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Scanning tunneling microscopy study of superconducting pairing symmetry : application to LiFeAs Chi, Shun 2014

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Scanning Tunneling MicroscopyStudy of SuperconductingPairing Symmetry: Applicationto LiFeAsbyShun ChiB.Sc., Zhejiang University, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2014c© Shun Chi 2014AbstractIdentifying the pairing symmetry is a crucial step towards uncovering the supercon-ducting mechanism. The pairing symmetry and interactions leading to pairing in theiron-based high-temperature superconductors are under debate. In this thesis work, thepairing symmetry of LiFeAs, a stoichiometric superconductor in the iron-based family,is studied by scanning tunneling microscopy. The tunneling conductance spectrum ina defect-free region shows two nodeless superconducting gaps. In addition, a dip-humpabove-gap structure was observed, indicating coupling between the superconducting car-riers and bosonic modes. Defect bound states were measured for iron-site defects. Thebound states are pinned to the gap edge of the small superconducting gap, consistentwith theoretical predictions for a sign-changing pairing symmetry. Finally, the observedBogoliubov quasiparticle interference associated with scattering from defects providescompelling evidence for an s±-wave pairing symmetry in LiFeAs.iiPrefaceThe work presented, with the exception of Chapter 6, was conducted in the Supercon-ductivity and LAIR STM labs at the University of British Columbia, Point Grey campus.The work in Chapter 6 was conducted in the Tunneling Spectroscopy of Strongly Cor-related Electron Materials lab at the Max-Planck Institute for Solid State Research, inStuttgart, Germany.Chapter 4 shows the work of single crystal growth. I was the lead investigator,responsible for most of the experimental design and sample growth. Ruixing Liang wasthe research supervisor, responsible for the early development of the experimental design,and involved throughout the project in method development. W. N. Hardy also playeda major role in the initial experimental setup.A version of chapter 5 has been published [Shun Chi, S. Grothe, Ruixing Liang, P.Dosanjh, W. N. Hardy, S. A. Burke, D. A. Bonn, and Y. Pennec, Scanning TunnelingSpectroscopy of Superconducting LiFeAs Single Crystals: Evidence for Two NodelessEnergy Gaps and Coupling to a Bosonic Mode, Phys. Rev. Lett. 109, 087002, 2012]. Iwas the lead investigator, responsible for the experimental design, data collection, andanalysis. S. Grothe played a major role in maintaining the experimental instruments andassisting the data collection. Ruixing Liang supervised the sample growth. P. Dosanjh,and W. N. Hardy both provided the technical support for the instruments. S. A. Burke,D. A. Bonn, and Y. Pennec were the research supervisors. All authors contributed ideas,discussed the results, and wrote the manuscript.For chapter 6, the work is original and unpublished. It was done by collaborationbetween the University of British Columbia and the Max-Planck Institute. I was the leadinvestigator, responsible for the majority of the experimental design, sample growth, andanalysis. R. Aluru and U. Singh played a major role in maintaining the experimentaliiiPrefaceinstruments at the Max-Planck Institute. R. Aluru, U. Singh and I collected the datatogether. D. A. Bonn and P. Wahl were the research supervisors.A version of chapter 7 has been published [Shun Chi, S. Johnston, G. Levy, S. Grothe,R. Szedlak, B. Ludbrook, Ruixing Liang, P. Dosanjh, S. A. Burke, A. Damascelli, D.A. Bonn, W. N. Hardy, and Y. Pennec, Sign inversion in the superconducting orderparameter of LiFeAs inferred from Bogoliubov quasiparticle interference, Phys. Rev. B89, 104522, 2014]. I was the lead investigator, responsible for the experimental design,data collection, and data analysis. S. Johnston performed the theoretical simulations.G. Levy, R. Szedlak, B. Ludbrook, and A. Damascelli provided the ARPES data forcomparison. S. Grothe played a major role in maintaining the experimental instrumentsand assisting the data collection. Ruixing Liang supervised the sample growth. P.Dosanjh and W. N. Hardy provided the technical support for the instruments. S. A.Burke, D. A. Bonn, and Y. Pennec were the research supervisors. All authors contributedideas, discussed the results and wrote the manuscript.Chapter 8 is an original, unpublished, independent discussion by the author.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Iron-based superconductors . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 LiFeAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Superconductivity and its pairing symmetry . . . . . . . . . . . . . . . 72.1 A new phase of material . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Cooper instability . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 BCS Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.3 Mean field approximation . . . . . . . . . . . . . . . . . . . . . . 152.2.4 Ground state wavefunction . . . . . . . . . . . . . . . . . . . . . 182.2.5 Gap equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20vTable of Contents2.2.6 Reduced gap ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.7 Pairing symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.8 Spin-singlet vs spin-triplet . . . . . . . . . . . . . . . . . . . . . . 272.2.9 Remarks on BCS theory . . . . . . . . . . . . . . . . . . . . . . . 282.3 How two electrons in a Cooper pair overcome the e-e repulsive interaction 292.3.1 Two cases for e-e interactions . . . . . . . . . . . . . . . . . . . . 302.3.2 Avoided on-site Coulomb repulsion . . . . . . . . . . . . . . . . . 332.3.3 The significance of sign change . . . . . . . . . . . . . . . . . . . 372.3.4 Pairing through repulsive interactions . . . . . . . . . . . . . . . 382.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Scanning tunneling microscopy . . . . . . . . . . . . . . . . . . . . . . . 413.1 Principle of STM and its operating modes . . . . . . . . . . . . . . . . . 413.2 Theory of STM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Remarks on Tersoff-Hamann’s theory . . . . . . . . . . . . . . . . . . . . 493.4 STM’s used in this thesis project . . . . . . . . . . . . . . . . . . . . . . 493.4.1 CreaTec STM at UBC . . . . . . . . . . . . . . . . . . . . . . . . 503.4.2 Home-built STM at MPI . . . . . . . . . . . . . . . . . . . . . . 524 Crystal growth of LiFeAs . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1 Synthesis of precursor materials . . . . . . . . . . . . . . . . . . . . . . . 554.1.1 Li3As . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1.2 FeAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Growth of LiFeAs single crystals . . . . . . . . . . . . . . . . . . . . . . 584.2.1 Modification #1: reusable molybdenum crucible . . . . . . . . . 584.2.2 Modification #2: temperature program for high purity samples . 614.3 Magnetization measurements . . . . . . . . . . . . . . . . . . . . . . . . 625 Properties of LiFeAs in a defect-free area . . . . . . . . . . . . . . . . . 645.1 Topography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.2 Local density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66viTable of Contents5.3 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.4 Superconducting gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.5 Temperature dependence of the superconducting gaps . . . . . . . . . . 735.6 Dip-hump structure above the superconducting gaps . . . . . . . . . . . 745.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796 Bound states of iron-site defects . . . . . . . . . . . . . . . . . . . . . . . 816.1 Impurity physics for identifying the pairing symmetry . . . . . . . . . . 826.2 Fe-site defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2.1 Native Fe-site defects . . . . . . . . . . . . . . . . . . . . . . . . 856.2.2 Engineered Fe-site impurities . . . . . . . . . . . . . . . . . . . . 866.3 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 897 Quasiparticle interference in LiFeAs . . . . . . . . . . . . . . . . . . . . 907.1 Introduction to quasiparticle interference . . . . . . . . . . . . . . . . . 907.1.1 A simple case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.1.2 Application to the iron-pnictide Fermi surface . . . . . . . . . . . 947.2 QPI in LiFeAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.2.1 Data processing methods . . . . . . . . . . . . . . . . . . . . . . 967.3 Identification of the scattering vectors . . . . . . . . . . . . . . . . . . . 1007.4 Reconciling the discrepancy between STM and ARPES measurements . 1027.5 Variation of the QPI intensity with energy: evidence for s±-wave pairingsymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047.5.1 Pairing symmetry sensitivity . . . . . . . . . . . . . . . . . . . . 1057.5.2 QPI intensity map vs energy . . . . . . . . . . . . . . . . . . . . 1077.5.3 Bogoliubov QPI . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 1128 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148.1 Another way to measure the pairing symmetry by QPI . . . . . . . . . . 1148.2 What does the s±-wave pairing symmetry tell us? . . . . . . . . . . . . 116viiTable of ContentsBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120AppendicesA QPI simulation using a T-matrix formalism . . . . . . . . . . . . . . . 136A.1 Multiorbital tightbinding model . . . . . . . . . . . . . . . . . . . . . . . 136A.2 Theory of multiband quasiparticle interference . . . . . . . . . . . . . . 137B ARPES measurement on LiFeAs . . . . . . . . . . . . . . . . . . . . . . 140viiiList of Tables5.1 Values of fitting parameters for the two-isotropic-gaps model andtwo-anisotropic-gaps model. The errors here are determined by the95% confidence bounds resulting from the fitting program. . . . . . . . . 727.1 A summary of the QPI selection rules expected for a pnictidesuperconductor with s++-wave or s±-wave. The QPI intensity of ascattering vector is either suppressed or enhanced when sweeping energiesfrom above to inside the superconducting gap. The four combinations oftwo pairing symmetries and two kinds of impurities result in completelydifferent selection rules. The intensity suppression or enhancement areestablished on the comparison of QPI intensities at energies between E >∆ (close to normal state QPI) and E ≤ ∆ (dominant by Bogoliubov QPI). 1068.1 A summary of the variation of the factor |u(ki)u∗(kf )±v(ki)v∗(kf )|2with temperature. Here the scattering vector q is equal to kf − ki.The “Suppressed Intensity” and “Enhanced and then Suppressed Intensity”correspond to the red and blue curves in Fig. 8.1(b), respectively. . . . . 115ixList of Figures1.1 Iron-based superconductors: the crystal structures and the phasediagram. The left side: the structure of a FeAs layer and three typ-ical crystal structures of iron pnictides: LiFeAs [19], LaFeAsO [14], andBaFe2As2 [20]. Iron chalcogenides have an FeSe layer, identical to theFeAs layered structure, as the central building block. The right side: aschematic phase diagram of iron-based superconductors. . . . . . . . . . 31.2 Band structure and Fermi surface of iron pnictides. (a) A simplifiedtwo-band model for the iron pnictides with a hole-like band centered atk = (0, 0) and an electron-like band centered at k = (pi/a, pi/a). (b) TheFermi surfaces of the bands in (a). A hole pocket and electron pocket arecentered at Γ and M points, respectively, in the first Brillouin zone (thedashed square). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Discovery of superconductivity. Zero resistivity below 4.15 K in mer-cury, measured in 1911 [66]. . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Tc records vs time. The yellow circles show Tc of conventional super-conductors. The Tc increases over time, but slowly. The blue diamondsshow Tc of cuprates, which gave a sudden leap in the 1980s. The greensquares show Tc of heavy fermion superconductors. The red stars showthe newly discovered Fe-based superconductors in 2008, another high-Tcfamily with maximum Tc > 50 K. The Tc data shown in the graph arecollected from Ref. [68] and Ref. [15]. . . . . . . . . . . . . . . . . . . . . 8xList of Figures2.3 Meissner-Ochsenfeld effect. (a) Schematic illustration of expellingmagnetic field by a sphere shape superconductor. (b) The phase diagramof a type-I superconductor in H-T phase space. Hc(T ) separates the H-Tplane into two regions: the superconducting state and the normal metallicstate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Cooper instability. (a) Schematic illustration of two electrons addedto an occupied Fermi sea with momentum k and −k. (b) With an effec-tive attractive interaction U , two electrons bind together as a spin-singletCooper pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 BCS theory: example of electron and hole bands coupling. (a)a parabolic electron-like band, (b) add the excitation hole-like band, (c)superconducting gap opens when the electron-like and hole-like bands arecoupled by an interaction ∆. . . . . . . . . . . . . . . . . . . . . . . . . 172.6 e-e interaction in a square lattice. The interactions are U on thesame site, J on the nearest-neighbor (NN) sites, and W on the next-nearest-neighbor (NNN) sites, respectively. . . . . . . . . . . . . . . . . 232.7 Pairing symmetry (square lattice): s-wave, s±-wave and dx2−y2-waveare plotted from left to right. The red color region has positive sign andthe blue color region has negative sign. In the transition from positive tonegative, there are nodal lines required by symmetry, where the supercon-ducting gap is zero, indicated by the trick black lines. . . . . . . . . . . 262.8 Spin configurations of a Cooper pair. (a) s±-wave as an example.In the first Brillouin zone, the spatial wavefunction is symmetric because∆k = ∆−k, so the spins have to adopt the antisymmetric spin-singletconfiguration. (b) p-wave is the converse case, ∆k = −∆−k, so it mustadopt spin-triplet. The black lines in (a) and (b) indicate the node linesin the two pairing symmetries. . . . . . . . . . . . . . . . . . . . . . . . 27xiList of Figures2.9 Case #1 Pairing under the on-site electron-phonon (e-p) attrac-tive and e-e Coulomb repulsive interactions. (a) the energy de-pendence of e-p with −Uph (shade red with cutoff at ~ωD) and e-e with−Ucoul (shade blue with cutoff at EF ). (b) superconducting gap as a func-tion of energy. ∆(E) changes sign from the region with the presence of anattractive interaction to the region with only the repulsive interaction. . 312.10 Example of pairing through on-site attractive interactions. Att = 0, a spin up electron with momentum k1 creates an phonon at locationr with a momentum transfer k2−k1. Then it travels away with velocity vF .After about t ∼ 1/ωD, another electron with spin down and momentum−k1 is attracted to location r and absorbs the phonon by momentumtransfer −k2 − (−k1). The interaction is on the same location r, but thetwo electrons avoid each other in time. . . . . . . . . . . . . . . . . . . . 352.11 Example of pairing through NN attractive interactions. (a) Theinteraction terms: on-site repulsive Ucoul and NN attractive −J0. (b) Realspace pairing for s±-wave. In both δx and δy, the pairing has a positivesign, indicated by the red oval. (c) dx2−y2-wave. Pairing in δx has apositive sign (red oval) and in δy has a negative sign (blue oval). . . . . 362.12 The pairing interaction vs sign change in momentum space. (a)For the s±-wave case, the interaction between k1 and k2 is Vk1k2 < 0,attractive, so there is no sign change. While there is a sign change betweenk1 and k3 because Vk1k3 > 0, the interaction is repulsive. (b) The samefor dx2−y2-wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.13 Example of the attractive channels from a repulsive interaction.Due to the screening effect, the repulsive interaction Vkk′ between electronsmay have attractive regions (red shade) at certain e-e distances. Thisattractive interaction, usually not on-site, tends to give an unconventionalpairing symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39xiiList of Figures3.1 Core components of an STM. The left side is a schematic plot of thekey components of an STM. In order to keep a stable tunneling barrier,two major components are required: a feedback control unit to adjust thetip-sample distance d and a scanning unit to control the tip position r onthe sample surface. The right side is a zoomed in image of the tip-sampletunneling junction: the tip and sample is separated by a distance of theorder of 1 nm. The quantum tunneling events primarily happen betweenthe apex atom of the tip and the top atomic layers of the sample surface. 423.2 STM operating modes. (a) STM adjusts its tip height in the topog-raphy mode. STM tip changes its height when the tip passes over animpurity (green ball) on the surface. (b) A topography of LiFeAs withSTM settings (Vs = 25 mV, It = 250 pA). A few impurities, shown inbright contrast, are present on the sample surface. (c) STS taken by nu-merical differentiation at location of 1© in (b): the upper panel shows aIt-Vs spectrum of LiFeAs at 4.2 K and the lower panel shows the dIt/dVsspectrum by numerical differentiating It-Vs spectrum above. (d) A tun-neling conductance map g(r,E = eVs) at Vs= 8 meV was measured overthe same area shown in topography (b). . . . . . . . . . . . . . . . . . . 443.3 Electron tunneling diagram. The colored regions indicate the occupiedstates of the tip (red) and the sample (blue). Upon applying a bias voltageVs to the sample, its Fermi energy EF shifts down to  = −eVs while EFof the tip is still at  = 0 (grounded). The gray region is the tunnelingbarrier, usually vacuum, originated from the work functions of the tip φT ipand the sample φSample plus the potential −eVs applied to the sample.Electrons tunnel through the barrier from the occupied states in the tipto the unoccupied states in the sample and vice versa, giving rise to atunneling current It. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47xiiiList of Figures3.4 STM head of the Createc system. The STM head is hanging fromthe cryostat by three springs (left picture). The details of the parts aredescribed in Zo¨phel’s thesis [113]. A thermal link is connected betweenthe STM head and the helium bath, as indicated on the right picture.The STM has a Zener diode attached on the head, enabling temperaturedependent measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Comparison of the dIt/dVs spectra before and after eliminatingelectrical noise sources. (a) A dIt/dVs spectrum taken at 4.2 K by Cre-atec STM before optimizing the system. (b) A dIt/dVs spectrum takenat 4.2 K by the same STM after eliminating 60 Hz and its higher har-monics noise. The coherence peaks after eliminating noise become signif-icantly sharper than them in (a). The spectrum measured at 2 K (whichis shown in Fig. 5.5) shows explicitly the second gap at 3 meV, indicatingthe limiting factor for energy resolution primarily origins from the thermalbroadening effect above 2 K. . . . . . . . . . . . . . . . . . . . . . . . . . 524.1 Synthesis of Li3As. (a) The iron crucible. Iron is used because it isan ingredient of LiFeAs. (b) The sealed quartz ampoule for synthesizingLi3As. When heating the ampoule, the lithium ingots melt and diffuseinto the As powder at the bottom. (c) The temperature program forsynthesizing Li3As. The two dwells at 443 K and 473 K are just belowand above the lithium melting temperature of 454 K. . . . . . . . . . . . 564.2 Synthesis of FeAs. (a) The temperature program for synthesizing FeAs.The dwell at 773 K, below the arsenic sublimation temperature 887 K, aimsto pre-react the iron and arsenic mixture and hence reduce the internalpressure when passing the arsenic sublimation temperature. (b) A batchof FeAs (after reaction) in the sealed quartz ampoule. . . . . . . . . . . 57xivList of Figures4.3 Single crystal growth of LiFeAs. (a) The temperature program. Thegrowth window (slow cooling) is between 1333 K and 873 K. (b) The ingotafter growth. Platelet-like single crystals are randomly distributed insidethe alumina crucible. (The alumina crucible was broken using a hammer)(c) A LiFeAs single crystal after cleavage. The typical single crystal sizeis 3×3×0.2 mm3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.4 Assembly of the crucibles in the LiFeAs growth. (a) The crucibles,caps and three quartz discs. The two molybdenum caps are used to confinethe lithium vapor. The quartz discs are used to absorb lithium vaporcoming out of the alumina crucible. (b) Illustration of the assembly ofcrucibles and caps. For each new growth, only the alumina crucible andthe quartz discs needed to be replaced. In this setup, the quartz ampouleis well protected from lithium vapor. . . . . . . . . . . . . . . . . . . . . 604.5 Li-Mo binary phase diagram [119]. Lithium and molybdenum do notreact to form alloys below 1615 K. Therefore, the only role of the Mo-crucible is to confine lithium vapor. In turn, Mo-crucible is also free fromcorrosion, allowing it to be kept for multiple uses. . . . . . . . . . . . . 614.6 LiFeAs crystals that have been grown. All crystals are preserved in aglovebox filled with inert argon gas. Each batch of LiFeAs crystals is keptwith a piece of lithium ingot inside a glass bottle. The lithium ingot isused to protect LiFeAs crystals by absorbing water, oxygen, and nitrogenthat enter into the glovebox accidentally. . . . . . . . . . . . . . . . . . 624.7 Magnetization measurement of LiFeAs: Magnetic susceptibility wasmeasured every 0.2 K from 5 K to 20 K in a 1 Oe magnetic field withH⊥ab crystallographic direction. The superconducting transition startsat 17.2 K. The transition width from 90% to 10%, indicated by verticaldashed lines, is 1 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63xvList of Figures5.1 A 38 × 38 nm2 topographic image (Vs = 205 mV, It = 50 pA)of LiFeAs measured at 4.2 K. The bright and dark objects are de-fects in the surface. Note that the black dots are absent in a freshlycleaved surface and their number increases with time. They can also beremoved through large bias voltage pulses (∼ 1000 mV). Therefore, theyare atoms/molecules either adsorbed or desorbed during the measurement. 655.2 Topography image of LiFeAs with atomic resolution. (a) 6.8× 6.8nm2 atomic resolution topographic image (Vs = 40 mV, It = 100 pA),showing an inter-atomic spacing of 3.74 ± 0.03 A˚. (b) Schematic of thecrystal structure of LiFeAs. A cyan plane indicates the cleavage plane (seethe text for details). (c) The lattice viewed in the ab-plane after cleavage.The lattice shown in (a) is either the lithium or arsenic lattice whose latticeconstant is 3.77 A˚ determined by neutron powder scattering method [122].(d) The lattice viewed in the ac-plane. The top-most lithium lattice isfollowed by the arsenic lattice with 0.7 A˚ beneath and then the iron lattice2.2 A˚ deep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3 dIt/dVs spectrum of LiFeAs in a range from -550 mV to +550 mVmeasured at 4.2 K. The Fermi level, indicated by the dashed verticalline at the zero bias voltage, locates at the center of two enhancements ofthe LDOS in the positive and negative bias voltages, as indicated by bluecolor. Superconducting gaps open at the Fermi level (red color). . . . . 675.4 Homogeneity of superconducting properties. (a) 90× 90 nm2 topo-graphic image (Vs = 15 mV, It = 100 pA) of LiFeAs. (b) 150 individualdI/dV spectra (T = 4.2 K) from the defect-free 33 nm long line markedin (a). (c) Gap map of the same region in (a). ∆hpp corresponds to halfthe energy separation between coherence peaks. Both, the gap map andthe histogram of ∆hpp shown in (d) reveal a high degree of homogeneityin the superconducting properties in LiFeAs. . . . . . . . . . . . . . . . 69xviList of Figures5.5 Superconducting gaps measured by dIt/dVs spectrum. A dIt/dVsspectrum on a defect-free area taken at 2 K. The blue dashed lines at ±6meV and ±3 meV indicate the positions of the coherence peaks of thelarge and small gaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.6 Fit to the superconducting gaps. The dashed vertical lines indicatethe fitting window [-6.8 mV, 6.8 mV]. (a) A fit using two isotropic gapsand a constant Γ = 0.5 meV. The fitting curve near zero bias voltage doesnot reproduce the observed spectrum. (b) A fit using two isotropic gapsand Γ = αE. (c) A fit using two anisotropic gaps and Γ = αE. The lattertwo fits reproduce the experimental spectrum very well within the fittingwindow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.7 Superconducting gaps at elevated temperatures. (a) The tunnelingspectra between 2 and 20 K in a defect-free area. (b) The gap ampli-tudes (median value) determined by fitting of a two-isotropic-gaps model.(c) The gap amplitudes (median value) determined by fitting of a two-anisotropic-gaps model. The errorbars are determined from the 95% confi-dence bounds resulting from the fitting program. The large gap determinedby fitting to either of the two models generally follows the temperaturedependence predicted by the mean-field BCS theory (dashed black line in(b) and (c)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.8 Examples of above-gap structures in superconductors. (a) Exam-ple of the LDOS of a weak-coupling phonon-mediated superconductor [1].No obvious above superconducting gap structure is resolved. (b) Exampleof the LDOS of a strong-coupling phonon-mediated superconductor, forexample lead [134]. Two shoulder-like structures, as indicated by greenarrows, appear at energies corresponding to two phonon modes shifted bythe superconducting gap, E1 = ∆+Ω1phonon and E2 = ∆+Ω2phonon. (c) Ex-ample of the LDOS of a strong-coupling high-Tc cuprate superconductor,for example Bi2Sr2Ca2Cu3O10+δ [135]. A pronounced dip-hump structure,indicated by the green ovals, appears just above the superconducting gap. 75xviiList of Figures5.9 The superconducting dIt/dVs spectrum (black) (T = 1.6 K, B =0 T) plots over the normal state dIt/dVs spectrum (blue) (T =12 K, B = 10 T). By comparison, we can see hump structures at ±18mV followed by dip structures at ±10 mV as approaching EF from bothsides. In particular, the hump structures appear as kinks in the LDOS asindicated by the black arrows. . . . . . . . . . . . . . . . . . . . . . . . 765.10 The superconducting dIt/dVs spectrum (red), taken at T = 12K, B = 0 T, is plotted with the normal state dIt/dVs spectrum(blue), taken at T = 12 K, B = 10 T. At the same temperature,namely with the same thermal broadening effect, the kinks is still visiblein the superconducting spectrum, as indicated by the black arrows. Thisobservation confirms that the kink features are associated with the openingof the superconducting gaps. . . . . . . . . . . . . . . . . . . . . . . . . 775.11 Dip-hump structure in LiFeAs. The upper panel is the normalizeddIt/dVs spectrum by using Eq. 5.1, not the direct ratio, between two spec-tra shown in Fig. 5.9. Dip-hump features are clearly visible at energiesE > ∆max1 . The lower panel shows the derivative of the normalized dIt/dVsspectrum, which gives the inflection points (EI) between the dip and hump. 786.1 Schematic examples of the theoretical predictions for impuritybound states in LiFeAs with s±-wave pairing symmetry. The bluespectrum is the LDOS without impurities. Each red spectrum is the LDOSon the nearest neighbor site of the defect site. (a) shows the effect of animpurity with a relatively weak non-magnetic potential Vimp < 1 eV [166].The impurity bound state pins to the gap-edge of the smaller supercon-ducting gap ∆2, as shown in the enhanced LDOS peak at E = ∆2. (b)shows the effects of a non-magnetic defect with a strong defect potentialVimp > 1 eV [166]. In-gap bound states are present as strong LDOS peaksat E < ∆2. (c) shows the effects of magnetic defects, which always tendto create in-gap bound states [156]. . . . . . . . . . . . . . . . . . . . . . 83xviiiList of Figures6.2 Example of Fe-site defects in the topography of a Mn-substitutedLiFeAs sample. The topography is measured with It = 300 pA, Vs = 50mV, T = 12 K, and a magnetic field B = 10 T. Three types of Fe-site defects are present in the surface, representative examples of whichare highlighted in the red circles. They all have a two-fold bow-tie-likeshape with two orientations, horizontal or vertical. A-type is an engineeredmanganese impurity, and B- and C-type are native Fe-D2 and Fe-D2-2defects, respectively. There are also defects at the lattice sites other thaniron, which are described by Grothe et al. in Ref [49]. . . . . . . . . . . 846.3 Geometrical shape of an Fe-site impurity. (a) shows the structure ofthe Fe-As tetrahedron: an iron is in the center of the tetrahedron formedby four arsenic atoms with two arsenic atoms above and the other twobelow. (b) shows the charge transfer effect due to an Fe-site defect. (c)In an STM measurement, the tip placed on top of the sample surface ismost sensitive to the LDOS from only top layers of atoms. (d) shows thelattice as seen by an STM tip. Only the top lithium, arsenic and ironlattice layers are shown. Two example Fe-site defects, depicted as blackdots with red lobes, are shown in two iron sites. . . . . . . . . . . . . . . 856.4 Superconducting bound states of native Fe-site defects. (a) To-pography of an Fe-D2 defect (It = 50 pA, Vs = 25 mV). (b) dIt/dVs spectrataken on the defect site: the blue and red spectra are taken at the blueand red dots in (a), respectively. The black curve is a reference spectrummeasured in a defect-free area of the same sample, with the same tip.Two superconducting gaps ∆1 = 6 meV and ∆2 = 3 meV are indicatedby the dashed lines. (c) Topography of an Fe-D2-2 defect (It = 100 pA,Vs = 50 mV). (d) The dIt/dVs spectrum on the defect (red) together withthe reference spectrum (black). . . . . . . . . . . . . . . . . . . . . . . . 87xixList of Figures6.5 Defect bound states of engineered Fe-site impurities. (a), (c), and(e) are topographic images of manganese (It = 200 pA, Vs = −100 mV),cobalt (It = 400 pA, Vs = 15 mV), and nickel (It = 100 pA, Vs = 12mV) impurities, respectively. (b), (d), and (f) are dIt/dVs spectra on theimpurity sites: the red spectra are taken at the locations indicated by thered dot in (a), (b), and (c), respectively. The reference spectra taken in adefect-free region are plotted in black. The gap amplitudes ∆1 = 6 meVand ∆2 = 3 meV are indicated by the dashed lines. . . . . . . . . . . . . 