing to ±3σ (σ is the standard deviation). (c) Te extracted along thenear-nodal direction (green circles) and HS (red circles), HF regime.The black line is the phenomenological bi-exponential fit as in (b). (d)dEDCs at HS for different time delays τ, HF regime. The black lines83(from the previous page): are thermal fits given by the difference of twoFermi-Dirac distributions at different temperature (base temperature60 K). A non-thermal tail is observed in the dEDC at τ = 0.08 ps for ω> 50 meV. In light of this, grey points and shadow in panels (b) - (c)indicate time delays for which a pure thermal-fitting is not accurate.dEDCs for removing extrinsic electron background which may affect a cor-rect estimate of thermal and non-thermal features). dEDCs have been fitby a thermal function given by the difference of two Fermi-Dirac distribu-tions and while dEDCs at 0.3 ps and 0.8 ps resemble well a pure thermalelectronic distribution, for τ = 0.08 ps (when pump and probe pulsed areoverlapped in time) non-thermal features appear on top of the thermal fitfor ω> 50 meV. Following these results, we identified the system well ther-malized for pump-probe delays τ > 0.3ps, which timescale is comparableto the temporal resolution of our TR-ARPES system (' 0.25ps). Therefore,in Figure 4.7b-c we mark in gray the time delays exhibiting non-thermalfeatures.Figure 4.8a displays the transient enhancement of the photoemissionintensity at the HS in a 20 meV energy window about ω=−50 meV, IPGω=−50,given by the momentum-integrated EDC along ϕ '26.5o (momentum di-rection indicated by the black solid line in Figure 4.2a). This particularchoice of energy window was motivated by the dEDCs at the HS (modeledand experimental) shown in Figure 4.6c and Figure 4.10b. The temporal re-sponse of IPGω=−50 exhibits different behaviour depending on the pump flu-ence: while the enhancement of IPGω=−50 recovers exponentially within'2 psfor the LF regime, in the HF regime it saturates for approximately 2 ps anddoes not recover within the domain of pump-probe delay studied in thiswork. This saturation of IPGω=−50 in the HF regime suggests a full suppres-sion of the PG. To further investigate the origin of the suppression of the PGspectral weight, we plot IPGω=−50 directly as a function of Te in Figure 4.8b(the non-thermal points – grey circles in Figure 4.7b-c and panel (a) – areomitted). The observed enhancement of photoemission intensity resemblesthe temperature dependence of the inverse of the spin-correlation lenght84w=-50 meV    Hot SpotIPG (a.u.) LF HF1/xspin (1/a 10-3)T* 1/xspin LF HFaDelay (ps)86420 40 80 120 160Te (K)04812IPG (a.u.)051015507090bFigure 4.8: Pseudogap spectral weight vs. temperature in optimally-doped NCCO. (a) Temporal evolution of the photoemission intensityat the HS (see black line in Figure 4.2a) for ω=-50±10 meV, IPGω=−50,for both the employed pump fluences LF and HF. The solid black lineis a phenomenological single exponential-decay fit for the LF curve,while the HF curve saturates in the first 2 ps after the pump excita-tion. Error bars represent ±σ. (b) Photoemission intensity at the HS,ω=-50±10 meV, as a function of the electronic temperature Te (blackand red circles for LF and HF, respectively; data points are plotted vsthe electronic temperature determined from the phenomenological fitsin Figure 4.7b). Non-thermal time delays – grey points and shadowin panels (a) – have been omitted. The green line and transparentshadow represent the inverse of the spin-correlation length ξspin fromneutron scattering studies for optimal doping Tc '24 K [169], appro-priately scaled and offset. We identify T? as the temperature at whichthe PG is completely filled, in agreement with the observed saturationof IPGω=−50 for HF. Error bars as defined in (a) and Figure 4.7b for IPGω=−50and Te, respectively.ξ−1spin reported in Ref. [169] for optimally doped NCCO (superimposed inFigure 4.8b as a green line and shadow, with appropriate offset and scal-ing). However, for temperatures Te >110 K, IPGω=−50(Te) is found to saturate(red shadow in Figure 4.8b) marking a departure from ξ−1spin. Despite thenovel dynamical approach employed here to perform a temperature de-pendent study at the HS, the observed onset of the saturation in Figure 4.8bis in good agreement with T? reported by other spectroscopy and transport85probes [42, 166–168]. Note that the unfolded deviation of IPGω=−50(Te >T?)from ξ−1spin is not driven by a phase transition, but is rather a consequenceof the PG filling, as discussed in the next Section 4.4.Before moving to a comprehensive analysis and modeling of the pho-toinduced thermal modification of the PG for both LF and HF pump regimes,we substantiate the analysis of the momentum-integrated dEDCs at theHS as a procedure to extract valuable information regarding the transientevolution of the pseudogap. As shown in Figure 4.3, the spectral fea-tures of NCCO are inherently broad, precluding the sort of detailed anal-ysis of the transient spectral function which has been achieved for hole-doped cuprates [114]. In this regard, an alternative approach involvesthe analysis of the temporal evolution of dEDCs, defined as the differencein the photoemission intensity after and before the pump perturbation,IPump − INoPump. If both the matrix-elements and the electronic distribu-tion are momentum-independent, the photoemission intensity integratedalong a well-defined momentum direction (ki → k f ) is proportional to thelocal DOS:∫ki→k f I(ω,k)dk ∝DOS(ω) · f (ω). We can then express dEDCas:dEDC ∝∫ki→k fIPump(k,ω)dk−∫ki→k fINoPump(k,ω)dk ' ∆DOS · f +DOS · ∆ f(4.3)where ∆DOS and ∆ f are the relative variations of the DOS and the distri-bution function, respectively. In order to robustly identify and disentangle∆DOS and ∆ f at least two different momentum directions need to be ex-amined. In this work the near-nodal direction and the HS were explored(see the green and black cut in Figure 4.2a, respectively), which showcasethe two contributions to the dEDCs. In fact, along the near-nodal direction– where the PG is expected to be pushed well above the Fermi energy inthe unoccupied states (see Figure 4.5a) – ∆ f term of Equation 4.3 domi-nates and a mere thermal broadening is expected. Contrarily, at the HS thepseudogap is predicted to be particle-hole symmetric, thus facilitating thedisclosure of the transient modification of the DOS (i.e. the ∆DOS term ofEquation 4.3 gives a substantial contribution, especially for ω <0). The dif-ferent contributions of ∆DOS and ∆ f to the dEDCs are highlighted in Fig-86-5005086420NearNode -50 meV +30 meV -15 meV +15 meV -30 meV +50 meV 10050NearNodeDelay (ps) Te (K) -15 meV +15 meVIntensity (a.u.)a20100-1086420840+15 meV NearNode HS10050HS HSDelay (ps) Te (K) -15 meV +15 meV -50 meVIntensity (a.u.)bFigure 4.9: Energy- and momentum-resolved TR-ARPES dynamics.(a) Left panel: transient evolution of the momentum-integrated pho-toemission intensity at different binding energies (10 meV integrationwindow) along the near-nodal direction, LF regime. Right panel: map-ping of the transient photoemission intensity as a function of the elec-tronic temperature Te reported in Figure 4.7b. (b) Same as in (a), but atthe HS. The inset shows how the transient photoemission intensity atω =+15 meV along the near-nodal direction and at the HS are compa-rable. The green dashed line in the right panel is the inverse of ξspin,appropriately scaled and offset, as discussed Figure 4.8b.ure 4.9 which displays the transient photoemission intensity as a functionof the binding energy and the momentum direction (near node and HS)in the LF regime. Along the near-nodal direction the transient evolutionof the photoemission intensity is approximately symmetric with respectto the Fermi level ω=0 (Figure 4.9a, left panel), as we are mainly track-ing the evolution of ∆ f in Equation 4.3. In addition, the right panel of Fig-ure 4.9a maps the change of the photoemission intensity atω=±15 meV as afunction of the electronic temperature, which displays a linear trend, as ex-pected within this energy- and temperature-range. Both these observationsconfirm that along the near-nodal direction the dynamics of the ARPES sig-nal is mainly driven by thermal effects. Otherwise, at the HS the transientphotoemission intensity is strongly not symmetric with respect to ω=0, asdisplayed in Figure 4.9b: while the excitation signal above the Fermi level iscomparable to that from along near-nodal direction (see inset Figure 4.9b),87the signal for ω<0 displays a different behaviour, as the depletion (i.e. sup-pression of photoemission intensity) is less pronounced at the HS than itscounterpart along the near-nodal