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Investigations in Flatland : scanning tunnelling microscopy measurements of noble metal surface states… Macdonald, Andrew James 2017

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Investigations in FlatlandScanning Tunnelling Microscopy Measurements of NobleMetal Surface States and Magnetic Atoms on MagnesiumOxidebyAndrew James MacdonaldB.Sc. (Hon. Mathematical Physics Co-op), The University of Waterloo, 2011A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2017c© Andrew James Macdonald, 2017Abstract iiAbstractAt the atomic scale, surfaces exhibit a rich variety of physical phenomena thatcan be probed using a scanning tunnelling microscope (STM). The STM mea-sures the quantum tunnelling of electrons between a metallic tip and conduct-ing sample and can be used to characterize the nanoscale surface. This thesispresents STM measurements taken at low-temperature in ultra-high vacuum,which are used to characterize two different nanoscale environments: the two-dimensional surface states of Ag(111) and Cu(111) and the magnetic momentsof iron and cobalt atoms deposited on a thin-film of magnesium oxide.Fourier-transform scanning tunnelling spectroscopy (FT-STS) analysis ofquasiparticle interference, created by impurity scattering on the surfaces of thenoble metals Ag(111) and Cu(111), is used to compare the most common modesof acquiring FT-STS data and shows, through both experiment and simulations,that artifact features can arise that depend on how the STM tip height is sta-bilized throughout the course of the measurement. Such artifact features aresimilar to those arising from physical processes in the sample and are suscepti-ble to misinterpretation in the analysis of FT-STS in a wide range of importantmaterials. A prescription for characterizing and avoiding these artifacts is pro-posed, which details how to check for artifacts using measurement acquisitionmodes that do not depend on tip height as a function of lateral position andcareful selection of the tunnelling energy.In a separate set of experiments a spin resonance technique is coupled to anSTM to probe the spin states of individual iron atoms on a magnesium oxidebilayer. The magnetic interaction between the iron atoms and surroundingspin centres shows an inverse-cubic distance dependence at distances greaterthan one nanometre. This distance-dependence demonstrates that the spinsare coupled via a magnetic dipole-dipole interaction. By characterizing thisinteraction and combining it with atomic manipulation techniques a new formof nanoscale magnetometry is invented. This nanoscale magnetometer can becombined with trilateration to probe the spin structure of individual atoms andnanoscale structures. The information gained characterizing these new forms ofmagnetic sensing sets the stage for the study of complex magnetic systems likemolecular magnets.Lay Summary iiiLay SummaryThis thesis gives an account of two different forays into the nanoscale flatlandof atomic surfaces using the scanning tunnelling microscope (STM). The STMoperates by measuring a current between a metallic tip and a conducting surface.The first study focuses on measurement artifacts that arise when probing theelectronic states of the surfaces of silver and copper. A theoretical frameworkis developed that demonstrates that these artifacts are a product of how theSTM measurement is performed, and provides insight into how to identify andavoid these artifacts in measurements of more complex materials. The secondset of experiments explores the magnetism of single iron atoms deposited on aninsulating thin film. The iron atoms magnetic interaction is fully characterizedusing spin resonant STM measurements. This characterization allows for a newtype of nanoscale magnetic sensor that can be used to locate nearby magneticatoms and determine their magnetic moment.Preface ivPrefaceThe study of the set-point effect in measurements of the Ag(111) surface statepresented in Chapter 4 has been published in IOP Nanotechnology, 27(41):17,September 2016. The experimental work was performed at the University ofBritish Columbia’s Laboratory for Atomic Imaging Research under the supervi-sion of Prof. S. A. Burke and Prof. Doug Bonn. Sample preparation and STMdata acquisition was led by A. J. Macdonald working with Y.-S. Tremblay-Johnston, S. Grothe, and Dr. S. Chi. Theoretical modelling of the modulationin the density of states was done by Prof. Steve Johnston, with modificationsmade by A. J. Macdonald. FT-STS analysis code was developed in Pythonby A. J. Macdonald and Y.-S. Tremblay-Johnston. The set-point model ofthe tunnelling current and differential tunnelling conductance was developed inMATLAB by A. J. Macdonald. Prof. S. A. Burke oversaw all aspects of thedata acquisition and analysis and wrote the resulting paper with A. J. Macdon-ald and Y.-S. Tremblay-Johnston. Technical support was provided by PinderDosanjh.The characterization of the atomic-scale dipole-dipole interaction of ironatoms on magnesium oxide in Chapter 6 was performed at the IBM AlmadenResearch Centre and the results have been published in Nature Nanotechnology,2017 12, 420-424. The project was done under the supervision of Dr. A. Hein-rich and experimental runs were led by Dr. T. Choi, with guidance providedby C. Lutz and Dr. W. Paul. Dr. T. Choi, Dr. W. Paul, A. J. Macdonald, S.Rolf-Pissarczyk, Dr. F. D. Natterer, Dr. K. Yang, and P. Willke ran the ex-perimental measurements shifts, analyzed the initial results, spin-polarized tips,and used atomic manipulation to design and measure nanoscale structures. A.J. Macdonald, S. Rolf-Pissarczyk, W. Paul and Taeyoung Choi collected thebulk of the measurements used in the dipole-dipole characterization curves as afunction of frequency and performed sample preparation of the Ag(100) singlecrystal, growth of the MgO thin-film, and deposition of the magnetic species.Dr. T. Choi wrote the resulting manuscript with comments, edits, and revisionsprovided by all authors. Technical support was provided by Bruce Meloir.Table of Contents vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvI Introduction to Scanning Tunnelling Microscopy 11 Welcome to Flatland . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Flatland: The Nanoscale Surface . . . . . . . . . . . . . . . . . . 21.1.1 Measuring Electrons at the Nanoscale Surface . . . . . . . 41.2 Surface Science at the Atomic Scale . . . . . . . . . . . . . . . . 51.2.1 Quantum Tunnelling . . . . . . . . . . . . . . . . . . . . . 51.2.2 Early History . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 The Scanning Tunnelling Microscope . . . . . . . . . . . . 71.2.4 Theory of the STM Tunnelling Junction . . . . . . . . . . 91.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 The Laboratory for Atomic Imaging Research . . . . . . . . . . . 192.1.1 The CreaTec LT-STM . . . . . . . . . . . . . . . . . . . . 202.2 The IBM-Almaden Nanoscience Laboratory . . . . . . . . . . . . 232.2.1 The IBM 1-K STM . . . . . . . . . . . . . . . . . . . . . . 23Table of Contents vi2.3 Comparison Between the CreaTec and the 1-K STM . . . . . . . 252.4 Data Acquisition Modes of the STM . . . . . . . . . . . . . . . . 262.4.1 Topography . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.2 Spectroscopic Imaging using a Lock-In Amplifier . . . . . 272.4.3 Scanning Tunnelling Spectroscopy (STS) . . . . . . . . . 302.4.4 Spectroscopic Imaging via Spectroscopic Grid . . . . . . . 312.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33II Quasiparticle Interference in Noble Metal Surfaces 343 Quasiparticle Interference and Fourier-Transform Scanning Tun-nelling Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 Introduction: Quasiparticle Interference as Measured by Scan-ning Tunnelling Microscopy . . . . . . . . . . . . . . . . . . . . . 353.1.1 Derivation of Friedel Oscillations in the Electronic Densityof States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Fourier-Transform Scanning Tunnelling Spectroscopy Measure-ments of Quasiparticle Interference . . . . . . . . . . . . . . . . . 433.2.1 History of FT-STS Measurements . . . . . . . . . . . . . 433.3 FT-STS Measurements of the Noble Metals Surface States . . . . 433.3.1 Acquiring and Analyzing FT-STS Data . . . . . . . . . . 463.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Sur-face State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.1 Topography and Electronic Character of Ag(111) . . . . . . . . . 534.1.1 The Ag(111) Surface State . . . . . . . . . . . . . . . . . 534.1.2 Sample Preparation and Measurement Protocol . . . . . . 544.2 Experimentally Observed Set-Point Effects in Different Acquisi-tion Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.1 Spectroscopic Grids as a Function of Set-Point Parameters 564.2.2 Spectroscopic Grids and Constant-Current dI/dV maps . 594.2.3 Constant-Height dI/dV Maps . . . . . . . . . . . . . . . . 624.2.4 Comparison Between all Acquisition Modes . . . . . . . . 634.2.5 Experimental Conclusions . . . . . . . . . . . . . . . . . . 654.3 Modelling the Set-point Effect . . . . . . . . . . . . . . . . . . . . 694.3.1 Sample Density of States via T-matrix . . . . . . . . . . . 694.3.2 Analytic Set-Point Theory . . . . . . . . . . . . . . . . . . 714.3.3 Set-Point Simulations . . . . . . . . . . . . . . . . . . . . 734.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 Characterization of the Cu(111) Surface with Multiple FT-STSAcquisition Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.1 Characterizing the Set-Point Effect in Cu(111) . . . . . . . . . . 815.1.1 The Cu(111) Surface State . . . . . . . . . . . . . . . . . 81Table of Contents vii5.1.2 Measured Set-Point Effects in Cu(111) . . . . . . . . . . . 835.2 Discussion and Outlook . . . . . . . . . . . . . . . . . . . . . . . 86III Magnetic Sensing and Control of Single Atoms onMgO 896 Electron Spin Resonance Scanning Tunnelling Microcopy . . 906.1 Electron Spin Resonance in Bulk Materials . . . . . . . . . . . . 906.2 Electron Spin Resonance and Scanning Tunnelling Microscopy . 916.2.1 Energy Resolution, STS, and ESR-STM . . . . . . . . . . 916.2.2 Previous Work to Pair ESR with STM . . . . . . . . . . . 926.2.3 ESR-STM at IBM . . . . . . . . . . . . . . . . . . . . . . 926.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997 Single Atom Magnetic Sensing on the Surface of MgO . . . . 1027.1 Characterization of an Atomic-Scale Magnetic Dipole-Dipole Sensor1027.1.1 Sample Preparation and Measurement Protocol . . . . . . 1037.1.2 Magnetic Dipole-Dipole Sensing using ESR-STM . . . . . 1057.1.3 Single-Atom Magnetometry from ESR Dipole Sensing . . 1097.2 Engineering a Nano-Scale Magnetometer Array . . . . . . . . . . 1137.2.1 Nano-scale Magnetic Trilateration . . . . . . . . . . . . . 1167.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119IV Future Direction and Open Questions 1208 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . 1218.2 The Future of the Set-Point Model . . . . . . . . . . . . . . . . . 1238.3 Proposed Experiments with Magnetic Dipole-Dipole Sensing . . . 1258.4 Leaving Flatland . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128A Upgrades and Operation for the UBC CreaTec LT-STM . . . 150A.1 A Brief History of the CreaTec LT-STM . . . . . . . . . . . . . . 150A.2 Components of the CreaTec . . . . . . . . . . . . . . . . . . . . . 150B Real Space Simulation of Multiple Scatterers . . . . . . . . . . 156B.1 Using Bessel Functions to Simulate QPI Data . . . . . . . . . . . 156C A Scheme for Pulsed ESR-STM . . . . . . . . . . . . . . . . . . . 158C.1 Development of a Pulsed ESR-STM Technique . . . . . . . . . . 158C.1.1 Introduction to Pulsed ESR . . . . . . . . . . . . . . . . . 159C.1.2 A Scheme for Tip-Oscillating Pulsed (TOP) ESR-STM . 161C.1.3 Benchmarking the TOP ESR-STM Technique . . . . . . . 162Table of Contents viiiC.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165List of Tables ixList of Tables2.1 A comparison of the technical specifications of the CreaTec atthe LAIR and the 1-K STM at the Nanoscience Laboratory. . . . 25A.1 History of the CreaTec highlighting major upgrades. . . . . . . . 150List of Figures xList of Figures1.1 Complex structure from base units and well-describedinteractions. From left to right: the works of Shakespeare stemfrom the alphabet and the rules of grammar; the superconductingMeissner effect is a result of the interaction of electrons and nu-clei within a superconductor; Canis lupus familiaris, the moderndog, arises from specific arrangements of the four base proteinsin DNA, according to the rules of genetics. Image credit for DNAin Reference [17] and for Shakespeare in Reference [18]. . . . . . 31.2 Nanoscale Flatland (a) Ripples in the electronic density ofCu(111) caused by impurity scattering. (b) Iron atoms sittingatop the oxygen site of a magnesium oxide bilayer. . . . . . . . . 41.3 Scaling from the size of everyday objects to the nanoscale.A competition level chessboard is roughly 50 cm across and theindividual pieces can be easily manipulated by human fingers. Anindividual hydrogen atom has an atomic radius ten billion timessmaller. To manipulate the hydrogen atom with any dexterityrequires scientific tools capable of operating on this length scale.Image credit for chessboard in Reference [28]. . . . . . . . . . . . 51.4 A visual interpretation of quantum tunnelling. A tiger,trapped behind a classically impenetrable barrier at the top isable to use quantum tunnelling to penetrate the barrier and givechase to a bystander on the other side. Inspired by Reference [34]. 61.5 Essential components of an STM. 1) The tip-sample junc-tion, 2) a current amplifier, 3) piezoelectric motors, 4) a feedbackcontrol system, and 5) a DC applied bias. The tip-sample junc-tion is shown magnified through a telescopic camera and then inan artistic interpretation at the nanoscale, with electrons cross-ing the vacuum gap. The DC bias in this case is applied to thesample but could instead be applied to the tip. . . . . . . . . . . 81.6 Atomic resolution on the surface of graphite. The tun-nelling current measured on highly oriented pyrolytic graphite.The brightest intensity corresponds to the centre of the hexagonsformed by rings of six carbon atoms. Imaging conditions: Is = 15pA, Vb = 600 mV, and T = 4.5 K. . . . . . . . . . . . . . . . . . 11List of Figures xi1.7 Measuring ρs(E) with STS. (a) In the low-temperature limit,electrons from a metallic tip held at a bias of EF + eVb tunnelinto a sample which has the onset of a state at (0). (b) Thesample can be characterized by measurements of It and dIt/dVb.The particular bias EF + eVb leads to the data point denoted bythe black circles. Sweeping the bias and measuring gives the twocurves in red. The tunnelling current shows the onset of the sam-ple state as a kink while the differential tunnelling conductanceexhibits a step that reflects the step in ρs(E). . . . . . . . . . . . 142.1 The CreaTec LT-STM in c-pod. (a) The vacuum chamberconsists of the (1) manipulator arm, (2) preparation chamber,(3) cryostat, and (4) pumping system. (b) The STM/AFM head.The visible components are the (5) tunnelling current cold finger,(6) damping springs, (7) walking disc, and (8) sample holdermounted in the measurement position. . . . . . . . . . . . . . . . 202.2 The CreaTec Baseline Noise Characterization. (a) Am-plitude spectral density of the CreaTec tunnelling junction mea-sured in a 1 kHz range with 977 mHz resolution. The curveslabelled FEMTO DLPCA-200 and FEMTO LCA-4K-1C are thenoise baselines of the tunnelling current pre-amplifiers when theirinputs are grounded. The out-of-tunnelling curve is the noisebaseline of the FEMTO LCA-4K-1C amplifier when its input isconnected to the STM tip but the tip is not close enough to thesample to produce a measurable tunnelling current. The tun-nelling curve shows the noise spectrum when tunnelling is oc-curring with a 100 mV bias and 100 pA tunnelling current. (b)Accelerometer measurements of the linear spectral density of theCreatec inertial block. Spikes in the linear spectral density areindicative of mechanical resonances within c-pod. For more de-tails on the vibration and acoustic noise level in the LAIR seeReferences [123, 124]. . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 The 1-K STM system at the Nanoscience Laboratory. (a)(1) The vertical manipulator arm for sample transfer (2) Fe andCo evaporators (3) rotary feedthrough flange and sample prepa-ration chamber (4) 3He pumping system. (b) Internal schematicof the 1-K STM cooling stages and magnet alignment. Certaindetails, such as heat exchangers, have been omitted for clarity. . 242.4 Topography of the Ag(111) surface. (a) A 282×282 nm areaof the surface of the noble metal Ag(111). A plane subtractionhas been performed to flatten the image. (b) An apparent heightprofile following the black line shown in (a) shows a double stepedge, which can be used to vertically calibrate the STM againstthe atomic lattice. Imaging conditions: Vb = −40 mV, Is = 540pA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27List of Figures xii2.5 Constant-height and constant-current dI/dV maps. (a)Measurement schematic, including AC modulation on the biasand lock-in amplifier acquisition of the dI/dV signal. In constant-current mode the tip adjusts the tip–sample gap via the feedbackas it rasters over step edges and adsorbates while in constant-height mode the tip height stays constant. (b) Constant-currentdI/dV maps acquired at various different energies on the Ag(111)surface show ripples of intensity off of surface impurities. (c)Constant-height dI/dV maps on the Ag(111) surface also showripples off of impurities but with a lower signal-to-noise ratio. . . 282.6 Acquisition of the dIt/dVb by modulating the applied bias.The signal measured by the lock-in amplifier is proportional to thechange in the tunnelling current ∆It caused by the modulationof the applied bias with strength and frequency given by VAC . . . 302.7 STS of Ag(111) (a) STS point spectra averaged from a 100 nm2clean area of the Ag(111) surface. (b) The corresponding differ-ential tunnelling conductance by way of the numerical derivative. 312.8 Acquisition of a spectroscopic grid on Ag(111) At everypixel of the grid the feedback stabilizes the tunnelling junctionat some apparent height, which yields a topography for the grid.Once stabilized, the feedback is disengaged and the bias is sweptover a specified range, allowing acquisition of I(V ) curves. OnceI(V ) curves have been acquired at every pixel the analysis pro-ceeds by Gaussian smoothing each curve and taking the numericalderivative. This gives a dI/dV curve at every pixel allowing forreconstruction in real space of the differential tunnelling conduc-tance at each bias. . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Quasiparticle interference in Cu(111). (a) Constant cur-rent image of QPI from impurities and step edges taken at theNanoscience Laboratory. Imaging conditions: T = 4 K, Vb = 100mV, Is = 1000 pA, and 50 × 50 nm2. Adapted with permissionfrom Macmillan Publishers Ltd: Nature [154], copyright (1993).(b) Differential tunnelling conductance of Cu(111) acquired froma constant current dI/dV map taken using the CreaTec in 2017.Imaging conditions: T = 4.5 K, Vb = −100 mV, Is = 900 pA,and 80× 80 nm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 The T-matrix as a sum of Feynman diagrams. . . . . . . 403.3 Quasiparticle interference in real-space, momentum-space,and scattering space. (a) In the absence of an impurity thedensity of states of a two-dimensional electron gas is flat with aparabolic dispersion relation (k). No signal in scattering spaceis observed. (b) In the presence of an impurity the sample den-sity of states acquires a dependence on the back scattering vector|q| = 2|k|. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42List of Figures xiii3.4 Fourier transform analysis of quasiparticle interference.(a) (i) A constant current STM image of QPI in Be(0001). Imag-ing conditions: T = 150 K, Vb = 4 mV, Is = 1.5 nA, and 4 × 4nm2. (ii) The associated |S(q,E)| intensity exhibits a ring of in-tensity associated with scattering across the Fermi surface andlattice Bragg peaks from the atomic lattice. Adapted from Ref-erence [85]. Reprinted with permission from AAAS. (b) (i) Dif-ferential tunnelling conductance image of LiFeAs. (ii) The corre-sponding Fourier transform shows the complexity of the FT-STSsignal in a multi-band system. Imaging conditions: T = 4.5 K,Vb = 10 mV, and 26 × 26 nm. Adapted figure with permissionfrom S. Chi, S. Johnston, G. Levy et al., Physical Review B, 89,1–10, 2014 [97]. Copyright 2014 by American Physical Society. . 443.5 Extracting scattering information from dI/dV by analyz-ing QPI. (a) Differential tunnelling conductance of Ag(111) atthe Fermi energy, Vb = 0 mV. (b) The absolute value of theFourier transform of the differential tunnelling conductance, ie.the FT-STS scattering intensity |S(q,E)|. The bright ring denotesthe preferred scattering vector for quasiparticles in the surface.(c) A radial projection of the scattering intensity |S(qr,E)|. Theprimary feature at qr = 2kF corresponds to the scattering vectorof the Ag(111) surface state at the Fermi energy. . . . . . . . . . 473.6 Scattering dispersion (qr) from a spectroscopic grid. (a)|S(qr),E)| line cuts for a spectroscopic grid on Ag(111). Theprimary feature, corresponding to scattering of a surface state,changes position or disperses as a function of energy. This figureonly shows three percent of the data obtained in the grid mea-surement shown in (b). To represent all the |S(qr),E)| data takenin the grid the scattering intensity can be plotted as function ofenergy and qr, where the maximum in intensity corresponds tothe peaks seen in (a). . . . . . . . . . . . . . . . . . . . . . . . . 483.7 Filtering processes for FT-STS data. (a) No filtering isapplied to the real-space differential tunnelling conductance im-age. Step edges and carbon monoxide scattering centres are bothvisible on the surface. The |S(q),E)| intensity exhibits a ring as-sociated with the carbon monoxide QPI and a near vertical lineof intensity associated with the step edge. (b) Real-space filter-ing averages out the step edges, filtered region denoted by dashedwhite lines. The |S(q),E)| intensity associated with the step edgeis reduced. (c) Angular filtering is performed before taking theradial projection. (d) Comparison of the radially projected scat-tering intensity |S(qr),E)| between the three methods. . . . . . . 51List of Figures xiv4.1 Lattice structure of the Ag(111) surface. (a) STM topo-graphic image of the Ag(111) surface showing multiple step edges.Imaging conditions: Vb = −40 mV, Is = 540 pA, and apparentheight 0− 2.3 nm. Inset shows atomic resolution on the Ag(111)surface. Imaging conditions: Vb = 5 mV, Is = 90 nA, and ap-parent height 0− 16 pm. (b) LEED measurement of the Ag(111)surface (Energy = 152 V, Current = 0.08 mA) after sputteringand annealing cycles. (c) Model which illustrates the Ag(111)lattice parameter and face-centred cubic lattice. . . . . . . . . . . 534.2 Characterization of the Ag(111) surface state band. (a)The k-space dispersion extracted from dI/dV tunnelling spec-troscopy (solid points). A parabolic fit to these data is plot-ted along the solid line. The dashed curve is the dispersion asmeasured by photoelectron spectroscopy. The inset is a dI/dVspectrum showing the surface state onset. Reprinted figure withpermission from L. Jiutao, W.-D Schneider, R. Berndt, PhysicalReview B, 56, 7656–7659, 1997 [81]. Copyright 1997 by AmericanPhysical Society. (b) ARPES data of the surface state dispersionin k-space shows the parabolic band. Reprinted figure with per-mission from F. Reinert, G. Nicolay, S. Schmidt, D. Ehm, and S.Hu¨fner, Physical Review B, 63, 1–7, 2001 [167]. Copyright 2001by the American Physical Society. . . . . . . . . . . . . . . . . . 554.3 The three different energy regions in which the stabiliza-tion bias Vs can be set when measuring Ag(111). . . . . . 574.4 Ag(111) spectroscopic grids with different set-point pa-rameters. Top panels are the differential tunnelling conductanceevaluated at the Fermi energy Vb = 0. The bottom panels are thecorresponding FT-STS scattering intensity |S(q, E)|. Verticallyaligned features of high intensity in the bottom panels are causedby the step edges present in the top panels. (a) Vb = (−100, 120)mV, 239×239 nm2 with 380×380 pixels. (b) Vb = (−40, 40) mV280 × 280 nm2 with 400 × 400 pixels. (c) Vb = (−100, 100) mV,240× 240 nm2 with 350× 350 pixels. . . . . . . . . . . . . . . . . 584.5 Scattering dispersion calculated from three spectroscopicgrids with different set-point conditions. Horizontal andvertical lines indicate E = EF = 0 meV and qr = 2kF re-spectively. (a) No secondary features observed. (b) Broad non-dispersing feature above EF and below 2kF . (c) Broad non-dispersing feature below EF and above 2kF . (d) Radial projec-tion of the FT-STS signal at EF for each grid. Imaging conditionsmatch Figure 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60List of Figures xv4.6 Energy dispersion from spectroscopic grids and constant-current dI/dV maps. Horizontal and vertical lines indicateE = −40 meV and qr = 2kF respectively and the parabola rep-resents a free electron model. (a) Spectroscopic grid with imagingconditions: 60 × 60 nm2, 230 × 230 pixels. (b) Dispersion con-structed from constant-current dI/dV maps with imaging condi-tions: 60× 60 nm2, 512× 512 pixels. (c) The scattering intensityat −40 meV for both grid and map measurements scaled so thatthe surface state feature has the same intensity. . . . . . . . . . . 624.7 Constant-height dI/dV maps compared with constant-current dI/dV maps and spectroscopic grids. (a) Constant-height dI/dV map radial projections of |S(q, E)| show only thesurface state feature. Imaging conditions: 65×65 nm2, 512×512pixels. (b) Constant-current dI/dV maps show two features awayfrom EF . The imaging conditions for (b) are the same as in Fig-ure 4.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.8 The effect of stabilization bias on the observed FT-STSpattern. (a) |S(qr, E = 50 meV)| comparing two grids withdifferent stabilization biases, a constant-current dI/dV map, anda constant-height dI/dV map. (b-e) the corresponding FT-STSpattern at E = 50 meV from (b) a spectroscopic grid with Vs = 50mV, Is = 100 pA, (c) a spectroscopic grid with Vs = −50 mV,Is = 100 pA, (d) constant-current map at Vs = 50 mV, Is = 100pA, and (e) a constant-height map at Vs = 50 mV, and initialcurrent Is = 100 pA. For the constant-height data a restrictedq-space angular filter was used to reduce the influence of a stepedge running across the top of the image. . . . . . . . . . . . . . 664.9 Comparison of all features |S(qr,E)| in different acqui-sition modes. Vertical and horizontal lines indicate EF andqF = 2kF respectively. Only the constant-current maps show astrongly dispersing set-point peak. All three modes map out thesurface state intensity, agreeing well with a free electron model. . 674.10 Full-width at half-maximum of the fits for all featuresin |S(qr,E)| in different acquisition modes. (a) FWHMvalues resulting from fits of the surface state peaks in FT-STSfor all three measurement modes. (b) FWHM values resultingfrom fits of the set-point peaks in FT-STS for constant-currentdI/dV maps and spectroscopic grids. . . . . . . . . . . . . . . . . 684.11 Optimizing the scattering phase δ. (a) Theoretical |S(qr, E)|intensity for a strong scattering impurity (b) Theoretical |S(qr, E)|intensity for a weak scattering impurity. (c) Comparison of the-oretical |S(qr, EF )| with experimental data from a spectroscopicgrid. (d) Residuals of the theoretical |S(qr, EF )| with the experi-mental data, the mix of two phases does the best job minimizingthe residuals around the surface state peak. . . . . . . . . . . . . 70List of Figures xvi4.12 Simulation of a grid with stabilization bias Vs = −100mV. (a) Simulated |δρ| in scattering-space. (b) The calculatedT (zs) of this |δρ| with Vs = −100 meV stabilization bias. (c) Theproduct of |δρ| with T (zs) gives the first term in Equation 4.11(d) Experimental grid data with stabilization bias Vs = −100 mV. 754.13 Simulation of a grid with stabilization bias Vs = 100 mV.(a) Simulated |δρ| in scattering-space. (b) The calculated T (zs)of this |δρ| with Vs = 100 meV stabilization bias, (c) The productof |δρ| with T (zs) with Vs = −100 meV stabilization bias. (d)Experimental grid data shown with stabilization bias Vs = 100mV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.14 Simulation of constant-current dI/dV maps. (a) Simulated|δρ| in scattering-space. (b) The calculated T (zs) of this |δρ| fora stabilization bias that matches the bias being probed Vs = Vb.(c) The product of |δρ| with T (zs). (d) Experimental FT-STSfrom constant-current dI/dV maps. . . . . . . . . . . . . . . . . . 774.15 |S(qr,E)| comparison between simulated constant-heightmaps, spectroscopic grids, and constant-current maps.(a) Simulated constant-height dI/dV maps at Is = 100 pA showonly the surface state feature. (b) Simulated constant-currentdI/dV maps at Is = 100 pA show a dispersing secondary feature.(c) Simulated spectroscopic grid data with Vs = 100 mV, Is = 100pA shows a non-dispersing secondary feature. . . . . . . . . . . . 795.1 Spectroscopic grid measurement of Cu(111) (a) The dif-ferential tunnelling conductance exhibits two step edges and anumber of tip changes, corresponding to the higher degree ofnoise in the top half of the image. The inset at top right showsoscillations around individual point defects and a step edge. (b)FT-STS shows the expected intra-band scattering intensity. (c)The scattering dispersion of the entire surface state band. Verti-cal line denotes qF = 2kF = 0.42 A˚−1 and horizontal line showsE = −250 mV for reference to (a) and (b). Imaging conditions:Vs = −520 meV, Is = 150 pA, 325× 325 pixels. . . . . . . . . . 825.2 Set-point effects in grids on Cu(111). (a) A spectroscopicgrid measurement exhibits a secondary non-dispersing featurejust above the Fermi scattering vector qF = 2kF . Imaging con-ditions: Vs = 30 meV, Is = 150 pA, 124 × 124 nm, 176 × 176pixels. (b) A spectroscopic grid with stabilization bias below theband onset shows no secondary features. Imaging conditions:Vs = −520 meV, Is = 150 pA, 225× 225 nm, 325× 325 pixels. . 845.3 Constant height and constant current dI/dV maps ofCu(111). (a) |S(qr, E)| extracted from constant height dI/dVmaps over areas ranging from 80− 140 nm in length at Is = 900pA. (b) |S(qr, E)| extracted from constant current dI/dV mapsover areas ranging from 80− 140 nm in length. Is = 900 pA. . . 85List of Figures xvii5.4 Comparison of Cu(111) dispersing features with Sessi etal. [113] Vertical and horizontal lines indicate EF and qF = 2kFrespectively. (a) Peak maxima extracted from constant currentand constant height maps both show the surface state disper-sion in good agreement with a free electron model from Equa-tion 4.1. The constant-current dI/dV maps show the set-pointfeature while constant-height dI/dV maps exhibit a previouslyunobserved feature. (b) Constant-current dI/dV maps by Sessimap out the surface state dispersion. The theoretical dispersionof an acoustic surface plasmon agrees well with secondary featuresbelow EF but deviates above. . . . . . . . . . . . . . . . . . . . 876.1 Logarithmic Energy Scale Above the energy scale a selectnumber of energies important in condensed matter are listed whilebelow the energy scale the limit of STS resolution is shown atvarious tip–sample temperatures. References: Hyperfine energyof hydrogen (H) water [222], superconducting (SC) gap energies[223], work function of silver [81], and quantum limit of STSenergy resolution [221]. . . . . . . . . . . . . . . . . . . . . . . . 926.2 STS spectra of a single Fe atom with and without a spin-polarized tip. STS spectra of the same Fe atom on MgO witha spin-polarized tip and a normal tip which averages over allavailable spin channels. Spectra were acquired using a lock-inamplifier to extract the differential tunnelling conductance. Set-point parameters: Vs = 10 mV, Is = 50 pA for the normal tipand Vs = 10 mV, Is = 100 pA for the spin-polarized tip. . . . . . 946.3 ESR-STM measurement scheme. (a) A magnetic atom sit-ting on an insulating thin film on a bulk conducting crystal hasis sent into resonance by application of an microwave bias exci-tation along the DC bias line. A spin-polarized tip is used toread-out the change in tunnelling current. (b) The microwavepulse train for this method, dubbed continuous wave ESR. (c)The microwave frequency is swept until a resonance peak in thetunnel current signal is detected via the lock-in amplifier. . . . . 956.4 STS spectra of Fe on MgO The jump in differential tunnellingconductance at Vb = 14 meV corresponds to the transition for theground state to the first excited state. Topographic height of Featom in inset is h = 0.17 nm. (b) The five lowest lying states ofan Fe atom on MgO [86]. . . . . . . . . . . . . . . . . . . . . . . 986.5 Single Fe atom ESR resonance. (a) Thermal population ofthe low-energy quantum states of an Fe atom on MgO in a mag-netic field. (b) Resonant population in continuous wave mode (c)ESR-STM spectra showing resonance at various magnetic fieldstrengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100List of Figures xviii6.6 ESR-STM spectra of the same Fe atom with two oppo-sitely spin-polarized tips. The ESR peak (dip) is caused bya lower (higher) tunnelling magnetoresistance at resonance. Thefrequency splitting between the observed features is independentof tip, though the absolute frequency at which each resonance isobserved is tip dependent. The rich structure of the ESR spectrais caused by the proximity of nearby magnetic atoms on the sur-face, which will be discussed in detail in Chapter 7. The abovespectra were taken in a magnetic field of B = 4.8 T and at atemperature of T = 1.2 K. . . . . . . . . . . . . . . . . . . . . . . 1017.1 ESR-STM for magnetic dipole detection. (a) Schematic forsensing the dipole-dipole interaction between two iron atoms. (b)Constant-current STM image of four Fe atoms (0.17 nm apparentheight) on the surface. Imaging conditions are VDC = 0.1 V,I = 10 pA, Bz = 0.18 T, Bx = 5.7 T and T = 1.2 K. . . . . . . . 1057.2 Magnetic dipole-dipole interaction detected via ESR. (a)ESR spectrum (black curve) of an Fe atom when another Fe atom(target) is positioned 2.46 nm away (Bz = 0.17 T, T = 1.2 K,VDC = 5 mV, IDC = 1 pA, VRF = 10 mVpp). A fit to twoLorentzian functions (red curve) yields the frequency splitting(∆f). The difference in amplitude between the two observedfeatures can be attributed to the ratio in thermal occupationprobability of the |0〉 and |1〉 states. (b) Topography of the sensorFe atom (outlined in black) and the Fe target. (c) The underlyingMgO lattice provides a metric to measure the distance betweenthe two Fe atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.3 ESR spectra of multi-atom structures. (a) ESR spectrumtaken on the middle atom (Fe) in (c) shows four ESR peaks. (b)ESR spectrum taken on the left atom (Fe) in (c) also exhibits fourESR peaks. (c) STM image of the three atom arrangement, withtwo Fe atoms and one Co atom. The binding-site assignmentmodel (lower panel) shows distances between atoms in units ofthe atomic lattice spacing (0.2877 nm for T < 20 K). Imagingcondition are 10 mV, 10 pA, Bz = 0.2 T, and T = 1.2 K. . . . . 1077.4 Log-log plot of ∆f ∝ rα. The measured splitting as a functionof the distance r of atom pairs of Fe-Fe (red squares), Fe-Co (bluecircles), and Fe-‘tall’ Fe (green diamonds). Data from the shadedregion below 1 nm are excluded from the fits. Error bars in ther and ∆f axes represent uncertainty in the determination of theinteratomic distance and the measured ∆f error due to frequencydrift. Distances are obtained using the Ag lattice constant mea-sured at low temperature by x-ray diffraction [186, 260]. . . . . . 108List of Figures xix7.5 Distance-dependent ESR splitting (∆f) for atoms sepa-rated by less than 1.0 nm (a) ∆f vs distance r for close atoms,zoomed in from Figure 7.4 on the Fe-Fe curve. (b) STM imagesof two close Fe atoms before and after placing a third (“remote”)Fe atom in the vicinity. The imaging conditions are 10 mV, 10pA, Bz = 0.17 T, and T = 1.2 K. (c) ESR splitting before andafter adding the remote atom. . . . . . . . . . . . . . . . . . . . . 1097.6 Magnetic dipole-dipole interaction detected via ESR Aschematic of the dipole-dipole interaction. The resonant fre-quency of the isolated sensor atom (f0) is split into two frequen-cies (f↑ and f↓) corresponding to the two spin states of the targetatom. ∆f = f↓ − f↑ is the measured splitting. . . . . . . . . . . . 1117.7 Construction of the dipole sensing array. (a) Five Fe atomswere gathered and arranged to form an array of four sensors andone target in the array centre. (b) The tip atom, another Fe,is placed outside the array for safekeeping. (c) The target Featom is picked up from the centre of the sensor array. (d) Thetarget is placed at a lower symmetry position, corresponding to(-3,1) oxygen lattice sites from the array centre. (e) The entirestructure is imaged in order to find the tip atom again. (f) Thetip is spin-polarized by picking up the Fe tip atom. . . . . . . . . 1147.8 Measuring a target atom at a low symmetry site with anano-sensor arrray. Five Fe atom structure with four equallyspaced corner atoms and a middle Fe (sensor atom) which ispositioned at (−3, 1) lattice sites from the centre of the square(oxygen binding sites indicated by white open circles). For clarity,the 1.2 K spectra is offset by −60 fA. Imaging conditions areVDC = 10 mV, I = 10 pA, and Bz = 0.17 T. . . . . . . . . . . . . 1157.9 Measuring a target atom at a low symmetry site with anano-sensor arrray. (b) The Fe sensor atom is placed in theexact centre of the square at a distance of 2.034 nm from thefour target atoms. This creates a four-fold symmetry which leadsto degeneracy of the excited spin states. Imaging conditions areVDC = 10 mV, I = 10 pA, and Bz = 0.17 T. . . . . . . . . . . . . 1167.10 Magnetic imaging by using trilateration. (a) MeasuredESR spectra (T = 0.6 K) from each sensor atom (black) andthe predicted unbroadened spectrum (red) due to all atoms ex-cluding the target. (b) ESR spectra of (a) after deconvolving theeffect of the other sensor atoms. The results show ESR spectra(peaks are indicated by black arrows) due solely to dipolar inter-action of the target atom and the sensor atom under the tip. (c)Predicted location of the target by trilateration. (d) Agreementof the target location between STM topography and nanoGPS. . 1188.1 Artistic depiction of nanoscale Flatland as a chessboardwith electronic pieces. . . . . . . . . . . . . . . . . . . . . . . 127List of Figures xxA.1 The CreaTec LT-STM in c-pod. (a) The vacuum chamberconsists of (1) the manipulator arm, (2) the preparation cham-ber, (3) the cryostat, and (4) the pumping system. (b) TheSTM/AFM head being installed outside of vacuum. The visiblecomponents are (5) tunnelling current cold finger, (6) dampingsprings, (7) walking disc, and (8) a sample holder mounted in themeasurement position. (c) The manipulator is used to move sam-ples within the vacuum space. It consists of (9) sample bias andannealing contacts and (10) sample clamping mechanism. (d)The sample holder (11) with a graphite sample mounted, coinshown for scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151A.2 The CreaTec STM head. (a) The upgraded UBC STM headused for the bulk of the measurements in Part II. The wiringis designed to be slack to avoid coupling vibration into the tip-sample junction and is extremely fine in order to prevent heatleaks from room temperature to the STM head. (B) A model ofthe original CreaTec head when mounted inside the heat shields. 153A.3 Operation of a beetle/Besocke style STM head. (a) Thevoltage profile applied to the quadrants of a piezo tube to make itmove. The voltage is slowly ramped from a constant value up toa maximum before being quickly inverted. This produces a saw-tooth waveform that moves the walking plate. (b) A basic modelof a beetle-style STM head such as the one used in the CreaTecSTM. (c) Forces generated from the piezo tubes on the walkingplate for X and Y motion. (d) Torque on the walking plate fromthe piezo tubes in Z motion. When all three piezo tubes produceclockwise forces a net torque is produced that moves the walkingplate and STM tip towards the sample. Counterclockwise forcesproduce a net torque that moves the walking plate and STM tipaway from the sample. . . . . . . . . . . . . . . . . . . . . . . . . 154B.1 Spectrscopic grid simulated in real-space by Bessel func-tions. (a) Experimental measurement of the differential tun-nelling conductance at EF from a spectroscopic grid with mea-surement conditions: Vs = −100 mV, Is = 100 pA, Vb = (−100, 120)mV, 239× 239 nm2 with 380× 380 pixels. (b) Simulation of theexperimental data using Equation B.2 and identifying the pixelscorresponding to the centre of each CO. (c) FT-STS scatteringintensity of both experiment and simulation projected onto theqr axis. The simulation results deviate from the experiment atlow-qr, possibly indicating the need to add a degree of noise tothe simulation data. . . . . . . . . . . . . . . . . . . . . . . . . . 