887.1 Example of QPI: one-dimensional electron gas. (a) An electrontravels with momentum ki, whose quantum state is shown in (c). Withoutthe presence of impurities, |ψ|2 is constant and therefore the LDOS is flat,as shown in (b). (d) shows that with an impurity, a finite density ofelectrons is back-scattered. (e) The LDOS shows an interference patternsbetween the incident electron wavefunction with momentum ki and thescattered electron wavefunction with momentum kf . The wavelength ofthe modulation is λ = 2pi/|q|, where q = kf−ki = 2kf as indicated in (f).(g) shows the quantum states before and after the electron being scatteredin the band structure. (h) Fourier transforming the LDOS in (e) producesa peak at a vector q = kf − ki = 2kf in q-space. (i) The dispersion of qin turn can recover the band dispersion shown in (g). Generally, in a realQPI measurement, the analysis steps start from (g), to (h), then to (i),and then back to (g). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.2 Possible QPI vectors in iron pnictides near the Fermi level. (a)The scattering vectors that connect the segments of the Fermi surface (k-space). qh−h and qe−e show intraband scattering within the hole and elec-tron pockets, respectively, while qh−e shows interband scattering betweenthe two. (b) An autocorrelation image of the Fermi surface (q-space).QPI features corresponding to the three QPI vectors are marked by blackarrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95xxList of Figures7.3 Area for performing QPI measurement. (a) Topography (It = 250pA, Vs = 25 mV) of the area. There are 14 native defects and one ad-sorbed defect, among which ten are Fe-D2 defects. (b) shows a typicalIt-Vs spectrum at position 1© and (c) shows its numerical differentiationfollowing a Gaussian filter with δ ∼ 0.5meV . . . . . . . . . . . . . . . . 967.4 Example of the data processing techniques: raw Fourier trans-form. (a) The unprocessed tunneling conductance map g(r, 8 meV). (b)The Fourier transform graw(q, 8meV). The direct Fourier transform of theraw data yields strong intensity near q = (0, 0). . . . . . . . . . . . . . . 977.5 Example of the data processing techniques: Gaussian mask ofthe defect centers. (a) The tunneling conductance map after defect-center masking gM(r, 8 meV) and (b) its Fourier transform gM(q, 8 meV).Masking the defect centers removes the strong and asymmetric intensitybackground, giving more symmetric patterns. (c) The tunneling conduc-tance map of the defect centers, the portion removed from the raw data,and (d) its Fourier transform. Here (b), (d) and Fig. 7.4(b) are plottedin the same color scale. By comparing (d) and Fig. 7.4(b), we can seethat the major contribution to the direct Fourier transform stems fromthe defect-center background. . . . . . . . . . . . . . . . . . . . . . . . . 997.6 Example of the data processing techniques: suppression the cen-ter peak in q-space. (a) The Fourier transform after the Gaussian maskof defect centers gM(q, 8 meV), the same image of Fig. 7.5(b). (b) The fi-nal QPI map after the additional application of the Gaussian suppressionmethod of Ref. [35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100xxiList of Figures7.7 QPI features in the QPI intensity map. There are two major QPIfeatures. One is the scattering among the hole bands, which appears asrings centered at q = (0, 0). The other is the interband scattering be-tween the hole and electron bands, which appears as arcs centered aroundq = (±pi/a,±pi/a), reminiscent of the autocorrelation image in Fig. 7.2(b).The symbols indicate the location of the QPI vectors whose dispersion istracked in Fig. 7.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.8 Dispersion of QPI vectors and the comparison with Green’s func-tion calculation using T -matrix formalism. (a) The Fermi surfacesobtained from ARPES (blue intensity map) and the model (red and bluecurves). (b) The ARPES and model dispersions along the high-symmetrycuts of the first Brillouin zone. In (a) and (b) the ARPES spectra areshown for a photon energy that selects kz values near zero. The modeldispersions are shown for kz = 0 (red) and kz = 0.4pi/c (blue). (c) Thecalculated QPI at Vs = 8 mV obtained assuming that electrons tunnelinto states with a non-zero kz = 0.4pi/c. Features associated with intra-and interband transitions are indicated by the open symbols. (d) For qh-hQPI and (e) for qh-e QPI: the experimental (blue points with error bars)and theoretical (solid symbols) dispersion of the QPI vectors indicated in(c). The error bars are determined approximately by the full width at halfmaximum of the QPI features plus one additional pixel uncertainty. Thesolid lines show the dispersion expected from the model dispersion. . . . 1037.9 Energy dependence of the superconducting coherence factors.(a) The modulus and (b) the modulus squared values of coherence factorsv(k) and u(k). The amplitudes of v(k) and u(k) are comparable nearthe Fermi level (E = 0). Moving away from the Fermi level, one of thembecomes dominant, depending on the sign of the energy. . . . . . . . . . 105xxiiList of Figures7.10 Example of QPI intensity variation: the case of s±-wave. For sim-plicity, we use the autocorrelation image from Fig. 7.2(b) for demonstra-tion, as shown in (a). Under s±-wave pairing symmetry with nonmagneticimpurities and at T  Tc, when E > ∆, the QPI intensity map is close tonormal state in (a). When approaching the superconducting gap, intensityof qe−e and qh−h are suppressed while the intensity of qh−e is enhanced orpreserves the original amplitude, as shown in (b). . . . . . . . . . . . . . 1077.11 The intensity variations of h-h and h-e scattering vectors. Abovethe superconducting gaps, from 20 meV to 6 meV, the intensities of bothh-h and h-e do not vary significantly. Inside the superconducting gap,from 6 meV to 3 meV, the intensity of the h-h QPI feature, highlightedinside the red circle, is highly suppressed, while the intensity of the h-eQPI feature, highlighted inside the black circle, is enhanced. . . . . . . . 1087.12 The energy dependence of the integrated QPI intensity. (a) Thered sector and blue circle are the integration windows of h2-h2 and h2-h3and h2-e1,2 scattering vectors, respectively. A noise background signal isintegrated in the grey rectangular area and subtracted. Here one quarter ofthe windows are shown for simplicity but the integration is performed overthe equivalent areas in all four quadrants of the image. (b) The integratedintensity of the QPI signal for the intra- and interband h2-h2 and h2-h3(red) and interband h2-e1,2 (blue) scattering vectors. The curves were thennormalized to the value at 12 meV and the h-e intensity has been offsetfor clarity. The dashed lines indicate the values of the superconductinggaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097.13 Observation of Bogoliubov QPI in real space. dIt/dVs maps in thevicinity of an Fe-D2 defect at (a) Vs = +3 mV and (b) Vs = −3 mV.The anti-phase relationship of the LDOS modulations is highlighted bythe locations with high intensities inside the white circles of (a) and thecorresponding low intensities inside black circles of (b), and vice versa. . 110xxiiiList of Figures7.14 Zmap QPI in LiFeAs. The real space Z-maps are defined as Z(r, eV ) =g(r, eV )/g(r,−eV ), which highlights the anti-phase Bogoliubov QPI. (a)The real-space Zmap at 3 meV shows short wavelength oscillations in theLDOS. (b) The Fourier transform of (a), using the method in Sec. 7.2.1,indicates that the h-e QPI feature dominates the anti-phase BogoliubovQPI at 3 meV. (c) The real-space Zmap at 10 meV, above the supercon-ducting gaps, and (d) its Fourier transform do not show strong h-e QPIfeature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118.1 Variation of Bogoliubov QPI intensity with temperature. (a)QPI of LiFeAs at 8 meV. The h-h and h-e QPI features are highlightedin red and blue circles, respectively. (b) Red curve: the variation of|u(ki)u∗(kf ) − |v(ki)v∗(kf )||2 as a function of temperature; blue curve:the variation of |u(ki)u∗(kf ) + |v(ki)v∗(kf )||2 as a function of tempera-ture. With a s±-wave intensities of h-h and h-e QPI features follow thered and blue curves, respectively. . . . . . . . . . . . . . . . . . . . . . . 1158.2 Pairing in the iron lattice of LiFeAs. (a) The Fermi surface of LiFeAsfrom a tight-binding model (Appendix A). The sign of the superconductinggaps are marked near the hole pockets (+) and the electron pockets (-),respectively. (b) The lattice with two “inequivalent” iron sites, indicatedby gray and red dots. The electron pairing occurs between NN’s in eitherthe gray or the red lattice. . . . . . . . . . . . . . . . . . . . . . . . . . 1178.3 Pairing interaction in LiFeAs. (a) The FeAs structure: a iron squarelattice with arsenic atoms locating alternatively above and beneath thesquare centers of the iron lattice. The NN interaction J and the NNNinteraction W are indicated by the dashed curves. (b) The antiferromag-netic stripe order in the iron square lattice. (c) Electrons pair on NNNsites under s±-wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118xxivGlossaryARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . angle-resolved photoemission spectroscopyBCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bardeen, Cooper, and SchriefferBEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bose-Einstein condensateB-qp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bogoliubov quasiparticlesDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . density functional theoryDOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . density of statesg(q, eVs) . . . . . . . . . . . . . . . . . . . . . . . . tunneling conductance map in scattering vector spaceg(r, eVs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tunneling conductance map in real spaceh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . holee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electronEF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermi levelINS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inelastic neutron scatteringIt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tunneling currentk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . momentumkB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . boltzmann constantkF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermi momentumLDOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . local density of statesMo-crucible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . molybdenum crucibleMPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Max-Plank InstituteNMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nuclear magnetic resonanceNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nearest-neighborNNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . next-nearest-neighborRPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . random phase approximationSTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scanning tunneling microscopyxxvGlossaryq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QPI scattering vectorQPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . quasiparticle interferenceTc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . superconducting transition temperatureUBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .University of British ColumbiaUHV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ultra-high vacuumvk, uk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . superconducting coherence factorsVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .bias voltage on the sample∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . superconducting gap|ψG〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BCS ground stateµSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . muon spin resonanceρs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DOS of the sampleρt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DOS of the tip~ωc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .cutoff energy for attractive interaction~ωD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Debye energyxxviAcknowledgementsSix years of Ph.D. study and research is a long journey. However, my journey has beenboth fruitful and colorful thanks to the wonderful folks with whom I have worked.First and foremost, I would like to thank Doug Bonn for offering me the opportunityto work in his famous superconductivity group at UBC as a Ph.D. student. Doug, as myprinciple research supervisor, has not only guided my study but has also provided mewith great freedom in my research. Under his direction I have become a researcher withvaluable critical thinking skills, capable of generating independent ideas. In addition, Ihave learned his precise and rigorous attitude toward science, which is a great treasure.I also want to thank my co-supervisor, Ruixing Liang. During my first three yearsat UBC, he taught me the way to grow very high-quality, single crystals. Having onlyacquired a fraction of his expertise in materials science, I found success in growing thehigh-purity LiFeAs single crystals that were the foundation of this thesis work.Furthermore, I would like to acknowledge my collaborators at UBC-without theirinput, this thesis project would not have reached completion. Stephanie Grothe workedtogether with me on most of the data collection at UBC. Her rigorous, precise, andresponsible personality was an indispensable factor for the success of these experiments.Yan Pennec introduced STM to me. His expertise on STM greatly boosted this researchproject. Sarah Burke, an expert on scanning probes, provided many deep insights toexperiment design and the data analysis. Steve Johnston, a theoretician who worksclosely with experimentalists, provided many theoretical supports in understanding mydata. He also patiently answered all of my many dumb physics questions, from whichmany new ideas were generated. Pinder Dosanjh, a magician on instruments, spent manydays and nights directing my quest to reduce or eliminate the noise in the instruments.Without his input, I could not have been able to present such beautiful data in my thesis.xxviiAcknowledgementsGiorgio Levy and Andrea Damascelli, experts on the ARPES technique, gave me manyprofound visions of the electronic structure of LiFeAs.In addition, I would like to acknowledge my collaborators at the Max-Plank Institute(MPI), in Germany. Peter Wahl hosted me in his lab during my visit to MPI. He is avery knowledgeable and insightful physicist. We had many discussions which triggeredmany of the new ideas in this thesis work. Ram Aluru, and Udai Singh worked togetherwith me on the impurity project. They provided a pleasant workspace for me. Our teamefforts have yielded great physical results.I am truly indebted to George Sawatzky and Ilya Elfimov. They opened the door ofdensity functional theory to me, and have deepened my understanding of physics. Theyalso gave me a lot valuable advice on the experimental work in this thesis, in particularthat shown in chapter 6.I also really appreciate the help from my co-workers in the superconductivity lab itself.James Day, the unofficial manager of the lab, helped with my integration into the groupand also to solve many technical problems. Jordan Baglo-who is like an encydopedia-was always available for assistance. Whenever I had a question, whether it was aboutphysics or instrumentation, Jordan-pedia was always the first person I consulted. BradRamshaw, like an elder brother, taught me many skills towards doing good research andhelped to direct me along the path to success. I also want to thank my colleagues inthe LAIR STM lab. I had many discussions with Andrew Macdonald, Agustin Schiffrin,Ben MacLeod, Katherine Cochrane, Martina Capsoni, and many others, ranging fromphysics to life. The friendly working environment was most valuable.I would like to give great thank to my parents, Xianping Chi and Shufang Huang.They did their best to provide me with a good education, no matter what the burden orthe many difficulties they needed to overcome. More importantly, they always showedtheir love and support to me, and have always encouraged me to pursue my dreams.Finally, of course, I would like to thank my fiance Janice Shen: the best gift to meduring my Ph.D. study. Her love and support has always been, and will always be, thedriving force for me to proceed in study, research, and life.xxviiidedicated to my lovely wife, Janice Shen.xxixChapter 1IntroductionSuperconductivity is one of the few macroscopic quantum phenomena in nature, describ-ing the ability of certain materials to carry electrical current without energy dissipationbelow their transition temperature Tc [1]. Its quantum physics as well as its potentialfor applications attract intensive research. In particular, high-temperature (high-Tc) su-perconductors have puzzled physicists for decades since its discovery in 1986 [2]. It iswell agreed that superconductivity originates from a superfluid of paired electrons. Inconventional superconductors, electron-phonon coupling provides the pairing interaction.In high-Tc superconductors, the origin of the pairing interaction is still under debate [3].Identifying the pairing interactions, and then using this knowledge to design new low-costhigh-Tc superconductors for practical applications is a central goal of the superconduc-tivity community.A recurring theme in the study of high-Tc superconductors is the pairing of elec-trons via repulsive interactions [3–7], rather than the attractive interaction mediated byphonons as in conventional superconductors. Pairing driven by a repulsive interaction,such as exchange of antiferromagnetic spin fluctuations, carries a distinguishing featurethat the superconducting gap, ∆(k), has changes in sign in different parts of the Bril-louin zone. This means that determining the pairing symmetry (the symmetry of ∆(k))in high-Tc superconductors is an essential step towards understanding the origin of su-perconductivity in these systems [3–7].Scanning tunneling microscopy (STM) is one of the key tools that can be used todiscern pairing symmetry [8, 9]. STM provides unique capabilities to image the atomicand electronic structure with sub-angstrom spatial resolution and sub-meV energy reso-lution. In particular, scanning tunneling spectroscopy, which measures the local densityof states, gives direct insight to the superconducting gap structure [8, 9]. In addition, via11.1. Iron-based superconductorsimaging quasiparticle interference (QPI), STM is able to resolve the pairing symmetryin momentum space [10, 11]. As a result, STM has achieved great successes in the studyof the pairing symmetry in high-Tc superconductors, including both cuprates [8, 10] andiron-based superconductors [9, 11]. In this thesis work, I have applied STM to study thepairing symmetry of LiFeAs, a particular member in the iron-based high-Tc family.1.1 Iron-based superconductorsIn 2008, Kamihara et al. discovered the second high-Tc family, iron-based pnictide andchalcogenide superconductors with Tc up to 56 K [12, 13]. This discovery excited thesuperconductivity community not only because of their high transition temperaturesbut also by the opportunity they offer to advance the existing understanding of high-Tcsuperconductivity. The functional heart of iron-based superconductors is an iron squarelattice with each iron nested at the center of an arsenic/selenium tetrahedron. The centraliron-arsenic/selenium blocks form 2-dimensional layered crystal structures along withpositive ionic layers, as shown in the left side of Fig. 1.1 [14, 15]. Most parent compounds(without chemical substitutions) possess an antiferromagnetic ground state and becomesuperconducting upon chemical substitutions or by exerting pressure [14–16], reminiscentof the phase diagram of high-Tc cuprates (see the right side of Fig. 1.1) [17]. However,iron-based superconductors clearly distinguish themselves in a number of ways, such astheir metallic parent phase, up to 5 pieces of Fermi surface, and a large arsenic/seleniumelectronic polarizability [15, 18].The electronic structure of most iron-based superconductors consists of hole (h) bandscentered at k = (0, 0) and electron (e) bands centered at k = (±pi/a,±pi/a) (Fig. 1.2).In many systems, the h and e bands are strongly nested, which is considered to be theroot cause of antiferromagnetic instabilities [4, 21]. Superconductivity emerges after thesuppression of this antiferromagnetic order. This nested multiband structure opens thepossibility of an s±-wave pairing symmetry mediated by antiferromagnetic spin fluctu-ations with ∆(k) having a sign change between the e and h bands. A number of otherpairing symmetries were also proposed, such as s++-wave [22, 23], p-wave [24–26], and21.1. Iron-based superconductorsFeAsLiFeAs LaFeAsO BaFe2As2 DopingyoryPressure xTemperaturey(K)0 AntiferromagnetismSuperconductivityFeAs1-Figure 1.1: Iron-based superconductors: the crystal structures and the phase di-agram. The left side: the structure of a FeAs layer and three typical crystal structures ofiron pnictides: LiFeAs [19], LaFeAsO [14], and BaFe2As2 [20]. Iron chalcogenides have an FeSelayer, identical to the FeAs layered structure, as the central building block. The right side: aschematic phase diagram of iron-based superconductors.d-wave [27, 28], depending on the choice of the dominant pairing interaction in thesetheoretical models.EF (0,0) (π,π)Ek(a) (b)ΓMπ,π))Figure 1.2: Band structure and Fermi surface of iron pnictides. (a) A simplifiedtwo-band model for the iron pnictides with a hole-like band centered at k = (0, 0) and anelectron-like band centered at k = (pi/a, pi/a). (b) The Fermi surfaces of the bands in (a).A hole pocket and electron pocket are centered at Γ and M points, respectively, in the firstBrillouin zone (the dashed square).To date there has been considerable progress in measuring the pairing symmetryin this family [6, 11, 29–41], including many measurements that can discern amplitudevariations of different gaps associated with the various bands [32–35, 38, 39], and a few31.2. LiFeAsthat are sensitive to whether or not there is a sign change between the e and h bands [11,31, 36, 37, 40, 41]. Even though most results suggested an s±-wave pairing symmetry forthe majority of materials, a consensus on a universal pairing symmetry for all iron-basedsuperconductors has not been reached, and there is strong evidence that small differencesin electronic structure can lead to large differences in superconducting gap structures andparing symmetries [6].1.2 LiFeAsA particularly interesting compound among the pnictides is LiFeAs, which is supercon-ducting (Tc = 17 K) without the need for cation substitution [19, 42, 43]. This potentiallyplaces it in the same position that YBa2Cu3O7−x (YBCO) holds in the cuprates [44], astoichiometric superconductor that can be chemically and structurally perfect enough toavoid artifacts arising from disorder. LiFeAs has the additional advantage of a naturalcleavage plane, which exposes a non-polar surface that does not undergo reconstruc-tion [45], and makes it well suited to surface sensitive spectroscopic studies such as angleresolved photoemission spectroscopy (ARPES) [32, 34, 46] and STM [31, 35, 47–50], muchlike the cuprate Bi2Sr2CaCu2O8+x (BSCCO) [51, 52]. These properties make LiFeAs anideal system for studying the underlying mechanism of superconductivity in the iron-based high-Tc materials.It should be noted that LiFeAs differs from the other iron-based compounds in somesignificant aspects. The band structures calculated from density functional theory (DFT)indicate a nesting between the hole and electron pockets [53], which is supported by anearly de Haas-van Alphen oscillation study [54]. However, an angle-resolved photoe-mission spectroscopy (ARPES) study [46] and a recent de Haas-van Alphen oscillationstudy [55] shows that, in LiFeAs, the two hole pockets centered at the Brillouin zone cen-ter are significantly smaller than DFT predicted. In particular, the electronic structure ofLiFeAs lacks the strong nesting conditions observed in other families, which is likely thereason for the absence of a magnetic phase [46, 55]. One way to resolve the discrepancy isthrough adding the electronic correlation to the DFT band structure which quantitatively41.2. LiFeAsreproduces the electronic structure measured by ARPES and quantum oscillation [56].Moreover, the underlying pairing symmetry in LiFeAs is under debate and it is unclearwhether the nature of superconductivity in this material is the same as in the other iron-based superconductors.The lack of nesting between the h and e pockets weakens the traditional argumentfor s±-wave pairing. This, coupled with the observation of multiple dispersion renormal-izations, has led to proposals for an s++-wave pairing symmetry [34] driven by phononassisted orbital fluctuations [22, 23, 57]. Alternatively, ARPES indicates the presenceof a van Hove singularity at the top of the inner hole pockets [46], which can enhanceferromagnetic fluctuations and lead to a p-wave pairing symmetry. This is supportedby a recent STM study [31] as well as theory based on the random phase approximation(RPA) and a two-dimensional (2D) three-band model [26]. However, high purity LiFeAssamples do not show any signature of triplet pairing from nuclear magnetic resonance(NMR) or muon spin resonance (µSR) measurements [58–61]. The possibility of p-waveis ruled out. Nevertheless, there are indications that an s±-wave symmetry is realizedin LiFeAs despite the lack of strong nesting between the h and e bands. From a the-oretical point of view, both an early functional renormalization group study based ondensity functional theory (DFT) bandstructure [62] and a more recent RPA study basedon an ARPES-derived bandstructure [63] find a leading s±-wave superconducting insta-bility. This scenario also has experimental support from a number of indirect probingtechniques [38, 39, 64].Given these open issues regarding the pairing symmetry in LiFeAs, it is desirable tohave a direct, phase-sensitive measurement of the superconducting gap. Here, I employedSTM to study the gap structure and pairing symmetry in LiFeAs. Using this approach,compelling evidence is presented in this thesis for an s±-wave pairing symmetry in LiFeAs,bringing LiFeAs back into the general iron-based family with a universal s±-wave pairingsymmetry.This thesis is organized as follows. In chapter 2, I will give a brief introductionto superconductivity and then describe in detail the importance of pairing symmetryin understanding the superconducting mechanism. In chapter 3, I will introduce the51.2. LiFeAsprinciple of STM and the instruments used for this work. In chapter 4, the single crystalgrowth of LiFeAs will be outlined. Then I will discuss the point spectroscopy of defect-free regions in chapter 5 and on iron-site defects in chapter 6. In chapter 7, I will showthe QPI data in LiFeAs. Finally, in the concluding chapter 8, I will summarize this thesiswork and discuss its contribution to the understanding of the pairing symmetry and theinteraction in LiFeAs. Chapter 5 and chapter 7 are primarily based on two publicationsduring my Ph.D. study: Ref. [48] and Ref. [65].6Chapter 2Superconductivity and its pairingsymmetryThe electrical resistivity of many metallic materials drops abruptly to zero when these ma-terials are cooled below their transition temperature Tc. This phenomenon, named super-conductivity, was first observed by Kammerlingh Onnes in mercury samples in 1911 [66].Fig. 2.1 shows the superconducting transition of mercury with Tc = 4.15 K. Thereafter,hundreds of superconductors have been discovered. Tc increased slowly until 1986 whenhigh-Tc superconductors were discovered (see Fig. 2.2) [67].Figure 2.1: Discovery of superconductivity. Zero resistivity below 4.15 K in mercury,measured in 1911 [66].In 1986, Mu¨ller and Bednorz discovered that polycrystalline samples of La2−xBaxCuO4become superconducting with Tc ∼ 30 K [2]. Because the active superconducting layer isa CuO2 plane, this superconductor was called a cuprate. A year later, another cupratesuperconductor was discovered with Tc well above 77 K, the boiling temperature of liquidnitrogen [69, 70]. This was a true milestone because it was the first time that a super-conducting phase could be established using liquid nitrogen, instead of the much more7Chapter 2. Superconductivity and its pairing symmetryTlMBaMCaMCuFO?IHgBaMCaMCuFO.