direction. Moreover, an enhancement ofthe photoemission intensity is observed for ω '-50 meV (black curve), inagreement with the schematics of Figure 4.6. These results attest the signif-icant contribution of ∆DOS to the dEDC signal at the HS, especially evidentfor ω '-50 meV where the thermal contribution is negligible (i.e. f '1 as50 meV'4 kBT for T=150 K). Indeed, for ω =-15 meV, even if a contribu-tion from ∆DOS is expected, the depletion at the HS (blue markers in Fig-ure 4.9b, right panel) does not resemble the trend reported for ω =-50 meV(black markers) as a consequence of the sizable thermal contribution. Alltogether these observations validate our approach in tracking the transientphotoemission intensity at the HS for ω '-50 meV as a direct signature ofthe evolution of the underlying PG.4.4 Relation of the pseudogap to the spin-correlationlength: theoretical modelIn Figure 4.10 we present a comparison between experimental (panels a1-b1) and simulated dEDCs (panels a2-b2) as a function of the pump-probedelay (τ) and binding energy (ω). The TR-ARPES data reported here can besimulated remarkably well using the simple model of Equation 4.1 througha substantial increase of the broadening term Γ alone, which phenomeno-logically describes the filling of the PG due to the reduction of the spin-correlation length [173, 186]. The qualitative and quantitative agreementbetween single experimental and simulated dEDCs along the near-nodaldirection and HS for τ=+0.6 ps (Figure 4.10b) attests the capability of thismodel in grasping the modifications of the DOS due to the PG filling. Itis important to remark that a full closure of the gap fails to reproduce ourTR-ARPES data. Figure 4.11a compares simulated dEDCs at the HS for acomplete gap closure (∆PG→ 0, black line) and gap filling (Γ = 2.5 · Γ0, redline), and only the latter reproduces well the experimental curves shown inFigure 4.10b1. This observation is in agreement with the SEDCs presented88Near Node LF Hot Spot LF Hot Spot HFExp.+0.6 ps86420Delay (ps)86420Delay (ps)86420Delay (ps)0.0-50 50Energy (meV)Sim.+0.6 psSim.Sim. Sim.Exp.Exp. Exp.-50-500.00.05050Energy (meV)Energy (meV)0011-1-1dEDC (a.u.)dEDC (a.u.)a1 b1a2 b2Figure 4.10: Comparison of experimental and simulated TR-ARPES data. (a) Momentum-integrateddEDCs, experimental (a1) and simulated (a2), along the near-nodal direction (left panel, LF) and at theHS (middle and right panels for LF and HF, respectively). Simulated panels have been generated using thefit of the experimental transient Te shown in Figure 4.7b, and assuming Γ(Te) = C · ξ−1spin(Te) , as in Ref. [178](C '1.9 a eV, where a is the unit cell size). (b), Experimental (b1) and simulated (b2) dEDCs along the near-nodal direction (green line, LF), and at the HS (black and red lines for LF and HF, respectively), at τ=+0.6 ps.89 1/xspin-50 0 50Energy (meV)10.5dEDC (a.u.)015010050Te (K)1/xspin (1/a 10-3)ClosureDPG→0907050Sim. IPG (a.u.)0612G=2.5G0FillingT*a bFigure 4.11: Filling of the PG through a reduction of ξspin. (a) Sim-ulated dEDCs at the HS assuming values ∆PG=η=85meV and initialand final temperature 50 K and 130 K, respectively. dEDCs were simu-lated assuming ∆PG→ 0 (gap closure, black line) and Γ = 2.5 · Γ0 (gapfilling, red line). (b) Simulated photoemission intensity at the HS, ω=-50±10 meV, as a function of the electronic temperature (black dashedline). The green line is the inverse of ξspin, obtained from Ref. [169] asdiscussed in Figure 4.8b.in Figure 4.6b and scattering studies [169, 174], which show that the spectralweight associated with magnetic excitations displays a much weaker tem-perature dependence than the one of the spin-correlation length. Finally,we plot in Figure 4.11b the simulated analog to Figure 4.8b, noting a re-markable correspondence between the two figures. In particular, assumingthe direct relationship between the filling of the PG and ξ−1spin (as predictedfor 2D spin-fluctuations [178]), the simulated filling of the PG saturates fortemperatures T'T? when the broadening Γ(T?)' 2∆PG '170 meV. This em-pirical observation agrees with a recent theoretical study of the vanishingof the PG in the electron-doped cuprate Pr1.3−xLa0.7CexCuO4 [187].We conclude this section by providing more details on the different con-tributions of the various parameters in Equation 4.1 (Γ, η and ∆PG) to thesimulations shown in Figure 4.10 and Figure 4.11. Indeed, in our simula-tions only the spin-fluctuation-induced broadening term Γ is assumed to90100.60.50.4 30 K 80 KNearNodeNear HSk (Å-1)Intensity (a.u.)b10-50 0 50h values 65 meV 85 meV 105 meVEnergy (meV)dEDC (a.u.)c30020010015010050Te (K)G (meV)G0aHSFigure 4.12: Contribution of Γ and η to the pseudogap modeling. (a)Temperature dependence of Γ(Te) used for simulations shown in Fig-ure 4.10 and Figure 4.11. (b) Momentum distribution curves (MDCs)along the near-nodal direction (ϕ '38o) and near the HS (ϕ '31o) fortwo different transient electronic temperatures. The horizontal linesgives an estimate of the full-width-half-maximum (FWHM). (d) η-dependence of the simulated dEDCs at the HS.display a temperature dependence, while the single-particle scattering rateη and the gap amplitude ∆PG remain unchanged. In particular, we haveimposed Γ(Te) = C · ξ−1spin(Te) as shown in Figure 4.12a, where ξspin is thespin-correlation length and C a constant, in agreement with theoretical pre-dictions by Vilk and Tremblay [178]. Neutron scattering experiments havereported a value of ξspin '20a (with a lattice constant) for optimally-dopedNCCO at low temperature (see Figure 4.1b) [169]. We set Γ0=85 meV asinitial value of the broadening term, which qualitatively reproduces thelow-temperature partial filling of the pseudogap at the HS due to the ab-sence of a long-range AF order. TR-ARPES data are well reproduced bydefining the constant C '1.9 eV·A˚, as shown in Figure 4.10. In addition,in our simulations we set ∆PG=η=85 meV, both parameters independent onthe temperature. The value of ∆PG derives from previous spectroscopicstudies [42, 166, 167, 170, 171, 183] and agrees well with our experimentalresults (see Figure 4.5). As for the single-particle scattering rate η, it canbe estimated by fitting the momentum distribution curves (MDCs) FWHM(Figure 4.12b) : η= FWHM2 · vF, where vF is the bare dispersion velocity from91the tight-binding model of Ref.[181]. In particular, we extracted a FWHMof∼0.066 A˚−1 and∼0.076 A˚−1 for the near-nodal and near-HS directions atlow temperature, in agreement with MDCs FWHM reported from a recenthigh-resolution ARPES study of NCCO [183]. These values correspond toη ∼72 meV and η ∼85 meV for the near-nodal and near-HS directions, re-spectively. Note that for both momentum directions we do not observe anyvariation of η larger than±5 meV within the range of transient temperatureexplored in this work. Following these observations, Figure 4.12c displayssimulated dEDCs at the HS (50 K and 120 K initial and final temperature,respectively, ∆PG=85 meV, and Γ tuned from 85 meV to 220 meV) as a func-tion of different η values, demonstrating how different values of η (evenwell beyond the limits of our experimental estimate) alter only minimallythe outcome of the simulations, thus not affecting our conclusions.4.5 ConclusionsIn conclusion, we have reported the direct relation between the partial sup-pression of the electronic spectral weight, a.k.a pseudogap, and short-rangemagnetic correlations in electron-doped cuprates. In particular, by per-forming a detailed ultrafast ARPES study at the hot-spot of the optimally-doped NCCO electron-doped cuprate, we have demonstrated that the tem-perature dependence of the low-energy DOS is closely related to the spin-correlation length ξspin. We identified two different temperature regimesfor the PG, moving from low to high temperature: i) T<T?, in which thePG begins to fill alongside with the reduction of ξspin; ii) T>T?, whereξspin ∼10-15 a and the PG is completely filled-up. Our results show thatthe PG phenomenology in optimally-doped NCCO originates from short-range AF correlations, parametrized by ξspin(T), and T? is a crossover tem-perature above which the spectral broadening driven by the reduction ofξspin overcomes the PG amplitude ∆PG. This suggests that the frequentlyreported onset temperature T? does not represent a thermodynamic phasetransition, i.e. a sharp quenching of a well-defined order parameter; rather,T? is associated with the weakening of the short-range AF correlations and92incipient (pi,pi)-folding [173, 186]. In addition, the observation of the filling– not closure – of the PG suggests that the energy scale associated withthe PG survives to temperatures well above T?, possibly