157List of Figures xxiC.1 Continuous wave versus pulsed ESR-STM (a) Continuouswave ESR-STM uses (i) a chopped microwave frequency sweep to(ii) randomly walk the spin state along the Bloch sphere, resultingin (iii) an increase in the tunnel current. (b) Pulsed ESR-STMuses a (i) series of resonant pulses to (ii) coherently manipulatethe spin state around the Bloch sphere, resulting in (iii) observa-tion of Rabi oscillations in the tunnelling current signal. . . . . . 160C.2 Tip Oscillating Scheme (a) A single set of TOP ESR-STMsignals over one period of the tip oscillation. The Vµ pulse istriggered at the maximum tip sample separation and VDC is usedfor initialization of the spin and read-out using a positive polar-ity DC bias pump. (b) Many tip shaking cycles occur duringone cycle of the lock-in measurement. The lock-in measurementcompares the average tunnelling current during sequences withand without the spin-manipulation microwave pulses. . . . . . . . 163C.3 Characterization of the tip modulation amplitude. (a)It(z) spectroscopy can be used to fit for the κ factor at a particu-lar point, giving a relation between the tunnelling current and therelative tip–sample gap. (b) The tunnelling current is measuredas a function of tip modulation amplitude. . . . . . . . . . . . . . 164C.4 Tip response as a function of modulation frequency. Arelatively stable area, away from mechanical resonances, is foundat 17241 Hz and so this frequency is chosen for TOP ESR-STM.The inset is a STM topograph of an Fe atom taken while theSTM tip is being modulated with a peak-to-peak height of 2 A˚.Imaging conditions are Is = 25 pA, Vb = 100 mV, and 30× 30 A˚. 166List of Acronyms xxiiList of Acronyms2DEG Two-dimensional Electron GasAFM Atomic Force MicroscopyARC Almaden Research CentreARPES Angle-resolved Photoemission SpectroscopyASP Acoustic Surface PlasmonESR-STM Electron Spin Resonance Scanning Tunnelling MicroscopyFT-STS Fourier-transform Scanning Tunnelling SpectroscopyFWHM Full-width half-maximumIBM International Business MachinesLAIR Laboratory for Atomic Imaging ResearchLEED Low-energy Electron DiffractionQPI Quasiparticle InterferenceSTM Scanning Tunnelling MicroscopeSTS Scanning Tunnelling SpectroscopyUBC University of British ColumbiaXMCD X-ray Magnetic Circular DichroismList of Symbols xxiiiList of SymbolsS(q, E) FT-STS scattering intensityS(qr, E) FT-STS scattering intensity projected onto the q radial axisρs Sample local density of statesδρs Impurity induced modulation in the local density of statesρt Tip local density of statesIt [A] Tunnelling currentVb [V] Applied bias (DC)Is [A] Set-point currentVs [V] Stabilization biasdIt/dVb or dI/dV [S] Differential tunnelling conductanceEdd Magnetic dipole-dipole energyMts Bardeen’s tunnelling matrix elementft,s Thermal factorT (z, E, Vb) Tunnelling transmission probability or transmission functionz Tip–sample gaph Apparent heightG Green’s functionT1 Energy relaxation timeT2 Quantum phase coherence timem∗ Effective electron masse Free electron chargeEF Fermi energyq Scattering-space vectork Momentum-space vectorkF Fermi wave vectorqF Fermi scattering vectorkB Boltzmann constanth¯ Dirac constantAcknowledgements xxivAcknowledgementsFirst and foremost, I want to acknowledge my supervisors Professor Sarah Burkeand Professor Doug Bonn, who have provided mentorship, guidance, and in-struction from the beginning of my time at the LAIR. I want to thank them forgoing the extra mile to provide me with opportunities to perform interestingwork, the chance to learn from failing, and the support necessary to work withresearchers internationally. They have shaped who I am as a researcher todayand I am very grateful for that.This thesis couldn’t have been completed without the support of the peopleat the UBC LAIR, the UBC Superconductivity Group, and the IBM NanoscienceLaboratory. At the LAIR I want to acknowledge the role of the CreaTec team:Dr. Stephanie Grothe, Dr. Shun Chi, Gelareh Farahi, and Yann-SebastienTremblay-Johnston. I want to thank Katherine Cochrane for being my deskmate for these past six years, for the long weekend fills, and of course for thegummy bears. I also want to thank Dr. Agustin Schiffrin, Martina Capsoni,Tanya Roussy, Rob Delaney, Damien Quentin, and Ben MacLeod for makingthe LAIR a rich scientific environment to work in. In the SuperconductivityGroup, I want to acknowledge the support of Pinder Dosanjh, without whomthe CreaTec would still be lying in pieces on a table somewhere. Thank you toJames Day for the mentorship, the conversations over coffee, and for the con-stant inspiration. Walter Hardy for always having time to discuss the details ofa scientific problem with me, particularly if it involved a grounding problem —I hope I was able to return the favour. Thank you to Ellen Schelew and NathanEvetts for entertaining my ideas about topics completely outside of my field.For the extraordinary experience I had visiting the IBM Almaden ResearchCentre I want to thank Andreas Heinrich, who was my host, running coach, andscientific role model. I want to thank William Paul, and his family Sara, Juliet,and Teddy, for welcoming a fellow Canadian to the Bay area and putting me upwhen my car was stolen. I learned more about electronics from Will in a weekthen I did in my entire undergraduate education. I want to thank TaeyoungChoi, and his family Lei, Alex, and Julian, for their hospitality. Taeyoung ledthe charge on the dipole coupling work and I learned a great deal about man-aging a scientific project working alongside him. Christoper Lutz is the mostinspirational scientist I have had the honour of working alongside; I try my bestto think about problems the way he does. Thank you to Susanne Baumann,Fabian Natterer, Steffen Rolf-Pissarczyk, Kai Yang, and Philip Willke for trad-ing ideas on the white board and discussing scientific data. Bruce Meloir wasan excellent colleague to discuss design problems and trade stories with. TheAcknowledgements xxvIBM lunch crowd for the excellent lunchtime conversations.The Natural Sciences and Engineering Research Council of Canada CRE-ATE QuEST program, PGS D program, and UBC Four-Year Fellowship pro-vided the financial support necessary for me to undertake the research presentedherein. My supervisory committee of Prof. Rob Kiefl, Prof. Scott Oser, andProf. Marcel Franz provided insightful comments and direction throughout thethesis work.My friends and scientific colleagues who helped me grow outside the lab-oratory. Particularly Adam Shaw, Stephanie Grothe, Natasha Holmes, LukeGovia, Brad Ramshaw, Tim Cox, Amanda Parker, Samara Pillay, Dorotheavon Bruch, and Mohammad Samani for being good friends and putting everyproblem in context.Lastly, I want to thank my family, my parents, grandparents, and brothers,for encouraging me to be a scientist and to take this as far as I could go. Andof course I want to thank my wife Tegan, who was not only there to see methrough the worst and best parts of this degree, but is the only person who hasread and edited and re-read every section of this thesis more diligently than Ithought possible.Dedication xxviDedicationTo Mom and Dad, for teaching me persistence.To my wife Tegan, who makes me a better scientist and a better person.To Stephanie, who helps me keep in mind what is important.And to the four-legged creatures that keep my sense of wonder close at hand.1Part IIntroduction to ScanningTunnelling MicroscopyChapter 1. Welcome to Flatland 2Chapter 1Welcome to FlatlandFacts are not science - as thedictionary is not literature.Martin H. Fischer - 1944 [1]Quantum tunnelling, a classically forbidden process, can be harnessed byinstruments like the scanning tunnelling microscope (STM) to characterize andengineer atomic-scale surface properties. In this chapter, the central conceptsof the nanoscale surface, quantum tunnelling, and the STM are introduced, andthe basic components, operation, and theory of an STM are discussed. Thischapter lays the groundwork for making STM measurements in two regimes thatthe STM cannot normally access: momentum-space electronic properties andmicro-electronvolt electronic transitions. Measurements of these two regimesare performed on two different nanoscale surfaces: the (111)-terminated noblemetals and magnesium oxide thin films.1.1 Flatland: The Nanoscale SurfaceIn 1884 Edwin A. Abbott published the short novel Flatland: A Romance ofMany Dimensions [2]. The story’s main character, A. Square, lives in a two-dimensional space known as Flatland and dreams of Lineland, a one-dimensionalequivalent to Flatland, and Spaceland, a three-dimensional space similar to ourown. Though Abbott was not a scientist, Flatland as a theoretical concept hasproven extremely useful to physicists. In relativity the warping of our four-dimensional space-time by matter and energy can be visualized as a thoughtexperiment by the stretching of a two-dimensional elastic sheet in the presenceof heavy objects [3]. In condensed matter physics there exist materials withprotected two-dimensional states on the edges of three-dimensional bulk mate-rials [4, 5]. What is more, in nanotechnology research there is a direct physicalanalogy to Abbott’s Flatland: the nanoscale surface.The surface of a material is the simplest type of two-dimensional interfacein condensed matter physics; it delineates the bulk of a material from vacuum.Inside the bulk, the electromagnetic field created by the atomic nuclei interactswith the electrons to determine the material’s properties. The physics of thisbulk region is described with great success by single-electron band theory [6] andLandau Fermi-liquid theory [7] in many classes of materials. Outside the bulk,particles maintain their free-space atomic properties like spin, mass, and chargeChapter 1. Welcome to Flatland 3as measured in particle physics experiments. The surface is a global defect [8]where the worlds of the bulk and the vacuum collide. The surface breaks thetranslational and inversion symmetry of the bulk crystal, and combined withthe interplay between bulk and vacuum electronic states can lead to electronicstates of matter that cannot be realized in either alone [4, 5, 9]. It is possible forthe physical properties and excitations at the surface to be radically differentfrom those of the bulk or the vacuum [10–15].Condensed matter physicists and surface scientists already know that thebase chemical unit that creates the surface is atoms, composed of electrons andnuclei, interacting in a way described extremely well by quantum electrodynam-ics [16]. However, knowing the constituents and interactions of the surface is notenough to predict all of the weird and wonderful behaviour that emerges. Statesof matter like superconductivity, ferromagnetism, or ferroelectricity stem fromthe complex interaction of ∼ 1023 of these base units in a macroscopic piece ofmatter. This is an example of simple, well-understood rules and constituentsgiving rise to macroscopically complex behaviour: like literature from lettersand grammar, Meissner diamagnetism from electrons and quantum electrody-namics, or life from base proteins and genetics (Figure 1.1).Figure 1.1: Complex structure from base units and well-described in-teractions. From left to right: the works of Shakespeare stem fromthe alphabet and the rules of grammar; the superconducting Meiss-ner effect is a result of the interaction of electrons and nuclei withina superconductor; Canis lupus familiaris, the modern dog, arisesfrom specific arrangements of the four base proteins in DNA, ac-cording to the rules of genetics. Image credit for DNA in Reference[17] and for Shakespeare in Reference [18].Nanoscale engineering of the atomic surface makes it possible for physiciststo design surfaces that give insight into fundamental physics and provide proto-types for new technological platforms. Atomic-scale control has allowed surfaceChapter 1. Welcome to Flatland 4scientists to measure new fundamental excitations that have only been predictedto exist in particle physics [19] and to test the predictions of quantum electrody-namics for elements far heavier than have ever been synthesized in acceleratorfacilities [20]. Current theoretical proposals rely on atomic manipulation tech-niques to implement new quantum computing architectures [9] and to betterthe efficiency of existing technologies, like organic solar cells [21].1.1.1 Measuring Electrons at the Nanoscale SurfaceOver one hundred years after the publication of Abbott’s work, this thesischaracterizes the behaviour of A. N. Electron in two different Nanoscale Flat-land environments using the scanning tunnelling microscope (STM): the (111)-terminated noble metals and magnetic atoms atop a magnesium oxide thin-film(Figure 1.2). The (111)-terminated surfaces of the noble metals offer the op-portunity to probe the physics of a two-dimensional electron gas (2DEG), aconducting medium strictly localized on the metallic surface. Scattering of this2DEG off of localized impurities on the surface creates a striking pattern in theelectronic density known as quasiparticle interference, which will be probed ex-tensively in Part II. On the other end of the conductivity spectrum, monolayerand bilayer insulators serve as a two-dimensional test bed for studying mag-netic interactions in the limit of discrete electronic energy levels. By growingthin films of magnesium oxide (MgO) on a silver crystal, an experimental Flat-land is engineered where it is possible to probe the magnetic dipolar interactionbetween individual iron atoms. Fully characterizing this interaction allows forthe invention of a new form of nanoscale magnetometry in Part III. Before mov-ing onto the details of these experiments, the remainder of Part I introduces theexperimental technique of STM, its history, and its limitations.1 nm(a) (111)-terminated Noble Metals (b) Magnetic Atoms on Insulating MgO10 nmFigure 1.2: Nanoscale Flatland (a) Ripples in the electronic density ofCu(111) caused by impurity scattering. (b) Iron atoms sitting atopthe oxygen site of a magnesium oxide bilayer.Chapter 1. Welcome to Flatland 51.2 Surface Science at the Atomic ScaleIn order to customize nanoscale surfaces to explore new science and create novelmaterials, scientists must be able to manipulate and measure individual surfaceatoms. If successful, this amounts to a game of atomic-scale chess where thepieces and the board can be tailored to the scientists’ individual needs. Thecaveat is that scientific tools are required that can measure and manipulate in-dividual atoms and molecules. The chief difficulty to overcome in this regardis that an atom of hydrogen is roughly a factor of ten billion smaller than thelength of a chessboard, as illustrated in Figure 1.3. This change in scale repre-sents a significant technical hurdle. Over the last forty years, a number of toolscapable of measuring the atomic scale properties of solids have been developed.Techniques such as transmission electron microscopy [22, 23], atomic force mi-croscopy [24, 25], and scanning tunnelling microscopy [26, 27] are all capable ofmeasuring atomic scale properties of a material’s surface. These techniques relyon the quantum properties of matter for their basic operation. Central to thisthesis work is the scanning tunnelling microscope, which takes advantage of thewave-particle duality of the electron to perform quantum tunnelling.Figure 1.3: Scaling from the size of everyday objects to the nanoscale.A competition level chessboard is roughly 50 cm across and the indi-vidual pieces can be easily manipulated by human fingers. An indi-vidual hydrogen atom has an atomic radius ten billion times smaller.To manipulate the hydrogen atom with any dexterity requires sci-entific tools capable of operating on this length scale. Image creditfor chessboard in Reference [28].1.2.1 Quantum TunnellingQuantum tunnelling describes the process by which a quantum object can passthrough a potential barrier that would be impenetrable to a classical object. TheChapter 1. Welcome to Flatland 6quantum tunnelling of light and matter is a direct result of the wave characterof quantum objects [29]. Figure 1.4 illustrates the essence of tunnelling byvisualizing a tiger tunnelling through a classically impenetrable barrier, muchto the chagrin of the character on the other side. In the mathematical languageof quantum mechanics, tunnelling describes the process by which a plane waveincident on a barrier has a finite probability of travelling through the barriereven if the barrier’s energy is greater than the wave’s own. Inside the barrier, aclassically forbidden region, the plane wave exponentially decays in magnitude.Without quantum tunnelling nuclear fusion could not occur in the heart ofstars [30], alpha decay of radioactive nuclei would not be possible [31–33], andthe field of surface science would be missing one of its central tools [26]. Noless than five Noble prizes have been awarded for work on quantum tunnellingin semiconductors, superconductors, and the invention of new devices like thescanning tunnelling microscope [29].Classically impenetrable barrierQuantumTunnellingFigure 1.4: A visual interpretation of quantum tunnelling. A tiger,trapped behind a classically impenetrable barrier at the top is ableto use quantum tunnelling to penetrate the barrier and give chaseto a bystander on the other side. Inspired by Reference [34].1.2.2 Early HistoryThe first theoretical work to employ the concept of quantum tunnelling wasperformed in the late 1920’s with the then newly minted theory of quantumChapter 1. Welcome to Flatland 7mechanics. Frederich Hund calculated the quantum mechanical barrier pene-tration – what is now called quantum tunnelling – in the context of molecularspectroscopy [35, 36]. His work was followed up by Fowler and Nordheim’s ex-planation of field emission [37, 38], Oppenheimer’s work analyzing the tunnellingin the continuous portion of the hydrogen atom spectrum [39, 40], and Gamow,Gurney, and Condon’s work that showed that tunnelling inside the nucleus wasresponsible for alpha decay [31–33]. In solid state physics, early applications ofthe theory of tunnelling in the 1930’s led to Zener’s theory of interband tun-nelling in solids [41] and the first tunnelling device, the field-emission microscopeinvented by Mu¨ller [42, 43].Over the next thirty years the theory of quantum tunnelling was put onfirmer theoretical footing by theorists [44, 45] while experimentalists used thetunnelling properties of matter to design tunnelling junctions and study super-conductivity [46, 47]. Building upon previous tunnelling devices in 1972 Young,Ward, and Scire designed the topographiner [48], in many ways the STM’s di-rect predecessor. Then, in 1982, Binnig and Rohrer at IBM Zurich invented thescanning tunnelling microscope [26, 49]. In 1983, they published the first im-ages of the surface of Si(7×7), solving a longstanding mystery about its surfacecharacter [50]. By 1986 they had won the Nobel Prize for their invention anda decade later there were thousands of papers per year being published in thenew field of STM studies [27].1.2.3 The Scanning Tunnelling MicroscopeThe STM measures a convolution of integrated electronic density of states andapparent atomic height by taking ‘topographs’ of a surface. Through carefulexperimentation and analysis, STM experiments have opened new vistas in sur-face science, such as untangling surface reconstructions [50, 51], observing newforms of charge order [52], and cataloguing step-by-step molecular reactionson surfaces [53]. STMs can be built to operate in liquid, in air, in ultra-highvacuum, and from room temperature to milliKelvin temperatures. Regardlessof the construction details every STM has a number of essential components,shown in Figure 1.5 and enumerated below:1. Tip: A sharp metallic tip, held in close proximity to a sample of interest.2. Motors: A motor system that can drive the tip in three-dimensions withsub-nanometer control over a millimetre range, typically accomplished bythe use of piezoelectric devices.3. Current Amplifier: A means of detecting a tunnelling current acrossthe tip-sample junction, typically involving a high-gain current amplifieror electrometer.4. Feedback Control: An electronic feedback system that can control thetip–sample distance as the tip is rastered across the surface.5. DC Bias: A direct current (DC) bias applied between tip and sample.Chapter 1. Welcome to Flatland 8Researchers can use an STM to do more than just image a nanoscale surface.The tip of the STM can be used to perform feats of nano-engineering: manipu-lating individual atoms into specific patterns to probe electronic states [54–56],store information [57], and even develop the world’s smallest stop action movie[58]. The STM can also be used for spectroscopic characterization, via scanningtunnelling spectroscopy (STS), that delves into the electronic energy levels ofmolecules [21, 59] and electronic dispersion of solids [60], providing complemen-tary information to other modern spectroscopic techniques like angle-resolvedphotoemission (ARPES) [61], but with a far more localized real-space probe.The STM is capable of probing and manipulating a large variety of surfaces.STMs have been combined with magnetic fields, low temperature, and laserpulses to probe all manner of physics from solar cells [21, 62] to superconduc-tors [63] to magnetism [64].Figure 1.5: Essential components of an STM. 1) The tip-sample junction,2) a current amplifier, 3) piezoelectric motors, 4) a feedback controlsystem, and 5) a DC applied bias. The tip-sample junction is shownmagnified through a telescopic camera and then in an artistic inter-pretation at the nanoscale, with electrons crossing the vacuum gap.The DC bias in this case is applied to the sample but could insteadbe applied to the tip.Chapter 1. Welcome to Flatland 91.2.4 Theory of the STM Tunnelling JunctionThe Tunnelling CurrentAt its core, the STM relies on bringing the tip and conducting sample closeenough together (≈ 10 A˚) that there is significant overlap between their elec-tronic wave functions [49]. At this distance the tunnelling probability becomesnon-trivial. Combined with a DC bias, applied to either tip or sample whilethe other remains grounded, electrons will preferentially travel from one to theother, thus creating a measurable tunnelling current. The magnitude of thetunnelling current depends on the tip–sample gap, the relative density of elec-tronic states of tip and sample, and the magnitude of the applied bias. Relatingthe tunnelling current to the physical quantity of interest, be it the local elec-tronic density of states of a sample (ρs) or sample work function (φ), requires atheoretical framework.Tersoff and Haman [65, 66] developed the first model of the STM tunnellingcurrent shortly after the invention of the STM. Based on Bardeen’s transferHamiltonian method [44], the Tersoff-Haman model is remarkably useful, inspite of a number of approximations explicitly built into the theory. In manycases, the Tersoff-Haman model accurately describes the observed physics andbecause of its straightforward relation between the tunnelling current and thesample local density of states it has proved popular in the interpretation of STMdata.In the Tersoff-Haman model the relation between the tunnelling current Itand the local density of states of tip ρt and sample ρs is derived in the contextof the interaction of two weakly bound metallic electrodes. Applying first-orderperturbation theory [65–67] the tunnelling current is given byIt =4pieh¯∫ ∞−∞ρs(r, E)ρt(E − eVb)|Mts|2fts(E, Vb)dE, (1.1)where e is the electron charge, h¯ is the reduced Planck constant, r denotes lateralposition of the tip over the sample, ρs is the sample local density of states, ρt isthe tip density of states, Vb is the applied bias on the sample with reference to agrounded tip (as in Figure 1.5), E is the electron energy, and Mt,s is Bardeen’smatrix element describing tunnelling between the states of the tip and those ofthe sample [68, 69]. The thermal factor,fts =11 + e(E−eVb−µt)/kBT− 11 + e(E−µs)/kBT, (1.2)is a function of the Fermi-Dirac distribution of the tip and sample and is re-sponsible for temperature-related energy broadening. Here T is the temperatureof the tip and the sample, kB is Boltzmann’s constant, and µs and µt are thechemical potential of the sample and tip respectively.Equation 1.1 shows that the tunnelling current is a measure of the integratedChapter 1. Welcome to Flatland 10density of states of the sample, convoluted with effects from the tip densityof states, the tunnelling barrier, and finite temperature. In practice, a greatdeal of effort is made to ensure that the tip density of states is independent ofenergy, i.e. flat, over the energy range being measured so that it can be safelyignored when analyzing the STM data. Furthermore, STM research focused onelectronic states as a function of energy will typically be cryogenically cooled toreduce thermal energy broadening.Under the approximations of a a spherical tip with a flat density of states, asquare tunnelling barrier treated semi-classically, and low temperature, Equa-tion 1.1 can be simplified toIt ∝ e−κz∫ eVb0ρs(x, y, E)dE (1.3)where κ is a constant related to the work function of the tip and the sampleand z is the distance between the sample surface and the last atom on the tip.This equation makes explicit the tunnelling current’s exponential dependenceon the tip-sample gap and shows that the tunnelling current is a function of theintegrated local density of states of the sample. The exponential dependenceof the tunnelling current on the tip–sample gap is responsible for the STM’satomic level resolution; it makes the tunnelling current exponentially sensitiveto any change in the tip–sample gap created by moving the tip over defects,step edges, or individual atoms.An image of the tunnelling current measured on highly oriented pyrolyticgraphite is shown in Figure 1.6. The ordered structure illustrates the atomiccorrugation associated with the ordering of the underlying carbon lattice. It isimportant to note that the tunnelling current is a measure of electronic density ofstates, and not a direct measure of the position of atomic nuclei1. Nevertheless,the hexagonal structure of the graphite lattice can be observed in the image.An assumption of the Tersoff-Haman model is that the tip has an ideal,spherically-symmetric wave function and that both the tip and sample are metal-lic. A considerable amount of theoretical work has been done to derive moregeneralized forms that do not require as many assumptions: Appelbaum et al.examined how many-body effects change the tunnelling current [73] while Chenexamined different non-spherical orbital characters for the tip [70, 71]. Feucht-wang developed what is perhaps the most comprehensive theory of the tunneljunction independent of the transfer-matrix Hamiltonian relied upon by Tersoffand Haman [74–78]. The Tersoff-Haman model will be used to interpret datain this thesis due to its ability to correctly describe the observed experimentaldata and its relative ease of use.1A more complete understanding of the STM’s atomic resolution on graphite requiresabandoning some of the assumptions of the Tersoff-Haman model and treating the tip wavefunction with more care [70–72].Chapter 1. Welcome to Flatland 11x (nm)y(nm)  10.89 pA23.74 pA0 2 4 6 8 100246810Figure 1.6: Atomic resolution on the surface of graphite. The tunnellingcurrent measured on highly oriented pyrolytic graphite. The bright-est intensity corresponds to the centre of the hexagons formed byrings of six carbon atoms. Imaging conditions: Is = 15 pA, Vb = 600mV, and T = 4.5 K.Chapter 1. Welcome to Flatland 12The Tunnelling Transmission ProbabilityThe Bardeen matrix element Mts can be expressed in terms of a tunnellingtransmission probability T (z, E, Vb), ie. |Mts|2 = T (z, E, Vb) [69, 73]. Under theapproximation that the tunnelling barrier is trapezoidal it can be treated semi-classically in the Wentzel-Kramers-Brillouin (WKB) approximation to yield [27]T (z, Vb, E) = exp(−[2√2mh¯2(φ+eVb2− E)+ k2||]z)= e−κz, (1.4)where φ is the effective amplitude of the tunnelling barrier, m is the free electronmass, and k|| is the component of tunnelling electron momentum parallel tothe tunnelling junction interface [69]. The exponential decay of the tunnellingelectron wave function inside the tunnel barrier has an inverse length scalegiven by the factor κ ≈ 1 − 2 A˚−1 [49, 70, 72]. The magnitude of κ meansthat a one Angstrom change in the tip–sample separation produces roughly ae−(1 A˚−1)(1 A˚) = 1/e change in the tunnelling current signal. As a consequence,the tunnelling current is dominated by the current flowing through the atom onthe tip closest to the sample surface2. It is this Angstrom-scale sensitivity of thetunnelling current to changes in the tip–sample separation z that gives the STMa vertical sensitivity on the order of 0.1 A˚. The lateral sensitivity of the STMis strongly dependent on the orbital character of the last tip atom, as shown byChen [70–72]. Ideal, metallic single atom tips achieve a lateral resolution of 1A˚[72] or better, allowing the STM to measure atomic sized surface corrugations.Results reported in Part II and Part III of this thesis were taken using low-temperature STMs with a maximum measurement temperature of T = 4.8 K.At these temperatures the thermal energy E = kBT = 0.41 meV or smaller.Features of interest in the tunnelling current signal are larger than 1 meV andso it is reasonable to collapse the Fermi-Dirac temperature distributions insidethe function fts in Equation 1.3 into step functions so that fts = 1. Suppressingthe constant pre-factors, assuming small positive3 bias (Vb < 1 V) applied tothe tip, and substituting the tunnelling transmission probability into Equation1.1 the tunnelling current becomesIt(r, z, Vb) =∫ EF+eVbEFρs(r, E)ρt(E − eVb)T (z, E, Vb)dE. (1.5)The integral runs from the Fermi energy of the sample EF , to the energy ofthe Fermi energy plus the energy of the applied bias EF + eVb. It is commonnotation to use a change of variables in the integral so that the bounds run from0 to eVb [79, 80]. Equation 1.5 is useful for interpreting STM data gathered byrastering the tip over a sample and measuring the change in the tunnelling cur-2Leading to the STM practitioners idiom: “The last atom wins.”3For negative bias the integral runs from −eVb to zero.Chapter 1. Welcome to Flatland 13rent. Changes in the measured tunnelling current are a result of either changesin the integrated tip local density of states, the integrated sample density ofstates, or the tip–sample distance. In practice, a great deal of effort is put intoensuring that the tip is metallic with a local density of states that is constantas a function of energy ρt(E) = ρt. This ensures that the tip’s density of statesbecomes merely a factor of proportionality.The Differential Tunnelling ConductanceEquation 1.5 illustrates that the tunnelling current is sensitive to the integratedsample local density of states∫ eVb0ρs(r, E)dE. It is often desirable to probe thesample local density of states ρs(r, E) directly, as important physics often occursat a single energy, e.g. the energy onset of a surface state. A more direct measureof the sample local density of states is the differential tunnelling conductancedIt/dVb. The differential tunnelling conductance measures the local density ofstates at a single energy corresponding to the applied bias energy eVb, acquisitionmodes that measure it are classified as scanning tunnelling spectroscopy (STS).The differential tunnelling conductance can be derived by taking the deriva-tive of the tunnelling current with respect to the applied bias [69]. Taking thederivative of Equation 1.5 yieldsdItdVb= ρs(r, eVb)ρt(0)T (z, eVb, Vb) (1.6)+∫ eV0ρs(r, E)dρt(E − eVb)dVbT (z, E, Vb)dE+∫ eV0ρs(r, E)ρt(E − eVb)dT (z, E, Vb)dVbdE.The second and third terms in Equation 1.6 are often, though not always, neg-ligible in comparison to the first term [81, 82]. These latter terms can alsobe reformulated so that they are expressed in terms of the tunnelling current[79, 80]. If the second and third terms can be neglected and the tip densityof states is constant as a function of energy, then the differential tunnellingconductance isdItdVb∝ ρs(r, eVb)T (z, eVb, Vb), (1.7)an approximation that is so useful that it is commonly taken as the startingpoint for analysis of STS data [83].Figure 1.7 illustrates the relation between the sample local density of states,the tunnelling current, and the differential tunnelling conductance for a simplemodel. The sample local density of states exhibits a sharp step, similar to thatexhibited by noble metal surfaces at their surface state onset. This is measuredChapter 1. Welcome to Flatland 14as a kink in the tunnelling current, which otherwise has a constant slope. Theconstant slope follows from Ohm’s Law for the tunnelling current between twometals, each with flat density of states. The differential tunnelling conductanceexhibits a step at the energy of the onset (0), allowing an experimentalist tomeasure its energy via STS.ItVbVbdIt/dVb⇢s(E)⇢t(E)EEF EF + eVbE✏(0)(a)(b)Figure 1.7: Measuring ρs(E) with STS. (a) In the low-temperature limit,electrons from a metallic tip held at a bias of EF + eVb tunnel intoa sample which has the onset of a state at (0). (b) The sample canbe characterized by measurements of It and dIt/dVb .The particularbias EF + eVb leads to the data point denoted by the black circles.Sweeping the bias and measuring gives the two curves in red. Thetunnelling current shows the onset of the sample state as a kinkwhile the differential tunnelling conductance exhibits a step thatreflects the step in ρs(E).Equation 1.7 establishes a proportionality between the differential tunnellingconductance and the density of states of the sample multiplied by the tunnellingtransmission probability. In order to remove the dependence on the tunnellingtransmission probability in STS data there are number of normalization schemesused by experimentalists and theorists [79, 80, 84]. These schemes are necessary,as without them features in the STS spectra do not necessarily correspondto those in the local density of states of the sample. Separating features ofthe sample local density of states from those of the tunnelling transmissionprobability, particularly in the previously unexplored context of Fourier analysisChapter 1. Welcome to Flatland 15of the dIt/dVb signal, will be the main focus of Part II.Thermally-Limited Energy ResolutionThe temperature of the tip–sample junction plays a crucial role in the stabilityof the STM tip, adatom mobility on the surface, and the energy resolution of anySTS measurement. To achieve the most stable measurement conditions possiblethe experimental data presented in this thesis was acquired at temperatures ofT = 4.8 K or lower. Such low temperatures rob the surface atoms of kineticenergy and ensure that the Flatland under study remains effectively frozen whenprobed by the STM.When considering the effect of temperature on STS measurements it is il-lustrative to briefly examine the effect of finite temperature on the differentialtunnelling conductance within the Tersoff-Haman model. The effect of finitetemperature on the energy resolution can be observed by returning to Equation1.1. Writing the bias dependence into the sample local density of states insteadof the tip local density of states this takes the formIt(r, z, Vb) =4pieh¯∫ ∞−∞ρs(r, E + eVb)ρt(E)|Mts|2fts(E, Vb)dE. (1.8)Assuming ρt and Mts are independent of the electronic energy, which holds truefor energies on the order of a few hundred millielectronvolts, Equation 1.8 canbe rewritten as followsIt = A∫ ∞−∞ρs(r, E + eVb)fts(E, Vb)dE, (1.9)where A is a constant that contains all of the factors independent of the elec-tronic energy and applied bias.When performing FT-STS, the differential tunnelling conductance, dIt/dVb,is the measured quantity, which will be affected by the presence of the func-tion fts(E, Vb) at finite temperature. Taking the derivative of Equation 1.9with respect to the applied bias while suppressing functional dependencies notexplicitly related to the bias or energy yieldsdItdVb∝∫ ∞−∞[dρs(E + eVb)dVbfts(E, Vb) +dfts(E, Vb)dVbρs(E + eVb)]dE. (1.10)Taking these derivatives and introducing the change of variable η = E + eVb,the differential tunnelling conductance can be writtendItdVb∝∫ ∞−∞ρs(η)(− ddη11 + e(η−eVb)/kBT)dη. (1.11)Chapter 1. Welcome to Flatland 16This expressions shows that the sample density of states is multiplied by afunction that is the derivative of the Fermi-Dirac distribution. In the limit ofzero temperature, this function collapses to a delta function δ(η − eVb), andevaluating the integral produces the expected result shown in Equation 1.7(where here the tunnelling transmission probability has been suppressed). Atfinite temperature, however, the dIt/dVb signal is a convolution of the samplelocal density of states and the derivative of the Fermi-Dirac distribution. Thislatter contribution is a peaked function with a full-width at half maximum equalto 3.5kBT , which proves a good way to quantify the thermally limited energyresolution of an STS measurement at temperature T . For the reminder of thisthesis, the Fermi-Dirac factors will be suppressed when discussing the tunnellingcurrent, however quoted measurements of the energy resolution in a particularmeasurement will be calculated as 3.5kBT .1.3 OutlookPart I: Flatland and the Scanning Tunnelling MicroscopeThe STM is a probe of electronic density with nanoscale resolution and theability to engineer devices atom-by-atom. The scientific results presented inthis thesis have all been obtained using low-temperature, ultra-high vacuumscanning tunnelling microscopes. The operation and characterization of theseinstruments, the CreaTec LT-STM at the UBC Laboratory for Atomic ImagingResearch and the 1-K STM at the IBM Nanoscience Laboratory will be discussedin detail in Chapter 2 before delving into specific science cases in Parts II andIII.One of the biggest strengths of the STM technique is its versatility, which willbe demonstrated in this thesis by examining two markedly different nanoscaleFlatlands in Part II and Part III. The physical properties of these two radicallydifferent nanoscale environments will be exploited to obtain information whichcannot be accessed in traditional STM data: the momentum-space energy dis-persion of the electronic states and the electromagnetic interactions betweenmagnetic atoms at an energy scale below the energy resolution of thermally-limited scanning probe techniques. In Part II this constitutes revisiting thebasics of an STM analysis technique invented twenty years ago [85]: Fourier-transform scanning tunnelling spectroscopy (FT-STS) of quasiparticle interfer-ence. In Part III new interaction regimes will be measured using a technique thatis still in its infancy: electron spin resonance scanning tunnelling microscopy(ESR-STM) [86].Part II: Quasiparticle Interference and Momentum Space-PropertiesPart II is dedicated to the study of the momentum-space properties of the two-dimensional free electron gases at the (111)-terminated surfaces of the noblemetals silver and copper. Chapter 3 describes how the FT-STS technique can beChapter 1. Welcome to Flatland 17used to obtain detailed information about the momentum-space electronic prop-erties via measurement of scattering events on the surface. This momentum-space information complements the real-space local density of states informationcollected by conventional STM measurements. The FT-STS technique has beenused extensively in the last two decades to study complex materials like su-perconducting cuprates [87–93], superconducting iron arsenides [94–98], heavyfermion compounds [99–101], topological insulators/materials [102–109], andgraphene [110–112]. Despite its extensive use, no systematic comparison of ar-tifact features across different STM acquisition modes used in FT-STS has everbeen performed.Chapter 4 presents the first rigorous comparison between acquisition modesusing FT-STS measurements of the Ag(111) surface state. Measurements frommultiple different STM acquisition modes are presented and analyzed for fea-tures related to the set-point parameters, which are used to stabilize the tip–sample distance. These set-point features are then simulated using a modelbased on the Tersoff-Haman theory backed up by T-matrix simulations of thesample local density of states. The conclusions drawn from this work give acomprehensive catalogue of set-point related artifacts caused by modulation ofthe tunnelling junction and offer possible solutions to minimize such effects.Chapter 5 demonstrates that the conclusions drawn from the Ag(111) set-point model are applicable for FT-STS measurements of the Cu(111) surface.Multiple acquisition modes are used to confirm the presence of set-point relatedartifacts in the FT-STS analysis. Correctly identifying