(δYBaMCuFOd(xLaM)xBaxCuOANbFGeNbFSnVFSiNbNNbPbHgliquid nitrogen 77K?9?I ?9FI ?9mI ?9hI ?99IYear5of5Discovery MI?IIMIAIdI.I?II?MI?AIT ccKvCeCuMSiM UPtF CeCoInmPuCoGam?Ba?)xKxBiO MgBM?LaFePO?)xFxLaFeAsO?)xFxLiFeAsSmFeAsO?)xFxGd?)xThxFeAsO Sr?)xSmxFeAsFKIUhmFe?UhmSeMFigure 2.2: Tc records vs time. The yellow circles show Tc of conventional superconductors.The Tc increases over time, but slowly. The blue diamonds show Tc of cuprates, which gave asudden leap in the 1980s. The green squares show Tc of heavy fermion superconductors. Thered stars show the newly discovered Fe-based superconductors in 2008, another high-Tc familywith maximum Tc > 50 K. The Tc data shown in the graph are collected from Ref. [68] andRef. [15].expensive and rare liquid helium. Although scientists placed very high hopes on usingcuprate superconductors in real world applications, poor mechanical properties and highmanufacturing costs have limited large-scale applications of high-Tc cuprates. In addi-tion, after more than twenty years intensive research, the superconducting mechanism ofcuprate superconductors is still controversial.In 2008, Hosono discovered a second family of high-Tc superconductors, the iron-basedsuperconductors, whose highest Tc to date is 56 K [13, 14]. Even though their Tc is lowerthan cuprates, their softer mechanical properties give renewed hope for applications.Moreover, the discovery of a brand-new high-Tc family is invaluable in the sense that itbroadens the horizon of people’s understanding of high-Tc superconductivity.From this brief history, we can see that superconductivity always surprises us. In82.1. A new phase of materialparticular, the discovery of high-Tc superconductors gives us a hope to take advantageof superconducting materials in industries. However there are also challenges to findsuperconductors that are both usable and low cost. Many physicists are working hard touncover the underlying mechanism of high-Tc superconductivity. In turn, they can usenew insights as guidelines in the search for low-cost high-Tc superconductors with goodmechanical properties. Such guidelines continue to evolve, so it is always worth lookingback at the periodic table from time to time, since most of the natural elements play arole in superconducting materials, from the pure elements themselves, to alloys, to morecomplicated ceramic compounds.2.1 A new phase of materialSuperconductivity is a distinct thermodynamical phase of materials. Even though thephenomenon of zero resistivity is the origin of the term superconductivity, it does notuniquely define a new phase of materials. This is because superconductors are metalsabove the transition temperature Tc, so one may think the phenomenon of zero resistivityis merely a consequence of all freedom of electrons to scatter being frozen below Tc.In this case, metals would become ideal zero-resistance conductors without involving athermodynamic phase transition to a new state of matter.The property that really marks superconductivity as a distinct thermodynamic phaseis the Meissner-Ochsenfeld effect. In an hypothetical zero-resistance conductor (no suchsolid state conductor really exists)1, whether or not the external magnetic field is ex-pelled depends on the history of when the external magnetic field is turned on [1]. Incontrast, the Meissner-Ochsenfeld effect demonstrates that a superconductor in an ex-ternal magnetic field H (< Hc(T )) expels magnetic flux from its interior below Tc nomatter when this magnetic field is turned on (see Fig. 2.3(a)) [71]. Hc(T ) is a criticalmagnetic field above which superconductivity disappears.2 The Hc(T ) curve separates1A special case is the low density gas phase plasma2The different responses to external magnetic field classify superconductors into two categories: typeI superconductors and type II superconductors. Here I discuss only type I superconductor for simplicity,but the concept can be generalized to type II superconductors which have two critical fields.92.2. BCS theorySupercondutingstatenormal metallic stateHHcTc T(a) (b)BsuperconductorFigure 2.3: Meissner-Ochsenfeld effect. (a) Schematic illustration of expelling magneticfield by a sphere shape superconductor. (b) The phase diagram of a type-I superconductor inH-T phase space. Hc(T ) separates the H-T plane into two regions: the superconducting stateand the normal metallic state.the H-T phase space into two regions. Below the Hc(T ) curve, the superconductingGibbs energy Gs(T,H) is less than the normal state Gn(T,H), and the material is in asuperconducting phase. In this phase, any external magnetic field is expelled, indepen-dent of its history. When one crosses the Hc(T ) curve from below to above, the systemhas Gn(T,H) > Gs(T,H), so the material prefers a normal metallic state to lower thefree energy. The curve Hc(T ) where Gs(T,H) = Gn(T,H) defines a phase transitionboundary in H-T phase space (see Fig. 2.3(b)). This establishes superconductivity as anew thermodynamic phase of the electronic state in the material.2.2 BCS theoryWhat causes electrons to become superconducting? To answer this question, we firstneed to know what causes resistivity. Here, we limit our discussion to the frame work ofthe Landau Fermi liquid approximation. In the normal metallic state, a dressed electron,namely a quasiparticle, can be scattered by the ionic lattice vibrations (phonons), by im-purities, and by electron-electron interactions. These scattering processes cause changesin the state of individual quasiparticles, which dissipates energy. In contrast, zero re-sistivity in superconductors indicates that electrons move in the crystal lattice without102.2. BCS theoryenergy dissipation. There are two ways to achieve this. One way is to freeze all scatteringprocesses, in which case the metal becomes an ideal zero-resistance conductor. But wehave ruled out the possibility of superconductors being ideal zero-resistance conductors(see Sec. 2.1). The other option is to consider the change in the state of the electronsthemselves below Tc. In the superconducting phase, could electrons self-organize into anew ground state that sets a high threshold for creating excitations, which could not beeasily reached by the scattering processes in the crystal?An analog phenomenon is superfluidity. In 1937, Kapitza and Allen discovered thatthe viscosity of He4 liquid drops to zero below Tλ = 2.17 K, forming a dissipationless fluid,called a superfluid [72, 73]. Ten years later, Bogoliubov and others realized that in He4,a degenerate Boson gas with strong interactions, a Bose-Einstein condensate (BEC) canform [74]. Below Tλ = 2.17 K, bosonic He4 atoms form a coherent macroscopic many-bodywavefunction which has only collective mode excitations (phonons and rotons). Thesephonons cannot be excited below a critical velocity and therefore liquid He4 becomes adissipationless superfluid [75].Is it possible that electrons in a superconductor also form such a coherent groundstate? Intuitively, this is impossible because electrons are fermions, and the Pauli exclu-sion principle forbids a BEC-like condensation. In 1956, Cooper made a key breakthroughby showing that electrons can be paired together by certain attractive interactions [76].These paired electrons are composite bosons that can form a BEC-like state. This insightled to the birth of BCS theory [77].2.2.1 Cooper instabilityCooper worked out the problem of two electrons with an attractive interaction placedjust outside an occupied Fermi sea [76]. He found that the two electrons bind together ifthere is an effective attractive interaction between them.Imagine two interacting electrons added to an occupied Fermi sea, as shown in Fig. 2.4112.2. BCS theory(a). The ground state wavefunction has to satisfy the Schro¨dinger equation(−~22m(521 +522) + V (r1, r2))︸ ︷︷ ︸Hamiltonianψ(r1, r2) = Eψ(r1, r2). (2.1)Here, EF is set equal to zero. In the center of mass frame, Eq. 2.1 simplifies to(−~22m52 +V (r))ψ(r) = Eψ(r), (2.2)where r = r1 − r2. Taking the Fourier transformφ(k) =∫ψ(r)eik·rdr,V (k) =∫V (r)eik·rdr. (2.3)Eq. 2.3 becomes2kφ(k) +∑k′>kFV (k− k′)φ(k′) = Eφ(k), (2.4)where k = ~2k22m is the unperturbed single electron kinetic energy. If a solution withE < 0 can be found, then a bound pairing state exists.Let’s first consider a special case: a pure isotropic attractive interaction with a cutoffenergy ~ωcV (k− k′) = −U (V ≤ ~ωc)V (k− k′) = 0 (V > ~ωc) (2.5)where U > 0. Inserting Eq. 2.5 into Eq. 2.4 and rearranging, we haveφk = U∑′ φk′2k − E, (2.6)where∑′ is a sum of states in 0 < Ek′ < ~ωc. Summing both sides and canceling∑′ φk,122.2. BCS theorywe obtain1U=Ek′<~ωc∑Ek′>012k − E=∫ ~ωc0N0d2− E=12N0 ln2~ωc − E−E. (2.7)Here we have assumed a constant density of state (DOS) N0 ≡ N(EF ) in the backgroundsystem. The eigen-energy isEeigen = −2~ωc11 + e2N0U< 0. (2.8)Thus, the eigenenergy is always negative, no matter how small the attractive interactionis.e- e-Occupied Fermi Seae--ke-k(a)(b)Ur1 r2Binding PotentialFigure 2.4: Cooper instability. (a) Schematic illustration of two electrons added to anoccupied Fermi sea with momentum k and −k. (b) With an effective attractive interaction U ,two electrons bind together as a spin-singlet Cooper pair.Then eigenfunction for the isotropic attractive interaction case isψ(r) =∑|k|>kFcos(k · r)2k − Eeigen=∑|k|>kFeik·r + e−ik·r2[2k − Eeigen]. (2.9)132.2. BCS theoryAssuming a spherical Fermi surface, the sum of k′ over the first Brillouin zone makes thewavefunction ψ(r) independent of the direction of the vector r ≡ r1 − r2. That meansthat the pair is bound in an s-wave state (like the s-wave state of the hydrogen atom).In fact, it is also possible to construct a two-electron bound state with angular mo-mentum different from zero, for exampleψ(r1 − r2) = f(|r1 − r2|)Ylm(θ, φ) (2.10)where Ylm is a spherical harmonic function. These solutions may exist when the interac-tion adopts a form that shares their symmetry, i.e. V (k − k′) ∝ Ylm(θk−k′ , φk−k′). Fordifferent quantum number l = 0, 1, 2, · · · , the wavefunction has the symmetry type ofs-wave, p-wave, d-wave, · · · . The symmetry type, known as pairing symmetry, will bediscussed in detail in Sec. 2.2.7.2.2.2 BCS HamiltonianIn 1957, Bardeen, Cooper, and Schrieffer published the first truly microscopic theoryof superconductivity, famously known as BCS theory [77]. Using the insight from theCooper instability, BCS realized that the whole Fermi surface would be unstable if thereis an effective attractive interaction between electrons. As a result, electrons near theFermi level tend to be bound into Cooper pairs. They built an effective HamiltonianH =∑k,σkc†k,σck,σ +∑k,k′Vk,k′ c†k,↑c†−k,↓c−k′,↓ck′,↑, (2.11)where c†k,σ is a quasiparticle creation operator with k denoting momentum and σ denotingspin. The first term is the kinetic energy which gives band dispersion k in the normalstate. The second term describes the major interaction between electron pairs, namelyCooper pairs, in a superconductor. The configuration of opposite momenta and spins ischosen for a spin singlet wavefunction. The spin-triplet case will be discussed in Sec. 2.2.8.142.2. BCS theory2.2.3 Mean field approximationThe solution of the BCS Hamiltonian 2.11 can be obtained by employing a Bogoliubovtransformation using a mean field approximation [78]. Because of the large numbers ofparticles involved, the fluctuations in the number of Cooper pairs should be small.3 Thissuggests that we can use a mean field approximation. Here I follow the derivations inchapter 3 in Ref. [1] and chapter 6 in Ref. [79]. We define the average number of Cooperpairsb∗k ≡ 〈c†k,↑c†−k,↓〉. (2.12)We can rewritec†k,↑c†−k,↓ = b∗k︸︷︷︸average+ (c†k,↑c†−k,↓ − b∗k)︸ ︷︷ ︸small fluctuation. (2.13)Substituting Eq. 2.13 into the BCS Hamiltonian and neglecting quantities which arebilinear in the small fluctuation term, we obtain the mean-field BCS HamiltonianH =∑k,σkc†k,σck,σ +∑k,k′Vk,k′ (c†k,↑c†−k,↓bk′ + b∗kc−k′,↓ck′,↑ − b∗kbk′). (2.14)We define a new parameter∆k = −∑k′Vk,k′ bk′ = −∑k′Vk,k′ 〈c−k′,↓ck′,↑〉. (2.15)In terms of ∆k, the BCS Hamiltonian becomes (after relabeling some subscripts)H =∑k,σkc†k,σck,σ +∑k(∆kc†k,↑c†−k,↓ + ∆∗kc−k,↓ck,↑ + ∆kb∗k). (2.16)3In the large N limit, where N is the quasiparticle number in the system, the value of δN/N¯ , evaluatedusing the ground state wavefunction constructed by BCS, is very small [1].152.2. BCS theoryEq. 2.16 becomes a sum of a few bilinear terms of creation and annihilation operators.Such a bilinear form can always be diagonalized by a Bogoliubov canonical transformationck,↑ = u∗kγk0 + vkγ†k1, (2.17)c†−k,↓ = −v∗kγk0 + ukγ†k1. (2.18)where the coefficients uk and vk arevk =∆k|∆k|√12(1−kEk), (2.19)uk =√1− |vk|2 (2.20)with Ek =√2k + |∆k|2. We should note that γki (i = 0, 1) are genuinely fermionicoperators, since they obey the standard anti-commutation laws{γki, γ†kj} = δkk′δij,{γ†ki, γ†kj} = 0,{γki, γkj} = 0, (2.21)Then the mean-field BCS Hamiltonian becomesH =∑k(k − Ek + ∆kb∗k)︸ ︷︷ ︸condensation energy+∑kEk(γ†k0γk0 + γ†k1γk1)︸ ︷︷ ︸Bogoliubovquasiparticle dispersion. (2.22)The first term is the ground state energy, which is the condensation energy of Cooperpairs. When the condensation energy is less than zero, superconductivity is favored. Thesecond sum gives the increase in energy above the ground state in terms of the numberoperators γ†kγk for γ†k fermonic quasiparticles. Thus, these γ†k describe the elementaryexcitations of the system, known as Bogoliubov quasiparticles (B-qp). The dispersionof B-qp is Ek =√2k + |∆k|2. To create such a B-qp, one needs an energy of Ek. Theminimum energy required for such an excitation is E = |∆| (when k = 0). Thus |∆k|162.2. BCS theoryplays the role of an energy gap in the superconducting spectrum.EFEkEFEk EFEk(a) (b) (c)Δadd excitationspectrumadd coupling between twospectrumFigure 2.5: BCS theory: example of electron and hole bands coupling. (a) a parabolicelectron-like band, (b) add the excitation hole-like band, (c) superconducting gap opens whenthe electron-like and hole-like bands are coupled by an interaction ∆.To further perceive the physical meaning of the BCS Hamiltonian, we rewrite Eq. 2.16in the matrix formH −∑k(k + ∆kb∗k) =∑k[c†k,↑, c−k,↓]k ∆k∆∗k −kck,↑c†−k,↓ . (2.23)Here we moved the constant sum∑k(k + ∆kb∗k) to the left of the equation. Now theHamiltonian looks like a two level system with energies ±k and an interaction (or mixingstrength) ∆k. The hole band −k can be viewed as excitations of electrons which areidentical to the creation of holes, as shown in the red curve in Fig. 2.5(b). In this sense,the mean-field BCS Hamiltonian describes the interaction between an electron band (withdispersion k and quasiparticle operator c†k,↑) and a hole band (with dispersion −k andquasiparticle operator ck,↓). ∆k is the coupling strength between the two bands. Fig. 2.5shows the physical process. From Eq. 2.17 to Eq. 2.22, all that is done is to diagonalizethe matrix( k ∆k∆∗k −k)by finding a linear combination of the electron band and hole band.The Bogoliubov transformation in Eq. 2.17 and Eq. 2.18 are the eigenvectors of the 2×2matrix. Now the Bogoliubov quasiparticle dispersion Ek is simply the determinant ofthe matrix.172.2. BCS theory2.2.4 Ground state wavefunctionThe ground state, according to Hamiltonian 2.22, should have zero Bogoliubov quasipar-ticles (excitations), which meansγki|ϕG〉 = 0 (2.24)where i = (0, 1) and |ϕG〉 denotes the BCS ground state. One obvious way to constructsuch a wavefunction is|ϕG〉 = A∏kγk1γk0|F 〉 (2.25)where A is a normalization factor and |F 〉 =∏k≤kFc†k|0〉 is the Fock state in a normalmetallic state. Manifestly γki|ϕG〉 = 0 is indeed satisfied since (γki)2 = 0, a natural factfor fermonic operators.Let’s rewrite Eq. 2.17 and 2.18γk0 = ukck,↑ − vkc†−k,↓, (2.26)γk1 = ukc−k,↓ + vkc†k,↑. (2.27)Substituting γk1 and γk0 into γk1γk0|F 〉, we findγk1γk0|F 〉 = −uk × (uk + vkc†k,↑c†−k,↓)|0〉 (k ≤ kF)γk1γk0|F 〉 = −vk × (uk + vkc†k,↑c†−k,↓)|0〉 (k > kF).(2.28)Therefore, the BCS ground state wavefunction must take the form|ϕG〉 =∏k(uk + vkc†k,↑c†−k,↓)|0〉, (2.29)which is already normalized because |uk|2 + |vk|2 = 1. This wavefunction implies that theprobability of a pair (k ↑,−k ↓) being occupied is |vk|2, whereas the probability beingunoccupied is |uk|2 = 1− |vk|2.The BCS ground state is a coherent wavefunction of Cooper pairs (see Chapter 4 in182.2. BCS theoryRef. [80]). A coherent state has an uncertain number of quasiparticles N but a definitephase φ. Rewriting ∆k = |∆k|eiφk , we find that vk carries the same phase factor eiφk as∆k, whereas uk is a real number (see Eqs. 2.19 and 2.20). The BCS wavefunction can bewritten [80]|ϕG〉 =∏k(|uk|+ |vk|eiφkc†k,↑c†−k,↓)|0〉. (2.30)The creation of each Cooper pair c†k,↑c†−k,↓|0〉 is associated with a fixed phase factor eiφk .In physics, the phase factor represents a gauge field. Note that the BCS Hamiltonian isgauge invariant. In a normal state, each electron takes a random gauge in its wavefunc-tion, independent of each other, preserving the gauge symmetry. (Note that the normalstate is a solution to the BCS Hamiltonian with ∆k = 0). In contrast, in the supercon-ducting state, the many-body wavefunction experiences a spontaneous gauge symmetrybreaking. Cooper pairs coherently choose gauge phases. The relative phase betweentwo Cooper pairs is locked. The coherent phenomenon is obvious when we expand |ϕG〉.Assuming a 2N-state system, we have|ϕG〉 =kN∏k(|uk|+ |vk|eiφkc†k,↑c†−k,↓)|0〉=kN∏k|uk||0〉︸ ︷︷ ︸0 particles+kN−1∏k 6=k1|uk||vk1|eiφk1c†k1,↑c†−k1,↓|0〉︸ ︷︷ ︸two particles/one Cooper pair+ · · ·+kN−n∏k 6=k′|uk|kn∏k′=k1|vk′ |eiφk′c†k′,↑c†−k′,↓|0〉︸ ︷︷ ︸2n particles/n Cooper pairs+ · · · .The expansion produces a sum of many terms. Each term has a definite particle number2n, where n is an integer. Each creation of a Cooper pair in state (k,−k) adds a definitephase eiφk . Therefore, |ϕG〉 is a coherent combination of states from zero particle numberto full occupation. Thus the particle number in the ground state |ϕG〉 fluctuates. Thisgauge-phase coherence between Cooper pairs is a result of BEC-like condensation. Inthis sense, superconductivity can be viewed as a superfluid of Cooper pairs, composite192.2. BCS theorybosons constructed from fermionic electrons.One thing we should notice is that condensation of Cooper pairs in BCS theory isnot identical to BEC condensation of pure bosonic particles [79]. In general, a pair offermions is not equivalent to a boson. In the BCS case, the spatial extension of a Cooperpair is very large. The average distance between two electrons in a Cooper pair is of theorder of the coherence length, defined asξ =2~ vFpi∆, (2.31)where vF is Fermi velocity and ∆ is the superconducting gap. ξ of a conventional s-wavesuperconductor is of the order a few hundred nanometers. Thus, in such superconductors,Cooper pairs strongly overlap with each other in real space, in contrast to generic bosonsystems, such as He4, where bosons barely overlap in real space. Instead, Cooper pairsare more localized in momentum space. In this sense, BCS theory describes a BEC-likestate with overlapping composite bosons.2.2.5 Gap equationWe have defined a parameter ∆k in Eq. 2.15 and found that it behaved like a gap for theelementary excitations. Rewriting ck,σ in terms of γki, we obtain∆k = −∑k′Vkk′u∗k′vk′〈1− γ∗k′0γk′0 − γ∗k′1γk′1〉. (2.32)At T = 0, when no Bogoliubov quasiparticle is excited, this reduces to∆k = −∑k′Vkk′u∗k′vk′= −∑k′Vkk′∆k′2Ek′. (2.33)Eq. 2.33 is a gap equation that needs to be solved self-consistently. One obvious solutionis ∆k = 0 for all momenta k, which is just the trivial normal metallic state. When anontrivial solution makes the condensation energy less than zero, superconductivity is202.2. BCS theoryrealized. To find such a nontrivial solution, we first need to find the dominant attractiveinteraction between electrons.2.2.6 Reduced gap ratioAt finite temperature, because γ∗k′iγk′i are fermionic particle number operators, the oc-cupation number in thermal equlibrium obeys a Fermi-Dirac distribution〈γ†kiγki〉 = f(Ek) =1eEkkBT + 1. (2.34)Then Eq. 2.32 becomes∆k = −∑k′Vkk′u∗k′vk′(1− 2f(Ek′))= −∑k′Vkk′∆k′2Ek′tanh[Ek′2kBT]. (2.35)The superconducting transition temperature Tc is the temperature at which ∆k = 0for all momenta k. In this case, Ek becomes k. Thus, Tc can be determined from theself-consistent Eq. 2.35 by replacing Ek with k∆k = −∑k′Vkk′∆k′2k′tanh[k′2kBTc]. (2.36)Let us consider a simple case, Vkk′ = −U when k < ~ωc, otherwise, Vkk′ = 0. FromEq. 2.33, we have∆k = ∆0 =~ωcsinh[1/N0U ](2.37)Then Tc can be determined from Eq. 2.36 [1],kBTc = 1.13~ωce−1/N0U . (2.38)212.2. BCS theoryNow we can define the reduced gap ratio as the following form2∆0kBTc= 3.53e1/N0Ue1/N0U − e−1/N0U(2.39)Assumed a weak-coupling condition N0U  1, then 2∆0kBTc = 3.53, which is the ”clas-sical” BCS value in the weak-coupling limit value. This value, resulting from the weak-coupling approximation and a constant pairing interaction U with a cutoff energy ~ωc,has been tested in many experiments and found to be consistent with many conventionallow-Tc superconductors. When increasing the coupling strength N0U , 2∆0kBTc becomeslarger than 3.53. In such cases, Eliashberg theory [81], a strong-coupling version of theBCS theory, is required to determine the exact ratios, which is beyond the content ofthis thesis. However, the value is always larger than 3.53 in the strong coupling cases.Therefore, the reduced gap ratio is a measure of the coupling strength of the pairinginteraction.2.2.7 Pairing symmetryWhen an electron moves in a crystal lattice, any physical quantity associated with theelectron usually reflects the symmetry of the crystal structure in certain ways. Thesymmetry of ∆k, known as the pairing symmetry, should obey this rule.I take a 2D square lattice structure as an example, because it is a good approximationto the superconductor studied in this thesis. The lattice constant is set to unity forsimplicity. The derivations here are developed from chapter 4 in Ref. [68]. Assume that222.2. BCS theorythe electron-electron (e-e) interactions are of the following formV (ri − rj) = U δ(ri − rj)︸ ︷︷ ︸on-site interaction+ J [δ(ri − (rj − δx)) + δ(ri − (rj + δx))+δ(ri − (rj − δy)) + δ(ri − (rj + δy))]︸ ︷︷ ︸nearest-neighbor interaction+W [δ(ri − (rj − δx − δy)) + δ(ri − (rj + δx − δy))+δ(ri − (rj − δx + δy)) + δ(ri − (rj + δx + δy))],︸ ︷︷ ︸next-nearest-neighbor interaction(2.40)where δx and δy are the unit lattice spacing in the xˆ and yˆ directions, respectively. Herethe e-e interaction U is on the same site, J is on the nearest-neighbor (NN) sites, andW is on the next-nearest-neighbor (NNN) sites.WJUδxδyFigure 2.6: e-e interaction in a square lattice. The interactions are U on the same site, J onthe nearest-neighbor (NN) sites, and W on the next-nearest-neighbor (NNN) sites, respectively.232.2. BCS theoryFourier-transforming Eq. 2.404, we obtain the interaction in momentum spaceVkk′ = U︸︷︷︸from the on-site interaction+ J [cos(kx) cos(k′x) + sin(kx) sin(k′x) + cos(ky) cos(k′y) + sin(ky) sin(k′y)]︸ ︷︷ ︸from the nearest-neighbor interaction+W [cos(kx) cos(k′x) + sin(kx) sin(k′x)]× [cos(ky) cos(k′y) + sin(ky) sin(k′y)]︸ ︷︷ ︸from the next-nearest-neighbor interaction.(2.41)We see that the on-site, NN, and NNN interactions are in terms of 0th, 2nd, and 4th orderof sin(ki) and cos(ki), respectively. This allow us to re-express Vkk′ in terms of squarelattice harmonicsVkk′ =9∑i=1λigi(k)gi(k′), (2.42)where λ1 = 4pi2U , λ2 = λ3 = λ6 = λ7 = 2pi2J , λ4 = λ5 = λ8 = λ9 = pi2W , and gi(k) aregiven byg1(k) =12pig2(k) =12pi[cos(kx) + cos(ky)]g3(k) =12pi[cos(kx)− cos(ky)]g4(k) =12picos(kx) cos(ky)g5(k) =12pisin(kx) sin(ky)g6(k) =12pi[sin(kx) + sin(ky)]g7(k) =12pi[sin(kx)− sin(ky)]g8(k) =12picos(kx) sin(ky)g9(k) =12pisin(kx) cos(ky) (2.43)4The Fourier transform Vkk′ = 1/Ω∑i,j∑m,n ei(k−k′)·(ri,j−rm,n)V (ri,j − rm,n)242.2. BCS theoryThese harmonic functions reflect the underlying symmetries of the square lattice. Allphysical quantities adhering to the symmetry of the square lattice can be expressed asan expansion of these harmonics.5Now we can substitute this factorized Vkk′ into the gap Eq. 2.33∆k = −∑k′Vkk′∆k′2Ek′= −9∑i=1gi(k)∑k′λigi(k′)∆k′2Ek′. (2.44)Defining∆i ≡∑k′λigi(k′)∆k′2Ek′, (2.45)∆i (i = 1, · · · , 9) are momentum independent constants because the sum runs over thewhole Brillouin zone. Therefore, we have∆k = −9∑i=1∆igi(k), (2.46)The significance of Eq. 2.46 is that the gap itself is also expressible as a linear combinationof a set of square lattice harmonics in Eq. 2.43. Each harmonic is a gap channel withthe corresponding symmetry. In general, Eqs. 2.45 and 2.46 are two coupled equationsthat need to be solved self-consistently. Usually, one of the gap channels wins, leadingto the corresponding pairing symmetry. The choice is made by the detailed form of theeffective e-e interaction.The pairing symmetry of each gap channel is named according to its changes underthe symmetry operations of the square lattice. In a spherical isotropic system, sphericalharmonics are a complete basis set for angle dependence. Spherical harmonics Ylm areclassified by quantum numbers l = 0, 1, 2, · · · , named as s-wave, p-wave, d-wave, · · · ,respectively (see the discussion in the end of Sec. 2.2.1). The symmetry of gi(k) is alsoclassified by the operations included in the symmetry group of the square lattice C4v.5These nine harmonics are the first few lowest orders in the complete set of the square harmonics.For more complicated interactions, an expansion with the complete basis set is required.252.2. BCS theoryFor g1(k) and g2(k), they transform as the identity under all symmetry operations, sothey are referred to as s-wave. For g6(k), g7(k), g8(k) and g9(k), they change sign underpi rotations, so they are referred to as p-wave. For g3(k), g4(k), and g5(k), they changesign under pi/2 rotations, so they are referred to as d-wave. Here, s, p, d are used to beconsistent with the labeling tradition in a spherical isotropic system. However, the strictsymmetry is defined by the harmonics of the crystal lattice.Pairing symmetry is closely related to the Fourier-transform of the Cooper-pair wave-function. Fig. 2.7 shows the pairing symmetry of the first three harmonics in the firstBrillouin zone. For g1(k), it is constant in the Brillouin zone, so we call it conventionals-wave or simply s-wave. For g2(k), there is a sign change from zone center (0,0) tozone corner (±pi,±pi). If we define the region near (0,0) to have a positive sign, theregion near (±pi,±pi) has a negative sign. Therefore, this s-wave symmetry is namedas extended s-wave or s±-wave. For g3(k), it has a sign change between the kx and kydirections, similar to the dx2−y2 orbital of an atom in a square lattice. So it is calleddx2−y2-wave. Other d-wave harmonics are also possible, such as dxy-wave for g5(k), whichrequires NNN interaction. Pairing symmetries other than conventional s-wave are knownas unconventional pairing symmetries [82].02-2s-wave s  -wave dx2-y22-wave+- -- -++- --π 0 π -π 0 π -π 0 πkkxy+Figure 2.7: Pairing symmetry (square lattice): s-wave, s±-wave and dx2−y2-wave areplotted from left to right. The red color region has positive sign and the blue color region hasnegative sign. In the transition from positive to negative, there are nodal lines required bysymmetry, where the superconducting gap is zero, indicated by the trick black lines.262.2. BCS theory2.2.8 Spin-singlet vs spin-tripletThe pairing symmetry discussed in the previous section refers to the symmetry of thespatial wavefunction. Because it is based on fermionic particles, the ground state wave-function has to be antisymmetric. If the spatial wavefunction is symmetric, the spinconfiguration has to be antisymmetric and vice versa. In addition, each Cooper pair iscomposed of two electrons, so the total spin of a Cooper pair is either spin-singlet (anti-symmetric) or spin triplet (symmetric). According to these principles, one can categorizesuperconductors according to the total spin of a Cooper pair (See Fig. 2.8 for example).In s-wave and d-wave channels, the spatial wavefunction is symmetric, so the spin con-figuration is singlet to keep the whole wavefunction antisymmetric. In a p-wave channel,the spatial wavefunction is antisymmetric, so the spin configuration is spin triplet.s  -wave+- -- --π 0 π -π 0 πp-wave---+++e--ke-ke--ke-k(a) (b)spin-singlet spin-tripletFigure 2.8: Spin configurations of a Cooper pair. (a) s±-wave as an example. In thefirst Brillouin zone, the spatial wavefunction is symmetric because ∆k = ∆−k, so the spinshave to adopt the antisymmetric spin-singlet configuration. (b) p-wave is the converse case,∆k = −∆−k, so it must adopt spin-triplet. The black lines in (a) and (b) indicate the nodelines in the two pairing symmetries.Most superconductors are spin-singlet superconductors. There are some candidatesthat possibly possess p-wave spin-triplet pairing, for example Sr2RuO4. The best knownspin-triplet pairing example is superfluid 3He. The research in this thesis concentrates272.2. BCS theoryon studying the pairing symmetry of LiFeAs, in which p-wave pairing symmetries areruled out by experiments [58–61]. In the following discussion, I am going to focus onspin-singlet pairing. Nevertheless, the fundamental ideas can be easily generalized top-wave cases.2.2.9 Remarks on BCS theoryThe BCS theory introduced above is valid for general e-e interactions. In principle, byidentifying different Vkk′ , BCS theory could describe different types of superconductors.However, this conclusion is not completely true.Usually, superconductors are categorized into two families: conventional and uncon-ventional superconductors. In a conventional superconductor, pairing is induced by anattractive on-site interaction Vkk′ = −V0 assisted by phonons. Phonons have a naturalcutoff energy ~ωD, where ωD is the Debye frequency. In addition, most conventionalsuperconductors are three dimensional materials with well-defined Fermi liquid behav-ior. In these superconductors, BCS theory with a mean-field approximation has achievedtremendous successes and its extension via Eliashberg theory has made it more quanti-tative and rooted in the actual electron-phonon interaction.Unconventional superconductors refer to all superconductors that cannot be under-stood in the framework of BCS theory with electron-phonon coupling, for example thehigh-Tc and heavy fermion superconductors. Two of the major issues that hinder theapplication of BCS theory to unconventional superconductivity are strong-correlationand fluctuation effects. Most cases of high-Tc superconductivity occurs in strongly corre-lated materials, beyond the approximation of Fermi liquid theory. The attractive pairinginteraction in such systems is related to the strongly correlated interactions. From ex-perimental findings, unconventional superconductivity is also likely to appear in layeredquasi-two dimensional materials [67], where fluctuation effects are stronger than in threedimensional materials. In addition, a large amount of evidence indicate that most un-conventional superconductors are close to a quantum critical point where quantum fluc-tuations become remarkably strong. A proper consideration of these fluctuation effectsis required to explain unconventional superconductivity.282.3. How two electrons in a Cooper pair overcome the e-e repulsive interactionIn addition, as mentioned in Sec.2.2.4, BCS theory describes the condensate of Cooperpairs localized in momentum space, which is the case for conventional superconductors.For high-Tc superconductors, their coherence lengths are short, from a few nanometersto a few angstroms [15, 83]. In this regime, Cooper pairs do not strongly overlap in realspace, and are closer to the case for interactive generic bosons. Recently, theories basedon BCS-BEC crossover have been proposed and applied to high-Tc superconductors [84].Nevertheless, there is little doubt that all known superconductors stem from the con-densation of Cooper pairs. Despite the great challenges in unconventional superconduc-tivity, BCS theory always provides us with a fundamental guideline. Superconductivityphysicist Lev P. Gor’kov wrote in his paper “While most of superconductors of “the newgeneration” present serious challenges of the fundamental character in understanding thesymmetry and the mechanisms behind the observed SC in them, the consensus is yet that,the basic notions and at least qualitative results of the BCS-like scheme remain applicablefor them as well” [85].2.3 How two electrons in a Cooper pair overcomethe e-e repulsive interactionThe concept of pairing symmetry has been introduced in Sec. 2.2.7. In this section, Iwould like to further quantify the significance of pairing symmetry. I shall continue withthe simple 2D square lattice model. I also limit the discussion to spin-singlet pairing bydropping the p-wave channels (from λ6-channel to λ9-channel in Eq. 2.42).BCS theory has shown that an attractive interaction can induce superconductivity.However, intuitively, there is always strong Coulomb repulsion when two electrons inter-act with each other. How do electrons overcome repulsion and bind into pairs? Let usconsider two examples.292.3. How two electrons in a Cooper pair overcome the e-e repulsive interaction2.3.1 Two cases for e-e interactionsCase #1First, consider a simple situation proposed by Bogoliubov [78]: Vkk′ has only on-siteinteractionU =−Uph + Ucoul (V ≤ ~ωD)Ucoul (~ωD < V < EF )J = 0W = 0 (2.47)where Uph and Ucoul are two positive constants describing phonon-assisted attractiveand Coulomb repulsive interactions, respectively (see Fig. 2.9(a)). This type of on-siteinteraction usually occurs in phonon-assisted conventional superconductors. Uph has anatural cutoff energy ~ωD. We have chosen the cutoff EF for the Coulomb interactionbecause only momenta of order kF enter into it. Now Vkk′ has an energy dependence, astep function at ~ωD. We need to use the energy dependent gap equation∆k(k) = −∑k′Vkk′(k, k′)∆k′√2k′ + ∆2k′(2.48)Assuming again for simplicity that the DOS is the constant N0, the integral equationis reduced to the two equations∆1 = N0(Uph − Ucoul)∫ ~ωD0∆1 d√2 + ∆21−N0Ucoul∫ EF~ωD∆2 d√2 + ∆22∆2 = −N0Ucoul∫ ~ωD0∆1 d√2 + ∆21−N0Ucoul∫ EF~ωD∆2 d√2 + ∆22. (2.49)where ∆k = ∆1 for  < ~ωD and ∆k = ∆2 for ~ωD <  < EF . It is obvious that ∆1and ∆2 are constants. We make one more assumption: the couplings are weak, namely302.3. How two electrons in a Cooper pair overcome the e-e repulsive interactionhωD EFUcol-UphInteraction strengthE(a)0Gap size  Δ EhωD EF0(b) Δ1Δ2Figure 2.9: Case #1 Pairing under the on-site electron-phonon (e-p) attractive ande-e Coulomb repulsive interactions. (a) the energy dependence of e-p with −Uph (shadered with cutoff at ~ωD) and e-e with −Ucoul (shade blue with cutoff at EF ). (b) superconductinggap as a function of energy. ∆(E) changes sign from the region with the presence of an attractiveinteraction to the region with only the repulsive interaction.