reminiscent ofscattering results [169, 174] which show that the spectral weight associatedwith magnetic excitations remains finite up to much higher temperaturesthan the spin-correlation length itself. This phenomenology may bear arelation to the underlying Mott physics [178], or to the recent proposal ofa crossover of the SU(2) gauge theory for fluctuating spin-density-wavesnear optimal-doping [188]. Finally, we note that our transient momentum-resolved study demonstrates that even underlying orders with correlationlengths of about ten unit cells may play a significant role in shaping theFermi surface topology and associated transport properties of complex ma-terials [41, 42, 166, 167, 170–173, 183, 189, 190].93Chapter 5Exploring the mixed-valentstate in SmxLa1−xB6Strong electron correlations originating from localized 4 f -states are crucialin describing the electronic structure and associated properties of rare-earthcompounds. An example of occurring phenomena in these materials is theemergence of the so-called mixed-valence, a state characterized by the ex-istence of a given element in the system in more than one oxidation state.The precise determination of the value of the fractional valence, its role inshaping the electronic properties and the pursuit to tune it by means of ex-ternal parameters such as temperature and pressure, have attracted a lot ofattention from the scientific community in the last few decades. Here wereport on the combined ARPES and XAS study of rare-earth SmxLa1−xB6series, employing chemical substitution as a tool to directly resolve thevalence fluctuations of Sm ions as a function of correlations. By gradu-ally varying the stochiometry, we track the evolution of the mixed-valentcharacter in the series, and resolve the emergence of an impurity regime ofintermediate-valent Sm ions for x < 0.2.945.1 IntroductionRare-earth hexaborides have been studied for many years due to the widerange of exotic phenomena arising from the correlations among localized4 f electrons. These include (but are not limited to) heavy fermions, super-conductivity, hidden order phases and mixed-valence (MV) [191–194]. Inparticular, the characteristic to exhibit a MV state is one of the most ex-tensively studied [9, 195, 196]. In a homogenous MV system two valenceconfigurations have nearly degenerate energies, so the ground state canbe expressed as a linear superposition of the different configurations, eachcorresponding to a distinct oxidation state [9, 43]. In the specific case ofrare-earth hexaborides, the two degenerate configurations correspond todifferent f occupation numbers, namely 4 f n and 4 f n−1. Therefore, the in-terplay between correlated 4 f -states and band-like states, deriving from the5d6s shell and crossing the Fermi level, plays an important role in shapingthe electronic structure of rare-earth hexaborides. This makes the investiga-tion of fluctuations between different valence configurations crucial for theunderstanding of the low-energy physics of these compounds. It has beenshown that external control parameters, such as temperature or pressure,may be used to tune the intermediate valence of the rare-earth ions and theassociated properties of the hexaborides [66–68, 197, 198]. As an alternativeapproach, chemical substitution on the rare-earth element is a chemistrytool acting directly on the occupation of the 4 f states. This approach mayallow the investigation of the modifications induced in the electronic struc-ture as a function of electronic correlations, from the very dilute regime tothe lattice limit. This approach becomes particularly intriguing in the con-text of an early theoretical work by Haldane [199], which speculates on thepotential occurrence of a MV state for a single impurity, thus raising thequestion on whether the MV concept can be extended to that limiting caseand if it can be observed experimentally.SmB6 represents a prototypical example of rare-earth MV compounds.Nearly fifty years after its characterization, its ground state still remainsunsolved. The description in terms of Kondo insulator is challenged by95fdMomentum (Å-1)Energy (eV)0.5 MГLa/SmBa b c02-20.0 0.5 1.0-0.5-1.0Figure 5.1: Crystal structure of SmxLa1−xB6. (a) Crystal structurecharacterizing the SmxLa1−xB6 series: rare-earth ions (La or Sm) andB-octhaedra are located at the corners and at the center of the cubic lat-tice structure, respectively. (b) Corresponding bulk Brillouin zone (BZ)with high-symmetry points highlighted in red. (c) Schematic of thebandstructure of a mixed-valent rare-earth hexaborides: the localizedf -state crosses the broad d conduction band opening a gap via hy-bridization. The wavefunction character is indicated by the color scale.Inset: Zoom-in of the momentum region where the bands cross. Pan-els (c) adapted from [4].the resistivity plateau at low temperature and the observation of quantumoscillations [49, 62, 63]. Furthermore, in-gap electronic states have beenreported by ARPES, in relation to the recent proposal of the possible real-ization of TKI in SmB6. In this scenario, the study of the intermediate va-lence state may be crucial for the understanding of the gap formation in thismaterial. Here we report an Angle-Resolved Photoemission Spectroscopy(ARPES) and X-Ray Absorption Spectroscopy (XAS) study of the chemical-substitution-induced valence fluctuations in the SmxLa1−xB6 series. All thecompounds of the series share the same CsCl-type crystal structure withthe rare-earth atoms at the cube corners and the B-octhaedra at its bodycenter (Figure 5.1a). The corresponding bulk Brillouin zone (BZ) is shownin Figure 5.1b, along with Γ, X and M high-symmetry points representingthe center, face-center and edge-center of the cube, respectively. Despite thesimilarity in the crystal structure, the two ends of the series exhibit very dif-ferent physics: LaB6 (x = 0) is characterized by a metallic ground state, incontrast to the rich ground state driven by MV in SmB6 (x = 1). Indeed,96Sm2+ [Xe]4 f 6 and Sm3+ [Xe]4 f 55d1 configurations are nearly degenerateand dominates the low-energy excitation spectrum of SmB6 (see Section 1.2and Figure 1.5). The reported value of the mean valency at low tempera-ture is ∼ +2.505, which increases to a plateau around ∼ +2.58 above 100K[66, 200]. As shown in Figure 5.1c, a small gap opens at the Fermi level asa result of the hybridization between the localized 4 f -states and the broadd-band. A strong admixture characterizes then the wavefunction near thecrossing points (i.e. where the small gap opens), while far away a definedcharacter is retained (color scale in Figure 5.1c). In this work, by graduallyvarying the stochiometry in the SmxLa1−xB6 series, we track the fluctua-tions of the mean valence of Sm ion, vSm, going from one end of the seriesto the other as 4 f electrons are progressively introduced in the system. Ourresults reveal the persistence of the MV character in SmxLa1−xB6 even inthe very dilute regime, pointing towards the limit of mixed-valent impu-rity state.The high-quality single crystals of SmxLa1−xB6 studied in this inves-tigation were grown using the aluminum flux method in a continuous Arpurged vertical high-temperature tube furnace [53]. Post-growth character-ization by energy dispersive x-ray spectroscopy for the actual Sm concen-tration was carried out in Los Alamos National Laboratory. ARPES exper-iments were performed in Damascelli Laboratory at UBC using a photonenergy of hν = 21.2eV, with a base pressure lower than 3 · 10−11 Torr anda base temperature of 10K. The electrons were collected using a SPECSPhoibos 150 hemisperical analyzer, with energy and momentum resolutionof 25meV and 0.02A˚ respectively. Additional ARPES measurements werecarried out at the SGM3 endstation at the ASTRID2 synchrotron radiationfacility [201], using a photon energy of hν = 67eV, with base temperature35K and energy resolution 35meV. All samples were cleaved in-situ andmeasured along the (001) surface. XAS measurements were performed us-ing the four-circle UHV diffractometer at the REIXS 10ID-2 beamline at theCanadian Light Source in Saskatoon [202], with base pressure and temper-ature of 5 · 10−10 Torr and 22K, respectively. DFT calculations for LaB6 wereperformed with the WIEN2K software package [203].975.2 LaB6 and SmB6: ARPES comparisonBefore moving to a detailed analysis and discussion of the SmxLa1−xB6 se-ries, we provide a comprehensive ARPES comparison between the twoends of the series: LaB6 and SmB6. LaB6 is the reference system for allrare-earth hexaborides compounds, with the absence of 4 f electrons and ametallic ground state. The fundamental B6 block can be described in termsof a series