set-point artifacts andchoosing the appropriate acquisition mode is particularly important in mea-surements of the Cu(111) surface state, as it is revealed that a set-point artifactrelated to the most commonly used measurement mode for FT-STS acquisitionobscures a secondary feature that may be caused by many-body interactions inthe sample surface [113].Part III: Characterizing the Magnetic Dipolar Interaction of SingleAtoms with ESR-STMPart III employs a newly characterized STM measurement mode, ESR-STM[86], to study the magnetic dipolar interaction between iron and cobalt atomson a magnesium oxide bilayer. Chapter 6 details the utility of spin-resonancetechniques, showing how ESR-STM is capable of measuring magnetic interac-tions with an energy resolution that exceeds the limits imposed on traditionalSTS, while maintaining the nanoscale resolution of the STM. A discussion ofspin-polarized STM and previous attempts at measurements of ESR-STM sig-nals illustrate the technical elements needed to perform this technique.Chapter 7 presents a study using the atomic manipulation properties ofthe STM to characterize the dipolar interaction between individual iron andcobalt atoms on the MgO surface. Fitting the frequency splitting observed inESR-STM spectra to the dipole-dipole curve derived from theory allows forextraction of the magnetic moments of atoms on the surface close to an ironatom under the STM tip. The correspondence with theory and the insensitiv-Chapter 1. Welcome to Flatland 18ity of the signal to changes in the tip, temperature, or magnetic field allowsfor the commissioning of a new form of magnetometry capable of sensing thenanoscale magnetic environment in a four nanometre radius around each ironatom. A magnetic sensor array of iron atoms is constructed and used to performnanoscale trilateration (dubbed nanoGPS) by combining this sensing techniquewith the nano-engineering capabilities of the STM. A proof of principle of thecapabilities of nanoGPS is demonstrated, showing that the position and mag-netic moment of a nearby target iron atom can be extracted solely based off theESR-STM signal of three iron atoms in the sensor array.Part IV: Future Directions and Open QuestionsPart IV summarizes the main results presented in Part II and Part III. Futureexperiments in the noble metals employing the knowledge of set-point artifactsgained in Part II are proposed, with some thought given to the expected tech-nical challenges and results from each experiment. The technique of nanoscalemagnetic sensing developed in Chapter 7 can potentially be used to measurea variety of physical systems, like nuclear spins or molecular magnets. A fewcandidate systems for these experiments are suggested.Appendix A details the technical details and upgrades of the CreaTec STMthroughout the course of the thesis. Appendix B shows the results of real-spacesimulations of impurity scattering in Ag(111). Appendix C details a measure-ment scheme for performing pulsed ESR-STM. Pulsed ESR-STM, unlike thecontinuous wave mode used in Chapter 7, could allow for coherent manipula-tion of the atomic spin, with potential applications in quantum computing. Byoscillating the STM tip in-phase with microwave bias pulses, the first measure-ments using this technique are characterized.Chapter 2. Experimental Methods 19Chapter 2Experimental Methods...quantum phenomena do notoccur in a Hilbert space, theyoccur in a laboratory.Asher Peres - 1995 [114]Modern STMs used to probe complex materials and many-body physics aredesigned with specifications that exceed the five essential components listed inChapter 1. STM laboratories have been constructed that employ low-vibrationconstruction methods for tunnel junction stability, cryogenic cooling to increaseenergy resolution, and ultra-high vacuum to preserve surface integrity for days toweeks [115–122]. Two such laboratories are the Laboratory for Atomic ImagingResearch at the University of British Columbia and the Nanoscience Laboratoryat the IBM Almaden Research Center. This chapter delves into the details ofthese two laboratories and the specific instruments used for the work presentedin Part II and III of this thesis. A comparison is made between the construc-tion, modes of operation, and relative strengths and weaknesses of the STMsat each laboratory. This is done in conjunction with further development ofthe theoretical framework necessary to interpret STM data acquired in differentmeasurement modes.2.1 The Laboratory for Atomic ImagingResearchThe Laboratory for Atomic Imaging Research (LAIR) at the University ofBritish Columbia (UBC) is a low-vibration facility that houses three distinctlow-temperature, ultra-high vacuum (UHV) STMs. Each STM floats on pneu-matic isolators during measurement. The individual STMs are contained insideacoustically-isolated pods, which are separate from each other and the buildingfoundation. The construction of the LAIR, thoroughly documented by MacLeodin references [123, 124], allows for the STM tunnelling junction stability thatis necessary to probe quantum materials and develop new STM measurementtechniques.The three separate instruments at the LAIR are the Omicron LT-STM/AFM,the CreaTec LT-STM, and a custom-built STM that includes a dilution refrig-erator to access milliKelvin temperatures, a vector magnet for studies of surfaceChapter 2. Experimental Methods 20magnetism, and a molecular beam epitaxy system for in-situ growth of thinfilms. The oldest of the STMs, the CreaTec LT-STM, was used to acquire thedata presented in Part II. The operation and technical specifications of the Cre-aTec are introduced below while a more detailed discussion of its componentsand upgrades can be found in Appendix A.2.1.1 The CreaTec LT-STMThe Createc LT-STM at the LAIR, referred to as the CreaTec henceforth, isbased on the design of Sven So¨phel and Gerard Meyer [125]. Shown in Figure2.1 (a), it is one of the first commercialized versions of this system sold. Anin-depth description of its previous measurement configurations can be found inReferences [126–128]. Since 2012, the CreaTec has been operated within a low-vibration pod, called c-pod, in the LAIR. Between 2012 and 2017 the CreaTecreceived several major upgrades, which are detailed in Appendix A. The mostrecent upgrade was a complete retrofit of the STM head to allow for atomicforce microscopy capability.Figure 2.1: The CreaTec LT-STM in c-pod. (a) The vacuum chamber con-sists of the (1) manipulator arm, (2) preparation chamber, (3) cryo-stat, and (4) pumping system. (b) The STM/AFM head. Thevisible components are the (5) tunnelling current cold finger, (6)damping springs, (7) walking disc, and (8) sample holder mountedin the measurement position.The CreaTec STM head, shown in Figure 2.1 (b), is a beetle-style1 design1The term beetle-style STM originates from the loose resemblance of the STM head to aninsect in this design [129, 130].Chapter 2. Experimental Methods 21[129, 131] that hangs off the bottom of a liquid helium cryogenic bath. Thebath is coupled directly to the STM head via a series of damping springs and acustom-built thermal braid [128], giving a base temperature of T = 4.5 K duringmeasurement. This beetle-style STM head design allows for coarse, millimetre-scale positioning of the STM tip over the sample followed by fine nanometrescale control when scanning the surface. The STM head is encased in a series oftwo heat shields, one coupled to the liquid helium bath and the other coupledto a liquid nitrogen shroud. The heat shields and the STM head are mountedwithin an ultra-high vacuum chamber that has a base pressure of 10−10 mbar.Ultra-high vacuum is a necessary condition for measurement of pristine atomicsurfaces over timescales exceeding hours. The vacuum chamber housing theSTM head is coupled to an ultra-high vacuum sample preparation chamber thatcan be used to prepare samples through sputtering with argon ions, annealing,or cleaving. Specific sample preparation procedures for noble metal samples aredetailed in Chapter 4 and Chapter 5.In order to acquire the long-duration, high-resolution spectroscopy data pre-sented in this thesis, considerable work was performed to optimize the electrical,vibration, and acoustic noise in the CreaTec’s experimental set-up. The resultsof this optimization are shown in Figure 2.2, which characterizes the tunnellingcurrent, amplifier electrical noise, and ambient vibration noise of the CreaTecunder ideal measurement conditions. The CreaTec is equipped with two differ-ent high-gain current amplifiers, both produced by FEMTO Messtechnik GmbH,each with slightly different noise characteristics and bandwidths (see AppendixA for details). The in-tunnelling curves and out-of-tunnelling curves shown inFigure 2.2 were acquired using the higher bandwidth LCA-4K-1G amplifier,which was used for the measurements presented in Part II.As detailed in Chapter 1 the energy resolution of spectroscopic measure-ments made with the CreaTec is set by the tip and sample temperature, andtheir associated Fermi-Dirac thermal distributions. The STM tip-sample junc-tion sits at a temperature of T = 4.5 − 4.8 K during measurement and has athermally limited energy resolution of 1.3 − 1.5 meV. The maximum uninter-rupted measurement period possible on the CreaTec is 72 hours, at which pointthe cryogens used to cool the STM head must be replenished. The CreaTectunnelling junction demonstrates a high level mechanical stability – under 2pm peak-to-peak noise in the tip height when stabilized by the STM feedbackcircuit – and good isolation of the tip–sample junction from the environment.Combining these strengths with ultra-high vacuum conditions it is possible tomaintain the tip’s position over the same area on the surface while refilling cryo-gens, allowing for repeated measurements of the same area of the sample overthe course of weeks. The ability to return to the same spot on the sample andthe high degree of mechanical stability allows researchers using the CreaTec toprobe the spatial and energy landscape of a surface at a level that can measuresubtle many-body interactions [132].Chapter 2. Experimental Methods 22200 400 600 800 100010−310−210−1Frequency (Hz )AmplitudeSpectralDensity(pA/√ Hz)  (a)FEMTO DLPCA- 200FEMTO LCA- 4K- 1GOut of Tunne l l in gTunne l in g 100 mV , 100 pA100 10210−1210−1110−1010−910−810−710−610−510−4Frequency (Hz )VelocityLinearSpectralDensity(m/s/√ Hz)  (b)C re ate c In e rt ial b lo ckAc c e l e rom e te r Noi se F loorFigure 2.2: The CreaTec Baseline Noise Characterization. (a) Ampli-tude spectral density of the CreaTec tunnelling junction measuredin a 1 kHz range with 977 mHz resolution. The curves labelledFEMTO DLPCA-200 and FEMTO LCA-4K-1C are the noise base-lines of the tunnelling current pre-amplifiers when their inputs aregrounded. The out-of-tunnelling curve is the noise baseline of theFEMTO LCA-4K-1C amplifier when its input is connected to theSTM tip but the tip is not close enough to the sample to producea measurable tunnelling current. The tunnelling curve shows thenoise spectrum when tunnelling is occurring with a 100 mV biasand 100 pA tunnelling current. (b) Accelerometer measurements ofthe linear spectral density of the Createc inertial block. Spikes inthe linear spectral density are indicative of mechanical resonanceswithin c-pod. For more details on the vibration and acoustic noiselevel in the LAIR see References [123, 124].Chapter 2. Experimental Methods 232.2 The IBM-Almaden NanoscienceLaboratoryThe Nanoscience Laboratory at the IBM Almaden Research Centre (ARC) hasplayed a significant role in the history and development of STM [133–135]. In1990, only eight years after the invention of the STM by Binnig and Rohrer,Donald Eigler and his team successfully commissioned the first STM operationalat liquid helium temperatures [136] and used this STM to demonstrate the firstuse of atomic manipulation by spelling the letters IBM using xenon atoms onthe surface of Ni(110) [136]. Researchers at the Nanoscience Laboratory havebeen world experts at lateral and vertical atomic manipulation for the past 27years, manipulating individual atoms to form quantum corrals [55], quantummirages [56], atomic scale magnetic bytes [137], and an atomically small movieset [58]. These experiments have served not only to increase understanding ofnanoscale physical processes but have also been the STM works best suited tocapture public imagination2.As of 2014, the Nanoscience Laboratory includes three STM systems inacoustically isolated and electromagnetically shielded rooms. The STM used toacquire the data presented in Part III is an ultra-high vacuum STM capableof reaching temperatures below 1 K by the use of a 3He refrigerator. ThisSTM, henceforth referred to as the 1-K STM, was designed by Andreas Heinrich[138] to perform studies of single-atom magnetism, and was built with a 7 Tsuperconducting magnet centred on the STM tunnelling junction.2.2.1 The IBM 1-K STMThe 1-K STM has been used to perform a number of landmark studies in thefield of single atom magnetism on surfaces [86, 137–145]. In the design of the1-K STM, the STM head is thermally coupled but vibrationally isolated from a3He reservoir, both of which are surrounded by a 4.2 K set of heat shields cooledby a 4He reservoir. The 3He is continuously circulated through a Joule-Thomsonstage, keeping the STM head at a temperature of T = 1.2 K and eliminatingthe need for a pumped 4He supply [146]. By pumping on the 3He reservoir theSTM head can be cooled further to a temperature of 600 mK for up to ten hours[138]. Magnetic fields of up to 7 T can be applied in the direction parallel tothe surface inside the STM head, as illustrated in Figure 2.3. High-frequencycabling built into the bias line allows for electrical pump-probe measurements[143] and electron spin resonance experiments at microwave frequencies [86].The 1-K STM uses a vertically oriented manipulator arm to transfer sam-ples from the preparation chamber down into the STM head and inside the3He fridge. The nominal pressure in the preparation chamber is 10−9 mbar.Mounted to the preparation chamber are a series of electron beam evaporatorsthat can be used to deposit metallic atoms on a sample while it is mounted2At last count the STM choreographed stop motion animation “A Boy and his Atom” hasover six million views on YouTube [58].Chapter 2. Experimental Methods 24in the STM head. In the data presented in Chapter 6, these evaporators wereused to deposit the magnetic atoms iron and cobalt onto the surface of the sam-ple. The preparation chamber has also been equipped with the tools necessaryfor oxide thin film growth and characterization. Single crystals samples canbe cleaned through the use of an argon ion sputtering gun and electron-beamsample heater. Growth of thin film oxides on single crystals can be measuredin real-time by an Auger spectrometer mounted on the preparation chamber.These features allow the 1-K STM to measure and study the properties of singlemagnetic atoms on oxide thin films.Figure 2.3: The 1-K STM system at the Nanoscience Laboratory. (a)(1) The vertical manipulator arm for sample transfer (2) Fe and Coevaporators (3) rotary feedthrough flange and sample preparationchamber (4) 3He pumping system. (b) Internal schematic of the 1-KSTM cooling stages and magnet alignment. Certain details, such asheat exchangers, have been omitted for clarity.Chapter 2. Experimental Methods 252.3 Comparison Between the CreaTec and the1-K STMThe CreaTec LT-STM at the LAIR and the 1-K STM at the Nanoscience Lab-oratory are optimized to perform different types of nanoscale science. The Cre-aTec has a standard set of features for a commercial 4 K instrument, and has theadvantage of being optimized specifically to perform scanning tunnelling spec-troscopy measurements of energy levels in different materials [97, 132, 147, 148].In comparison to the 1-K STM, the CreaTec offers a higher throughput forpreparing and transferring samples between the preparation and STM cham-ber, ideal for experiments that require many cycles of preparation of metalsamples. The CreaTec data acquisition system, the Nanonis RC4/SC4 detailedin Appendix A, allows for interfacing with LabVIEW scripts, which is particu-larly useful for automating measurements that require a series of bias energiesover the same location. The CreaTec was used to gather the spectroscopic dataexamined in Part II.The 1 K-STM was used to gather the data presented in Part III. The 1-KSTM is one of the few systems in the world equipped with all the necessary fea-tures needed to perform electron spin resonance scanning tunnelling microscopyexperiments: a magnetic field, oxide growth capability, a bias line that cantransmit high-frequency signals, a base temperature below 1 K, and softwarespecifically designed for atomic manipulation. The 1-K STM head’s construc-tion differs substantially from the CreaTec in terms of material design, as the 7T magnet system precludes the use of magnetic materials. The data acquisitionsystem, originally written in MS DOS by D. Eigler, has recently been upgradedto interface with MATLAB. MATLAB scripts were used to execute the timingof voltage pulses needed for ESR experiments. A direct comparison betweenthe two STM’s features is shown in Table 2.1.Table 2.1: A comparison of the technical specifications of the CreaTec at theLAIR and the 1-K STM at the Nanoscience Laboratory.Parameter CreaTec 1-K STMFeedback Digital AnalogBias Sample TipCryogen Hold Time 72 hrs 120 hrsEnergy Resolution 1.3 meV 0.2 meVIn-situ Deposition Yes YesSample Preparation Yes YesOxide Growth No YesHigh-Frequency Cabling No YesMagnetic Field No YesChapter 2. Experimental Methods 262.4 Data Acquisition Modes of the STMDifferent STM acquisition modes are used to gather information about the tun-nelling current, surface height, and differential tunnelling conductance. Mea-surement with an STM can be performed in a variety of ways: at a single spatiallocation on the sample while varying the bias (e.g. STS point spectroscopy, ESR-STM), moving the tip across the sample (e.g. topography, spectroscopic maps),or a combination of both (e.g. spectroscopic grids). These different acquisitionmodes, introduced and described below, depend on when and how the STMfeedback circuit is used to change the height of the tip during the measurement,a factor that will be crucial in Part II.2.4.1 TopographyThe first acquisition mode invented for the STM was the topographic scan [26].In this measurement mode, the STM feedback circuit attempts to keep thetunnelling current constant at a user-defined set-point tunnelling current andbias voltage by varying the tip height as the tip is moved over the sample. Datain this measurement mode is acquired in the form of an image, which is collectedby rastering the tip along a slow scan and a fast scan axis. The resulting imageis composed of individual pixels, with the number of pixels per nanometre setby the experimentalist using the control software.Topographic measurements are used to record the apparent height of featureson the sample. Apparent height, h, refers to the amount that the STM feedbackcircuit moves the tip in order to keep the tunnelling current constant from onepixel to the next. The feedback circuit operates by comparing a user specifiedset-point tunnelling current, Is, with the measured tunnelling current at a givenlocation or pixel Ii. The difference ∆I = Ii − Is is converted to a change in tipheight by inverting Equation 1.3 to solve for the change in the tip–sample gapnecessary to make the measured current equal to the set-point current. Thefeedback circuit uses a proportional/integral controller to calculate the voltagesignal necessary to make the measured current closer to the set point currentand applies that voltage to the tip piezo actuator. By recording this outputsignal as the tip moves from pixel to pixel an apparent height measurement ismade. It is important to note that topography mode measures changes in theheight of the tip and not the tip–sample gap directly.The amount the tip moves is based on a convolution of electronic density andthe physical height of features on the sample surface, thus care must be takenwhen interpreting the apparent height. Adsorbed atoms or molecules, which siton top of the surface, can appear as depressions due to a lower electronic densitythan the surrounding surface eg. carbon monoxide on Ag(111) or Cu(111).Despite this complication, the topographic mode is extremely useful for imagingsurface structure, allowing glimpses into surface reconstructions [50], probing thearrangement of deposited molecules or atoms on a surface [149], and measuringthe height of atomic steps. Figure 2.4 shows a measurement of the topographyof an Ag(111) surface with five distinct step edges visible in the image. ByChapter 2. Experimental Methods 27taking measurements of the topography that show single atomic step edges andatomically resolved surface corrugations, the topography mode can be used tocalibrate the STM piezoelectric motors against x-ray diffraction measurementsof the interatomic spacing. Topographic data can therefore provide a means tomeasure the size and distance between nanoscale features.(a)  0 nm2.3 nm100 nm 80 100 12011.21.41.61.8x (nm)h(nm)(b)Figure 2.4: Topography of the Ag(111) surface. (a) A 282× 282 nm areaof the surface of the noble metal Ag(111). A plane subtractionhas been performed to flatten the image. (b) An apparent heightprofile following the black line shown in (a) shows a double stepedge, which can be used to vertically calibrate the STM against theatomic lattice. Imaging conditions: Vb = −40 mV, Is = 540 pA.2.4.2 Spectroscopic Imaging using a Lock-In AmplifierA lock-in amplifier can be used to extract information about the differentialtunnelling conductance of a sample while the regular data acquisition systemsimultaneously acquires information about the tunnelling current and apparentheight. This acquisition mode, dubbed spectroscopic-imaging STM, is useful forextracting information about the spatial properties of the sample local densityof states. The lock-in measurement of the differential tunnelling conductance inspectroscopic-imaging STM requires adding a small AC bias modulation to theDC applied bias. The experimental set-up is illustrated in Figure 2.5 (a).In spectroscopic imaging maps, the tip is rastered across a surface gatheringinformation pixel-by-pixel just like in topography mode. However, unlike to-pography mode, spectroscopic imaging can be performed either with the STMfeedback circuit engaged or disengaged. In Figure 2.5 (a) the path of the tip withthe feedback engaged is noticeably different from the path of the tip with thefeedback disengaged. Throughout this thesis, when the STM feedback circuit isengaged the measured quantity will be referred to as a constant-current dI/dVChapter 2. Experimental Methods 28Figure 2.5: Constant-height and constant-current dI/dV maps. (a) Mea-surement schematic, including AC modulation on the bias and lock-in amplifier acquisition of the dI/dV signal. In constant-currentmode the tip adjusts the tip–sample gap via the feedback as it rastersover step edges and adsorbates while in constant-height mode the tipheight stays constant. (b) Constant-current dI/dV maps acquiredat various different energies on the Ag(111) surface show ripples ofintensity off of surface impurities. (c) Constant-height dI/dV mapson the Ag(111) surface also show ripples off of impurities but witha lower signal-to-noise ratio.Chapter 2. Experimental Methods 29map. The data in this mode is recorded as a two-dimensional image whichcontains information about the tunnelling current, apparent height, and differ-ential tunnelling conductance. When the STM feedback circuit is disengagedthe measurement will be called a constant-height dI/dV map. Like a constant-current dI/dV map, it contains information about the tunnelling current anddifferential tunnelling conductance; however, it does not provide informationabout the apparent height since the feedback is not adjusting the tip heightbased on surface features. In Figure 2.5 (a) and (b) the differential tunnellingconductance of the Ag(111) as a function of tip position is plotted at differentenergies, corresponding to different applied biases.Constant-height dI/dV maps are considerably more difficult to acquire andanalyze than constant-current dI/dV maps. The lack of feedback means that thetunnel junction must maintain its stability without any correction for changesin the surface height3. As such, there is no way for the feedback to stop thetip from contacting the sample if there is a sudden change in surface height. Ifthe STM tip crashes in this way, the remaining data is lost and both tip andsample may be damaged. Despite this risk, constant-height dI/dV maps are auseful method of ensuring that features seen in constant-current dI/dV mapsare reproducible independent of the acquisition mode.The differential tunnelling conductance in constant-current and constant-height dI/dV maps is extracted by using a lock-in amplifier. A lock-in amplifieroffers phase-sensitive detection at a specified frequency. The lock-in measure-ment of the differential tunnelling conductance requires adding a small AC biasmodulation, VAC = V0ei(ωt+φ) to the DC bias, Vb = VDC . In Figure 2.6 themodulation of the bias around the DC value causes the tunnelling current sig-nal to sample the tunnelling current in a small regime around the applied bias,giving a ∆It signal as a function of Vb. The lock-in amplifier is used to comparethe frequency of this tunnelling current signal to a reference signal, extractingthe contribution related to the differential tunnelling conductance as derivedbelow.By ensuring that V0 << VDC , it is valid to perform a Taylor expansion ofthe tunnel current around the DC bias as followsIt = IDCt +dItdVb∣∣∣∣VDCV0ei(ωt+Φ) +O((VAC)2). (2.1)Inside the lock-in amplifier a band-pass filter is applied to the tunnelling currentsignal centred around the bias excitation frequency ω. The signal is then mixedto zero and a low-pass filter is applied so that the final signal extracted bythe lock-in amplifier is proportional to the differential tunnelling conductanceat the applied DC bias, the second term in Equation 2.1. The constant ofproportionality is given by the magnitude of the AC bias, V0,3The tunnelling current, measured concurrently, can be used to assess the degree of tip–sample stability throughout the measurement.Chapter 2. Experimental Methods 30It(VDC)ItVDCVACVbSlockin / ItFigure 2.6: Acquisition of the dIt/dVb by modulating the applied bias.The signal measured by the lock-in amplifier is proportional to thechange in the tunnelling current ∆It caused by the modulation ofthe applied bias with strength and frequency given by VAC .Slock−in =dItdVb∣∣∣∣VDCV0. (2.2)High-quality lock-in map acquisition requires measuring at each pixel ofthe map for several times the lock-in amplifier time constant, meaning that ahigh resolution map takes longer to acquire than a topography scan. The datapresented in Part II required 6− 9 hours for a single map with 512× 512 pixels.2.4.3 Scanning Tunnelling Spectroscopy (STS)Scanning tunnelling spectroscopy is used to obtain the differential tunnellingconductance as a function of applied bias with the STM tip stabilized over asingle spatial location. The acquisition of the differential tunnelling conduc-tance can be performed either by measuring It(Vb) as the bias is swept overa range, then taking the numerical derivative, or by simultaneously acquiringthe tunnelling current and differential tunnelling conductance using a lock-inamplifier. In either case, the measurement process consists of using the STMfeedback circuit to stabilize the tip at the set-point parameters and then disen-gaging the feedback. The bias is then swept over a user specified range whilethe tunnelling current and/or the lock-in signal are measured. STS is useful forresolving the energy structure of a sample in a localized area, with point spectraconveying information about the density of states of individual molecules [21],superconducting gaps [148], and magnetic transitions in single atoms [138, 139].In Figure 2.7 the STS spectra obtained from an Ag(111) surface exhibit fea-Chapter 2. Experimental Methods 31tures corresponding to the onset of a surface state. The surface state signature,which can be seen as a kink in the tunnelling current at a bias around Vb = −65mV, appears as a sharp increase in the differential tunnelling conductance. Theincrease in the differential tunnelling conductance is reflective of the increase inthe sample density of states as described by Equation 1.7. Note that the biasis swept from a negative to a positive regime. The bias voltage is referencedto the Fermi energy of the sample. At positive bias electrons tunnel from filledstates of the tip to empty states of the sample. At negative bias electrons tunnelfrom filled states of the sample to empty states of the tip. Due to the structureof the tunnelling transmission probability, this means that with a negative biasthe tunnelling current is more sensitive to the electronic structure of the tip, aneffect which must be carefully experimentally controlled for when attempting toextract only the sample density of states.100 50 0 50 100Vb (mV)3002001000100200300I t(pA)50 0 50Vb (mV)0.00.51.01.52.02.53.03.5dIt/dVb(nS)(a) (b)Figure 2.7: STS of Ag(111) (a) STS point spectra averaged from a 100 nm2clean area of the Ag(111) surface. (b) The corresponding differentialtunnelling conductance by way of the numerical derivative.2.4.4 Spectroscopic Imaging via Spectroscopic GridSpectroscopic grid acquisition combines the spatial measurements of topogra-phy with the bias measurements of STS. Acquiring a spectroscopic grid involvesmeasuring the STS spectra at each pixel of an image. In the case of the Cre-aTec, the most efficient way to gather STS spectra at each pixel is to measureIt(Vb) at each pixel of an image and then take the derivative of each of thesespectra numerically. The resulting data set contains spatial and energy infor-mation equivalent to n spectroscopic maps (where n is the number of points inthe bias sweep). Spectroscopic grid acquisition allows for an order of magnitudemore energies to be gathered per unit time than spectroscopic imaging with alock-in amplifier, because in grid acquisition the time per pixel is limited onlyby how fast the bias can be changed (the slew rate) and the bandwidth of thetunnel current amplifier. However, the numerical derivative of the tunnellingChapter 2. Experimental Methods 32current can often be quite noisy. For this reason, high-resolution grid measure-ments are only possible in an STM that has been optimized for low electronic,acoustic, and vibrational noise and even then numerical smoothing of the It(V )is required. The methodology for acquiring a spectroscopic grid and extractingthe differential tunnelling conductance at each pixel is shown in Figure 2.8.Bias (mV)Current(pA) Gaussian SmoothingNumericalDifferentiationTopographyI(V) dI/dVDifferential Tunneling Conductance dI/dVBias (mV)Bias (mV)Current (pA)dI/dV(nS)Vb = -50 mV Vb = 0 mV Vb = +50 mVFigure 2.8: Acquisition of a spectroscopic grid on Ag(111) At every pixelof the grid the feedback stabilizes the tunnelling junction at someapparent height, which yields a topography for the grid. Once sta-bilized, the feedback is disengaged and the bias is swept over a spec-ified range, allowing acquisition of I(V ) curves. Once I(V ) curveshave been acquired at every pixel the analysis proceeds by Gaus-sian smoothing each curve and taking the numerical derivative. Thisgives a dI/dV curve at every pixel allowing for reconstruction in realspace of the differential tunnelling conductance at each bias.Grid measurements offer a powerful way to probe the local density of statesas a function of energy and spatial location. A single, sufficiently large, gridmeasurement can take several days and involve the acquisition of over one mil-lion individual point spectra. A good grid measurement yields data on thesurface topography and a slice of the differential tunnelling conductance at eachenergy in the bias sweep. Acquiring such a grid is a high stakes endeavour;if the STM tip interacts with the surface or changes electronically during thecourse of the measurement the grid will typically be rendered unsalvageable foranalysis.Tip interaction with the sample occurs frequently enough that crafting atip capable of producing reproducible atomic-resolved images and STS pointspectra without changing its state is a major component of performing STMexperiments. Experimental results presented in this thesis were acquired withtips that underwent some level of nanoscale shaping through voltage pulsesand controlled contact with a metallic surface. In order to control for selec-Chapter 2. Experimental Methods 33tion bias introduced by only analyzing data acquired with the minority of tips,ie. those that were stable throughout days of measurement, every result wasreproduced using at least one other macroscopically different tip and multipledifferent preparations of the sample.2.5 ConclusionsThis chapter introduced the STMs at the LAIR and Nanoscience Laboratoryand the data acquisition techniques used in this thesis. This knowledge will beapplied in the examination of the physics of two different surface systems. Spec-troscopic measurements of the noble metal surface states using constant currentdI/dV maps, constant height dI/dV maps, and spectroscopic grids taken withthe CreaTec are presented in Part II. By taking the Fourier-transform of theresulting data a comparison will be made between the expected electronic be-haviour in scattering-space and artifacts features that are dependent upon theacquisition mode. The development of electron spin resonance techniques usingthe 1-K STM is described in Part III and relies on characterization of the surfaceusing topography mode and STS.34Part IIQuasiparticle Interferencein Noble Metal SurfacesChapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 35Chapter 3Quasiparticle Interferenceand Fourier-TransformScanning TunnellingSpectroscopyA good idea has a way ofbecoming simpler and solvingproblems other than that forwhich it was intended.Robert Tarjan [150]Fourier-transform scanning tunnelling spectroscopy (FT-STS) is used to studya wide range of complex materials in condensed matter physics by analyzing thesignatures of quasiparticle interference in a material’s electronic density. Quasi-particle interference, originally derived in the context of Friedel oscillations, isa physical phenomenon that can be used to provide access to the momentum-space properties of the electrons in a material. FT-STS provides a means toextract information about these properties in scattering-space, provided thereexists some initial knowledge of the material’s band structure and the properfiltering techniques are applied. The history of FT-STS measurements illus-trate its importance in the field of quantum materials: revealing the underlyingphysics of superconductors, heavy fermion materials, and topological materials.3.1 Introduction: Quasiparticle Interference asMeasured by Scanning TunnellingMicroscopyWhen an impurity is placed inside a Fermi liquid, like an atom deposited ona metallic surface, the Fermi liquid rearranges to screen the electromagneticpotential created by the presence of the impurity [6]. Friedel was the firstChapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 36to theoretically calculate the resulting fluctuations surrounding the impurity1,which lead to standing waves in the electronic density [152]. These so-calledFriedel oscillations, which are known as quasiparticle interference (QPI) whenthey occur at energies other than the Fermi energy2 [153], can be observed inreal-space using an STM. This effect was first reported simultaneously by twodifferent groups in 1993 [154, 155]. In Figure 3.1 the very first real-space imageof QPI on Cu(111), taken at the Nanoscience Laboratory in 1993, is shown nextto an image of QPI on Cu(111) taken at the LAIR in 2017. In both imagescarbon monoxide molecules adsorbed on the surface create spherical waves inthe measured signal, while in Figure 3.1 (a) step edges produce plane waveoscillations. Observation of QPI is a compelling demonstration of the ability ofthe STM to image quantum phenomena in real-space [156].Figure 3.1: Quasiparticle interference in Cu(111). (a) Constant cur-rent image of QPI from impurities and step edges taken at theNanoscience Laboratory. Imaging conditions: T = 4 K, Vb = 100mV, Is = 1000 pA, and 50 × 50 nm2. Adapted with permissionfrom Macmillan Publishers Ltd: Nature [154], copyright (1993). (b)Differential tunnelling conductance of Cu(111) acquired from a con-stant current dI/dV map taken using the CreaTec in 2017. Imagingconditions: T = 4.5 K, Vb = −100 mV, Is = 900 pA, and 80 × 80nm2.1The theoretical developments leading to the discovery of Friedel oscillations throughoutthe 1950s are quite interesting. For a synopsis by Friedel’s student E´mile Daniel see reference[151].2Friedel’s original derivation was not specific to the Fermi level [152]; however, this defini-tion of Friedel oscillations and QPI has been adopted in the intervening decades [153].Chapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 373.1.1 Derivation of Friedel Oscillations in the ElectronicDensity of StatesFriedel oscillations play an important role in the physical properties of metallicsolids, and have been extensively studied in the last sixty years [156, 157]. Overthe course of these six decades the mathematical language used to describe theobserved phenomenon has varied, often in a way that provides new perspectivesbut also makes it difficult to understand the entire body of literature in context[157]. This section briefly reviews Friedel oscillations: first in the context ofusing Friedel’s original method of applying the scattering theory of quantummechanics to understand impurity screening in a gas of electrons and secondusing a Green’s function treatment of scattering from a localized impurity.Screening and Phase ShiftsIn classical electromagnetism, screening of a charged impurity embedded in acharged fluid occurs when the fluid rearranges itself to screen the impurity. Thisrearrangement of the fluid can be quantified as a change in the charge distribu-tion δρ(r) and it falls off exponentially with distance from the impurity centre,δρ(r) ∝ e−αr/r. This screening distribution can be observed experimentally influids of electrolytes [158].In a metal, the charged fluid is, to a good approximation, a gas of electronsand screening of a charged impurity must be calculated in the context of quan-tum mechanics. The first attempt to calculate the fluctuations in the chargedistribution eδρ(r), or correspondingly the modulation of the density of statesδρ(r), was performed by Thomas [159] and Fermi [160]. This theory makesthe important approximation that the screened potential of the impurity δV (r)and the screening charge δρ(r) are locally proportional, as they are in elec-trolytes. A result of this approximation is that the decay is again exponential:δρ(r) ∝ e−κr/r. The Thomas-Fermi screening model is useful because it showsthat the characteristic screening length for electrons is smaller than or of orderof the electron-electron distance in many materials. This explains why simpletheories that neglect electron-electron correlations, like the Hartree-Fock treat-ment [6, 161], still accurately predict the properties of many materials. However,the Thomas-Fermi screening model does not predict oscillatory behaviour in theelectronic density, as the assumption that δρ(r) is locally proportional to δV (r)is only true for long-wavelength charge variations [162].Friedel’s approach abandoned the assumption of local proportionality be-tween the screened potential and the charge distribution. He instead appliedscattering theory, by that point a well-developed component of quantum me-chanics, to scattering of electrons within a metal [152]. A simplified version ofthe original derivation, following reference [157], is given below.In the absence of any impurities and suppressing the effect of the periodiclattice an electron wave function in a metal can be described by a plane wavecharacterized by momentum kiChapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 38Ψki(r) = eiki·r, (3.1)which satisfies the time-independent Schro¨dinger equationHk |Ψk〉 = (k) |Ψk〉 . (3.2)The energy eigenstates are described by the free-electron dispersion(k) = h¯2|k|2/2m∗ (3.3)where m∗ denotes the quasiparticle effective mass and h¯ is the Dirac constant.The wave functions |Ψk〉 have accompanying density of statesρ(r, ) =∑k|Ψk(r)|2δ(− k). (3.4)Adding a spherically-symmetric charged impurity at the origin changes thesolution to the Schro¨dinger equation. The wave functions that satisfy theSchro¨dinger equation in the presence of a radially symmetric potential can bewritten in terms of Bessel functions of the first jl(kr) and second nl(kr) kindand the Legendre polynomials Pl(cos(θ))Ψk(r) =∑l(aljl(kr) + blnl(kr))Pl(cos(θ)), (3.5)where the sum is over angular momentum denoted by l. The coefficients al, blcan be re-parameterized in terms of an amplitude Al and a phase δlal = Al cos(δl) (3.6)bl = −Al sin(δl). (3.7)This parametrization is useful because the angular momentum-dependent phaseshift can be interpreted as a measure of how far the solution at the origin of theimpurity is displaced from the free particle solution, for which δl = 0 ∀ l.Making the assumption of radially symmetric s-wave scattering, all the am-plitudes for l > 0 become zero so that only the zeroth order scattering phaseshift δ0 is relevant to the solution. The scattered wave function then simplifiestoΨkf (r) = αeiδ0eikf ·rkfr∝ j0(kfr) cos(δ0)− n0(kfr) sin(δ0) (3.8)Chapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 39where α is an amplitude related to the scattering cross-section. The initial planewave, in the absence of the impurity, can also be expressed in terms of Besselfunctions. Taking the difference between the amplitude of the initial wave andscattered wave the density fluctuations caused by the presence of the impurityare given byδρ(r, k) = |Ψkf (r)|2 − |Ψki(r)|2 (3.9)δρ(r, k) ∝ |j0(kfr) cos(δ0)− n0(kfr) sin(δ0)|2 − |j0(kir)|2 (3.10)Employing the identities j0(x) = sin(x)/x, n0(x) = − cos(x)/x and assumingthat the energy dispersion (k) depends only on the magnitude of |k| thenδρ(r, k) ∝ cos(2k()r + δ0)r2. (3.11)This result, unlike the classical and Thomas-Fermi results, demonstratesthat oscillations in the density of states are expected upon introduction ofan impurity. The function 2k() is the inverse of the energy dispersion (k)and it is responsible for the wavelength of observed oscillations in experimentalmeasurements using the STM. For the simple band structure and dispersionassumed here, one free-electron band with dispersion (k) = h¯2|k|2/2m∗, the2k() function is heavily weighted towards the backscattering vector3 2k() =|q| = |kf |− |ki|. For elastic scattering at the Fermi level this corresponds to thevector q = 2kF . The damping function 1/r2 in three-dimensions, is dependenton the dimension of the electron gas, such that for a two-dimensional electrongas it has a 1/r dependence [157, 163, 164] and takes the following form at theFermi energyδρ(r, EF ) ∝ cos(2kF r + δ0)r. (3.12)The surface states of Ag(111) and Cu(111) are very good approximation toa two-dimensional electron gas [8, 12, 81, 165–167] and Friedel oscillations ontheir surface are well described by Equation 3.12.Quasiparticle interference and the T-matrixModern condensed matter theory treats the problem of Friedel oscillations, andthe more general problem of QPI, using the formalism of Green’s functions. Thisformalism comes from quantum electrodynamics and provides a more generalway to derive Friedel oscillations in the presence of many-body interactions3To understand why this vector is strongly preferred it is beneficial to look at Friedeloscillations in the context of linear response theory [157].Chapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 40[168]. Rather than giving the full derivation here, the interested reader is insteaddirected towards reference [169]. This section will state the form of the electronicdensity changes caused by a localized impurity and explain it in the context ofGreen’s functions and the T-matrix. This formalism will be used to simulateexperimental results in Ag(111) in Chapter 4.The T-matrix formalism attempts to solve the problem of how an initialwave function |ψ0〉 is scattered by the presence of an impurity potential V intoa new state |ψ〉. The scattered wave function can be solved for iteratively inthe form of the Lippmann-Schwinger Equation|ψ〉 = |ψ0〉+GV |ψ〉 (3.13)where G is the Green’s function of the initial wave function |ψ0〉. The Green’sfunction G is introduced as a function with the property(E −H0)G(E) = δ(x− x′) (3.14)where H0 is the Hamiltonian for the time independent Schro¨dinger Equation inthe absence of any impurity potential and E is the corresponding eigenenergy.The Green’s function can be written in terms of the Hamiltonian as followsG =1E −H0 + iη , (3.15)where η is a small number that is set to zero before calculating any physicallymeasurable property. The Green’s function is useful because it contains the fullinformation necessary to solve the Schro¨dinger equation in the absence of theimpurity scattering centre and acts as the propagator for particles from x to x′(or k to k′ in momentum-space).Returning to the Lippmann-Schwinger Equation of Equation 3.13, the scat-tered wave function can be substituted into itself iteratively to give the Bornseries, which is represented visually in Figure 3.2. The T-matrix is introducedas a succinct form of the Born series that takes the initial wave function to thescattered wave function. For more details about the derivation and role of theT-matrix in condensed matter physics see Callaway [170, 171].