|∆1|, |∆2|  ~ωD, EF . Then we have∆1 = 2~ωD e−1/(N0Ue) (2.50)∆2 = −∆1UcoulUe[1 +N0Ucoul ln(EF/~ωD)](2.51)where Ue is the effective attractive interaction.Ue = Uph −Ucoul1 +N0Ucoul ln(EF/~ωD). (2.52)312.3. How two electrons in a Cooper pair overcome the e-e repulsive interactionThe criterion for a nontrivial solution and, hence, for superconductivity is Ue > 0, i.e.,Uph >Ucoul1 +N0Ucoul ln(EF/~ωD)(2.53)The term on the right hand side is the renormalized Coulomb repulsion. The attractiveUph only needs to overcome a fraction of Ucoul to achieve superconductivity. When thiscriterion is satisfied, there is a sign change between ∆1 and ∆2. The superconducting gap∆ changes from a positive sign in regime  < ~ωD to a negative sign in regime  > ~ωD(see Fig. 2.9(b)).Case #2Now we consider another case: Vkk′ has the following formU = UcoulJ = −J0W = 0 (2.54)where Ucoul and J0 are two positive constants. We find that there is no attraction in theλ1-channel (on-site), and attraction in the λ2-channel (NN) and λ3-channel (NN). λ4 andλ5 are not involved because of the absence of the NNN interaction. λ6 and λ7 will notbe considered due to the symmetry constraints of spin-singlet pairing.There are three harmonics left in Eq. 2.46∆k = −∆1g1(k)−∆2g2(k)−∆3g3(k). (2.55)Inserting ∆k above into gap equation 2.46, we have∆1 = −U [∆1∑k′12Ek′+ ∆2∑k′cos(k′x) + cos(k′y)2Ek′+ ∆3∑k′cos(k′x)− cos(k′y)2Ek′].The second and third terms on the right side are zero because of the sum of cos(kx) over322.3. How two electrons in a Cooper pair overcome the e-e repulsive interactionthe whole first Brillouin zone. Therefore, the equation above becomes∆1 = −U∆1∑k′12Ek′= −U∆1∑k′12√2k′ + ∆2k′.Because U∑k′12Ek′> 0, the only solution is ∆1 = 0.Doing the same calculation, we find∆2 =J2∆2∑k′(cos(k′x) + cos(k′y))22Ek′(2.56)∆3 =J2∆3∑k′(cos(k′x)− cos(k′y))22Ek′(2.57)Since both side are positive, we can cancel ∆i on both sides and solve the gap equationswith ∆k contained in Ek =√2k + |∆k|2. Both s±-wave and dx2−y2-wave have non-trivialsolutions for the superconducting gaps. The exact form of interaction and Fermi surfacetopology determine which channel is favored.2.3.2 Avoided on-site Coulomb repulsionIn order to inspect the implications of different pairing symmetries, it is instructive towrite down the real-space representations of ∆. The original definition of the supercon-ducting gap in Eq. 2.15 demonstrates that the gap is proportional to the average numberof Cooper pairs.332.3. How two electrons in a Cooper pair overcome the e-e repulsive interactionAvoided Coulomb repulsion through time – Case #1First, consider the s-wave case (Ucoul = 0 for simplicity),∆k = −∑k′Vk,k′ 〈c−k′,↓ck′,↑〉= −Uph∑k′〈c−k′,↓ck′,↑〉= −Uph∑k′〈∑rie−ik′ricri,↓∑rjeik′rjcrj ,↑〉= −Uph∑ri〈cri,↓cri,↑〉 (2.58)Here we define a real space pairing parameter [86]∆(r) = U〈cr,↓cr,↑〉, (2.59)or∆∗(r) = U〈c†r,↑c†r,↓〉. (2.60)Eq. 2.60 indicates that a Cooper pair consists of two electrons that are created at thesame location, as a result of on-site interaction. Even though the pairing is on-site, theyavoid each other through time. The fact that there is a natural cutoff for Uph indicatesa retarded interaction between two electrons with retarded time τ ∼ ~/~ωD = 1/ωD.As shown in Fig. 2.10, the interaction process via phonon is the following. An electroncreates a phonon at a lattice site r, then it travels away with distance d ∼ vF τ . A secondelectron comes to the lattice site and experiences attraction from the phonon at r. Inthis way, even though the interaction is on-site, these two bound electrons avoid eachother in time to eliminate the adverse effects of Coulomb repulsion.342.3. How two electrons in a Cooper pair overcome the e-e repulsive interactione-t=0, electron       creates a phonon at location rt ~ 1/ωD, electron     travels away, electron     is attracted to location r k1-k1 e-k2k2-k1k2e-e-Figure 2.10: Example of pairing through on-site attractive interactions. At t = 0, aspin up electron with momentum k1 creates an phonon at location r with a momentum transferk2 − k1. Then it travels away with velocity vF . After about t ∼ 1/ωD, another electron withspin down and momentum −k1 is attracted to location r and absorbs the phonon by momentumtransfer −k2 − (−k1). The interaction is on the same location r, but the two electrons avoideach other in time.Avoided Coulomb repulsion through space – Case #2We do the same transformation for s±-wave and dx2−y2-wave∆s±k = −∑ri∆s±ri (cos(kx) + cos(ky)) (s± − wave) (2.61)∆dk = −∑ri∆dri(cos(kx)− cos(ky)) (dx2−y2 − wave) (2.62)352.3. How two electrons in a Cooper pair overcome the e-e repulsive interactionwith the definitions [86](∆s±(r))∗ =J4[〈c†r,↑c†r−δx,↓〉+ 〈c†r,↑c†r+δx,↓〉+ 〈c†r,↑c†r−δy ,↓〉+ 〈c†r,↑c†r+δy ,↓〉](2.63)(∆d(r))∗ =J4[〈c†r,↑c†r−δx,↓〉+ 〈c†r,↑c†r+δx,↓〉 − 〈c†r,↑c†r−δy ,↓〉 − 〈c†r,↑c†r+δy ,↓〉] (2.64)Here, δx and δy are vectors of unit lattice space in xˆ and yˆ direction, respectively.-J0δxδyUcoule-e-e- e-e-e-e- e-(a)    e-e interactions (b)    s  -wave pairing (c)    dx2-y22-wave pairingFigure 2.11: Example of pairing through NN attractive interactions. (a) The interactionterms: on-site repulsive Ucoul and NN attractive −J0. (b) Real space pairing for s±-wave. Inboth δx and δy, the pairing has a positive sign, indicated by the red oval. (c) dx2−y2-wave.Pairing in δx has a positive sign (red oval) and in δy has a negative sign (blue oval).The real-space gap equations 2.63 and 2.64 demonstrate that s±-wave and dx2−y2-wavepairing symmetries are a consequence of electrons pairing on nearby sites. We see thatthe gap function adjusts itself so as to be zero in channels, on-site s-wave channel in thiscase, where the repulsive interaction dominates. By choosing s±-wave or dx2−y2-wave,the electrons in a Cooper pair avoid the on-site Coulomb interaction.Unconventional pairing symmetry does not necessarily mean there is no retardation ofthe pairing interaction. Even though two electrons have already avoided meeting in space,they may also want to reduce the opportunity to be close. In fact, the coherence lengthof a superconductor is a good indication of whether the paring interaction is retardedbecause it is a measure of the spatial extension of the Cooper pair wavefunction. Thelonger the coherence length is, the further apart the two electron are and so the longer362.3. How two electrons in a Cooper pair overcome the e-e repulsive interactionthe retarded time. In conventional s-wave superconductors, the coherence lengths areof the order thousands of lattice constant, indicating very long retardation. In the caseof superconductors with unconventional pairing symmetries, the coherence lengths aresignificantly shorter but still larger than the spacing of two nearby pairing sites, implyingsome retardation. The retardation of the latter case is much shorter than in conventionalsuperconductors (comparing the time between an electron traveling a few nanometersand an electron traveling a few hundred nanometers). There are some unconventionalsuperconductors with extremely short coherence length, of the order of the spacing oftwo nearby paring sites. Retardation may be negligible in such systems. Nevertheless,the description above is phenomenological and whether retardation is always required forsuperconductivity is also controversial.In summary, avoiding each other in time (retarded on-site interaction), and in space(NN interaction) are the basic two ways electrons can avoid the repulsive Coulomb inter-action. In the retarded on-site interaction case, the superconducting gap is scaled by thecutoff energy, i.e. |∆| ∝ ~ωD. This limits the Tc of superconductors because Tc ∝ |∆|.The benefit of having unconventional pairing symmetry is that the electrons avoid eachother in space, so there is no need for long retardation time. The lack of long retardationis manifest in the lack of an energy cutoff around the Fermi-surface. Remember that∆ ∝ Ec where Ec a characteristic energy for an unconventional pairing interaction. IfEc  ~ωD, high-Tc superconductivity can appear.2.3.3 The significance of sign changeA change in sign of the superconducting gap is a consequence of minimizing the conden-sation energy.In case #1, single-particle states with energies k such that |k| > ~ωD enter thewavefunction with a negative weight. Hence, despite the fact that the interaction isrepulsive, the matrix element of the interaction between a pair with |k| > ~ωD and onewith |k′ | > ~ωD is negative and tends to lower the total energy. Because of the signchange at higher energy, the effective repulsion below ~ωD is highly reduced. The criterionto have superconductivity is not Uph > Ucoul but Uph > U renormalizedcoul , where Urenormalizedcoul372.3. How two electrons in a Cooper pair overcome the e-e repulsive interactionis the renormalized repulsive interaction below ~ωD (see Eq. 2.53). Changing sign above~ωD helps the electrons overcome repulsive interaction below ~ωD.In case #2, instead of energy, the superconducting gap changes sign in momentumspace. For higher harmonics, the interaction potential Vkk′ , as defined in Eq. 2.42, alsohas a repulsive part for certain vectors q = k−k′. Even though, Vkk′ is repulsive betweenstate k and k′, the matrix element of the interaction between the pair (k,−k) with +sign and the pair (k′,−k′) with − sign becomes negative (see Fig. 2.12). Thus, this signchange allows the system to lower the total energy. Here, I ignore the energy dependence.However, the energy dependence of ∆k for NN pairing could also play an important role,as in case #1.s  -wave-π 0 πdx2-y22-wave-π 0 π02-2+-k2 k1k3 k1k2k3(a) (b)Figure 2.12: The pairing interaction vs sign change in momentum space. (a) For thes±-wave case, the interaction between k1 and k2 is Vk1k2 < 0, attractive, so there is no signchange. While there is a sign change between k1 and k3 because Vk1k3 > 0, the interaction isrepulsive. (b) The same for dx2−y2-wave.2.3.4 Pairing through repulsive interactionsIn general, the effective e-e interaction in a real superconductor is complicated. However,one empirical principle to find the pairing interaction is to look at the pairing symmetry.In a superconductor that has an unconventional pairing, if there is a strong repulsiveinteraction Vkk′ with a vector q = k − k′ that links the opposite sign of the supercon-ducting gap, it is a good clue that this repulsive interaction is likely related to the pairinginteraction.As early as 1965, Kohn and Luttinger pointed out that repulsive interactions could382.3. How two electrons in a Cooper pair overcome the e-e repulsive interactionbe responsible for pairing, at least theoretically [87]. The way to understand what isinvolved is to consider the screening of a charge placed in a metal. It is well knownthat if the fermion system is degenerate, the screening produces an oscillatory potentialwhich has attractive regions (i.e. Friedel oscillations, see Fig. 2.13) [81, 88]. Through theattractive interaction due to screening, unconventional Cooper pairs can form with signchange of the pairing symmetry.0 r   r1-r2VkkFigure 2.13: Example of the attractive channels from a repulsive interaction. Dueto the screening effect, the repulsive interaction Vkk′ between electrons may have attractiveregions (red shade) at certain e-e distances. This attractive interaction, usually not on-site,tends to give an unconventional pairing symmetry.In systems such as cuprate and iron-based superconductors where e-e correlations aresignificant, the condensate wavefunction would prefer an unconventional pairing symme-try in order to minimize the energy associated with large on-site Coulomb repulsion.Now it is firmly established that High-Tc cuprate superconductors have a dx2−y2-wavesymmetry [89, 90] and it is generally believed that most iron-based superconductors haves±-wave symmetry [6, 7]. These unconventional pairing symmetries have served to re-duce the strong on-site Coulomb repulsion. In addition, both cuprates and iron-basedsuperconductors have spin-fluctuation (repulsive) interactions that link states of oppositesign [3, 6, 7]. Therefore, spin-fluctuations are strongly suspected to be the origin of thepairing interaction.392.4. Final remarks2.4 Final remarksHere in this chapter, all the discussions were originated from the BCS effective Hamilto-nian in Eq. 2.11 and kept in the framework of BCS theory. I would like to present a simplepicture of the close relation between the pairing interaction and the pairing symmetry,in particular, what we can know of the pairing interaction from the pairing symmetry.Nevertheless, more comprehensive theories with the considerations of strong correlation,strong coupling are required to describe the exact nature of the pairing symmetry inunconventional superconductors. For current developments of such theories, please referto the reviews by Hirschfeld et al. [6], Chubukov [7], Sawatzky et al. [18, 91], Scalapino [3]and the references therein.40Chapter 3Scanning tunneling microscopyQuantum tunneling has been used as a tool to probe superconductivity since 1960. Planartunneling junctions have been employed to measure superconducting gaps [92] as well assignatures of electron-phonon coupling [93]. These experiments played a central role inexamining the BCS theory for conventional superconductors. Adding to these tools,scanning tunneling microscopy (STM), based on an atomic scale tunneling junction, wasquickly applied to study superconductivity after its invention [94]. In contrast to planartunneling junctions, STM is not only able to measure both the superconducting gap andthe bosonic coupling features [95, 96] but is also capable of imaging the spectroscopicfeatures of superconductors locally, such as Abrikosov vortices [97], pair breaking effectsof single impurities [98], and quasiparticle interference [99]. The sub-angstrom spatialresolution, together with sub-meV energy resolution, make STM an important tool forunderstanding the superconducting mechanism of unconventional superconductors [8, 29].3.1 Principle of STM and its operating modesSTM offers an unrivaled way to image in real space both the crystallography and theelectronic structure down to the atomic level. It was invented in 1982 by Binning andRohrer [94] and further developed thereafter [51, 100]. Nowadays, STM is able to probethe surface electronic properties, to microvolt and picometer precision, owing to theinstallation of ultra-low temperature refrigerators and the construction of state-of-artultra-low vibration labs [51, 101, 102].The central core of an STM is shown in Fig. 3.1. An atomically sharp tip approachesa sample surface, using a piezo motor, with the tip-sample-distance d of the order ofone nanometer (see the zoomed-in image). The space between the tip and the sample,413.1. Principle of STM and its operating modesVFeedback ControlScanning unitsample e-ItsVxVyVzz xysampletipFigure 3.1: Core components of an STM. The left side is a schematic plot of the keycomponents of an STM. In order to keep a stable tunneling barrier, two major components arerequired: a feedback control unit to adjust the tip-sample distance d and a scanning unit tocontrol the tip position r on the sample surface. The right side is a zoomed in image of thetip-sample tunneling junction: the tip and sample is separated by a distance of the order of 1nm. The quantum tunneling events primarily happen between the apex atom of the tip andthe top atomic layers of the sample surface.usually vacuum, forms a natural tunneling barrier with a tunneling resistance in therange of a few giga-ohm. Upon applying a bias voltage V between the tip and thesample, a tunneling current It flows through the tunneling barrier. The most widelyused convention for the polarity of the bias voltage is that the tip is grounded6. In thisconvention, the bias voltage V is then the electric potential applied on the sample Vs. Apiezoelectric scanner is used to control tip position r = (x, y) and the tip-sample-distanced = z(r)− z0(r), where z(r) is the height of the tip and z0(r) is the profile of the samplesurface. In total, four variables are allowed to be tuned in a measurement: the tip heightz, the sample bias Vs, the tunneling current It, and the tip position r. These variablesare not independent from each other. At low temperature where the thermal broadening6In fact, the tip is virtually grounded by the current amplifier. For details, please refer to Ref. [103].423.1. Principle of STM and its operating modesof the Fermi distribution is negligible, It is given by [104, 105]It(r, z, Vs) =4pie~× ρt(EF )× e−2κdEF+eVs∫EFρs(r, )d, (3.1)where e is the unit electron charge, ρt and ρs are the local density of states (LDOS)of the tip and sample, respectively, κ is the inverse of the decay length of the samplewavefunction into the tunneling barrier, and EF is the Fermi level which is set to zeroin the remainder of this thesis. Here I assume a metallic tip with a structureless LDOSρt() near its Fermi level EtipF . Derivations from Eq. 3.1 are described in Sec. 3.2. Eq. 3.1indicates that, through controlling the set of variables (Vs, It, z, r), one can have anumber of different operating modes in STM. Three common modes are used in thisthesis.The Topography mode is used to image the surface corrugations of the sample. Inthis mode, the tunneling current It and sample bias Vs are kept constant. From Eq. 3.1,the tip height z can be written as a function of the tip position rz(r)− z0(r) = d =12κln0+eVs∫0ρs(r, )d+ ln(4pieρt~It) . (3.2)z(r) is a measure of the sample surface corrugation. On a perfect and homogeneoussurface, the LDOS of the sample is constant and spatially homogeneous. Therefore,the right side of Eq. 3.2 is a constant, and the tip height z(r) records the profile of thesurface z0(r) which has corrugations of the atomic lattice. In addition, any variationof the LDOS can also lead to corrugations. For example, when an impurity is presenton the surface, the LDOS on the impurity site is modified and so the right side ofEq. 3.2 changes accordingly. As a result, the tip adjusts its height when passing overan impurity, as shown in Fig. 3.2(a). Fig. 3.2(b) shows a topography of a LiFeAs samplesurface measured by STM, in which a few impurities are present. It should be notedthat z0(r) is not a well-defined variable. Usually, z0(r), as indicated in the dashed curvein Fig. 3.2(a), is assumed to be a constant profile of the integrated LDOS between the433.1. Principle of STM and its operating modeszxVIts 10rnm1highlow(a) (b)-0.10.10.2-20 -10 0 10 20051015200SamplerBiasr(mV)TunnelingConductancer(nS)TunnelingCurrentr(nA)r(c)10rnm(d) Er=reVsr=r8rmeVhighlowFigure 3.2: STM operating modes. (a) STM adjusts its tip height in the topography mode.STM tip changes its height when the tip passes over an impurity (green ball) on the surface. (b)A topography of LiFeAs with STM settings (Vs = 25 mV, It = 250 pA). A few impurities, shownin bright contrast, are present on the sample surface. (c) STS taken by numerical differentiationat location of 1© in (b): the upper panel shows a It-Vs spectrum of LiFeAs at 4.2 K and thelower panel shows the dIt/dVs spectrum by numerical differentiating It-Vs spectrum above. (d)A tunneling conductance map g(r,E = eVs) at Vs= 8 meV was measured over the same areashown in topography (b).Fermi level (eVs = 0) and sample potential energy eVs. This assumption is based on thefact that a constant It is a constant integration of the LDOS (see Eq. 3.1) [8]. In a realtunnel junction, z0(r) should also be strongly correlated to the atomic wavefunction of443.1. Principle of STM and its operating modesthe tip apex and the local wavefunction on the sample surface. Thus the exact nature ofthe surface profile depends on the properties of the sample surface and the tip apex.The Scanning tunneling spectroscopy mode enables the investigation of theelectronic LDOS of the sample. By taking the derivative of It in Eq. 3.1 with respect toVs, one obtainsdIt(r, E = eVs)dVs= C × ρs(r, E = eVs), (3.3)where C ≡ 4pie~ ×ρt(EF )×e−2κd. The factor C is a constant when the values of d and r arefixed, namely a frozen tip height z and lateral position r. In this case, one can directlyevaluate the LDOS of the sample through measuring dIt/dVs, the tunneling conductance,while sweeping Vs.dIt/dVs spectra can be obtained either by numerical differentiation of It-Vs spectraor by a lock-in amplifier technique. In the latter case, Vs adopts a small AC modulationV ACs sin(ωt) with the condition that VACs  VDCs . Expanding It using a Taylor series [51,104]It = IDCt +dItdVsV ACs sin(ωt) +O((VACs )2), (3.4)one finds that the component of It at frequency ω is proportional to dIt/dVs. The twotechniques have different advantages and trade-offs. The lock-in technique allows one tochoose a frequency ω away from frequency ranges with electrical and mechanical noise,enhancing the measurement sensitivity. In this technique, the data acquisition speed islimited by the time constant of the Lock-in used in the experiment, which is of the order oftens of milliseconds per data point. The advantage offered by numerical differentiationof It is that the data acquisition rate is significantly faster. The speed is primarilylimited by the bandwidth of the STM electronics, namely the time for each (It, Vs) datapoint to be acquired, which is typically on the order of a few hundred microseconds.Nevertheless, the drawback of numerical differentiation is that one is more susceptibleto electrical and mechanical noise. In this thesis, both methods are used depending onconditions. Fig. 3.2(c) shows an example of It-Vs (upper panel) and its numerical dIt/dVs(lower panel) on a LiFeAs sample.The spectroscopic imaging mode combines the topography and STS modes, mea-453.2. Theory of STMsuring the spatial variation of the LDOS [29]. As a result, the SI mode provides a veryrich set of information. In this mode, It and Vs are set to record the topographic infor-mation, the same as the topography mode. Meanwhile, at each pixel of the topographicmap, the tip stops, freezes its position (r, z), and the sample bias Vs is swept to measuredIt/dVs. Thus spatial variation of the LDOS is recorded by the spatial mapping of thedIt/dVs at certain energies. The dIt/dVs map is also called a tunneling conductance mapg(r, E = eVs). In summary, a dIt/dVs map is a map measured by varying rat a constantenergy E in Eq. 3.3. Fig. 3.2(d) shows the tunneling conductance map g(r, E = 8 meV)measured in the area shown in Fig. 3.2(b). In this tunneling conductance map, prominentquasiparticle interference (QPI) patterns are visible near all impurities. The method toextract physical information from such an image will be discussed in Chapter 7.3.2 Theory of STMThe expression for the tunneling current, Eq. 3.1, is calculated from Fermi’s goldenrule [104, 105]. The theory was first proposed by Bardeen in 1961 for a planar tun-neling junction [106] and then developed by Tersoff and Hamann in 1985 [104] for a sharptip and flat sample geometry in STM. Fig. 3.3 gives a schematic drawing of the tunnel-ing events between the tip and the sample. Upon applying a small bias voltage Vs, theprobability of electrons elastically tunneling from the tip to the sample is given byP 1t =2pi~∞∫−∞|Mts|2 ρs(+ eVs) · [1− f(+ eVs)]︸ ︷︷ ︸#of sample empty states× ρt() · f()︸ ︷︷ ︸#of tip filled statesd, (3.5)and from the sample to the tip is given byP 2t =2pi~∞∫−∞|Mts|2 ρs(+ eVs) · f(+ eVs)︸ ︷︷ ︸#of sample filled states× ρt() · [1− f()]︸ ︷︷ ︸#of tip empty statesd,463.2. Theory of STMDOS DOSTip SampleεtunnelingbarrierItEFEF0-eVse-TipSampleφTipφSample-eVsFigure 3.3: Electron tunneling diagram. The colored regions indicate the occupied statesof the tip (red) and the sample (blue). Upon applying a bias voltage Vs to the sample, its Fermienergy EF shifts down to  = −eVs while EF of the tip is still at  = 0 (grounded). The grayregion is the tunneling barrier, usually vacuum, originated from the work functions of the tipφT ip and the sample φSample plus the potential −eVs applied to the sample. Electrons tunnelthrough the barrier from the occupied states in the tip to the unoccupied states in the sampleand vice versa, giving rise to a tunneling current It.where Mts is the tunneling matrix between the tip and the sample and f() is the Fermi-Dirac distribution function. The total current, It, can then be written asIt = 2e× (P1t − P2t )= 2e×2pi~∞∫−∞|Mts|2ρt() · ρs(+ eVs)× [f()− f(+ eVs)]d, (3.6)473.2. Theory of STMwhere the factor of 2 comes from the spin degeneracy. In the low temperature limitkBT  Vs, the Fermi-Dirac distribution is a sharp step function. Eq. 3.6 is then simplifiedtoIt = 2e×2pi~0∫−eVs|Mts|2ρt() · ρs(+ eV )d= 2e×2pi~eVs∫0|Mts|2ρt(− eVs) · ρs()d. (3.7)Eq. 3.7 shows that the tunneling current is proportional to the convolution of ρt and ρs.Because we are interested in probing the sample LDOS, we choose a tip material suchthat its LDOS in the energy range of interest is constant. This can be approximatelymet by some elemental metals and their alloys. If we consider ρt ≈ constant near EF ,Eq. 3.7 becomesIt =4pie~× ρt(EF )×eVs∫0|Mts|2 × ρs()d. (3.8)So far, we have not discussed the tunneling matrix Mts. It is this factor that actuallyenables the building of an STM. As illustrated in Fig. 3.3, the tunnel barrier is determinedby the work function φ of the tip and the sample. For most cases, the work functions forboth the tip and the sample are about 4 eV. In the limit of small Vs ∼ 100 meV whichsatisfies φT ip ≈ φSample  |φT ip−φSample+eVs|, Mts is given by the WKB approximationfor the transimission probability through a square barrier of height φ¯ = (φT ip+φSample)/2:|Mts|2 = e−2κd, (3.9)where κ =√2mφ¯/~ is on the order of ∼ 1 A˚−1 with φ¯ ∼ 4 eV. The exponentialdependence of |Mts| on the sample-tip distance d with a decay length 1/2κ ∼ 0.5 A˚allows extremely fine control of the sample-tip tunneling junction through a feedbacksystem.483.3. Remarks on Tersoff-Hamann’s theoryFinally, the tunneling current is given byIt(r, z, Vs) =4pie~× ρt(EF )× e−2κdEF+eVs∫EFρs(r, )d, (3.10)which is Eq. 3.1.3.3 Remarks on Tersoff-Hamann’s theoryTersoff and Hamann assumed an idealized spherical tip apex to derive Eq. 3.1 [104]. How-ever, in a real STM tunnel junction, the geometry of the tip is the most uncontrollableelement. The tip shape easily changes from time to time and the species of the last fewatoms are hard to determine. Consequently, the exact symmetry of the tip wavefunctionis usually unknown. This issue was investigated by Chen, who assumed that a singleatom at the tip apex is responsible for the interaction with the sample surface [107–109].The major correction to the model is the tunneling matrix Mts. Depending on the angu-lar momentum of the wavefunction in the last single atom, Mts has different sensitivitiesto the surface corrugations. This also explains why some tips give atomic resolution andsome others do not. Otherwise, the general approximation of Eq. 3.1 is valid for mostlow bias tunneling conditions.3.4 STM’s used in this thesis projectIn this study, two STM’s have been used. At UBC, we used a commercially availableSTM from SPS-CreaTec GmbH.In Germany, we used a home-built STM at the Max-Plank-Institute (MPI) [110]. BothSTM’s are constructed from the same core units as shown in Fig. 3.1.493.4. STM’s used in this thesis project3.4.1 CreaTec STM at UBCThe Createc LT-STM system is designed for working at the atomic scale, low temperatureand ultra-high vacuum (UHV). The STM head adopts a Besocke beetle design, ensuringvery high stability and small drift [111, 112]. Fig. 3.4 shows the picture of the STMhead. The full details of the STM are described by Zo¨phel [113]. During the period ofthis thesis project, I worked with my colleagues to maintain the instruments in a goodworking condition. In particular, I worked with Pinder Dosanjh to eliminate the 60 Hznoise, which was a key step for obtaining the high quality data set shown in this thesiswork.Thermal broadening, ∼3.5 kBT , and electrical noise, 60 Hz and its harmonics, aretwo primary factors limiting energy resolution. For the measurements of LiFeAs whichhas a Tc = 17 K (kBTc ∼ 1.5 meV), it is desirable to have sub 1 meV energy resolution.For this purpose, there are two major modifications made on this STM:• Lowering the base temperature. During the measurements, the STM headhangs freely from the inner cryostat with three springs, with minimal thermalcontact to the bath. This sets a base temperature of about 10 K. A lower basetemperature is desired to resolve the details of the superconducting gaps of LiFeAs.To achieve this, an additional thermal link from the helium bath was anchored tothe STM body, as shown in Fig. 3.4. After this modification, a base temperature of4.2 K was achieved for the STM, and this drops to 1.8 K if one pumps on the liquidhelium bath. (This work was completed before the start of this thesis project.)