Visualizing the Interplay of Stru ural and Ele ronic Disorders in High-Temperature Supercondu ors using Scanning Tunneling Microscopy I T D Z P D P H C M P U ,M © A -I Z . esis advisor: Jennifer E. Hoffman Ilija Zeljkovic Visualizing the Interplay of Stru ural and Ele ronic Disorders in High-Temperature Supercondu ors using Scanning Tunneling Microscopy A e discovery of high-Tc superconductivity in over generated tremendous excitement. However, despite years of intense research efforts, many properties of these complex materials are still poorly understood. For example, the cuprate phase diagram is dominated by a mysterious “pseudogap” state, a depletion in the Fermi level density of states which persists above the superconducting critical temperature Tc . Furthermore, these materials are typically electronically inhomogeneous at the atomic scale, but to what extent the intrinsic chemical or structural disorder is responsible for electronic inhomogeneity, and whether the inhomogeneity is relevant to pseudogap or superconductivity, are unresolved questions. In this thesis, I will describe scanning tunneling microscopy experiments which probe the interplay of structural, chemical and electronic disorder in high-Tc superconductors. First, I will present the imaging of a picoscale orthorhombic structural distortion in Bi-based cuprates. Based on insensitivity of this structural distortion to temperature, magnetic eld, and doping level we conclude that it is an omnipresent background not related to the pseudogap state. I will also present the discovery of three types of oxygen disorder in the high-Tc superconductor Bi Sr CaCu O +x : two different interstitials as well as vacancies at the apical oxygen site. We nd a strong correlation between the positions of these defects and the nanoscale inhomogeneity in the pseudogap phase, which highlights the importance of chemical disorder in these compounds. Furthermore, I will show the determination of the exact intra-unit-cell positions of these dopants and the effect of different types of intrinsic strain on their placement. I will also describe the identi cation of chemical disorder in another cuprate Y −x Cax Ba Cu O −x , and the rst observation of electronic inhomogeneity of the spectral gap in this iii esis advisor: Jennifer E. Hoffman Ilija Zeljkovic material. Finally, I will present de nitive identi cation of the cleavage surfaces in Prx Ca −x Fe As , and imaging of Pr dopants which exhibit lack of clustering, thus ruling out Pr inhomogeneity as the likely source of the high-Tc volume fraction. To achieve the aforementioned results, we employ novel analytical and experimental tools such as an average supercell algorithm, high-bias dI/dV dopant mapping, and local barrier height mapping. iv Contents I . . . . S . . . R . . . . . . O . . Conventional Superconductors . . . Cu-based High-Tc Superconductors Fe-based High-Tc Superconductors Dissertation Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T M Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Home-built STM System in the Hoffman Lab . . . . . . . . . . . . . . . . . . . . . . . D O D BC Pseudogap as a Broken-symmetry State . . . . . . . . . . . . . . . . . . . . . . . . Detection of the Orthorhombic Distortion . . . . . . . . . . . . . . . . . . . . . . . Quantifying the local magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthorhombic Distortion as a Function of Doping, Temperature and Magnetic Field Imaging the Mirror Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D D P BNanoscale Electronic Inhomogeneity in Bi. . . . . . . . . . . . . . . . . . . . . . Imaging of Atomic-scale Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v . . . . O . . . . . . I . . . . . . S . . . A A A. A. A. Correlation of Oxygen Dopants with the Pseudogap State . . . . . Correlation of Oxygen Dopants with Checkerboard Charge Order Comparison of Oxygen Disorder to Other Types of Disorder . . . Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D D S BIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Previous Experimental and eoretical Studies . . . . . . . . . . . . . . . . . . Intra-unit-cell Position Determination . . . . . . . . . . . . . . . . . . . . . . . Relationship between Oxygen Disorder and Supermodulation Structural Buckling Nature of Oxygen Dopant Distributions . . . . . . . . . . . . . . . . . . . . . . Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B S E Introduction . . . . . . . . . . . . . . . . . . Surface Morphology . . . . . . . . . . . . . . Spectral Gap Dependence . . . . . . . . . . . Imaging of Resonances in the CuO Chain Plane Spectral Gap Inhomogeneity . . . . . . . . . . Conclusion and Discussion . . . . . . . . . . . I . . . . . . Y −x C . . . . . . . . . . . . . . . . . . . . . . . . −x F xB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . . . . . . . . . . . . . . . . . . O −x . . . . . . . . . . . . E I D H P Identi cation of Surface Chemical Composition . . . . . . . . . . Dopant Homogeneity in Prx Ca −x Fe As . . . . . . . . . . . . . Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xC Dri -correction Algorithm Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Details of Pseudogap vs. Dopants Correlations . . . . . . . . . . . . . . . . . . . . . . . Details of “Checkerboard” vs. Dopants Correlations . . . . . . . . . . . . . . . . . . . . vi B A B. B. R Dopant Distribution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dopant Locator Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Listing of gures . . Timeline of the discovery of new superconductors. . . . . . . . . . . . . . . . . . . . . . . Phase diagrams for high-Tc superconductors. . . . . . . . . . . . . . . . . . . . . . . . . . Crystal structure and typical STM topographs of Bi-based cuprates and YBa Cu O −x . . . . . Basic principles of STM and STS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of measurements using STM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Home-built STM system in the Hoffman Lab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . STM topographs of Bi-based cuprates. . . . . . . . . . . . . . . . . . . . . . . . BiO la ice in real and momentum space. . . . . . . . . . . . . . . . . . . . . . . Illustration of average unit cell algorithm. . . . . . . . . . . . . . . . . . . . . . Orthorhombic distortion as a function of doping, temperature, and magnetic eld. dI/dV spectra at the two inequivalent Bi sites. . . . . . . . . . . . . . . . . . . . Orthorhombic domain boundary. . . . . . . . . . . . . . . . . . . . . . . . . . dI/dV signature and density of O dopants. . . . . . . . . . Identi cation of intra-unit-cell position of + V impurities. Comparison of dopant locations and PG map. . . . . . . . Local PG versus local dopant density. . . . . . . . . . . . Comparison of dopant locations and the checkerboard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Previous studies on interstitial O dopant locations. . . . . . . . . . . . . . . . . . . . . . . Sca er plots of dopant locations within the orthorhombic Biunit cell. . . . . . . . viii . . Distribution of dopants with respect to the SM. . . . . . . . . . . . . . . . . . . . . . . . . Correlation between different types of dopants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. A. A. A. A. A. A. A. A. . . . . . . . . . . . . . . . . . . . . . . Transport measurements of Ca-YBCO single-crystals and STM topographs. . STM topographs at different temperature and magnetic eld. . . . . . . . . . e evolution of average dI/dV spectra with temperature and magnetic eld. . Identi cation of dopant positions . . . . . . . . . . . . . . . . . . . . . . . Spectroscopic imaging of resonances. . . . . . . . . . . . . . . . . . . . . . . Spatial signature of resonances at different bias. . . . . . . . . . . . . . . . . Spatial inhomogeneity in the spectral gap. . . . . . . . . . . . . . . . . . . . Spatial dependence of the bound-state peak energy and NDC minima energy. Surface morphologies of cold-cleaved Pr . Ca . Fe As . . . . . LBH comparison between × and × surface morphologies. . LBH comparison between “web-like” and × surface structures. Characteristic transport measurements of Prx Ca −x Fe As . . . . . Mapping of Pr dopants in Pr . Ca . Fe As . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier transforms of STM topographs before and a er the process of dri correction. Typical dI/dV spectra from Tc = K sample, exemplifying the extraction of Δ . . . . Binned dI/dV spectra from Tc = K sample. . . . . . . . . . . . . . . . . . . . . . . Gapmaps acquired across different dopings. . . . . . . . . . . . . . . . . . . . . . . Cross-correlation of PG and O disorder. . . . . . . . . . . . . . . . . . . . . . . . . Average PG vs. distance from O defects. . . . . . . . . . . . . . . . . . . . . . . . . Non-dispersing CB pa ern in dI/dV. . . . . . . . . . . . . . . . . . . . . . . . . . . Raw and ltered differential conductance. . . . . . . . . . . . . . . . . . . . . . . . Dopant proximity to the peaks of the CB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. . Robustness of dopant locator algorithm used to locate the dopants. . . . . . . . . . . . . ix D C , . x Acknowledgments I am forever indebted to my family for their unconditional support. To my wife Courtney for being my best friend and partner in crime who helped me make it to the end when, at times, it seemed impossible. To my parents, for their unwavering love and support throughout the years and for making sacri ces to help me achieve my dreams. To my brother, Mihajlo for helping me see the lighter side in life and not take myself too seriously. I would also like to thank my labmates and friends who helped, both professionally and personally, on numerous occasions. To Liz Main, for being an amazing mentor who generously invested much of her time and energy into training me. To Dennis Huang, for being the best mentee I could have hoped for, who made the nal stretch of my Ph.D. so much more manageable. To Can-Li Song, for his positive a itude and enthusiasm for science. To Nick Litombe who was always a phone call away ready to help, working on achieving greater things from BNL. To Anjan Somyanarayanan, for his insight over the years, and for making the long days and weekends feel a li le shorter. To Michael Yee, for all the good times we had, both in the lab and in the sun, and more good times to come. To Yang He, for his passion for physics and many exciting exchanges of scienti c ideas. To Magdalena Huefner, for her willingness to listen when I needed to vent and for our many conversations about life and physics. To Jeehoon Kim, Yi Yin, and Martin Zech for invaluable advice and guidance when rst starting up in the lab. To Alex Frenzel, Cun Ye, Sarah Schlo er and Harry Mickalide for shared laughs and making the lab a more pleasant place. I would like to thank my friends outside of the world of science, John Kim and Leanna Work for sticking with me from day one and helping me feel at home in Boston. To my tennis partners and friends throughout the years Erin Fleming, Tarek Austin, Greg Leya, Jake Dockterman, Anthony Carpenter, Ma hew Anstey, and Ronald Kamdem who kept me sane and were always ready to hit when I was stressed, or just sit down and talk on the side of the court about nearly anything. xi Finally, I would like thank my thesis commi ee, Prof. Subir Sachdev and Prof. Markus Greiner for valuable feedback and interesting ideas about future scienti c projects. To Prof. Eric Hudson, for his immense contributions to nearly every project I was involved with, invaluable advice and patience no ma er how busy he was. Most of all, I’d like to thank my thesis advisor, Prof. Jenny Hoffman for her wisdom, guidance and generous nancial support over the past ve and a half years. e opportunities and equipment she provided and the environment she created helped me excel and achieve the lo y goals I set for myself when I rst started graduate school. I have been lucky enough to learn from her about all aspects of a scienti c project, from acquiring data to completing a manuscript, and none of my academic success would be possible without her guidance. ank you all again so much. I could not have done this without you. xii Introdu ion 1 . C S Conventional superconductors exhibit two de ning properties: (I) vanishing of electrical resistivity below a critical temperature Tc , and (II) expulsion of magnetic ux below a critical eld Hc . e former was discovered by Kamerlingh-Onnes in [ ], and the la er by Meissner and Oschsenfeld in [ ]. Almost years a er the initial discovery, Bardeen, Cooper and Schrieffer proposed a theory explaining the mechanism of superconductivity (BCS theory) [ ]. According to BCS theory, a small a ractive force between electrons, arising from their retarded interaction with the la ice (electron-phonon coupling), can pair the otherwise repulsive electrons, opening up a spectral gap in the electronic density of states (DOS) corresponding to the pairing energy. ese “Cooper pairs” form an aggregate ground state which facilitates movement of charge through the crystal without loss of energy from sca ering. 140 120 HgBaCaCuO TlSrBaCuO BiCaSrCu2O9 YBa2Cu3O7 Temperature (K) 100 80 60 40 20 Hg Nb NbN V3Si NbC LaBaCuO4 Nb3Ge Nb3Sn 77K LN2 Gd1-xThx FeAsO La(O1-x Fx )FeAs Pb 0 1900 4.2K LHe 1920 1940 1960 1980 2000 2020 Year of Discovery Figure 1.2.1: Timeline of the discovery of new superconductors. Orange circles represent Cu-based high-Tc superconductors, and green circles represent Fe-based high-Tc superconductors. . C - H -Tc S By the s, superconductivity was considered theoretically solved but technologically marginal as the highest Tc was only K for Nb Ge. Furthermore, McMillan had predicted the maximum Tc for phonon-induced Cooper pairing to be around K [ ], which is still inconveniently low for widespread commercial application. en in came the discovery of an oxide superconductor with Tc of K (nominal composition Bax La −x Cu O ( −y) ) [ ], whose properties could not be explained by the BCS theory. Within one year, Tc in this new class of superconductors had already broken the liquid nitrogen barrier with the discovery of YBa Cu O −x (YBCO) with Tc of K [ ] (Fig. . . ). Since the common property of all these materials was the layered crystal structure that contained one or more CuO planes, they were called “cuprates.” In contrast to the conventional superconductivity of elements, simple alloys, and stoichiometric compounds, high-Tc superconductivity in cuprates typically arises by off-stoichiometric doping of an a T* T Pseudogap T* b T Structural transition AFM SC Hole doping SC Electron doping SC SDW SC Hole doping Electron doping Figure 1.2.2: Phase diagrams for (a) Cu-based and (b) Fe-based high-Tc superconductors. The horizontal axis in both cases represents charge carrier doping. antiferromagnetic parent compound. e parent compound is a Mo insulator due to the Coulomb repulsion that prevents double occupancy of Cu la ice sites. Upon the introduction of extra charge carriers into the system (hole or electron doping), the charges begin to hop between sites, and the electronic properties of these materials change dramatically [ ]. As antiferromagnetism is weakened by doping, the material becomes superconducting, with Tc rising to a maximum and falling again, while the decreasing spectral gap indicates weakening pairing in the “overdoped” regime. Another prominent feature in the cuprate phase diagram is the mysterious “pseudogap” (PG) state characterized by the opening of a gap in the electronic DOS even above Tc , predominantly on the underdoped side (Fig. . . a). A er more than years of cuprate research, neither superconductivity nor the PG are well understood, and it is still debated whether the PG is a manifestation of a competing/coexisting order of charge- or spin-density wave (CDW)[ ], or a Cooper-paired precursor to the superconducting state [ ]. . . C S Crystal structures of some of the prominent members of the cuprates family are shown in Fig. . . . e experiments described in Chapters - have been performed on Bi-based cuprates (Bishown in Fig. . . a), and the measurements presented in Chapter have been acquired on Ca-doped YBCO (Fig. . . b). d Figure 1.2.3: (a) Unit cell of Bi-2212. The crystal cleaves between two BiO layers. STM topographs of the BiO layer in (b) Bi-2212 and (c) Pb-doped Bi-2201. (d) Unit cell of YBCO. The crystal does not have a natural cleavage plane, and typically reveals either (e) CuO chain plane, or (f) BaO plane in STM topographs. b a BiO SrO CuO2 Ca CuO2 SrO BiO BiO SrO CuO2 Ca CuO2 SrO BiO CuO BaO CuO2 Y CuO2 BaO CuO b c 5 nm 5 nm e f 10 nm 1 nm e family tree of Bi-based cuprates, discovered in [ , ], boosted Tc up to ∼ K, and constituted the rst generation of commercially produced high-Tc cables [ ]. Since adjacent BiO layers are weakly bonded by the van