| 0 > | > = +V+V V+	…= | 0 > | 0 > + T| 0 > | >Figure 3.2: The T-matrix as a sum of Feynman diagrams.Chapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 41Using the T-matrix the scattered state can be expressed in terms of theinitial state and the Green’s function|ψ〉 = |ψ0〉+GT |ψ0〉 . (3.16)This equation allows for the solution of the wave function in the presence of animpurity scatterer. It is a more general form than Friedel’s original derivation,better able to simulate QPI with fewer necessary assumptions. The fluctuationsin the local density of states can be formulated in terms of the T-matrix andGreen’s functions and take the formδρ(q, E) = − i2pi∑kIm[G(k, E)TG(k + q, E)]. (3.17)This expression will be used in the modelling section of Chapter 4 to simulatethe QPI in Ag(111) introduced by the presence of carbon monoxide moleculeson its surface [132].QPI in a 2D Electron GasFigure 3.3 illustrates how the presence of QPI allows access to the band dis-persion of a two-dimensional electron gas in scattering-space using an STM.Measurements of the differential tunnelling conductance using an STM provideinformation about the sample local density of states through the relation be-tween dIt/dVb and ρs derived in Chapter 1. In the absence of a scatteringimpurity, the situation is as shown in Figure 3.3 (a), and an STM measurementof the differential tunnelling conductance cannot provide information about themomentum-space electronic properties or electronic dispersion (k).Figure 3.3 (b) illustrates how the sample local density of states exhibits QPIin the presence of an impurity. Images of the differential tunnelling conductancethat measure this QPI signature in the density of states contain informationabout the electronic dispersion relation because the oscillations in ρs are pro-portional to cos(2|k|r+δ)/r, and so depend on a scattering vector q = 2|k| thatconnects two pieces of the allowed momentum states (at the Fermi energy thisis called the Fermi surface). Measurement of the wave vector of the QPI patternobserved in the differential tunnelling conductance corresponds to a measure-ment of this scattering vector. When the scattering vector q is extracted atdifferent energies the dispersion in scattering space (q) can be constructedas shown in Figure 3.3 (b) (iii). The scattering-space dispersion can then berelated back to the dispersion in momentum-space, providing information onimportant electronic properties that can’t be measured in the absence of QPI.The scattering-space intensity |S(q,E)| is calculated by taking the Fourier trans-form of the differential tunnelling conductance, an analysis technique known asFourier transform scanning tunnelling microscopy (FT-STS).Chapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 42(a) No Impurity Potential(i) Real-Space Local Density of States(ii) Momentum-Space Dispersion(iii) Scattering-Space Intensity |S(q,E)|qxqyQPI Intensity Dispersion|q|✏(q)(b) Impurity      Potential ✏(k)qxqy2|q||q|✏(q)yx⇢syx⇢s⇢s / |eik·r|2 = 1⇢s / cos(2|k| · r+ )/rkk|q| = 2|k| |q| = 2|k|✏(k) =~2|k|22mFigure 3.3: Quasiparticle interference in real-space, momentum-space,and scattering space. (a) In the absence of an impurity thedensity of states of a two-dimensional electron gas is flat with aparabolic dispersion relation (k). No signal in scattering space isobserved. (b) In the presence of an impurity the sample density ofstates acquires a dependence on the back scattering vector |q| =2|k|.Chapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 433.2 Fourier-Transform Scanning TunnellingSpectroscopy Measurements ofQuasiparticle Interference3.2.1 History of FT-STS MeasurementsFour years after the observation of the first QPI, independently by Crommie [54]and Hasegawa [155], it was realized that the Fourier-transform of images thatexhibited QPI could be used to infer information about the momentum-spaceproperties of the underlying material being imaged. The first Fourier-transformSTM/STS measurements were performed on metal surfaces [85, 172–178]. Inthese early publications, there was some contention about how directly the FT-STS intensity corresponded to the material’s momentum-space properties [176].Despite difficulties in exactly relating the observed FT-STS scattering intensitywith the quasiparticle momentum, the authors of these early works predictedthat FT-STS would prove a powerful technique in the study of the momentum-space properties of quantum materials. This prediction wound up being ex-tremely accurate, as over the next two decades FT-STS was used to study su-perconducting cuprates [87–93], superconducting iron arsenides [94–98], heavyfermion compounds [99–101], topological insulators/materials [102–109], andgraphene [110–112]. With the stability of modern STM instrumentation, largeFT-STS data sets can be acquired yielding resolution in energy and momentum-space rivalling that of state-of-the-art ARPES [132] and allowing new insightinto physical processes like electron-boson coupling [128, 132, 179].Figure 3.4 shows FT-STS measurements, both the very first measurementreported by Sprunger et al. on the surface of Be(0001), and a more recentexample from the LAIR of the surface of the superconductor LiFeAs. TheSTM constant current image of Be(0001) in Figure 3.4 (a) exhibits strong QPI.The absolute value of this image is shown in Figure 3.4 (b). It shows a ringof intensity that is associated with scattering of quasiparticles across the Fermisurface and Bragg peaks that correspond to the atomic lattice. QPI from a morecomplicated material is shown in Figure 3.4 (c); where a differential tunnellingconductance image of the iron pnictide superconductor LiFeAs exhibits QPIcaused by different lattice defects. The FT-STS signal corresponding to thisimage, shown in Figure 3.4 (d), gives a much richer pattern, as well as showingthe Bragg peaks associated with the atomic lattice. The more complicatedFT-STS pattern can be attributed to a more complicated quasiparticle bandstructure in momentum-space, involving scattering between multiple bands [97].3.3 FT-STS Measurements of the NobleMetals Surface StatesNoble metal (111)-terminated surfaces have been a subject of intense studyusing FT-STS [113, 180, 181]. They were the first surfaces upon which QPIChapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 44Figure 3.4: Fourier transform analysis of quasiparticle interference. (a)(i) A constant current STM image of QPI in Be(0001). Imaging con-ditions: T = 150 K, Vb = 4 mV, Is = 1.5 nA, and 4×4 nm2. (ii) Theassociated |S(q,E)| intensity exhibits a ring of intensity associatedwith scattering across the Fermi surface and lattice Bragg peaksfrom the atomic lattice. Adapted from Reference [85]. Reprintedwith permission from AAAS. (b) (i) Differential tunnelling conduc-tance image of LiFeAs. (ii) The corresponding Fourier transformshows the complexity of the FT-STS signal in a multi-band system.Imaging conditions: T = 4.5 K, Vb = 10 mV, and 26 × 26 nm.Adapted figure with permission from S. Chi, S. Johnston, G. Levyet al., Physical Review B, 89, 1–10, 2014 [97]. Copyright 2014 byAmerican Physical Society.Chapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 45was measured [54, 155] and exhibit large QPI signatures because their surfacetermination supports the existence of a surface state that functions as a two-dimensional electron gas (2DEG). Following from Equation 3.12, QPI in a 2DEGis described byδρ(r, ) ∝ cos(2k()r + δ0)r. (3.18)The lower order damping compared to materials with three-dimensional elec-tronic character means that the oscillations are more long-lived on these surfacesand easier to measure with an STM.Surface states are ubiquitous among the noble metals. They arise becauseat the surface of a crystal the discrete translational symmetry of the crystallattice is broken, and Bloch’s theorem no longer holds. Bloch electronic wavefunctions are therefore no longer valid solutions to the Schro¨dinger Equation.Instead, there exist valid solutions that are localized within several atomic radiiof the surface, decaying exponentially into the bulk and into the vacuum. Suchstates were first characterized by Shockley in 1939 [182] and appear in manydifferent terminations of face-centred cubic noble metals, such as (100), (111),and (110).In (111) terminated noble metals the broken translational symmetry at thesurface gives rise to sp derived electronic wave functions that are localized nearthe crystal surface [183], and form a parabolic band with onset below the Fermienergy [12, 165, 167]. This surface state, which is the source of the QPI in the(111)-terminated noble metals, has a band structure that agrees very well withthe single-band free electron model at energies near the Fermi energy. This sin-gle, parabolic band supports only one intra-band back scattering vector that canbe related to the momentum-space structure by a single factor of 12 at a givenenergy. This band structure provides an easy way to infer momentum-spaceproperties from FT-STS, unlike materials such as the cuprate superconductors,where there are eight primary scattering vectors at any given energy [88, 89, 91].In the (111) noble metals, QPI measured by FT-STS is reflective of the surfaceproperties rather than the bulk, as the surface state band across which quasi-particles scatter is caused by, and localized to, the surface. The combination oflarge QPI oscillations combined with a simple band structure made the noblemetals some of the first materials characterized using FT-STS [180].Despite their simple band structure, secondary scattering features that arenot compatible with the expected intra-band scattering vector have a long his-tory in FT-STS measurements of the noble metals [113, 132, 173, 176, 180, 181].Petersen et al. first reported a secondary scattering feature in FT-STS inAu(111) and Cu(111) and ascribed this to scattering across a neck in the bulkFermi surface [180]. Schouteden et al. performed further measurements of thesesecondary features and attributed their behaviour above the Fermi energy (EF )to inelastic electron relaxation [181]. Sessi et al. demonstrated that the sec-ondary features are not compatible with the positions of the bulk bands inChapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 46Au(111), Cu(111), or Ag(111). They instead attributed the secondary featuresto an acoustic surface plasmon dispersion [113]. Data presented in Chapter4 and Chapter 5 will provide another possible explanation for the presence ofsecondary scattering features in the FT-STS scattering intensity: measurementartifacts caused by modulation of the STM tunnelling barrier during data ac-quisition.3.3.1 Acquiring and Analyzing FT-STS DataA low-temperature, ultra-high-vacuum STM is a probe of real-space electronicproperties at surfaces with sub-nanometre resolution of electronic density; how-ever, in most cases it is not possible to access the momentum-space electronicproperties of a sample using an STM. An exception occurs when quasiparticleswithin the sample scatter from defects, adsorbates, or step edges on a surface.By taking the absolute value of the Fourier transform of images of the differ-ential tunnelling conductance that exhibit QPI a large amount of informationrelating to the material’s band structure can be extracted.Fourier-transform scanning tunnelling spectroscopy (FT-STS) can be per-formed using differential tunnelling conductance data acquired either from dI/dVmaps using a lock-in amplifier or from spectroscopic grids. In either case, theanalysis proceeds by taking the Fourier transform of an image of the differentialtunnelling conductance of the sample, as shown in Figure 3.5 (a) and (b). Theextracted scattering intensity S(q,E), is dependent on the modulation in thesample density of states caused by the presence of impuritiesS(q, E) ≡ FT(dItdVb(r, E))∝ δρs(q, E). (3.19)The scattering intensity is a complex quantity, to represent the real and imag-inary parts it is common practice to plot |S(q,E)|. At a single energy inscattering-space |S(q,E)| shows the backscattering vectors between allowed kvectors at that energy. If |S(q,E)| is spherically symmetric, a further analysisstep can be taken: the data can be projected onto a single radial vector in scat-tering space qr, as shown in Figure 3.5 (c). Radial projection is performed bycalculating the Euclidean distance of each pixel in the scattering intensity imagefrom q = (0, 0) and then summing the intensity contribution from each pixel foreach discrete distance, normalized by the number of pixels. Radial projectionallows for an increase in the signal-to-noise background and for one-dimensionalfitting of observed features in the FT-STS line cut.For a single constant-current or constant-height dI/dV map Figure 3.5 en-capsulates the full scope of FT-STS analysis. For a spectroscopic grid, however,the steps shown in Figure 3.5 can be repeated for every energy in the grid’s biasrange. The scattering intensity can therefore be constructed at each energy inthe grid, along with a radial projection for each of these energies, as shown inFigure 3.6 (a). Grid measurements can probe hundreds of bias energies and soChapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 479 pS18 pS(a) dIdV (r, E = 0)  (b) |S(q,E=0)|qx (A˚− 1)qy(A˚−1 )−0.2 0 0.2−0.200.20 0.1 0.2 0.300.30.6q r=2kFqr (A˚− 1)|S(qr,E=0)|(c) |S(qr,E)|100 nmFigure 3.5: Extracting scattering information from dI/dV by analyz-ing QPI. (a) Differential tunnelling conductance of Ag(111) at theFermi energy, Vb = 0 mV. (b) The absolute value of the Fouriertransform of the differential tunnelling conductance, ie. the FT-STSscattering intensity |S(q,E)|. The bright ring denotes the preferredscattering vector for quasiparticles in the surface. (c) A radial pro-jection of the scattering intensity |S(qr,E)|. The primary feature atqr = 2kF corresponds to the scattering vector of the Ag(111) surfacestate at the Fermi energy.a succinct way to represent all of the projected FT-STS line cuts is to constructan image of the scattering dispersion (qr) as function of energy, as shown inFigure 3.6 (b). Plots such as Figure 3.6 (b) are extremely useful, as they exhibitthe full scattering-space and energy resolution obtained by the grid, which insome cases match the resolution of state-of-the-art ARPES [132]. Dispersionplots can also be constructed from map data, it just requires taking many mapsover the same area, typically a task that is too experimentally challenging toachieve with the same energy resolution as a grid measurement over the samearea.The resolution of FT-STS data in scattering space is set by the real-spacesize of the differential tunnelling conductance image. If an image has a realspace size of L× L in nanometres then its resolution in scattering space ∆q ininverse Angstroms is given by∆q =2pi10L(3.20)The size of the Fourier-transformed image in scattering space Q is given by thescattering-space resolution multiplied by the number of pixels P in length L inthe real-space imageChapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 480 0.1 0.2 0.3012345|S(qr,E)|qr (A˚−1)(a)-38.6 m eV- 33.9 m eV- 29.2 m eV- 24.5 m eV- 19.8 m eV- 15.1 m eV- 10.4 m eV- 5.7 m eV- 1.0 m eV3.7 m eV8.4 m eV13.1 m eV17.8 m eV22.5 m eV27.2 m eV31.9 m eV36.6 m eVqr (A˚−1)Energy(meV)(b)MinMax  0 0.1 0.2 0.3−30−20−100102030Figure 3.6: Scattering dispersion (qr) from a spectroscopic grid. (a)|S(qr),E)| line cuts for a spectroscopic grid on Ag(111). The pri-mary feature, corresponding to scattering of a surface state, changesposition or disperses as a function of energy. This figure only showsthree percent of the data obtained in the grid measurement shownin (b). To represent all the |S(qr),E)| data taken in the grid thescattering intensity can be plotted as function of energy and qr,where the maximum in intensity corresponds to the peaks seen in(a).Chapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 49Q = (∆q)P. (3.21)Equations 3.20 and 3.21 imply that the best FT-STS data, with the largest andhighest-resolution view of scattering space, can be obtained by collecting thelargest possible real-space image of the differential tunnelling conductance withthe maximum number of pixels experimentally attainable with the experimentalapparatus. The desire to maximize Q while simultaneously minimizing ∆q leadsto data acquisition over the largest areas of the sample accessible with theSTM. For the Createc, such areas are on the order of 200 − 300 nm in lineardimension. Despite sample preparation cycles using sputtering and annealing itis often the case that there will be multiple scattering impurities in images of thesurface. The most common type of scattering centres on the noble metal surfacesdiscussed in Chapter 4 and Chapter 5 are step edge dislocations in the crystalstructure and carbon monoxide (CO) adsorbates. Carbon monoxide adsorbsonto the noble metal surfaces from the ultra-high vacuum environment, whereit is present at a higher partial pressure than other gases due to its presencein the steel used in the chamber walls. In practice, the Createc is optimized totake a single, high-resolution spectroscopic grid every 72 hours. In this time,it can gather real-space tunnelling conductance information over hundreds ofnanometers and with pixel densities high enough to obtain a scattering-spaceresolution ∆q = 0.0026 A˚−1.The energy resolution for spectroscopic grids is often comparable to the ther-mal limit imposed by the Fermi-Dirac distributions of tip and sample (∆E = 1.5meV for T = 4.2 K) while for maps it depends on the number of maps acquiredand their spacing in bias. In practice this often means that the energy resolutionof a scattering dispersion constructed from maps is an order of magnitude worsethan the thermally-limited energy resolution.Filtering in FT-STS analysisRegardless of which acquisition mode is used to gather the differential tunnellingconductance data, FT-STS analysis often requires the application of filters, inboth real-space and scattering-space. Filtering helps to suppress signals un-related to the QPI that saturate the Fourier-transform signal. These filteringtechniques are introduced in Figure 3.7 and are used heavily in the analysis ofthe data presented in Chapter 4.Both steps edges and carbon monoxide produce QPI in the noble metalsurface states. Step edges act like a line of one-dimensional scatterers and pro-duce a scattering intensity with a preferred direction. Carbon monoxide actsas a radially symmetric scattering centre, which produces a radially isotropicpattern in scattering-space. When performing a radial projection of |S(q),E)|to produce |S(qr),E)|, isolating the signal from the carbon monoxide scatter-ing centres is preferred, since they do not produce a signal with a preferreddirection. Focusing on the carbon monoxide scattering centres allows for moreChapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 50direct, quantitative comparisons between images taken of different portions ofthe sample surface, since features in |S(q),E)| are more easily separated fromthe topographic details of the real-space image.Filtering of the step edge signals on the surface of Ag(111) is shown in Figure3.7. Figure 3.7 (a) shows the differential tunnelling conductance and scatteringintensity with no filters. The most intense signal in |S(q),E)| is caused by theseries of step edges in the real-space image. To isolate the ring of intensity causedby the carbon monoxide scattering centres, visible in the real-space image, thefirst step is to set the differential tunnelling conductance over the step edges tothe average value of the differential tunnelling conductance in the image. This isperformed over each step edge area, denoted by the white dashed lines in Figure3.7 (b). The scattering intensity shows a corresponding drop in the asymmetricintensity associated with the step edges, though the pattern is still not isotropic.Before taking the radial projection, a further step of angular filtering inscattering-space is performed. Angular filtering takes the radial projection fo-cusing over an area of the |S(q),E)| devoid of step edge induced signal. In Figure3.7 (c) the filtered area is denoted by the white arrow between the dashed lines.A comparison of |S(qr),E)| for the case of no filters, real-space filter, andreal-space and scattering-space filter is shown in Figure 3.7 (d). The recoveredsignal with both sets of filter can be attributed more readily to a single typeof scattering centre, rather than a mix of step edges and point scattering cen-tres. This makes the theoretical modelling of the scattering centres as radiallyisotropic in Chapter 4 apt and allows for better comparison between experimentand theory.3.4 OutlookFT-STS analysis of QPI is a powerful technique for extracting momentum-space information about quasiparticles imaged with atomic-scale resolution inreal-space. As has been demonstrated in this chapter, the complexity of themomentum-space quasiparticle structure in real materials requires extreme carewhen interpreting FT-STS data. Interpretation of FT-STS can be further com-plicated by effects related to the acquisition mode used to measure the differen-tial tunnelling conductance. Chapter 4 and Chapter 5 will examine how FT-STSdata, even in measurements of a very simple band structure, are susceptible tothe presence of secondary features in |S(q,E)| related not to the quasiparticledispersion, but to the measurement mode.Chapter 3. Quasiparticle Interference and Fourier-Transform Scanning Tunnelling Spectroscopy 51( a ) N o fi l t e r s( b ) r - s p a c e fi l t e r s( c ) r - s p a c e an d q - s p a c e fi l t e r sθ0 0.1 0.2 0.300.30.60.91.2 q r = 2k Fq r ( A˚ − 1)|S(qr,E=0)|( d )  N o fi l t e r sr - s p a c e fi l t e r sr - s p a c e an dq - s p a c e fi l t e r s100 nm100 nm100 nm0.1 Å-10.1 Å-10.1 Å-1Figure 3.7: Filtering processes for FT-STS data. (a) No filtering is ap-plied to the real-space differential tunnelling conductance image.Step edges and carbon monoxide scattering centres are both visibleon the surface. The |S(q),E)| intensity exhibits a ring associatedwith the carbon monoxide QPI and a near vertical line of intensityassociated with the step edge. (b) Real-space filtering averages outthe step edges, filtered region denoted by dashed white lines. The|S(q),E)| intensity associated with the step edge is reduced. (c) An-gular filtering is performed before taking the radial projection. (d)Comparison of the radially projected scattering intensity |S(qr),E)|between the three methods.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 52Chapter 4Acquisition-DependentArtifacts in FT-STS of theAg(111) Surface StateWhat we see depends mainly onwhat we look for.Jonathan Lubbock [184]The following Chapter contains text and figures adapted from “Dispers-ing Artifacts in FT-STS: a comparison of set point effects across acquisitionmodes” IOP Nanotechnology, 27(41):1-7 [185]. Figures are altered unless oth-erwise noted.FT-STS is an important technique in the study of complex materials becauseit gives experimentalists the ability to map the electronic dispersion of both oc-cupied and unoccupied bands, and locally correlate the electronic dispersion withthe surface structure. There is a catch: some a priori knowledge of the under-lying band structure is required to assign meaning to the features observed inFT-STS measurements. This chapter presents a comparison of the most com-mon modes of acquiring FT-STS data on the well-characterized Ag(111) surfaceand demonstrates, through both experiment and simulations, that artifact fea-tures1 can arise that depend on how the STM tip height is stabilized. The mostdramatic effect occurs when a series of constant-current dI/dV maps at differ-ent energies are acquired; in this acquisition mode a feature that disperses inenergy appears that is not observed in other measurement modes. Such artifactfeatures are similar to those arising from real physical processes in the sampleand are susceptible to misinterpretation.1Artifact vs artefact: The language in this thesis adheres to British written English insteadof American, hence the use of tunnelling instead of tunneling. Artifact proves to be the oneexception as the spelling ‘artefact’ only gained traction in British English around 1990.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 534.1 Topography and Electronic Character ofAg(111)4.1.1 The Ag(111) Surface StateCrystalline silver has a close-packed, face-centred-cubic crystal structure witha lattice constant of a0 = 0.409 nm [186]. At a (111)-terminated surface,the atomic lattice is hexagonal with an atomic nearest-neighbour distance ofa0√2/2 = 0.288 nm. Figure 4.1 shows the surface structure: (a) imaged viaan STM topograph, (b) measured by low-energy electron diffraction (LEED),and (c) drawn using a graphic model. The height of single step edges, predictedto be 2.4 A˚, are used to calibrate the vertical scale of the STM piezoelectricmotion. Atomically resolved images of the Ag(111) surface, as shown in theinset of Figure 4.1 (a), are used to calibrate the x and y motion of the STMpiezos.Figure 4.1: Lattice structure of the Ag(111) surface. (a) STM topo-graphic image of the Ag(111) surface showing multiple step edges.Imaging conditions: Vb = −40 mV, Is = 540 pA, and apparentheight 0 − 2.3 nm. Inset shows atomic resolution on the Ag(111)surface. Imaging conditions: Vb = 5 mV, Is = 90 nA, and apparentheight 0 − 16 pm. (b) LEED measurement of the Ag(111) surface(Energy = 152 V, Current = 0.08 mA) after sputtering and anneal-ing cycles. (c) Model which illustrates the Ag(111) lattice parameterand face-centred cubic lattice.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 54The (111) termination of a silver crystal surface supports a two-dimensionalsurface state, known as a Shockley state [182], that has been extensively studiedexperimentally and theoretically [8, 12, 60, 81, 128, 132, 165, 166, 187–194]. TheAg(111) surface state is an ideal candidate for the comparison of different FT-STS acquisition modes as it is a well-characterized, theoretically understoodmaterial with a single band in momentum-space. The Shockley surface statearises from the breaking of crystalline symmetry between the bulk crystal statesand the vacuum level [182]. This broken symmetry gives rise to sp derivedelectronic wave functions that are localized near the crystal surface [183], andform a parabolic band with onset below the Fermi energy [12, 165, 167]. Withinthe bias regime2 probed in this chapter (−100 mV,100 mV) the Ag(111) surfacestate band is well-described by a single free-electron band [81, 128, 132, 166,167, 183, 188, 195] of the form(k) =h¯2k2‖2m∗− µ, (4.1)where (k) is the band dispersion, h¯ is the reduced Planck constant, m∗ is theeffective electron mass, which for the Ag(111) surface state is approximatelyequal to 0.4 times the free electron mass, and µ = 65 meV is the chemicalpotential of the band. The surface state band in the noble metals occupies aregion of momentum-space devoid of bulk bands, known as the L-gap [167].Within the surface state band the electric field of the surface state electrons iseffectively screened by bulk electrons, reducing electron-electron interactions sothat the free-electron model of Equation 4.1 works very well [193].Figure 4.2 shows spectroscopic characterization of the Ag(111) surface stateband using the STM and ARPES. The surface state band is centred at the Γpoint in k-space, corresponding to the point where k|| = 0 [183]. An advan-tage of the Γ point is that it is preferentially probed by STM tunnelling elec-trons. Modifications to the surface state band due to many-body effects suchas electron-phonon coupling or plasmon formation have been probed by bothARPES [167, 195] and STM measurements [60, 113, 132, 187, 189, 191, 194]and do not compromise the conclusions drawn in this chapter.4.1.2 Sample Preparation and Measurement ProtocolMeasurements of the Ag(111) surface state were made at a temperature of T =4.5 K using the CreaTec. The STM tip was made from electrochemically etchedtungsten wire, which was further prepared in situ by sputtering and annealing toremove the oxide layer. Initial contact with the silver crystal resulted in a silver-terminated tip. The Ag(111) crystal was cleaned by three cycles of sputteringunder 2.0 × 10−5 mbar argon atmosphere and annealed at 500◦C to producelarge, clean terraces with a low density of CO adsorbates that act as scatteringcentres. Spatial calibration was performed once prior to all measurements by2The applied bias Vb in mV characterizes the electronic energy E being probed in meV.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 55Figure 4.2: Characterization of the Ag(111) surface state band. (a) Thek-space dispersion extracted from dI/dV tunnelling spectroscopy(solid points). A parabolic fit to these data is plotted along thesolid line. The dashed curve is the dispersion as measured by pho-toelectron spectroscopy. The inset is a dI/dV spectrum showingthe surface state onset. Reprinted figure with permission from L.Jiutao, W.-D Schneider, R. Berndt, Physical Review B, 56, 7656–7659, 1997 [81]. Copyright 1997 by American Physical Society. (b)ARPES data of the surface state dispersion in k-space shows theparabolic band. Reprinted figure with permission from F. Reinert,G. Nicolay, S. Schmidt, D. Ehm, and S. Hu¨fner, Physical ReviewB, 63, 1–7, 2001 [167]. Copyright 2001 by the American PhysicalSociety.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 56obtaining atomic resolution of the Ag(111) surface, as seen in Figure 4.1 (a), toensure accurate real and scattering-space measurements.Spatially-resolved spectroscopic measurements of the differential tunnellingconductance were performed using three different acquisition modes: spectro-scopic grids, constant-current dI/dV maps, and constant-height dI/dV maps.Spectroscopic grids were measured with varying size and spatial resolution (giv-ing different pixel densities) and a thermally-limited energy resolution of 1.5meV. Each tunnelling current spectrum It(V ) in the spectroscopic grids con-sisted of 512 data points in bias, which were Gaussian smoothed over 3 adjacentpoints in energy and averaged over 8−12 repeated measurements at each spatiallocation. Typical grid measurements took between 50 and 80 hours to complete.Constant-current and constant-height dI/dV maps were taken using a lock-inamplifier with a bias modulation frequency of 1.017 kHz and an amplitude of5 mV. Spatial resolution in pixels for each map was set to 512 × 512. As eachmap yields spectroscopic information at only one energy, a series of maps wasacquired to investigate the scattering dispersion (q).4.2 Experimentally Observed Set-Point Effectsin Different Acquisition ModesA “set-point effect” in scanning tunnelling microscopy describes a feature thatis dependent on the stabilization bias Vs and set-point tunnelling current Is,and not one characteristic of the surface under study. This section is dedicatedto identifying set-point effects in FT-STS measurements of the Ag(111) surfacestate using spectroscopic grids, constant-current dI/dV maps, and constant-height dI/dV maps.There are three separate bias or energy regions of the Ag(111) surface stateband where set-point parameters can be probed. These regions are distinguishedby where the set-point bias Vs is in relation to the band onset (0) = −65 meVand the Fermi Energy EF = 0 meV. The different energy regimes are illustratedin Figure 4.3. It is important to note that the stabilization bias only sets theenergy at which the STM tip is stabilized by the feedback, and, depending onmeasurement mode it can be independent from the energy range probed by themeasurement, which corresponds to the values of the tunnelling bias Vb. Forexample, a spectroscopic grid measurement could have a stabilization bias inRegion I at Vs = −100 mV while probing an energy range in Region III, suchas Vb = 20− 60 mV.4.2.1 Spectroscopic Grids as a Function of Set-PointParametersSpectroscopic grids measured with different set-point parameters exhibit differ-ent features in the FT-STS scattering intensity |S(q, E)|. Figure 4.4 shows thedifferential tunnelling conductance and FT-STS scattering intensity evaluatedChapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 57✏(k)kRegion III: eVs > EFRegion II: EF > eVs > ✏(0)Region I: ✏(0) > eVs✏(0) = 65 meVEF = 0 meVFigure 4.3: The three different energy regions in which the stabilizationbias Vs can be set when measuring Ag(111).at zero bias (the Fermi energy EF ) for three grids with different stabilizationbiases and set-point currents. The real-space images show different spatial re-gions with distributions of step edges and carbon monoxide molecules on theAg(111) surface. These grids probe the three different energy regimes possiblefor the stabilization bias around the surface state band: Region I below thesurface state onset (0) > eVs, Region II between the onset and Fermi energyEF > eVs > (0), and Region III above the Fermi energy eVs > EF . In all threecases step edges in the real-space images have been filtered using the prescriptiongiven in Figure 3.7.All three grids in Figure 4.4 exhibit QPI; decaying plane waves can be ob-served near the step edges while the circular ripples surround the carbon monox-ide molecules. FT-STS analysis of the real-space images produces a scatteringintensity that has a ring of intensity, of radius q = 2kF = 0.168 ± 0.003 A˚−1,corresponding to quasiparticles back-scattering across the surface state band atthe Fermi energy EF . High intensity features that run vertically through theFT-STS data are caused by signal from the step edges that remains after thereal-space filtering process. For the grid measurement with stabilization biasVs = −40 mV in Region II, shown in Figure 4.4 (b), a second ring of intensity ofsmaller radius than 2kF is present in the scattering intensity, indicated by theblack arrow in the figure. Similarly, for the grid measurement with stabilizationbias Vb = 100 mV in Region III, shown in Figure 4.4 (c), there is additional in-tensity beyond the 2kF radius of the surface state scattering, again shown witha black arrow. Both the grids taken in Region II and Region III correspond tothe grid measurements with stabilization bias above the band onset.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 58(a) Region IVs = −100 mV, I s = 100 pA(b) Region IIVs = −40 mV, I s = 540 pA(c) Region IIIVs = 100 mV, I s = 100 pA0.1 Å-1 0.1 Å-1 0.1 Å-150 nm 50 nm 50 nm q = 2kF q = 2kF q = 2kFFigure 4.4: Ag(111) spectroscopic grids with different set-point param-eters. Top panels are the differential tunnelling conductance evalu-ated at the Fermi energy Vb = 0. The bottom panels are the corre-sponding FT-STS scattering intensity |S(q, E)|. Vertically alignedfeatures of high intensity in the bottom panels are caused by the stepedges present in the top panels. (a) Vb = (−100, 120) mV, 239×239nm2 with 380 × 380 pixels. (b) Vb = (−40, 40) mV 280 × 280 nm2with 400×400 pixels. (c) Vb = (−100, 100) mV, 240×240 nm2 with350× 350 pixels.