• Eliminating electrical noise. For an STM system with many electrical com-ponents connected, the It-Vs spectrum can easily pick up electrical noise from 60Hz AC power sources. A great effort had been made to eliminate electrical noiseto achieve maximum sensitivity. First, a proper grounding scheme was introducedto minimize the ground loop current which is one of the principal noise sources.Second, all the AC to DC voltage converters were checked. One of the problemswe had came from a leak of AC noise from an AC to DC voltage converter. Aftereliminating electrical noise sources, the STM produced very high quality data on503.4. STM’s used in this thesis projectSample holderSpringThermal linkFigure 3.4: STM head of the Createc system. The STM head is hanging from the cryostatby three springs (left picture). The details of the parts are described in Zo¨phel’s thesis [113]. Athermal link is connected between the STM head and the helium bath, as indicated on the rightpicture. The STM has a Zener diode attached on the head, enabling temperature dependentmeasurement.topographies, STS, and QPI which constitutes the major datasets shown in thisthesis. Fig. 3.5 shows the comparison of the dIt/dVs spectra taken by the Cre-atec STM at 4.2 K before and after eliminating electrical noise sources. They bothshow the superconducting gaps. However, the coherence peaks of the superconduct-ing gaps after eliminating noise (Fig. 3.5(b)) are much sharper than those before(Fig. 3.5(a)), confirming the significant improvement of the energy resolution. (Thiswork was completed by Pinder Dosanjh and me.)For each measurement, we used a home-made electrochemically etched tungsten tip.The tip was Ar+ sputtered, and thermally annealed under UHV conditions prior tomeasuring. LiFeAs single-crystal samples were cleaved in-situ at cryogenic temperaturebelow 20 K and inserted into the STM head operating under UHV with pressure P <513.4. STM’s used in this thesis project−10 −5 0 5 10Sample Bias (mV)dI t/dV s (arb. unit)−10 −5 0 5 10Sample Bias (mV)dI t /dVs  (arb. unit)(a) (b)Before eliminating electrical noise After eliminating electrical noiseFigure 3.5: Comparison of the dIt/dVs spectra before and after eliminating electricalnoise sources. (a) A dIt/dVs spectrum taken at 4.2 K by Createc STM before optimizing thesystem. (b) A dIt/dVs spectrum taken at 4.2 K by the same STM after eliminating 60 Hz andits higher harmonics noise. The coherence peaks after eliminating noise become significantlysharper than them in (a). The spectrum measured at 2 K (which is shown in Fig. 5.5) showsexplicitly the second gap at 3 meV, indicating the limiting factor for energy resolution primarilyorigins from the thermal broadening effect above 2 K.1 × 10−9 Torr. The results presented here are reproducible among different tips andsamples.3.4.2 Home-built STM at MPIThe home-built STM at MPI was designed and constructed by White et al. [110]. TheSTM head, whose body is made of sapphire, is a derivative of Pan’s head design, aimingfor a high level of stability [110]. The STM head is attached to a cryogenic vacuum insert,which is mounted in a commercial vapor shielded magnet dewar. The superconductingmagnet provides a 14 T magnetic field that is perpendicular to the sample cleavage plane.The insert is equipped with a 1 K pot, giving a base temperature of 1.6 K on the STMhead. The sample is cleaved at about 4 K and transferred to the STM head.Two types of tips were used in this STM. The first was an IrPt alloy tip and the otherwas a vanadium tip. Both types of tips are prepared by a wire cutter. Each tip is fieldemitted on gold before approaching to LiFeAs samples. In particular, a vanadium tip, asuperconducting materials with Tc = 5.38 K, was field emitted on gold with both voltage523.4. STM’s used in this thesis projectpolarities, resulting in a non-superconducting gold-covered metallic tip.We used the home-built STM at MPI for three reasons. First, its base temperature(1.6 K) is lower than it of the Createc STM (4.2 K). Even though the Createc STM canreach 1.8 K by pumping its liquid helium bath, this temperature is not stable for a longtime window (∼hours) because of a lack of temperature feedback control. Second, theSTM at MPI enables the study of samples in magnetic field.During the period I worked in Germany, I played a major role in data acquisition andinvolved partly the maintenance of the instrument.53Chapter 4Crystal growth of LiFeAsSamples are the foundation of experimental research in condensed matter physics. Ihave grown high purity LiFeAs single crystals in-house under the direction of Dr. Liang,who is a world leader in the growth of cuprate crystals. The in-house growth servestwo purposes. First, it is traditional for the Superconductivity Group at UBC to pro-duce its own high-quality single crystals, enabling us to bring the sample quality to adesired level. Second, in order to engineer specific impurities, it is crucial to controlthe impurity substitution concentrations, either one is studying the single impurity ef-fects (extremely low impurity-substituted samples) or the multiple-impurity interferenceeffects (high impurity-substituted samples).LiFeAs was first synthesized as far back as 1968 [114]. It did not attract much at-tention until the recent discovery of the iron-based high-Tc superconductors [19, 42, 43].As explained in the introduction, LiFeAs is a stoichiometric superconductor, enablingstudies of the superconducting state in the absence of chemical substitutions which tendto produce unwanted scattering. Furthermore, it cleaves between two weakly bonded Lilayers, resulting in a charge neutral surface without surface reconstruction, which is idealfor study by surface sensitive probes.In spite of its clean properties, LiFeAs is less studied than other iron-based com-pounds, owing to the complexities involved in the crystal growth and sample handling.First, LiFeAs contains arsenic, whose oxide is among the most toxic of materials. Second,lithium is very reactive and volatile at elevated temperatures. This causes great technicaldifficulties in the growth of LiFeAs crystals.LiFeAs crystals have been obtained using several growth methods. One of the earlygrowth methods is a Sn-flux method, carried out by Lee et al. [115]. Large plate-like sin-gle crystals were produced. However, the experimental data obtained on these crystals544.1. Synthesis of precursor materialsindicated relatively poor sample quality, possibly because of Sn substitution in LiFeAs, aside effect that had been observed in the growth of other iron pnictides via Sn-flux [116].Therefore, it was desirable to find a growth method without the presence of materialsother than Li, Fe, and As. The first attempt in this direction was made by Song et al.using the Bridgeman technique [117]. This method, yielding large and thick single crys-tals, indeed increased the sample quality. However, the growth requires sealing LiFeAswith an excess of Li in a tungsten capsule and heating it up to 1773 K. This is not onlytechnically challenging but also hazardous. Later, Morozov et al. developed a self-fluxmethod using LiAs as the flux [118]. In this method, the growth is carried out between1363 K and 873 K. The low growth temperatures make the containment of Li and Asmuch easier than the Bridgman method.We chose the self-flux technique from Morozov et al. because it was easier to set up,and it also had produced the best quality of samples among all existing methods. Themethod was further modified by us in order to grow high-purity crystals at low cost.In the growth, the toxic arsenic and reactive lithium are the two major factors to betaken care of. Hence, the entire room-temperature processing, including both precursormaterials synthesis and LiFeAs growth, were carried out in an Ar-filled glove-box. Thetube furnace used was also located in a fumehood.4.1 Synthesis of precursor materialsFor the considerations of safety, reproducibility and crystal quality, two precursor mate-rials are synthesized for the LiFeAs growth. They are Li3As and FeAs.4.1.1 Li3AsThe chemical reaction between lithium and arsenic is very violent. If they are directlyused for single crystal growth, the alumina crucible can be damaged by the sudden heatgenerated by the highly exothermic combination reaction of lithium and arsenic. Toavoid this situation, Li3As is pre-synthesized as a precursor material.To synthesize Li3As, vacuum sealing of lithium and arsenic is required to prevent554.1. Synthesis of precursor materialsoxidization. Fused quartz tubing is commonly used for sealing. However, Li reacts withquartz, causing cracks in the sealed quartz ampoule and hence leaking of arsenic. An ironcrucible, shown in Fig. 4.1(a), was fabricated to contain the lithium and arsenic. Usingiron as the crucible material serves the purpose of avoiding contamination in LiFeAssingle crystals.Li ingotAs powderfusedquartzironvacuumsealed(a) (b)(c)Figure 4.1: Synthesis of Li3As. (a) The iron crucible. Iron is used because it is an ingredientof LiFeAs. (b) The sealed quartz ampoule for synthesizing Li3As. When heating the ampoule,the lithium ingots melt and diffuse into the As powder at the bottom. (c) The temperatureprogram for synthesizing Li3As. The two dwells at 443 K and 473 K are just below and abovethe lithium melting temperature of 454 K.The synthesis procedures are the following: arsenic lumps (Alfa Aesar: purity 99.9999%)and lithium ingot (Alfa Aesar: purity 99.9%) were weighed in molar ratio 1:1. The ar-senic was ground and then placed at the bottom of the iron crucible. The lithium ingotwas placed on top of the arsenic powder (see Fig. 4.1(b)). Then the quartz ampoule wassealed under vacuum and inserted into the tube furnace. The temperature was gradually564.1. Synthesis of precursor materialsraised to 773 K at a ramp rate of 30 K/h, followed by a dwell time of 10 hours. Thedetailed temperature program is shown in Fig. 4.1(c).4.1.2 FeAsSam Spmle hFigure 4.2: Synthesis of FeAs. (a) The temperature program for synthesizing FeAs. Thedwell at 773 K, below the arsenic sublimation temperature 887 K, aims to pre-react the iron andarsenic mixture and hence reduce the internal pressure when passing the arsenic sublimationtemperature. (b) A batch of FeAs (after reaction) in the sealed quartz ampoule.FeAs is also pre-synthesized for safety considerations. Elemental As sublimes at 887K, well below the LiFeAs crystal growth temperature (∼ 1300 K). The pressure in thesealed quartz ampoule could become very high when the temperature is far above thearsenic sublimation temperature, potentially leading to explosion of the quartz ampoule.Pre-synthesis of FeAs can avoid this safety issue.To react arsenic and iron, As lumps (Alfa Aesar: purity 99.9999%) and Fe podwer(Alfa Aesar: purity 99.995%) were weighed in 1:1 ratio, mixed and ground using an agatemortar. The mixture was directly placed into a quartz tube and sealed in high vacuum (<10−5 torr). It was subsequently heated, in a tube furnace, to 773 K at a rate of 120 K/hand followed by a dwell time of 5 hours for a pre-reaction below the arsenic sublimation574.2. Growth of LiFeAs single crystalstemperature. Then the mixture was heated to 973 K at a rate of 30 K/h, kept at thistemperature for 20 hours, and then cooled down to room temperature. Fig. 4.2 showsthe detailed temperature program and the resulting FeAs.4.2 Growth of LiFeAs single crystalsLi3As and FeAs were mixed in a composition of 1:2 and sealed in a quartz ampoule under0.3 bar of argon gas. For each growth, a total of 1.9 grams of the precursor materialswas put inside a cylindrical alumina crucible which itself was placed inside a cylindricalmolybdenum crucible (Mo-crucible). The precursor materials were then heated accordingto the temperature program shown in Fig. 4.3(a). Thin millimeter sized platelet-like singlecrystals were extracted mechanically from the ingot (see Fig. 4.3(b)). All crystals grew ina layered morphology with thickness of the order of a few hundred micrometers. They areeasily cleaved along the ab-plane. Fig. 4.3(c) shows a LiFeAs single crystal after cleaving.Single crystals with different transition metal substitutions were also grown using thesame method. For Co-substitution, CoAs precursor was synthesized and used. For Mn,Ni, and Zn substitutions, metal powder could be used directly due to the low substitutionconcentration required.There are two modifications to the method first proposed by Morozov et al.. One isthe use of a home-made molybdenum crucible and the other is the improvement of thetemperature program for crystal growth.4.2.1 Modification #1: reusable molybdenum crucibleA Mo-crucible is used to protect the quartz ampoule from lithium vapor. Lithium vapor,from Li3As, can diffuse into fused quartz at high temperature, softening the quartzampoule. In addition, the diffusion of lithium produces changes in the thermal expansioncoefficient and therefore causes cracks when the quartz ampoule is cooled down. Thecommonly used method of protecting quartz ampoules is to use sealed crucibles, usuallymade of niobium or tantalum; the mixture of precursor materials is placed into theniobium crucible and welded under pressure of argon gas in an arc-melting facility. In584.2. Growth of LiFeAs single crystalstime (hour)(a)(b) (c)plate like single crystals11 mmFigure 4.3: Single crystal growth of LiFeAs. (a) The temperature program. The growthwindow (slow cooling) is between 1333 K and 873 K. (b) The ingot after growth. Platelet-likesingle crystals are randomly distributed inside the alumina crucible. (The alumina crucible wasbroken using a hammer) (c) A LiFeAs single crystal after cleavage. The typical single crystalsize is 3×3×0.2 mm3.this way, lithium vapor is confined by the niobium crucible. After the growth, one breaksthe niobium crucible to retrieve the crystals. There are two disadvantages to usingniobium crucibles. One is that a new niobium crucible is required for each growth. Theother is that the arc-melting facility is usually placed outside of the glovebox, so thatone has to expose the unsealed precursor materials to air for a short time.We have designed a new vapor-confining crucible. The crucible, made of molybdenum,uses an end cap to confine lithium vapor instead of welding, as shown in Fig. 4.4. Theuse of molybdenum is because it does not absorb lithium to form a Li-Mo alloy (see the594.2. Growth of LiFeAs single crystalsMo-crucible alumina crucibleMo-cap-1 Mo-cap-2quartz discsAr 0.3 barsealed(a)(b)fusedquartzMo-cruciblealumina crucibleMo-cap-1Mo-cap-2quartz discsFigure 4.4: Assembly of the crucibles in the LiFeAs growth. (a) The crucibles, caps andthree quartz discs. The two molybdenum caps are used to confine the lithium vapor. The quartzdiscs are used to absorb lithium vapor coming out of the alumina crucible. (b) Illustration ofthe assembly of crucibles and caps. For each new growth, only the alumina crucible and thequartz discs needed to be replaced. In this setup, the quartz ampoule is well protected fromlithium vapor.phase diagram in Fig. 4.5). This fact helps to keep the crucible free from corrosion aswell as confine lithium for crystal growth. The assembly for each growth is the following.The alumina crucible with precursor materials is placed inside the Mo-crucible. A smallMo-made cap plugs the end of the alumina crucible. Then a few quartz discs which canabsorb most of the lithium vapor in high temperature are placed next to the aluminacrucible. Finally, a fairly long cylinder of Mo is used to plug the Mo-crucible. This typeof Mo-crucible was reusable for more than twenty growths, reducing the cost of crucibles.604.2. Growth of LiFeAs single crystalsIn each growth, we only needed to replace the alumina crucible and the quartz sliceswhich are much less expensive than crucibles made of niobium.Figure 4.5: Li-Mo binary phase diagram [119]. Lithium and molybdenum do not reactto form alloys below 1615 K. Therefore, the only role of the Mo-crucible is to confine lithiumvapor. In turn, Mo-crucible is also free from corrosion, allowing it to be kept for multiple uses.4.2.2 Modification #2: temperature program for high puritysamplesCrystal quality is sensitive to the growth temperature program. The superconductingTc and the transition width of LiFeAs varies from group to group, even when the sameself-flux method is used, indicating different crystal quality [117, 118, 120]. The majorquantitative difference in the growth methods used by these groups is the temperatureprogram [115, 117, 118, 120]. We made an effort to find a growth temperature programthat could reproducibly yield high-quality samples.After systematic testing, we found that the temperature program, shown in Fig. 4.3(c),614.3. Magnetization measurementsconsistently produces high-quality LiFeAs crystals. The primary difference from previouswork is the addition of a long annealing process. In the crystal growth, before the ampoulecools to room temperature, it was annealed at 873 K for 8 hours and subsequently at 673K for 12 hours. After this process, the growth usually yields high-quality single crystals.The annealing process probably helps to reduce the number of defects, in particulardefects of the very light and mobile lithium. However, the exact role of the annealingprocess is still under investigation.Figure 4.6: LiFeAs crystals that have been grown. All crystals are preserved in a gloveboxfilled with inert argon gas. Each batch of LiFeAs crystals is kept with a piece of lithium ingotinside a glass bottle. The lithium ingot is used to protect LiFeAs crystals by absorbing water,oxygen, and nitrogen that enter into the glovebox accidentally.4.3 Magnetization measurementsBefore the STM measurement of the LiFeAs samples, we performed magnetization mea-surements using a Magnetic Properties Measurement System (MPMS) from QuantumDesign. In addition to establishing the value of Tc, the superconducting transition widthdetermined from magnetization measurements is a good measure of the crystal quality.The sharper the transition, the more homogeneous are the superconducting propertiesof the sample. Fig. 4.7 shows the zero-field-cooled magnetic susceptibility of an as-grownstoichiometric LiFeAs. T onsetc is 17.2 K and the transition width ∆Tc is 1 K. This narrowtransition width, the sharpest among all published results so far [115, 117, 118, 120],624.3. Magnetization measurementsindicates the high crystal quality.0 2 4 6 8 10 12 14 16 18 20T (K)0Magnetic MomentFigure 4.7: Magnetization measurement of LiFeAs: Magnetic susceptibility was measuredevery 0.2 K from 5 K to 20 K in a 1 Oe magnetic field with H⊥ab crystallographic direction. Thesuperconducting transition starts at 17.2 K. The transition width from 90% to 10%, indicatedby vertical dashed lines, is 1 K.63Chapter 5Properties of LiFeAs in a defect-freeareaLiFeAs is an ideal system to study the intrinsic superconducting properties of iron-basedsuperconductors. In this chapter, I will show that, from the results of STM measure-ments, LiFeAs is indeed a clean system. Therefore, it serves as a clean testbed for theunderstanding of high-Tc superconductivity in the iron-based family.Furthermore, dIt/dVs spectrum acquired at 2 K reveals two nodeless gaps, exclud-ing the possibility of d-wave pairing symmetry. A pronounced dip-hump structure justabove the coherence peaks indicates strong coupling to bosonic modes. The energy ofthe modes is consistent with the energy of magnetic excitations (spin fluctuations) de-tected by inelastic neutron scattering (INS) [57, 121], consistent with the expectation ofspin fluctuations as the pairing interaction [3]. Thus, the observation of the dip-humpstructure in LiFeAs indirectly implies s±-wave pairing symmetry resulting from super-conductivity driven by spin-fluctuation interaction [4, 6, 7].5.1 TopographyA topographic image of a 38×38 nm2 area is shown in Fig. 5.1. There are several types ofdefects visible. A statistical analysis is made on the observed defects native to the LiFeAssingle crystal. The density of native defects was found to be 0.0020± 0.0005 per LiFeAsformula unit. Such a low defect density further establishes the stoichiometric compositionof our LiFeAs crystals, indicating high-quality of the in-house grown samples.Fig. 5.2(a) shows an atomic resolution topographic image with size 6.8×6.8 nm2 (Vs =40 mV, It = 100 pA). The inter-atomic spacing of the square lattice is (3.74 ± 0.03) A˚.645.1. TopographyFigure 5.1: A 38 × 38 nm2 topographic image (Vs = 205 mV, It = 50 pA) of LiFeAsmeasured at 4.2 K. The bright and dark objects are defects in the surface. Note thatthe black dots are absent in a freshly cleaved surface and their number increases with time.They can also be removed through large bias voltage pulses (∼ 1000 mV). Therefore, they areatoms/molecules either adsorbed or desorbed during the measurement.In LiFeAs, cleavage occurs between two weakly bonded lithium layers [45], shown inFig. 5.2(b), resulting in a surface that consists of a lithium layer, with arsenic and theniron beneath. Fig. 5.2(c) shows the top-view after cleavage: the top-most lithium squarelattice (yellow block) and the arsenic square lattice (blue block, translated by [1/2a,1/2b]) have the same lattice constant a = 3.77 A˚ [122]. The iron square lattice (redblock) fills the edge centers of the arsenic lattice in the ab-plane, giving the latticeconstant a/√2 = 2.67 A˚. The observed atomic lattice is consistent with the periodicityof either lithium or arsenic at the surface, but not with the iron lattice underneath.From density functional theory, the DOS within a few hundred millivolts of the Fermilevel consists primarily of iron orbitals [53]. However, this does not necessary mean theiron lattice is being observed at these bias voltages. Because of the hybridization ofatomic orbitals, there is still a finite DOS on the lithium and arsenic ions. In addition,the tunneling tip sensitivity exponentially decays with increasing the tip-sample distancewith a decay length ∼ 0.5 A˚. Thus, even though the iron DOS dominates near the Fermilevel, the DOS on lithium and arsenic, through hybridization, still plays an essential role655.2. Local density of statesin the tunneling matrix because these atoms are closer to the tip (see Fig. 5.2(d)).(a)2 nmabLiFeAsa bc(b)(c)0 A, cleavage plane-0.7 A-2.2 Aac(d) lowhighFigure 5.2: Topography image of LiFeAs with atomic resolution. (a) 6.8 × 6.8 nm2atomic resolution topographic image (Vs = 40 mV, It = 100 pA), showing an inter-atomicspacing of 3.74±0.03 A˚. (b) Schematic of the crystal structure of LiFeAs. A cyan plane indicatesthe cleavage plane (see the text for details). (c) The lattice viewed in the ab-plane after cleavage.The lattice shown in (a) is either the lithium or arsenic lattice whose lattice constant is 3.77 A˚determined by neutron powder scattering method [122]. (d) The lattice viewed in the ac-plane.The top-most lithium lattice is followed by the arsenic lattice with 0.7 A˚ beneath and then theiron lattice 2.2 A˚ deep.5.2 Local density of statesFig. 5.3 shows a dIt/dVs spectrum, measured at a defect-free area, in a bias range from−550 mV to 550 mV. The spectrum was acquired at 4.2 K, at which temperature a∼ 6 meV superconducting gap is visible near the zero bias voltage (see red section inFig. 5.3). Irrespective of the opening of superconducting gaps, the Fermi level EF (thezero bias voltage) is located in a region of low local density of states (LDOS) enclosedby enhancements below about −250 meV and above 10 meV. The sharp increase of theLDOS just above the Fermi level reaches a plateau at 30 meV. Similar features havebeen explained by a surface state, as summarized in Ref. [29]. However, a surface state665.2. Local density of states−450 −350 −250 −150 −50 50 150 250 350 45000.020.040.06Sample Bias (mV)dIt/dVs (nS) EF0Figure 5.3: dIt/dVs spectrum of LiFeAs in a range from -550 mV to +550 mVmeasured at 4.2 K. The Fermi level, indicated by the dashed vertical line at the zero biasvoltage, locates at the center of two enhancements of the LDOS in the positive and negativebias voltages, as indicated by blue color. Superconducting gaps open at the Fermi level (redcolor).in LiFeAs is unlikely according to density functional theory [45, 46]. The overall U orV shaped background DOS is a generic feature of iron-based superconductors. Thisfeature has been attributed to the semimetallic character caused by a small overlap ofthe bottom of electron bands and the top of hole bands close to EF [29]. Recently, in theBaFe2−xCoxAs2 system, the filtering effects from surface Ba atoms was proposed to giverise to the asymmetric dip-like tunneling spectrum [123]. We should note that, unlikeLiFeAs, BaFe2−xCoxAs2 has a polar surface with surface reconstruction. Theoreticalsimulations incorporating the LiFeAs surface lattice will be required to test whether ornot the filtering effect applies to LiFeAs.The suppression of the LDOS around the Fermi level is reminiscent of the tunnelingspectra observed in some underdoped cuprates [124, 125]. This phenomenon also occursin NaFe1−xCoxAs compounds whose surface is also non-polar with no surface reconstruc-675.3. Homogeneitytion [126]. Zhou et al. showed that, as a function of doping, the dip pins to EF fromthe underdoped regime to the overdoped regime, followed by a sudden shift after fullsuppression of superconductivity [126]. In underdoped cuprates, this dip feature near EFhas been attributed to the opening of a pseudogap. However, the situation is still unclearfor LiFeAs and NaFeAs. This pseudogap-like feature in pnictides can be generated by theinteraction of local magnetic moments and itinerant electrons [127], or nematicity [128].However these theories are built on an ideal two dimensional iron lattice, ignoring the realcrystal lattice. An alternative understanding without invoking local magnetic moments ornematicity is the filter effects of surface atoms applied to BaFe2−xCoxAs2 materials [123].To understand the pinning of the DOS dip to EF , a theoretical model is required thatcan take account of the tunneling matrix for the actual cleaved LiFeAs surface.5.3 HomogeneityA core advantage of STM over other gap measurement techniques is its access to gapvariations on the nanoscale. It has been employed to produce gapmaps of many cupratesand iron pnictides. In Bi2Sr2CaCu2O8+x, a much-studied cuprate, local inhomogeneitiesin doping have been shown to result in strong variations in the gap magnitude. In mostiron-based superconductors, for which superconductivity is generally induced by chemicalsubstitution, modest nanoscale variations have been found in the gap magnitude [29, 129].LiFeAs is one of the few exceptions among high-Tc superconductors that shows a highlyhomogeneous gap magnitude.To study the homogeneity of the gap in LiFeAs, we recorded 400 × 400 dIt/dVsspectra within a bias range of ±15 mV over an area of 90× 90 nm2 (see the topographyin Fig. 5.4(a)). Fig. 5.4(b) shows the 150 measured dIt/dVs spectra along the 33 nm whiteline indicated in Fig. 5.4(a), demonstrating a striking homogeneity of the superconductinggap in defect-free areas. A gap map from the whole image is extracted from ∆hpp, whichis half the energy separation between apparent coherence peak maxima. ∆hpp generallyoverestimates the magnitude of nodeless gaps due to thermal broadening (here T = 4.2K). However, ∆hpp does allow for an analysis of the spatial variation, independent of685.3. Homogeneity5.75.96.16.36.533 nm(a) (b)∆hpp[meV](c)z [pm]5.7 5.9 6.1 6.3 6.501234∆hpp[meV]Frequency [104 ]Sample Bias [mV](d)010203040Figure 5.4: Homogeneity of superconducting properties. (a) 90 × 90 nm2 topographicimage (Vs = 15 mV, It = 100 pA) of LiFeAs. (b) 150 individual dI/dV spectra (T = 4.2 K)from the defect-free 33 nm long line marked in (a). (c) Gap map of the same region in (a). ∆hppcorresponds to half the energy separation between coherence peaks. Both, the gap map andthe histogram of ∆hpp shown in (d) reveal a high degree of homogeneity in the superconductingproperties in LiFeAs.any specific model. The gap map Fig. 5.4(c) and the corresponding histogram Fig. 5.4(d)indicate a consistent magnitude with a mean value ∆hpp= 6.07 meV and a small standarddeviation σ = 0.08 meV. Much of the variation comes from spectra near defects, where∆hpp is reduced to a minimum of 5.7 meV.The homogeneity of the superconducting gap measured in LiFeAs by dIt/dVs withσ∆/∆¯ ' 1.3% is in contrast to most other high temperature superconductors. For com-parison, the iron pnictide compound BaFe1.8Co0.2As2 was found to have a less homoge-neous gap with σ∆/∆¯ ' 12% [129]. An extreme example is the much-studied cuprate,BSCCO [130],where local inhomogeneities in doping result in strong variation in the gapmagnitude. Hence, free from surface reconstruction or disorder induced by local dopants,LiFeAs presents an ideal system for surface sensitive spectroscopic investigations.695.4. Superconducting gap5.4 Superconducting gapThe superconducting gaps of LiFeAs have been measured by many techniques. Recently,careful ARPES and STM measurements, including this work, found fully open gaps withgap amplitudes that vary from 2.5 meV to 6 meV on different pockets of the Fermisurface [32, 35, 48, 50]. In this section, the superconducting gaps measured throughtunneling spectra will be discussed.The observed homogeneity ensures that a spectrum at a defect-free area is a goodrepresentative of the intrinsic superconducting properties in LiFeAs. Fig. 5.5 shows thedIt/dVs spectrum in the vicinity of the superconducting gaps measured in a defect freearea. With the sample at 2 K, two nodeless gaps are clearly resolved with the coherencepeaks locating at about ±6 meV and ±3 meV, respectively, consistent with recentlyreported values obtained from STM measurements [35, 50].