of molecular orbitals (MOs) originating from the 2s and 2p B-atomic orbitals [204]. The trivalency of La ions leads to the partial occupa-tion of one B6 antibonding MOs (after entirely filling all the bonding MOs),which hybridizes – in a strongly momentum-dependent way – with the La-5d band. Figure 5.2a illustrates the main features of the electronic structureof LaB6 as obtained from DFT calculations: dispersing bands stemming pri-marily from Bp-orbitals can be observed at higher binding energies, whilethe low energy electronic structure is dominated by large electron pocketscentered at the X-high-symmetry points of the BZ and exhibiting a strongadmixture of La5d and B2p character. For simplicity, we will denominatesuch electronic feature as ”5d-pocket” or ”X-pocket” throughout the rest ofthe chapter. Figure 5.2b presents the experimental ARPES spectra of LaB6acquired along two different crystallographic directions with 67 eV probelight, showing a good agreement with the DFT dispersion (white dots). Inparticular, we extract a Fermi velocity vF characterizing the half-filled X-pocket equal to (5.4 ± 0.2)eV · A˚ and (6.5 ± 0.2)eV · A˚ along X-Γ-X andM-X-M high-symmetry directions, respectively.Moving right in the lanthanide series, the 4 f shell of the rare-earth ele-ment gets filled (4 f n, with n varying from 1 for Ce3+, to 14 for Lu3+). De-spite the variety of rich physics arising from the interaction of f -electrons,the electronic structure of all rare-earth hexaborides share the key-role playedby the X-pocket, with primarily rare-earth 5d and B2p character, in dictat-ing the low energy physics. In Figure 5.3 we compare the ARPES spectra ofLaB6 and SmB6, as acquired along X-Γ-X (top) and M-X-M (bottom). Non-dispersing 4 f -states are clearly visible in the SmB6 spectrum (right side ofpanels a-b) at 15meV, 150meV and 1eV, consistent to what was reported98-2.5-2.0-1.5-1.0-0.50.0Energy (eV)ГX X X MM-5-4-3-2-10Energy (eV)1Г X MLaBa bFigure 5.2: Electronic structure of LaB6. (a) Electronic dispersion ofLaB6 obtained by DFT. The color scale shows the atomic character as-sociated to each band. (b) ARPES spectra of LaB6 along ΓX and XMhigh-symmetry directions, measured with hν = 67 eV and 35 K basetemperature. The white dots represent the dispersion as obtained byDFT shown in (a). The green dashed line is the dispersion of the X-pocket for SmB6 from slab-DFT calculations reported by Zhu et al. [60].in previous ARPES studies [56–58, 64], and associated to the 6H and 6FSm 4 f -multiplets. As anticipated, both compounds exhibit bulk electronpockets centered at X, which dispersion is shown in Figure 5.3c-d. It is im-portant to point out that, despite the hybridization between the X-pocketand the 4 f -states in SmB6 (red circles) leading to band-renormalization, thebare pocket dispersion can still be described by the same effective massof LaB6 (as also shown by the comparison of the DFT dispersion for LaB6(white dots) and SmB6 (green dashed line) in Figure 5.2b). Moreover, thesize of the X-pocket contours in SmB6 is much smaller than in LaB6. Notethat this distinction does not originate from a difference in the crystal struc-ture, as the lattice parameters differ by less than 1% (aLa = 4.156 A˚−1 andaSm = 4.134 A˚−1 [205]). This substantial variation indicates a change in theelectronic occupation of the X-pocket, which arises from the charge com-pensation between f and d electrons in the presence of intermediate va-lence of the Sm ions in SmB6. Indeed, if the system was entirely Sm3+([Xe]4 f 55d1), the X-pocket would be exactly half-filled (as is the case forLaB6), while it would be totally empty (i.e. above EF), if only Sm2+ ([Xe]4 f 6)was present. In the case of a MV system as SmB6, the Sm2+/Sm3+ ratio99-2.5-2.0-1.5-1.0-0.50.0Energy (eV)-0.4 0.0 0.4Momentum (Å-1)-0.4 0.0 0.4Momentum (Å-1)-2.5-2.0-1.5-1.0-0.50.0Energy (eV)-0.4 0.0 0.4Momentum (Å-1)-0.4 0.0 0.4Momentum (Å-1)-1.5-1.0-0.50.0Energy (eV)-1.5-1.0-0.50.0Energy (eV)kXM (Å-1)0.0 0.5kГX (Å-1)0.0 0.5LaB6LaB6SmB6SmB6M      X      M   X      M   X      Г      X   Г      X   badcFigure 5.3: ARPES dispersion of LaB6 and SmB6. (a) ARPES bandmapping along X-Γ-X direction of LaB6 (left) and SmB6 (right), mea-sured with 67 eV probe photon energy, at a base temperature of 35 K.(b) Same as panel (a), but along M-X-M direction. (c)-(d) Compari-son of the X-pocket dispersion as extracted from the ARPES spectraof LaB6 (black circles) and SmB6 (red circles) along X-Γ-X and M-X-Mdirections, respectively.determines the exact electronic occupation of the 5d band at the X-point,n5de , as an intermediate value between 0 and 1 (where n5de = 0, 0.5 representthe two specific cases for all Sm2+ and all Sm3+, respectively). This closerelation between n5de and the fractional percentage of Sm3+ will be funda-mental to evaluate the possible fluctuations of the valence of the Sm ionsin the SmxLa1−xB6 series, as discussed in Section 5.3. Figure 5.4 shows theevolution of the ARPES iso-energy maps as a function of binding energy,highlighting the different size of the X-pocket ellipsoid contours in the twocompounds. In addition, signatures of a (1×2) reconstruction are detected1000.00.5-0.5-1.0-1.5 -0.5 1.5XMГ ГXM0.50.0-0.5-1.0-1.5 -0.5 1.50.0-0.5-1.0-1.50.0-0.5-1.0-1.5kx (Å-1)ky  (Å-1)ky  (Å-1)kx (Å-1)Energy (eV)a bГ' Г'Figure 5.4: ARPES iso-energy contours in LaB6 and SmB6. (a) ARPESiso-energy maps at various binding energies of LaB6. (b) Same as inpanel (a), but for SmB6. Dashed green lines mark the crystallographichigh-symmetry directions. All energy maps are taken at hν =21.2 eV,base temperature 10 K. The integration window in energy is 20 meV.at Γ′ in both cases, in accordance to what is reported in previous studies forSmB6 [56, 57, 206, 207].In order to substantiate a direct comparison between LaB6 and SmB6,we must discuss the kz-dispersion of the electronic structure of the twocompounds, given the three-dimensional (3D) nature of the crystal struc-ture. In Figure 5.5a we present the photon energy dependence of the pho-toemission intensity along X-Γ-X in LaB6 and SmB6. Note that the em-ployed range of photon energy allows the mapping over a full kz period ofthe 3D BZ, as shown in Figure 5.5b. Given the cubic symmetry of both com-101807060504030200.0 0.5-0.5-1.0 1.0 0.0 0.5-0.5-1.0 1.0Photon energy (eV)Momentum (Å-1) Momentum (Å-1)LaB6 SmB6aX        Г        X   surface 210-1-223456kx (Å-1)k z (Å-1)b1521304050606780100120150MXMMXXXXXГГГFigure 5.5: Photon energy dependence of LaB6 and SmB6. (a) Photoe-mission intensity map at 500 meV (integration window 40 meV) alongX-Γ-X as a function of the probe photon energy for LaB6 (left) andSmB6 (right). All data taken at a base temperature of 35 K. The greenand blue solid lines mark the two photon energies employed in thiswork, hν=67 eV and hν=21.2 eV, respectively. We associated the bulkΓ high-symmetry point with 67 eV probe energy. (b) Illustration of themomentum cuts probed with different photon energies (inner poten-tial of 14 eV). Note that the range of photon energy used in (a) coversa full Brillouin zone in kz. Panel (b) adapted from Ref.[64].pounds, the kz-dispersion of bulk bands, such as the X-pocket here underconsideration, is expected to resemble that along kx and ky. Indeed, ellip-soidal contours associated to the X-pocket can be observed in Figure 5.5aand they exhibit an analogous photon energy dispersion in LaB6 and SmB6.Two different photon energies were employed for the ARPES measure-ments presented in this work: hν =21.2 eV (blue cut) and hν =67 eV (greencut), the latter identified as the photon energy corresponding to the bulkΓ high-symmetry point. Figure 5.6a illustrates how the photoemission in-tensity pattern across the X-point of SmB6 varies by adopting 21.2 eV (left)and 67 eV (right) as probe photon energy light. As shown in Figure 5.6b,while at low photon energies the photoionization cross-section for Sm4 fis comparable to that of Sm5d and B2p, for 67 eV (close to the 4d-4 f reso-102-1.5-1.0-0.50.0-0.4 0.0 0.4 -0.4 0.0 0.421eV 67eVM      X      M   Momentum (Å-1) Momentum (Å-1)Energy (eV) B2p La5d Sm4f0.010.111020 40 60 80Energy (eV)Cross-section (Mbarn)a bFigure 5.6: Dispersion of SmB6 probed with different photon ener-gies. (a) M-X-M ARPES spectrum of SmB6 acquired with hν = 21.2eV (left) and hν = 67 eV (right). The insets show EDCs extractedat 0.5 A˚−1 (white dashed lines), highlighting the enhancement of 4 f -states photoemission intensity observed with 67 eV. (b) Photoioniza-tion cross-section of the relevant orbitals defining the low-energy elec-tronic states of LaB6 and SmB6. Data in panel (b) taken from [208].nance) there is a substantial difference 1. As a