der Waals force, Bi-based cuprates typically cleave on a charge neutral plane )), and has thus been the main between the two BiO layers (see dashed line in Fig. . . a for Bitarget among cuprates for STM studies. STM experiments reveal a clear Bi la ice in topographs, as well as bright and dark “patches” arising from dI/dV spectral inhomogeneity (Fig. . . b). An additional complication in many Bi-based cuprates is the structural “supermodulation” (SM), a ∼ Å incommensurate sinusoidal modulation that pervades the bulk of the material. It is seen in STM topographs with additional “ ” quasiperiodic distortions at the modulation crests (Fig. . . b). Pb dopants substituted at the Bi sites can be used to partially or completely suppress this superstructure [ ] as seen for Pb-doped Biin Fig. . . c. Bi-based cuprates are non-stoichiometric, with several types of interstitial atoms, la ice site vacancies, and isovalent or aliovalent substitutions introduced into these compounds to achieve superconductivity. In contrast to Bi-based cuprates, YBCO lacks a natural cleavage plane (Fig. . . d), but cryogenically-cleaved surfaces usually reveal either the BaO plane [ – ] (Fig. . . e) or CuO chain plane [ , – ] (Fig. . . f). Although these surfaces may be imaged with atomic resolution, neither show a superconducting gap. . F - H -Tc S e discovery of high-Tc superconductivity in Fe-based materials (Fe-SCs) in [ ] provided a new opportunity for comparison with cuprates [ ]. Similar to cuprates, Fe-SCs exhibit a dome-shaped superconducting phase diagram with a layered crystal structure and an antiferromagnetic parent compound (Fig. . . b). However, the parent state antiferromagnetism is itinerant, and of a different ordering wavevector than cuprates. Superconductivity can be induced or enhanced in Fe-based parent compounds by a wider variety of mechanisms [ , ], including electron or hole doping, chemical or external pressure, and apparently even treatment with alcoholic beverages [ ]. Almost ve years since the initial discovery, the maximum Tc in bulk Fe-SCs remains capped at ∼ K for Sm . La . O . F . FeAs [ ], while superconductivity up to K has very recently been reported in single layer FeSe [ ]. . D S High-Tc superconductors are typically electronically inhomogeneous at the atomic scale, but to what extent the intrinsic chemical or structural disorder is responsible for electronic inhomogeneity, and whether the inhomogeneity is relevant to superconductivity, are unresolved questions. In contrast to bulk-probe techniques that measure average properties over large areas of the sample, scanning tunneling microscopy (STM) is a real-space technique able to measure the electronic density of states with atomic resolution. In this thesis, I will talk about the experiments I have done on investigating the interplay of chemical disorder, strain, and electronic inhomogeneity in high-Tc superconductors. In Chapter , I will give a brief theoretical background on scanning tunneling microscopy technique, and talk about different types of measurements possible. In Chapter , I will discuss the rst real-space detection of structural distortion and its relation to the PG state in Bi-based cuprates. In Chapter , I will talk about the rst direct imaging of all oxygen interstitial atoms, as well as oxygen vacancies in Bi, and the correlation between their positions and the observed inhomogeneity in electronic DOS. In Chapter , I will provide a detailed determination of interstitial oxygen dopant intra-unit-cell locations and their dependence on strain due to both SM buckling and orthorhombic distortion in Bi. In Chapter , I will describe the experiments on Ca-doped YBCO samples involving the observed inhomogeneity in spectral gap, as well as an unknown set of C -symmetric impurities. In Chapter , I will show the utility of local barrier height mapping in determining the nature of different surface morphologies observed in an Fe-SC Prx Ca −x Fe As . Scanning Tunneling Microscopy 2 . B P Scanning tunneling microscopy (STM) was pioneered by Binnig and Rohrer in [ ], a remarkable discovery subsequently awarded a Nobel Prize in Physics in . e experimental technique is based on the principles of quantum tunneling of electrons between two electrodes separated by a potential barrier. e experimental setup consists of a sharp metallic tip which is brought within several Å of a conducting metallic surface using a -dimensional piezoelectric scanner (Fig. . . a). is scanner can position the tip both laterally (in the xy-plane) and vertically (in the z-direction) with sub-Å precision. Application of voltage between the tip and the sample allows electrons to quantum-mechanically tunnel between the two (Fig. . . b). e resulting tunneling current can be calculated using the time-dependent perturbation theory. If a positive V is applied to the sample, the Fermi level of the sample shi s down with respect to the Fermi level of the tip, and electrons tunnel from the occupied states of the a Piezoelectric scanner +X +Y Feedback loop b electrons tunneling Iset Imeas 0 Tip V -eV Sample d Tip Sample Figure 2.1.1: (a) Schematic representation of STM. A voltage V is applied between the tip and the sample. The tip is rastered across the surface in the xy plane and its z coordinate is adjusted using the three-dimensional piezoelectric scanner controlled by a feedback loop. (b) Quantum tunneling of electrons between the tip and the sample across a vacuum barrier of width d upon the application of a voltage bias V between the two. If a positive V is applied to the sample, the Fermi level of the sample shifts down with respect to the Fermi level of the tip, and electrons tunnel from the occupied states of the tip into the empty states of the sample. tip into the empty states of the sample (Fig. . . b). e contribution to the current from tunneling of electrons from sample to tip, and from tip to sample at energy ε is: isample→tip = − e π |M| (ρs (ε) · f(ε)) · (ρt (ε − eV) · [ − f(ε − eV)]) ( . ) itip→sample = − e π |M| (ρt (ε − eV) · f(ε − eV)) · (ρs (ε) · [ − f(ε)]) ( . ) where |M| is the matrix element, ρs (ε) and ρt (ε) are density of states (DOS) of the sample and the tip respectively, and f(ε) is the Fermi distribution given by: f(ε) = ε + e kB T ( . ) Summing the two contributions, and integrating over all energies gives: I=− πe ∫ ∞ −∞ |M| ρs (ε)ρt (ε − eV)[f(ε) · [ − f(ε − eV)]] − f(ε − eV) · [ − f(ε)]dε ( . ) Since thermal broadening kB T ∼ . meV at T∼ K where most of the data presented in this thesis have been acquired, the above integral can be reduced to: I≈− πe ∫ eV |M| ρs (ε)ρt (ε − eV)dε ( . ) In most conventional STM studies, tip consists of W or PtIr alloy which has a at density of states around Fermi level (this is con rmed during the process of high-bias eld-emission on a Au substrate by acquiring I vs. V curves that should be at). In this scenario, ρt (ε − eV) ≈ ρt ( ), and: I≈− πe ∫ ρt ( ) eV |M| ρs (ε)dε ( . ) Furthermore, the matrix element |M| is nearly energy-independent [ ], and can be taken out of the integral to obtain: I≈− πe ∫ ρt ( )|M| eV ρs (ε)dε ( . ) Using the assumption that the vacuum barrier between the tip and the sample is a simple square barrier, and applying WKB approximation, the matrix element |M| can be wri en as: |M| ≈ e− s √ mφ ( . ) where m is the electron mass, s is the width of the vacuum barrier, and φ is the effective local barrier height (LBH), which represent some mixture of the tip and sample work functions [ ]. By combining Eqns. . and . , the expression for the tunneling current becomes: I≈− πe e −s √ mφ ∫ ρt ( ) eV ρs (ε)dε ( . ) In summary, tunneling current measured in STM studies at bias V is proportional to the integral of the density of states from the Fermi level to eV. a Energy x y dI/dV (arb. units) b 100 d I (pA) 10 0 -150 -100 -50 0 50 100 150 Bias (mV) 20 40 60 s (pm) 80 100 c e Figure 2.2.1: (a) Schematic of 3-dimensional data sets obtained on a pixel grid (DOS-maps). Each yellow sheet represents a single dI/dV map acquired at a different energy. Red line denotes a single dI/dV spectrum. (b) Single dI/dV spectrum obtained on Bi-2212. (c) An example of a dI/dV map obtained at ∼30 mV showing “checkerboard” charge order in Bi-2212. (d) Example of an I−Z spectrum which can be used to extract LBH. (e) An example of a typical topographic measurement obtained on Bi-2212. Data shown in (b-e) have all been acquired using the STM shown in Fig. 2.3.1a. . . . T T M STM is most commonly used in the constant-current topographic mode. As the tip is rastered across the surface in the xy-plane at a xed bias voltage V, the feedback loop adjusts the position of the tip in the z-direction to maintain the measured current Imeas at a xed setpoint value Iset (Fig. . . a). e z trajectory of the tip effectively maps the surface contour. However, as seen from Eqn. . , tunneling current is dependent on tip sample separation as well as the integral of the density of states from Fermi level to eV. erefore, for a sample with a homogenous DOS, this contour purely corresponds to the geometric surface corrugations. However, most materials exhibit a spatially heterogeneous DOS, which means that STM topographs of these compounds represent a combination of effects due to geometric corrugations and electronic density of states. Example of a topograph of Biacquired with the STM built in the Hoffman lab (Fig. . . a) is shown in (Fig. . . e). . . dI/dV In addition to acquiring information about the surface geometry, STM can obtain electronic DOS at energies of up to several electron volts, both in occupied and unoccupied sample states. e measurement is done by turning off the feedback loop (which xes the tip-sample distance d), sweeping the bias voltage V, and measuring the current response I(V). As seen from from Eqn. . , I will be proportional to the integral of the density of states from Fermi level to eV. Taking a numerical derivative, it can be shown that: dI ∝ ρs (eV) ( . ) dV From a practical stand point, taking a numerical derivative of I vs. V curve, to obtain the conductance dI/dV is extremely noisy. To circumvent this problem, dI/dV is usually measured using a lock-in ampli er technique, where a small bias voltage modulation dV (typically a few millivolts) is added to V, and the change in the tunneling current dI is measured to obtain dI/dV. An example of a typical dI/dV spectrum on Biacquired using the home-built STM in Fig. . . a is shown in Fig. . . b. . . dI/dV dI/dV spectra can be obtained on a densely-spaced pixel grid, which results in a -dimensional data set o en referred to as a “DOS map” (Fig. . . a). Extracting a single “slice” (dI/dV map) at any desired energy essentially reveals spatial distribution of the density of states. An example of a dI/dV map can be seen in Fig. . . c, which shows “checkerboard” charge modulation in Bi. In the experiments presented in this thesis, square grids of to pixel length have been used over ∼ nm areas of the sample. . . I−Z From Eqn. . , it can be calculated that: √ ln(I) ≈ −s mφ ∫ + ln( eV ρs (ε)dε) ( . ) erefore, by measuring the tunneling current I as a function of tip-sample separation s (“I − Z spectrum”), we can obtain φ which is the effective local barrier height (LBH). In practice, this process is done by stabilizing the tip in tunneling using the feedback loop (typically at pA and mV STM setup condition), and turning off the feedback loop. en, by varying the voltage applied to the scantube, tip is moved away from the sample surface, and this distance s is recorded simultaneously with measuring the tunneling current I. As it can be seen from a typical I − Z curve shown in Fig. . . d, ln(I) vs. s is linear, as expected from Eqn. . . Furthermore, the slope a of this curve can be extracted to obtain the LBH: φ= ( a) m ( . ) . H - STM S H L Scanning probes used in the Hoffman lab are all home-built. e particular STM system used to acquire most of the data presented in this thesis has been built by my colleagues, Elizabeth Main and Adam Pivonka, and myself (Fig. . . ). e microscope itself is only ∼ inches tall, and ∼ inches in diameter, and the Z-walker range is ∼ inch (Fig. . . a). Tip is pointing upwards and the sample is pointing downwards as shown in Fig. . . b. Both the tip and the sample can be changed in situ using a ∼ -foot long, vertical magnetic manipulator. e system (Fig. . . c) is designed to be ultra-high vacuum (UHV) compatible, which in turn constrained the choice of the materials (e.g. no brass). ere are several levels of vibration isolation present– both the room and the experiment table are oated on air springs to reduce any external vibrations coupling into the measurements. Although no magnetic eld data will be presented in this thesis, the system is equipped with a T vertical, and T horizontal magnetic eld. a rotor scan tube c Z-walker b sample holder tip Figure 2.3.1: (a) STM (∼6 inches tall) built by Elizabeth Main. (b) Zoom-in on the sample holder and the tip. Prominent features in (a) and (b) are denoted by arrows. (c) The experimental setup built by Elizabeth Main, Adam Pivonka and Ilija Zeljkovic. The position of the microscope is denoted by red ellipse. Real- ace Dete ion of Orthorhombic Distortion in Bi-based Cuprates 3 . P B - S Numerous symmetry-breaking electronic states have been theoretically proposed to explain the PG in the cuprate phase diagram (Fig. . . a). A two-dimensional CB charge order, may break translational but not rotational symmetry [ ], while coexisting spin and charge density wave “stripes” [ ] break both (though a precursor nematic state may break only rotational symmetry [ ]). More exotic states, like the d-density wave, similarly break translational symmetry, but also time-reversal symmetry, while preserving their product [ ]. Intra-unit-cell orbital current loops break time-reversal and inversion symmetry but also preserve their product [ ]. e experimental realization of these symmetry breaking states, and their identi cation with the PG, remain highly debated. Electronic states in Bi-based cuprates have been heavily investigated by a variety of techniques. Angle-resolved photoemission spectroscopy (ARPES) has shown electronic states breaking time-reversal symmetry [ ], as well as particle-hole and translational symmetry [ ]. STM has shown evidence for electronic CB states breaking translational symmetry [ – ] and nematic states breaking rotational symmetry [ ]. Furthermore, STM has found nanoscale variations in these symmetry breaking states [ – , , ]. us, to investigate the role of structure in these broken symmetry electronic states, it is imperative to make atomic scale measurements of the structural symmetry. Since strong electronic distortions are typically accompanied by structural distortions, determining the relationship between these orders can be complicated. However, a clue comes from their (co-)dependence on other parameters. In cuprates, both superconductivity and the PG are highly dependent on doping, temperature, and magnetic eld. In this chapter, I will show the rst imaging of structural symmetry breaking due to orthorhombic distortion in Bi-based cuprates, and investigate whether this structural symmetry breaking is similarly dependent on these parameters, or whether it is an omnipresent background within which the electronic states evolve. . D O D Structural symmetries are traditionally measured by sca ering experiments, such as X-ray or neutron sca ering to determine bulk symmetries, or low energy electron diffraction (LEED) to determine surface symmetries. Although nearly tetragonal, a ∼ . % difference between a and b axes [ ] makes the true structure orthorhombic. We apply the Lawler-Fujita dri -correction algorithm on all our data to correct for temperature dri (typically < mK), piezo hysteresis, and piezo nonlinearity present during the process of data acquisition (see Appendix A. ). e global structural symmetry can be seen in the Fourier transforms of STM topographies (Figs. . . d-f). e dominant features are the tetragonal la ice vectors ⃗ ⃗ Qx and Qy , and, in elds of view where the supermodulation has not been suppressed by Pb, the ⃗ wavevector Qsm . However, in each case we also nd one, and only one, of the orthorhombic wavevectors ⃗ ⃗ Qa or Qb , shown schematically in Fig. . . a. e presence of an orthorhombic wavevector con rms the in-plane relative shi of the two Bi atoms already noted in the average supercells (see Fig. . . a). is shi reduces the symmetry of the crystal by removing mirror planes perpendicular to the b and c axes and replacing them with glide planes, thus changing the space group of the crystal from orthorhombic Fmmm to orthorhombic Amaa. Crystals in this space group still have three-dimensional inversion symmetry; for a b c d Qb Qsm Qy Qa e Qb Qy Qa f Qb Qy Qa Qx Qx Qx Figure 3.2.1: (a-c) Topographic images of 30 nm square regions of the BiO