-15 -10 -5 0 5 10 150123Sample Bias (mV)dI t/dV s (nS)Figure 5.5: Superconducting gaps measured by dIt/dVs spectrum. A dIt/dVs spectrumon a defect-free area taken at 2 K. The blue dashed lines at ±6 meV and ±3 meV indicate thepositions of the coherence peaks of the large and small gaps.The two-gap superconducting excitation spectrum can be fitted within the frameworkof BCS, where the normalized quasiparticle density of states N̂(E) of a superconductoris defined as [1]N̂(E) =∑kNs(E(k))Nn(√E(k)2 −∆(k)2) (5.1)with the superconducting DOS, Ns, and normal states DOS, Nn, being a function ofthe Bogoliubov quasiparticle energy E(k) and the normal state single particle energy705.4. Superconducting gap√E(k)2 −∆(k)2, respectively. If measured by the tunneling method, this can be de-scribed by the phenomenological Dynes’ formula [131],N̂(eV ) =∑k∫Re[2∑i=1wi × (E − iΓ)×∂f(E−eV )∂E√(E − iΓ)2 − |∆i(k)|2]dE (5.2)where f(E − eV ) is the Fermi-Dirac distribution function, wi is the weight of the con-tribution from the ith gap with the constraint of w1 + w2 = 1,7 and Γ is an effectivedamping term describing the lifetime of the quasiparticles. Note that wi is influenced bythe tunneling matrix elements between the tip and sample. Hence, the fitting to Eq. 5.2provides only a qualitative assessment of wi, the superconducting DOS from band i.To obtain an actual fit, I employ a few constraints and assumptions. First, the 2 KdIt/dVs spectrum is fitted over a bias range from -6.8 mV to 6.8 mV. This procedure ismotivated by the appearance of the dip-hump structure above the superconducting gapswhich cannot be properly described by the weak-coupling Dynes’ formula. However,the fit inside the superconducting gaps is valid as long as the superconducting gaps arenot strongly energy-dependent in the small energy window [-6.8 meV, 6.8 meV]. Second,the normal state DOS is assumed to be linear over the small energy range examined:Nn(E) = a + b × E. This assumption can be rationalized from the fact that Eq. 5.1is the ratio between the DOS of Bogoliubov quasiparticles at E(k) and normal statequasiparticles at√E(k)2 −∆(k)2. Right near the gap coherence peaks (∼ 6 meV forthe large gap), only the DOS of the normal state quasiparticles at EF is involved. Alinear function is therefore a good approximation for the fit within the selected energywindow. Third, an energy dependent damping term Γ(E) = αE is used to properlyrepresent zero DOS at EF [132]. Fig. 5.6(a) shows the fit with a constant Γ(E) = 0.5meV, which fails in the region of zero bias voltage.8The Dynes’ formula is sensitive to the gap amplitude |∆(k)|, not the sign. There-fore, two different ∆(k) models are considered to represent the gap amplitudes and gapanisotropy. One model consists of two isotropic gaps, yielding ∆1 = 5.33 ± 0.10 meV,7Namely, wi is the fraction of DOS in the i-th band.8Note that the choice of the linear scattering rate Γ(E) = αE was not because of some intrinsicphysics, but simply for the purpose of a proper fit to the experimental spectra.715.4. Superconducting gap-15 -10 -5 0 5 10 150123Sample Bias (mV)dI t/dV s (nS)-15 -10 -5 0 5 10 150123Sample Bias (mV)dI t/dV s (nS)-15 -10 -5 0 5 10 150123Sample Bias (mV)dI t/dV s (nS)(a) (b) (a)Figure 5.6: Fit to the superconducting gaps. The dashed vertical lines indicate the fittingwindow [-6.8 mV, 6.8 mV]. (a) A fit using two isotropic gaps and a constant Γ = 0.5 meV. Thefitting curve near zero bias voltage does not reproduce the observed spectrum. (b) A fit usingtwo isotropic gaps and Γ = αE. (c) A fit using two anisotropic gaps and Γ = αE. The lattertwo fits reproduce the experimental spectrum very well within the fitting window.w1 α a btwo isotropic gaps model 0.890±0.005 0.131±0.002 8.353±0.04 -0.081±0.004two anisotropic gaps model 0.870±0.005 0.080±0.005 8.185±0.04 -0.0282±0.004Table 5.1: Values of fitting parameters for the two-isotropic-gaps model and two-anisotropic-gaps model. The errors here are determined by the 95% confidence boundsresulting from the fitting program.∆2 = 2.50 ± 0.15 meV. The other model consists of two anisotropic gaps with four foldsymmetry, yielding ∆1 = 5.33 × (1 + 0.09 × cos(4θ)) ± 0.1 meV (∆max1 = 5.8 meV), ∆2= 2.50 × (1 + 0.20 × cos(4θ′)) ± 0.20 meV. The four fold symmetry is chosen accordingto s++ and s±-wave pairing symmetries. However, the fitting results are independent ofthe selection of cos(nθ) as long as n is an integer. The values of the fitting parametersare listed in Table. 5.1.Both gap models fit the 2 K dIt/dVs spectrum very well within the fitting range (seeFigs. 5.6(b)&(c)). However, both fittings clearly fail to represent the measured spectraoutside the fitting range due to additional above-gap features, which will be discussed inSec. 5.6.The gap magnitudes obtained from fitting agree well with the gap amplitudes on thetwo hole pockets at the Brillouin zone center, as determined by ARPES [32]. In addition,quasiparticle interference measurement by Allen et al. confirms that the two gaps openat hole like bands [35]. Therefore, STM measures the LDOS of the two hole pockets at725.5. Temperature dependence of the superconducting gapsthe Brillouin zone center. Tunneling into the two electron pockets, containing the othertwo gaps, is expected to be strongly suppressed because of the larger in-plane momentum|k||| [29, 104]. The small anisotropy factors obtained from the fit to two anisotropic gapsalso agree well with ARPES results [32], reinforcing the consistency of the two surface-sensitive measurements in LiFeAs.5.5 Temperature dependence of thesuperconducting gaps(a)−20 −15 −10 −5 0 5 10 15 200123420K 17.5K 17K 16.5K 16K 15K 14K 13K12K 11K 10K 8K 6K 4.2K 2KdI t/dV s (nS)0 5 10 15 200246T (K)∆(meV)0 5 10 15 200246T (K)∆(meV)(b) (c)Figure 5.7: Superconducting gaps at elevated temperatures. (a) The tunneling spectrabetween 2 and 20 K in a defect-free area. (b) The gap amplitudes (median value) determinedby fitting of a two-isotropic-gaps model. (c) The gap amplitudes (median value) determined byfitting of a two-anisotropic-gaps model. The errorbars are determined from the 95% confidencebounds resulting from the fitting program. The large gap determined by fitting to either ofthe two models generally follows the temperature dependence predicted by the mean-field BCStheory (dashed black line in (b) and (c)).Fig. 5.7(a) shows the temperature dependence of the dIt/dVs spectra from 2 K to 20 Kin the same defect-free region. Each spectrum is the average of 16 spectra acquired froma 4×4 nm2 area. The superconducting gaps are visible up to 16.5 K and disappear at 17735.6. Dip-hump structure above the superconducting gapsK. The bulk Tc of 17 K, as determined by magnetization measurement (see Sec. 4.3), is inagreement with the surface critical temperature, demonstrating agreement between sur-face and bulk properties. The temperature dependence of the gap amplitudes, extractedfrom fitting to the two models, are plotted in Fig. 5.7(b) and (c). The two models agreevery well with each other. The temperature dependence of ∆1 follows BCS theory andthe gap closes at the bulk Tc value. These results and the apparent absence of a surfacestate or electronic reconstruction [45, 46] suggest that the surface behavior echoes thebulk.As shown in Figs. 5.7(b) and (c), ∆2 seems to decrease to zero faster approaching Tc.This independent temperature dependence of ∆1 and ∆2 is reminiscent of the case forweak interband coupling in multiband superconductors [133]. Our result implies that thecoupling constants satisfy λ12  λ11, λ22, where λ12 is the interband coupling strengthbetween band 1 and 2, and λ11 and λ22 are the intraband coupling strength withinband 1 and 2, respectively. Nevertheless, the accurate gap amplitude of ∆2 is obscuredby thermal broadening at temperatures above 10 K. To confirm the weak interbandcoupling effect, it requires a combination of techniques to better determine ∆2 at elevatedtemperatures.5.6 Dip-hump structure above the superconductinggapsThe reduced gap ratio is a measure of the coupling strength (see Sec. 2.2.6). In LiFeAs,the reduced gaps ratios are 2∆max1kBTc= 7.9 and 2∆max2kBTc= 4.0. Here, the maximum gapamplitudes, ∆max1 = 5.81 meV and ∆max2 = 3 meV, are taken from the fitting of thetwo-anisotropic-gaps model, because this model is better fit to the 2 K spectrum andmore consistent with gap values and anisotropies measured by other techniques [32, 35].Both ratios are larger than the weak-coupling s-wave BCS value 3.53. In particular,2∆max1kBTc= 7.9 is close to typical values in cuprates [8], placing LiFeAs in the strong-couplingregime.745.6. Dip-hump structure above the superconducting gapsIn strong-coupling superconductors, an above-gap structure always shows up withthe opening of the superconducting gaps in the LDOS. The above-gap structure ap-pears as a shoulder-dip in strong-coupling phonon-mediated superconductors, as shownin Fig. 5.8(b), and a dip-hump in strong-coupling cuprates, as shown in Fig. 5.8(c). InLiFeAs, the failure of the weak-coupling Dynes’ formula outside the superconductinggaps suggests the inadequacy of the weak-coupling picture.10-Δ Δ0EnergyNormalized DOS1010(a)-Δ Δ0Energy -Δ Δ0Energy(b) (c)Δ+Ω1Δ+Ω2Figure 5.8: Examples of above-gap structures in superconductors. (a) Example ofthe LDOS of a weak-coupling phonon-mediated superconductor [1]. No obvious above super-conducting gap structure is resolved. (b) Example of the LDOS of a strong-coupling phonon-mediated superconductor, for example lead [134]. Two shoulder-like structures, as indicatedby green arrows, appear at energies corresponding to two phonon modes shifted by the su-perconducting gap, E1 = ∆ + Ω1phonon and E2 = ∆ + Ω2phonon. (c) Example of the LDOS ofa strong-coupling high-Tc cuprate superconductor, for example Bi2Sr2Ca2Cu3O10+δ [135]. Apronounced dip-hump structure, indicated by the green ovals, appears just above the supercon-ducting gap.Fig. 5.9 shows a comparison between the superconducting spectrum (black) measuredat T = 1.6 K and the normal state spectrum (blue) measured at T = 12 K and B = 10T. Here, the normal state was measured in high magnetic field to reduce the thermalbroadening effect, using the fact that Tc of LiFeAs is suppressed from 17 K to 12 K in 10 Tmagnetic field. We note that there is no evident difference between the dIt/dVs spectrumacquired at 17 K and the dIt/dVs spectrum acquired 12 K in 10 T magnetic field. In thesuperconducting state, the LDOS shows two kinks symmetrically at ±18 meV followedby two dips at ±10 mV as approaching EF . To confirm that the kinks disappear onlywith the suppression of superconductivity, we compare the superconducting spectrum755.6. Dip-hump structure above the superconducting gaps−40 −30 −20 −10 0 10 20 30 4000.20.40.60.81Sample Bias  (mV)Tunnelling Conductance (nS)Figure 5.9: The superconducting dIt/dVs spectrum (black) (T = 1.6 K, B = 0 T)plots over the normal state dIt/dVs spectrum (blue) (T = 12 K, B = 10 T). Bycomparison, we can see hump structures at ±18 mV followed by dip structures at ±10 mV asapproaching EF from both sides. In particular, the hump structures appear as kinks in theLDOS as indicated by the black arrows.taken at 12 K and 0 T with the normal-state spectrum taken at 12 K and 10 T, asshown in Fig. 5.10. At the same temperature, the kinks at ±18 meV are present in thesuperconducting spectrum but not in the normal-state spectrum, indicating that theyarise with the appearance of superconductivity.The appearance of kinks in the LDOS usually interpreted in terms of the couplingbetween quasiparticles and certain bosonic modes, such as electron-phonon coupling. Theenergies of the kinks indicate the energies of the bosonic modes. The kinks at 18 meVin the superconducting state of LiFeAs indicate the strong coupling between Bogoliubovquasiparticles and certain bosonic modes.To more carefully examine the structure surrounding the superconducting gaps, thesuperconducting dIt/dVs spectrum is normalized by the normal state spectrum accordingto Eq. 5.1, and shown in Fig. 5.11. The above-gap structure in the normalized spec-trum consists of two dips at ±10 meV, followed by two broad humps that peak atabout ±18 meV, at the energies of the kinks in the un-normalized spectrum. Thesefeatures can be characterized by three different energies: ED ' 10 meV, EI ' 12 meV,and EH ' 18 meV, the energy of the dip, inflection point between dip and hump, and765.6. Dip-hump structure above the superconducting gaps−40 −30 −20 −10 0 10 20 30 4000.20.40.60.81Sample Bias (mV)Tunnelling Conductance (nS)Figure 5.10: The superconducting dIt/dVs spectrum (red), taken at T = 12 K, B = 0T, is plotted with the normal state dIt/dVs spectrum (blue), taken at T = 12 K, B =10 T. At the same temperature, namely with the same thermal broadening effect, the kinks isstill visible in the superconducting spectrum, as indicated by the black arrows. This observationconfirms that the kink features are associated with the opening of the superconducting gaps.the hump respectively.The features observed in LiFeAs, characterized by a dip between ∆max1 and 2∆max1 fol-lowed by a broader hump, bear striking resemblance to those observed in the cuprates [51,136–138]. In past studies on superconducting cuprate materials, these features havebeen attributed to several different origins due to the large variety of competing ef-fects at similar energy scales. These include inelastic tunneling effects [139, 140], bandstructure effects [141], although most attribute these spectral features to either pair-ing [135–137, 142–144] or non-pairing boson interactions [145]. Other explanations havebeen based on the pseudogap observed in the cuprates [146, 147], but these are likelyabsent in the iron pnictides[29]. In our data, the fact that these features are present onlyin the superconducting state indicates that inelastic tunneling effects and band structureeffects, which induce changes of LDOS in both the normal state and the superconductingstate, are unlikely to be the cause.Thus, we turn our attention to possible boson interactions. In the well-establishedframework of phonon-based pairing in an s-wave superconductor, the energy dependenceof the gap leads to an initial peak or shoulder at E = ∆ + Ωphonon due to increased775.6. Dip-hump structure above the superconducting gaps00.511.522.53Normalized conductance−40 −30 −20 −10 0 10 20 30 40−1.5−1−0.500.51Sample Bias (mV)dI t2 /dV s2 12 mV15 mV(Normalized)EHEDEIFigure 5.11: Dip-hump structure in LiFeAs. The upper panel is the normalized dIt/dVsspectrum by using Eq. 5.1, not the direct ratio, between two spectra shown in Fig. 5.9. Dip-hump features are clearly visible at energies E > ∆max1 . The lower panel shows the derivativeof the normalized dIt/dVs spectrum, which gives the inflection points (EI) between the dip andhump.pairing strength as one approaches the phonon mode at energy Ωphonon, followed by adrop owing to the strong enhancement of the phonon emission rate when passing the modeenergy (see Fig. 5.8(b)) [134]. Our spectra do not show the initial peak or the shoulder.Regardless, features caused by coupling to a bosonic mode are expected to appear at themode energy shifted by the gap (∆+Ωboson) [136, 148]. Given the differences between ourspectra and the classic phonon coupling case, combined with the lack of strong featuresin the phonon spectrum of LiFeAs below ∼ 14 meV [149, 150],9 it seems unlikely thatthe features we observe arise from electron-phonon coupling because the bosonic modeenergies Ωboson = E − ∆ are all below 12 meV (ED-∆max1 = 4.2 meV, EI-∆max1 = 6.29Li et al. calculated the phonon spectra by density functional theory in which only smooth acousticmodes are present below 14 meV [149]. Um et al. experimentally measured the optical phonons whichare not present below 14 meV [150], consistent with theoretical calculations.785.7. SummarymeV, EH-∆max1 = 12.2 meV).Spin-fluctuation mediated superconductivity has been suggested in the pnictides [4],a theory also supported by dIt/dVs data of SmFeAsO1−xFx [151]. Indeed, recent reportsfrom neutron scattering have indicated a broad magnetic excitation peaked around 5-10meV [57, 121], corresponding well with the energy between the dip and the hump shiftedby the large gap, ED −∆max1 ' 4.2 meV and EH −∆max1 ' 12.2 meV.The observed energy scale of spin-fluctuation modes is consistent with the dip-humpstructure in LiFeAs’s LDOS [48], indicating that this mode may be related to the pairingglue [3], in analogy to the case of phonon-mediated pairing. If the spin-fluctuation inter-action is the pairing interaction, LiFeAs would possess s±-wave pairing symmetry [4, 63].Therefore, the consistency of the energies of a spin-fluctuation mode and the dip-humpstructure implies an s±-wave pairing symmetry mediated by spin-fluctuation in LiFeAs.There are, however, open questions regarding this interpretation. First, the observedspin resonance is rather broad in comparison to the sharp LDOS modulations [48]. Sec-ond, in the case of the high-Tc cuprates it was shown that LDOS modulation in the formof a dip-hump feature is indicative of pair breaking modes within the Eliashberg formal-ism [152]. Third, no corresponding feature has been observed in the ARPES spectra aswould be expected if the mode was coupling strongly to carriers. In light of these issueswe conclude that, while the existence of a dip-hump structure is indicative of a strongcoupling to spin-fluctuation bosonic modes at the same energy range, and therefore a signchange in the order parameter, its role in establishing superconductivity is not fully un-derstood, and its presence can only be considered as circumstantial evidence for s±-wavepairing symmetry.5.7 SummaryThe characteristics of LiFeAs presented here demonstrate that this material providesa comparatively simple system in which to study high-Tc superconductivity. In starkcontrast to most cuprate materials, the superconducting gap remains remarkably homo-geneous over large areas in LiFeAs. Additionally, the presence of a non-polar cleaved795.7. Summarysurface, one that does not reconstruct and accurately represents the bulk properties,makes it ideal for surface sensitive studies.Although this material shows multiple superconducting gaps, the gaps are withoutnodes, and exhibit a temperature dependence as predicted by mean-field BCS theory.The observation of two nodeless gaps, that open on the hole pockets in the Brillouin zonecenter, exclude the possibility of d-wave pairing symmetry that would require symmetryprotected nodes on these pockets.In addition, this material shows all the signs of strong coupling with a relatively largereduced-gap-ratio of 7.9, and strong above-gap features, corresponding closely in energywith a magnetic resonance recently reported, are consistent with the prediction of a spin-fluctuation mediated pairing with the s±-wave pairing symmetry in LiFeAs. However,unambiguous assignment of the spectral features will require a proper microscopic theoryincluding the pairing symmetry of LiFeAs.More intriguingly, the cuprates and the iron pnictides share two features that suggesta common origin for the dip-hump structure. Both have large reduced-gap ratios. Amagnetic mode develops in the spin-fluctuation spectrum in both systems when thesuperconducting gap appears below Tc [15, 153]. The energy scale of the spectroscopicfeatures found here also draws a parallel to the cuprates. LiFeAs, as opposed to mostother high-Tc compounds, offers a clearer view of the quasiparticle phenomenology andcan serve as a testbed for the understanding of high-Tc superconductivity.80Chapter 6Bound states of iron-site defectsFrom the previous chapter, we learned that the properties at defect-free regions did notprovide direct information on the pairing phase. Therefore, it is desirable to slightlyperturb the system and measure the response, since perturbations can lead to new viewsof the system, uncovering the nature of the ground state. Here, we use point defectsas the perturbations to examine the pairing phase. Defects can cause pair breakingand therefore create in-gap bound states. The energies at which the superconductingbound states are located are determined by the phase of the superconducting orderparameter [154–158]. Thus, defect bound states offer direct evidence of the sign changein the pairing symmetry.In this chapter, I will show the bound states of Fe-site defects in LiFeAs measuredby dIt/dVs spectra. Defect bound states are LDOS peaks localized at the defect sites,present as high-intensity peaks in the dIt/dVs tunneling spectra. Five Fe-site defects arestudied: two of them are native defects in as-grown LiFeAs single crystals and the otherthree are manganese, cobalt, and nickel that were deliberately incorporated into the ironlattice. In contrast to the defects at the other lattice sites [49], the Fe-site defects allcreate bound states at the gap edge of the small superconducting gap ∆2 = 3 meV.According to theoretical predictions, our observation indicates that a sign change occursin the superconducting gaps, consistent with s±-wave pairing symmetry.Fe-site defects are measured at MPI using the STM with a 1.6 K base temperature.10Except where otherwise declared in the text, the data shown in this chapter were acquiredat T = 1.6 K. LiFeAs single crystals with engineered impurities were grown with a self-flux method as described in Chapter 4. In order to study single impurity effects, theamount of impurities was deliberately controlled in the crystal growth. The samples10For the details of the instrument, please refer to Sec. 3.4.2 and Ref. [110].816.1. Impurity physics for identifying the pairing symmetryshown in this chapter have 3∼5 engineered impurities per one thousand iron lattice sites.6.1 Impurity physics for identifying the pairingsymmetryImpurity bound states have been used previously in identifying the nature of the pair-ing state in superconductors. For a conventional superconductor with an s-wave pairingsymmetry, non-magnetic impurities have no impact inside the superconducting gap [154],while magnetic defects cause pair breaking and in-gap states [98, 159, 160]. In contrast,non-magnetic impurities also lead to pair breaking and create in-gap bound states ifthere is a sign change in the pairing symmetry, a result of time-reversal symmetry break-ing [156–158]. For example, both potential and magnetic impurities can induce in-gapstates in d-wave [161–164] and multi-band sign reversal s±-wave superconductors [156–158, 165].Since the superconducting gaps are nodeless, the most probable candidates of pairingsymmetry in LiFeAs are s++-wave and s±-wave. s++-wave has the same sign over thewhole Brillouin zone, whereas s±-wave has a sign change between the hole pockets and theelectron pockets of the Fermi surface. In the iron-pnictide superconductors, theoreticalcalculations show distinct bound-state features between s++-wave and s±-wave [156–158].With s++-wave, only magnetic impurities can create in-gap bound states. With s±-wave, both non-magnetic impurities and magnetic impurities can produce in-gap boundstates. In particular, in LiFeAs–as summarized in Fig. 6.1 for the case of s±-wave pairingsymmetry–non-magnetic defects can produce bound states that are either pinned to thegap-edge of the small superconducting gap or inside the superconducting gaps [158, 166]depending on the strength of the defect scattering potential, while magnetic defectsalways generate in-gap bound states [156]. Therefore, the study of impurity bound statesin LiFeAs can help determine whether or not there is a sign change in k-space in itssuperconducting order parameter.826.2. Fe-site defects(a) (b) (c)defect-free areadefect siteweak nonmagneticVimp strong nonmagneticVimp magneticLDOSLDOSLDOSEnergy 0 Δ2 Δ1-Δ2-Δ1 0 Δ2 Δ1-Δ2-Δ1 0 Δ2 Δ1-Δ2-Δ1Energy Energy Figure 6.1: Schematic examples of the theoretical predictions for impurity boundstates in LiFeAs with s±-wave pairing symmetry. The blue spectrum is the LDOSwithout impurities. Each red spectrum is the LDOS on the nearest neighbor site of the defectsite. (a) shows the effect of an impurity with a relatively weak non-magnetic potential Vimp < 1eV [166]. The impurity bound state pins to the gap-edge of the smaller superconducting gap∆2, as shown in the enhanced LDOS peak at E = ∆2. (b) shows the effects of a non-magneticdefect with a strong defect potential Vimp > 1 eV [166]. In-gap bound states are present asstrong LDOS peaks at E < ∆2. (c) shows the effects of magnetic defects, which always tendto create in-gap bound states [156].6.2 Fe-site defectsFe-site defects in LiFeAs are characterized by a bow-tie like geometrical shape in STMtopographies [49]. Fig. 6.2 shows some Fe-site defects in a Mn-substituted LiFeAs singlecrystal. Here, we use “defect” because the native defects could be either impurities orvacancies in the iron lattice.The bow-tie like shape originates from the lattice structure of LiFeAs. Each Fe isnested at the center of an arsenic tetrahedron, as shown in Fig. 6.3(a). There are twonearest-neighboring (NN) arsenic atoms above the iron site and two NN arsenic atomsbeneath it with 90◦ rotation. When an impurity is present on an iron lattice site, thefour strongly polarizable NN arsenic atoms would be most significantly affected throughcharge transfer along Fe-As bonds, as depicted in Fig. 6.3(b).STM topography provides a weighted 2D projection of the 3D crystal because thetip is most sensitive to corrugation on the upper rather than the buried planes. Thus836.2. Fe-site defectsFigure 6.2: Example of Fe-site defects in the topography of a Mn-substituted LiFeAssample. The topography is measured with It = 300 pA, Vs = 50 mV, T = 12 K, and a magneticfield B = 10 T. Three types of Fe-site defects are present in the surface, representative examplesof which are highlighted in the red circles. They all have a two-fold bow-tie-like shape with twoorientations, horizontal or vertical. A-type is an engineered manganese impurity, and B- andC-type are native Fe-D2 and Fe-D2-2 defects, respectively. There are also defects at the latticesites other than iron, which are described by Grothe et al. in Ref [49].a STM tip primarily images the top three lithium, arsenic, and iron layers, as shown inFig. 6.3(d). In this projected lattice, each iron site has a two-fold symmetry. Hence, anFe-site defect appears as a two-fold D2-symmetry shape in topographic images and thestrongly corrugated lobes are most likely aligning along the Fe-As bonding directions.The D2 symmetry discussed here is the Scho¨nfließ notation adapted for two dimensionalpoint groups. In addition, the defect is present in two orientations that are perpendicularto each other because of two inequivalent iron sites, as shown in Fig. 6.3(d).846.2. Fe-site defectsa bc(c) STMtipLiFeAs(d)ab(a)Defect(b)a bcFigure 6.3: Geometrical shape of an Fe-site impurity. (a) shows the structure of theFe-As tetrahedron: an iron is in the center of the tetrahedron formed by four arsenic atomswith two arsenic atoms above and the other two below. (b) shows the charge transfer effect dueto an Fe-site defect. (c) In an STM measurement, the tip placed on top of the sample surface ismost sensitive to the LDOS from only top layers of atoms. (d) shows the lattice as seen by anSTM tip. Only the top lithium, arsenic and iron lattice layers are shown. Two example Fe-sitedefects, depicted as black dots with red lobes, are shown in two iron sites.6.2.1 Native Fe-site defectsIn an as-grown LiFeAs single crystal, two types of Fe-site defects are observed: the Fe-D2defect (Fig. 6.4(a)) and the Fe-D2-2 defect (Fig. 6.4(c)) [49]. An Fe-D2 defect exhibits twobright lobes, while an Fe-D2-2 defect appears as a deep trench with two dim lobes.Fig. 6.4(b) shows the dIt/dVs spectra of an Fe-D2 defect. The blue and red spectraare taken on the defect center and lobe, respectively. For reference, the spectrum ata defect-free area is also plotted in black, which shows two superconducting gaps. Forsimplicity, we use the energies of the coherence peaks as the gap amplitudes ∆1 = 6 meVand ∆2 = 3 meV, which are consistent with fitted gap amplitudes within error bars.856.2. Fe-site defectsComparing to the black spectrum, the red and blue spectra show a pronounced peakat the gap edge of ∆2. This is a bound state corresponding to the case of a relativelyweak scattering potential according to Ref. [166]. Here we note that the spectrum onthe bright lobes has the strongest bound state feature. This fact is also true for otherFe-site defects. Therefore, for the other four Fe-site defects, only the dIt/dVs spectra onthe defect lobes will be shown and discussed.Before showing the data of other defects, we should note that, for each defect, thereference and on-defect dIt/dVs spectra were measured with the same tip. For differentdefects, the tip varies, giving slightly different weight of the superconducting gaps.The dIt/dVs spectrum on an Fe-D2-2 (the red curve in Fig. 6.4(d)) also shows anenhanced spectral weight right at the gap edge of ∆2. However, the bound state of anFe-D2-2 is weaker than that of an Fe-D2 defect. Comparing the two defects (types B andC) in Fig. 6.2, we find that an Fe-D2 defect is much brighter than Fe-D2-2. This indicatesthat the Fe-D2 defect has a stronger impact on the LDOS, and therefore probably has alarger scattering potential, inducing a stronger bound state than an Fe-D2-2 defect.The bound states of the two native Fe-site defects pin to the energy of ∆2. This isconsistent with the scenario of s±-wave with weak non-magnetic scatterers [158, 166].6.2.2 Engineered Fe-site impuritiesThe disadvantage of native defects is that we are blind to their nature. From their boundstates, we infer that they are most likely non-magnetic defects. To further examine theeffect of Fe-site defects, we engineered three known impurities, manganese, cobalt andnickel, into the iron lattice in LiFeAs. Figs. 6.5(a), 6.5(c) and 6.5(e) show the topographicimages of manganese, cobalt and nickel impurities, respectively. All engineered impuritieshave the characteristic bow-tie shape, similar to the native Fe-site defects.The pair-breaking effects of the engineered impurities are weaker than that of thenative Fe-site defects. As shown in Fig. 6.5(b), a single manganese impurity does notshow evident bound states within the energy resolution of our instrument. The majorinfluence on the superconducting gaps is the slight suppression of the coherence peaks.Fig. 6.5(d) shows the effect of a cobalt impurity. A tiny bound state is present at the866.2. Fe-site defects(a) (b)−10 0 1000.050.10.150.20.250.3dIt/dVs  (nS) Sample Bias (mV)Sample Bias (mV)highlow2 nm2 nmhighlow −10 0 1000.10.20.30.4dIt/dVs  (nS) (c) (d)3 6 Δ1Δ2Figure 6.4: Superconducting bound states of native Fe-site defects. (a) Topographyof an Fe-D2 defect (It = 50 pA, Vs = 25 mV). (b) dIt/dVs spectra taken on the defect site: theblue and red spectra are taken at the blue and red dots in (a), respectively. The black curveis a reference spectrum measured in a defect-free area of the same sample, with the same tip.Two superconducting gaps ∆1 = 6 meV and ∆2 = 3 meV are indicated by the dashed lines.