result, we clearly recognizethe X-pocket and valence band associated to Sm5d and B2p in the ARPESspectrum at 21.2 eV, while the photoemission intensity acquired at 67 eV ismainly dominated by the non-dispersing 4 f states. This same dichotomy inthe photoemission pattern collected by employing the two different photonenergies characterizes the ARPES spectra of the SmxLa1−xB6 compoundspresented in this work.To conclude this section, we point out that the lack of a natural cleavageplane, along with the kz disorder likely characterizing the strong 3D disper-sion in these SmxLa1−xB6 compounds, leads to an extrinsic broadening ofthe spectral features. This precludes the sort of detailed lineshape analysisin terms of intrinsic scattering rates achieved in the previous chapters forhigh-temperature cuprate superconductors. Instead, here we focus our in-vestigation on the overall evolution of the electronic structure as a functionof x, with particular emphasis on the large electron pockets centered at X.1The resonance corresponds to a virtual transition 4d→ 4 f followed by an Auger decayand a subsequent emission of a 4 f photoelectron; this process would then have the samefinal state as the direct photoemission process (i.e. removal of one 4 f electron).1035.3 Evolution of the mixed-valence character in theSmxLa1−xB6 seriesOnce we established a solid base for a direct comparison between LaB6 andSmB6, we now present a detailed analysis of the SmxLa1−xB6 series. Fig-ure 5.7a illustrates the overall modification of the electronic structure uponSm substitution. ARPES spectra were acquired with 21.2eV along ΓX direc-tion (green solid line in inset of Figure 5.7c) for x = (0, 0.2, 0.55, 0.7, 0.8, 1).As x increases, the non-dispersing Sm 4 f -states emerge in the spectra andgain intensity, as highlighted in the EDCs displayed in Figure 5.7b. Clearlyvisible for all compounds, the B2p-character valence band (VB) centered atΓ exhibits a gradual shift to deeper binding energies as a function of theSm concentration x, with a maximum shift of ∼ 320 meV across the series(see Figure 5.7d). This effect is related to the change in the attractive poten-tial when trivalent ions, as La ions, are introduced in the 5d-band (i.e. bydecreasing x). Note that the bottom of the large X-pocket is instead char-acterized by a much larger shift of the order of ∼ 0.9eV (the bottom of theband is observed at ∼ −2.5eV and ∼ −1.6eV for LaB6 and SmB6, respec-tively). This discrepancy may stem from the stronger effect that a changein the attractive interaction for the 5d-band would exert on the X-pocket,which partially retains 5d-character, with respect to the pure B2p-characterVB.As discussed in Section 5.2, the dispersion of the X-pocket in SmB6 em-bodies information about the interplay between the rare-earth 5d- and 4 f -electrons close to the Fermi level, stemming from the mixed-valent charac-ter of the Sm ions. This observation makes the investigation of the evolu-tion of the bulk 5d pockets centered at X instrumental to get insights on thepossible fluctuations of the valence of the Sm ions in the SmxLa1−xB6 series.Figure 5.8 offers an overview of the evolution of the X-pocket as probed byARPES. By visual inspection, the size of the X-pocket progressively de-creases as x increases in the system (see ARPES maps in Figure 5.8b), asexpected from the removal of the trivalent La ions, which contribute to theoccupation of the 5d-band. However, we note that the observed behaviour104-0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4-2.5-2.0-1.5-1.0-0.50.0Energy (eV)-2.5-2.0-1.5-1.0-0.50.0Energy (eV)a LaB6 Sm0.2La0.8B6 Sm0.55La0.45B6Sm0.7La0.3B6 Sm0.8La0.2B6 SmB6-2.0 -1.0 0.0Energy (eV)Sm x1.ГX (Å-1) kГX (Å-1) kГX (Å-1)Intensity-1.7-1.8-1.9-2.0VB peak energy (eV) x=0  0.2  0.55  0.7  0.8  x=1  B2p VB  Sm4f  X  M  Г bcFigure 5.7: ARPES spectra of SmxLa1−xB6. (a) Evolution of ARPESspectrum along X-Γ-X (green line in inset of panel (c)) in SmxLa1−xB6series. Data acquired for x = (0, 0.2, 0.55, 0.7, 0.8, 1), with 21.2eV pho-ton energy at temperature 10K. (b) Energy distribution curves (EDCs)extracted at Γ (dashed white line in panel (a)) for the selected x values.The two peaks close to EF, and associated to Sm4 f states, are visible,as well as the B2p valence band (VB) at higher energy. (c) Bindingenergy of the B2p VB centered at Γ as a function of x. The overallshift is ∼ 320 meV. Inset: Iso-energy map of LaB6 at EF illustrating themomentum direction along which the ARPES spectra in panel (a) areacquired (green line).departs from a constant-rate reduction, as the change in the ARPES disper-sion is more pronounced for x ≤ 0.55 than for higher Sm concentrations, asshown inFigure 5.8c. In order to quantify the observed variation and thusestablish a direct relation between the ARPES dispersion and the fractionalpercentage of Sm3+, we must convert the size of the X-pocket contours,as extracted from the ARPES data, to the electronic occupation n5de . Thisis done via application of the Luttinger’s theorem. Here we stress that adirect comparison among different compounds in the series is made pos-105-1.0-Å-1) kx(Å-1) kx(Å-1) kx(Å-1) kx(Å-1) kx(Å-1) kx(Å-1)LaB6 Sm0.2La0.8B6 Sm0.55La0.45B6 Sm0.7La0.3B6 Sm0.8La0.2B6 SmB6-1.5-1.0-0.50.0Energy (eV)-0.4 0.0 0.4kXM (Å-1)-0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4kXM (Å-1) kXM (Å-1) kXM (Å-1) kXM (Å-1) kXM (Å-1) x=0.0  x=0.2 x=0.55 x=0.7 x=0.8 x=1.0 LaB6SmB6X MГEnergy (eV)-1.2-1.0-0.8-0.6-0.4-0.20.0Momentum (Å-1)-0.4 -0.2 0.0 0.2 0.4abcFigure 5.8: X-pocket dispersion in SmxLa1−xB6. (a) ARPES spectra along XM high-symmetry direction ofthe Brillouin zone (black dashed line in panel (b), left), for x = (0, 0.2, 0.55, 0.7, 0.8, 1). (b) ARPES iso-energycontours close to EF for the same samples shown in (a). The integration window in energy is 15 meV. (c)Evolution of the X-pocket dispersion along ΓX and XM directions. All data acquired with hν = 21.2eV atT = 10K.106sible by the fact that the bare X-pocket dispersion in SmxLa1−xB6 can bedescribed, in first approximation, by the same effective mass of LaB6, withthe only Fermi momentum k5dF changing to accommodate for the differentoccupation (see Figure 5.3). This observation allows us to make use of theFermi velocity as extracted for LaB6 to estimate the value of k5dF for sampleswith x 6= 0 in the series: that would be extrapolated in the limit of no hy-bridization between the 4 f and 5d bands, where the Fermi surface definedby the crossing points in k-space would be consistent with the Luttinger’stheorem [209, 210].Figure 5.9a displays the values of n5de (x) as extracted from ARPES spec-tra acquired with 21.2eV and 67eV. The total enclosed volume of the X-pocket was calculated by taking into account all the three X-point ellipsoidsincluded in the 3D Brillouin zone (BZ). For LaB6 (i.e. x = 0), the X-pocketscover∼ 50% of the bulk BZ (thus n5de = 0.5), corresponding to having 1 elec-tron in the 5d band and thus consistent with the picture of a metallic groundstate. As x increases, n5de (x) clearly deviates from the linear reduction ex-pected in the case of a constant 1:1 ratio of Sm2+ and Sm3+, represented bythe black dashed line in Figure 5.9a. In addition, the different slopes char-acterizing the evolution of n5de for x≤ 0.2 and x≥ 0.55 suggest the presenceof two distinct regimes as a function of x, rather than a single monotonicincrement or reduction of the Sm2+/Sm3+ ratio. In order to quantify thistransition, we compute the fractional percentage of Sm2+ and Sm3+ (nor-malized over the total amount of Sm, i.e. x) from n5de (x) as follow:Fraction Sm2+ (%) =1− 2n5de (x)xFraction Sm3+ (%) = 1+2n5de (x)− 1x.(5.1)The resulting values are plotted in Figure 5.9b. At x = 1, the fraction ofboth Sm2+ and Sm3+ is ∼ 50%, in accordance with the 1:1 ratio reportedin the literature for SmB6 at low temperature [66, 68, 200]. As x decreases,the amount of Sm2+ (top panel in Figure 5.9b) gradually increases uponreaching a maximum of ∼ 80% at x = 0.2, followed by a reduction at even107Sm x1. e5d (x) 0.400.450.250.300.350.50 21eV 67eVSm x1. Sm3+ (%)103050506070100Fraction Sm2+ (%) 21eV 67eV 21eV 67eVa b2. (x)Sm x1. 5.9: Evolution of the Sm valence from the ARPES dispersion.