lattice from: optimally doped Bi-2212 with supermodulation, optimal Tc = 35 K Bi-2201 with Pb doping completely suppressing the supermodulation, and overdoped Tc ∼0 K Bi-2201, also doped with Pb but with remnants of the supermodulation respectively. Each was cropped from larger (up to 80 nm) fields of view, for ease of viewing. Insets show the average supercell with arrows denoting the shifts of the ⃗ ⃗ Bi atoms. (d-f) Fourier transforms of (a-c), showing peaks at the tetragonal Bragg vectors Qx and Qy ⃗ ⃗ ⃗ ⃗ as well as at the orthorhombic Qa = (Qx + Qy )/ (but not at the equivalent Qb ). In (d) the supermod⃗ ulation creates an additional incommensurate peak at Qsm . a Qb Q y Qa Qx b b a ⃗ Figure 3.2.2: (a) In momentum space, atomic Bragg vectors (black circles) at Qx = ( π/a , ), ⃗y = ( , π/a ), define the tetragonal Brillouin zone (black square). Orthorhombic vectors (circles) Q ⃗ ⃗ Qa = (π/a , π/a ), Qb = (−π/a , π/a ) define the orthorhombic Brillouin zone (dashed square). (The average Bi-Bi nearest neighbor distance is a = . nm.) The orthorhombic distortion appears as a density wave along the a-axis, with alternate Bi rows moving closer/farther apart, leading ⃗ ⃗ to strength at Qa in the FTs. Qb (open circle) is notably absent. (b) In real space, the distortion in the surface layer of Bi (blue) and O (green) atoms appears as shifts of two Bi sub-lattices in opposite directions along the orthorhombic a-axis. This distortion breaks inversion symmetry at the Cu site, but preserves a single mirror plane along the a-axis. The undistorted Cu lattice (orange), two layers beneath the Bi atoms, is shown for reference (but does not appear in topographic images). The displacement d, of one Bi atom from its undistorted position, is marked here and reported on the left axis of Figs. 3.4.1b-d as a fraction of the orthorhombic lattice constant a , and on the right axis of Figs. 3.4.1b-d in picometers. instance in Bi, the inversion center lies in the calcium layer. However, distortion removes the two-fold rotation symmetry along the c axis, effectively breaking two-dimensional inversion symmetry in the BiO plane. e presence of only one of Qa or Qb indicates that, in Figs. . . a-c, the Bi shi is along the same orthorhombic la ice axis throughout the eld of view. is Bi shi is consistent with a distortion observed in supermodulated and un-supermodulated Biand Bi, by multiple bulk techniques [ , ]. STM now becomes the rst technique to measure this distortion locally. . . . Q R - Starting with a dri -corrected topograph (schematic shown in Fig. . . a), we create an average single-Bi unit cell with more pixels per atom than our raw data (typically x pixels per unit cell). For each pixel in the actual topograph, the location can be calculated with picoscale precision in relation to the nearest Bi atom. e data from that pixel is then placed in the appropriate bin in the average unit cell (Fig. . . b). is process builds up a histogram of weight at each sub-unit-cell-resolved position in the average unit cell, ultimately showing a high-resolution map of an average Bi atom (Fig. . . c). An analogous process can be used to create an average supercell of any size. e creation of an average supercell allows an improvement of intra-unit-cell spatial resolution, at the expense of inter-unit-cell variation. is method allows us to detect variations in atomic positions as small as a few thousandths of a unit cell. For example, from a x supercell, we t a peak to each of the four “atoms” in order to calculate their exact positions, and nd orthorhombic distortion shi s as small as . % of a unit cell. e x supercells for several different Bi-based cuprates are shown as insets in Fig. . . a-c. We compare our technique to the original supercell averaging technique used by Lee et al. [ ]. Lee’s starting point for aligning the different unit cells to be averaged was a real-space Gaussian t to each atomic peak in the topograph. eir ing procedure allowed location of the atom with picometer accuracy, but proved sufficiently time consuming that only unit cells were ultimately averaged. Our algorithm which combines Lawler’s dri correction and supercell averaging can locate individual atoms in un-supermodulated BSCCO with typical ∼ picometer absolute accuracy and ∼ picometer relative accuracy within a single eld of view [ ]. Furthermore, the centers of these atoms can be automatically located, and the resultant average supercell computed, for > , unit cells in just a few seconds of run time. . . F - We derive here the relationship between the two observed quantities: the value of the complex Fourier component at Qa (shown in Fig. . . a), and the displacement d (shown in red in Fig. . . b) directly measured in the average supercell. b a0 c a Figure 3.3.1: (a) Schematic of a drift-corrected topographic image with Bi atoms represented as blue circles. The data is acquired in a pixel grid outlined in black; the center of each pixel is marked with a red dot. The resolution of this image is only slightly better than the Nyquist frequency for atoms. (b) Schematic of the average 1 x 1 unit cell we will create with 15 x 15 pixels. We calculate the exact distance of every pixel in (a) to the nearest Bi atom (4 example distances are shown in green in (a)), then place it in the average unit cell in (b) to create a weighted histogram at each sub-unit-cell pixel. (c) Example of the final average unit cell obtained from a real topographic image. We start by representing the topographic signal T(⃗ and its spatial derivative ⃗ (T(⃗ in terms of the r) r)) r ⃗ ⃗ ⃗ Bragg peak wavevectors Qx and Qy , the wavevector of the orthorhombic distortion Qa , and the complex ⃗ Fourier component A at wavevector Qa . (We normalize the expression so that a Cu atom sits at the origin, ⃗ ⃗ and the strength of the signal at wavevectors Qx and Qy is equal to .) r r r r r r T(⃗ = eıQx⃗ + e−ıQx⃗ + eıQy⃗ + e−ıQy⃗ + AeıQa⃗ + A∗ e−ıQa⃗ r) ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ( . ) r r r r ⃗ ⃗ ⃗ ⃗ (T(⃗ = ıQx eıQx⃗ − ıQx e−ıQx⃗ + ıQy eıQy⃗ − ıQy e−ıQy⃗ + r)) ⃗ ⃗ ⃗ ⃗ ⃗r ⃗ ⃗r ⃗ AıQa eıQa⃗ − A∗ ıQa e−ıQa⃗ ( . ) Our objective is to nd the value of⃗ at which T(⃗ reaches its maximum. is is the position in space to r r) which the distorted Bi atom moves, with respect to an undistorted ( , ) position. Se ing the gradient of ⃗ ⃗ ⃗ T(⃗ to , and using the fact that Qa = (Qx + Qy )/ , we nd that: r) A ⃗ r A∗ ⃗ r ⃗r ⃗ ⃗r Qx (eıQx⃗ − e−ıQx⃗ + eıQa⃗ − e−ıQa⃗ ) + A ⃗ r A∗ ⃗ r ⃗r ⃗ ⃗r Qy (eıQy⃗ − e−ıQy⃗ + eıQa⃗ − e−ıQa⃗ ) = ( . ) ⃗ ⃗ Since Qx and Qy are orthogonal, both parenthetic expressions must vanish. Because the parenthetic ⃗ ⃗ ⃗ ⃗ ⃗ expressions are equal, we see that Qx · r = Qy · r. Because Qa = (Qx + Qy )/ , we see that ⃗ ⃗ ⃗ Qa · r = Qx · r = Qy · r. In full generality, we express the complex distortion strength A in terms of its half amplitude and phase, A = A eıθ . erefore, ⃗ r) ⃗ r sin(Qx ·⃗ + A sin(Qa ·⃗ + θ) = ⃗ r) ⃗ r sin(Qy ·⃗ + A sin(Qa ·⃗ + θ) = ⃗ r) ⃗ r ⇒ sin(Qa ·⃗ + A sin(Qa ·⃗ + θ) = From this expression, we nd the distortion d as a percent of the orthorhombic unit cell. ⃗ r πd = Qa ·⃗ = arctan( −A sin(θ) ) + A cos(θ) ( . ) ( . ) π One can simplify this expression considerably by noting that the orthorhombic distortion is odd about each Cu site. In other words, the ideal topography T(⃗ could be expressed more simply as: r) ⃗ r) ⃗ r) ⃗ r) T(⃗ = cos(Qx ·⃗ + cos(Qy ·⃗ + A sin(Qa ·⃗ r) ( . ) (using the same normalization as in Eqn. . ). erefore, for the perfect la ice, θ = −π/ . In this case we nd simply that πd = arctan(A ). However, imperfections in the application of the dri correction algorithm, at most ∼ % of a unit cell [ ], may introduce a small phase error, so that θ = −π/ + ε. We then nd: πd = arctan( A cos(ε) ) + A sin(ε) ( . ) Because all quantities (d, A , and ε) are small, we expand the argument as: πd ≈ arctan(A ( − ε )( − A ε)) ≈ arctan(A ( − A ε − ε )) ≈ arctan(A ) ( . ) In conclusion, the complex value A of the Fourier transform at the Qa point is expected to be purely imaginary. Small (up to % unit cell) errors from imperfect dri correction may introduce a small real component to A, but such errors contribute only a second order correction to the relation πd = arctan (A ). . O F D F D ,T M - e question remains: what relationship, if any, does this structural distortion bear to the variety of predicted and observed electronic orders, particularly to highly-debated claims [ ] that the PG is characterized by intra-unit-cell inversion symmetry breaking? To investigate this, we characterize the dependence of the structural distortion on parameters which are known to heavily in uence electronic ordered states: doping, temperature, and magnetic eld. Fig. . . a locates in a three-dimensional phase diagram the datasets in which we measured the structure. e key results are summarized in Figs. . . b-d. We do not nd a dependence of the structural distortion on doping, temperature, or eld, across a wide range of values. We have measured the distortion both inside and outside the superconducting state, and in samples with vastly different PG energies, yet the structural distortion appears insensitive to these effects. We therefore conclude that the structural inversion symmetry breaking state is an omnipresent background against which the electronic states evolve, and not the long-sought PG state itself. In order to further investigate the spectral effect of strain from the inversion symmetry breaking orthorhombic distortion, we compare the gap at the two inequivalent Bi sites in the orthorhombic unit cell. We nd no evidence that the gap changes between these two sites. A representative example (chosen from our largest image) is shown in Fig. . . . . I M P In non Pb-doped Bi, we nd that the orthorhombic distortion lies along the a-axis (wavevector at ⃗ Qa ), orthogonal to the supermodulation in all samples investigated. e probability that this is a coincidence is less than . %. In Pb-doped samples, however, the supermodulation is suppressed to the point where it no longer forces the orthorhombic distortion along the a-axis. Although without the supermodulation it is impossible for us to differentiate between the a and b axes, we did, as shown in Fig. . . , nd one domain boundary, where the distortion rotates by ◦ . is is rare – in all other images, up to nm square, we nd, as shown in Fig. . . , a single orthorhombic distortion vector. e probability of nding a domain boundary is on the order of L/D, where L is the size of our image and D is the size of a domain; from images of average size nm, we estimate the size of a domain to be on the order of one micrometer. To nd domain boundaries we map the spatial dependence of the strength of the topographic signal A at the orthorhombic wavevectors. is is calculated by demodulating the signal (shi ing the Fourier transform so the desired wavevector⃗ is at zero frequency, low-pass ltering and then inverse q ⃗ r) ⃗ r) transforming), giving us a spatially dependent Fourier amplitude A(⃗ ,⃗ One of A(Qa ,⃗ and A(Qb ,⃗ q r). ⃗ r) ⃗ r), will be zero and the other non-zero, so that a map of the difference, D(⃗ = A(Qa ,⃗ − A(Qb ,⃗ will have r) a roughly uniform magnitude and sign. e spatial distributions of these maps are re ected in the error bars of Fig. . . . In Fig. . . , however, the sign of D(⃗ ips between the upper and lower half of the r) 1.2 b 50 6 Shift (% Ortho Unit Cell) a 40 30 T (K) 20 15 10 10 0 0.1 0.2 0.3 0 B (T) 5 0.8 4 0.4 2 0.0 0 3 6 9 0 Magnetic field (T) Tc(K) 1.2 4 0 c 35 0 Shift (% Ortho Unit Cell) c 0.8 6 d Shift (% Ortho Unit Cell) 20 15 10 5 0 0.30 Shift (picometers) 4 2 0.4 2 1 0.0 0 6 TC 18 Temperature (K) 12 0 24 0 0.05 0.10 0.15 0.20 0.25 Doping (p from TC) Figure 3.4.1: (a) Schematic phase diagram of Bi-2201, locating (red dots) the 15 experimental conditions under which the 21 datasets used in this figure were taken. (b-d) show orthorhombic distortion as a function of (b) magnetic field, (c) temperature, and (d) doping [49]. (b) and (c) each contain data from a single tip and region of overdoped Bi-2201 samples with Tc = 16 K and 15 K respectively. Additional scatter in (d) may reflect differences in tip quality or field of view across the 14 samples. Shift (picometers) 3 Shift (picometers) a dI/dV (nS) 6000 b 4000 2000 0 -150 -100 -50 0 50 Bias (mV) 100 150 Figure 3.4.2: (a) 2 x 2 average supercell from Tc =35 K underdoped Bi-2201 with Pb dopants used to suppress the supermodulation. (b) Average spectra from the two inequivalent Bi sites show identical gap energy. gures, highlighting the rotation of the distortion axis between these two regions. clear in Fourier transforms from the two different regions (Figs. . . c-d). e difference is also . C D In conclusion, we have demonstrated the rst local imaging of the orthorhombic distortion present across the phase diagram in Bi-based cuprates. Furthermore, we have devised a novel algorithm to extract the magnitude of this distortion from STM topographs, and have shown how to utilize two-dimensional Fourier-transforms to detect the local mirror plane. Based on lack of dependence of magnitude of the orthorhombic distortion on temperature, magnetic eld and doping, coupled with insensitivity of dI/dV spectra to the position at the two inequivalent Bi sites, we conclude that the PG state is not sensitive to the strain of the orthorhombic distortion. In claiming a broken electronic symmetry state, one should know whether the electronic order breaks or preserves the structural symmetry of the crystal. Bulk probes such as X-ray diffraction and neutron sca ering cannot determine whether electronic and structural orders choose the same symmetries a b Qb Qa c Qa d Qb Figure 3.5.1: (a) Topographic image of Tc = 25 K underdoped Bi-2201. (b) The distortion map, ⃗ r) ⃗ r), D(⃗ = A(Qa ,⃗ − A(Qb ,⃗ of the same field of view, where A(⃗ ,⃗ is the spatially dependent Fourier r) q r) amplitude. A change in sign (color) indicates the rotation of the distortion axis and associated mirror plane across the domain boundary separating top and bottom of the figure. (c-d), Fourier transforms of the top and bottom of (a) respectively, highlight the rotation of the orthorhombic distortion vector ⃗ ⃗ from Qa to Qb . locally; in contrast, STM can investigate these symmetries on atomic length scales. For example, we suggest that future STM studies should search for modi cation of the local electronic symmetry across structural domain boundaries, such as that shown in Fig. . . . Furthermore, as the structural inversion symmetry breaking may lead to the appearance of electronic inversion symmetry breaking, we suggest that any reports identifying this inversion symmetry breaking with the PG phase should explicitly track the broken symmetry with temperature, magnetic eld, and doping, as we have done, to ensure that it changes as would be expected with the changing PG energy scale. Finally, implementation of our average supercell algorithm, coupled with the ability of STM to measure structural and electronic symmetries locally, will become an important part of future STM studies, not only on cuprates, but on many important materials, such as the new iron-based superconductors, where the cleaved surface structure remains controversial [ ]. Our real-space algorithm can even enable atom-speci c high-resolution unit cell mapping not possible with Fourier techniques, in cases where two visibly different species are stochastically mixed at the sample surface, such as Se and Te site in FeTe −x Sex which can be sorted automatically based on their bimodal distribution of topographic height [ ]. Oxygen Dopant Disorder and Pseudogap in Bi- 4 . N E I B- STM studies of Bidensity of states. have revealed several types of spatial nanoscale inhomogeneity in the electronic • Spectral gap magnitude. e Fermi level gap in the density of states is found to vary by ∼ % in width, on a - nm length scale, across a wide range of doping [ , ]. Because of the similar energy scales near optimal doping [ ], there has been persistent confusion between the identi cation of the superconducting gap and the pseudogap (PG) – a mysterious depression in the near-εF density of states which surprisingly persists far above the superconducting Tc on the underdoped side of the phase diagram . . a. Many early STM studies discuss superconducting gap inhomogeneity, but more recent normalization [ ] and ing [ ] techniques reinforce that the PG represents a distinct energy scale and phase [ ], and suggest that the observed spectral gap inhomogeneity is due primarily to variations in the PG. is is signi cant, because it is believed that the PG phase competes with superconductivity [ , ], so the nanoscale variations in PG suggest the possibility to control and mitigate its competition, if only the hidden variable determining its local strength can be found. • Checkerboard charge order. A second type of