(c) Topography of an Fe-D2-2 defect (It = 100 pA, Vs = 50 mV). (d) The dIt/dVs spectrum onthe defect (red) together with the reference spectrum (black).gap-edge of the small gap, as highlighted in the green circle in Fig. 6.5(d). For the caseof a single nickel impurity, a clear bound state shows up in the dIt/dVs spectrum, ashighlighted by the green ellipse in Fig. 6.5(f). In the periodic table, manganese andcobalt are the nearest neighbors of iron, and nickel is the second nearest neighbor ofiron. Therefore, the impurity potential probably becomes stronger as one moves frommanganese and cobalt to nickel, consistent with the observation of stronger bound stateson a nickel impurity than on the other two impurities.The bound states of the engineered impurities are again reminiscent of the theoreticalprediction for s±-wave pairing symmetry with weak non-magnetic impurities.876.2. Fe-site defectsdIt/dVs  (nS) highlow(a) (b)−10 0 1000.20.40.60.81Sample Bias (mV)dIt/dVs  (nS) highlow(c) (d)−10 0 1000.050.10.150.20.250.3Sample Bias (mV)dIt/dVs  (nS) highlow(e) (f)2 nm2 nm2 nm−10 0 1000.050.10.150.2Sample Bias (mV)Δ1Δ2Figure 6.5: Defect bound states of engineered Fe-site impurities. (a), (c), and (e) aretopographic images of manganese (It = 200 pA, Vs = −100 mV), cobalt (It = 400 pA, Vs = 15mV), and nickel (It = 100 pA, Vs = 12 mV) impurities, respectively. (b), (d), and (f) aredIt/dVs spectra on the impurity sites: the red spectra are taken at the locations indicated bythe red dot in (a), (b), and (c), respectively. The reference spectra taken in a defect-free regionare plotted in black. The gap amplitudes ∆1 = 6 meV and ∆2 = 3 meV are indicated by thedashed lines.886.3. Discussion and conclusions6.3 Discussion and conclusionsThe Fe-site defects consistently show bound states at the gap edge of the small gap.The only exception is manganese, which has little effect on the superconducting gaps.Owing to the variation of the scattering potential, the bound states appear with dif-ferent strengths. Our observations agree with the predicted phenomena for sign-changes±-wave pairing symmetry with weak non-magnetic impurities in which case the boundstates pin to the gap edge of the small gap in a wide range of nonmagnetic scatteringpotentials [158, 166]. In contrast, for magnetic point defects, the bound states usuallyappear inside both superconducting gaps [157, 166]. The nature and species of the nativedefects Fe-D2 and Fe-D2-2 are unknown, and hence the scattering potentials of themare undetermined. Likely these two defects arise from Fe vacancies or Li substitutionat iron sites, as the crystal was grown from a Li-rich flux (see Chapter 4). In the caseof Fe vacancies or Li substitution at iron sites, the nonmagnetic scattering potentialsare stronger than engineered transition metal impurities owing to the missing of d-shellelectrons. From Mn, Co to Ni, the nonmagnetic scattering potentials increase in theFeAs lattice environment from 0.28 eV, -0.35 eV to -0.87 eV according to ab initio calcu-lations [167]. These predictions are consistent with the fact that from Mn, Co, to Ni, andto Fe-D2 and Fe-D2-2, the defect bound states become stronger. We should note thatthe strength of bound states and the scattering potential in reality is considerably morecomplicated. In some cases, even strong potentials could produce bound states near thegap edge of the superconducting gaps [168]. For the defects studied in this thesis work,they are most likely in the regime of relative weak scattering potentials.However, the theoretical calculations so far are focused on s±-wave. Other pairingsymmetries with a sign change may also be able to give rise to such pinned bound states,which has not been examined by theory and hence requires further theoretical work.Thus, the impurity bound states shown here indicate a sign change in the superconductingorder parameter, but are not capable of revealing where the sign change happens in theBrillouin zone.89Chapter 7Quasiparticle interference in LiFeAsThrough examining the superconducting tunneling spectra of LiFeAs with local pertur-bations, we found evidence for a sign change in the superconducting pairing symmetry.However, the information as to where the sign change occurs is still missing. Thus, it isvery desirable to discern the pairing symmetry in momentum space. Quasiparticle inter-ference (QPI) is both a momentum- and phase-sensitive probe. Here, we combine STM,ARPES, and multi-orbital scattering theory to study QPI in LiFeAs. Using this coherentapproach to the electronic structure, we identify the relevant scattering vectors for thissystem and show that the energy dependence of the QPI intensity behaves as expectedfor an s±-wave superconductor with scattering from a non-magnetic impurity. In thisway we provide direct, phase-sensitive proof for s±-wave pairing symmetry in LiFeAs.7.1 Introduction to quasiparticle interferenceQPI are oscillations of the local density of states (LDOS) near defects, such as impuritiesand step edges. The LDOS oscillations are imaged in real space by measuring the differ-ential conductance dIt/dVs between the tip and sample as a function of position r andenergy E (see Sec. 3.1). A Fourier transform of this image produces a q-space QPI inten-sity map, where peaks occur at vectors connecting segments of the band structure [29].There are two approaches to understand the origin of QPI oscillations near a defect.One way is to view QPI as the screening of the defect charge and spin. The oscilla-tions are then the well-known Friedel oscillations [169], which provide direct observationof screening and of electron-electron interactions. The other way concentrates on thequasiparticles, in which QPI is viewed as a standing wave in the LDOS resulting fromthe interference of the quasiparticles wavefunctions before and after the quasiparticle has907.1. Introduction to quasiparticle interferencebeen scattered by an impurity [170]. This latter approach treats QPI as a two-particleinterference process, providing insight into the band dispersion and the gauge-field phaseof the quasiparticle wavefunction. The two approaches are essentially identical becausethey both deal with the linear response of the conducting quasiparticles to the localperturbation of an impurity.7.1.1 A simple caseIn this section, the second approach is used to demonstrate the physics of QPI. Forsimplicity, let’s take the one dimensional (1D) electron gas as an example. The Blochwavefunction of an electron in such system is ψk(r) = eik·r with eigen-energy k = ~k22m .The band dispersion is plotted in Fig. 7.1(c). When an electron travels with momentumki without being scattered, the LDOS at location r is given byN0(E, r) ∝ |ψki(r)|2 = |eiki·r|2 = 1.The LDOS is constant everywhere, as depicted in Fig. 7.1(b). When an impurity ispresent electrons can be scattered. The scattering process is elastic, if one assumes thatthe impurity does not have internal degrees of freedom to exchange energy. Therefore,the electron is scattered to the state with a momentum kf = −ki and an energy Ef = Ei(see Fig. 7.1(g)). Here i and f denote the initial and final states in the scattering event.Then the wavefunction becomes ψ(r) ∝ eiki·r + αeiδeikf ·r, where α is proportional to thescattering rate, and δ is the gauge-field phase difference between the initial and finalstates. The LDOS near the impurity becomesN(E, r) ∝ |ψ(r)|2 ∝ |eiki·r + αeiδeikf ·r|2∝ 1 + α2︸ ︷︷ ︸constant background+ 2α cos[(kf − ki) · r + δ]︸ ︷︷ ︸oscillation. (7.1)We define a scattering vector q = kf − ki which connects the states ki and kf in theband structure, as shown in Fig. 7.1(f). The LDOS in Eq. 7.1 contains an oscillationterm with a wavelength of λ = 2pi/|q| (see Fig. 7.1(e)). Fourier transforming this QPI917.1. Introduction to quasiparticle interferenceoscillation yields a peak at the scattering vector q in q-space, as shown in Fig. 7.1(h).Here, we define the q-space as scattering-vector space which can map to momentum k-space through the q-k relation. The energy dependence of the QPI-peak position givesa dispersion of the scattering vector q, which is directly related to the dispersion of kiand kf (see Fig. 7.1(i)). In the 1D-electron-gas example in which kf = q/2, the banddispersion in Fig. 7.1(g) can be recovered by replacing kf = q/2 in the dispersion of theQPI vector.This idea can be easily generalized to metallic materials in which Fermi-liquid theoryapplies. Quasiparticle states Ψk(r) are characterized by momentum k and eigenenergy(k). The dispersion relation (k) gives the band structure in momentum space. QPIoscillations are generated in the LDOS through the scattering of electrons by impurities.The Fourier transform of such oscillations yields a q-space QPI intensity map, wherepeaks occur at scattering vectors linking states in the band structure, namely q = kf−ki.The scattering rate can be calculated from Fermi’s golden rule:Wi→f (ki,kf ) ∝ |V (q)|2Ni(Ei,ki)Nf (Ef ,kf ), (7.2)where Ei = Ef for elastic scattering, V (q) is the Fourier component of the scatteringpotential at q, and Ni and Nf are the DOS of the initial and final states, respectively.Then for each q vector, the total intensity in q-space is given byI(E,q) ∝ |V (q)|2∑kNi(E,k)Nf (E,k + q)︸ ︷︷ ︸autocorrelation between momenta. (7.3)In the case of a constant |V (q)|2 for all q, the QPI intensity map in q-space is anautocorrelation of the DOS in momentum space at constant energy E. In general, |V (q)|2may enhance or suppress certain scattering channels according to orbital characters, spin,etc.A more rigorous understanding of QPI can be obtained from a Green’s functioncalculation with T -matrix formalism [171]. The full Green’s function treatment includesterms that take into account additional effects, such as multiple scattering. However, the927.1. Introduction to quasiparticle interference(a) no-impurityki(d) with-impuritykfkiLDOSr|ψ|2(b)EF(0 ,0 ) kkiE(c)λ=-2π/|q||ψ|2(e) kfkiq   kf-ki=(f)rkkfkiEEF(0 ,0 )(g)qq   kf-kiEEF(0 ,0 )=(i)qIntensity(0 ,0 )q   kf-ki=(h)band-structure-determinesscattering-vector-q Energy-dependence-of-qrecover-k-by-q-k-relationFigure 7.1: Example of QPI: one-dimensional electron gas. (a) An electron travelswith momentum ki, whose quantum state is shown in (c). Without the presence of impurities,|ψ|2 is constant and therefore the LDOS is flat, as shown in (b). (d) shows that with animpurity, a finite density of electrons is back-scattered. (e) The LDOS shows an interferencepatterns between the incident electron wavefunction with momentum ki and the scatteredelectron wavefunction with momentum kf . The wavelength of the modulation is λ = 2pi/|q|,where q = kf − ki = 2kf as indicated in (f). (g) shows the quantum states before and afterthe electron being scattered in the band structure. (h) Fourier transforming the LDOS in (e)produces a peak at a vector q = kf − ki = 2kf in q-space. (i) The dispersion of q in turn canrecover the band dispersion shown in (g). Generally, in a real QPI measurement, the analysissteps start from (g), to (h), then to (i), and then back to (g).relation between the scattering vector q and the momentum k is unchanged [29, 171].Thus, QPI allows us to study the momentum-space behavior of quasiparticles by tracing937.1. Introduction to quasiparticle interferencethe QPI vectors in q-space. In a real measurement, the band structure is obtained eitherfrom theory or other experimental techniques, and then used to determine q-k relationsin the QPI data. In turn, the dispersion in q is used to determine the details of the banddispersion (see the circulation of Figs. 7.1(g), 7.1(h) and 7.1(i)). The advantages of STM-QPI in measuring electronic structure lie in the sub-meV energy resolution, the ability tomeasure both occupied and unoccupied states, and the possibility to measure in magneticfield. A simple and demonstrative example of this method is the QPI measurement ofthe electron-phonon coupling in Ag-(111) Shockley surface states [172].Finally, I would like to emphasize the gauge-field phase sensitivity in QPI. In Eq. 7.1,the phase difference between the wavefunctions of the initial and final states enters intothe oscillation term. Imagining the case that the relative phase is locked among quasi-particle states, such as with Bogoliubov quasiparticles, selection rules can be establisheddepending on the nature of defects. Certain QPI intensity peaks can be suppressed ifthe phase shift due to scattering does not match the phase difference between the twostates. In turn, if the nature of the impurity is known, and hence the allowed scatteringphase shift is known, the phase differences among states can be extracted by observingthe variation of the amplitude of the QPI peaks. Thus, QPI is a gauge-phase sensitivemethod that enables the determination of the phase difference between states in the bandstructure, which is the foundation for probing the pairing symmetry of a superconductor.This phase sensitivity has been predicted and applied to cuprate high-Tc superconductorswith great success [10, 173–175].7.1.2 Application to the iron-pnictide Fermi surfaceThe electronic structure of iron pnictides generally consists of hole (h) bands centered atk = (0, 0) and electron (e) bands centered at k = (±pi/a,±pi/a) as shown in Fig. 7.2(a).Possible QPI vectors in the iron-pnictide band structure are shown in Fig. 7.2(a).Here, we are concerned with QPI features near the Fermi level, and therefore the bandstructure at E = EF (the Fermi surface) is used as being representative. qh−h and qe−eare intraband scattering vectors connecting segments within the hole and electron bands,respectively. qh−e represents interband scattering vectors connecting the segments of hole947.2. QPI in LiFeAsand electron bands. Fig. 7.2(b) shows the autocorrelation image of the Fermi surface inFig. 7.2(a). The QPI features corresponding to intraband qh−h and qe−e are two ringscentered at the (0, 0) point. The QPI feature corresponding to the interband qh−e is anannulus centered at the (pi, pi) point in q-space.qh-h(π, π)qh-ee-eq(a) (b) (π, π)π-πqh-hqh-ee-eqLowHighFigure 7.2: Possible QPI vectors in iron pnictides near the Fermi level. (a) Thescattering vectors that connect the segments of the Fermi surface (k-space). qh−h and qe−eshow intraband scattering within the hole and electron pockets, respectively, while qh−e showsinterband scattering between the two. (b) An autocorrelation image of the Fermi surface (q-space). QPI features corresponding to the three QPI vectors are marked by black arrows.7.2 QPI in LiFeAsA 26×26 nm2 area was chosen to perform the QPI measurement, as shown in the to-pography in Fig. 7.3(a). This area contains fourteen native defects and one adsorbedimpurity. In this area, defects are well-separated from each other, leaving adequate spacefor acquiring extended LDOS oscillations around each defect. The QPI data were ac-quired by numerical differentiation of the It-Vs sweep at each pixel at 4.2 K. The settingcurrent It and bias voltage Vs for STM feedback control are I0 = 250 pA and V0 = 25 mV,respectively. The bias voltage varied from 25 mV to -25 mV with a 0.1 mV interval perpoint. Fig. 7.3(b) shows a typical It-Vs sweep in a defect-free region, while Fig. 7.3(c)shows its numerical differentiation after applying a Gaussian filter. A clear ∆1 = 6 meVsuperconducting gap is resolved along with a subtle shoulder ∆2 ≈ 3 meV. These val-957.2. QPI in LiFeAsues are consistent with the full double gap structure found in the same sample at lowertemperature (T = 2 K) (see Sec. 5.4).10 nmhigh1(a)-0.10.10.2-20 -10 0 10 20051015200Sample Bias (mV)Tunneling Conductance (nS)Tunneling Current (nA)(b)(c)Figure 7.3: Area for performing QPI measurement. (a) Topography (It = 250 pA,Vs = 25 mV) of the area. There are 14 native defects and one adsorbed defect, among whichten are Fe-D2 defects. (b) shows a typical It-Vs spectrum at position 1© and (c) shows itsnumerical differentiation following a Gaussian filter with δ ∼ 0.5meV .7.2.1 Data processing methodsFig. 7.4(a) shows a tunneling conductance map g(r, eVs), i.e. a dIt/dVs map, in the areashown in Fig. 7.3(a), taken at Vs = 8 mV (E = eVs = 8 meV). A rapidly decayingoscillation is present around each defect, resulting from modulations of the LDOS dueto scattering. Fig. 7.4(b) shows the direct Fourier transform graw(q, 8 meV). Althoughthere are obvious oscillations in the tunneling conductance map, the Fourier transformedimage does not show all features of the QPI pattern due to a dominant background signalcentered at q = (0, 0).Here, we employ two methods to remove this background and recover the underlyingQPI patterns. As shown in Fig. 7.4(a), g(r, 8 meV) exhibits strong conductance peaks atthe defect centers that give rise to a strong background signal in q-space, overwhelming967.2. QPI in LiFeAsSample hampFigure 7.4: Example of the data processing techniques: raw Fourier transform.(a) The unprocessed tunneling conductance map g(r, 8 meV). (b) The Fourier transformgraw(q, 8 meV). The direct Fourier transform of the raw data yields strong intensity nearq = (0, 0).the QPI signal. In general, the tunneling conductance map g(r, E = eVs) is given byg (r, eVs) =eIt∫ eV00 N (r, E) dE×N(r, eVs), (7.4)where e is a unit charge, N(r, eVs) is the LDOS at r, and E = eVs [176]. V0, the biasvoltage for STM feedback control, is 25 mV in this QPI measurement. According toEq. 7.4 the variation of g(r, E) is directly proportional to the variation of the LDOS ifthe normalization∫ eV00 N(r, eVs)dE is spatially homogeneous. This condition, however,does not hold at the center of the defects where the defects strongly modify the localpotential. This is because N(r, E) can be dramatically modified by local changes inthe electronic structure and/or the creation of bound states. The LDOS of LiFeAs isstrongly energy dependent near EF (see Sec. 5.2), so the defect-induced changes in thelocal electronic structure cause a significant variation in the integral of the LDOS overthe energy range [0, eV0 = 25 meV]. In addition, all of the common defects in LiFeAsgenerate bound states inside the superconducting gaps [49]. Therefore, the behavior ofg(r, E) close to the defect centers cannot be simply interpreted as QPI oscillations inN(r, E) due to the inhomogeneity of the normalization factor.977.2. QPI in LiFeAsGaussian mask of defect centersA Gaussian masking method is used to eliminate the signal from the central conductancepeaks of these defects. For a defect located at r0, we chose a masked conductance mapgM(r, E) given bygM(r, E) = g(r, E)× (1−M(r− r0, σ)), (7.5)where M(r− r0, σ) is a truncated Gaussian function with maximum value = 0.99 and σis the standard deviation, taken to be approximately the half width of the defect center.This Gaussian masking method suppresses the local conductance peaks associated withthe defect centers yet preserves the sign of g(r, E) and produces a smooth transition fromthe masked regions to the QPI nearby.We apply the Gaussian mask to each of the defects. Fig. 7.5(a) and 7.5(b) show thereal space and q-space conductance maps after applying the Gaussian mask technique,respectively. Significantly more symmetric and regular patterns stand out in q-space.Fig. 7.5(c) and 7.5(d) show the portion of the real space conductance map removed bythe mask technique and its corresponding Fourier transform, respectively. By comparingFigs. 7.4(b) and 7.5(d), we find that the defect centers contribute the primary backgroundsignal centered at q = (0, 0) in the direct (unmasked) Fourier transform.Suppression of the center peak in q-spaceAs shown in Fig. 7.6(a), which is a duplicate of 7.5(b), the strong intensity around q =(0, 0) lowers the visibility of the QPI pattern at larger q. We therefore further appliedthe Gaussian suppression method of Allan et al. (Ref. [35]) to suppress the central peak.The final QPI intensity map g(q, E) is given byg(q, E) = gM(q, E)× (1− 0.95×G(q, σ)) (7.6)where G(q, σ) is a Gaussian function with peak value = 1 and σ ∼ 0.35pi/a. We chose toretain 5% of the signal at q = (0, 0) in order to not overly suppress the real QPI signalnear q = (0, 0). Fig. 7.6(b) shows the final QPI intensity map after applying both the987.2. QPI in LiFeAs7.65 nS4.65 503000g - gM7.65 nS4.65 503000gM(r,8 meV) gM(q,8 meV)(a) (b)(c) (d) g - gMFigure 7.5: Example of the data processing techniques: Gaussian mask of the defectcenters. (a) The tunneling conductance map after defect-center masking gM (r, 8 meV) and(b) its Fourier transform gM (q, 8 meV). Masking the defect centers removes the strong andasymmetric intensity background, giving more symmetric patterns. (c) The tunneling conduc-tance map of the defect centers, the portion removed from the raw data, and (d) its Fouriertransform. Here (b), (d) and Fig. 7.4(b) are plotted in the same color scale. By comparing (d)and Fig. 7.4(b), we can see that the major contribution to the direct Fourier transform stemsfrom the defect-center background.Gaussian mask in real space and Gaussian suppression in q-space.We emphasize that the same treatment with the same masking parameters was ap-plied to the QPI intensity maps at all energies. Thus the intensities of the QPI atdifferent energies, shown in the following sections, are directly comparable. In addition,no symmetrization has been applied to the data.11 Therefore, the symmetry of our QPI11In some materials, symmetrization of the QPI intensity maps according to the crystallographicsymmetries is used to increase the signal-to-noise ratio. However, the symmetrizing procedure can coverthe possible signature of symmetry breaking in the electronic state. In this dataset, the signal-to-noiseratio is very good, and hence no symmetrizing procedure is employed.997.3. Identification of the scattering vectorsFigure 7.6: Example of the data processing techniques: suppression the center peakin q-space. (a) The Fourier transform after the Gaussian mask of defect centers gM (q, 8meV),the same image of Fig. 7.5(b). (b) The final QPI map after the additional application of theGaussian suppression method of Ref. [35].intensity map reflects the original symmetry of the underlying electronic structure of thesample and the tip.7.3 Identification of the scattering vectorsFig. 7.7 shows the QPI intensity map after elimination of the overwhelming backgroundsignal. The Bragg diffraction peaks of the As/Li [(2pi/a, 0)] and Fe [(2pi/a, 2pi/a)] sub-lattices are clearly resolved at the outer edge of the QPI map. Here a is the latticeconstant of LiFeAs (two iron unit cell). In addition to the Bragg peaks, we find threefeatures centered on q = (0, 0); two small inner rings and a larger outer ring, in agree-ment with previous studies [31, 35]. We also observe a set of “arc” features located near(±pi/a,±pi/a) point. These rings and arcs originate from multiple inter- and intrabandscattering processes. Comparing to the autocorrelation image in Fig. 7.2(b), these ring-like features are QPI of h-h or e-e scatterings while the arc-like features are QPI of h-escattering.To identify the underlying bands associated with each of these vectors, QPI maps weremodeled using the T -matrix formalism outlined in Appendix A. In order to accuratelyidentify each of the vectors observed, it is important to anchor the model electronic1007.3. Identification of the scattering vectorsqx (2π/a )q y(2π/a)1-1 -1 1high(1/2,1/2)lowFigure 7.7: QPI features in the QPI intensity map. There are two major QPI features.One is the scattering among the hole bands, which appears as rings centered at q = (0, 0). Theother is the interband scattering between the hole and electron bands, which appears as arcscentered around q = (±pi/a,±pi/a), reminiscent of the autocorrelation image in Fig. 7.2(b).The symbols indicate the location of the QPI vectors whose dispersion is tracked in Fig. 7.8.structure to the empirical band structure observed by ARPES, as shown in Figs. 7.8(a)and 7.8(b). The details of the ARPES measurement are given in Appendix B. TheFermi surface along the k|| plane at kz ∼ 0 (Fig. 7.8(a)) is composed of two hole pocketscentered at Γ, denoted h2 and h3, and two electron pockets centered at each of the Mpoints, denoted e1 and e2. A momentum distribution curve analysis of the ARPES spectraindicates the presence of a third inner hole pocket h1. The tops of the h1 and h2 bandslocate within a ±6 meV window of EF . To model this electronic structure, we adoptedthe modified two-iron ten-orbital tight-binding model introduced in Appendix A.1. Theband dispersion for this model is shown in Figs. 7.8(a) and 7.8(b), reproducing the ARPESband structure at kz = 0.The calculated QPI intensity map at Vs = 8 mV, based on our model of the bandstructure, is shown in Fig. 7.8(c), where we have assumed that electrons tunnel into anon-zero kz = 0.4pi/c cut of the three-dimensional band structure. (We will return tothis point in Sec. 7.4.) In addition, the impurity potential in the T -matrix formalism onlyallows intra-atomic-orbital scattering (see Appendix A for details), according to the theo-1017.4. Reconciling the discrepancy between STM and ARPES measurementsretical prediction that intra-orbital scattering dominates [177]. A number of QPI vectorsare present in the calculation, and are highlighted by the open symbols. The calculationidentifies the two innermost rings (red © and black 5) and the outermost large ring(blue 2) with intraband scattering between three hole bands, h1-h1, h2-h2, and h3-h3,respectively. The third ring from the center (orange 3) is due to interband scatteringbetween the inner and outer hole bands h1-h3 and h2-h3. Our model also identifies thearc-like features (black©) centered on (±pi/a,±pi/a) with scattering between the h2 ande1,2 bands. Scattering between the h3 and e1,2 bands is suppressed due to a mismatch oforbital character in these two bands.Comparing to the experiment, we associate the smallest to largest of the three QPIrings in Fig. 7.7(g) with h2-h2, h2-h3, and h3-h3 scattering, respectively. Furthermore,we observe a distinct scattering process between the hole and electron pockets h2-e1,2,the arc-like features, which, unlike the similar feature in Fe(Se,Te) [11], is well separatedfrom the commensurate (pi, pi) point. This allows us to unambiguously disentangle QPIof h-e scattering from Bragg peaks of possible charge or magnetic ordering [178]. The e-eQPI feature is absent in our measurement. As shown in Sec. 5.4, the LDOS of electronpockets do not show in dIt/dVs spectra, therefore an e-e QPI feature is not expected inQPI intensity maps.7.4 Reconciling the discrepancy between STM andARPES measurementsSTM and ARPES are probes that are capable of measuring surface electronic structure.For a crystal with non-polar surface and without surface reconstruction, the band disper-sion obtained from STM-QPI and from ARPES measurements should agree with eachother. However, in LiFeAs, there is a controversy with regard to the band dispersionsacquired by the two techniques [35, 46, 47].The dispersion of the QPI vectors obtained in this thesis work quantitatively disagreeswith the ARPES band dispersion near the Γ-point. Notably, the ARPES measurements1027.4. Reconciling the discrepancy between STM and ARPES measurements)0,0()0,0( (1,1)−2000Momentum--(π/a)Energy-(meV)h1, h2h3e1, e2kz = 0kz = 0.4π/c(1,0)qx--(2π/a) 0 0.2 0.4 0.6 0.8−200200Energy-(meV) h1 - h1h2 - h2h3 - h3h2 - h3h2 - e1,2(a) (b)(c) (d)q-along-(0,0)---(π/a,0) (π/a,π/a)---(π/a,0)q y--(2π/a)10-110-1kx--(π/a) 10-10-11k y--(π/a)(e)0.2 0.4Figure 7.8: Dispersion of QPI vectors and the comparison with Green’s functioncalculation using T -matrix formalism. (a) The Fermi surfaces obtained from ARPES (blueintensity map) and the model (red and blue curves). (b) The ARPES and model dispersionsalong the high-symmetry cuts of the first Brillouin zone. In (a) and (b) the ARPES spectra areshown for a photon energy that selects kz values near zero. The model dispersions are shown forkz = 0 (red) and kz = 0.4pi/c (blue). (c) The calculated QPI at Vs = 8 mV obtained assumingthat electrons tunnel into states with a non-zero kz = 0.4pi/c. Features associated with intra-and interband transitions are indicated by the open symbols. (d) For qh-h QPI and (e) for qh-eQPI: the experimental (blue points with error bars) and theoretical (solid symbols) dispersionof the QPI vectors indicated in (c). The error bars are determined approximately by the fullwidth at half maximum of the QPI features plus one additional pixel uncertainty. The solidlines show the dispersion expected from the model dispersion.indicate that the top of h2 is no more than 6 meV above EF at Γ [32, 34, 179]; above thisenergy the h2-h2 and h2-h3 features should vanish due to phase space constraints, if STMis probing the band structure in the kz = 0 plane. This is inconsistent with the observedQPI dispersions, shown in Fig. 7.8(d) (data points with error bars) where all of the ringsdisperse to energies > 20 meV. In an ARPES experiment, the Fermi surface is measuredin the kx-ky plane at a constant kz. The value of kz is determined by the incident photon1037.5. Variation of the QPI intensity with energy: evidence for s±-wave pairing symmetryenergy. The photon energy used at UBC corresponds to kz ∼ 0. However, in the caseof STM, it is less clear which values of kz are actually probed in a bulk 3D system.Empirically we have found that kz = 0.4pi/c provides good agreement between our modeland the data (Fig. 7.8(d)). The solid symbols plot the dispersion of the calculated QPIfeatures from T -matrix formalism. The agreement between the model and the experimentis good and a non-zero value of kz reconciles differences in band structure inferred fromARPES and STM measurements [34, 35, 47]. We note that the agreement might befurther improved by integrating the signal over a range of kz values. However, this wouldrequire an explicit calculation of the tunneling matrix element and is left for future work.The fact that the inner hole pocket(s) disperse well above 20 meV at finite kz impliesthat a weak nesting condition exists between small inner hole pockets and comparativelylarge electron pockets at the Fermi level. This is consistent with a weak and incommensu-rate spin resonance mode revealed by inelastic neutron scattering (INS) at a wavevectorlinking the h and e pockets [57, 121]. Therefore, our work brings LiFeAs back into thegeneral iron-pnictide family whose members share the common feature of quasi-nestedFermi surface topology.7.5 Variation of the QPI intensity with energy:evidence for s±-wave pairing symmetryNow that the QPI vectors have been identified, we turn to identifying the symmetryof the order parameter. This is accomplished by an examination of the QPI of Bogoli-ubov quasiparticles near the superconducting gap, where the selection rules discussed inSec. 