(a) Electronic occupation number of the X-pocket, n5de (x), for differentSmxLa1−xB6 compounds. Data points were calculated using ARPESspectra acquired with 21.2eV and 67eV. Note the deviation fromthe case of constant 1:1 ratio of Sm2+ and Sm3+ (dashed black line),as in the case of pure SmB6. (b) Calculated fractional percentage ofSm2+ (top) and Sm3+ (bottom) using Equation 5.1 for the various com-pounds measured by ARPES. (c) Mean valence of Sm ions, vSm, in theSmxLa1−xB6 series as obtained from ARPES.lower Sm concentrations. Note that the difficulties associated with probingminimal modifications of the X-pocket ARPES dispersion for concentra-tions x ≤ 0.1, reflected in the large error bars in Figure 5.9b, preclude theexclusion of a saturation of the fractional percentage values rather than adecrease of Sm2+ for very low x. Nevertheless, Figure 5.9a-b indicate aclear distinction between the low (i.e. x ≤ 0.2) and high (x ≥ 0.55) Sm con-centration regimes, along with a substantial variation of the Sm2+/Sm3+ratio as a function of x. These results naturally imply an evolution of themean valence of Sm ions, vSm, in the SmxLa1−xB6 series, as illustrated inFigure 5.9c.108In order to verify the two regimes scenario and get further insights onthe low concentration regime, we performed a XAS study on the sameSmxLa1−xB6 series to complement the ARPES results illustrated above. Inparticular, in order to achieve a higher bulk sensitivity, and thus establishour results as an intrinsic bulk-property of the SmxLa1−xB6 compounds, weexploit Partial Fluorescence Yield (PFY) and Inverse Partial FluorescenceYield (IPFY), characterized by a probing depth of several tens of nm. Thisallows one to circumvent some of the challenges associated to a detailedARPES study of SmxLa1−xB6 samples, such as cleaving procedure and sur-face degradation. XAS has been shown to be a powerful technique to ex-plore the physics of MV systems, such as SmB6. The absorption spectrumcan be described in first approximation by the sum of two independentcomponents, in the present case corresponding to Sm2+ and Sm3+ ions.Here, by tuning the incident energy across the Sm M4 and M5 edges (i.e. ex-citing 3d core electrons into 4 f orbitals), the XAS spectrum of SmxLa1−xB6can be mapped into a specific Sm2+/Sm3+ ratio, providing us with a toolto determine the specific valence of Sm ions as a function of x. Figure 5.10apresents the PFY spectrum of SmB6 acquired by scanning the Sm reso-nances. Two different fluorescence lines are clearly distinguishable repre-senting the direct and indirect Sm fluorescence peaks at higher and lowerenergy, respectively. The absorption profiles integrated around 850eV (i.e.at the indirect Sm fluorescence peak, red dashed line in Figure 5.10a) areexploited in the present work to perform the detailed analysis. This choicewas adopted to avoid additional distortions characterizing the spectra atthe direct fluorescence peak energy due to selection rules [211]. Also notethat since several natural mechanisms such as self-absorption effects oftenlead to distortions in the PFY spectra, a correction is needed to disentanglethe intrinsic absorption signal [109, 212–214]. In Figure 5.10b we report theevolution of the XAS intensity at the Sm M5 edge in the SmxLa1−xB6 series.The experimental data is fit with a weighted sum (black lines) of the Sm2+(blue lines) and Sm3+ (green lines) components. While for high x (top partof Figure 5.10b) the Sm3+ component dominates, its contribution dramat-ically reduces at x = 0.2. This observation is fully consistent with the re-109XAS Intensity (a.u.)1090108510801075Energy (eV)x=1.000.9750.900.700.200.300.130.07  XAS Sm2+Fluorescence Energy (eV)  Sm3+ SumM52. SmB6v Sm(x)ΔSm2+(%)Sm xEnergy (eV)502502.6a bcFigure 5.10: XAS study of the SmxLa1−xB6 series. (a) PFY spectrumof SmB6 acquired at the Sm M4 and M5 edges. (b) Evolution of theXAS intensity at the Sm M5 edge for x = (0.07, 0.13, 0.2, 0.3, 0.7, 0.9,0.975, 1) at the red dashed line in panel (a). The total spectral weightcan be fit by a sum (black lines) of two independent components, as-sociated to Sm2+ and Sm3+ ions (blues and green lines, respectively).(c) Mean valence of Sm ions, vSm, calculated from the XAS spectra pre-sented in (b). The raw values obtained by the fit are represented bythe black open circles, while the orange diamonds are the values uponrenormalization to account for the different cross-section of Sm2+ andSm3+. Inset: Percentage variation of Sm2+ ions, ∆Sm2+, with respect tothe x = 1 case. All data taken at a base temperature of 20K.110ported evolution of the fractional percentages of Sm2+ and Sm3+ obtainedfrom ARPES (Figure 5.9b). Moreover, a clear inversion in the progressionof the XAS spectra occurs for x ≤ 0.2, as the Sm3+ component strengthensagain. By exploiting the specific weights associated to the Sm2+ and Sm3+components in each XAS spectrum, we compute vSm, in the SmxLa1−xB6series and plot it in Figure 5.10c. Note that in order to account for the differ-ence in the relative cross-sections of the two Sm components at the energiesthe XAS measurements were performed, the weights obtained from the fitsshown in Figure 5.9b were renormalized such that vSm(x = 1) = 2.505, asreported for SmB6 at T = 20K [66, 68], base temperature for the XAS mea-surements. The black open circles in Figure 5.10c represent then the rawvalues, while the orange diamonds the actual valence upon normalization.vSm as extracted via analysis of XAS spectra confirms the two regimes sce-nario already suggested by the ARPES results shown in Figure 5.9. Forx≤ 0.2, the average Sm valence rapidly decreases as Sm is introduced in thesystem, before increasing again for higher x. A minimum value of ∼ 2.07is observed at x = 0.2, which corresponds to an increment of Sm2+ ions inthe system, ∆Sm2+, as large as ∼ 45% respect to the case of x = 1 (top insetof Figure 5.10c). Note that this percentage variation of Sm2+ is much moresubstantial than what was reported for temperature (∼ 8%) and pressure(∼ 20%) dependent studies of SmB6 [66, 197, 200].5.4 Towards the limit of mixed-valent impurityregimeIn Section 5.3 we showed that both ARPES and XAS measurements ofSmxLa1−xB6 display an evolution of vSm. Here, we start the discussionof the experimental data in terms of trivalent La ions substituting on theSm site in SmB6, known to have an equal fractional percentage of Sm2+and Sm3+. In Figure 5.11 three ideal scenarios are presented for the La-substitution in SmB6: Case 1 and Case 2 where only Sm2+ and Sm3+ arereplaced first, respectively, and Case 3 representing the equal substitutionof Sm ions corresponding to the two configurations. Recalling that only111 Sm2+ Sm3+ Case 1  Case 2  Case 3 Case 1 Case 2 Case 3 Case 1 Case 2 Case (x)Sm x1. x1. e5d (x) 0.400.450.250.300.350.500.2 0.8Sm x0.2 0.8Sm x0.2 0.8Sm x60800.02040100Fraction Sm (%)abcFigure 5.11: Calculated Sm valence for three ideal cases. (a) Elec-tronic occupation number of the X-pocket, n5de , calculated for threeideal cases of La-substitution in SmB6: La ions replace only Sm2+ first(Case 1, light blue points), only Sm3+ (Case 2, pink points) and ionsof the two Sm configurations evenly (Case 3, purple points). In all thethree cases, a fix equal percentage of 50% is assumed as a starting pointfor Sm2+ and Sm3+ in SmB6. (b) Fractional percentage of Sm2+ (blue)and Sm3+ (green), calculated using Equation 5.1, for the three cases inpanel (a). (c) Mean valence of Sm ions, vSm, obtained for the differentscenarios.La and Sm3+ ions contribute to the occupation of the X-pockets, we cancompute the expected evolution of vSm as a function of x by following thesame procedure adopted for the experimental ARPES data (see Figure 5.9and Equation 5.1). As illustrated by the comparison of Figure 5.11 with theexperimental results of Section 5.3, Case 2 qualitatively captures the pro-gression from an evenly Sm2+/Sm3+ regime into a predominant presenceof Sm2+ as x decreases from 1. This result is consistent with the obser-vation of the average Sm valence tending towards 2+ upon trivalent ion112substitution (as La3+ or Y3+) reported in early studies exploiting XAS andthe analysis of the variation of the cubic lattice parameter in the solid so-lutions [215, 216]. Furthermore, recent transport studies on La-substitutedSmB6 have reported the complete closure of the d- f hybridization gap, andthe consequent emergence of a metallic-like behaviour, for La concentra-tions higher than 25% (x ≤ 0.75) [217, 218]. These observations corroboratethe substantial increase of Sm2+ in the system observed in this work as xdecreases from 1 (Figure 5.9b and inset in Figure 5.10c). Figure 5.11 repre-sents ideal cases, where a fix equal amount of Sm2+ and Sm3+ (i.e. 50%) isassumed available in the starting system SmB6. In practice, when trivalentLa ions are introduced in the system (i.e. by decreasing x), two competingprocesses may occur to preserve the system’s charge neutrality: electronscan be accommodated in the 5d-band, thus shifting the bottom of the con-duction band to deeper binding energy, or in the f -shell, thus facilitatingthe transformation of Sm3+ into Sm2+. Therefore, the delicate balance be-tween these two processes to maintain charge neutrality is dictated by thetotal energy resulting from the increase in the attractive potential as moretrivalent ions are added to the 5d-band and the conversion of Sm3+ ionsinto Sm2+ (recall that these two valence configurations are indeed nearlydegenerate in SmB6, making such transition available at a very low energycost). Given the small energy scales involved, it is not possible a priori todesignate either one as the dominating mechanism, but we are currentlyworking on a theoretical description of how such delicate balance may oc-cur.Instead, here we offer a simplified phenomenological model to describethe observed transition of vSm based on the two distinct regimes revealedexperimentally in the SmxLa1−xB6 series. For high x, we mimic the de-crease of vSm upon La-substitution by fitting the experimental data for x ≥0.2 with a linear fit (V0(x), red line in Figure 5.12a). To provide a phe-nomenological description of the increase of vSm observed for low x (i.e.x < 0.2), we refer to an early work by Haldane examining the dilute limitof a single rare-earth impurity in a d-band metal [199]. The derivation isbased on the addition to the Anderson-model Hamiltonian of a screening113 21eV 67eV XAS Model2. 