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 59Analysis of spectroscopic grid scattering dispersionIn order to examine how the scattering intensity for different set-point param-eters varies as a function of energy, Figure 4.5 shows the scattering dispersionfor the same spectroscopic grids shown in Figure 4.4. |S(qr, EF )| is plotted foreach grid in Figure 4.5 (d), with a vertical offset to make the curves more easilydistinguishable. In all cases, the primary feature is caused by the scatteringintensity corresponding to the surface state band [132]. A high background atlow qr ≈ 0 is visible in all three cases and is the product of broad low-frequencynoise in the measurement of dI/dV causing sharp intensity in scattering-spaceas a byproduct of the Fourier-transform. Differences in the intensity of the sur-face state band scattering are related to the change in set-point current becausehigher tunnelling current produces larger QPI signals in the differential tun-nelling conductance. Figure 4.5 (a) shows that there are no prominent featuresother than the surface state dispersion for the grid with stabilization bias belowthe onset of the band.For the grid with stabilization bias in Region II, shown in Figure 4.5 (b),there is a broad vertical feature below 2kF and above EF centred at qr =0.11 ± 0.03 A˚−1 and indicated by a black arrow in the figure. This featurediminishes in intensity as it nears the band intensity below EF . A second broadfeature appears on other side of the band at bias below Vb = −20 mV, centredat qr = 0.15 ± 0.02 A˚−1. Neither of these features are present in scatteringintensity of the Region I grid.For the grid with stabilization bias in Region III, shown in Figure 4.5 (c),there is a broad feature at fixed qr = 0.22±0.03 A˚−1 that appears (indicated bythe black arrow) in addition to the surface state back scattering intensity. Thisadditional feature can also be observed at EF in Figure 4.5 (d), where it appearas a broad peak smaller in amplitude than the surface state feature. This peakin intensity is unique for the grid with stabilization bias in Region III and isvisible even though this grid has the worst signal-to-noise ratio of the three datasets.Subsequent grid measurements demonstrated that changes to the tunnellingcurrent set-point Is produced changes in signal intensity at all energies andvalues of qr but did not cause the secondary scattering features observed inFigure 4.5 to change location. Therefore, for spectroscopic grid measurements,there is an additional feature in the scattering dispersion that depends on thebias used when the tip height is stabilized.4.2.2 Spectroscopic Grids and Constant-Current dI/dVmapsIn order to probe the nature of secondary features in the energy dispersion inmore detail, the scattering dispersion was calculated based on measurementsof the differential tunnelling conductance collected using the constant-currentdI/dV map acquisition mode. Constant-current dI/dV maps were collected overtwo different 60×60 nm2 areas of the sample over a bias range Vb = (−100, 180)Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 60(a) Vs = −100 mVq r = 2kFEnergy(meV)Region I0 0.1 0.2 0.3−40−2002040(b) Vs = −40 mVRegion II0 0.1 0.2 0.3(c) Vs = 100 mVMinMaxRegion III0 0.1 0.2 0.30 0.05 0.1 0.15 0.2 0.25 0.3 0.35012(d)qr (A˚−1)|S(qr,EF)|  Vs = −100 mVVs = −40 mVVs = 100 mVFigure 4.5: Scattering dispersion calculated from three spectroscopicgrids with different set-point conditions. Horizontal and ver-tical lines indicate E = EF = 0 meV and qr = 2kF respectively. (a)No secondary features observed. (b) Broad non-dispersing featureabove EF and below 2kF . (c) Broad non-dispersing feature belowEF and above 2kF . (d) Radial projection of the FT-STS signal atEF for each grid. Imaging conditions match Figure 4.4.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 61mV. Neither of these areas included step edges, and so the observed QPI wasdriven primarily by the presence of carbon monoxide molecules scattering sur-face state electrons.The FT-STS data derived from the constant-current dI/dV maps were usedto construct a scattering dispersion. A spectroscopic grid measurement with thesame set-point current Is = 100 pA and scattering-space resolution ∆q = 0.01A˚−1 was acquired for direct comparison between the two measurement modes.Figure 4.6 illustrates the scattering dispersion constructed from seventeen ofthe constant-current dI/dV maps and a grid with stabilization bias Vs = 100mV (Region III). A comparison between these two measurement modes is par-ticularly important, as they form the primary means of data acquisition in theFT-STS literature, with constant-current dI/dV maps being more common [87–93, 96–98, 132, 196]. The constant-current dI/dV maps have lower energy res-olution compared to grid measurements (∆Emaps = 10.6 meV vs ∆Egrid = 1.5meV) collected over the same time scale, due to the difference in the speed ofdata acquisition between the two measurement techniques mentioned in Chapter2.Both measurements show the expected parabolic dispersion of the surfacestate band, but differ in other features. The grid measurement shows significantintensity below the band onset at low-qr. As determined previously in Grotheet al. [128, 132], the scattering intensity below the onset of the band is stronglydependent on the nature of the scattering impurities. The scattering intensityin this region varies in between measurements, regardless of acquisition mode,depending on whether the dominant scatter is carbon monoxide, other impuri-ties, or if there is significant intensity from step edge scattering that remainsafter filtering. This intensity below the band is therefore ignored to focus on fea-tures that vary only with the acquisition mode, ie. dispersive or non-dispersiveFT-STS features.The grid measurement in Figure 4.6 (a) shows the same broad, verticalfeature slightly above qF = 2kF , in agreement with Figure 4.5 (c). The ap-pearance of the same feature during a different experimental run, including adifferent sample preparation and measured sample area, demonstrate the repro-ducibility of the observed features provided the set-point parameters remain thesame. The measurements made by acquiring constant-current dI/dV maps atdifferent energies, shown in Figure 4.6 (b), have an additional faint, and alsobroad feature appears that disperses, crossing qF = 2kF at EF . The constant-current dI/dV maps in Figure 4.6 (b) also show a strongly varying backgroundintensity as a function of energy, strongest near the Fermi energy, that is notobserved in the grid measurement. This varying background can be attributedto a change in signal intensity caused by variation in the tunnelling junctionresistance. Though the set-point current was a constant Is = 100 pA for allthe acquired maps the stabilization bias Vs varied exactly with the bias beingprobed Vb.Figure 4.6 (c) shows a comparison at a single bias Vb = −40 mV, wherethe primary surface state back-scattering feature occurs at the same qr for bothmeasurement modes but the secondary feature, indicated by the black arrowsChapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 62for both measurement modes, does not. This clearly illustrates that acquiringthe FT-STS dispersion using different acquisition methods yields quantitativelydifferent results.(a) Vs = 100 mVEnergy(meV)0 0.1 0.2 0.3−80−4004080  MinMax(b) Vs = Vbqr = 2kF0 0.1 0.2 0.3−80−40040800 0.1 0.2 0.3 0.400.51 (c)qr (A˚−1)|S(qr,E=−40meV)|  Grid Region II IMaps E = eVsFigure 4.6: Energy dispersion from spectroscopic grids and constant-current dI/dV maps. Horizontal and vertical lines indicate E =−40 meV and qr = 2kF respectively and the parabola represents afree electron model. (a) Spectroscopic grid with imaging conditions:60 × 60 nm2, 230 × 230 pixels. (b) Dispersion constructed fromconstant-current dI/dV maps with imaging conditions: 60×60 nm2,512× 512 pixels. (c) The scattering intensity at −40 meV for bothgrid and map measurements scaled so that the surface state featurehas the same intensity.4.2.3 Constant-Height dI/dV MapsConstant-height dI/dV maps, though experimentally more difficult to acquire,decouple the height of the tip set by the STM feedback from surface featuresthat vary with lateral position. They are the acquisition mode least sensitiveto effects caused by the values of the set-point parameters. A series of sevenChapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 63constant-height maps was acquired over a bias range Vb = (−90, 110) mV inorder to compare the scattering intensity with constant-current dI/dV mapresults. The set-point current was Is = 150 pA and the measured area was 65×65 nm2 with no step edges observed. Reproducibility of the acquired FT-STSsignal, independent of the detailed position of scattering centres, was checkedby performing constant-height maps with identical acquisition parameters overtwo different areas at an acquisition bias of Vb = 110 mV. Like constant-currentdI/dV maps, the stabilization bias in constant-height dI/dV maps is equivalentto the bias being measured Vb.The energy resolution of the constant-height dI/dV map data is too low toeffectively plot using a colour scale. Instead, a comparison of |S(qr, E)| betweenconstant-height dI/dV maps and constant-current dI/dV maps at individualenergies is plotted in Figure 4.7. In this figure, |S(qr, E)| curves at varyingenergies are plotted for both map acquisition modes, with offsets used to dis-tinguish between curves and each curve scaled so that the surface state featureis the same strength. The constant-height dI/dV map data exhibits only onepeak, corresponding to the expected scattering vector for the surface state band,at all values of Vb. The constant-current dI/dV maps exhibit secondary peaksshown by the black arrows, that below the Fermi energy are at larger qr thanthe surface state peak but cross over above the Fermi energy to appear at lowerqr. The comparison between FT-STS intensity derived from constant-heightdI/dV maps versus from constant-current dI/dV maps suggests that the sec-ondary features are a result of the STM feedback changing the tip height duringthe measurement.4.2.4 Comparison Between all Acquisition ModesIn order to make a more direct comparison between the different acquisitionmodes, FT-STS data were generated at the same energy, E = 50 meV, by thethree different acquisition modes with four different set-point conditions: a gridwith Vs corresponding to the same energy as the energy examined, eVs = E = 50meV, a grid with opposite polarity Vs from the energy examined, eVs = −E =−50 meV, a constant-current dI/dV map where Vs always corresponds to theenergy examined, eVs = E = 50 mV, and a constant-height dI/dV map wherethe tip height is stabilized only at the first pixel of the image. The resulting|S(qr, E = 50 meV)| signals are shown in Figure 4.8. All four measurementsexhibit a sharp peak at qr = 0.220±0.006 A˚−1, corresponding to the intra-bandscattering of the surface state at 50 meV. Away from this feature, the resultingsignal varies significantly based on acquisition mode.FT-STS of the grid with Vs = 50 mV produces a nearly identical signal tothe FT-STS derived from the constant-current dI/dV map; this is expected ifthe additional feature is a set-point effect since the feedback is stabilized at eachpoint with the same parameters for both measurements. Both exhibit a strongsecondary feature just below the Fermi back-scattering vector q = 2kF (indi-cated by the black arrow) and otherwise are featureless, with the lowest levelsof noise of the four measurements. The intensity |S(q, E = 50 meV)|, plottedChapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 640 0.2 0.40123456(a) Constant Height-90 meV-50 meV-10 meV30 meV70 meV110 meVq r (A˚−1)|S(qr,E)|0 0.2 0.4012345(b) Constant Current-85 meV-50 meV-6 meV27 meV71 meV111 meVq r (A˚−1)Figure 4.7: Constant-height dI/dV maps compared with constant-current dI/dV maps and spectroscopic grids. (a) Constant-height dI/dV map radial projections of |S(q, E)| show only the sur-face state feature. Imaging conditions: 65×65 nm2, 512×512 pixels.(b) Constant-current dI/dV maps show two features away from EF .The imaging conditions for (b) are the same as in Figure 4.6.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 65in Figure 4.8 (b) and (d) for these two measurements, shows two concentricrings of intensity corresponding to the primary back scattering vector and thesecondary feature.For the grid acquired with Vs = −50 mV, while the surface state back scat-tering vector at qr = 0.22 A˚−1 remains the same, there is now a much broaderfeature, shown by the red arrow, centred below qr = 2kF , and the feature seenwith Vs = +50 mV is no longer observed. The constant-height dI/dV measure-ment has a larger low-frequency background with no clear secondary features.Each of the secondary features appear above the background level of the otherline cuts, indicating that each are additional features tied only to the measure-ment mode and set-point parameters. The lack of any secondary features inthe constant-height measurement, along with the dependence of the additionalfeatures on the bias used to stabilize the tip height at each position for gridsand maps, provide further experimental evidence of the influence of the spatiallyvarying tip height on the FT-STS measurement.4.2.5 Experimental ConclusionsWhen the tip is stabilized at each pixel, a constant-current condition is metby the STM feedback circuit. That constant-current condition depends on theintegrated density of states, convolved with the transmission function of thetunnelling junction. Since the density of states varies with both position andenergy, the constant-current topography will contain spatial modulations due tothe electronic structure that depend on the bias applied, modulating the physicaltip-sample separation. As the dI/dV signal also depends on the tunnellingtransmission probability, which depends on tip-sample separation, it is perhapsnot surprising that extraneous features are observed in FT-STS that depend onthe energy used to stabilize the tip height. This dependence of the FT-STS onthe tip height explains the differences between grid and constant-current dI/dVmap measurements: for a grid, only one stabilization bias is used for all energiesprobed, so a non-dispersing feature either above (positive Vs) or below (negativeVs) 2kF is observed, but for constant-current dI/dV maps, the stabilization biasfollows the energy being probed, generating a secondary feature that disperses.The positions of the features observed in a grid measurement with V = 100mV and a series of constant-current and constant-height dI/dV maps are shownin Figure 4.9 to summarize the potential set-point artifacts. Fits were performedby windowing the experimental line cuts around the peak feature, subtracting alinear background fit and then fitting the peak with either a Lorentzian, for thesurface state peaks, or Gaussian functions, for the set-point dependent feature.Only one of every ten grid energies is plotted in Figure 4.9. For all threemeasurement modes the surface state peaks extracted from the data agree wellwith a free electron model of surface state scattering, reproducing the expectedparabolic intra-band scattering dispersion. Fits to the STS grid data producea vertical artifact feature, which appears at different scattering wave vectorsabove and below the Fermi level. The constant-current dI/dV maps generatea feature that disperses strongly below the Fermi level and then crosses theChapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 66Figure 4.8: The effect of stabilization bias on the observed FT-STSpattern. (a) |S(qr, E = 50 meV)| comparing two grids with dif-ferent stabilization biases, a constant-current dI/dV map, and aconstant-height dI/dV map. (b-e) the corresponding FT-STS pat-tern at E = 50 meV from (b) a spectroscopic grid with Vs = 50 mV,Is = 100 pA, (c) a spectroscopic grid with Vs = −50 mV, Is = 100pA, (d) constant-current map at Vs = 50 mV, Is = 100 pA, and (e)a constant-height map at Vs = 50 mV, and initial current Is = 100pA. For the constant-height data a restricted q-space angular filterwas used to reduce the influence of a step edge running across thetop of the image..Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 67band dispersion at EF . Above EF this feature appears at roughly the samewave vector independent of the applied bias energy. The constant-height dI/dVmaps showed no additional features beyond the surface state peak.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−50050100150qr (A˚−1)Energy(meV)Current Map Surface StateCurrent Map Set-PointGrid Surface StateGrid Set PointHeight Map Surface StateFree Electron ModelFigure 4.9: Comparison of all features |S(qr,E)| in different acquisitionmodes. Vertical and horizontal lines indicate EF and qF = 2kFrespectively. Only the constant-current maps show a strongly dis-persing set-point peak. All three modes map out the surface stateintensity, agreeing well with a free electron model.The full-width half-maximum (FWHM) values resulting from the fits plottedin Figure 4.9 are plotted in Figure 4.10 as a function of the applied bias energy.The FWHM of the fits of the surface state peak, shown in (a), are smaller forall three measurement modes than for the set-point related features identified in(b) for constant-current dI/dV maps and the spectroscopic grid measurement.This trend is expected based on the line cuts plotted in Figures 4.5-4.8, as thesurface state peak is always sharper and higher in amplitude than the set-pointdependent feature, regardless of whether the mode of acquisition is constant-current dI/dV maps or spectroscopic grid.As a function of energy the surface state peaks plotted in Figure 4.10 (a)Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 68exhibit a similar FWHM, except near the band onset, where increases in theFWHM values correspond to broadening of the intra-band scattering peak. TheFWHM of the set-point features plotted in Figure 4.10 (b) show that the set-point feature identified in grid measurements is broader below the onset of theband and the further away the energy is from the stabilization bias of Vs = 100mV. For the constant-current dI/dV map fits the FWHM is largest near theband onset and then exhibits an increase as a function of increasing energy. Thiscorresponds to a decreasing amplitude in the set-point peaks as the tunnellingenergy is increased away from EF , indicating a potential relationship betweenthe number of energies in the sample being probed by the tunnelling currentand the strength of the set-point peak in FT-STS.−50 0 50 100 15000.020.040.060.08Energy (meV)FWHM(A˚−1 )(a) Surface State−50 0 50 100 150Energy (meV)(b) Set-PointCurrent Map Surface StateGrid Surface StateHeight Map Surface StateCurrent Map Set-PointGrid Set PointFigure 4.10: Full-width at half-maximum of the fits for all features in|S(qr,E)| in different acquisition modes. (a) FWHM valuesresulting from fits of the surface state peaks in FT-STS for all threemeasurement modes. (b) FWHM values resulting from fits of theset-point peaks in FT-STS for constant-current dI/dV maps andspectroscopic grids.The experimental data presented in Figures 4.4 through 4.10 demonstratesfeatures in |S(qr, E)| on Ag(111) that have a strong dependence on the acqui-sition mode used to acquire the differential tunnelling conductance. The nextsection will focus on theoretically modelling the tunnelling transmission prob-ability, sample density of states, and acquisition modes in order to construct amodel that can predict the observed features based on set-point parameters.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 694.3 Modelling the Set-point Effect4.3.1 Sample Density of States via T-matrixAn accurate model of the Ag(111) local density of states in the presence of im-purities is necessary in order to model set-point effects in the various acquisitionmodes. Fortunately, a very detailed model of the Ag(111) surface state can beconstructed with the help of scattering theory. Following Grothe et al. [132]the QPI scattering intensity from a single impurity, |S(q, E)| = |δρ(q, E)|, canbe obtained from the Fourier transform of the impurity-induced local density ofstates modulationsδρ(q, E) = − i2pi∑kIm[G(k, E)TG(k + q, E)]. (4.2)where G(k, E) is the bare Green’s function and T is the T-matrix, which istaken to beT = −V0 sin(δ)eiδ, (4.3)where V0 is the strength of the scattering potential produced by an impurityand δ is the scattering phase shift [132]. The scattering phase shift can betaken in two separate limits: δ → pi/2 is the unitary limit of strong scatteringwhile δ → 0, pi represents the Born or weak limit. For more details on theGreen’s function approach used here, including self-energy effects see Grothe etal. [128, 132].A complicating factor in comparing the simulation results with experimentaldata is the presence of multiple types of scattering impurities in the experimentaldata, both step edges and carbon monoxide absorbates, that have differentscattering strengths. Bu¨rgi [187] simulated step edges with a phase shift ofδ → 0, corresponding to the Born limit, while Grothe [132] took the unitarylimit of δ → pi/2 when looking at carbon monoxide absorbates and step edges,but used masking techniques to suppress the effects of the step edges. The QPIdispersion in scattering-space is strongly affected by the choice of δ.Figure 4.11 shows how a total modulation in the density of states can beconstructed by forming a linear combination of the Born and unitary scatter-ing centres and optimizing the amplitudes in order to minimize the residualsbetween the theoretical QPI intensity and the experimental results from a spec-troscopic grid. Rather than strictly unitary scattering at δ = pi/2, mixing Bornand unitary character does the best job of capturing the peak shape, in agree-ment with Grothe [128]. The following ratio best replicates the experimentalline shape at all energiesδρ(q, E)total = 0.6δρ(q, E)δ→pi/2 − 0.4δρ(q, E)δ→0. (4.4)Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 70As illustrated in Figure 4.11 (a) and (b) the expected scattering dispersion fora purely unitary scattering centre has pronounced differences compared with apurely Born scattering centre. For the purely unitary scattering centre, signifi-cant intensity is redistributed below the onset of the surface state band, similarto the intensity observed in the grid measurement of Figure 4.6 (a). This fea-ture is completely absent for a purely Born scattering centre, which also has lessintensity overall. Both scattering centres are capable of reproducing the surfacestate back-scattering associated with the Shockley surface state.Energy(meV) (a) Unitary δ = π/20 0.1 0.2 0.3−50050100 (b) Born δ ≈ 0q=2kF0 0.1 0.2 0.300.51|S(q,EF)|  Exper imentρπ /2ρ 00.6ρπ /2− 0.4ρ 00 0.05 0.1 0.15 0.2 0.25 0.3−0.500.5q (A˚−1)ResidualsFigure 4.11: Optimizing the scattering phase δ. (a) Theoretical |S(qr, E)|intensity for a strong scattering impurity (b) Theoretical |S(qr, E)|intensity for a weak scattering impurity. (c) Comparison of the-oretical |S(qr, EF )| with experimental data from a spectroscopicgrid. (d) Residuals of the theoretical |S(qr, EF )| with the experi-mental data, the mix of two phases does the best job minimizingthe residuals around the surface state peak.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 71After performing optimzation of the scattering phase against experimentaldata sets, the calculated modulation in the local density of states δρ(q, E) isused to seed calculations of the differential tunnelling conductance and expected|S(q, E)| intensity with different set-point parameters.4.3.2 Analytic Set-Point TheoryT-matrix calculations provide a good approximation of the Ag(111) local densityof states in the presence of a scatterer. However, a direct comparison between lo-cal density of states and the measured differential tunnelling conductance is notan appropriate comparison. Instead, the QPI pattern generated by an impuritymust be interpreted in the context of a tunnelling measurement. This sectioncalculates the tunnelling current, z-dependent tunnelling transmission probabil-ity, and other derived quantities with specific focus on isolating the set-pointconditions Vs and Is to allow for direct comparison with STM measurements.As shown in Chapter 2, at low-temperature and positive applied bias, thetunnelling current It = I(x, y, z, Vb) isIt(x, y, z, Vb) =∫ eVb0ρ(x, y, E)ρt(E − eV )T (z, E, Vb)dE. (4.5)When the set-point conditions are fulfilled by the feedback then It = Is(x, y, zs, Vs)as followsIs(x, y, zs, Vs) =∫ eVs0ρ(x, y, E)ρt(E − eVs)T (zs, Vs, E)dE, (4.6)where Vs is the stabilization bias and zs is the tunnelling barrier gap that satis-fies the set-point conditions. Following previous works [79, 80], the transmissiontunnelling probability can be approximated using a trapezoidal tunnelling bar-rier and the WKB approximation to giveT (zs, Vs, E) = exp(− zs 2√2mh¯√φ+eVs2− E), (4.7)where φ is the effective amplitude of the tunnelling barrier, E is the energy ofthe tunnelling electron, and m is the free electron mass. The effective tunnellingbarrier can be calculated based on the average of the work function of the Wtip (φ = 4.55 eV) and the Ag sample (φ = 4.74 eV) [81] to be φ = 4.65 eV. Inthe low-bias approximation, Koslowski [197] let T (zs, Vs, E) ≈ T (zs). Insertingthis into the right side of Equation 4.6 for Is givesChapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 72Is(x, y, zs, Vs) =∫ eVs0ρ(x, y, E)ρt(E − eVs)T (zs)dE (4.8)orIs(x, y, zs, Vs) = e(−zs 2√2mφh¯) ∫ eVs0ρ(x, y, E)ρt(E − eVs)dE. (4.9)Equation 4.9 gives the condition necessary for the STM feedback to satisfythe set point current. Notably, it shows that this condition changes based on thelateral position of the tip over the sample because of potential lateral changes inmagnitude of the local integrated density of states. If the sample local densityof states is not featureless then the STM feedback will adjust the tip heightas it scans, leading to different set point heights zs at each pixel. RearrangingEquation 4.9 to solve for zs giveszs(x, y, Is, Vs) = − h¯2√2mφln(Is∫ eVs0ρ(x, y, E)ρt(E − eVs)dE). (4.10)Equation 4.10 holds the key to understanding how the set-point parametersinfluence the FT-STS results differently for different acquisition modes. In gridacquisition zs = z(x, y, Vs, Is) and is set by the feedback at each pixel basedon the values of Vs and Is. This makes the tip sensitive to lateral variationsin the local density of states, but since Vs remains the same at every pixel thisat most introduces a single spatial frequency corresponding to a non-dispersingfeature in scattering-space. This feature appears at approximately the averageof all scattering q values between 0 and eVs, as it is related to the integratedlocal density of states. This is not the case for constant-current dI/dV maps,where the stabilization bias is tied to the map energy E for each map. Thismeans that zs = z(x, y, Vs = E/e, Is) where Vs varies, changing the spatialfeatures convolved into the differential tunnelling conductance measurement.Since zs contains periodic spatial modulations at approximately the average ofall q values between 0 and Vs = E/e, this leads to the presence of dispersingfeatures in the FT-STS intensity. In contrast, constant-height dI/dV maps haveno sensitivity to lateral variations of the local density of states. In a constant-height dI/dV map the tip height is set at one position, (xs, ys), at the startof the map and then the feedback is disengaged, excluding the possibility ofspatially dependent, feedback induced artifacts ie. zs = z(xs, ys, Vs = E/e, Is).Obtaining the contribution to the FT-STS intensity from set-point effectsrequires calculating the differential tunnelling conductance of the theoreticallycalculated tunnelling current while accounting for the effect of the set-pointparameters. Taking the full derivative of the tunnelling current and assuming aconstant tip density of states gives two terms with a zs dependence [80]Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 73dIt(x, y, zs, Vb)dV∝ eρ(eVb)ρt(0)T (zs)−√2me2h¯√φzsIt(zs, Vb). (4.11)This expression puts particular emphasis on the role of the set point gap zs.Previous works have studied the effect of zs and T (zs) on the measured differ-ential tunnelling conductance [69, 79, 81, 166, 197–199]. In all of these worksit is made explicit that oscillations in dI/dV are not equivalent to oscillationsin the sample local density of states ρ. The reason the two oscillations are notequivalent is that even though the tip is ostensibly ‘fixed’ at a tip–sample gap ofzs this height is a function of Is and Vs which in turn are dependent on spatialvariations in x and y of the local density of states and the tunnelling barrier.Analyzing term-by-term it is clear that in the first term of Equation 4.11 ρs(eV )is weighted by T (zs), which varies exponentially with changes in zs. The secondterm of Equation 4.11 contains a direct proportionality to zs and It both ofwhich can demonstrate oscillatory behaviour as a function of lateral position ofthe tip.Comparison of the magnitude of the two terms of Equation 4.11 allows foridentification of which term is the most relevant perturbation to the QPI signalof ρs(eV ). Taking the ratio of the first term to the second term indicates that thefirst term of Equation 4.11 is larger than the second term by approximately twoorders of magnitude under typical experimental conditions. Therefore, the mostsuitable place to search for a modification to the QPI due to the stabilization biaseffect is within the first term, focusing on any effects produced by the tunnellingtransmission probability T (zs). This is in agreement with the approach takenby Li who concluded that for small applied bias integration over a small energyrange led to oscillations in zs [81]. Taken together with the QPI oscillations in ρthis leads to a ‘beating’ in the dI/dV signal. This phenomenon only occurs in thelow bias regime |Vb| < 1 V. Burgi showed that this effect can be ignored when thebias is sufficiently high, as integration over a large number of frequencies resultsin a zs that can be treated as constant as a function of spatial position [166].This is not the case for the experimental measurements of Ag(111) presentedhere because the stabilization bias is too close to the Fermi energy.Using these analytic results, the effect on the FT-STS pattern of the differentmeasurement modes and their set-point artifacts can be simulated, expandingon previous work describing the effect of real-space oscillations of T (zs) in one-dimension [69, 80, 81, 197–199], in semi-conductors [82], and on molecules intwo-dimensions [79].4.3.3 Set-Point SimulationsThe analytical framework built in the previous section allows each of the differ-ent STM acquisition modes to be simulated. Equation 4.2 was used to generate amodulation in the sample density of states due to a single impurity in scattering-space ρ(q, E) using T-matrix code written by S. Johnston. This modulation toChapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 74the local density of states was inverse Fourier-transformed into a modulationin the real-space density of states ρ(r, E) and used to calculate zs(x, y, Is, Vs),T (zs), I(x, y, zs, Vs), and dIt(x, y, zs, Vb)/dV under different set-point condi-tions from the analytic expressions derived in the previous section. T (zs) anddI(x, y, zs, Vs)/dV were then Fourier transformed back into scattering-space andcompared with the original local density of states modulation, δρ(q,E).Grid with set-point in Region IFigure 4.12 simulates the case of a spectroscopic grids measurement with sta-bilization bias below the band onset in Region I. Figure 4.12 (a) shows themodulated local density of states |δρ(q, E)| with the scattering phase detailedin Section 4.3.1. The corresponding tunnelling transmission probability shownin Figure 4.12 (b) does not show any particularly sharp features but has a higherintensity before qr = 2kF . The simulated FT-STS intensity in Figure 4.12 (c)is qualitatively very similar to the local density of states, with the largest dif-ference being a slight redistribution of intensity below the band onset. Thesimulated result agrees well with the experimental data in Figure 4.12 (d), bothare free of spurious set-point features that could be confused as an additionalscattering vector.Grid with set-point in Region IIIFigure 4.13 simulates a spectroscopic grid with stabilization bias above theFermi energy, in Region III. The best experimental data for this set point had apredominantly unitary character, as a result the modulation in local density ofstates in Figure 4.13 (a) was calculated with a scattering phase equal to that ofa unitary scatterer δ = pi/2. The tunnelling transmission probability in Figure4.13 (b) shows the strongest intensity at just above qr = 2kF and is constantas a function of energy. The simulated FT-STS intensity in Figure 4.13 (c) hassignificant intensity running vertically below the surface state band scatteringjust above qr = 2kF . This feature is not present in the modulated local densityof states. The simulated result agrees well with the experimental data in Figure4.13 (d): both exhibit high intensity below the band onset attributable to theunitary nature of the scattering centre(s). They also share a secondary feature,the vertical intensity at qr = 2kF below the Fermi energy, that is not presentbelow the band onset. The simulations suggest that this feature in the experi-mental data is caused by modulations in the transmission tunnelling probabilityT (zs) and not the sample local density of states.Constant-current dI/dV mapsFigure 4.14 simulates the constant-current dI/dV map acquisition mode. Thebehaviour of the transmission tunnelling probability T (zs) is markedly differentthan for the case of the two grid measurements. It is strongly peaked alongEF and has a complicated dispersing structure away from EF . Its lowest in-tensity feature runs along the scattering vector expected for the surface stateChapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 75(a) T he ory |δ ρ (q, E ) |Grid R e gion I : Vs = −100 mV−50050(b ) T (z s(x , y , I s, V s))Energy(meV)−50050(c ) T he ory |S (q r, E ) |−50050(d ) Ex p e r im ent |S (q r, E ) |q r = 2k Fq r (A˚− 1)0 0.15 0.3−50050Figure 4.12: Simulation of a grid with stabilization bias Vs = −100 mV.(a) Simulated |δρ| in scattering-space. (b) The calculated T (zs) ofthis |δρ| with Vs = −100 meV stabilization bias. (c) The prod-uct of |δρ| with T (zs) gives the first term in Equation 4.11 (d)Experimental grid data with stabilization bias Vs = −100 mV.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 76(a) T he ory |δ ρ (q, E ) |Grid R e gion I I I : Vs = 100 mV−50050(b ) T (z s(x , y , I s, V s))Energy(meV)−50050(c ) T he ory |S (q r, E ) |−50050(d ) Ex p e r im ent |S (q r, E ) |q r = 2k Fq r (A˚− 1)0 0.15 0.3−50050Figure 4.13: Simulation of a grid with stabilization bias Vs = 100 mV.(a) Simulated |δρ| in scattering-space. (b) The calculated T (zs) ofthis |δρ| with Vs = 100 meV stabilization bias, (c) The productof |δρ| with T (zs) with Vs = −100 meV stabilization bias. (d)Experimental grid data shown with stabilization bias Vs = 100mV.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 77(a) T he ory |δ ρ (q, E ) |Curre nt M ap s Vs = Vb−50050(b ) T (z s(x , y , I s, V s))Energy(meV)−50050(c ) T he ory |S (q r, E ) |−50050(d ) Ex p e r im ent |S (q r, E ) |q r = 2k Fq r (A˚− 1)0 0.15 0.3−50050Figure 4.14: Simulation of constant-current dI/dV maps. (a) Simulated|δρ| in scattering-space. (b) The calculated T (zs) of this |δρ| for astabilization bias that matches the bias being probed Vs = Vb. (c)The product of |δρ| with T (zs). (d) Experimental FT-STS fromconstant-current dI/dV maps.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 78intraband scattering. The product of the modulated density of states |δρ| withthis transmission function produces a scattering intensity that has a sharp peakin intensity at EF , a strong back scattering intensity, and a set of secondarydispersing features that are different than the scattering intensity associatedwith the band. Though the energy resolution of the experimental data makesa direct comparison of features difficult, there is good qualitative agreementbetween the features observed in theory in Figure 4.14 (c) and the experimentalconstant-current dI/dV maps in Figure 4.14 (d).Constant-height dI/dV mapsConstant-height dI/dV maps are the one acquisition mode that should remainunaffected by modulations in the tunnelling barrier probability caused by theSTM feedback responding to the presence of QPI in the local density of states.Figure 4.15 shows the simulated |S(qr, E)| for three different acquisition modes:constant-height dI/dV maps, constant-current dI/dV maps, and a spectroscopicgrid with stabilization bias above the band onset in Region III. This can be com-pared to the experimental data in Figure 4.7. As in the experimental constant-height dI/dV maps the only feature observed when the tip height is not adjustedby the feedback is that due to the intra-band scattering of quasiparticles acrossthe surface state band. In both the constant-current dI/dV maps and spec-troscopic grid simulations a secondary peak is clearly visible that crosses overwith the surface state peak at EF , as indicated by the black arrows. Quali-tative agreement with the experimental data is again very good. Quantitativeagreement with the experimental line cuts is not expected, as the T-matrix the-ory used to generate the theoretical local density of states takes only a singleimpurity scattering centre, while the experimental data features many differentscattering centres, with potentially varying scattering phases3.Figures 4.12-4.15 show that the secondary feature observed in certain FT-STS measurement modes can be reproduced by modelling the effect of the set-point parameters, particularly the stabilization bias Vs. The most striking fea-ture due to this effect appears in FT-STS of constant-current dI/dV map data,where a secondary dispersing feature appears that crosses the surface state dis-persion at EF . The best measurement modes to avoid set-point artifacts, causedby modulations in T (zs), are spectroscopic grids with stabilization biases belowthe surface state onset and constant-height dI/dV maps. Given the experimen-tal challenges associated with the acquisition of constant-height dI/dV mapsit is likely that spectroscopic grids would be the preferred acquisition modein most scenarios. For any subtle features observed in FT-STS data in whichonly constant-current dI/dV map acquisition is possible, it would be wise to at-tempt constant-height acquisition at at least a few energies, to rule out set-pointrelated artifacts.3For a simulation of multiple scattering centres in real-space see Appendix B.