7.5.1 become dominant. They are reflected in the intensity variations with respect toenergy in QPI maps. In this section, I will show that the intensity variations in LiFeAsare only consistent with s±-wave.1047.5. Variation of the QPI intensity with energy: evidence for s±-wave pairing symmetry7.5.1 Pairing symmetry sensitivityQPI provides access to the phase of the superconducting gap by imaging QPI of Bo-goliubov quasiparticles, which are a superposition of e and h excitations. Bogoliubovquasiparticles, exitations of Cooper pairs, inherit the pairing phase through the coher-ence factors v(k) and u(k) that are given byv(k) =∆(k)|∆(k)|√12(1−(k)E(k)),u(k) =√1− |v(k)|2. (7.7)Therefore, the phase of ∆(k) determines the phase of v(k). (To review Bogoliubovquasiparticles, please see Sec. 2.2.3.)In addition to Fermi’s golden rule for normal quasiparticles (Eq. 7.2), a prefactor isattached for the scattering rate in the superconducting state [1], which is given byWsc(ki,kf ) ∝ |u(ki)u∗(kf )± v(ki)v∗(kf )|2 ×Wnormal(ki,kf ). (7.8)The negative and positive signs in Eq. (7.8) correspond to scattering from a nonmagneticand a magnetic impurity, respectively [1, 29, 180]. 10-Δ Δ0.53Δ2Δ-2Δ-3Δ|v(k)|2 |u(k)|210-Δ Δ0.53Δ2Δ-2Δ-3Δ|v(k)| |u(k)|(a) (b)Figure 7.9: Energy dependence of the superconducting coherence factors. (a) Themodulus and (b) the modulus squared values of coherence factors v(k) and u(k). The ampli-tudes of v(k) and u(k) are comparable near the Fermi level (E = 0). Moving away from theFermi level, one of them becomes dominant, depending on the sign of the energy.For energies E > |∆|, the term |u(ki)u∗(kf )± v(ki)v∗(kf )|2 is close to unity becauseone of the two coherence factors becomes dominant (see Fig. 7.9). Hence, the QPI inten-1057.5. Variation of the QPI intensity with energy: evidence for s±-wave pairing symmetryScenario q qSuppressed Intensity Enhanced Intensitynon-mag. imp., s++ qh−h, qe−e, qh−e -mag. imp., s++ - qh−h, qe−e, qh−enon-mag. imp., s± qh−h, qe−e qh−emag. imp., s± qh−e qh−h, qe−eTable 7.1: A summary of the QPI selection rules expected for a pnictide super-conductor with s++-wave or s±-wave. The QPI intensity of a scattering vector is eithersuppressed or enhanced when sweeping energies from above to inside the superconducting gap.The four combinations of two pairing symmetries and two kinds of impurities result in com-pletely different selection rules. The intensity suppression or enhancement are established onthe comparison of QPI intensities at energies between E > ∆ (close to normal state QPI) andE ≤ ∆ (dominant by Bogoliubov QPI).sity maps at such energies are nearly the same as those in the normal state. However,near, and in particular inside, the superconducting gap where v(k) and u(k) becomecomparable in magnitude, the term |u(ki)u∗(kf ) ± v(ki)v∗(kf )|2 becomes significantlydifferent from one. This establishes a set of “selection rules” that will enhance or sup-press the scattering rate near the superconducting gap, relative to the rate away fromthe superconducting gap, and thus the QPI intensity, depending on the nature of theimpurity and the relative sign between ∆(ki) and ∆(kf ). Thus, comparing the QPIintensity variations between E > ∆ and E . ∆ is equivalent to comparing the normalstate QPI with the superconducting QPI. Note also that these selection rules can be morerigorously derived using the T -matrix formalism [180, 181].The selection rules for the pnictide band structure shown in Fig. 7.2 are summarizedin Table 7.1 for the cases of an s++-wave and s±-wave pairing symmetry. For example,as shown in Fig. 7.10, in the s±-wave scenario with non-magnetic impurities, one expectsthe intensity of QPI vectors associated with interband scattering between the hole andelectron bands qh−e to be enhanced while intraband scattering within the hole bands orthe electron bands qh−h and qe−e, respectively, will be suppressed when sweeping energiesfrom above to inside the superconducting gap (Fig. 7.10(b)). Finally, we emphasize herethat both the symmetry and nature of the impurity can be uniquely inferred from the1067.5. Variation of the QPI intensity with energy: evidence for s±-wave pairing symmetry(a)(π, π)π-πqh-hqh-ee-eqLowHigh(b)Figure 7.10: Example of QPI intensity variation: the case of s±-wave. For simplicity,we use the autocorrelation image from Fig. 7.2(b) for demonstration, as shown in (a). Unders±-wave pairing symmetry with nonmagnetic impurities and at T  Tc, when E > ∆, theQPI intensity map is close to normal state in (a). When approaching the superconducting gap,intensity of qe−e and qh−h are suppressed while the intensity of qh−e is enhanced or preservesthe original amplitude, as shown in (b).relative behavior of subsets of the QPI intensities as indicated in Table 7.1.7.5.2 QPI intensity map vs energyQPI intensity maps at different energies are shown in Fig. 7.11. For biases well above thesuperconducting gap, the intensities of different scattering vectors in the QPI maps arerelatively energy-independent; for example, one can compare the QPI maps at Vs = 12and 20 mV. In contrast, as the bias voltage sweeps from above the gap (Fig. 7.11 at 12 mV)to inside the gap (Fig. 7.11 at 3 mV), the intensity of the intraband hi-hi (i = 1, 2, 3) andinterband h2-h3 scattering is strongly suppressed while the interband h2-e1,2 scatteringis significantly enhanced. Comparing to the selection rules in Table 7.1, the intensityvariations are only consistent with s±-wave pairing symmetry.1077.5.VariationoftheQPIintensitywithenergy:evidencefors±-wavepairingsymmetryFigure 7.11: The intensity variations of h-h and h-e scattering vectors. Above the superconducting gaps, from 20 meV to 6meV, the intensities of both h-h and h-e do not vary significantly. Inside the superconducting gap, from 6 meV to 3 meV, the intensity ofthe h-h QPI feature, highlighted inside the red circle, is highly suppressed, while the intensity of the h-e QPI feature, highlighted insidethe black circle, is enhanced.1087.5. Variation of the QPI intensity with energy: evidence for s±-wave pairing symmetryWe further quantify these intensity variations shown in Fig. 7.11 by examining theintegrated weight of each QPI vector as a function of energy, as shown in Fig. 7.12(a).Here, the intra- and interband h-h and interband h-e scattering vectors are isolated bydefining appropriate integration windows shown in Fig. 7.12(b). The QPI of h3-h3 wasnot included in the h-h intensity integration because it is relatively weak and partiallyoverlaps the QPI of h2-e1,2. Ignoring h3-h3 will not affect our discussion since h2-h2 andh2-h3 are good representatives of the h-h feature.Integrated3Intensity3(Arb.3Units)-12 -6 0 6 12Sample3Bias3(mV)h2 -3e1,2h2 -3h2,3h2 -3h3(a)qx (2π/a)q y(2π/a)1-1 1-18 mV (b)Figure 7.12: The energy dependence of the integrated QPI intensity. (a) The redsector and blue circle are the integration windows of h2-h2 and h2-h3 and h2-e1,2 scatteringvectors, respectively. A noise background signal is integrated in the grey rectangular area andsubtracted. Here one quarter of the windows are shown for simplicity but the integration isperformed over the equivalent areas in all four quadrants of the image. (b) The integratedintensity of the QPI signal for the intra- and interband h2-h2 and h2-h3 (red) and interbandh2-e1,2 (blue) scattering vectors. The curves were then normalized to the value at 12 meVand the h-e intensity has been offset for clarity. The dashed lines indicate the values of thesuperconducting gaps.The integrated QPI intensity of h-h scattering basically follows the spectral shape ofthe dIt/dVs tunneling spectrum, which is reasonable in the sense that the QPI intensityis proportional to the LDOS (see Eq. 7.2 and 7.8). However, inside the large supercon-ducting gap (6 meV), the intensity is quickly suppressed. In contrast, the integratedQPI intensity of h-e keeps increasing until 3 meV. Below 3 meV, the LDOS of all bandsvanishes because of the opening of the superconducting gaps. The integrated intensityvariations are consistent with the scenario of s±-wave with a nonmagnetic impurity (see1097.5. Variation of the QPI intensity with energy: evidence for s±-wave pairing symmetryTable 7.1).7.5.3 Bogoliubov QPIQPI of Bogoliubov quasiparticles distinguishes itself from normal state QPI by an anti-phase relation of LDOS modulations at positive and negative energies [176, 182]. Weillustrate this in Fig. 7.13(a) and 7.13(b), which show oscillations of the LDOS near anFe-D2 defect at ±3 mV [49]. Note that, Fe-D2 defect is the majority defect in the areaused for the QPI measurement. The anti-phase relation is apparent in the contrastinversion highlighted inside the dashed circles in Fig. 7.13. This confirms the dominanceof Bogoliubov QPI inside the superconducting gap.(a)2 nm(b)highlow3 mV -3 mVFigure 7.13: Observation of Bogoliubov QPI in real space. dIt/dVs maps in the vicinityof an Fe-D2 defect at (a) Vs = +3 mV and (b) Vs = −3 mV. The anti-phase relationship of theLDOS modulations is highlighted by the locations with high intensities inside the white circlesof (a) and the corresponding low intensities inside black circles of (b), and vice versa.Z-map QPI, defined as the ratio of tunneling conductance maps at positive andnegative biases Z(r, E) = g(r, E)/g(r,−E) [176], emphasizes the anti-phase compo-nent of Bogoliubov QPI while suppressing the in-phase component of the normal stateQPI [176, 182]. Z(r, eV ) at 3 mV (see Fig. 7.14(a)) shows strong short-wavelength real-space oscillations near each impurity. As shown in Fig. 7.14(b), the Fourier transformZ(q, E = 3 meV) reveals strong intensity arcs near (±pi/a,±pi/a), corresponding to thepreviously identified h-e scattering vectors. The intensity of these arcs diminishes as thebias voltage sweeps from inside the large gap to above it (Fig. 7.14(c) and 7.14(d) at 10meV). Therefore the intensity variations observed in this energy range are indeed due tothe selection rules imposed by the symmetry of the order parameter.1107.5. Variation of the QPI intensity with energy: evidence for s±-wave pairing symmetry(a) (b)503000.53.010 meV0.51.550300(c) (d)3 meVFigure 7.14: Zmap QPI in LiFeAs. The real space Z-maps are defined as Z(r, eV ) =g(r, eV )/g(r,−eV ), which highlights the anti-phase Bogoliubov QPI. (a) The real-space Zmapat 3 meV shows short wavelength oscillations in the LDOS. (b) The Fourier transform of (a),using the method in Sec. 7.2.1, indicates that the h-e QPI feature dominates the anti-phaseBogoliubov QPI at 3 meV. (c) The real-space Zmap at 10 meV, above the superconductinggaps, and (d) its Fourier transform do not show strong h-e QPI feature.Comparing to the selection rules in Table 7.1, we find that the data is most consistentwith an s±-wave scenario with ∆(k) changing sign between the h and e bands; belowthe superconducting gap, h-h scattering intensities are suppressed while h-e scatteringintensities are enhanced for non-magnetic impurities [177, 180]. Magnetic impuritiesin the s±-wave scenario have the opposite effect. The observed selection rules are alsodistinct from the s++-wave scenario with either magnetic or non-magnetic impurities, seeTable 7.1. Our results indicate the non-magnetic nature of the most common defect. Thisis consistent with the fact that the Fe-D2 defect generates bound states only at the gapedge of the small superconducting gap (see Sec. 6.2.1) which implies a nonmagnetic natureof the defect. In the crystal growth, we used a self-flux method to avoid the contamination1117.6. Summary and conclusionsfrom other elements. Therefore, Fe-D2 defect is most likely a Li substitution on an Fesite, or an Fe vacancy, as a result of “self-contamination” [49], both of which are expectedto be non-magnetic. We therefore infer that the only candidate consistent with ourmeasurements is an s±-wave pairing symmetry.7.6 Summary and conclusionsWe have examined QPI in LiFeAs using a combination of STM, ARPES, and a tight bind-ing model. By anchoring our tight binding description of LiFeAs to the ARPES-derivedband dispersions we were able to unambiguously assign each of the scattering vectorsin the QPI maps. In this framework, we have reconciled, not only the discrepancies inthe assignments of scattering vectors in prior QPI studies, but also the disagreement onthe sizes of inner hole pockets between ARPES and STM techniques by recognizing anon-trivial kz dependence in the tunneling process. With the assignment of the scatter-ing vectors made, we then examined the detailed variations of the QPI intensity as afunction of bias voltage. The variations in intensity near the superconducting gap areonly consistent with an s±-wave pairing symmetry, where the change in sign occurs be-tween the electron and hole pockets. Together with the observation of a spin fluctuationresonance by INS [57, 121], this work presents compelling evidence of unconventional s±-wave pairing in LiFeAs, driven by repulsive spin fluctuation interactions. This impliesthat LiFeAs shares a common superconducting mechanism with the other members inthe iron pnictide family [3, 4, 6]. Hence LiFeAs is a simple and clean model material forprobing the common physics of iron pnictides.This work also demonstrates how Bogoliubov QPI from defect/impurity scatteringprovides a direct phase sensitive measurement of superconducting pairing symmetry.Bogoliubov QPI has been used to confirm the sign flip in the d-wave Ca2−xNaxCuO2Cl2cuprate superconductor [183] as well as the Fe(Se,Te) iron-based superconductor [11] un-der high magnetic field, where vortices behave as magnetic scattering centers. However,the latter method is only suitable for materials with very short superconducting coherencelengths, so that a vortex can be treated as a localized strong magnetic scattering center.1127.6. Summary and conclusionsHere Bogoliubov QPI is measured (without the application of a magnetic field) by takingadvantage of point defects/impurities inside the material, which has been proposed onlytheoretically [177, 180, 181]. This method can be generalized to other superconductors,provided the nature (magnetic vs nonmagnetic) of the impurities is known.113Chapter 8Concluding remarksThis thesis work demonstrates that STM is indeed a powerful technique in measuringthe pairing symmetry, both the gap amplitudes and the pairing phase. We have appliedthis technique to LiFeAs, one controversial compound in iron-based superconductors.Our observations show compelling evidence for s±-wave pairing symmetry in LiFeAs. Inthe final remarks, I first show an additional new method of STM-QPI that could bepotentially used to determine the pairing phase. Then I discuss what are the physicalimplications of s±-wave pairing symmetry on the pairing interaction and superconductingmechanism.8.1 Another way to measure the pairing symmetryby QPIIn addition to the method used in chapter 7, in which the pairing symmetry is identifiedvia comparing the QPI intensity at energies E > ∆ and E . ∆, I will demonstrate analternative QPI method for identifying the pairing symmetry in LiFeAs.Recall the Bogoliubov QPI scattering rate, which is given by (see Sec. 7.5.1)Wsc(ki,kf ) ∝ |u(ki)u∗(kf )± v(ki)v∗(kf )|2 ×Wnormal(ki,kf ), (8.1)where v(k) and u(k) are the the coherence factors. The negative and positive signs inthe interference factor |u(ki)u∗(kf ) ± v(ki)v∗(kf )|2 correspond to scattering from non-magnetic and magnetic impurities, respectively. Here, I continue to follow the previousconvention that u(k) is a real and v(k) changes sign with ∆(k), as defined in Eq. 7.7.Here, the interference factor |u(ki)u∗(kf ) ± v(ki)v∗(kf )|2 is temperature dependent.1148.1. Another way to measure the pairing symmetry by QPIqx (2π/a)q y(2π/a)1-1 1-18 mV(a) TTcIsIn0 112(b)h-hh-eFigure 8.1: Variation of Bogoliubov QPI intensity with temperature. (a) QPI of LiFeAsat 8 meV. The h-h and h-e QPI features are highlighted in red and blue circles, respectively.(b) Red curve: the variation of |u(ki)u∗(kf ) − |v(ki)v∗(kf )||2 as a function of temperature;blue curve: the variation of |u(ki)u∗(kf ) + |v(ki)v∗(kf )||2 as a function of temperature. Witha s±-wave intensities of h-h and h-e QPI features follow the red and blue curves, respectively.Scenario q qSuppressed Intensity Enhanced and then Suppressed Intensitynon-mag. imp., s++ qh−h, qe−e, qh−e -mag. imp., s++ - qh−h, qe−e, qh−enon-mag. imp., s± qh−h, qe−e qh−emag. imp., s± qh−e qh−h, qe−eTable 8.1: A summary of the variation of the factor |u(ki)u∗(kf )± v(ki)v∗(kf )|2 withtemperature. Here the scattering vector q is equal to kf−ki. The “Suppressed Intensity” and“Enhanced and then Suppressed Intensity” correspond to the red and blue curves in Fig. 8.1(b),respectively.Following the derivation in Chapter 3.9 in Ref. [1], we find that the QPI intensity insidethe superconducting gap (near EF ) varies with temperature in two ways, depending onthe sign difference between ∆(ki) and ∆(kf ). As shown in the blue curve in Fig. 8.1(b),the factor |u(ki)u∗(kf ) + |v(ki)v∗(kf )||2, where |v(ki)v∗(kf )| is the absolute value ofv(ki)v∗(kf ), is first enhanced approximately by a factor of 2 and then suppressed below Tc.While the factor |u(ki)u∗(kf )− |v(ki)v∗(kf )||2 is quickly suppressed below Tc, followingthe red curve in Fig. 8.1(b). This temperature dependence establishes a set of rules fordifferent pairing symmetries, as listed in Table 8.1.1158.2. What does the s±-wave pairing symmetry tell us?These coherence effects have been observed in conventional s-wave superconductorsby ultrasonic and nuclear magnetic resonance (NMR) measurements. The ultrasonicattenuation of an s-wave superconductor appears as the red curve in Fig. 8.1(b) [184].The nuclear spin relaxation rate in an s-wave superconductor follows the blue curve inFig. 8.1(b), which is also know as the Hebel-Slichter Effect [185].However, for an s±-wave superconductor, ultrasonic and NMR measurements maynot apply because they mix the scatterings of the same sign and of the opposite signs.QPI in LiFeAs perfectly avoids such issues since the h-h and h-e QPI intensities are wellseparated in q-space. Thus, the temperature dependence of the QPI intensity insidethe superconducting gaps can be used to determine the pairing symmetry. Based onTable 8.1, in the case of nonmagnetic impurities and a s±-wave pairing symmetry, theh-h and h-e QPI intensities follow the red and blue curve, respectively.In summary, I here demonstrated an alternative method using Bogoliubov QPI toidentify the pairing symmetry in LiFeAs. In combination with other methods that havebeen applied in LiFeAs in previous chapters, STM is shown to be a truly powerful tech-nique in determining the superconducting order parameter.8.2 What does the s±-wave pairing symmetry tellus?In a simplified picture, s±-wave pairing symmetry suggests a pairing on the nearest-neighboring (NN) sites in a square lattice, as demonstrated in Sec. 2.3.2. In LiFeAs,the first Brillioun zone shown in Fig. 8.2(a) corresponds to a unit cell with two ironatoms at two “inequivalent” lattice sites 12 which are shown in gray and red dots inFig. 8.2(b) [19, 42]. Thus, the electron pairing occurs on the NN sites in either the grayor the red iron lattice.The two “inequivalent” iron sites can translate to each other by symmetry operations.Hence, for simplicity, let’s view the two inequivalent iron sites as identical sites, namely12We note that the two atomic sites are essentially equivalent because they can translate to each otherby certain symmetry operations.1168.2. What does the s±-wave pairing symmetry tell us?WJ(a) (b)h3e1,e2-π π+--- -h1 h2, e-e-Figure 8.2: Pairing in the iron lattice of LiFeAs. (a) The Fermi surface of LiFeAsfrom a tight-binding model (Appendix A). The sign of the superconducting gaps are markednear the hole pockets (+) and the electron pockets (-), respectively. (b) The lattice with two“inequivalent” iron sites, indicated by gray and red dots. The electron pairing occurs betweenNN’s in either the gray or the red lattice.treating the gray and the red lattices as the same, resulting a blue lattice in Figs. 8.3(b)and 8.3(c). As a result, the NN sites in the gray lattice in Fig. 8.2(b) become the NNNsites in the blue lattice in Fig. 8.3(c) and 8.3(b). In the simplified iron lattice, electronspair together on the next-nearest-neighboring (NNN) sites, as depicted in Fig. 8.3(c).This indicates that the attractive channel in the overall repulsive interaction exists on twoNNN iron sites. As shown in Fig. 8.3(a), the NNN iron-iron interaction, W , is primarilymediated by the iron-arsenic-iron bonds, whereas the NN iron-iron interaction, J , ismainly mediated directly by the iron-iron bonds. Thus, the NNN iron-iron interactionthrough arsenic anions plays an important role in the pairing interaction in LiFeAs.From Fig. 8.3(a), one can see that As is out of the Fe lattice plane. Therefore, theNNN interaction for itinerant carriers mainly occurs through the out-of-plane iron dxzand dyz orbitals. Note also that the h1, h2 and e1, e2 pockets in LiFeAs primarily consistof carriers in iron dxz and dyz orbitals [186]. The crossing of h1, h2 with the Fermi level,which is demonstrated in chapter 7, is important for the understanding of the NNNinteraction in LiFeAs.The parent phase of the iron-pnictide superconductors has magnetic order with an-tiferromagnetic (AFM) stripes, as shown in Fig. 8.3(b) by the arrows [3, 15, 16]. ThisAFM-stripe state reminds us of the simple J1-J2 model describing the spin interactionsin a square lattice [187]. Here, J1 is the NN interaction J0, and J2 is the NNN interaction1178.2. What does the s±-wave pairing symmetry tell us?e-e-(b) (c)(a)JWFeAsWJW0J0e-e-Figure 8.3: Pairing interaction in LiFeAs. (a) The FeAs structure: a iron square latticewith arsenic atoms locating alternatively above and beneath the square centers of the ironlattice. The NN interaction J and the NNN interaction W are indicated by the dashed curves.(b) The antiferromagnetic stripe order in the iron square lattice. (c) Electrons pair on NNNsites under s±-wave.W0. When the interactions satisfy W0 > J0/2, the stripe AFM phase is realized [187]. Inthis sense, the NNN interaction also plays an important role in the magnetic state of theiron pnictides. Note that the antiferromagnetic order in the iron pnictides is an itiner-ant spin density wave and, therefore, the competition between J0 and W0 may be morecomplicated than the simple J1-J2 model. However, the J1-J2 model gives us qualitativeknowledge of the ratio of the NN and the NNN interactions.Following the discussion above, let’s take a look at the connections between the AFMphase and the superconducting phase. First, the NNN iron-iron interaction through iron-arsenic-iron bonds produces significant effects in both the AFM and the superconductingstates. Second, by comparing the red ovals in Fig. 8.3(b) and 8.3(c), the two NNN sites,in which a singlet Cooper pair is located in the superconducting state, are also siteswith an antiferromagnetic spin configuration in the AFM state. This fact suggests thatthe NNN spin interaction in AFM state also plays a role in the spin-singlet s±-wavepairing interaction. Third, the superconducting state emerges after the suppression ofAFM order in the general iron-based materials [14, 15]. Finally, antiferromagnetic spinfluctuations coexist with superconductivity in LiFeAs [38, 39] as well as in other iron-pnictide superconductors [36, 37, 41, 188]. The spin excitation occurs at a vector thatalso connects the h (∆ with positive sign) and e (∆ with negative sign) pockets in1188.2. What does the s±-wave pairing symmetry tell us?reciprocal space. These connections naturally lead to the conclusion that the pairing ofelectrons in LiFeAs is driven by the antiferromagnetic spin-fluctuation interaction.Nevertheless, the pairing interaction is likely determined by the attractive channelon NNN sites. Theoretical results indicate that the repulsive antiferromagnetic spin-fluctuation interaction possesses an attractive channel that likely to give rise to thesign-change s±-wave pairing symmetry [6, 7]. However, it might still be possible thatthe repulsive spin-fluctuation interaction only locks the phase of the pairing symmetryand the predominate attractive channel is provided through other interactions. Furtherinvestigations are required to address this question. Moreover, LiFeAs has a supercon-ducting coherence length of 4.5 nm [189], which is one order of magnitude longer thanthe NNN site distance of 0.38 nm. This indicates a retardation of the pairing interac-tion, further complicating the search for the pairing interaction. After all, the strongconnections between superconductivity and the AFM order provides a strong hint thatthe spin-fluctuation interaction is closely related to the pairing interaction in LiFeAs.119Bibliography[1] M. 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B 80, 104503 (2009). —[Cited §§. A.1and A.1][191] J. Ferber, K. Foyevtsova, R. Valent´ı, and H. O. Jeschke, Phys. Rev. B 85, 094505(2012). —[Cited §. A.1]135Appendix AQPI simulation using a T-matrixformalismDr. Steve Johnston calculated the QPI intensity maps through a T-matrix formalism.A.1 Multiorbital tightbinding modelTo model the electronic dispersion of LiFeAs we modified the ten orbital tight-bindingmodel of Ref. [190], formulated in two-Fe unit cell. In the normal state, the tight-bindingHamiltonian is given byHns(k) =∑k,σψ†k,σ hˆ (k) ψk,σ. (A.1)where ψ†k,σ = [c†k,1,σ, · · · , c†k,10,σ] is a row vector of creation operators for the ten Feorbitals. Here, we follow the notation of Ref. [190] and the matrix representation of thetight-binding Hamiltonian hˆ(k) is given therein.In order to obtain better agreement with the ARPES bandstructure at kz = 0, ahandful of the hopping parameters was adjusted. (A comparison with the ARPES dis-persion is shown in Fig. 7.8, and will be discussed in greater detail below.) Specifically(in the notation of Ref. [190]), we set 1 = -0.235, 3 = 4 = 0.23, t1018 = 0.211i, t1027 =-0.258, t1133 = 0.267, t1134 = 0.0225, t1049 = 0.377, t11001 = 0.0714, and t10118 = t10119 = 0 (in unitsof eV). The remaining parameters remain unchanged from those specified in the originalmodel. Finally, the resulting bands were renormalized by a factor of 2.17, which is typicalfor the iron pnictides [191].136A.2. Theory of multiband quasiparticle interferenceA.2 Theory of multiband quasiparticle interferenceThe QPI patterns are calculated using the usual T-matrix formalism for a single impurity,formulated for a multiorbital system [177]. The single impurity approach is justified bythe dilute concentration of impurities observed in our sample [48, 49].First, it is convenient to establish some notation by introducing the band repre-sentation for the tight-binding Hamiltonian. We define ˆ(k) = Uˆ(k)hˆ(k)Uˆ(k)†, whereˆ(k) is understood to be a 10 × 10 diagonal matrix whose diagonal elements are theeigenvalues of hˆ(k) and U(k) is the orthogonal transform diagonalizing hˆ(k), which isobtained numerically. We introduce superconductivity in band representation by assign-ing a momentum independent instantaneous intraband pairing potential ∆i(k) = ∆ito each band. The BCS Hamiltonian is then Hbcs =∑k Ψ˜†kB˜(k)Ψ˜k where Ψ˜†k =[c˜†k,1,↑, · · · , c˜†k,10,↑, c˜−k,1,↓, · · · , c˜−k,10,↓] andB˜(k) =ˆ(k) ∆ˆ∆ˆ −ˆ(−k) (A.2)is a 20 × 20 matrix. Here, operators decorated with a tilde A˜ denote operators in bandrepresentation and both ˆ(k) and ∆ˆ are 10 × 10 diagonal matrices whose i-th diagonalelement is the eigenenergy i(k) and pairing potential ∆i for band i, respectively. Sincethe impurity must be introduced at the orbital level it is convenient to return to orbitalrepresentation by reinserting the orthogonal transformation Uˆ(k)Bˆ(k) =Uˆ †(k)ˆ(k)Uˆ(k) Uˆ †(k)∆ˆUˆ∗(−k)UˆT (−k)∆ˆUˆ(k) −UˆT (−k)ˆ(−k)Uˆ∗(−k) .where T denotes the transpose, ∗ the complex conjugate, and † the hermitian conjugate.In orbital representation, the Green’s function for the clean system in the supercon-ducting state is given byGˆ0(k, ω) = [(ω + iδ)Iˆ − Bˆ(k)]−1 (A.3)137A.2. Theory of multiband quasiparticle interferencewhere δ is a broadening factor and Iˆ is the 20×20 identity matrix. The impurity inducedGreen’s function is given byGˆ(k,p, ω) = Gˆ0(k, ω)δk,p + Gˆ0(k, ω)Tˆ (k,p, ω)Gˆ0(p, ω)= Gˆ0(k, ω)δk,p + δGˆ(k,p, ω) (A.4)where Tˆ is the T-matrix obtained by solving the matrix equationTˆkp(ω) = Vˆkp +1N∑k′Vˆkk′Gˆ0(k′, ω)Tˆk′p(ω). (A.5)We consider the LDOS modulations induced by a single impurity that replaces oneof the Fe atoms in the two-Fe unit cell. For simplicity we assume that the potentialscatterer affects all orbitals on the Fe site in the same way. The impurity Hamiltonian isgiven byHimp =5∑i=1∑k,p,σV0c†i,k,σci,p,σ (A.6)where the sum over i runs over the five orbitals on one of the Fe sites. Under theseassumptions, the T -matrix is momentum independent and given byTˆ (ω) = [Iˆ − Vˆ gˆ(ω)]−1Vˆ (A.7)where gˆ(ω) = 1N∑k Gˆ0(k, ω) andVˆ = V0Iˆ 0ˆ 0ˆ 0ˆ0ˆ 0ˆ 0ˆ 0ˆ0ˆ 0ˆ −Iˆ 0ˆ0ˆ 0ˆ 0ˆ 0ˆ. (A.8)Here, each element of the matrix in Eq. (A.8) represents a 5 × 5 matrix. The Fouriertransform of the impurity induced LDOS modulations δρ(q, ω) is then given by the trace138A.2. Theory of multiband quasiparticle interferenceover the imaginary part of δGˆ(k,p, ω)δρ(q, ω) =iN∑k10∑i=1[δGˆii(k,k + q, ω)− δGˆ∗ii(k + q,k, ω)]. (A.9)For our calculations we took V0 = 50 meV, however our conclusions are not sensitive tothis value. Furthermore, we assumed superconducting gap values of ∆h1 = ∆h2 = 7 meV,∆h3 = 3 meV, and ∆e1,2 = −4 meV [32, 34]. Note that since we restrict our simulationsto energies above the gap edges, the precise choice in ∆i values is not critical to ouridentification of the QPI wavevectors.139Appendix BARPES measurement on LiFeAsDr. Giorgio Levy measured the electronic structure of these samples by using an in-houseARPES facility. The ARPES measurements were performed with a SPECS Phoibos 150analyzer and 21.218 eV linearly polarized photons from a monochromatized UVS300lamp. The LiFeAs single crystals were cleaved in-situ at a temperature of 6 K in anUHV environment with a base pressure of P = 5 × 10−11 Torr. The full width athalf maximum energy and angular resolutions were measured to be 22 meV and 0.025◦,respectively. This corresponds to a momentum resolution of 0.001pi/a.With a photon energy of 21.2 eV, and based on an inner potential V0 = 15.4 eV [186],ARPES maps the electronic excitations for k|| spanning the first Brillouin zone at theaverage perpendicular momentum kz = 2.93 × 2pi/c, where c = 6.31 A˚ is the latticeparameter perpendicular to the (100) surface [122]. This selects a k|| plane intersectingthe three-dimensional dispersion close to the Γ point (up to a reciprocal lattice vectorG = (0, 0, 6pi/c), or kz ∼ 0 in a higher Brillouin zone). The ARPES measured bandstructure provides a reference for constructing our tight-binding model that used for QPIsimulation.140

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