2.3 2.35 2.4vimp value1. 0 1 2 3 4 5 6 N valuePN(x)Sm xSm xV0(x)v Model(x)Sm x1. x2. Sm(x)a bcdLinear fit:a = 2.02 ± 0.02b = 0.48 ± 0.03Figure 5.12: Simplified model for the Sm valence in SmxLa1−xB6. (a)Linear fit, V0(x), to vSm for x ≥ 0.2 capturing the decrease in the va-lence due to formation of Sm2+ upon La-substitution. (b) Calculatedprobability for a Sm ions to have N other Sm as nearest-neighboursfrom Equation 5.2. P0(x) (black line) is used to define the fraction ofSm ions which assumes the impurity valence vimp = 2.35 at each x. (c)vimp-dependence of vModel as defined by Equation 5.3. (d) Compari-son between the model of Equation 5.3 (dashed blue line) and the ex-perimental values of vSm obtained from ARPES (light and dark greencircles, acquired with 21eV and 67eV, respectively) and XAS (orangediamonds).term, which describes the interaction between the impurity f -orbitals andthe d-orbitals (expressed in terms of Wannier functions) on the same site.If this extra f -d interaction is reasonably larger than the standard inter-sitehybridization term, it stabilizes the f -electrons fluctuations, enabling thesingle impurity to be in a mixed-valent state. Although we recognize thatthe realization of such scenario would be very coincidental (for the energyof the f -states to be as such to return mixed-valence), the system described114in Haldane’s work retains a lot of similarity with SmxLa1−xB6 for x 1,where the Sm ions can be described as impurities in the d-band metal LaB6.This observation stimulates us in considering the possibility of contribu-tions to vSm given by Sm ions acting as single impurities with a specificfixed fractional valence, vimp. In this regard, we compute the probabilityfor a Sm ion to have N Sm in the nearest-neighbour sites, PN (with N rang-ing from 0 to 6), as a function of x:PN(x) = xN (1− x)6−N(6N). (5.2)The calculated PN(x), N = [0,6], are plotted in Figure 5.12b. The functionP0(x) (black line) is used to define in first approximation the fraction ofSm ions which assumes the impurity valence vimp = 2.35 (as extrapolatedfrom the experimental results in Figure 5.9c and Figure 5.10c) at any givenconcentration x. Therefore, we can describe the evolution of the Sm valencein the SmxLa1−xB6 series by:vModel = P0(x) · vimp + (1− P0(x)) ·V0(x) . (5.3)Note that P0(x) rapidly decays below 0.1 within x = 0.3, thus leading tonegligible contributions to vSm at high x. Figure 5.12d compares the sim-plified model of Equation 5.3 with the experimental values of the averageSm valence obtained from ARPES (light and dark green circles) and XAS(orange diamonds), showing a good agreement. In particular, the steepincrease of vSm observed for x ≤ 0.2 is well described by the significantcontribution given by the emerging impurity regime with fixed fractionalvalence. Furthermore, we note that a variation of ±0.5 in vSm, associated tothe error in determining the extrapolated value, mainly reflects in a modi-fication of the steepness of the upturn for x < 0.2, not significantly affectingthe overall evolution of vModel and the distinctive crossover around x = 0.2(Figure 5.12c).In light of the fact that the phenomenological inclusion of an impurityregime with intermediate valence seems to grasp the evolution of the ex-115perimental data for low x, we conclude this section with a short discussionon the viability of having such a MV impurity regime in SmxLa1−xB6. Inthis regard, we illustrate here the case of one Sm impurity hosted in thed-band metallic LaB6, to verify whether or not an admixture of two differ-ent f -states is possible in the ground state wavefunction. According to theone-electron excitation spectrum presented in Figure 1.5, the high energycost (∼ 5− 6eV) associated to transitions of the form f 6→ f 7 and f 5→ f 4allows to discard them in the following discussion of the lowest energyexcitations, which are solely defined by the electron addition to 4 f 5 andremoval from 4 f 6 states. In order to preserve the system’s charge neutral-ity, such excitations must involve adding/removing an electron from thed conduction band. By considering only the lowest energy possible statesand by replacing the multiplet quantum number with an effective spin 2,we can write for the impurity a:f 5a↑→ f 6a dk↓→ f 5a↓ dk↓ d†k′↑ . (5.4)Visually, such transition resemble the Kondo-like interaction, where a con-duction electron is scattered from k to k′along with a flip in the local spin.However, in the present case all the states involved (both the two 4 f statesand the dk) are nearly degenerate at the Fermi energy. This aspect sets acrucial distinction from the conventional Kondo problem. Effectively, the Uincluded in the Anderson Hamiltonian-type way to describing the Kondophysics is nearly zero for the lowest energy 4 f states and the dk involvedin Equation 5.4. Therefore, the energy denominator in a perturbation-typetreatment is ≈ 0, making the Schrieffer−Wolff transformation – essentialstep in the description of the Kondo physics from the Anderson impu-rity model – not valid in this context. Despite this fundamental difference,Equation 5.4 shows the viability of having both f 5 and f 6 mixed into theground state wavefunction accompanied by virtual fluctuations from k tok′, spin-down to spin-up in the conduction electron sea. Although a highly2In particular, J = 0 for the 4 f 6 state while 4 f 5 has J = 5/2, with the latter further splitby the crystal field into a doublet and a quartet. Here we consider the doublet as the lowestenergy state and associate an ”effective spin” quantum number to such state.116improbable scenario, requiring a critical balance of interactions such thatthe two f -states and the dk are degenerate, that admixture could reflectinto a mixed-valent character of the system even in the impurity limit.5.5 ConclusionsTo summarize, we studied the valence fluctuations in the SmxLa1−xB6 se-ries by gradually changing the stochiometry. Combining ARPES and XAStechniques, we tracked the evolution of the mean valence of the Sm ions,vSm, revealing the persistence of the intermediate valence character evenin the very dilute limit. In particular, our results disclose the presence oftwo distinct regimes for vSm, moving from high to low Sm concentration x:for 0.2 ≤ x ≤ 1, vSm linearly converges towards a predominant Sm2+ state,consistent with the substitution on the Sm site by a trivalent ion as La3+;contrary, for very low Sm concentrations, i.e. x < 0.2, vSm increases again,marking a substantial deviation from a monotonic evolution across the en-tire series. The formulation of a phenomenological model based on theinclusion of a Sm impurity regime, in which Sm ions exhibit a fixed frac-tional valence of vimp = 2.35, succeeds in capturing the experimental resultsand the observed crossover at x ∼ 0.2. Our study suggests the possible re-alization of a MV impurity state in SmxLa1−xB6, a scenario speculated byHaldane few decades ago [199]. These results may pave the way for fur-ther theoretical and experimental considerations on the concept of MV andits influence on the macroscopic electronic and transport properties of rare-earth compounds in the impurity regime.117Chapter 6ConclusionsThis thesis brings together three projects, employing angle-resolved pho-toemission spectroscopy, both in its conventional and time-resolved fash-ion, to quantitatively investigate the interplay between different