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 790 0.2 0.40123456(a) Height Map-90 meV-50 meV-10 meV30 meV70 meV110 meVq r (A˚−1)|S(qr,E)|0 0.2 0.4012345678(b) Current Map-90 meV-50 meV-10 meV30 meV70 meV110 meVq r (A˚−1)0 0.2 0.40123456(c) Grid eVS > EF-90 meV-50 meV-10 meV30 meV70 meV110 meVq r (A˚−1)Figure 4.15: |S(qr,E)| comparison between simulated constant-heightmaps, spectroscopic grids, and constant-current maps. (a)Simulated constant-height dI/dV maps at Is = 100 pA show onlythe surface state feature. (b) Simulated constant-current dI/dVmaps at Is = 100 pA show a dispersing secondary feature. (c)Simulated spectroscopic grid data with Vs = 100 mV, Is = 100 pAshows a non-dispersing secondary feature.Chapter 4. Acquisition-Dependent Artifacts in FT-STS of the Ag(111) Surface State 804.4 ConclusionsThe comparison of the set-point simulations with the experimental data showsthat the dominant factor yielding experimental artifacts in the FT-STS arisesfrom the contribution of the tunnelling transmission probability T (zs) to thedifferential tunnelling conductance. This is in agreement with the early workperformed in real-space by Ho¨rmandinger [198, 200], Ukraintsev [69], and Li[81], who realized that in attempting to extract the local density of states fromthe differential tunnelling conductance, the effect of the tunnelling transmissionprobability cannot be ignored. In the combined dI/dV simulation of the FT-STS dispersion the main features introduced by modulations in T (zs) are asfollows: for grids with a Vs above EF there is a vertical, non-dispersing lineabove 2kF , for grids with a Vs below the onset of the band there is a weak verybroad non-dispersing feature in T (zs) with little influence on the S(qr, E), andfor the series of constant current dI/dV maps there is a dispersing feature whichcrosses 2kF at EF . The constant-current maps also show an overall increase inintensity near EF arising from strong variations in T (zs).In summary, artifact features in FT-STS derived dispersions on Ag(111) canoccur that depend on the momentum-space conditions used to stabilize the tipheight; as demonstrated through a combination of measurements in differentacquisition modes, and simulations of the expected FT-STS patterns for thesemodes. Simulations show that this arises from spatial modulations in the trans-mission function due to variations in zs at each (x, y) pixel that are dependenton the set-point conditions. This effect is most pronounced, and most con-cerning, for measurements acquired by taking dI/dV maps with a simultaneousconstant-current feedback at each energy, which produces a relatively strongdispersing feature.As mentioned at the outset of this chapter, dispersing features with similarq-dependence have been observed on (111) noble metal surfaces using constant-current maps and attributed to a number of different sources [113, 180, 181].Most recently, Sessi et al. [113] demonstrated that the secondary dispersionis not compatible with the position of the bulk bands in Au(111), Cu(111), orAg(111) and instead attributed the secondary features to an acoustic surfaceplasmon dispersion [113]. Work presented in this chapter casts doubt on thisconclusion, a feature following this same dispersion arises from an artifact of theconstant-current measurement mode, which was the measurement mode usedin the main results of Sessi’s work. Since the set-point effect due to modulationof the tunnel barrier is not present in constant-height dI/dV maps a secondarydispersing feature is not predicted to appear in dispersions constructed fromconstant-height data. However, a secondary feature was observed by Sessi onCu(111) at energies E = ±250 meV and reported in the supplementary materialof their work [113]. To address this concern and to show the broader applicabilityof the results presented here to other (111) noble metals the next section willfocus on FT-STS acquired on Cu(111).Chapter 5. Characterization of the Cu(111) Surface with Multiple FT-STS Acquisition Modes 81Chapter 5Characterization of theCu(111) Surface withMultiple FT-STSAcquisition ModesAn experiment is reproducibleuntil another laboratory tries torepeat it.Dr. Alexander Kohn [201]Like Ag(111), Cu(111) possesses a Shockley surface state with a free electron-like dispersion. The Cu(111) surface state has been studied using the FT-STStechnique and secondary features have been observed in constant-height dI/dVmaps [113], features not observed in Ag(111). This chapter presents FT-STSdata of the Cu(111) surface taken using spectroscopic grids, constant-currentmaps dI/dV maps, and constant-height dI/dV maps. A comparison between theacquisition modes allows for identification of the set-point effects and isolationof FT-STS features that cannot be attributed to the modulation of the tunnellingtransmission probability.5.1 Characterizing the Set-Point Effect inCu(111)5.1.1 The Cu(111) Surface StateA (111)-terminated copper surface supports the existence of a Shockley surfacestate, which has been thoroughly characterized by ARPES and STM [8, 12,113, 165–167, 190, 193, 198, 202, 203]. Close to the Fermi energy, this surfacestate forms a parabolic band in k-space well modelled by Equation 4.1, with achemical potential µ = 420 meV [190], but roughly the same effective electronmass as the band in Ag(111) [190]1. Cu(111) shares the same crystal structure1Low temperature values measured for the effective mass range from m∗ = 0.38−0.46 whilethe band onset has been measured anywhere from µ = 420− 440 meV [8, 167, 190, 198, 203].Chapter 5. Characterization of the Cu(111) Surface with Multiple FT-STS Acquisition Modes 82as Ag(111) with a lattice constant of a0 = 0.361 nm [186], giving an atomicspacing of on the (111) surface of a0√2/2 = 0.212 nm.The sample preparation and measurement details for data acquisition onCu(111) followed the same protocols used for Ag(111). All measurements wereperformed using the CreaTec at a temperature of T = 4.5 K. The Cu(111)crystal was prepared via three cycles of sputtering and annealing. The annealingtemperature was raised to 600◦C due to the higher melting temperature ofcopper. This preparation procedure produced large, flat terraces with a lowdensity of surface impurities. Impurities that were present were likely eitherCO molecules or sulphur atoms. Contact and sharpening of the STM tip on theCu(111) surface likely resulted in a copper terminated tip.Figure 5.1: Spectroscopic grid measurement of Cu(111) (a) The differ-ential tunnelling conductance exhibits two step edges and a num-ber of tip changes, corresponding to the higher degree of noise inthe top half of the image. The inset at top right shows oscilla-tions around individual point defects and a step edge. (b) FT-STSshows the expected intra-band scattering intensity. (c) The scatter-ing dispersion of the entire surface state band. Vertical line denotesqF = 2kF = 0.42 A˚−1 and horizontal line shows E = −250 mVfor reference to (a) and (b). Imaging conditions: Vs = −520 meV,Is = 150 pA, 325× 325 pixels.Stability issues between the STM tip and sample prevented the acquisition ofspectroscopic grids without tip changes due to mechanical issues in the CreaTecSTM head. This resulted in a higher degree of background noise in the gridmeasurements for Cu(111) than those presented for Ag(111). This higher noiselevel did not affect the ability to identify the surface state band from QPI, asshown in Figure 5.1. This figure shows data from a spectroscopic grid over a250 × 250 nm2 area of the Cu(111) surface. The grid set-point voltage is Vs =−520 meV, below the onset of the surface state band. The differential tunnellingconductance at −250 mV in Figure 5.1 (a) shows two step edges and a multitudeChapter 5. Characterization of the Cu(111) Surface with Multiple FT-STS Acquisition Modes 83of scattering centres sitting on large terraces. The FT-STS pattern exhibits aring of intensity with a q value that matches that expected from Equation 4.1.The vertical line of intensity in the centre of the FT-STS image results fromthe tip changes present in the image of the differential tunnelling conductance.STM tip changes in the image of the dI/dV data can be identified as spikes inintensity associated with the fast scan direction, causing an effect that resemblesstatic noise. The Fourier transform of the noise associated with the tip changesis the vertical line observed in Figure 5.1 (b). The surface state dispersionfrom the onset, at approximately (0) = −420 meV, to the Fermi energy isshown in Figure 5.1 (c) and fits the expected parabolic dispersion. No set-pointartifact features are readily apparent, as expected for a grid measurement withstabilization bias below the band onset.5.1.2 Measured Set-Point Effects in Cu(111)As in Ag(111), spectroscopic grids were measured with different set-point pa-rameters. Figure 5.2 shows a comparison between two spectroscopic grids, onewith the stabilization bias above the Fermi energy eVs > EF and one withthe stabilization bias below the surface state onset (0) > eVs. The dispersingfeature that appears in both data sets is the expected intra-band scattering in-tensity. A secondary feature is present at a constant q just above qF = 2kFfor the grid with stabilization bias above EF . This is analogous to the fea-ture observed in Ag(111) in Figure 4.5 (c) and serves to confirm the set-pointmodel’s results in Cu(111) grids. As in Ag(111), the grid with eVs < (0) showsno set-point related artifacts. The higher intensity at low-qr in Figure 5.2 (b)compared to Figure 5.2 (a) is due to the presence of a number of step edges.Constant-current dI/dV maps of the Cu(111) surface were measured to fur-ther demonstrate set-point artifacts in the measurement of surface state bands.Based on the model constructed for Ag(111) the constant-current dI/dV mapswere predicted to show a dispersing set-point artifact below the EF that be-comes non-dispersing and approximately tracks the qF value above EF . Thisprediction was confirmed by the experimental data shown in Figure 5.3 (b).Two peaks are visible at every energy measured, one behaving as predictedby set-point theory and one following the expected surface state dispersion ofEquation 4.1.The final measurements of Cu(111) were performed using the constant-heightdI/dV map acquisition mode. Based on the measured results in Ag(111) theexpectation was that only the surface state peak would be visible. However,previous work by Sessi et al. [113] had measured a weak secondary feature inaddition to the surface state band dispersion at E = ±250 meV.At low tunnelling currents (Is ≤ 300 pA) the constant height measurementsdemonstrated only one peak, in agreement with the constant height dI/dV mapresults on Ag(111). However, Equation 4.11 contains a secondary term whichis current dependent, which is present independent of the acquisition mode.Although it had been shown that this term did not contribute in a measurableway to Ag(111) measurements, it has had an impact on differential tunnellingChapter 5. Characterization of the Cu(111) Surface with Multiple FT-STS Acquisition Modes 84(a)qr (A˚−1)Energy(meV)eVs > EF0 0.1 0.2 0.3 0.4−150−100−500qr (A˚−1)ϵ(0) > eVs  MinMax(b)0 0.1 0.2 0.3 0.4Figure 5.2: Set-point effects in grids on Cu(111). (a) A spectroscopicgrid measurement exhibits a secondary non-dispersing feature justabove the Fermi scattering vector qF = 2kF . Imaging conditions:Vs = 30 meV, Is = 150 pA, 124 × 124 nm, 176 × 176 pixels. (b)A spectroscopic grid with stabilization bias below the band onsetshows no secondary features. Imaging conditions: Vs = −520 meV,Is = 150 pA, 225× 225 nm, 325× 325 pixels.Chapter 5. Characterization of the Cu(111) Surface with Multiple FT-STS Acquisition Modes 85conductance measurements of two-dimensional electron gases in semi-conductors[82, 204]. As Sessi measured constant height dI/dV maps at higher current itwas necessary to rule out this effect.0 0.2 0.4 0.600.20.40.60.811.21.41.61.8(a) Constant Height-250 meV-200 meV-100 meV100 meV250 meV300 meV350 meVq r (A˚−1)|S(q,E)|0 0.2 0.4 0.600.20.40.60.811.21.41.61.8(b) Constant Current-250 meV-150 meV-100 meV100 meV250 meV300 meV350 meVq r (A˚−1)Figure 5.3: Constant height and constant current dI/dV maps ofCu(111). (a) |S(qr, E)| extracted from constant height dI/dVmaps over areas ranging from 80 − 140 nm in length at Is = 900pA. (b) |S(qr, E)| extracted from constant current dI/dV maps overareas ranging from 80− 140 nm in length. Is = 900 pA.At tunnelling currents centred around 1000 pA the constant-height dI/dVmaps exhibited a secondary dispersing feature, indicated by the orange arrowsin Figure 5.3 (a). This feature was weaker than the set-point peak seen inconstant current dI/dV maps and exhibited roughly the same dispersion belowEF . However, above EF this feature continued dispersing with the surfacestate band and did not remain at qF like the set-point feature. This featureis visible in the constant height map data in Figure 5.3 in addition to theexpected surface state peak. In order to acquire constant height dI/dV maps atsuch high currents the data acquisition was changed to re-engage the feedbackmomentarily at the end of every line of the tip fast scan axis. This ensuredChapter 5. Characterization of the Cu(111) Surface with Multiple FT-STS Acquisition Modes 86tip stability throughout the measurement without allowing feedback inducedmodulations of T (zs).5.2 Discussion and OutlookMeasurements of spectroscopic grids, constant current dI/dV maps, and lowtunnelling current constant-height dI/dV maps confirmed the validity of theset-point model of FT-STS features on the noble metal surface states. As inAg(111) grid measurements with stabilization bias above the band onset ex-hibited non-dispersing artifacts in the scattering intensity of the surface state.Data on Cu(111) from constant-current dI/dV maps demonstrated an artifactthat disperses below EF and remains roughly constant at qF above EF , just asin Ag(111).Constant height dI/dV maps provided the first surprising result not ob-served in the analysis of the Ag(111) data. At high currents a new secondarydispersing feature was observed not seen in the other measurement modes. Asthe measurement of QPI by constant-height dI/dV maps is set-point indepen-dent this feature may be attributable to physics of the Cu(111) and not thatof the tunnelling transmission probability. This feature may be the effect ofthe second term in Equation 4.11, though order of magnitude calculations inthe set-point theory model indicate that this term would need an order of mag-nitude enhancement compared to the first term in order to be experimentallyaccessible.Interestingly, this feature appears to follow the dispersion of an acousticsurface plasmon (ASP) as predicted theoretically by Sessi. Sessi’s experimentalmeasurements, acquired using constant current dI/dV map acquisition, showtwo features in Cu(111), the expected surface state peaks and a second peak,which they identify as a signature of the predicted ASP. However, comparingtheir measurements to constant-current dI/dV maps taken at the LAIR, asis done in Figure 5.4, the secondary feature in their data agrees better withthe set-point feature in constant current dI/dV maps than the predicted ASPfeature. In particular, above the Fermi energy both constant-current dI/dVsecondary features become non-dispersing, while the ASP prediction continuesto disperse with the surface state back-scattering intensity. This causes theirtheoretical prediction to deviate from their experimental data, and this maybe because the surface state peak is masking any sign of the acoustic surfaceplasmon peak above EF . By measuring in constant height mode, as shown inFigure 5.3 (a), the set-point feature is suppressed and it is possible to measurea secondary feature that continues dispersing above EF . This secondary featureshould also be visible in high-current grid measurement with stabilization biasbelow the band onset, but due to the transient nature of the STM tip stabilityon this surface such a grid could not be acquired. The constant-height secondaryfeature was also more difficult to observe than the previously observed set-pointfeatures, with not every FT-STS measurement demonstrating a peak, even athigh tunnelling currents.Chapter 5. Characterization of the Cu(111) Surface with Multiple FT-STS Acquisition Modes 87Figure 5.4 shows the extracted maxima of intensity for the surface state, set-point, and possible acoustic surface plasmon features observed experimentallyon Cu(111), for both this work and Sessi et al. Figure 5.4 (b) also shows thetheoretically predicted dispersion of the acoustic surface plasmon. This showsthat the feature seen in constant height maps in Figure 5.4 (a) is a bettercandidate for the acoustic surface plasmon mode than the one seen by Sessiin constant current maps. More work, both theoretical and experimental, isneeded to verify this. If it is truly a feature of the Cu(111) then it should alsobe a detectable feature in a high resolution grid with set-point below the bandonset and if it is a universal feature of noble metal surface states then is shouldappear in Ag(111) constant height dI/dV maps at higher tunnelling currents.0 0.2 0.4 0.6 0.8−400−300−200−1000100200300400qr (A˚−1)(a)Energy(meV)0 0.2 0.4 0.6 0.8(b)qr (A˚−1)Current MapSurface StateCurrent MapSet PointHeight MapNew FeatureHeight Map Surface StateFree ElectronModelSessi SurfaceStateSessiPlasmonTheory PlasmonFree ElectronModelFigure 5.4: Comparison of Cu(111) dispersing features with Sessi etal. [113] Vertical and horizontal lines indicate EF and qF = 2kFrespectively. (a) Peak maxima extracted from constant currentand constant height maps both show the surface state dispersionin good agreement with a free electron model from Equation 4.1.The constant-current dI/dV maps show the set-point feature whileconstant-height dI/dV maps exhibit a previously unobserved fea-ture. (b) Constant-current dI/dV maps by Sessi map out the surfacestate dispersion. The theoretical dispersion of an acoustic surfaceplasmon agrees well with secondary features below EF but deviatesabove.Chapter 5. Characterization of the Cu(111) Surface with Multiple FT-STS Acquisition Modes 88This chapter has provided a detailed look at the features observed in FT-STS data on the surfaces of Ag(111) and Cu(111). By developing a set-pointtheory of the data acquisition that linked the experimental set-point parameterswith the modulation of the tunnelling barrier gap in each acquisition mode itwas demonstrated that artifacts will be introduced that are roughly constantfor grid measurements with stabilization bias’ above the band onset and dis-persing for constant current maps. This has implications for measurements ofFT-STS in any system that exhibits QPI. As a general rule, if apparent heightmeasurements exhibit QPI signatures then it should be expected that modula-tions in T (zs) will have a signature in the dI/dV . This can either be taken intoaccount by acquiring data using multiple different acquisition modes or usingan appropriate normalization scheme [82, 204, 205].In grid measurements of the noble metals a choice of stabilization bias belowthe onset of the band showed a very weak influence with no distinct features,providing a way to avoid these effects without resorting to demanding constantheight measurements, or a more elaborate program of returning the tip to thesame location to reset the height for each measurement pixel, as has been donefor AFM measurements [206]. These results urge caution in the field; featuresin QPI require careful consideration, and artifacts can arise depending on themeasurement mode that may obscure or masquerade as physical processes inthe sample.89Part IIIMagnetic Sensing andControl of Single Atoms onMgOChapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 90Chapter 6Electron Spin ResonanceScanning TunnellingMicrocopyAll of physics is either impossibleor trivial. It is impossible untilyou understand it, and then itbecomes trivial.Ernest RutherfordSpin resonance provides the high-energy resolution needed to examine bio-logical and material structures by sensing weak magnetic interactions [207]. Bypairing electron spin resonance techniques with scanning tunnelling microscopy,researchers at IBM have developed an experimental tool with the spatial resolu-tion of the STM and energy resolution of ESR. The ESR-STM technique relieson the use of microwave bias pulses to resonantly excite the spin state of aniron atom sitting on the surface of magnesium oxide. The spin state of the ironatom is probed by a spin-polarized STM tip. In this chapter the basis of thistechnique will be explored in preparation for its use in Chapter 7.6.1 Electron Spin Resonance in Bulk MaterialsIn spin resonance experiments, electromagnetic radiation is used to excite tran-sitions between electronic states (ESR) or nuclear states (NMR). Traditionalspin resonance experiments use electromagnetic radiation to interact resonantlywith an ensemble of electron or nuclear spins via absorption and emission ofphotons. When performed on bulk materials ESR, experiments can reveal theelectronic states of paramagnetic defects in solids [207, 208] and the presenceof spin centres in biomolecules [209]. An ensemble of 107 − 1010 spins [207] isrequired to achieve sufficient signal detection in conventional, bulk ESR exper-iments. The need to measure such a large number of spins leads to line-widthbroadening of the ESR signal, as at the atomic level each spin is in a slightlydifferent electromagnetic environment and so possesses a slightly different reso-nance frequency.Chapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 91There has been considerable effort expended to develop experimental toolsthat can perform spin resonance on much smaller ensembles of spins, with theultimate goal of single atom spin resonance. Single atom spin resonance isdesirable for the local control it offers, and the potential for implementations ofspin-based quantum computing architectures [210]. In certain systems magneticresonance can be detected in single spin centres via force microscopy [211].Couplings between itinerant and localized spins have been used to electricallydetect magnetic resonance in small ensembles [212], spins in quantum dots [213],in single P atoms in Si [214, 215], and in individual magnetic molecules [216].These works represent significant progress; however, none of these techniqueshave the atomic-scale control possible with the STM.6.2 Electron Spin Resonance and ScanningTunnelling Microscopy6.2.1 Energy Resolution, STS, and ESR-STMScanning tunnelling spectroscopy measurements can reveal a great deal of infor-mation about the spectroscopic character of a surface, particularly when com-bined with analysis techniques like FT-STS as demonstrated in Part II. However,the thermally-limited energy resolution of the STS signal, derived in Chapter 2,can obscure features below a certain energy. In order to examine surfaces wherebetter energy resolution can give important insight into the underlying physics,multiple ultra-low temperature STMs have been commissioned that achieve tip–sample temperatures below 100 mK [117, 217–220]. However, recent work byAst in Reference [221] has revealed that there is a quantum mechanical limit toSTS resolution that cannot be overcome by lowering the temperature of tip andsample.Spin resonant techniques do not measure the same physical quantity, thesample density of states, that STS measures. However, because resonant pro-cess are inherently out of thermal equilibrium, they potentially allow accessto energies below the quantum limit of conventional STS. In the case of weakmagnetic interactions, such as the dipole-dipole interactions between magneticatoms nanometres apart, electron spin resonance offers a way to measure therelevant physics beyond the scope of conventional STM techniques.Figure 6.1 shows a logarithmic energy scale in units of electronvolts. Theenergy scale of physical processes of interest are written above the line, highlight-ing quantities like metallic work functions, chemical bonding, superconductinggaps, and magnetic interactions. Below the energy scale in Figure 6.1 is theresolution of STS at various temperatures, which clearly demonstrates the needto cryogenically cool the STM in order to measure STS at the right energy scale.For the final interaction depicted the energy scale is beyond the quantum limitdetermined by Ast for STS [221]. Therefore, to probe this regime requires anew acquisition technique, motivating the development of ESR-STM as a wayto measure a new energy regime, in addition to the possibility of resonant spinChapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 92control of a single atom.Figure 6.1: Logarithmic Energy Scale Above the energy scale a select num-ber of energies important in condensed matter are listed while belowthe energy scale the limit of STS resolution is shown at various tip–sample temperatures. References: Hyperfine energy of hydrogen (H)water [222], superconducting (SC) gap energies [223], work functionof silver [81], and quantum limit of STS energy resolution [221].6.2.2 Previous Work to Pair ESR with STMSTM offers nanoscale spatial control and imaging while ESR provides high-energy resolution that is otherwise not possible with STM techniques. A num-ber of research groups have attempted to realize electron spin resonance scan-ning tunnelling microscopy (ESR-STM), combining the strengths of these twotechniques. The first attempts to pair ESR and STM measured the tunnellingcurrent noise at the spin precession frequency in a room temperature experi-ment [224–226]. The frequency-dependent signal in this case was sporadic andwas not widely adopted, though theoretical mechanisms for the observed effecthave been proposed [227, 228]. A more recent experiment by Mullegger et al.applied a microwave electric field between 0.5−3.0 GHz to a magnetic moleculein a milliTesla magnetic field and attributed the frequency-dependent dI/dVfeature to a spin resonance signal [229]. This work has yet to be reproducedand may potentially suffer from issues related to compensating the microwavepower at the tip–sample junction, as discussed by Paul [230].6.2.3 ESR-STM at IBMResearchers at the Nanoscience Laboratory have developed their own ESR-STM technique [86]. This version of ESR-STM works at low temperature (≈ 1K), in magnetic fields, on Fe atoms and has already been used in a numberof experiments [86, 231, 232]. Of the ESR-STM techniques currently in theliterature, this technique has been the most thoroughly documented [86, 230,233] and offers the nanoscale manipulation and imaging properties of the STMcombined with the energy resolution of ESR.Chapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 93Spin-polarized STMIn order to understand the ESR-STM measurement scheme, it is necessary tointroduce the basic theory of spin-polarized scanning tunnelling microscopy.When both the sample and tip possess a magnetic moment, the Tersoff-Hamantunnelling current acquires a dependence on the spin orientation of the two[64, 234, 235]. The density of states for both the tip and the sample must thenbe expressed as a spin resolved quantity, ρ↑,↓t,s . By defining the quantitiesnt = ρ↑t + ρ↓t , ns = ρ↑s + ρ↓s (6.1)mt = ρ↑t − ρ↓t , ms = ρ↑s − ρ↓s (6.2)it is possible to express the spin-polarized differential tunnelling conductance inthe low bias limit asdI/dV ∝ (ntns +mtms cos θ)T (z, E, V ) (6.3)where θ is the angle between the tip and sample magnetization [64, 235]. Equa-tion 6.3 reduces to Equation 1.7 in the event that the tip or sample is non-magnetic. When tip and sample are both magnetized the maximum tunnellingcurrent will be measured when the tip and sample magnetization are aligned(θ = 0) and the minimum tunnelling current when they are anti-aligned (θ = pi).This means that the STM tip can be used to measure tunnelling magnetoresis-tance via the tunnelling current, making it an atomic-scale magnetic probe.The degree of spin-polarization of the STM tip can be characterized by thebehaviour of the differential tunnelling conductance when measuring inelasticmagnetic transitions in STS. Figure 6.2 illustrates the difference between theSTS spectra of a single Fe atom on the surface of a MgO bilayer taken witha normal (spin-averaging) STM tip versus a spin-polarized tip in a magneticfield. An inelastic transition is visible as steps at a bias of VB = ±14 mVfor the spectrum obtained with the normal tip. The spectrum taken with thespin-polarized tip also exhibits features at a bias of VB = ±14 mV but thesefeatures are asymmetric as a function of the polarity of the bias and demonstratepeaks in addition to steps. As shown by Loth [144, 236], the asymmetry of theobserved features in the spin-polarized STS spectrum is due to differences in themajority versus minority spin population in the tip. Furthermore, the shape ofthe peaks can be related to the spin-momentum transfer imparted to the Featom electrons by the spin-polarized tunnelling electrons.Sensitivity to surface magnetic moments is of great experimental utility inthe study of magnetic information storage [143, 237] and magnetic quasiparti-cles [238]. For the purposes of this work spin-polarized tips were characterizedusing the SP-STS spectra of either Fe or Co atoms. The spin-polarized tipswere then used to detect the magnetic orientation of the electrons in ESR-STMexperiments.Chapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 94−30 −20 −10 0 10 20 300.511.522.53Bias Voltage (mV)dI/dV(nS)  Spin−Polarized TipSpin−Averaging (Normal) TipFigure 6.2: STS spectra of a single Fe atom with and without a spin-polarized tip. STS spectra of the same Fe atom on MgO with aspin-polarized tip and a normal tip which averages over all availablespin channels. Spectra were acquired using a lock-in amplifier toextract the differential tunnelling conductance. Set-point parame-ters: Vs = 10 mV, Is = 50 pA for the normal tip and Vs = 10 mV,Is = 100 pA for the spin-polarized tip.Chapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 95Experimental Set-up for ESR-STMThe experimental set-up for ESR-STM is illustrated in Figure 6.3. A spin-polarized tip is positioned over a magnetic atom on a surface in a magneticfield, as shown in Figure 6.3. The magnetic atom sits atop a thin-film insulatorin order to preserve the atomic magnetic moment and spin relaxation lifetimefrom interactions with conduction electrons of the bulk sample. A microwavefrequency electrical excitation is sent into the tunnelling junction via the tip.The frequency of this excitation is swept until it is equal to the energy differ-ence between the two lowest lying electronic spin states. When this occurs, aresonance condition is achieved and the spin state becomes a coherent mixtureof its two lowest lying states, changing the magnetization of the surface atom.This change in magnetization is detected via a change in the tunnelling magne-toresistance of the spin-polarized tunnel junction. By chopping the microwaveexcitation on and off, a scheme known as continuous wave ESR-STM as shownin Figure 6.3 (b), a lock-in amplifier can be used to take a difference measure-ment between the tunnelling current with and without microwave excitation.Measuring this lock-in signal as a function of microwave frequency produces themeasured change in tunnelling current due to ESR observed in 6.3 (c).Figure 6.3: ESR-STM measurement scheme. (a) A magnetic atom sit-ting on an insulating thin film on a bulk conducting crystal has issent into resonance by application of an microwave bias excitationalong the DC bias line. A spin-polarized tip is used to read-outthe change in tunnelling current. (b) The microwave pulse train forthis method, dubbed continuous wave ESR. (c) The microwave fre-quency is swept until a resonance peak in the tunnel current signalis detected via the lock-in amplifier.Chapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 96In the IBM experimental set-up for ESR-STM, Fe and Co atoms are de-posited on a thin-film of MgO grown on a Ag(100) crystal substrate in a cantedmagnetic field (tilted 2◦ from parallel to the sample plane). Both the Fe and Copreferentially adsorb on the O site, and each shares four Mg atoms as nearest-neighbours, giving the binding site C4v symmetry [239]. The ligand field of theneighbouring oxygens causes the Fe and Co atoms to exhibit a strong energeticpreference for developing magnetization perpendicular to the surface, ie. in thedirection of the easy-axis magnetic anisotropy [239]. An ESR-STM signal hasonly been observed when measuring Fe atoms on the MgO surface, and not Coatoms. This may be due to the threefold symmetry of the Co orbitals not beingcompatible with the fourfold symmetry of the MgO ligand field, and so the largetime varying electric field caused by the microwave bias does not address thespin component of the Co electronic wave function in a way that could excite aresonance [86].Spin-polarized tips are constructed by using vertical manipulation to pickup an Fe atom so that the tip is terminated by a single magnetic atom. Theprocedure for picking up single atoms consists of moving the tip close to thesample surface, corresponding to a tip–sample resistance of ≈ 1 MΩ, increasingthe bias voltage to 0.55 V, and then withdrawing the tip [86]. The same tech-nique can be used to position single atoms and build structures on the surfaceby placing them from the tip onto specific atomic binding sites, as will be shownin Chapter 7. Characterization of the degree of spin polarization of the tip canbe checked by using STS to measure inelastic magnetic transitions in individualFe or Co atoms, as shown for an Fe atom in Figure 6.2.The single atom on the tip apex has a sub-picosecond spin relaxation timedue to its strong interaction with the conduction electrons of the tip, so it actsonly as a spin filter and cannot be resonantly excited in the same way as atomson the MgO surface. The lifetime of the atoms on the surface is considerablylonger due to their decoupling from the conduction electrons of the Ag(100)crystal by the MgO thin film. This is crucial for the success of ESR-STM as along energy relaxation time, T1, for excited magnetic states is necessary in orderto observe spin resonance. Electrical pump-probe spectroscopy, in which a DCbias pulse called the pump is delivered to the tip–sample junction followed by alower voltage probe pulse at increasing time intervals [143], is used to establishthe magnitude of T1 before attempting ESR-STM frequency sweeps. The T1time is one of three important parameters used to describe a resonantly drivenmagnetic moment, and optimizing T1 is crucial for implementing quantum logicoperations, as is discussed in the development of pulsed ESR in Appendix Cand by Paul in Reference [233].Once a spin-polarized tip has been crafted and a suitable Fe atom candidateon the surface has been found then the STM feedback circuit is used to stabilizethe tip at low current and low DC voltage (Is = 1 pA, Vs = 5 mV). The feedbackis then disengaged and the microwave bias modulation scheme is applied throughthe tip, with Vµ = 5− 10 mV (peak-to-peak) producing a 106 − 107 V/m time-varying electric field at the tip and sample [86]. Before any microwave bias isapplied the microwave amplitude is calibrated to achieve constant power at allChapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 97frequencies in the sweep [230]. In order to keep the microwave power constantat the tip–sample junction at each frequency the source amplitude is varied byup to 40 dB, indicating that if such steps are not taken any measured signal willbe dominated by variations in the microwave power at the junction as a functionof frequency. The power-calibrated frequency is swept and the time-averagedtunnelling current is measured as the microwave bias is chopped at a 97 Hzaudio frequency. A lock-in amplifier is then used to demodulate this signal andcompare the tunnelling current when the microwave bias is applied and when itis not. Electron spin resonance of the Fe atom can be detected as a change1 inthe tunnelling current, as shown in Figure 6.5.Theory of ESR-STM on single Fe atomsThe low-energy quantum states of Fe on MgO have been characterized usingspin excitation spectroscopy [144, 236], x-ray absorption spectroscopy [239],and multiplet calculations [239–241]. These studies have shown that the spinand orbital angular momentum of the free Fe atom are largely preserved onthe MgO surface and that it is a reasonable approximation to treat the Featom as if it is in a d6 electronic configuration. Under this assumption thelowest Hund’s rule term dictates that Fe has an orbital moment L = 2 and spinS = 2 [86]. Detailed multiplet calculations of the multi-electron wave functionsinvolved in the Fe-MgO system show that it is possible to accurately capturethe essential physics needed to describe ESR-STM by using an effective spinHamiltonian [240, 241]. By choosing the z-axis to align with the direction ofthe easy-axis anisotropy dictated by the MgO ligand field (out of the sampleplane) the following Hamiltonian can be used to model the magnetic states andenergies of Fe atom on MgO [86]H0 = DL2z + F0(L4+ + L4−) + λ~L · ~S + µB(~L+ ~S) · ~B, (6.4)where D = −433 meV is the out-of-plane magnetic anisotropy, F0 = 2.19 meVdescribes the strength of the ligand field from the MgO lattice, and λ = −12.6meV is the spin-orbit coupling constant. The last term in the Hamiltonian rep-resents the Zeeman coupling of the electron spins to the external magnetic field.The operator Lz addresses the z component of the orbital angular momentum,while L+, and L− are the orbital angular momentum ladder operators.Equation 6.4 has a manifold of five lowest lying eigenstates, which can beexpressed as linear combinations of the z-axis orbital and spin quantum numbers|mL,mS〉. Transitions between the ground state and the excited states can beprobed experimentally by STS measurements, as shown in Figure 6.4 (a). Thestep in the differential tunnelling conductance at Vb = ±14 mV step correspondsto an inelastic excitation of a spin-flip transition [138] as the tunnelling electroncrosses the energy threshold needed to flip the Fe spin from the ground state1This change can be either an increase or a decrease depending on the majority spinpopulation in the spin-polarized tip.Chapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 98|0〉 to the excited state |2〉. This transition is illustrated with respect to the Featom low-energy spin states in Figure 6.4 (b).-30 -20 -10 0 10 20 301.61.71.8 dI/dV (arb.units)Voltage (mV)Energyms-2   -1    0   +1  +2V → = 14 meV |4i|3i|1i|0i |2i|2i|0i(a) (b)DC Bias Voltage (mV)dI/dV(nS) .Figure 6.4: STS spectra of Fe on MgO The jump in differential tunnellingconductance at Vb = 14 meV corresponds to the transition for theground state to the first excited state. Topographic height of Featom in inset is h = 0.17 nm. (b) The five lowest lying states of anFe atom on MgO [86].The two lowest lying states |0〉 and |1〉, are degenerate except for Zeemansplitting caused by the out-of-plane component of the external magnetic field,Bz. It is these states which are driven into resonance by application of the ESR-STM microwave bias. To leading order the |0〉 and |1〉 states can be expressedas|0〉 = 0.92 |+2,+2〉 − 0.40 |−2,+2〉 (6.5)|1〉 = 0.92 |−2,−2〉 − 0.40 |+2,−2〉 . (6.6)Both states are polarized in their spin but mixed in their orbital angular mo-mentum. They are separated by a large energy barrier, as illustrated in Figure6.4 (b), decreasing the possibility for quantum tunnelling of the magnetizationbetween the two states that would drastically shorten the lifetime of resonantspin states [240].Spin resonance between the |0〉 and the |1〉 state is driven using a microwavebias voltage. Figure 6.5 (a) and (b) show the population equalization of the|0〉 and |1〉 states when resonance is achieved. At T = 0.6 K the occupationprobability of the |0〉 state is 75% (Figure 6.5 (a)), as determined by a Boltzmanndistribution [242]. Sufficient mixing between the |mL,mS〉 quantum numbers ofthe |0〉 and |1〉 states is accomplished by use of a large in-plane magnetic fieldBx, while the out-of-plane component Bz sets the magnitude of the resonantfrequency (100 µeV splitting gives a 25 GHz resonance) and spin polarizes thetip. Note that although the applied experimental field used is large, on the orderChapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 99of several Tesla, it is still the weakest term in the spin Hamiltonian, justifyingits treatment as a perturbation that mixes |mL,mS〉 states.The microwave bias pulse introduces a time-dependent electric field thatmoves the Fe atom with respect to the MgO substrate. This modifies the lig-and field parameters and so acts as a time-dependent perturbation that can bemodelled byH1(t) = F1(t)(L4+ + L4−) (6.7)where the strength of the parameter F1(t) is related to the magnitude of thedriving microwave field. This term acts to drive coherent transitions between the|0〉 and the |1〉 states because it exhibits a non-vanishing coherent transition rate〈0|H1(t) |1〉, particularly if a strong in-plane Bx field mixes the spin componentsof both states.The microwave frequency is swept until it excites a resonant transition, whichin a continuous wave mode of operation results in a equalization of the occu-pation probability of the |0〉 and |1〉 states (Figure 6.5 (b)). The change inoccupation probability between a thermal and a resonant distribution changesthe tunnelling magnetoresistance between the STM tip and Fe atom. Figure 6.5(c) shows how this change can be measured in the spin-polarized tunnelling cur-rent as a function of the applied field. Whether the ESR-STM exhibits a peakor a dip at the resonance frequency depends upon the majority spin carrier ofthe spin-polarized tip, as illustrated in Figure 6.6. Individual spin-polarized tipswill produce different local electromagnetic environments, causing the resonantfrequency to shift even when measuring the same Fe atom. Among different Featoms the ESR-STM resonant frequency exhibits variations on the order of 5GHz, even with the same tip. This can be attributed to different local environ-ments creating different ligand fields, highlighting the need for an atomic-scaleexperimental approach. The ESR frequency also exhibits time-dependent drifton the order of minutes, which is likely caused by mechanical vibration of thetip–sample junction.6.3 OutlookThe ESR-STM technique provides a powerful combination of atomic-scale res-olution combined with high energy resolution of magnetic transitions. At first,it may appear to be limited in scope, as the experimental set-up works only forthe use of single Fe atoms (so far). No observation of spin resonance has beenobserved on other magnetic species. In the next chapter it will be demonstratedthat single Fe atom ESR is sensitive to the presence of magnetic moments in thenanoscale neighbourhood of the resonant atom. Using this sensitivity it is pos-sible to use ESR-STM as a magnetic dipole-dipole sensor with a broad range ofmagnetic targets, greatly increasing the scope of future ESR-STM experiments.Chapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 100Energyms-2   -1    0   +1  +2Energyms-2   -1    0   +1  +2Vµ(a) Thermally populated (b) On resonanceBZ = 0.170 TBZ = 0.183 TBZ = 0.196 T(c) ESR-STM Signal|0i|0i|1i|1i|2i|2i |3i|3i|4i |4i21 22 23 24 25 26 27Frequency (GHz)50 fAΔItFigure 6.5: Single Fe atom ESR resonance. (a) Thermal population of thelow-energy quantum states of an Fe atom on MgO in a magneticfield. (b) Resonant population in continuous wave mode (c) ESR-STM spectra showing resonance at various magnetic field strengths.Chapter 6. Electron Spin Resonance Scanning Tunnelling Microcopy 10125.4 25.6 25.8 26.0 26.2 26.4 26.60510DI	(fA)Frequency	(GHz)27.2 27.4 27.6 27.8 28.0 28.2-10-50Off-resonanceOff-resonanceOn-resonanceOn-resonanceTip TipTip Tip25.4 25.6 25.8 26.0 26.2ESR signal ΔI (fA)26.4 26.6Frequency (GHz)Figure 6.6: ESR-STM spectra of the same Fe atom with two oppositelyspin-polarized tips. The ESR peak (dip) is caused by a lower(higher) tunnelling magnetoresistance at resonance. The frequencysplitting between the observed features is independent of tip, thoughthe absolute frequency at which each resonance is observed is tipdependent. The rich structure of the ESR spectra is caused by theproximity of nearby magnetic atoms on the surface, which will bediscussed in detail in Chapter 7. The above spectra were taken in amagnetic field of B = 4.8 T and at a temperature of T = 1.2 K.Chapter 7. Single Atom Magnetic Sensing on the Surface of MgO 102Chapter 7Single Atom MagneticSensing on the Surface ofMgOBut I am not afraid to considerthe final question as to whether,ultimately — in the great future— we can arrange the atoms