inhomogeneity is a disordered periodic modulation of the spectral weight which is static but most noticeable at energies within and near the PG energy. e wavevector of this modulation tracks the antinodal nesting wavevector across a wide range of doping in single [ ] and double [ ] layer BSCCO. is modulation has been termed “checkerboard” [ ], “ uctuating stripes” [ , ], “electronic glass” [ ], or “charge density wave” (CDW) [ ]; for simplicity we will use the la er terminology. Field [ ], doping [ ], and temperature [ ] dependence suggest that this static spatial modulation is in fact the electronic ordered phase associated with the PG. • Quasiparticle interference. A third type of disordered periodic inhomogeneity arises from elastic sca ering between degenerate states; this dispersing ‘quasiparticle interference’ (QPI) may exist at similar wavevectors to the static checkerboard modulation, but only at a limited range of energies within the superconducting gap [ ]. Although several theoretical studies have predicted nanoscale electronic inhomogeneity [ , ], the roles of spontaneous electronic phase separation [ ] and chemical disorder [ ] remain unresolved. Since a combination of off-stoichiometric doping by oxygen intercalation and/or cation substitution is necessary to achieve superconductivity in Bi-based cuprates, this chemical disorder has been the main candidate to cause the observed spatial inhomogeneity in the electronic DOS. In particular, the random distribution of interstitial oxygen dopants within the crystal seems like an obvious explanation behind the inhomogeneity in the spectral gap [ ]. However, this hypothesis was difficult to test since typical, low-energy STM experiments were unable to locate these dopants, and more challenging experiments at higher energies were needed. In a pioneering study, McElroy et al. imaged a set of interstitial oxygen atoms in Bias atomic-scale resonances in dI/dV images at - . V bias (type-B Oxygen atoms) [ ]. However, the correlation between the locations of the observed O dopants and the magnitude of the spectral gap was relatively weak and of unexpected sign: although it is well-established that the dominant spectral gap, now known to be the PG, decreases monotonically with increasing O dopant concentration [ ], McElroys O dopants were more likely to be found in the areas of the high PG. To resolve the issue, one carefully tuned theory suggested that the dominant local effect of interstitial oxygens may be strain, which increases the local pairing strength [ , ]. A second proposal by Zhou et al. [ ] postulated the existence of two types of interstitial oxygen dopants: type-B Os observed by McElroy, which live around the BiO plane and contribute only delocalized charge, and type-A Oxygen atoms, which reside around the SrO plane and have an immediate electrostatic effect, locally hole-doping the adjacent CuO layer. erefore, Zhou et al. predicted that the type-A oxygens would follow the expected global trend: a strong anticorrelation with the PG. However, the type-A oxygens were predicted to have resonances even farther below Fermi level than the – . V type-B Os. is experimental challenge has prevented their observation to date. In this chapter, I will describe the rst imaging of type-A Os, as well as surprising vacancies at the apical O site (AOVs), and their correlation with the observed electronic inhomogeneity in Bi. . I A - D Fig. . . a shows a topograph of the BiO surface of Biwith Tc = K, demonstrating atomic resolution at + V bias. Fig. . . b shows dI/dV(− V) map acquired over the same area as Fig. . . a, containing atomic scale features of similar form and concentration to the previously observed type-B O dopants [ ]. Fig. . . c extends the energy range down to show dI/dV(− . V) map, resolving for the rst time a second set of atomic scale features presumed to be the predicted type-A interstitial oxygen [ ]. Fig. . . d extends the energy range above Fermi level to show dI/dV(+ V) map, revealing a third set of atomic scale features. No other distinct atomic-scale features emerge in differential conductance images at biases between - V and + . V. Typical dI/dV spectra at each of the three types of dopants, and the background, are shown in Fig. . . e. e de ning features used to identify the impurities, such as the peak around – V for a type-B oxygen, and the sudden changes of the slope around – . V and + . V for type-A and + V feature, respectively, are robust for different setup conditions, locations, and samples. In order to determine and verify the identity of + V features, which were previously unobserved and not theoretically predicted, we look at: (I) dependence of concentration with doping, and (II) lateral intra-unit-cell position. a b 5 nm Low High 5 nm Low High c d 5 nm 5 nm type-B oxygen type-A oxygen apical O vacancy background e dI/dV (arb. units.) 80 60 40 20 0 -1.5 f % per CuO2 5 4 3 2 1 -1V (type-B oxygen) -1.5V (type-A oxygen) +1V (apical O vacancy) -1.0 -0.5 0.0 0.5 Bias (V) 1.0 1.5 0 50 55 60 65 70 75 80 85 90 95 Tc (K) Figure 4.2.1: (a) Atomically resolved topographic image of Bi-2212 with Tc = 55 K, acquired at Vsample = +1 V and setpoint current I = 150 pA over a 35 nm area (inset, 3 times magnification). (b-d) dI/dV images in the same area as (a), acquired at Vsample = –1, –1.5, and +1 V, respectively. (e) dI/dV spectra over each dopant type, from the Tc =82 K sample. Each colored spectrum is the average of all imaged dopants of its type within a 30-nm field of view; the black trace shows the average spectrum far from dopants. Setup condition: 260 pA and -650 mV. (f) Density of each type of dopant imaged within four samples studied. . . O We repeat the same high-bias measurement, and map the three types of dopants on four different samples, and show that the density of each type of dopant vs. Tc in Fig. . . f. As expected, we observe a monotonic decrease in the number of both types of interstitial oxygen dopants with falling Tc . However, the + V features increase from virtually zero for optimally doped Bi, to almost % per CuO ◦ plaque e for the Tc = K sample. is rules out cation substitutions, since annealing up to C does not change the cation concentrations in Bi[ ], and it can be concluded that these defects are a direct consequence of oxygen vacancies at la ice sites. . . I + V ere are three distinct O la ice sites (all laterally offset from one another) where O vacancies could be created (Fig. . . a). To determine the exact positions of the atomic-scale features appearing in differential conductance maps at biases higher than + . V, we rst utilize the Lawler-Fujita dri -correction algorithm [ ] on each dataset to locate each Bi atom with picoscale accuracy (Appendix A. ). When this algorithm is used with correction length scale Λ greater than the supermodulation wavelength Λ SM ∼ a , the effect is to remove the inevitable slow thermal and piezo dri which occur during a multi-hour map, without altering the true structure of the surface. However, for the purpose of locating the dopant features within the unit cell, we need a “perfect” square grid. We therefore choose a short correction length scale Λ ∼ a which removes both the slow dri which is an experimental artifact, and also most of the real lateral distortion of the supermodulation, as seen in Fig. . . b. We next perform a Gaussian t to nd the center of each feature in the + V dI/dV map, and determine its lateral location with respect to the nearest atoms in the corrected topograph. We directly compare the + V feature locations to each of the three distinct O la ice sites, illustrated in Fig. . . a. Fig. . . c-e shows the lateral position of each + V feature within the unit cell, for two samples. For each sample, the same set of + V features is replo ed times, with respect to each layer, to emphasize its relation to the inequivalent oxygen la ice sites. Due to the large size of the dopant (full width at half max ∼ . nm, compared to the . nm unit cell size), and due to the difficulty in removing % of the supermodulation distortion, there remains some sca er in the lateral locations, but we are still able to distinguish clearly between the three possible O sites. We nd that the + V features are consistent only a 1 2 3 BiO SrO CuO2 Ca CuO2 SrO BiO b c 1. BiO cell d 2. SrO cell e 3. CuO2 cell Tc=55K Tc=82K +1V feature density (a.u.) f from BiO plane O Tc = 55K Tc = 82K g from apical O h from CuO2 plane O 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 Distance (a0) Distance (a0) Distance (a0) Figure 4.2.2: (a) Top half of Bi-2212 unit cell denoting the three inequivalent lattice oxygen sites. (b) Drift-corrected Bi-2212 topograph (Vsample =600 pA, Iset =+150 mV) over a 4 nm area, with a perfect square lattice of small yellow circles superimposed. The yellow cross denotes the position of one +1 V feature. The brightness of this spot suggests that the +1 V feature is not a missing Bi atom. (c-e) Positions of each localized +1V feature (blue dots) within a single unit cell in two samples. (c) is plotted with respect to the BiO layer (Bi: blue; O:green); (d) is plotted with respect to the SrO layer (Sr: purple; O: green); (e) is plotted with respect to the CuO layer (Cu: orange; O: green).(f-h) Areal density of +1 V features as a function of lateral distance from each of the three distinct oxygen lattice sites shown in (c-e). with the apical oxygen site. is conclusion is reinforced by plo ing the density of observed + V features vs. distance from each of the possible O sites, in Fig. . . f-h. . C O D P S Having identi ed the positions of three types of O dopants, I proceed to discuss the correlation between the local observation of these and the inhomogeneous PG state. Fig. . . a-c shows the three types of dopants (type-B O, type-A O and AOV respectively), superimposed on a PG map acquired for the Tc = K sample (see Appendix A. . for details of PG determination). Any correlation between type-B Os and PG is too weak to be seen by the naked eye. In contrast to Zhou’s prediction, the type-A Os are correlated with regions of large PG. One immediately sees the strongest correlation between AOVs and PG. Fig. . . d shows the crosscorrelations between the positions of the dopant atoms and the corresponding gap map (for crosscorrelation algorithm details, see Appendix A. . ). Fig. . . e shows the average PG as a function of distance from the nearest dopant. is behavior is consistent across the three underdoped samples studied (Appendix A. . ). We examined more carefully the surprising departure from Zhou’s prediction in Fig. . . , which plots the local PG energy versus the local concentration of each dopant type. For the type-B interstitial Os in Fig. . . a, the local trend within each sample shows correlation between increased local oxygen concentration and increased local PG; the global trend between samples shows anticorrelation between global oxygen concentration and global PG. In contrast to Zhou’s prediction, Fig. . . b shows that a qualitatively similar juxtaposition of local and global trends holds for the type-A interstitial oxygens in the three underdoped samples. However, Fig. . . c shows excellent alignment of local and global trends for the correlation between AOV density and PG. is suggests that, in underdoped samples, variations in the local hole concentration, and thus the local PG, are governed primarily by the local removal of holes by AOVs, rather than the strain or the donation of localized holes from the interstitial oxygens. e interstitial oxygens, previously predicted [ ] to determine the local gap, do contribute de-localized holes, but in underdoped samples even the type-A interstitial O positions remain correlated to regions of decreased local hole concentration, likely due to a tendency to remain close to the AOVs. In the optimally doped sample, the AOV concentration is negligible, so the local hole concentration may be set by the next-closest dopant to the CuO plane, the type-A interstitial O, as predicted by Zhou et al. [ ]. Indeed, a b 5 nm Correlation coefficient 5 nm 0.5 0.4 0.3 0.2 0.1 0.0 c d 0 1 type-B oxygen type-A oxygen apical O vacancy 2 3 Distance [nm] 4 5 5 nm 40 meV 100meV 120 110 100 90 80 70 60 Gap [meV] 0 e 1 2 3 Distance [nm] 4 5 Figure 4.3.1: Locations of (a) type-B interstitial Os (green circles), (b) type-A interstitial oxygens (red circles), and (c) AOVs (blue circles) superimposed on top of the PG map of the Tc =55 K sample. (d) Cross-correlation coefficient relating the PG to the distance to the nearest dopant. (e) Average PG versus distance from the nearest dopant of each type. In (d) and (e), green, red, and blue lines represent type-B oxygens, type-A oxygens, and AOVs, respectively. a 150 type-B oxygen T c=55K T c=68K T c=82K T c=91K b Local gap (meV) 150 type-A oxygen Local gap (meV) 100 Local gap (meV) 100 T c=55K Tc =55K T c=68K Tc =68K T c=82K Tc =82K T c=91K Tc =91K c 150 apical O vacancy 100 Tc =55K Tc =68K Tc =82K 50 2.5 4.0 3.0 3.5 Local % per CuO 2 50 1.5 2.5 2.0 Local % per CuO 2 50 0.0 1.0 Local % per CuO 2 2.0 Figure 4.3.2: For each sample, we create a local density map of the number of a given dopant type per region of radius x∼2 nm. The values in the density map are then binned based on the PG value at the corresponding pixel. The average PG value in each bin is then plotted versus the average dopant density value within each bin. (a) Local PG versus local density of type-B interstitial oxygen dopants, from the four different samples used in this study. (b) Local PG versus local density of typeA interstitial oxygen dopants, from the four different samples used in this study. (c) Local PG versus local density of apical oxygen vacancies, from the three underdoped samples used in this study. Trend lines in (a) to (c) are guides to the eye. Fig. . . b shows that the local PG decreases with increasing local type-A concentration in the Tc = K sample. We therefore reconcile the apparent contradiction between the global [ ] and local [ , ] doping trends for both types of interstitial oxygens. . C O D C C O Another manifestation of electronic inhomogeneity is a disordered CB modulation of the spectral weight that is static and possibly associated with the PG phase. A closely related disordered QPI periodic inhomogeneity may exist at similar wave vectors to the static CB, but only at a limited range of energies within the superconducting gap [ ]. It was previously claimed that type-B O dopants are found in the minima of the QPI pa erns [ ], at both positive and negative energies. However, QPI has opposite spatial phase for lled and empty states [ ], suggesting that the previously observed correlation [ ] relates instead to the CB. First, we have closely examined the energy dependence of the CB pa ern observed in our Bisamples. Due to insensitivity of CB wavevector to imaging bias (see Appendix A. . ), it can be concluded a b Low High Figure 4.4.1: (a) Fourier-filtered dI/dV image of the Tc =55 K sample at +36 mV, showing a clear CB (setup: Vsample =–150 mV; Iset =800 pA). Type-B Os, type-A Os, and AOVs are superimposed as green, red, and blue circles, respectively. (b) Dopant density of each dopant type versus distance from the center of the nearest CB maximum. that the phenomenon observed is indeed static CB modulation, and not energy-dispersive QPI. Figure . . a shows the distribution of the three types of observed atomic dopants on top of a low-pass ltered dI/dV( mV) image. A histogram of the distance from each dopant to the center of the nearest “checker” (Fig. . . b) demonstrates the strong tendency of AOVs to lie in the peaks of the imaged CB and a weaker tendency for interstitial O dopants to lie in the troughs (see Appendix A. . for algorithm details). We therefore conclude that the AOVs play the primary role in pinning the CB. . C O D O T D In addition to previously described O dopant disorder, there exist several other kinds of chemical disorder in Biinvestigated in literature. • Sr site defects. Since Bicrystals are difficult to synthesize in the perfectly stoichiometric form [ ], extra Bi atoms are commonly used to facilitate the growth process, and most crystals contain ∼ − % excess Bi substituted at the Sr site [ ]. Because the Sr site is located next to the apical O site (which is directly above Cu, see crystal structure in Fig. . . a), it might be expected that the substitution at this site would have a strong effect on the electronic properties of the crucial CuO plane. Kinoda et al. imaged Sr site defects as a set of bright features in STM topographs of Pb-doped Biat bias greater than + . V. e dI/dV spectrum on top of a Sr site defect shows a resonance peak at + . V. Even though no quantitative analysis was presented [ ], we have analyzed the published data to obtain only a small positive correlation between the two of ∼ . as compared to ∼ . for AOVs. Hence, although