phenom-ena characterizing the physics of strongly correlated materials, in particular3d and 4 f systems.Firstly, we extended the ability of ARPES to directly access the mo-mentum and energy-dependent spectral function onto the time-domain,to study the dynamical interplay of competing phenomena characterizingthe superconducting phase (SC) of hole-doped Bi-based cuprates. A sce-nario in which the macroscopic Tc in the underdoped regime is primarilydetermined by the onset of phase coherence, rather the pairing strength ∆,was proposed in the 1990s by Emery and Kivelson [113]. A recent time-and angle-resolved photoemission spectroscopy (TR-ARPES) study of theunderdoped bi-layer Bi2Sr2CaCu2O8+δ (Bi2212) confirmed such a scenarioby reporting an ultrafast melting of superconductivity driven by the en-hancement of phase fluctuations upon optical pumping, while leaving thepairing strength mainly unaffected [114]. Here, the same experimental ap-proach was employed to extend such a study of the spectral function dy-namics to the single-layer hole-doped Bi2Sr2CuO6+δ (Bi2201). By perturb-ing with a 1.55 eV probe pulse the overdoped Bi2201 OD24 (Tc ∼ 24K), weresolved an increase of the pair-breaking scattering rate associated to the118presence of phase fluctuations. This ultrafast response is comparable bothin magnitude and in timescale to what was previously reported for Bi2212,and is completely decoupled from the dynamics of the pairing strength andthe effective electronic temperature. These results testify to the importanceof phase coherence in determining the macroscopic stability of the SC con-densate in the underdoped and close to optimally-doped regimes of theBi-based cuprates, independently of the number of CuO2 planes within theunit cell. Whether or not this ultrafast experimental approach can be usedto directly map the transition to a regime in which the formation of the SCcondensate is instead solely ruled by the paring strength, as proposed forthe very overdoped regime, has still to be assessed.Next, we provided a comprehensive study of the temperature depen-dent suppression of the nodal coherent spectral weight in Bi-based cuprates.It has been proposed that such a quenching of the quasiparticle peak bearsdirect relation to the SC state and the superfluid density. Here, by com-bining data from different doping levels of both the single- and bi-layercompounds, we demonstrated that the nearly linear suppression of CSWis universal across the Bi-based cuprates’ family and it persists for TTc,therefore excluding an explicit link to SC. Instead, the observed nodal CSWmeltdown can be described in terms of the quadratic temperature depen-dence of the imaginary part of the self-energy within a Fermi-liquid picture.In addition, these findings suggest that such a T-evolution of the CSW canbe employed as a sensitive tool to explore the fragility of the Fermi-liquidbehaviour as a function of momentum, while going from the nodal direc-tion to the antinode. Given the low Tc, the single-layer Bi2201 representsan intriguing candidate to test such a momentum-dependent investigationof the normal-state self-energy in cuprates, with no complications arisingfrom the SC gap.The second project focused on the relation between pseudogap (PG)and antiferromagnetic (AF) correlations in electron-doped cuprates. In thesematerials, it is believed that the PG bears strong relation with the robust AForder, which extends for a wide doping range up to the SC dome. Whilethis picture is well established in the underdoped regime, where a true119long-range AF order is present, close to optimal doping only short-rangespin-fluctuations have been detected, arising the question on whether andhow such short-range order relates to the spectroscopically observed PG.In this regard, we made use of the capability of TR-ARPES to perform anextremely detailed temperature-dependent analysis of the PG spectral fea-tures in optimally-doped Nd2−xCe2CuO4 (NCCO). We showed that suchtemperature evolution is closely linked to the spin-correlation length, ξspin,and that a complete filling of the PG is observed when the spectral broad-ening due to the reduction of ξspin overcomes the paring amplitude ∆PG.These results demonstrate that the PG in electron-doped cuprates origi-nates from short-range AF correlations and that the onset temperature T?represents a crossover temperature related to the reduction of ξspin, more sothat a sharp phase transition. These findings also testify the importance oforders even with short-range correlation length in the shaping of the Fermisurface topology of correlated materials.Lastly, the rare-earth haxaboride SmxLa1−xB6 series was the subject ofthe third project. Among the multitude of novel phenomena driven by thehighly correlated 4 f electrons, the emergence of the mixed-valence (MV)state is a key element to account for in the description of the rich groundstate of SmB6. On the other hand, the lack of 4 f electrons and the half-filled 5d-band make LaB6 metallic. This provided us with the opportunityto investigate the evolution of the MV character as a function of 4 f elec-trons gradually introduced in the system. 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Philosophical Magazine, 96(31):3274–3283,2016.144Appendix APublicationsDuring the course of my PhD I was involved in a number of researchprojects, some of which have resulted in peer reviewed publications. Alist is provided here.• Emergence of the pseudogap from short-range spin-correlations inelectron doped cupratesF. Boschini*, M. Zonno*, E. Razzoli, R.P. Day, M. Michiardi, B. Zwart-senberg, P. Nigge, M. Schneider, E. H. da Silva Neto, A. Erb, S. Zh-danovich, A. K. Mills, G. Levy, C. Giannetti, M. Greven, D. J. Jonesand A. Damascellinpj Quantum Materials 5, 6 (2020)• Role of matrix elements in the time-resolved photoemission signalF. Boschini, D. Bugini, M. Zonno, M. Michiardi, R. P. Day, E. Razzoli,B. Zwartsenberg, E. H. da Silva Neto, S. dal Conte, S. K. Kushwaha,R. J. Cava, S. Zhdanovich, A. K. Mills, G. Levy, E. Carpene, C. Dallera,C. Giannetti, D. J. Jones, G. Cerullo and A. DamascelliNew J. Phys. 22, 023031 (2020)145• Room temperature strain-induced Landau levels in graphene on awafer-scale platformP. Nigge, A. C. Qu, E´. Lantagne-Hurtubise, E. Ma˚rsell, S. Link, G.Tom, M. Zonno, M. Michiardi, M. Schneider, S. Zhdanovich, G. Levy,U. Starke, C. Gutirrez, D. Bonn, S. A. Burke, M. Franz and A. Damas-celliScience Advances 5, 11 (2019)• Influence of Spin-Orbit Coupling in Iron-Based SuperconductorsR. P. Day, G. Levy, M. Michiardi, B. Zwartsenberg, M. Zonno, F. Ji, E.Razzoli, F. Boschini, S. Chi, R. Liang, P. K. Das, I. Vobornik, J. Fujii, W.N. Hardy, D. A. Bonn, I. S. Elfimov, and A. DamascelliPhys. Rev. Lett. 121, 076401 (2018)• Collapse of superconductivity in cuprates via ultrafast quenching ofphase coherenceF. Boschini, E. H. da Silva Neto, E. Razzoli, M. Zonno, S. Peli, R. P.Day, M. Michiardi, M. Schneider, B. Zwartsenberg, P. Nigge, R. D.Zhong, J. Schneeloch, G. D. Gu, S. Zhdanovich, A. K. Mills, G. Levy,D. J. Jones, C. Giannetti, and A. DamascelliNature Materials 17, 416 (2018)• Doping-dependent charge order correlations in electron-doped cupratesE. H. da Silva Neto, B. Yu, M. Minola, R. Sutarto, E. Schierle, F. Bos-chini, M. Zonno, M. Bluschke, J. Higgins, Y. Li, G. Yu, E. Weschke, F.He, M. Le Tacon, R. L. Greene, M. Greven, G. A. Sawatzky, B. Keimer,and A. DamascelliScience Advances 2, 8 (2016)146• Evidence for superconductivity in Li-decorated monolayer grapheneB.M. Ludbrook, G. Levy, P. Nigge, M. Zonno, M. Schneider, D.J. Dvo-rak, C.N. Veenstra, S. Zhdanovich, D. Wong, P. Dosanjh, C. Straer, A.Stohr, S. Forti, C.R. Ast, U. Starke and A. DamascelliPNAS 112, 11795 (2015)• Effect of Pt substitution on the electronic structure of AuTe2D. Ootsuki, K. Takubo, K. Kudo, H. Ishii, M. Nohara, N. L. Saini, R.Sutarto, F. He, T. Z. Regier, M. Zonno, M. Schneider, G. Levy, G. A.Sawatzky, A. Damascelli and T. MizokawaPhys. Rev. B 90, 144515 (2014)• The Thinnest Carpet on the Smallest Staircase: The Growth of Grapheneon Rh(533)B. Casarin, A. Cian, Z. Feng, E. Monachino, F. Randi, G. Zamborlini,M. Zonno, E. Minussi, P. Lacovig, S. Lizzit and A. BaraldiJ. Phys. Chem. C 118, 6242 (2014)• The momentum and photon energy dependence of the circular dichroicphotoemission in the bulk Rashba semiconductors BiTeX (X=I, Br, Cl)A. Crepaldi, F. Cilento, M. Zacchigna, M. Zonno, J. C. Johannsen,C. Tournier-Colletta, L. Moreschini, I. Vobornik, F. Bondino, E. Mag-nano, H. Berger, A. Magrez, Ph. Bugnon, G. Auts, O. V. Yazyev, M.Grioni and F. ParmigianiPhys. Rev. B 89, 125408 (2014)147


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