theway we want; the very atoms, allthe way down!Richard Feynamn - 1959 [243]The following Chapter contains text and figures adapted from “Atomic-scalesensing of the magnetic dipolar field from single atoms” Nature Nanotechnology,12:420–424 [231]. Figures are altered unless otherwise noted.Spin resonance provides the high-energy resolution needed to examine bio-logical and material structures by sensing weak magnetic interactions [207]. Bypairing electron spin resonance techniques with scanning tunnelling microscopyresearchers at IBM have developed an experimental tool with the spatial resolu-tion of the STM and energy resolution of ESR. In this chapter, this techniqueis used to develop a new form of nanoscale magnetometry, capable of determin-ing the position and magnitude of nearby spin centres. This culminates in thedemonstration of nanoscale trilateration, dubbed “nano-GPS”.7.1 Characterization of an Atomic-ScaleMagnetic Dipole-Dipole SensorExperimental strides in atomic-scale magnetometry have seen recent advancesin the detection [211] and coherent control [213, 216, 244–246] of individual spincentres for sensitive local magnetometry [247–249]. However, sub-nanometre po-sitioning and characterization of the spin centres remains a challenge [250, 251].The ESR-STM technique offers a means to perform atomic-scale magnetometryand spin-spin sensing by using the resonance of a target Fe atom to sense theChapter 7. Single Atom Magnetic Sensing on the Surface of MgO 103dipolar magnetic field emanating from nearby spin centres. This dipolar sen-sor can determine the magnetic moment of individual nearby atoms with highaccuracy and can be used to pinpoint their spatial location up to 4 nm away.Detecting individual spins and their interactions has been pursued by meansof atomic-scale spin centres such as optically trapped ions [246], nitrogen va-cancy centres in diamond [247–249], spins in quantum dots [213], dopant atomsin semiconductors [245, 252], and single-molecule magnets in break junctions[216]. A resolution of 10 nm has been achieved for the controlled positioningand spin-resonant imaging of spin centres in solid-state materials [248, 250].Atomic-scale control remains a challenging goal [251]. The STM excels atatomic scale positioning [54–56, 136], including of single atom magnets [137–139, 235, 237, 253–255] ,and so coupling this power to ESR-STM detectionbreaks new experimental ground and may ultimately allow for magnetic struc-tural imaging of complex magnetic molecules, nanostructures, or spin-labelledbiomolecules.For magnetic atoms deposited onto metallic surfaces the dipole-dipole in-teraction is dominated by other interactions, such as Ruderman-Kittel-Kasuya-Yosida (RKKY) [253, 256] or superexchange [255, 257]. However, for mag-netic atoms inside an insulator separated by nanometre distances the long-range dipole-dipole interaction dominates, and can lead to fascinating magneticground states [258] and excitations [259]. Fe and Co sitting atop the insula-tor MgO are expected to interact magnetically through the dipole-dipole inter-action. This dipole-dipole coupling is too weak to measure using traditionalscanning tunnelling spectroscopy techniques, even at low temperature. Thiscoupling regime is accessible using the energy-resolution of ESR-STM.7.1.1 Sample Preparation and Measurement ProtocolMeasurements presented in this chapter were performed using the 1 K STMoperating in ultra-high vacuum P < 10−10 mbar. The majority of these mea-surements were performed with the tip and sample at the base temperature ofT = 0.6 K but temperature dependence of the observed frequency splitting wasalso checked up to T = 2.1 K. A total magnetic field of B = 4.8 − 6.6 T wasapplied at an angle of 1.8◦ to the sample surface. This produced an out-of-planemagnetic field Bz ranging from B = 0.15− 0.21 T.Sample preparation began with repeated cycles of argon ion sputtering andannealing at T = 450◦C of the Ag(100) crystal. MgO films were grown ontop of the Ag(100) surface, which was heated to T = 320◦C during growth, bythermally evaporating Mg from a Knudsen cell evaporator in an O2 environ-ment of PO2 ≈ 10−6 mbar. The growth was monitored by Auger spectroscopyand characterized by STM/AFM measurements [86, 239, 240, 260] at 0.5 mono-layer/minute. Experiments reported here were done with Fe and Co atomsdeposited on bilayer MgO, where the MgO buffer layer is crucial to observingspin resonance [86] and also reduces scattering between conduction electrons ofthe Ag(100) and the deposited atoms, leading to longer spin relaxation times[233]. Fe and Co atoms were deposited using electron beam evaporators onceChapter 7. Single Atom Magnetic Sensing on the Surface of MgO 104the sample had been transferred in the STM and cooled to 10 K. The coveragecorresponded to a low dosage of Fe and Co atoms, roughly 1 atom/20 nm2.The STM tip was prepared from a mechanically cut Ir wire and preparedwith field emission and controlled contact with the exposed Ag(100) surfacein-situ. The tip was spin-polarized by using vertical atomic manipulation totransfer an Fe or Co atom from the MgO surface, following the prescriptiongiven in Chapter 6. The degree of tip spin-polarization was measured using STSon individual Fe and Co atoms [144] and magnetic pump-probe measurements[86, 143]. Systematic effects resulting from the tip magnetic field were accountedfor by confirming results using more than 18 individually crafted microscopictips, and performing repeated measurement of the same atom using differenttips.Calibration of the 1 K-STM piezo motion was accomplished using atomically-resolved images of the MgO surface. MgO has an epitaxial match with theAg(100) surface [261], so the Ag(100) low-temperature lattice spacing of a0 =0.4069 [186] was used to calibrate distances between adatoms on the MgO sur-face [262]. The nearest-neighbour binding site distance using this calibrationis d0 = a0√2 = 0.2877 nm. Previous studies using STM and DFT had estab-lished the preferential binding site on top of the oxygen for both Fe and Co[86, 145, 239, 240].ESR-STM was performed by sweeping microwave radiation at constant am-plitude in the STM tunnelling junction [230] between 15 − 35 GHz. Standardset point parameters were Vs = 5 mV and Is = 1 pA. The microwave bias am-plitude was set at 10 mV peak-to-peak, which ensured that the instantaneousvoltage between tip and sample was never more than 10 mV, so as to remain be-low the first inelastic spin excitation energy shown in Figure 6.4. Measurementswere performed in continuous-wave mode using a lock-in amplifier to measurethe time-averaged root-mean-square tunnelling current with a microwave biaschopping scheme at 95 Hz.Electron spin resonance of individual Fe atoms on bilayer MgO/Ag(100)was driven by a microwave electric field at the tunnelling junction due to Vµ,as shown in Figure 7.1. The out-of-plane magnetic field Bz sets the resonancefrequency. An in-plane magnetic field Bx mixes the spin states of Fe to increasethe ESR signal [86]. A spin-polarized (SP) STM tip measures the spin resonancesignal via changes in the tunnelling magnetoresistance.Deposited Fe atoms were distinguished from Co by their STS signature, Featoms exhibit an inelastic transition at Vb = 14 mV [236] while Co atoms showan inelastic transition at Vb = 58 mV [145]. Vertical atom manipulation wasused to position individual magnetic atoms at a prescribed distance in order toexplore the ESR frequency splitting as a function of distance. In roughly 10%of cases positioning an Fe atom using this method resulted in an atom thathad a measured apparent height taller than Fe atoms deposited by evaporation(0.23 nm versus 0.17 nm), with a different spectroscopic signature. This ‘tall’-Fe could be switched back to the as-deposited species by voltage pulses largerthan 0.8 V or repeated vertical atom manipulation, and so was deduced to bea meta-stable charge state of Fe on MgO.Chapter 7. Single Atom Magnetic Sensing on the Surface of MgO 105Figure 7.1: ESR-STM for magnetic dipole detection. (a) Schematic forsensing the dipole-dipole interaction between two iron atoms. (b)Constant-current STM image of four Fe atoms (0.17 nm apparentheight) on the surface. Imaging conditions are VDC = 0.1 V, I = 10pA, Bz = 0.18 T, Bx = 5.7 T and T = 1.2 K.7.1.2 Magnetic Dipole-Dipole Sensing using ESR-STMCharacterization of the dipole-dipole interaction between magnetic atoms onthe MgO surface began with the ESR active species, the Fe adatoms. Atomicmanipulation was used to precisely set the atomic spacing between Fe atomsin increments of the MgO lattice spacing and then ESR-STM spectra weremeasured at each spacing. This was performed by positioning the SP-STM tipover one Fe atom (the sensor) and detecting the signature of the nearby Fe atom(the target) using continuous wave ESR. Each ESR sweep in frequency showeda primary and a secondary feature, as shown in Figure 7.2 for two Fe atoms 2.46nm apart. The observed spectral features corresponded to the spin up |↑〉 andspin down |↓〉 state of the target atom. The frequency splitting between thesefeatures gave the distance dependent magnetic interaction. The ESR peaks werefit with Lorentzian functions and the frequency difference between the peaks wasrecorded as a function of inter-atomic distance. This process was performed formore than 10 atomic pairs of Fe and the measured frequency splitting wasreproducible regardless of microwave power, tip apex, temperature, or appliedmagnetic field.The interaction between all ESR active Fe atoms was catalogued by con-structing Fe-Fe pairs at different MgO lattice spaces and measuring the splittingin ESR frequency as a function of distance. Once this had been performed forover ten pairs of Fe atoms the dipole sensing technique was applied to differentmagnetic species by using an ESR active Fe atom as a sensor to detect a nearbymagnetic target. In this case, the target atom was either Co or the charge stateChapter 7. Single Atom Magnetic Sensing on the Surface of MgO 10623.0 23.1 23.2 23.3-15-10-50ESR	signal	DI	(fA)Frequency	(GHz)ESR Signal ΔI (fA)Frequency (GHz)Δf23.0 23.1 23.2 23.3---1 nmrFeFe 3.66 Å 0.00 Å2.7 Å0 Å8×3xy MgOFe(a) (b)(c)Figure 7.2: Magnetic dipole-dipole interaction detected via ESR. (a)ESR spectrum (black curve) of an Fe atom when another Fe atom(target) is positioned 2.46 nm away (Bz = 0.17 T, T = 1.2 K,VDC = 5 mV, IDC = 1 pA, VRF = 10 mVpp). A fit to two Lorentzianfunctions (red curve) yields the frequency splitting (∆f). The dif-ference in amplitude between the two observed features can be at-tributed to the ratio in thermal occupation probability of the |0〉and |1〉 states. (b) Topography of the sensor Fe atom (outlined inblack) and the Fe target. (c) The underlying MgO lattice providesa metric to measure the distance between the two Fe atoms.of Fe dubbed ‘tall’-Fe. Vertical atomic manipulation was used to move sensorFe atoms at different spacings from these targets, so that the ESR frequencysplitting could be measured. In many cases, more than one target atom wouldbe present in the detection range of the Fe sensor. Figure 7.3 shows the ESRmeasurements and STM topographs of a three-atom nanostructure consistingof two Fe atoms and one Co atom. ESR-STM was performed on both of theFe atoms in the structure, in both cases the ESR spectra exhibited four differ-ent spectral features. By tracking which frequency splittings remained constantusing different sensor Fe atoms, spectral features could be correlated with theFe-Fe interaction. The dotted vertical lines in Figure 7.3 connecting the spectrain (a) and (b) show the ESR frequency splitting that remained the same, indi-cating that this splitting is due to the magnetic interaction of the two Fe atoms.The other pairs of ESR splittings originate from magnetic interaction betweeneach Fe atom and the Co atom, which changes due to the different inter-atomicseparations involved.The results of measuring the ESR frequency splittings as a function of inter-atomic distance for all magnetic targets on the MgO are shown in a log-logplot in Figure 7.4. The data is fit to a power law δf ∝ rα for each type oftarget atom: Fe, ‘tall’ Fe, and Co. The slope of the fit gives α = −3.01 ± 0.04(Fe-Fe), −2.98± 0.04 (Fe-Co), and −2.94± 0.08 (Fe-tall Fe), in good agreementwith the power law exponent expected for a magnetic dipole-dipole interaction(α = −3) in each case. The intercept of the fits contains information about theChapter 7. Single Atom Magnetic Sensing on the Surface of MgO 10728.5 28.8 29.1 29.4 29.7-40-200	Experiment	Lorentzian	fitESR	signal	DI(fA)Frequency	(GHz)28.5 28.8 29.1 29.4 29.7-40-200	Experiment	Lorentzian	fitAFe-Fe (5,-2)Fe-Co (9,0)Fe-Fe (-5,2) Fe-Co (4,2)(b)1 nm FeFe CoFeFeCo(9,0)(-5,2) (4,2)(c)(a)ESR signal ΔI (fA)r  ( Hz)Experimentorentzian FitExperimentLorentzian Fit1 nm28.5 28.8 29.1 29.4 29.728.8 29.1 29.4 29.7028.5Figure 7.3: ESR spectra of multi-atom structures. (a) ESR spectrumtaken on the middle atom (Fe) in (c) shows four ESR peaks. (b)ESR spectrum taken on the left atom (Fe) in (c) also exhibits fourESR peaks. (c) STM image of the three atom arrangement, withtwo Fe atoms and one Co atom. The binding-site assignment model(lower panel) shows distances between atoms in units of the atomiclattice spacing (0.2877 nm for T < 20 K). Imaging condition are 10mV, 10 pA, Bz = 0.2 T, and T = 1.2 K.Chapter 7. Single Atom Magnetic Sensing on the Surface of MgO 108magnetic moment of the sensor and target, information that can be extractedwith the theory developed in the next section. For inter-atomic distances lessthan 1 nm significant deviation from the inverse cubic power law is observed.The short-range data (r < 1 nm, gray region in Figure 7.4) were not includedin the power law fit.1 1.5 2 2.5 3 3.5 40.010.11Df(GHz)Distance	(nm)    Δf (GHz)Distance r (nm)Fe-Fer-3 fitFe-Cor-3 fitFe-‘tall’ Fer-3 fit0.00.1.0 1.5 2.0 2.5 3.0 3.5 4.0Figure 7.4: Log-log plot of ∆f ∝ rα. The measured splitting as a functionof the distance r of atom pairs of Fe-Fe (red squares), Fe-Co (bluecircles), and Fe-‘tall’ Fe (green diamonds). Data from the shadedregion below 1 nm are excluded from the fits. Error bars in ther and ∆f axes represent uncertainty in the determination of theinteratomic distance and the measured ∆f error due to frequencydrift. Distances are obtained using the Ag lattice constant measuredat low temperature by x-ray diffraction [186, 260].Deviation from the power law behaviour at small atomic separations may in-dicate an additional interaction, such as exchange coupling, is becoming promi-nent or classical point-dipole approximation, which scales like r−3, is no longervalid because at this distance the atomic size becomes comparable to the inter-atomic spacing. To investigate the discrepancy from the dipole interaction ob-served at short distances further experiments were performed for sensor andtarget atoms less than 1 nm apart. By using a sensor Fe atom situated in closeproximity to a target Fe atom, but using it to probe targets at larger distances,it was possible to check whether the breakdown in the inverse cubic fit at smalldistance was caused by canting of the sensor magnetic moment. Figure 7.5shows the ESR spectra and topography of a three Fe atom nanostructure. TheChapter 7. Single Atom Magnetic Sensing on the Surface of MgO 109sensor Fe atom is indicated by a white dotted circle and is less than 1 nm awayfrom one of the Fe targets. The ESR splitting measured on the Fe sensor dueto the remote Fe target atom quantitatively follows the expected dipole-dipoleESR splitting from Figure 7.4, indicating that the magnetic moment of the sen-sor atom is not significantly changed due to the presence of the close Fe target.Thus, the deviation of ∆f for close-spaced Fe-Fe pairs is not due to a changein magnetic moment but rather a change in the interaction between magneticmoments.24.0 24.3 26.4 26.7 27.002040DI	(fA)24.6 24.9 27.0 27.3 27.602040Frequency	(GHz)(b)(0,-6)(3,1)1nmFe Fe Fe FeFe(3,1)(a)0.8 0.9 112345Df(GHz)Distance	(nm)  (2,-2) (3,0)(3,1)(3,-2)(0,-6) ≈ 1.7 nm(3,1) ≈ 0.9 nm(c)Distance (nm)Δf (GHz)Frequency (GHz)ΔI (fA)02024.6 24.9 27.9 27.3 27.624.0 (3,1) ≈ 0.9 nm 27.026.726.424.3Figure 7.5: Distance-dependent ESR splitting (∆f) for atoms sepa-rated by less than 1.0 nm (a) ∆f vs distance r for close atoms,zoomed in from Figure 7.4 on the Fe-Fe curve. (b) STM images oftwo close Fe atoms before and after placing a third (“remote”) Featom in the vicinity. The imaging conditions are 10 mV, 10 pA,Bz = 0.17 T, and T = 1.2 K. (c) ESR splitting before and afteradding the remote atom.7.1.3 Single-Atom Magnetometry from ESR DipoleSensingThe magnetic dipole-dipole energy between two magnetic moments is given byChapter 7. Single Atom Magnetic Sensing on the Surface of MgO 110Edd =µ04pi1r3[(~m(1) · ~m(2))− 3(~m(1) · rˆ)(~m(2) · rˆ)](7.1)where µ0 is the vacuum permeability, r is the separation between the magneticmoments, and rˆ is a unit vector pointing in the direction linking the moments,and ~mi is the magnetic moment of atom i. The strong magnetic anisotropy forFe on the O binding site means that the Fe atoms are fully polarized out-of-plane [86, 239]. This means that canting of the magnetic moment due to thein-plane field is minimal and so Equation 7.1 can be simplified toEdd =µ04pi1r3(m(1)z m(2)z). (7.2)Equation 7.2 gives the magnetic dipole-dipole energy for two magnetic mo-ments in the same plane with magnetic moments fully perpendicular to thatplane. It shows that the expected dipole-dipole energy follows a power law withan inverse cubic dependency Edd ∝ rα where α = 3, in good agreement withthe experimental fits. The sensor and target atom’s magnetic moments canbe represented in Dirac notation, ie. |↑s〉, |↓s〉, |↑t〉, |↓t〉, where the subscriptss and t denote sensor and target respectively. When on resonance the sensoris in a time-varying, coherent superposition of the up and down spin states:α |↑s〉 + β |↓s〉. In this state, the sensor’s resonant frequency is affected by thespin state of the target, which has a thermally weighted probability of beingeither |↑t〉 or |↓t〉. This leads to four dipole-coupled microstates possible for theESR to sample, as illustrated in Figure 7.6. As a result, the resonant frequencyof the sensor in the absence of the target f0 is split into two frequencies (f↑ andf↓) corresponding to the two spin states of the target atom and giving a totalchange in frequency ∆f = f↓ − f↑ that is related to the magnetic dipole-dipoleenergy by∆f =4Eddh. (7.3)where h is Planck’s constant.By relating the measured frequency splitting to the magnetic dipole-dipoleenergy it is possible to rearrange Equation 7.3 to solve for the magnetic momentof sensor and target. In the case where both target and sensor are Fe atomsthis takes the formmFez =√hpiµ0r3∆f. (7.4)Using this equation to fit Fe experimental data in Figure 7.4, with a fixedChapter 7. Single Atom Magnetic Sensing on the Surface of MgO 111Sensoratom Targetatom Dipole coupled𝑓" 𝑓↑	2𝐸''/ℎ𝑓↓2𝐸''/ℎSensor onlySensoratomFigure 7.6: Magnetic dipole-dipole interaction detected via ESR Aschematic of the dipole-dipole interaction. The resonant frequencyof the isolated sensor atom (f0) is split into two frequencies (f↑and f↓) corresponding to the two spin states of the target atom.∆f = f↓ − f↑ is the measured splitting.Chapter 7. Single Atom Magnetic Sensing on the Surface of MgO 112exponent of α = −3, gives a magnetic moment of mFez = 5.44 ± 0.03µB . Theuncertainty in the magnetic moment comes from a combination of uncertaintyin the atomic lattice spacing d = 0.2877±0.003 (1%) nm, frequency drift whichhas an upper bound of 1% of the measured ∆f , and the uncertainty in the fitparameters. The magnetic moment of Fe on MgO has also been determinedby fitting multiplet simulations to X-ray magnetic circular dichroism (XMCD)spectra, giving 5.2µB [239]. A comparison between the value of the momentextracted from fitting the dipole curve in ESR-STM and multiplet simulations ofXMCD data is difficult, as no error bar is given for the XMCD fit. Furthermore,XMCD is an ensemble measurement technique which averages over< 1010 atoms[145, 239], and potentially averages different charge states of Fe. The ESR-STM technique presented here has a unique advantage in this regard, as it candistinguish between measurements of Fe and ‘tall’ Fe.With the magnetic moment of the sensor thus determined, ESR-STM pro-vides a way to measure the magnetic moments of other atomic species with highprecision. Knowledge of the Fe atom magnetic moment can be used, along withthe frequency-splitting data as a function of atomic separation in Figure 7.4, tofit for the magnetic moment. For the data presented in Figure 7.4 the magneticmoments were determined to be (5.88± 0.06µB) for Co and (4.35± 0.08µB) for‘tall’ Fe atoms usingmtargetz =hpiµ0mFezr3∆f. (7.5)This nanoscale form of magnetometry can be used to characterize target atomsto within one MgO lattice site for sensor–target separations of up to 4 nm. Theunique strengths of this form of magnetometry are its ability to measure themagnetic moments of atoms on different binding sites [232] or different chargestates like Fe and ‘tall’ Fe, something not possible with an ensemble measure-ment technique. In order to calculate the magnetic moment of an unknowntarget a plot of distance dependent frequency splittings must be constructedbetween the target and an Fe sensor, as was done for Fe, Co, and ‘tall’ Fe inFigure 7.4.The relative amplitude of the two observed ESR peaks also provides infor-mation about the thermal occupation probability of the target atom. For thecase of an Fe target the |↓t〉 and |↑t〉 states correspond to the |0〉 and |1〉 statesdescribed in single-atom ESR. At the applied out-of-plane field Bz = 0.15 Tused here these states are separated by a Zeeman energy of 95 µeV (23 GHz).The larger of the two ESR peaks corresponds to a measurement of the reso-nance frequency of the sensor when the target is in its ground state. This peakis the larger of the two as the target’s ground state is thermally populated with ahigher thermal occupation probability. The smaller, secondary peak correspondsto a measurement of the resonance frequency of the sensor when the target is inits first excited state, which has a lower thermal probability of occupation. Theratio of the peak heights is given by the Boltzmann distribution e−∆E/kBT , andChapter 7. Single Atom Magnetic Sensing on the Surface of MgO 113can be used to quantitatively extract the electronic temperature. Sampling ofboth states is made possible by the relatively long spin relaxation time of thetarget atoms (for Fe pump-probe spectroscopy gives 100 µs [86, 230]), whichis much faster than the data point averaging time for the ESR spectra (≈ 1s). This means that the ESR spectra on the Fe sensor statistically samples thethermal population of the target Fe spin at each data point.Development of a single atom magnetometer with such high spatial reso-lution offers a large step forward in the structural imaging of individual mo-ments within magnetic molecules. Having already characterized a previouslyunmeasured magnetic species in ‘tall’ Fe the next section will illustrate how thisnanoscale magnetometry technique could be used to spatially and magneticallycharacterize a completely unknown, complex target.7.2 Engineering a Nano-Scale MagnetometerArrayRather than using a single sensor Fe atom for dipole field detection, a sensorarray of Fe atoms allows for multiple confirmations of the position and magni-tude of target magnetic moments. Using vertical atom manipulation, as shownin Figure 7.7, a square sensor array was built using four Fe atoms. The sidesof the square array were designed to be ten MgO lattice spacings apart so thatthe total size was 2.877× 2.877 nm2. An additional Fe atom was chosen as thetarget, and positioned first at the very centre of the array and then at a positionthat was (-3,1) from the centre. This manipulation was achieved in conjunctionwith dropping off a sixth Fe atom that was used to spin-polarize the STM tipthis atom is shown in Figure 7.7 (b) and (e).Though the target atom in this case has already been fully characterized, thissensor array provides the opportunity to probe how the symmetry of a nano-engineered array is reflected in the degeneracy of the associated spin states.Using the corner atoms of the array as ESR dipole sensors, each ESR spec-trum exhibits signals from the four Fe targets in range, reflecting the 24 = 16possible magnetic configurations of the target atoms. The ESR spectrum forthe low symmetry configuration is shown in Figure 7.8 and the high symme-try configuration is shown in Figure 7.9. The dominant feature in both casescorresponds to the resonant frequency of the sensor when all target atoms arein their magnetic ground state, as this is the microstate most heavily favouredby the Boltzmann distribution. This spectral weight is redistributed to otherstates when the temperature is raised, as shown in the contrast between theESR spectra at T = 0.6 K versus that at T = 1.2 K. At both temperatures theexpected spectral features with the lowest Boltzmann probability, those corre-sponding to three or more of the Fe atoms are in the |1〉 state, are either notvisible or at the level of the current noise.In the low-symmetry position of Figure 7.8 there are no degenerate ESRtransitions. The various states and the associated orientation of the sensor andChapter 7. Single Atom Magnetic Sensing on the Surface of MgO 114Tip atomTip atom(a) (b) (c)(d) (e) (f)Figure 7.7: Construction of the dipole sensing array. (a) Five Fe atomswere gathered and arranged to form an array of four sensors and onetarget in the array centre. (b) The tip atom, another Fe, is placedoutside the array for safekeeping. (c) The target Fe atom is pickedup from the centre of the sensor array. (d) The target is placed ata lower symmetry position, corresponding to (-3,1) oxygen latticesites from the array centre. (e) The entire structure is imaged inorder to find the tip atom again. (f) The tip is spin-polarized bypicking up the Fe tip atom.Chapter 7. Single Atom Magnetic Sensing on the Surface of MgO 115target spins are illustrated through the use of circular symbols in the figure.A filled circle indicates that a spin on that atom is in its excited state at thatfrequency. The grey line indicates the ESR frequency of the Fe sensor with allcorner (target) atoms in their magnetic ground state. Pink lines indicate ESRfrequencies with one of the four target atoms in the excited state. AdditionalESR lines arise (with less intensity) when two or more of the corner atoms are inthe excited state.The ESR spectra were taken at two different sample tempera-tures: 0.6 K (upper spectra) and 1.2 K (lower) measured by a thermometer onthe 1-K STM. The red and blue solid curves are modelled based on the dipoleenergy, the degeneracy of the states, and the thermal population of the states.1 nmT=1.2	KT = 0.6 K0.0 0.2 0.4 0.6 0.8 1.0-120-90-60-300DI	(fA)Df	(GHz)T = 1.2 KΔI (fA)Δf ( z). . . . .Figure 7.8: Measuring a target atom at a low symmetry site with anano-sensor arrray. Five Fe atom structure with four equallyspaced corner atoms and a middle Fe (sensor atom) which is posi-tioned at (−3, 1) lattice sites from the centre of the square (oxygenbinding sites indicated by white open circles). For clarity, the 1.2 Kspectra is offset by −60 fA. Imaging conditions are VDC = 10 mV,I = 10 pA, and Bz = 0.17 T.When atomic manipulation is used to manipulate the target from the low-symmetry position in Figure 7.8 to the high-symmetry position in Figure 7.9 theESR spectrum exhibits changes that reflect the higher degree of symmetry inthe magnetic states being probed. As the separation between the target and allfour sensor atoms is now the same many of the weaker amplitude peaks observedin Figure 7.8 become degenerate, as highlighted by the coloured vertical lines.For example, the four pink lines are now degenerate at ∆f ≈ 0.183 Hz. Theintensity of the ESR peaks in Figure 7.9 follows the degeneracy of the sensoratoms states1, weighted by the Boltzmann distribution [242]. The Boltzmanndistribution weighting is again evident when comparing the ESR spectra atT = 0.6 K versus at T = 1.2 K, where spectral weight is transferred to higher1The degeneracy can be calculated using Pascal’s triangle (1:4:6:4:1).Chapter 7. Single Atom Magnetic Sensing on the Surface of MgO 116energy states.T=1.2	KT=0.6	K0.0 0.2 0.4 0.6 0.8 1.0-120-90-60-300DI	(fA)Df	(GHz)1 nmT = 1.2 KT = 0.6 KΔf (G z).ΔI (fA)Figure 7.9: Measuring a target atom at a low symmetry site with anano-sensor arrray. (b) The Fe sensor atom is placed in the exactcentre of the square at a distance of 2.034 nm from the four targetatoms. This creates a four-fold symmetry which leads to degeneracyof the excited spin states. Imaging conditions are VDC = 10 mV,I = 10 pA, and Bz = 0.17 T.The difference in the observed ESR spectra as a function of temperature al-lows the spin array to be used as a local thermometer, in addition to its functionas a magnetometer. Since the relative amplitudes of the observed peaks are givenby the Boltzmann ratio e−∆E/kBT and the dipole-dipole energy can be measuredfrom Equation 7.3 it is possible to extract temperatures of T = 0.72 ± 0.02Kand T = 1.43± 0.13 K for the measurements of Figure 7.9. These temperaturesdo not agree within statistical error with measurements of the temperature froma thermometer on the 1-K STM head, which reports T = 0.6 K and T = 1.2 K.There are a number of systematic issues that may cause such a disagreement,such as the ESR temperature measurement predominantly sampling the elec-tronic temperature [221] while the head mounted thermometer is more sensitiveto the phonon temperature. To use ESR thermometry in a quantitative wayrequires more exploration of systematics to develop a calibration.7.2.1 Nano-scale Magnetic TrilaterationThe nanoscale sensor array can be used to develop a new form of magneticimaging based on trilateration of the ESR signal between the different cornersensors. Trilateration is the geometric technique used to calculate macroscopicpositions on the surface of the Earth by the Global Positioning System (GPS)and so this technique is dubbed ‘nano-GPS’. This imaging technique allows foridentification of the spatial location and magnitude of the magnetic momentChapter 7. Single Atom Magnetic Sensing on the Surface of MgO 117of the target independent of the topography mode of the STM. This is illus-trated by extracting the position of the Fe target atom in the low symmetryconfiguration shown in Figure 7.8, as illustrated in Figure 7.10.Measuring the ESR signal at three of the sensor atoms of the array givesthree different frequency splittings ∆f shown in Figure 7.10 (a). The goal ofthis experiment is to demonstrate that the sensor array can be used to charac-terize a target with an unknown magnetic moment. For this reason the target’smagnetic moment is considered to be unknown, and is solved for numericallyby finding the value for the moment for which all three ESR signals identifya unique point in space, rather than using Equation 7.4 to extract the knownFe magnetic moment. The problem consists of using the measured frequencysplitting to simultaneously solve for three unknowns: the unknown target’s xposition, unknown target’s y position, and the unknown target’s magnetic mo-ment.Numbering the spin sensors as 1, 2, and 3, as shown in Figure 7.10, themeasured ESR splittings from each sensor can be related to the three unknowns.For sensor 1 this takes the formmtargetz =hpiµ0mFezr31∆f1 (7.6)where the distance between the sensor and the unknown target can be writtenr1 =√(xFesensor − xunknown)2 + (yFesensor − yunknown)2. (7.7)The deconvolved energy splittings extracted from the ESR signal three sets ofparameters which can be used to solve the unknown coordinates and moment.The solution is illustrated by a single point of overlap for the three circles drawnaround the sensor atoms in Figure 7.10 (c), which gives the position of the targetatom, in agreement with the atom location measured by STM topography shownoverlaid in Figure 7.10 (d).A complication to the nano-GPS method is the presence of ESR peaks causedby the sensor atoms measuring each other. Fortunately, the Fe-Fe interactionbetween the sensors has been fully characterized in Figure 7.4. It is thereforepossible to predict the position and magnitude of peaks caused by nearby sensoratoms and deconvolve the ESR signal to focus just on signals emanating from thetarget. The predicted ESR peaks due to the other sensors for Sensor 1 are shownFigure 7.10 (b) as a red curve. The peaks associated with the other sensors aresubtracted off, with a magnitude determined by the expected ratio of the peakheights from the Boltzmann distribution. The spatial resolution derived fromnano-GPS is better than 0.1 nm, as illustrated by the size of the black symbolin Figure 7.10 (d), and the magnetic moment resolution better than 0.1µB .Nano-GPS offers the ability to spatially pinpoint spin centres in complicatedChapter 7. Single Atom Magnetic Sensing on the Surface of MgO 118bio-molecules2, in which normal STM topography would not necessarily be ableto clearly resolve individual atoms due to the delocalized nature of the electrons.Nano-GPS also offers chemical sensitivity, a weakness of traditional STM, viasensing of the target’s magnetic moment.Figure 7.10: Magnetic imaging by using trilateration. (a) Measured ESRspectra (T = 0.6 K) from each sensor atom (black) and the pre-dicted unbroadened spectrum (red) due to all atoms excluding thetarget. (b) ESR spectra of (a) after deconvolving the effect of theother sensor atoms. The results show ESR spectra (peaks are in-dicated by black arrows) due solely to dipolar interaction of thetarget atom and the sensor atom under the tip. (c) Predicted lo-cation of the target by trilateration. (d) Agreement of the targetlocation between STM topography and nanoGPS.2Subject to size constraints imposed by the 4 nm detection limit of the dipole sensingtechnique and practical limitations on the number of sensors atoms that would be required totrilaterate a large number of spin centres in the same nanoscale area.Chapter 7. Single Atom Magnetic Sensing on the Surface of MgO 1197.3 ConclusionsBy fully characterizing the magnetic dipole-dipole interaction a new form ofatomic-scale magnetometry was developed that can identify the magnetic mo-ment of unknown spin centres up to 4 nm away. Frequency splitting of theESR-STM spectra into two peaks was investigated and it was determined thatthe presence of nearby magnetic moments caused changes in the local magneticenvironment for the ESR-STM sensor atom that pushed the resonance frequencyeither up or down. The frequency splitting caused by specific magnetic specieswas shown to be robust against changes in STM tip, magnetic field, microwavepower, and temperature.Furthermore, when combined with the atomic manipulation techniques ofthe STM, nano-sensor arrays of Fe atoms could be engineered with controllabledegrees of macroscopic symmetry. These arrays make it possible to performtrilateration on unknown spin centres to determine their position and moment.This provides a new route to atomic-scale structural imaging of nanostructuresthat can be assembled or placed on a surface, magnetic molecules or spin labelledbiomolecules.120Part IVFuture Direction and OpenQuestionsChapter 8. Conclusions 121Chapter 8ConclusionsDistress not yourself if you cannotat first understand the deepermysteries of Spaceland. Bydegrees they will dawn upon you.Edwin A. Abbott - 1884 [2]8.1 Summary of ResultsThis thesis began by introducing the concept of the nanoscale surface and howit can be measured using the scanning tunnelling microscope. The importanceof ultra-high-vacuum, low-temperature STMs to this thesis work was developedin Chapter 2, detailing the acquisition modes used for data collection, and thespecific instruments: the Createc and 1-K STM.In Part II, experimental results were presented demonstrating quasiparti-cle interference in the metals Ag(111) and Cu(111). The differential tunnellingconductance was acquired using multiple different acquisition modes and the ef-fect of the tunnelling transmission probability was isolated and compared withsimulation results based off of T-matrix scattering theory of the sample densityof states and a Tersoff-Haman model of the tunnelling current. The simulationresults agreed qualitatively with experimental measurements of the set-pointartifacts in Ag(111) and Cu(111). The most dramatic set-point effect occurredwhen a series of constant-current dI/dV maps were acquired at different ener-gies. An FT-STS feature appeared that dispersed in energy and was not ob-served in other measurement modes. This artifact is similar to features arisingfrom real physical processes in the sample and is susceptible to misinterpreta-tion and masking of other signals, as observed in the comparison of Cu(111)constant-height dI/dV maps to other acquisition modes.Characterizing the set-point effect in FT-STS measurements has significantramifications for future measurements of QPI in both the noble metals and inmaterials with more complex band structure. In the noble metals, the increas-ing scattering space resolution available with better instrumentation and morecomputational power means that researchers are examining noble metals QPIfor subtle many-body effects [113, 132] not experimentally accessible when thesemetals were first characterized with FT-STS two decades ago. The signaturesof these many-body effects, whether they are electron-phonon coupling or plas-Chapter 8. Conclusions 122mon effects, are of the same order or smaller than the signal from the set-pointeffect, and so it is paramount that the set-point effect is taken into accountwhen attempting future measurements. This thesis offers a prescription to ruleout set-point effects in future noble metal FT-STS measurements: for spectro-scopic grid measurements ensure the stabilization bias is set at a bias energythat does not show quasiparticle interference in the tunnelling current signal,such as below the onset of a surface state; and for constant-current dI/dV mapsperform a number of checks using constant-height dI/dV maps using the sameparameters in order to ensure that the features of interest are not caused bymodulations of the tunnelling junction by the feedback.For QPI in more complex materials, it is often not possible to follow theprescription of setting the stabilization bias below the onset of the band beingprobed; in many cases there are multiple bands being probed and these bandscan be either hole or electron-like [88, 97, 111]. The greatest impact that this set-point work will have for studies in these materials is to serve as an importantnote of caution. Previously, features in the FT-STS intensity that did notchange as a function of energy were attributed to the atomic lattice, noise, orstatic surface features while features that did change as a function of energywere considered to be caused by QPI and could be linked to the band structure.These results present a second possible source for dispersing features in the FT-STS intensity: set-point effects. A careful analysis of the expected set-pointfeatures will vary from material to material but a few general guidelines canbe given: the appearance or disappearance of FT-STS features as a functionof set-point conditions should be checked, at least two measurement acquisitionmodes should be used to verify any FT-STS intensity as each measurement modeexhibits slightly different set-point effects, and the FT-STS results should becompared with other techniques, such as ARPES, to verify that both techniquesobserve features that match the predicted band structure.In Part III the dipole-dipole interaction of individual iron atoms on the sur-face of MgO was characterized over a length scale of 1.0 − 4.0 nm. In thisregime, the distance-dependent frequency splitting of ESR-STM features wasconstant, independent of magnetic field, temperature, or STM tip. Fitting theiron ESR-STM frequency splitting as a function of the separation between Fe-Fepairs showed that the splitting followed an inverse cubic distance dependence(δf ∝ rα with α = −3.01 ± 0.04). Relating the measured frequency splittingto the magnetic dipole-dipole energy this power law fitting allowed for the ex-traction of the Fe magnetic moment mFez = 5.44 ± 0.03µB . With the Fe-Feinteraction fully characterized the Fe atoms on the surface could be used assensors of nearby magnetic moments. A new form of nanoscale magnetome-try was invented that allowed for the extraction of the magnetic moment ofCo (5.88 ± 0.06µB) and ‘tall’ Fe (4.35 ± 0.08µB). To demonstrate the poten-tial of this technique when combined with atomic manipulation, a four atomnano-sensor array was constructed to trilaterate the position and measure themagnetic moment of an Fe atom situated inside the array. This trilaterationtechnique, nanoGPS, was used to locate a target magnetic moment with a pre-cision of ±0.1µB and in agreement with STM topograph measurements of theChapter 8. Conclusions 123atom’s position with a spatial resolution better than one MgO lattice spacing.Dipole sensing ESR-STM represents