not the strongest correlation, Sr site defects also seem to be correlated with the observed PG state inhomogeneity. • Cu site defects. It has been reported that Cu site impurities (Ni, Zn dopants and intrinsic crystal defects), have a tendency to appear in the areas of small spectral gap (< meV) [ ]. However, since spectral gap inhomogeneity is observed in the absence of Zn or Ni doping (or intrinsic crystal defects which we rarely observe in our extensive studies of Bi), we rule these out as the cause of the spectral gap variation. Nevertheless, due to the correlation of their distributions with the areas of the small spectral gap, it seems plausible that the electronic inhomogeneity could be pinned by (some of) these impurities. • Pb substitutions. Kinoda et al. studied heavily-overdoped Bicrystals with Pb substitutions [ ] and claimed to observe no correlation between the Pb dopant positions and the magnitude of the spectral gap, therefore ruling out these dopants as the cause or the pinning sites of the nanoscale heterogeneity. e lack of correlation between Pb dopants and the spectral gap was consequently con rmed in Pb-doped Bicrystals [ ]. Both experiments seem to agree that although Pb dopants add carriers and alter structure, their effect on the electronic properties is not localized. . C D In conclusion, we have extended the energy range of local spectroscopy on Biby ∼ to image for the rst time two types of interstitial O dopant atoms, as well as AOVs. We have shown that AOVs have the strongest local correlation with the areas of the large PG, whereas the correlation of type-A O and type-B O dopants with PG is comparatively much weaker. It can be concluded that AOVs enhance the magnitude of PG phase locally and pin the maxima of CB charge modulation, while the interstitial O atoms contribute mostly delocalized charge and may exhibit weak tendency to occur in the minima of CB charge modulation. Based on other experimental studies, the correlation of other types of defects such as Sr and Cu site defects, and Pb substitutions seem to be weaker compared to the AOVs. Finally, the discovery of AOVs and type-A O resolves an outstanding mystery of opposing local vs. global correlations between interstitial O dopants and the PG. But a question remains – does dopant inhomogeneity cause PG inhomogeneity or merely pin intrinsic inhomogeneity, caused perhaps by strong correlations? Because the - to - nm length scale of PG inhomogeneity remains the same across a wide range of dopant concentrations (Fig. A. . a, inset), intrinsic PG inhomogeneity seems plausible. However, our experiment shows clearly that the dopant locations that are xed at room temperature, particularly the apical oxygen vacancies, do enhance the local PG strength that is subsequently determined on cooling through the PG transition temperature T∗ . Finally, is there a relationship between apical oxygens and superconductivity itself? In fact, apical oxygen effects may be credited for the discovery of high-Tc superconductivity, as M¨ller was originally u driven to explore the LaBaCuO system by the expectation of strong electron-phonon coupling due to the Jahn-Teller effect in the CuO octahedral environment, speci cally the displacement of the Cu-apical-O bond [ ]. However, subsequent isotope effect measurements suggested that any phonon contribution to superconductivity is dominated by CuO plane oxygen [ ]. We conjecture that AOVs in uence the superconductivity indirectly in underdoped samples by reducing the local hole concentration, which locally strengthens the PG, tying up antinodal states [ , ], and locally decreasing the Fermi level carriers that would otherwise be available for coherent pairing. Calculations also showed a correlation between Tc and apical oxygen height and emphasized the importance of the axial orbital for the hopping and phase coherence necessary for superconductivity [ , ]. us, AOVs may lower the local superconducting Tc and/or critical current Jc . We propose a possible route to increase Tc in Bi: underdope to increase the pairing potential [ ] but explore different annealing recipes to allow interstitial oxygen removal without creating AOVs, the defects most favorable to the PG and possibly competitive to superconductivity. Recently, a ∼ % increase in maximum Tc has been predicted from alternate dopant arrangements [ ]. Oxygen Dopant Disorder and Strain in Bi- 5 . I As discussed in previous chapters, Bi-based cuprates are an exceptionally complex family of materials. Even though STM has proven to be the front-runner technique in studying these materials at the atomic scale [ , ], the interpretation of the results has sometimes been hindered by the intricacy of the crystal structure. e ideal la ice of Biis complicated due to the presence of commensurate (see Chapter ) and incommensurate supermodulation (SM) structural distortions [ ] that pervade the bulk of the crystal, off-stoichiometric doping of oxygen atoms and cation-substitutions [ , ]. In Chapter , we have shown the rst direct imaging of all oxygen dopants introduced in the parent compound to induce superconductivity in this material. However, the exact intra-unit-cell positions and electronic effects of these dopants remain controversial. With recent advances in algorithms used to analyze STM data [ ], it is now possible to determine the positions of atoms in STM topographs with picometer precision. Using the Lawler–Fujita dri -correction Figure 5.2.1: Gray shaded area represents a vertical “cut” through the crystal structure of Bi-2212 to emphasize the positions of type-B Os. Yellow spheres represent the type-B O positions predicted by theory, and light blue sphere is the position extracted from X-ray experiments. Pink vertical line shows the position of type-B O atoms obtained by McElroy et al. Only four layers (BiO, SrO, CuO and Ca) of the Bi-2212 unit cell are shown for simplicity. Black arrows denote the direction of orthorhombic distortion of Bi and O atoms in the BiO layer. algorithm [ ], we are able to pinpoint the exact positions of both types of interstitial oxygen dopants within the unit cell, and show how their distributions are related to AOVs. We also discovered that the spatial distribution of one type of interstitial oxygen atoms is tied to the periodicity of the SM. . P E T S Early X-ray studies on dopant disorder in Bi[ – ] have found extra oxygen atoms inserted into the BiO plane, but their exact positions within the unit cell have been complicated to extract due to the difficulty of analyzing higher harmonics of the SM. In an X-ray study done almost years ago, Levin et al. determined that the interstitial oxygen atoms are laterally located one half of the distance between neighboring Bi atoms [ ] (blue sphere in Fig. . . ). No real-space probe has been able to evaluate this observation until an STM study imaged one subset of oxygen dopants as atomic-scale resonances in dI/dV maps at - V bias (type-B O) [ ], and determined their position to be approximately . Å away from the O in the BiO layer, along the BiO direction (purple line in Fig. . . ). A theoretical study used density-functional calculations to determine several stable positions of type-B O atoms within the unit cell that reproduced the resonance in dI/dV at - V bias [ ]. e most energetically favorable position was found to be in a vertical plane connecting two neighboring Bi atoms (“He ” yellow sphere in Fig. . . ). However, none of these theories or experiments took into account the orthorhombic distortion of the unit cell (Chapter ). . I - - P D To determine the positions of both type-A and type-B O dopants within the Biunit cell, we acquire topographs and simultaneous dI/dV images of BiO layer at - and - . V bias (Chapter ), and locate the exact positions of oxygen dopants with respect to the Bi la ice seen in the topographs. We use the Lawler-Fujita dri -correction algorithm [ ] with short correction length scale ∼a on all our data that removes the artifact slow dri due to experimental conditions and most of the true lateral distortion due to incommensurate SM, and places Bi atoms on a nearly perfect square la ice (Appendix A. ). To determine the centers of individual dopant atoms, we use a simple two-dimensional Gaussian ing method. Since apparent positions of dopants determined by this method can be in uenced by other dopants in the vicinity, we determine the positions of only the dopants where we can choose a rectangular ing window (FW) that completely encompasses exactly one atom. e dopant position determined by this procedure is insensitive to the size of the FW that we choose, as it varies by less than . % of a unit cell with the varying size of FW. Fig. . . a shows the sca erplot of the lateral positions of type-B O dopants within two different orthorhombic BiO unit cells. In a perfect tetragonal cell, type-B O locations would appear to be directly above the O site in the BiO layer (O(Bi)), which would be impossible due to the la ice O already occupying the site. However, we resolve this apparent discrepancy by separating the distribution of dopant locations into two distinct positions within the orthorhombic unit cell. Each la ice O shi s ∼ % away from the high symmetry point [ ], and we now nd that type-B interstitial Os are located on the opposite side of the unit cell from the shi ed la ice O. Furthermore, the distribution of type-B Os is “stretched” along one particular Bi-O direction which is parallel to the SM wavevector in both samples a 1.0 0.5 0.0 Lattice site Bi O Interstitial O T =55K c T =82K c Lattice site Sr O Interstitial O T =55K c T =73K c Lattice site Sr O AOV T =55K c T =82K c 0.0 0.5 1.0 1.5 2.0 d b 1.0 0.5 e 0.0 c 1.0 0.5 f 0.0 Figure 5.3.1: (a-c) Distributions of type-B O, type-A O and AOV within the orthorhombic unit cell for two different samples. Coordinates of the lattice atoms in all panels have been taken from a neutron diffraction study by Miles et al. [83]. (d-f) dI/dV map of 1 nm square region containing a single type-B O (-1 V), type-A O (-1.5 V) and AOV (+1 V) respectively, for Tc =73 K sample. Setup conditions for (d-f) are -1 V, -1.5 V, +1 V and 300 pA, 50 pA, 200 pA respectively. Open white circles show the idealized Bi lattice in the simultaneously acquired BiO topographs. studied. It can be concluded that type-B O atoms lie directly below O(Bi), consistent with the second most energetically favorable position “He ” shown in Figure . . . We repeat the same process to locate the positions of type-A Os, place them within two distinct orthorhombic SrO unit cells, and nd that they lie at the la ice O site in the SrO layer (O(Sr)) (Fig. . . b). No discernible difference can be seen between the distributions in the two orthorhombic unit cells, which is somewhat expected as the orthorhombic distortion affecting O(Sr) is much smaller than the corresponding distortion of O(Bi). However, O(Sr) is vertically displaced by ∼ . Å compared to the plane containing Sr atoms, thus possibly leaving enough space for type-A Os to position themselves just below O(Sr). For comparison, we plot again the sca erplot of AOVs in Fig. . . c (shown in Chapter ). Due to the apparent larger size of the AOVs (compared to interstitial Os), a greater sca er of points exists between Fig. . . a and Figs. . . a-b. In order to illustrate the difference in apparent sizes between different dopants in dI/dV images, Figs. . . d-f show a nm dI/dV map with exactly one type-B O, type-A and AOV respectively. e surprising conclusion of the analysis of positions of both types of interstitial oxygen atoms is that the two clearly different positions we observe are both inconsistent with the theoretically predicted position “He ” and X-ray position “Levin”, both shown in Fig. . . . . R B O D S S We further investigate the relationship between the distribution of interstitial oxygen atoms, AOVs, and the incommensurate structural SM. e period of the SM in Biis ∼ unit cells, and is oriented ◦ with respect to the Cu-O bond. We use an algorithm described in detail by Slezak et al. [ ] to determine the phase of the SM for every pixel in the eld-of-view (FOV), and calculate the average density of the dopants of each type as a function of the SM phase. Fig. . . a shows a . nm wide atomically-resolved topograph of Bi. Type-A O dopant density as a function of the SM phase is shown in Fig. . . b, and it reveals a strong inclination of type-A O dopants to sit in the peaks of the SM. is observation is in contrast to previously investigated type-B O dopant spatial dependence [ ] that might exhibit a weak bimodal distribution for some samples, peaking at both the crests and the troughs of the SM (Fig. . . c). So for the rst time we have identi ed a set of oxygen dopants that are spatially distributed with the SM periodicity. AOVs may also be more likely to appear in the peaks of the SM (Fig. . . d), but this a 1.5 1.0 0.5 0.0 b Density of type-A O [nm-2] 91K 82K 68K 55K 1.0 Density of type-B O [nm-2] c 91K 82K 68K 55K 0.5 0.0 d 0.2 82K 68K 55K Density of AOV [nm-2] 0.1 0.0 0 60 120 180 240 Phase (degrees) 300 360 Figure 5.4.1: (a) BiO layer STM topograph acquired at 200 mV and 50 pA, ∼2.5 nm wide, showing distortions in the Bi lattice over one period of the SM. (b-d) Histograms of densities of type-A O, type-B O and AOVs respectively as a function of the phase of the SM. Different shades of red, green and blue indicate data obtained on samples of different doping concentrations. Peak of the SM is at 180 degrees as emphasized by the topograph in (a). tendency is much weaker compared to the sharply peaked distribution of type-A Os in Fig. . . b. . N O D D Having identi ed the positions of all interstitial oxygen atoms and AOVs within the same FOV, we proceed to determine if different sets of dopants are prone to clustering, or if their distributions are uncorrelated with one another. e algorithm used to extract this information essentially histograms and compares pair-wise distances in experimental and random datasets to extract their ratio as a function of distance (described in detail in Appendix B. ). If the obtained ratio quotient is: • > then the dopants a ract (prefer clustering) • ∼ then the dopant distribution is random • < then the dopants repel and are even more homogenous than you would expect from a random distribution Both type-A and type-B Os show a tendency to repel each other (Fig. . . a-b) in all samples except for type-A Os in the most underdoped Tc = K sample. is tendency is expected from particles of like-charge that prefer keeping themselves as far as possible from one another. e distribution of type-A Os in the Tc = K sample is likely affected by AOVs, as the two have a strong positive correlation in all samples (Fig. . . c), with the correlation being the strongest in the Tc = K sample. e interstitial atom – la ice vacancy pairs, known as Frenkel defects that were rst theoretically predicted to exist in [ ], have since also been observed as displaced oxygen atoms near la ice vacancies in TiO [ ]. Furthermore, since AOVs create a cloud of local positive charge, one would expect negatively charged interstitial oxygen atoms to be a racted towards the vacancies. We also observe a positive correlation between type-B O atoms and AOVs (Fig. . . d), but this correlation seems to be much weaker than the correlation between type-A Os and AOVs. Since AOVs have a well-de ned position in the SrO layer, this information provides us with a clue that type-A O atoms are indeed in the SrO layer, as it has been previously predicted [ ]. Finally, we nd that the AOVs like to cluster together, with a preferable separation of ∼ - nm (Fig. . . e). Correlation between type-A O and type-B O is inconsistent between different samples, exhibiting a weak negative correlation in the most underdoped sample, and a positive correlation in the optimally doped sample in the absence of AOVs (Fig. . . f). a Ratio of experimental and random distributions 1.0 b Ratio of experimental and random distributions 55K 68K 82K 91K 1.0 0.5 0.5 55K 68K 82K 91K 0.0 0 2 type-B O – type-B O 4 Distance (nm) 6 55K 68K 82K 0.0 8 0 2 type-A O – type-A O 4 Distance (nm) 6 55K 68K 82K 8 c Ratio of experimental and random distributions 2.0 d Ratio of experimental and random distributions 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0 2 type-A O – AOV 4 Distance (nm) 6 8 0.0 0 2 type-B O – AOV 4 Distance (nm) 6 8 e Ratio of experimental and random distributions 4.0 f Ratio of experimental and random distributions 55K 68K 82K 1.5 3.0 1.0 2.0 0.5 1.0 55K 68K 82K 91K 0.0 0 2 4 Distance (nm) AOV – AOV 6 8 0.0 0 2 type-A O – type-B O 4 Distance (nm) 6 8 Figure 5.5.1: (a-f) Ratios of experimental and random distributions of dopants as a function of distance for type-B O – type-B O, type-A O – type-A O, type-A O – AOV, type-B O – AOV, AOV – AOV, and type-A O – type-B O respectively. Values of the ratio greater than 1 indicate inclination of two types of dopants to appear closer to each other compared to a random distribution, whereas values that are less than 1 signal tendency of the two types of dopants to “repel” each other. . C D In this Chapter, we have presented a comprehensive investigation of the role of strain on the intra-unit-cell placement of both types of interstitial oxygen dopants, and discussed the correlation of interstitial Os and AOVs. We have shown that type-B O dopant lateral position is located at the O site in the BiO plane (consistent with “He ” position, see Fig. . . ), and that the strain of the orthorhombic distortion plays an important part in allowing this intra-unit-cell location. Furthermore, type-A O dopants are located closer to the apical O site, likely just below O(Sr) in the SrO layer as previously predicted [ ] due to their strong tendency to occur in the vicinity of AOVs (which have a well-de ned position in the SrO plane). Type-A O dopants also exhibit a strong tendency to lie in the peaks of the incommensurate SM. We hypothesize that SM strain, which “pushes” atoms further apart in its crest, creates extra space for type-A O dopants to t within the crystal. We hope that the precise identi cation of the exact positions of all interstitial oxygen atoms within the unit cell will allow more computational modeling and theoretical work to be done to further the understanding of the impact of strain on the electronic states in Bi. Impurity-induced Bound States and Ele ronic Inhomogeneity in Y −xCaxBa Cu O −x 6 . I e main obstacle in studying YBCO single-crystals using surface-sensitive probes is the surface “overdoping” catastrophe produced during the process of cleaving [ ], which in turn results in a non-superconducting topmost bilayer. Furthermore, YBCO lacks a natural cleavage plane, and cryogenically-cleaved surfaces usually reveal either the BaO plane or CuO chain plane. e BaO termination plane is observed in STM topographs as a clean square la ice of Ba atoms (Fig. . . f). e cleaved CuO chain surface exhibits a complicated one-dimensional modulation (Fig. . . e) initially a ributed to a CDW state [ , , ], but later proposed to be a consequence of superconducting quasiparticle sca ering, likely due to O vacancies in the CuO chain plane [ ]. Although these surfaces may be imaged with atomic resolution, neither show a gap which is proven to be of superconducting origin by temperature or magnetic eld dependence. As-grown surfaces of YBCO do retain their bulk superconducting properties as demonstrated by direct vortex imaging [ ], but lack the atomic resolution of cryogenically-cleaved YBCO. Recently, Ca substitution at the Y site in YBCO has been shown to circumvent the overdoping problem [ ]. Furthermore, Zabolotnyy et al. hypothesized that this substitution resulted in an alternate cleavage plane due to the strain introduced by the Ca dopants. To test this hypothesis, a local probe is naturally needed. . S M Single-crystals of underdoped Y −x Cax Ba Cu O −x (Ca-YBCO) used in the study contain ∼ Ca substitution at the Y site, which has been checked with energy-dispersive X-ray measurements. Each Ca + substitution at the Y + donates one hole to the bulk. Since Ca-YBCO is a double-layer system, ∼ Ca substitutions leads to ∼ hole doping per CuO . For the equivalent Ca-free YBa Cu O −x (YBCO) crystals with oxygen content + x = . , we estimated a hole doping level of ∼ based on the c-axis la ice parameter of the crystal [ ]. erefore, we annealed the Ca-doped crystals to this oxygen doping level, which should give a total hole doping level of ∼ . e Ca-YBCO crystals have a superconducting Tc of about K, de ned as the midpoint of the transition (Fig. . . a). Single crystals of Ca-YBCO with Tc of ∼ K are cleaved in ultra-high vacuum at cryogenic temperature, and immediately inserted into the STM head where they are imaged with a PtIr tip, cleaned by eld emission on polycrystalline Au foil. Fig. . . b shows a typical STM topograph obtained on cryogenically-cleaved surfaces of Ca-YBCO samples. In contrast to the atomically-resolved CuO chain plane or BaO surface seen in optimally doped YBCO (without Ca substitutions), no atomic resolution has been achieved in Ca-YBCO. e surface is disordered but atomically at, with no step edges or valleys appearing over several hundred nanometer areas (Fig. . . c). Our observation is consistent with the hypothesis that Ca doping in YBCO results in an alternate cleavage plane, which is distinct from either CuO or BaO planes and is presumed to be the Y/Ca layer [ ]. We deem the CuO surface termination unlikely to explain the disordered surface seen in Figs. . . b-c, as atomically-resolved CuO surfaces have been imaged in Bi[ ], and we observe no atomic signature. We further note that similar observations of a disordered surface has been reported in Prx Ca −x Fe As where reconstruction of a fraction of the Ca-layer results in a “web-like” surface morphology (Chapter ). erefore, we suggest that a Magnetic susceptibility 0.0 -0.5 -1.0 0 10 20 30 Temperature (K) 40 50 b c 3 nm 0 pm 81 pm 50 nm Low High Figure 6.2.1: (a) Magnetic susceptibility measurements of Ca-YBCO samples with Tc ∼30 K as determined by the midpoint of the superconducting transition. STM topographs of Ca-YBCO, revealing a new, disordered surface morphology over (b) 15 nm, and (c)∼300 nm square regions. Setup conditions in (b): Iset =500 pA and Vset =-60 mV; in (c): Iset =10 pA and Vset =-500 mV. a 10 nm 0T b 10 nm 3T c 10 nm 6T d 10 nm 9T e 5 nm 15 K f 5 nm 20 K g 5 nm 25 K h 5 nm High 30 K i 5 nm 34 K Low Figure 6.3.1: (a-d) STM topographs acquired over the same region of the sample at 7 K and 0, 3, 6, and 9 T magnetic fields respectively. (e-i) STM topographs acquired over the same square region of the sample at 0 T magnetic field and 15 K, 20 K, 25 K, 30 K, and 34 K respectively. Ca-YBCO crystals indeed cleave at the Y/Ca layer (see crystal structure in Fig. . . d). . S G D Typical spectroscopic measurements on the surface shown in Fig. . . b consistently show a gap in the density of states. To determine the nature of this gap, we investigate its dependence on temperature and the applied magnetic eld. In order to show the ability to track the same area of the sample at different temperatures and magnetic elds, we show the regions of the sample where average dI/dV spectra have been obtained (Fig. . . ). a5 dI/dV (a. u.) 4 3 2 1 0 -40 -20 0 20 Bias (mV) 34K 30K 25K 20K 15K b 1.0 Ratio of dI/dV 0.5 15K/34K 20K/34K 25K/34K 30K/34K c 1.0 Ratio of dI/dV 0.5 0.0 40 -40 -20 0.0 15K/20K d5 dI/dV (a. u.) 4 3 2 1 0 e 1.0 Ratio of dI/dV 0.5 0T 3T 6T 9T 0 20 Bias (mV) 40 -40 -20 0 20 Bias (mV) 40 f Ratio of dI/dV 1.0 0.5 0T/3T 0T/6T 0T/9T -40 -20 0 20 Bias (mV) 40 0.0 -40 -20 0 20 Bias (mV) 40 0.0 -40 -20 0 20 Bias (mV) 3T/6T 40 Figure 6.3.2: (a) Average dI/dV spectra from region in Figs. 6.3.1e-i at different temperatures, all thermally-broadened to 34 K. (b) Ratio of dI/dV spectra below Tc (15 K, 20 K, 25 K, and 30 K) and above Tc (34 K). (c) Division of average dI/dV spectra from (a) at temperatures of 15 K and 20 K showing less than 1% difference. (d) Average dI/dV spectra from a 50 nm square region in Figs. 6.3.1a-d at different magnetic fields. (e) Ratio of dI/dV spectrum at 0 T, and the ones acquired at 3 T, 6 T, and 9 T. (f) Ratio of average dI/dV spectra from (d) for 3 T and 6 T, exemplifying no difference in spectra to 1 %. Setup in (a,b,c): Iset =50 pA, Vset =-50 mV; (d,e,f): Iset =100 pA, Vset =100 mV, 7 K. . . T dI/dV Average dI/dV spectra acquired over the same nm square region of the sample and thermally broadened to K are shown in Fig. . . a. ermal broadening represents convolution with the derivative of the Fermi function. is procedure accounts for the decrease in STM energy resolution at higher temperatures, and allows direct comparison of dI/dV curves obtained at different temperatures. e spectral gap observed is insensitive to the change in temperature from below to above Tc , which implies that the prominent gap observed is not of superconducting origin. In a related cuprate Bi, Boyer et al. have shown that the prominent gap observed by STM is also not of superconducting origin, but is actually the mysterious PG [ ]. However, they were able to extract the hidden superconducting gap of ∼ % by the division of dI/dV spectra acquired at different temperatures. In Ca-YBCO, the ratios of dI/dV spectra acquired below and above Tc (Figs. . . b-c) do not show a gap. We emphasize that the spectral division technique is very sensitive to subtle STM tip changes (even though average dI/dV spectra in Fig. . . a appear nearly identical), and con rm the lack of emergence of a superconducting gap by the division of dI/dV spectra at K and K, which shows no gap to less than (Fig. . . c). erefore we conclude that the spectral gap observed in Ca-YBCO using STM is not superconducting, although ARPES has been able to detect superconducting signatures in the same material [ ]. We hypothesize that the contradiction stems from the difference in experimental techniques– ARPES might be able to probe deeper into the crystal compared to STM, which is likely only sensitive to a few atomic layers, far less than one bilayer unit cell. erefore, we can conclude that the Ca-YBCO surface also suffers the overdoping catastrophe of non-Ca-doped YBCO samples, at least as far as STM studies are concerned. . . T dI/dV To further con rm the origin of the spectral gap, we investigate its dependence on the applied magnetic eld. When a magnetic eld is applied to a type-I superconductor, magnetic ux is expelled to a thin layer on the surface. In contrast, in type-II superconductors such as high-Tc superconductors, magnetic ux penetrates as quantized ux tubes called vortices. Previous STM studies have imaged disordered vortex la ices in several cuprates, such as Bi[ , ], and YBCO [ ]. Vortices in these studies appear as regions of: (I) higher differential conductance in dI/dV maps near the Fermi level, or (II) lower differential conductance in dI/dV maps close to the spectral gap energy. In contrast to vortex imaging in these compounds, no vortices have been detected in Biunder the application of magnetic eld using either of these methods [ ]. However, simple division of average dI/dV spectra in and out of magnetic eld revealed the opening of a gap, signaling the existence of superconductivity [ ]. In ten approaches on two different samples, we have been unable to image the vortex la ice in Ca-YBCO. Fig. . . d shows average dI/dV spectra over the same region of the sample under the application of different magnetic elds which look qualitatively very similar. Furthermore, the division of a b 3 nm 3 nm Figure 6.4.1: Dopant locations identified by the algorithm denoted as light-blue circles superimposed on top of (a) topograph and (b) integral of dI/dV from -9 mV to 9 mV. Setup: Iset =50 pA, Vset = -60 mV, and 7 K. average dI/dV spectra at different magnetic elds con rm our previous conclusion (Figs. . . e-f), as the division of dI/dV spectra at T and T reveals no gap to less than . is is another indication of suppression of superconductivity at the surface of Ca-YBCO. . I R C OC P Having identi ed the prominent gap in dI/dV as the PG, we proceed to investigate a set of impurities revealed in dI/dV maps. Fig. . . a shows an integral of dI/dV from - mV to mV, which was acquired simultaneously with the topograph shown in Fig. . . b and reveals a set of atomic-scale features as regions of high differential conductance. Using the dopant locator algorithm with DD= . Å, NS= . Å, and SW= . Å (Appendix B. ) we count approximately ± resonances in the eld-of-view (Fig. . . ), which corresponds to a density of ∼ . % ± . % per unit cell . Let us compare this quantity to the expected number of O vacancies and Ca dopants in the crystal: • O vacancies. e particular batch of samples we studied is determined to have ∼ . O atoms per unit cell ( makes the crystal structure perfectly stoichiometric). Since O vacancies are thought to occur primarily in the CuO chain plane [ ], of all unit cells imaged would contain an O vacancy (or ∼ of them should be detected in a nm square region). e number of unknown features that we observe is one order of magnitude fewer. us, the observed resonances are unlikely to be O vacancies. • Ca dopants. Based on energy-dispersive X-ray measurements, our samples contain ∼ Ca substitutions at the Y sites. is density is a factor of different from that of the unknown features we observe, which might suggest that these atomic-scale regions of high-conductance around Fermi level are Ca dopants. Figs. . . b-c show a series of dI/dV spectra along a line starting at the center of one representative impurity outwards. ese spectroscopic measurements reveal two bound states, symmetric around the Fermi level, that result in negative differential conductance at energies of ∼ ± meV. A series of dI/dV maps across different energies encompassing a single impurity are shown in Fig. . . . e observed spatial signature is C - but not C -symmetric, which allows us to pinpoint the location of these features to possibly be in the CuO chain plane, as it is the only layer that structurally breaks the C symmetry. Since the density of these resonances is not consistent with the expected density of O vacancies as discussed earlier, we believe that Cu-site defects (likely Cu vacancies) in the CuO chain plane are the culprits behind the structural origin of these resonances. Our identi cation is strengthened by similar spectroscopic signatures in dI/dV that originate as a consequence of Cu-site defects in other cuprates. In Bi, Zn [ ] and Ni [ ] substitutions for Cu, as well as unknown features likely to be Cu vacancies [ , ] induce bound states at the Fermi level and suppress gap-edge peaks. e fact that Zn substitutions and native defects, both presumed to be non-magnetic impurities directly in the crucial CuO superconducting layer, showed strong sub-gap resonances gave strong support to the sign-changing d-wave superconducting order parameter. In Bi, a set of intrinsic Cu-site defects (likely vacancies) induce two peaks symmetric around the Fermi level [ ], qualitatively very similar to what we observe in Fig. . . c. Furthermore, since these features maintained their spectroscopic signature well above Tc , Cha erjee et al. concluded that the spatial and energy distribution of the impurity state is not determined by the superconducting gap, but rather by the a b 1234567 3 nm 200 pm c dI/dV (a. u.) Low 15 10 5 0 -5 -10 -100 -50 1 2 3 4 5 6 7 High 0 50 Bias (mV) 100 Figure 6.4.2: (a) Integral of dI/dV maps from -9 mV to +9 mV bias, simultaneously acquired with the topograph in Fig. 6.2.1b, showing a set of resonances as bright, atomic-scale features. (b) dI/dV(0 mV) map, zooming-in on a resonance denoted by white square in (a). (c) dI/dV spectra obtained at the positions denoted in (b), starting from the center of the impurity outwards. Setup in (a): Iset =500 pA and Vset = -60 mV; in (b,c): Iset =100 pA and Vset = -120 mV. 48 mV 50 mV 52mV 54 mV 56 mV 58 mV 60 mV 62 mV 32 mV 34 mV 36 mV 38 mV 40 mV 42 mV 44 mV 46 mV 16 mV 18 mV 20 mV 22 mV 24 mV 26 mV 28 mV 30 mV 0 mV 2mV 4 mV 6 mV 8 mV 10 mV 12 mV 14 mV Low High Figure 6.4.3: Series of dI/dV maps over a 800 pm square region with a single impurity at a bias ranging from 0 mV to 62 mV, showing C -symmetry breaking. This observation is consistent for several different impurities we closely investigated. Similar spatial dependence is observed at corresponding negative bias. Setup: Iset =100 pA and Vset = -120 mV. pseudogap. As discussed in the previous section, the spectral gap we observe in Ca-YBCO is not of superconducting origin. us, our observation of impurity-induced bound states in this material provides additional support to the claim that these impurity states are not necessarily a signature of the superconducting state. Many theoretical efforts have proposed different ways of extracting the strength of a potential sca erer based on the energy of an impurity-induced bound state [ ]. We use the theoretical treatment by Salkola et al. used by other STM studies to extract the strength of the sca erer in terms of the phase shi δ [ ]: π/ Ω = cΔ ( . ) ln( /(πc)) where Ω is the resonance energy, Δ is the spectral gap magnitude, and c = cot(δ ). In our studies of Ca-YBCO, Ω ≈ meV and Δ ≈ meV, which allows us to determine the phase shi δ to be . π. For direct comparison, δ is calculated to be . π, . π, and . π for Zn dopants, Ni dopants and vacancies in Bi[ – ] and . π for intrinsic defects in Bi[ ]. In the strongest sca ering (unitary) limit, δ = π/ . us, the resonances we observed in Ca-YBCO are induced by strong, but not unitary sca erers. . S G I In contrast to the consistent observation of spectral gap inhomogeneity in Bi-based