a new way to characterize the magneticproperties of single atoms on surfaces. The work presented in this thesis servesto characterize this emerging technique for the larger field of researchers work-ing with atomic magnets on surfaces. Experiments using Fe atoms as remotemagnetic sensors have the potential to probe aspects of magnetism previouslyinaccessible to experimentalists. For example, during the writing of this the-sis Natterer and collaborators [232] demonstrated that the magnetic states ofholmium atoms placed on MgO can be ‘written’ using voltage pulses with theSTM tip and ‘read’ by measuring the tunnelling magnetoresistance. Dipolesensing ESR-STM on nearby Fe atoms was used to definitely show that the Hoatom behaviour had a magnetic origin and determine its magnetic moment onthe MgO surface (10.1± 0.1µB).The ESR experimental scheme presented herein also offers a template for re-searchers to invent other experimental set-ups that may work for electron spinresonance STM. For example, other commonly used insulating substrates, suchas NaCl [21] or Cu2N [137], have lattice sites that share the fourfold symmetryof the oxygen binding site in MgO, and could potentially be used to produceelectron spin resonance in deposited iron atoms. Dipole-sensing ESR-STM onan iron atom on a Cu2N thin film would be a particularly interesting exper-iment, as such measurements could prove complementary to the pump-probemagnetic sensing technique recently detailed in Reference [263]. There is alsothe possibility for significant collaboration with experts of other techniques, likeX-ray magnetic circular dichroism (XMCD). Just as with FT-STS and ARPES,ESR-STM and XMCD are complementary techniques that together form a morecomplete picture of the system under study than possible with either alone.8.2 The Future of the Set-Point ModelA number of experiments can also be suggested based on the results of Part II.These are elaborated below, along with comments on potential difficulties thatmay need to be overcome.High-resolution spectroscopic grid measurements of the Cu(111)surface stateThe spectroscopic grid data presented in this thesis contained noise related tochanges of the STM tip that occurred during the measurement. A new STMhead has recently been installed on the CreaTec that has better mechanicaland electrical isolation. In principle, this should allow for measurements ofthe Cu(111) surface state with resolution on par with the best measurementsperformed on the Ag(111) surface state. These improved grid measurementsshould reveal the electron-phonon coupling kink near the Fermi energy, similarto the kink observed in Ag(111) [132]. Furthermore, they should have sufficientresolution and signal-to-noise that they should be able to observe the secondaryChapter 8. Conclusions 124features observed in constant-height dI/dV maps. This is a necessary exper-iment because it would demonstrate whether these features are dependent onacquisition mode. When attempting these measurements the stabilization biasshould be set in Region I, below the band onset of −420 meV, to avoid set-pointartifacts masking the signal of interest.High set-point current measurements of Ag(111)The secondary dispersing feature observed in constant-height dI/dV maps of theCu(111) surface has not been studied in the context of other (111) surface states.A return to the Ag(111) surface state to make constant-height measurementsat set-point currents at or above 1 nA could provide useful information onthe ubiquity of this feature in the (111)-terminated noble metals. The acousticsurface plasmon predicted by Sessi [113] should appear in Cu(111), Ag(111), andAu(111). The set-point current range in measurements of Ag(111) presentedhere was from 100 pA to 540 pA. Even in the event that the same feature seenin Cu(111) is not observed, an examination of FT-STS features observed atdifferent set-point currents could reveal the presence of the second and thirdterms in the differential tunnelling conductance [80]:dItdVb= eρs(eVb)ρt(0)T (z, eVb, Vb)−√2mez(Vb)2h¯√φIs − 2√2mh¯∂z(Vb)∂Vb√φIs. (8.1)The second and third terms in this expression are dependent upon the set-pointcurrent. These terms should exhibit increased signal at higher current, andmay already have been observed in the highest current constant-height dI/dVmeasurements of Cu(111) presented here.Development of T-matrix theory of Cu(111) for comparisonThe numerous features in the FT-STS signal derived from constant-height dI/dVmaps of Cu(111), which should be independent of effects of modulation of thetunnelling transmission probability, suggest the need for a rigorous theory of theCu(111) surface state. The framework already developed to describe intra-bandsurface state scattering in Ag(111) could serve as a starting point. To adaptthe same model for Cu(111) modifications would be required to the band on-set, effective mass, and electron-phonon coupling according to experimentallydetermined values for the Cu(111) surface. Such a model provides a complete de-scription of the intra-band scattering but does not include other processes, suchas coupling to plasmons, electron-electron interactions, or spin-orbit coupling.These physical processes require different treatment, but could be developedwithin the context of the existing model.Chapter 8. Conclusions 125Real-space simulation of multiple scattering impuritiesThe set-point model does a good qualitative job of predicting the observed set-point artifacts in grid and map measurements of Ag(111) and Cu(111). It doesnot, however, provide a quantitative comparison. This is expected, as the T-matrix theory considers only a single scattering impurity, while the experimentaldata measures many scattering impurities of different types. An attempt tosimulate multiple scattering centres in real-space is explored in Appendix B.Simulating each experimental data set in this way may allow for a more precisefit of the entire experimental FT-STS line shape. This would be useful fortwo reasons: it would potentially allow a way to quantitatively characterizethe effect of systematic uncertainties on the FT-STS scattering intensity andit would allow for extraction of physical quantities of interest, such as lifetimeand scattering phase, from the entire FT-STS line shape and not just the peakposition. Results gleaned from attempting a quantitative fit to the line shapecould then be applied to the study of 2DEG QPI in semiconductors [82], while athorough characterization of systematic uncertainties could be applied broadlyto QPI studies of complex, strongly correlated materials.8.3 Proposed Experiments with MagneticDipole-Dipole SensingThe dipole-dipole iron sensing technique complements the atomic manipulationstrengths of STM coupled with the high-energy resolution of ESR. At present,the detection range (4 nm) of the Fe sensor is mainly limited by the spectral linewidth of the ESR signal, which could be improved by lowering the magnitude ofthe applied microwave bias. With the fully characterized Fe sensing techniquea number of exciting experiments are possible, a few of which are suggestedbelow.Dipole Sensing of Long-Lived Atomic MomentsIron and cobalt both derive their magnetism from the orbital character of theird electrons. Magnetic species with f orbital character have larger magneticmoments and longer magnetic relaxation times, making them more suitable forsingle-atom magnetic information storage. Holmium atoms on magnesium oxideare a prime candidate in this regard [264] and during the writing of this thesisthe Fe dipole sensing technique was used to remotely read out the magneticstates of two holmium atoms which had their magnetic states written by theSTM tip [232]. These experiments could be scaled up to include arrays of Hoatoms, as was done for clusters of 5-12 atoms by Loth [137] and Khajetoorians[265] to examine the interface between single atom-magnetism and collectivemagnetic dynamics of magnetic structures.Chapter 8. Conclusions 126Measuring molecular magnetismThe nano-GPS scheme presented here provides a new route to atomic-scalestructural imaging of nanostructures that can be assembled or placed on thesurface, such as magnetic molecules. An excellent candidate for the first experi-ments of molecular magnets with this technique is terbium (III) phthalocyanine(TbPc2) as it has already demonstrated interesting magnetic behaviour whendeposited on MgO [266]. The terbium atom has an electronic spin state of J = 6with an unpaired electron delocalized over the phthalocyanine ligands. Dipolesensing the magnetic field from this unpaired electron could illuminate aspectsof the internal magnetic structure not available to other techniques.Probing nuclear spinsESR-STM could allow for the measurement of nuclear spin via the hyperfineinteraction between the electrons and the nuclei. The most common isotope ofiron is 56Fe, which has nuclear spin-0. As a result, 56Fe does not have a nuclearspin with a non-zero magnetic moment and so it is not a suitable candidate forexploring hyperfine coupling with ESR-STM. However, 57Fe has an additionalneutron and a nuclear spin of 1/2. By depositing isotopically pure 57Fe on theMgO surface and performing ESR-STM frequency sweeps it may be possibleto read out the nuclear spin state via the ESR signal. This is potentially veryinteresting as an architecture for engineering a qubit, as the nuclear spin shouldhave a long relaxation time and quantum phase coherence in comparison to theelectronic spin.Developing a pulsed ESR-STM techniqueThe results reported in Part III relied on continuous wave ESR-STM, a tech-nique which is not capable of phase coherent measurement of the electron spin.This limits continuous wave ESR-STM’s usefulness for use in quantum comput-ing. Pulsed ESR-STM, in which microwave bias pulses are delivered in shortbursts on resonance, should allow quantum manipulation of the electron spin,if the technique can be realized. A potential method for doing so, involving os-cillating the tip in phase with microwave pulse delivery and DC probe read-outis presented in Appendix C.8.4 Leaving FlatlandThis thesis has characterized two tools for exploring Flatland: FT-STS anddipole sensing ESR-STM. FT-STS is a heavily-used, important experimentalmethod for probing complicated materials. It was demonstrated that set-pointrelated artifacts that affect this measurement mode are not as well understood asneeded even in the simplest of physical systems that exhibit QPI. Dipole sensingESR-STM is an emerging experimental method that shows great potential butso far has only been performed by a single experimental group. By carefullyChapter 8. 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Lan, S. Klyatskaya, M. Ruben, H. Brune, andJ. Dreiser. Giant Hysteresis of Single-Molecule Magnets Adsorbed on aNonmagnetic Insulator. Advanced Materials, 28(26):5195–5199, 2016.[267] N. Pertaya, K.-F. Braun, and K.-H. Rieder. On the stability of Besocke-type scanners. Review of Scientific Instruments, 75(8):2608–2612, 2004.Appendix A. Upgrades and Operation for the UBC CreaTec LT-STM 150Appendix AUpgrades and Operation forthe UBC CreaTec LT-STMProgress is made by trial andfailure; the failures are generally ahundred times more numerousthan the successes; yet they areusually left unchronicled.William RamsayA.1 A Brief History of the CreaTec LT-STMThe CreaTec LT-STM in the LAIR has a fifteen-year operating history thatincludes a number of major upgrades and part replacements. The purpose ofthis appendix is to provide details of the operational status of this STM as of2017. A short timeline of the instrument is shown below in Table A.1.Table A.1: History of the CreaTec highlighting major upgrades.Year Event2003 CreaTec purchased by Professor Johannes Barth.2010-2011 This cryostat is replaced and a thermal braid is added.2013 The CreaTec STM is moved into c-pod, part of the LAIR.2014 The tunnelling current line is replaced.2015 Addition of a Nanonis controller with high voltage amplifier.2016-2017 The STM head is upgraded to an AFM-STM model.A.2 Components of the CreaTecThe Createc is composed of a large number of electrical, mechanical, and cryo-genic systems that work together to create and maintain a low-temperature,UHV environment at the STM tip-sample junction. These systems and theirindividual components are described in detail in this section, with many of themshown in Figure A.1.Appendix A. Upgrades and Operation for the UBC CreaTec LT-STM 151Figure A.1: The CreaTec LT-STM in c-pod. (a) The vacuum chamberconsists of (1) the manipulator arm, (2) the preparation chamber,(3) the cryostat, and (4) the pumping system. (b) The STM/AFMhead being installed outside of vacuum. The visible components are(5) tunnelling current cold finger, (6) damping springs, (7) walkingdisc, and (8) a sample holder mounted in the measurement position.(c) The manipulator is used to move samples within the vacuumspace. It consists of (9) sample bias and annealing contacts and(10) sample clamping mechanism. (d) The sample holder (11) witha graphite sample mounted, coin shown for scale.Appendix A. Upgrades and Operation for the UBC CreaTec LT-STM 152The Manipulator Arm: The manipulator arm allows transfer and controlof samples inside the UHV environment of the preparation and STM chamber.The manipulator is capable of motion in x, y, and z directions as well as 360degrees of rotation. It is mounted on a rotary feedthrough that allows it to bedifferentially pumped and it has a cooling line through which liquid helium orliquid nitrogen can be circulated to pre-cool a sample holder to as low as 20 Kbefore transfer inside the STM cryogenic shields.The Preparation Chamber: The preparation chamber houses samples inultra-high vacuum, between 10−10 − 10−9 mbar and allows for sample prepa-ration and characterization of samples before they are inserted into the STMchamber. The preparation chamber is equipped with state-of-the-art surface sci-ence technique tools for in-situ sample preparation and characterization, suchas a low-energy electron diffraction (LEED) setup1, a quadrupole mass spec-trometer2, electron beam tip heating tool3, and argon ion sputtering gun4. Thepreparation chamber is also connected to a load lock that can be used to trans-fer in samples from ambient conditions without breaking vacuum and containsa sample garage theoretically capable of holding six samples in-situ.The STM Chamber: The STM chamber houses the STM head, includingthe electrical wiring used for measurement and control, within a 77 K nitrogenheat shield and 4.2 K helium heat shield. The nominal pressure in the chamberoutside of the heat shields is 3 − 5 × 10−10 mbar during measurement, andpressure inside the 4.2 K cold space where the sample is measured is less than10−12 mbar due to cryogenic pumping by the heat shields.The Cryostat: The cryostat is a two-layer system of heat shields and cryo-genic reservoirs mounted on the top of the STM chamber. The outer reservoirconsists of 13.3 L shroud filled with liquid nitrogen while the inner reservoir isfilled with 4.3 L of liquid helium and coupled directly to the STM head [125].The maximum time that cryogenic liquids last before boiling off, known as thehold time, is approximately 72 hours, making this the longest time possible fora single measurement.Pumping System: The pumping system keeps the STM chamber, prepa-ration chamber, and manipulator arm under ultra-high vacuum. The systemconsists of a series of interconnected turbo molecular vacuum pumps 5, a rough-ing pump6, a titanium sublimation pump7, and ion pumps mounted on thebottom of each chamber. The pressure is monitored at each pumping stage bya set of vacuum gauges8 and a mass spectrometer in the preparation chamber9.The STM head: The STM head is a beetle-style STM head [129, 131]. Abeetle design uses a walking plate sitting atop three piezo tubes to achieve lateral1Specs ErLEED 100/1502SRS RGA3003Ferrovac Gmbh.4Specs IQE 11/355Varian TV 301 Navigator Turbo Pump andTMU 071 P Turbo Pump6TriScroll Dry Vacuum Pump7Varian Model 929-00228IK 270 Compact Cold Cathode Gauges9SRS RGA300Appendix A. Upgrades and Operation for the UBC CreaTec LT-STM 153and vertical motion on the order of millimetres. Coarse positioning of the STMtip, which hangs off of a piezo tube mounted in the centre of the walking plate,is possible with a maximum radial lateral range of 3 mm and a vertical range of0.9 mm. The coarse controls can be used to position the tip within micrometresof the sample surface. Fine motion of the tip, on the scale of Angstroms tohundreds of nanometres, is generally performed using the main piezo tube uponwhich the tip is mounted, but can also be split between the outer piezo tubesand the main piezo tube. The beetle-style head offers relatively large lateralmotion at the potential expense of reduced mechanical stability [267]. Work hasbeen done to find the optimum configuration for minimal mechanical noise onthe LAIR CreaTec. The set-up and voltage profiles used to move a beetle-styleSTM head are shown in Figure A.3. The current incarnation of the CreaTec hasa head capable of performing STM/AFM measurements, though this upgradewas only completed at the beginning of 2017 and AFM measurements are notfeatured in this thesis.Figure A.2: The CreaTec STM head. (a) The upgraded UBC STM headused for the bulk of the measurements in Part II. The wiring isdesigned to be slack to avoid coupling vibration into the tip-samplejunction and is extremely fine in order to prevent heat leaks fromroom temperature to the STM head. (B) A model of the originalCreaTec head when mounted inside the heat shields.The Sample Holder: The sample holder is a copper piece which mechan-ically holds the sample atop an electrically isolated heating stage. Bias andannealing contacts are run from the sample to electrical contacts on the back ofthe sample holder. The height profile of the sample holder can be manually ad-Appendix A. Upgrades and Operation for the UBC CreaTec LT-STM 154tVtVtV(a) Piezoelectric Actuation(d) Z Motion(c) XY Motion(b) Besocke HeadFigure A.3: Operation of a beetle/Besocke style STM head. (a) Thevoltage profile applied to the quadrants of a piezo tube to make itmove. The voltage is slowly ramped from a constant value up to amaximum before being quickly inverted. This produces a sawtoothwaveform that moves the walking plate. (b) A basic model of abeetle-style STM head such as the one used in the CreaTec STM.(c) Forces generated from the piezo tubes on the walking plate forX and Y motion. (d) Torque on the walking plate from the piezotubes in Z motion. When all three piezo tubes produce clockwiseforces a net torque is produced that moves the walking plate andSTM tip towards the sample. Counterclockwise forces produce anet torque that moves the walking plate and STM tip away fromthe sample.Appendix A. Upgrades and Operation for the UBC CreaTec LT-STM 155justed and additional contacts can be applied to measure temperature or applya back gate voltage.The Isolation Pod (c-pod): The frame housing the CreaTec sits atoppneumatic isolation legs10 on a 22 tonne concrete block that is positioned ona set of even larger set of pneumatic isolators. This setup ensures a very highdegree of passive damping of ambient seismic and acoustic noise [123]. Thebaseline noise spectrum of c-pod is shown in Figure 2.2.Electronic Measurement: For measurement of the tunnel current theCreaTec is paired with either a FEMTO DLPCA-200 or FEMTO LCA-4K-1Gamplifier. Both offer roughly 4 − 5 fA/√Hz of electrical noise. The DLPCA-200 offers manually adjustable gain with 1.2 kHz bandwidth at 109 V/A gainwhile the LCA-4K-1G offers 1.1 kHz bandwidth at 109 V/A gain, making itpreferred for fast measurement acquisition. The baseline noise spectrum ofboth amplifiers, expressed as amplitude spectral density, is shown in Figure 2.2.Data Acquisition: Data acquisition is performed by a Nanonis SPM RC5Control System11. The Nanonis is responsible for applying the direct currentbias between tip and sample and generating the high voltage sawtooth wave-forms sent to the piezoelectric tubes in order to move the tip over the sample,as shown in Figure A.3. The Nanonis is also used to measure the tunnellingcurrent signal from the FEMTO amplifiers and contains an internal lock-in am-plifier that can be used to add AC excitations to the bias line and measurederived quantities. It has an 18-bit analog-to-digital converter, and measuressignals differentially with a ±10 V range. Electrical grounding of all STM linesis done through the Nanonis ground in order to prevent the formation of groundloops, which can pose a significant measurement problem if left unaddressed.Ideal tunnelling noise spectra, both in and out of tunnelling, are shown in Figure2.2.10Newport11This controller runs National Instruments LabVIEW Real-Time OS and routes signalsthrough a National Instruments Field Programmable Gate Array (FPGA)Appendix B. Real Space Simulation of Multiple Scatterers 156Appendix BReal Space Simulation ofMultiple ScatterersScience is not built on success.It’s built on failure. It’s built onfrustration. Sometimes it’s builton catastrophe.Sumner RedstoneB.1 Using Bessel Functions to Simulate QPIDataThe T-matrix scattering calculations combined with set point theory does avery good job qualitatively reproducing the measurement mode dependent fea-ture seen in the experimental data. It does not, however, produce quantitativeagreement away from the surface state peak in the line cuts. To obtain furtherinformation one must take into account both the presence of multiple scatteringcentres within a single experimental image and also the different phase of eachscattering centre. One of the reasons the T-matrix theory fails to reproduce thefull QPI line shape is that it takes as its starting point only a single scatteringcentre, while the experimental data features many scattering sites.To verify that the differences between the theoretical and experimental FT-STS radially projected data could be accounted for by the presence of multiplescatterers a real space model was developed. This model does not have theability to simulate many-body effects, such as electron-phonon coupling [132],but in sacrificing this it is able simulate many scattering centres.Following the real-space fitting of QPI signals undertaken by Crommie et al.[154] the scattering off of both step edges and point defects (CO molecules) canbe performed using Bessel functionsρstep(E, x) = (1− J0(2k0x))L0 (B.1)ρCO(E, r) ∝ 1k0r[cos2(k0r − pi4+ δ)− cos2(kr − pi4)] (B.2)Appendix B. Real Space Simulation of Multiple Scatterers 157where δ is the phase shift discussed in Chapter 4 and L0 = m∗/pih¯2 is anapproximate form for the density of states of the surface state in the absence ofscattering.Figure B.1 shows a comparison at the Fermi level between an experimentalgrid and the Bessel function model. Unlike the T-matrix theory the real spacesimulation does a much better job of replicating the features of the FT-STSradial projection away from the surface state peak. This result establishes thegeneral shape of a QPI line cut on Ag(111) independent of the surface state andset point effects is determined by the spatial position of the scattering centresand the image analysis used to correct for step edges and z drift.(a) (b)0 0.05 0.1 0.15 0.2 0.25 0.3 0.3500.511.52 (c)qr (A˚−1)|S(qr,E=EF)|  DataS imu lationFigure B.1: Spectrscopic grid simulated in real-space by Bessel func-tions. (a) Experimental measurement of the differential tunnellingconductance at EF from a spectroscopic grid with measurementconditions: Vs = −100 mV, Is = 100 pA, Vb = (−100, 120) mV,239× 239 nm2 with 380× 380 pixels. (b) Simulation of the exper-imental data using Equation B.2 and identifying the pixels corre-sponding to the centre of each CO. (c) FT-STS scattering intensityof both experiment and simulation projected onto the qr axis. Thesimulation results deviate from the experiment at low-qr, possiblyindicating the need to add a degree of noise to the simulation data.Appendix C. A Scheme for Pulsed ESR-STM 158Appendix CA Scheme for PulsedESR-STMUpward, not Northward.Edwin A. Abbott - 1884 [2]C.1 Development of a Pulsed ESR-STMTechniqueThe ESR-STM results presented in Part III were achieved using the continuouswave (CW) mode of ESR-STM described in Chapter 6. This technique hasproven extremely fruitful in achieving spin resonance, providing a new STMacquisition mode with unprecedented energy resolution, and developing a newform of magnetometry. By sweeping the microwave frequency over a single atomthe resonance frequency can be measured via the SP-STM tip as the tunnellingmagnetoresistance changes.One weakness of CW ESR-STM is its inability to coherently manipulate thequantum spin of an atom in a way that would allow for applications in eitherquantum computing or spintronics. This shortcoming stems from the ratiobetween the energy relaxation time (T1), the quantum phase coherence time(T2), and the Rabi flop time (TRabi). The energy relaxation time quantifiesthe time it takes for the resonant spin to return to the thermally distributedequilibrium population. This can be characterized by pump-probe spectroscopyfor Fe and is found to be T1 ≈ 100 µs [86]. The quantum phase coherence timedescribes the amount of time that a resonant spin maintains its phase, a crucialtime scale as it sets the decoherence time of spin-based qubits [210]. For ESR-STM on Fe the T2 can be deduced by fitting the ESR resonance peak width asa function of microwave drive amplitude, which gives a T2 = 100 ns [86]. TheRabi flop time is the time it takes to coherently reverse the magnetic state, andit depends on the strength of the driving field. For CW ESR-STM this time isTRabi = 1.2± 0.1 µs [86].In CW ESR-STM the T1 time is three orders of magnitude larger than the T2time. From results in traditional ESR [207], this implies that when on resonancethe spin is performing a random walk consisting of many periods of coherentevolution (each one lasting roughly one T2 period) over the course of a singleAppendix C. A Scheme for Pulsed ESR-STM 159T1 period. Though this is useful for observing resonance, it does not allow forcoherent control of the spin because the spin loses its phase orientation on tooshort a timescale compared to the manipulation time, which is characterized byTRabi.The three order of magnitude difference between TRabi and T2 means thatthe spin cannot be coherently driven hard enough to reverse the magnetizationbefore the phase is lost. One potential route to avoid this difficulty is a pulsedESR-STM set-up.C.1.1 Introduction to Pulsed ESRIn pulsed ESR, a microwave pulse is used to coherently manipulate spin centresusing either a single pulse or series of pulses on resonance. The differencebetween continuous wave resonance and pulsed resonance is illustrated in FigureC.1. In Figure C.1 (a)(i) the familiar microwave chopping scheme is shownfor CW-ESR. To understand how this affects the resonant spin the conceptof the Bloch sphere is introduced in (a) (ii). The Bloch sphere is a usefultheoretical concept that visually represents an arbitrary superposition state ofa spin or qubit [210]. When on resonance the spin wave function is a coherentsuperposition of two spin eigenstates|ψ〉 = α |0〉+ β |1〉 . (C.1)It is a requirement of quantum mechanics that |α|2 + |β|2 = 1 to ensure thatprobabilities sum to one. The conservation of probability allows the superposi-tion state prefactors to be rewritten in terms of spherical coordinates|ψ〉 = eiφ( cos(θ/2) |0〉+ sin(θ/2) |1〉 ). (C.2)This expression illustrates that the superposition state of the resonant spin canbe represented by a vector of unit length that lives on the unit sphere. Thissphere has the |0〉 state aligned with +z axis and the |1〉 state aligned with −zaxis.The Bloch sphere representation of a spin in CW ESR-STM is shown inFigure C.1 (a)(ii). The red arrow evolves coherently, with set phase, for the du-ration of one T2 period, before losing that phase and evolving coherently with anew phase. This is a visual representation of the random walk described in theprevious section. The optimal case for CW ESR-STM consists of the superpo-sition state being located in the x− y plane of the Bloch sphere, correspondingto an evenly weighted coherent state and leading to the increase in tunnellingcurrent on resonance observed in the experiment.Pulsed ESR, in contrast, is shown in Figure C.1 (b). In this measurementmode a microwave pulse is delivered at the resonant frequency that causes thatspin to precess. The spin is detected using a DC electrical probe which collapsesit’s state to either |0〉 or |1〉. It is then hit with another resonant RF pulseAppendix C. A Scheme for Pulsed ESR-STM 160(a) Continuous Wave ESRtimeΔI100.5timeVtip~5ms(b) Pulsed ESRVtippulse Δtdetectpulse Δtdetect…pulse ΔtdetecttimePulse width ΔtΔI100.5Rabi oscillation (i) (ii) (iii)(i) (ii) (iii)|1i|1i|0i|0iFigure C.1: Continuous wave versus pulsed ESR-STM (a) Continuouswave ESR-STM uses (i) a chopped microwave frequency sweep to(ii) randomly walk the spin state along the Bloch sphere, resultingin (iii) an increase in the tunnel current. (b) Pulsed ESR-STM usesa (i) series of resonant pulses to (ii) coherently manipulate the spinstate around the Bloch sphere, resulting in (iii) observation of Rabioscillations in the tunnelling current signal.Appendix C. A Scheme for Pulsed ESR-STM 161that restarts the coherent evolution. Instead of a continuous increase in thetunnelling current signal on resonance the observed signal instead correspondsto a Rabi oscillation, as the coherent superposition moves from a completely |0〉state to a |1〉 state and back (called a Rabi flop) [210].Pulsed ESR-STM experiments did not show signatures of the expected Rabioscillations. Instead only a small change in tunnelling magnetoresistance wasobserved. The lack of Rabi oscillations can be attributed to the small T2 time,in particular when compared to the time needed for the microwave pulse toperform a full coherent rotation of the spin state (the Rabi rate Ω). This posesa technical problem, as the Rabi flop rate can only be increased by increasing themagnitude of the driving pulse Vµ, and this cannot be increased significantlywithout exceeding the 14 meV excitation threshold to state |2〉. Thus, theonly way to achieve fully coherent control is via increasing the T2 time, whichnecessitates better isolation of the spin from its environment.One method of increasing the T2 time is by engineering the spin environment.This has been used to successfully optimize the T1 time [230], resulting in a longand tunable lifetime of the excited spin. One potential method is to examineFe atoms on thicker MgO thin films, such as three or four monolayer films, asthicker films should have a smaller scattering rate from conduction electrons.Thicker films have the disadvantage that it increases the tunnelling junctionresistance, meaning that the tip must move closer the the Fe atom in order toachieve the same tunnelling current. This potentially causes a large relaxationrate due to scattering of the spin state by tunnelling electrons.Interaction with the tunnelling electrons is the limiting factor that deter-mines T2, as the limit imposed by substrate electron scattering is greater than100 ns [230]. This leads to a proposal for a pulsed ESR scheme that works inconjunction with oscillations of the tip. This method, as detailed in the nextsection, has the potential to increase the quantum phase coherence time enoughto observe a Rabi oscillation.C.1.2 A Scheme for Tip-Oscillating Pulsed (TOP)ESR-STMThe number of tunnelling current electrons passing through the Fe atom de-pends exponentially on the tip–sample vacuum gap, as discussed in Chapter 2.Withdrawing the tip on the order of a nanometre therefore causes a drop in thenumber of tunnelling electrons probing the Fe electronic spin, and reduces theprobability of a tunnelling electron spin scattering from the Fe spin and causinga loss of phase coherence. At a tunnelling current of 1 pA, roughly 10 milliontunnelling electrons cross the tunnel barrier every second. If the tip is retractedby 5 Angstroms than the tunnelling current falls by roughly e−5 to roughly 1fA and the number of tunnelling electrons that can perturb the spin state of theatom falls by three orders of magnitude. Unfortunately, the tunnelling electronsare needed to read out the spin state, and withdrawing the tip causes a loss ofthe read-out signal. By oscillating the STM tip in phase with the microwavepulsed sequence a compromise can be made between minimizing the interactionAppendix C. A Scheme for Pulsed ESR-STM 162of the spin via tunnelling electrons and allowing high enough tunnelling currentsto ensure read-out of the spin state.The set-up for this measurement scheme, dubbed Tip-Oscillating Pulsed(TOP) ESR-STM, is shown in Figure C.2. The tip height is modulated at afrequency from 10−25 kHz by applying an AC voltage to the piezomotor holdingthe tip. The ESR microwave pulse that sets the spin into resonance is deliveredwhen the tip is at the maximum distance away from the Fe atom. This setsthe Fe atom spin into resonance when the tunnelling current is at a minimum,making it the least likely that tunnelling electrons scatter the spin and cause lossof phase. At the point of closest tip–atom approach a DC electrical bias probeis applied to the tip which causes a large tunnelling current to read-out the spinstate. This likely causes loss of phase coherence but only after a period of phaseevolution free of tunnelling electron induced phase loss. The Fe spin state isthen reset to a known initial configuration using a DC pump voltage. The entiresequence is repeated many times and demodulated by the lock-in amplifier at 95Hz. Every half-period of the lock-in amplifier frequency the microwave triggeris turned off, in order to measure the change in tunnelling current caused bythe microwave excitation, as in CW ERS-STM. The lock-in amplifier measuresthe tunnelling current at the chopping frequency, detecting the spin state ofthe Fe atom. In addition to limiting the possibility of tunnelling electron phasedecoherence this method also has the advantage that the microwave power thatcan be delivered is much higher, as the DC bias is 0 meV when the microwavepulse is applied and the tunnelling electrons are exponentially suppressed as thetip moves away from the sample surface.C.1.3 Benchmarking the TOP ESR-STM TechniqueIn order to successfully measure a Rabi oscillation and achieve coherent controlover the Fe spin state the TOP ESR-STM technique had to be optimized forthe 1K-STM. This involved the following series of steps:• Calibration of the applied AC piezomotor voltage used to oscillate the tipagainst the physical amplitude of tip oscillations observed.• Characterization of the magnitude of the tip oscillations as a function oftip shaking frequency.• Testing whether the tip can still image in topography mode while the tipis oscillating.• Tuning experimental parameters and phases to optimize the T2 period.TOP ESR-STM begins by using continuous wave ESR-STM to find theresonance frequency of the Fe atom located under the STM tip. This Fe atom ischaracterized using pump-probe spectroscopy to ensure a T1 time on the orderof 100 µs. The T1 time can be optimized by careful control of the tip–sample gapand tunnelling current set-point [233]. Before attempting any tip oscillationsAppendix C. A Scheme for Pulsed ESR-STM 163Figure C.2: Tip Oscillating Scheme (a) A single set of TOP ESR-STM sig-nals over one period of the tip oscillation. The Vµ pulse is triggeredat the maximum tip sample separation and VDC is used for ini-tialization of the spin and read-out using a positive polarity DCbias pump. (b) Many tip shaking cycles occur during one cycle ofthe lock-in measurement. The lock-in measurement compares theaverage tunnelling current during sequences with and without thespin-manipulation microwave pulses.Appendix C. A Scheme for Pulsed ESR-STM 164on top of the target Fe It(z) spectroscopy is used to characterize the relationbetween the tip height and tunnelling current. It(z) spectroscopy is similar inmethodology to the STS point spectroscopy introduced in Chapter 2. insteadof sweeping the bias voltage the tip height is adjusted via a linear voltage rampapplied to the z piezo motor. As shown in Chapter 2, It(z) ∝ e−κz, whichmeans that a linear ramp of z while measuring the tunnelling current allowsfor extraction of κ and a conversion from height in A˚ to pA. The change inheight must be measured with respect to a reference as the absolute size of thetip-sample gap is not known. In TOP ESR-STM that reference is chosen to bethe height at which the set-point condition is met by the feedback, typically atvalues of It = 1 pA and Vs = 5 mV. Measured It(z) spectroscopy over an Featom is shown in Figure C.3 (a) on a semi-log plot. An accompanying linear fitallows for extraction of κ, which is equal to 1.704 A˚−1.Tip oscillation at a single frequency can be characterized in a similar wayto It(z) spectroscopy, except in this case the feedback remains engaged. Bydelivering an AC voltage to the tip piezo motor and ramping the magnitudeof this voltage while recording the tunnelling current a relation between It andthe excitation voltage can be established. The excitation voltage applied can beconverted into an effective change in height of the tip using the κ factor extractedfrom conventional It(z) spectroscopy. An example of the measured tunnellingcurrent as a function of the excitation (converted to A˚) is shown in Figure C.3(b) at a frequency of fz = 17241 Hz. The DC tunnelling current increases asa function of increased excitation amplitude because the tunnelling current issampling smaller tunnelling gaps, which correspond to higher tunnelling currentsat a fixed bias.−1 0 110−2100102I t = e−κz, κ = 1.704(a)h (A˚)I t(nA)0 0.5 1 1.500.20.40.60.8fz = 17241 Hz(b)Excitat ion (A˚)I t(nA)Figure C.3: Characterization of the tip modulation amplitude. (a) It(z)spectroscopy can be used to fit for the κ factor at a particular point,giving a relation between the tunnelling current and the relative tip–sample gap. (b) The tunnelling current is measured as a functionof tip modulation amplitude.Appendix C. A Scheme for Pulsed ESR-STM 165In order to successfully perform the TOP ESR-STM technique the oscilla-tion magnitude of the tip must be stable. This is not trivial, as unlike atomicforce microscopy instruments, STMs are not optimized for stable mechanicaloscillation of the tip–sample junction at kHz frequencies. To find a stable set offrequencies that demonstrates a flat response as a function of constant AC volt-age applied to the tip piezo the frequency (fz) was swept while the tunnellingcurrent was recorded. Figure C.4 illustrates the response of the feedback loopunder constant-current feedback as a function of modulation frequency deliveredto the z piezo motor at constant power. The degree of modulation demonstratesa non-monotonic response due to frequency-dependent mechanical resonancesbeing excited in the STM head. The frequency sweep characterization allowsfor identification of an area with a relatively flat response, centred at 17241 Hz.The relation between the applied shaking voltage and the change in tip heightis deduced by the previously measured tunnelling current spectroscopy curve atthis frequency shown in Figure C.3.A major milestone in establishing the feasibility of the TOP ESR-STM modeis the ability to image individual Fe atoms on the MgO surface while oscillatingthe tip at kHz frequencies. The inset of Figure C.4 shows STM topography of anFe atom while the tip is being modulated at 17241 Hz with a 2 A˚ peak-to-peakamplitude. This demonstrates a proof of concept for one of the largest technicalhurdles to the TOP ESR-STM method, and marks the first time topographyhas been measured with the 1-K STM using this technique.C.2 ConclusionA scheme for performing successful pulsed ESR-STM was introduced. TOPESR-STM should be able to measure Rabi oscillations of an Fe spin by deliver-ing a kHz AC voltage to the tip piezomotor and synchronizing the tip motionwith microwave excitation and pump-probe readout of the spin state. A mile-stone was achieved in benchmarking this new technique by demonstrating thatit is possible to perform stable STM topography measurements over single Featoms while oscillating the tip at over 17 kHz. Additional work to ensure phasesynchronization between the microwave pulse sequences, DC pump-probe sig-nals, and tip oscillation is needed to ensure the best possible chance of measuringa coherent Rabi oscillation of the Fe spin.Appendix C. A Scheme for Pulsed ESR-STM 16615 15.5 16 16.5 17 17.5 18−0.200.20.40.60.811.2fz = 17241 HzFrequency (kHz)TipHeight(A˚)XY  20 40 60 8010203040506070802323.223.423.623.82424.2Figure C.4: Tip response as a function of modulation frequency. Arelatively stable area, away from mechanical resonances, is foundat 17241 Hz and so this frequency is chosen for TOP ESR-STM.The inset is a STM topograph of an Fe atom taken while the STMtip is being modulated with a peak-to-peak height of 2 A˚. Imagingconditions are Is = 25 pA, Vb = 100 mV, and 30× 30 A˚.

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