cuprates, no such reports have been made in YBCO. To address this issue in Ca-YBCO samples, we acquire dI/dV spectra on a square, densely-spaced pixel grid, and calculate the magnitude of the gap at each point by ing a Gaussian curve to the peak on the negative side of the spectrum. Fig. . . a shows a topograph, and Fig. . . b shows an inhomogeneous spectral gap map acquired simultaneously. e average gap value is . ± . meV, which represents ∼ % variation within our FOV, comparable to ∼ % variation we observe in optimally doped Tc = K Bisample. Fig. . . c shows average dI/dV spectra binned by the magnitude of the gap. Azimuthally averaged autocorrelation of the gapmap in Fig. . . b shows a length scale of about - nm, which is slightly different from the gap maps of optimally-doped Biwith characteristic length scales of - nm (Fig. . . d). Cross-correlation coefficients between topograph and zero-bias conductance (ZBC), and topograph and the gap map are very weak (Fig. . . d), a b 2 nm 5 4 dI/dV (a. u.) 3 2 1 15.8 meV 20.7 meV 23.9 meV 28.3 meV 2 nm Cross-correlation coefficient 9 mV 1.0 0.5 0.0 -0.5 -1.0 0 35 mV 0 -60 -40 -20 0 20 40 60 Bias (mV) c d Ca-YBCO: gapmap - ZBC Ca-YBCO: gapmap - topograph Ca-YBCO: topograph - ZBC Ca-YBCO: gapmap - gapmap Bi2212: gapmap - gapmap 1 2 Distance (nm) 3 Figure 6.5.1: (a) Topograph and (b) spectral gap map acquired simultaneously over a ∼10 nm square region. (c) Average dI/dV spectra binned by the magnitude of the gap as determined by Gaussian fit to the negative side of the spectra. (d) Cross-correlation coefficients relating gapmap, ZBC, and topograph in Ca-YBCO (black, red and blue lines). Azimuthally averaged autocorrelation for CaYBCO with Tc =30 K and optimally-doped Bi-2212 with Tc =91 K are shown in green and orange respectively. Setup: Iset =50 pA, Vset =-60 mV, 7 K. suggesting that the spatial distribution of the spectral gap and ZBC is not related to the surface disorder. We also map the spatial distribution of the energy of the NDC minimum (Fig. . . c) and the bound-state peak in the vicinity of a single impurity above Fermi level (Fig. . . d). Both show C - but not C -symmetric signatures. . C D In summary, our detailed STM studies of Ca-YBCO have addressed several important issues. First, STM topographs con rm the hypothesis that Ca doping results in an alternative cleavage plane in YBCO distinct from previously known BaO or CuO surface terminations, likely to be the disordered Y/Ca layer. Second, based on insensitivity to temperature and magnetic eld, the dominant spectral gap in dI/dV is not the superconducting gap. We also observe a set of impurity-induced resonances that break C symmetry we a ribute to Cu-site vacancies in the CuO chain plane. is observation provides further evidence that the impurity-induced bound states near the Fermi level are not necessarily a signature of the superconducting state, and that any depletion of states near the Fermi level would allow a resonance-like bound state near a non-magnetic impurity, as theory had predicted [ ]. Finally, we have for the rst time observed nanoscale inhomogeneity in the spectral gap on the surface of YBCO, with characteristic length scales of - nm, which is slightly shorter compared to the optimally-doped Bisamples. a b 200 pm 200 pm Low High Low High c d 200 pm 200 pm 25 meV 50 meV 8 meV 35 meV Figure 6.5.2: Simultaneous (a) topograph and (b) integral of dI/dV from -8 mV to +8 mV showing one impurity. Energy of the (c) NDC minima, and (d) the bound-state peak on the positive side of the dI/dV spectrum. Setup: Iset =100 pA and Vset = -120 mV. Surface Element Identi cation and Dopant Homogeneity in PrxCa −xFe As In this chapter, we will show (I) the rst de nitive identi cation of the cleaved surface termination and (II) the rst image of all individual dopants in any CaFe As system which suggests that dopant inhomogeneity is unlikely to be responsible for the low volume fraction of high-Tc superconductivity in Prx Ca −x Fe As . 7 . . . I S S C AF A C F - Like cuprates, Fe-based superconductors (Fe-SCs) are layered, with Fe-based superconducting planes separated by buffer layers. However, in contrast to those of cuprates, the nature of the cleavage surfaces in many Fe-SCs has been difficult to pin down. Furthermore, surface cleanliness, atness, and charge neutrality are all necessary, but not sufficient conditions to conclude that the surface is representative of intrinsic bulk properties of a material– surface can have different carrier concentration compared to the bulk, or it may exhibit structural or electronic reconstructions. erefore, the rst challenge in STM imaging of any new class of materials is to identify the surface structure and evaluate to what extent it is representative of the bulk. e surface structure of the AFe As system has been particularly controversial [ ]. Due to the stronger bonding within the FeAs layer (Fig. . . a), the FeAs layer is expected to remain intact, leaving half a complete A layer on each surface [ ]. However, a number of experiments have claimed that the [ ], and Ca [ ] systems. cleaved surface is As-terminated in the Ba [ ], Sr . . T STM We encounter three different surface morphologies in our STM topographs of Prx Ca −x Fe As . e majority of the observed sample surface displays a × structure (Fig. . . b) frequently observed in other STM studies of materials [ ]. We occasionally observe a disordered, “web-like” structure (Fig. . . c), which smoothly merges with the × structure (Fig. . . d). e third type of surface, observed rarely, shows a × square la ice with ∼ Å periodicity (Fig. . . e). . . S e tunneling current In order to identify these surfaces, we proceed to map the tunneling barrier height. I is expected to decay exponentially with the tip-sample separation z as I ∝ e− √ me Φ z where Φ is the local barrier height (LBH), approximately equal to the average of the tip and sample work functions [ ] (see Section . . for more details). However, the LBH is sensitive not only to the elemental composition of the tip, but also the geometric con guration of the tip’s terminal atoms, and the tip-sample angle (which reduces the LBH by cos θ, where θ is the deviation between the sample surface perpendicular and the z direction of tip piezo motion [ ]). Moreover, the LBH depends on the sample topograph through two opposing mechanisms. On the one hand, protruding atoms or clusters may a Ca/Pr Fe As b c 4nm Low High 4nm d e 4nm 4nm Figure 7.1.1: (a) Crystal structure of Prx Ca −x Fe As . Topographs of (b) × surface structure (250 pA, +300 mV, 7 K) (c) disordered, “web-like” surface structure (15 pA, 100 mV, 7 K), (d) smooth transition between × and “web-like” structures (20pA, -100mV, 7K), and (e) × square lattice with ∼ Å lattice constant (5 pA, 50 mV, 25 K). Inset in (b) shows an average × pixel, × supercell tiled × times. Inset in (e) shows an enlarged, 4 nm topograph of × square lattice acquired at 50 mV and 50 pA. (Due to an external noise source present during the acquisition of the data in panel (e), the images in (e) have been filtered to remove all spurious spatial frequencies higher than the Å periodicity.) Table 7.1.1: Work functions for several elements[114]. Atom Fe ϕ (eV) . As . Ca . Pr . Sr . Ba . Au . Pt . Ir . stretch out as the tip is retracted, reducing the effective rate at which the tip-sample distance decreases, and thus suppressing the measured LBH above the protrusion [ ]. On the other hand, the topographic corrugation appears smoothed out at distances far from the surface; this implies that the wave function decays faster above a protrusion than a depression, thus enhancing the measured LBH above a protrusion [ ]. Without accounting for these factors, previous studies found the LBH on the × surface of BaFe As to be much lower than the expected work functions for either Ba or As [ ]. In contrast, the comparison of LBH measurements with the same tip (i.e. the same microscopic con guration of terminating atoms) across different at regions of the same cleaved surface (i.e. the same tip-sample angle) can yield a robust measure of relative work functions, and can be utilized for element identi cation in cases where the sample consists of two different surfaces [ ]. Here, we directly compare LBH values measured with the same STM tip across the different morphologies of Fig. . . on the same cleaved sample. S “ × ” vs. “ × ” To extract the LBH at each point (x, y) in a eld of view (FOV), a feedback loop rst adjusts z (x, y) to maintain I = pA at Vset = − mV; the current I(z) is then measured as the tip is retracted from z . Figure . . shows simultaneous topographs and work function maps for the × square la ice across a step edge (Figs. . . a,c) and the × structure in a nearby at area (Figs. . . b,d). Figure . . e shows two sets of representative I(z) curves from the square regions in Figs. . . a-b, which are clearly distinct from one another. A er correcting for the surface slope, the average LBH values is calculated to be Φ × = . ± . eV and Φ × = . ± . eV. Taking Table . . and the tip work function into account, these values suggest that the × surface is a complete As layer, while the × surface is a half Ca layer. We note that charge redistribution at a surface may give rise to an additional dipole barrier, which can a c 3nm 0 nm 0.7 nm 3nm 2 eV 6.5 eV b d 3nm 0 nm 0.23 nm 3nm 2 eV 2x1 1x1 6.5 eV e 100 Current (pA) 10 0 20 40 60 80 100 Relative distance δz (pm) Figure 7.1.2: Topographs acquired at 25 K of (a) × (4 Å × 4 Å) square lattice appearing on both sides of a step edge and (b) × (8 Å × 4 Å) lattice. Simultaneously acquired LBH maps are shown in (c) and (d). Furthermore, sparse topographic protrusions on the × surface (e.g. marked by yellow arrow) show a lower LBH close to that of the flat × surface, suggesting that they are scattered remaining Ca or Pr atoms. Both datasets were acquired at Iset =100 pA and Vset =-100 mV. (e) Representative sets of I(z) curves from square regions in (a) and (b) are shown as thin red and green lines respectively. Darker red and green lines represent linear fits to the average I(z) curves from boxes in (a) and (b). increase or decrease the measured LBH according to the dipole orientation [ ]. We would therefore expect ϕAs + Edip + ϕtip ϕCa + Edip + ϕtip As Ca Φ× = ; Φ× = where ϕAs and ϕCa are the element work functions, and Edip and Edip are the additional energies for an As Ca electron to escape the dipole layers at the As- and Ca-terminated surfaces. Assuming ϕtip ϕPtIr , the values of Edip , Edip , and their difference Edip − Edip = . eV are all of the correct magnitude for such As As Ca Ca dipole layers[ ]. e sign of the difference, which indicates that it is harder to remove an electron from the dipole barrier of the As surface than that of the half-Ca surface, is physically justi ed because the half-Ca surface is nonpolar, whereas the As surface is de cient of electrons from the stripped Ca, and thus more electronegative. e inferred electron-de ciency of this As surface is consistent with the failure to observe even proximity-induced superconductivity on the As-terminated surface of the related Sr . K . Fe As [ ]. We also note the reduced LBH along the step edge in Fig. . . c, which may be a ributed to two mechanisms [ ]. First, the step edge is effectively an angled surface, so the LBH is reduced by cos θ. Second, the Smoluchowski smoothing of the electron wave functions along the step edge results in an additional dipole moment which reduces the LBH [ ]. To further support the identi cation of the × surface, we show a high-resolution map of the intra-unit cell structure to rule out the possibility of “hidden” surface atoms. We correct for small piezoelectric and thermal dri by placing the Ca/Pr atoms of Fig. . . b on a perfect la ice [ ]. We then use the whole FOV to create the average × supercell in the inset to Fig. . . b. We do not observe ]), but rather a atom dimerization (as seen in Ca . La . Fe As [ ] and Sr −x Kx Fe As [ , single row of atoms, similar to the CaFe As parent compound [ ]. S “ ” vs. “ × ” For completeness, we investigate the nature of the “web-like” surface. Because it merges smoothly into the × surface without any evident step edges (Fig. . . d), it is also likely a reconstruction of the Ca layer. Simultaneous topograph and LBH map of the “web-like” surface are shown in Figs. . . a,c, with analogous maps for the × surface, acquired with the same tip for direct comparison, shown in Figs. . . b,d. Bright spots in the topograph of Fig. . . a exhibit anomalously high LBH, highlighting the a c 4nm 0 nm 0.15 nm 4nm 2 eV 6.5 eV b d 4nm 0 nm 0.11 nm 4nm 2 eV 6.5 eV Figure 7.1.3: Topographs acquired at 7 K of (a) “web-like” surface and (b) × surface. Simultaneously acquired LBH maps are shown in (c) and (d). Both datasets were acquired at Iset =105 pA and Vset =-100 mV. The z calibration used here was obtained by assuming that the average LBH for the × surface in the boxed region of (b) here is the same as that in Fig. 7.1.2(d). importance of the complex geometric effects of protrusions previously mentioned. is reinforces the necessity of at atomic planes in order to extract a reliable LBH comparison. We reiterate that our identi cation of the × surface as a complete As layer and the × surface as a half-Ca layer is robustly drawn from the at surfaces in Fig. . . . . . . D R - H AF A P xC −x F A F - In the rst generation of AFe As (A ) Fe-SCs, hole doping resulted in higher maximum Tc ( K in Kx Ba −x Fe As [ ]) than electron doping ( K in Ba(Fe −x Cox ) As [ ]). However, the highest Tc among all Fe-SCs was K in electron-doped Sm −x Lax O −y Fy FeAs [ ], prompting the suggestion that Tc could be enhanced in electron-doped A ’s by removing the damaging dopant disorder from the crucial Fe layer, and doping the buffer layer instead. e strategy was successful in the rare-earth-doped Ca family [ , ], with Tc reaching K in Prx Ca −x Fe As . However, the high Tc appeared in only ∼ % of the volume, while the bulk of the material showed Tc ∼ − K. Saha et al. performed a thorough search for the origin of the low volume fraction high-Tc phase, using bulk experimental probes. First, the high Tc was found to be impervious to etching or oxidation, arguing against surface superconductivity. Second, high-Tc resistive transitions were never observed for dopant concentrations below those necessary to suppress the parent antiferromagnetic phase, arguing against random inclusions as the origin. Furthermore, no such contaminant phases were observed in over samples examined by X-ray diffraction. ird, the high Tc was unaffected by the global structural collapse phase transition (the abrupt ∼ % shrinkage of the c-axis la ice constant that occurs in the Ca family under external or chemical pressure), arguing against any relationship to the collapsed phase or to interfaces between collapsed and non-collapsed phases. In fact, aliovalently-doped CaFe (As −x Px ) also shows the structural collapse but no high-Tc volume fraction [ ]. Saha et al. therefore concluded that the charge doping is an essential ingredient to the high-Tc phase, and speculated that it has “a localized nature tied to the low percentage of rare earth substitution.” Given the challenges in identifying the origin of the low volume fraction high-Tc phase from bulk experiments, a local probe is naturally required. Here we use scanning tunneling microscopy (STM) to investigate dopant clustering as a possible source of electronic inhomogeneity in Prx Ca −x Fe As . Single a cm) 0.8 Resistance (m b0 -1 c 43.2 K 0 4πχ ZFC FC 4πχ 0.4 -2 -3 -0.16 50 20 T(K) ZFC FC Field: 2 Oe Field: 2 Oe 0.0 0 50 100 150 200 250 T(K) 0 10 20 30 T(K) 40 40 Figure 7.2.1: (a) Resistive measurements for the batch of Pr . Ca . Fe As samples used in this study (b) Magnetic susceptibility measurements denoting the onset of superconductivity. (c) Zoom-in on the transition in (b) showing Tc =43.2 K onset. crystals of Prx Ca −x Fe As are grown via self- ux with measured x = . % and Tc = . K (Fig. . . ). e crystals are handled exclusively in Ar environment, and cleaved in ultra-high vacuum at cryogenic temperature. . . D Since ϕCa and ϕPr differ by less than (Table . . ), LBH mapping cannot be used to identify Pr atoms in the Ca surface layer. However, STM can image dopants using the differential conductance dI/dV, which is proportional to the local density of states. Substituting Pr + for Ca + creates a localized positive charge, so the impurity state is expected above the Fermi level. We therefore search for Pr dopants in dI/dV images at high bias. Figure . . a shows a dI/dV image obtained simultaneously with the topograph in Fig. . . b at + mV, revealing a set of bright, atomic-scale features. ese features, which start to appear in dI/dV at biases higher than + mV, comprise ∼ . of the total number of visible atoms in this FOV (Appendix B. ), matching the macroscopically measured x = . % and con rming the half-Ca termination. Although a subset of Co dopants were previously imaged in Ca(Fe −x Cox ) As [ ], this is the rst time that all dopants have been imaged in a Ca system. Because we have imaged all dopants, we can investigate the possibility of clustering, which was suggested as the origin of the inhomogeneous high-Tc phase [ ]. We compute a “radial distribution Radial distribution ratio a b 1.0 0.5 4nm Low High 0.0 0 2 4 6 Distance (nm) 8 Figure 7.2.2: (a) dI/dV image at +300 mV showing Pr dopants as bright atomic-scale features from the same FOV as Fig. 7.1.1(b). (b) Radial distribution ratios for two sets of Pr dopants. Full squares represent the distribution of Pr dopants shown in (a), while open squares represent a different dataset used to confirm the conclusions. Inset shows a 2.5 nm × 2.5 nm region in which surface Ca positions (white dots) and Pr dopants (yellow dots) have been marked, demonstrating our ability to resolve individual Pr dopants even at adjacent Ca sites. ratio” (RDR) by histogramming all observed Pr-Pr distances within a FOV, then dividing this observed histogram by an average histogram of simulated random dopant distributions at the same concentration. e RDR in Fig. . . b shows no clustering, and in fact slight repulsion of the Pr dopants at short distances, possibly due to their like charges. e repulsion is not an artifact of poor dopant identi cation, as illustrated by clear detection of two adjacent Pr dopants in the inset to Fig. . . b. e lack of dopant clustering in Prx Ca −x Fe As contrasts with the Se dopants in FeTe −x Sex that are prone to forming patches of ∼ nm size [ ]. is contrast may arise from the ∼ size mismatch of Se ( pm) and Te ( pm) vs. the similar sizes of Ca ( pm) and Pr ( . pm) [ , ]. Our observation of the expected number of Pr dopants, more homogeneously distributed than would be expected for a random distribution, suggests that dopant clustering is unlikely to be responsible for the small volume fraction high-Tc superconducting state. . C D In conclusion, STM images of Prx Ca −x Fe As have addressed its surface structure and dopant distribution, with bearing on its high-Tc volume fraction. First, we used LBH mapping to identify the × surface as a half-Ca termination, and the × surface as an As termination. is LBH mapping method could be used to resolve debated cleaved surface terminations in a wide variety of materials, such as other Fe-SCs [ ] or heavy fermion materials [ , ]. Second, we demonstrated by direct imaging that the Pr dopants responsible for superconductivity do not cluster, and in fact show a slight repulsion at very short length scales. e ndings suggest that Pr inhomogeneity is unlikely to be the source of the high-Tc volume fraction, in contrast to previous speculation [ ]. Appendix A A. D - A D In this section I will describe the algorithm, rst implemented by Lawler et al. [ ], used to correct the small thermal and piezoelectric dri present during the course of data acquisition (typically several hours to - days). Typical topographs of Bishow several different modulations: atomic Bragg wave vectors Qx and Qy , and supermodulation structural buckling with wave vector QSM . is idealized la ice structure is distorted due to thermal and piezoelectric dri ; we can represent this picoscale “displacement” of atoms as a slowly-varying eld u(⃗ such that atomic positions⃗ − u(⃗ form a perfect r) r r) square la ice with Bi/Cu atom at ⃗ = . We can write idealized topograph as: d ˙r ˙r ˙r T(⃗ = T [cos(Qx (⃗ − u(⃗ + cos(Qy (⃗ − u(⃗ + Tsup cos(QSM (⃗ − u(⃗ r) r))) r))) r))) In order to extract the dri (A. ) eld u(⃗ we can rst assume that it is constant over length scale of /Λ, r), a Qy b Qy Qa Qa Qx Qx Figure A.1.1: Fourier transforms of STM topographs (a) before, and (b) after the application of drift-correction procedure. Atomic Bragg peaks Qx and Qy , and the orthorhombic peak Qa that are blurred out in (a) are significantly sharper in (b). where Λ is much smaller compared to magnitudes of Qx , Qy and QSM . relatively sharp, we can simplify the expressions: Tx (⃗ = r) Ty (⃗ = r) ∑ ∑ ⃗′ r ⃗′ r r T(⃗′ )e−ıQx⃗ ( r r T(⃗′ )e−ıQy⃗ ( r ′ ′ en, since atomic Bragg peaks are Λ −Λ (⃗ r′ ) / r−⃗ r) e ) ≈ (T / )e−ıQx u(⃗ π Λ −Λ (⃗ r′ ) / r) r−⃗ e ) ≈ (T / )e−ıQy u(⃗ π (A. ) (A. ) Using these two equations, we can extract the dri eld u(⃗ An example of a two dimensional Fourier r). transform before and a er the process of dri -correction has been applied to the corresponding STM topograph can be seen in Fig. A. . . For more detailed description of the dependence of this algorithm on relevant parameters, see [ ]. A. A. . D P P vs. D C A number of different ing techniques have been employed to extract the PG energy Δ from raw STM , ]. However, without an accepted microscopic model for the PG, spectra in previous studies [ , it is hard to justify the computationally intensive effort required for the use of any speci c ing function. We therefore use a simple, two part algorithm to extract the apparent PG energy Δ from the positive bias conductance (empty states) between εF and + meV. For such spectra in which the global maximum in conductance lies between εF and + meV, this typically corresponds to a clear gap-edge peak, so we set Δ equal to this energy. For spectra in which the global maximum in conductance occurs at energy greater than meV, there is typically no clear gap-edge peak, so we instead set Δ equal to the energy of the kink where the conductance slope abruptly a ens. To demonstrate the reliability of this simple PG extraction algorithm, we focus on the most challenging sample, with Tc = K, where we identify clear gap-edge peaks at Δ meV in only % of the spectra. From this sample, we show examples of typical spectra and their extracted Δ in Fig. A. . . As a further demonstration, we bin all spectra by Δ , with bin size determined by the energy spacing at which the data was acquired. Fig. A. . shows the complete set of spectra in each of bins, exemplifying the quantitative spread but qualitative similarity of most spectra within each bin. Because the slope-compare algorithm used for Δ > meV is more sensitive to the noise level than the global maximum algorithm for Δ meV, we estimate the error of our Δ extraction by repeating the same algorithm on increasingly smoothed data, and comparing results. Our nal Δ maps (shown in Fig. A. . ) are determined from spectral surveys that have been smoothed with a pixel boxcar average (corresponding to . - . nm, depending on the dataset, which is still far less than the superconducting coherence length ε ∼ . nm [ ]). e pixel-by-pixel RMS differences between these Δ maps and analogous Δ maps from spectra surveys smoothed with pixel boxcar average (∼ nm) are . , . , . , and . meV from the Tc = , , , and K samples, respectively. In summary, we estimate that the PG extraction error is largest (up to ∼ meV) in the Tc = K sample, and decreases to < meV in the Tc = K sample. a 25 dI/dV (arb. units) 20 15 10 5 =60 meV b 25 dI/dV (arb. units) 20 15 10 5 =90 meV c 30 dI/dV (arb. units) =120 meV 20 10 0 -150 -100 -50 0 50 100 150 Bias (mV) 0 -150 -100 -50 0 50 100 150 Bias (mV) 0 -150 -100 -50 0 50 100 150 Bias (mV) Figure A.2.1: (a) Δ =60 meV is obtained as the location of the global maximum on the positive side of the spectrum. (b-c) Δ =90 and Δ =120 meV are each obtained as the position of the first local maximum beyond 80 meV, which is typically an effective marker of the kink where the conductance slope flattens. All spectra have been 3 pixel boxcar averaged (9 spectra are averaged together, effectively smoothing on a 0.48 nm length scale). (Setup Vsample =-150 meV and Iset =800 pA.) A. . C To correlate the PG map Δ (⃗ with the locations of the dopants, one possibility would be to correlate it r) directly with the dI/dV map at the relevant energy. However, some dopants of the same type appear with slightly different size or brightness which might weight the cross-correlation unintentionally in favor of a subset of dopants. Instead, we create two-dimensional maps DA (⃗ DB (⃗ and Dv (⃗ of the distance from r), r), r) the nearest type-A O, type-B O, and AOV, respectively. We compute the cross-correlation C(⃗ ) between R the PG map Δ (⃗ and each distance map D(⃗ using the standard formula: r) r) ∫ C(⃗ ) = − R ¯ [D(⃗ − D] × [Δ (⃗ + ⃗ ) − Δ ]d r r) ¯ r R √ AD,D ( )AΔ,Δ ( ) ¯ [D(⃗ − D] × [D(⃗ + ⃗ ) − D]d r r) ¯ r R (A. ) ∫ AD,D (⃗ ) = R A. . R PG .O (A. ) Fig. A. . shows large positive correlation between AOVs and the PG for the three underdoped samples. Type-A O dopants are correlated to a lesser extent, while type-B O dopants exhibit very weak correlation 30 dI/dV (arb. units) a dI/dV (arb. units) 40 b =90 meV =60 meV 20 30 20 10 0 -150 -100 -50 0 50 100 150 Bias (mV) 40 10 0 -150 -100 -50 0 50 Bias (mV) 40 dI/dV (arb. units) 30 100 150 c dI/dV (arb. units) d 150 meV 30 20 10 0 -150 -100 -50 =120 meV 20 10 0 -150 -100 -50 0 50 100 150 Bias (mV) 0 50 100 150 Bias (mV) Figure A.2.2: Binned dI/dV spectra assigned a common Δ of: (a) 60 meV, (b) 90 meV, (c) 120 meV, and (d) 150 meV. All black spectra have been 3 pixel boxcar averaged (9 spectra are averaged together, effectively smoothing on a 0.48 nm length scale). The bold, red curve in each panel represents the average of all the individual black spectra shown in that panel. (Setup Vsample =-150 meV and Iset =800 pA.) with the PG. From the correlation length scales shown in Fig. A. . , we see that the AOVs in uence the PG on a ∼ - nm length scale, and we speculate that they will also impact the superconducting state on this length scale. is is consistent with the PG map correlation length εPG ∼ - nm (Fig. A. . a inset), and with measurements of the superconducting coherence length εSC = . ± . nm [ ]. We also employ a second method to view the same information as in Fig. A. . . From a given distance map Δ(⃗ we bin the pixels based on their distance, then apply this binning to the corresponding PG map r), Δ (⃗ and compute the average gap within each distance bin. Fig. A. . shows the average gap as a r), function of distance from the nearest impurity of each type. Correlation coefficient a 1.0 0.8 0.6 0.4 0.2 0.0 0 Tc=55K Tc=68K Tc=82K Tc=91K b 4 1 2 3 Distance (nm) 5 nm c d ∆1: 30mV 100 mV Figure A.2.3: Extracted Δ map for (a) Tc =55 K, (b) Tc =68 K, (c) Tc =82 K, (d) Tc =91 K. All maps are 30 nm x 30 nm, and all are displayed using the same colorscale. Inset to (a) shows the azimuthally averaged autocorrelation of each gapmap, demonstrating a length scale ∼2-3 nm. This justifies our initial smoothing of the spectral survey on the ∼0.5-0.7 nm length scale, to reduce noise in each spectrum and ensure the reliability of the Δ extraction algorithm. a Correlation coefficient Correlation coefficient 0.4 0.3 0.2 0.1 0.0 0 1 4 2 3 Distance [nm] 5 6 0.4 0.3 0.2 0.1 0.0 0 1 4 2 3 Distance [nm] 5 6 Correlation coefficient 0.5 b 0.5 c 0.5 0.4 0.3 0.2 0.1 0.0 0 1 Tc=55K Tc=68K Tc=82K type-B oxygen type-A oxygen apical O vacancy 4 2 3 Distance [nm] 5 6 Figure A.2.4: Azimuthally averaged cross-correlation of the PG map Δ and the map of distance to the nearest dopant D(⃗ of (a) type-B O, (b) type-A O, and (c) AOV. Different curves within each r) panel represent data taken on different underdoped samples (solid: Tc =55 K, dashed: Tc =68 K, and dotted: Tc =82 K). a 120 Gap [meV] b 120 c 120 Gap [meV] Gap [meV] 100 80 60 0.0 0.5 1.0 1.5 2.0 Distance [nm] 2.5 100 80 60 100 80 60 Tc=55K Tc=68K Tc=82K type-B oxygen type-A oxygen apical O vacancy 3.0 0.0 0.5 1.0 1.5 2.0 Distance [nm] 2.5 3.0 0.0 0.5 1.0 1.5 2.0 Distance [nm] 2.5 3.0 Figure A.2.5: Average PG as a function of distance from (a) type-B Os, (b) type-A Os, and (c) AOVs, respectively. Different curves within each panel represent data taken on different underdoped samples (solid: Tc =55 K, dashed: Tc =68 K, and dotted: Tc =82 K). 0.4 66mV 63mV 60mV 57mV 54mV 51mV 48mV 45mV 42mV 39mV 36mV 33mV 30mV 27mV 24mV 21mV 18mV 15mV 12mV 9mV 6mV 3mV 0mV 0.3 Intensity (a. u.) 0.2 0.1 0.0 0.2 0.4 0.6 0.8 1.0 Distance from the center of Fourier transfom (2π/a0) Figure A.3.1: Energy dependent linecut through the Fourier transform of dI/dV, along the Cu-O-Cu direction from ⃗ =0 to the atomic Bragg peak, demonstrating the non-dispersing, incommensurate CB q feature around ⃗ ∼0.3. Data was acquired on the Tc =55 K sample, with setup Vsample =800 pA and q Iset =-150 mV. A. A. . D E “C “ ” vs. D ” C Utilizing two-dimensional Fourier transforms (FTs) of BidI/dV images from mV to mV, and extracting linecuts along the Cu-O-Cu direction, from the center of each FT to the atomic Bragg peak (Fig. A. . ), it can be seen that the CB peak does not disperse with energy. a b c 3 nm 3 nm Low High 3 nm Figure A.3.2: (a) Raw dI/dV image from Tc =55 K, at energy +36 mV, with setup Vsample =-150 mV and Iset =800 pA. (b) Fourier-filtered image that contains only the wavevectors smaller than π/( a ). (c) Quotient: image (a) divided by image (b). A. . D “ ” In order to determine whether the dopants play a role in pinning the checkerboard, we rst disentangle the long-range DOS variation (length scale > a ) in the dI/dV maps from the checkerboard itself (typical wavelength λc ∼ a ), as demonstrated in Fig. A. . . From the Fourier- ltered dI/dV image, it is easy to determine the centers of the ‘checkers’ (Fig. A. . a), and calculate the distance from each impurity to the center of the nearest ‘checker’ (Fig. A. . b). To produce Fig. . . b, I histogrammed the number of occurrences of each dopant type with respect to the distance d from the nearest ‘checker’, and normalized the count by the probability that any pixel would lie a distance d from a ‘checker’. Because the CB does not disperse with energy, as shown in Fig. A. . , the analysis and conclusions about CB pinning are insensitive to the exact energy used for imaging the CB. a b d d type-B oxygen type-A oxygen apical O vacancy center of a ‘checker’ High 5nm 1 nm Low Figure A.3.3: (a) Fourier-filtered dI/dV image from Tc =55 K sample, at energy +35 mV, with setup Vsample =-150 mV and Iset =750 pA. The center of each peak is shown with a superimposed black circle. (b) Zoom on 4nm sub-FOV of the image in (a), exemplifying the distance of the dopants to the nearest CB peak. Appendix B B. D D A e algorithm used in Chapters and calculates how likely it is to have a dopant of one type (X) within a certain radius of another dopant of the same or different type (Y) compared to a completely random distribution of all dopants. More precisely, Ni (r) is a number of dopants of type Y within a radius r of ∑ ¯ dopant Xi . We compute Ni (r) for every Xi in the FOV and calculate the average N (r) = n n Ni (r), i= where n is the total number of dopants of type X. en we repeat the process for , completely ¯ random distributions of both type X and type Y dopants to get Nj (r), where j takes values from to , . Finally, the value we plot on the y-axis in Fig. . . is computed as: ∑ ¯ N (r)/( If the obtained ratio quotient is: , j= ¯ Nj (r) , ) (B. ) • > then the dopants a ract (prefer clustering) • ∼ then the dopant distribution is random • < then the dopants repel and are even more homogenous than you would expect from a random distribution B. D L A e dopant locator algorithm consists of four user-tunable parameters: neighborhood size (NS), duplicate distance (DD), smoothing window (SW), and local threshold (LT). For each pixel (x, y) in the eld of view (FOV), the algorithm nds the position of the local maximum (LM) within a square of side NS centered at (x, y). From the set of LMs (xi , yi ), the algorithm eliminates those within DD of a brighter LM, and those whose relative brightness compared to the regional average, de ned as the average value within a square of side SW+ centered at (xi , yi ), is less than LT. To identify the dopants in Fig. . . , we used DD= . Å (cf. Ca-Ca nearest-neighbor distance Å), NS= . Å, and SW= Å. Qualitative conclusions about the absence of clustering and short length scale repulsion of Pr dopants were unaffected by small errors in locating the centers of dopants (Fig. B. . (b)), variation of LT (Fig. B. . (d)). and were robust up to variation of SW (Fig. B. . (c)) and a b 1.0 Radial distribution ratio 0.8 0.6 0.4 0.2 0.0 0 2 Exact atomic lattice locations Raw pixel locations DD=2.4A NS=4.8A 4 Distance (nm) 6 8 c Radial distribution ratio 1.0 0.8 0.6 0.4 0.2 0.0 0 2 SW=108 A (210) SW=124 A (210) SW=93 A (215) SW=78 A (217) SW=139 A (209) d 1.0 Radial distribution ratio 0.8 0.6 0.4 0.2 0.0 0 2 4 Distance (nm) LT=94 (210) LT=101 (200) LT=106 (194) LT=108 (192) LT=90 (214) LT=85 (221) LT=80 (229) LT=79 (230) 4 Distance (nm) 6 8 6 8 Figure B.2.1: (a) Schematic of a perfect 2 x 1 reconstruction, where the small black circles represent the locations of surface Ca/Pr atoms. 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