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Probing many-body scattering in Cu(111) via FT-STS : understanding local perturbations from the collective… Farahi, Gelareh 2017

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Probing many-body scattering in Cu(111) viaFT-STSUnderstanding local perturbations from the collectivesignatures of a 2D electron gasbyGelareh FarahiB.Sc. Honours Physics, University of British Columbia, 2015a thesis submitted in partial fulfillmentof the requirements for the degree ofMaster of Scienceinthe faculty of graduate and postdoctoralstudies(Physics)The University of British Columbia(Vancouver)August 2017© Gelareh Farahi, 2017AbstractSurface states of close-packed noble metals form an approximate two-dimensionalelectron gas whose many-body signatures can be locally probed using ascanning tunneling microscope (STM). In this work I present a study ofthe Cu(111) surface state with high-resolution Fourier Transform ScanningTunneling Spectroscopy (FT-STS), and for the first time demonstrate thatthe energy dispersion and quasi-particle lifetime of the surface states can beaccurately quantified in both the occupied and unoccupied states. The scat-tering phase-shift imposed by defect potentials is then extracted in Fourierspace, which is consistent with previous real space analyses. This result islater used in the T-matrix simulation of the density of states that gives anaccurate description of our data. Finally, I report that in dilute Co/Cu(111)where the absence of time-reversal symmetry allows for spin-flip scattering,spin-conserving scattering dominates the FT-STS signal.iiLay SummaryOwing to their wave-like properties, electrons on surfaces of materials gen-erate standing waves by scattering from defects and impurities, where theirstates before and after scattering lead to an interference pattern resemblingthat of water waves. Because of the many-body nature of the electronic in-teractions, quantifying the collective signatures of electrons is a fundamen-tal challenge in condensed matter physics. Nonetheless metals like Coppercan host electronic states that exhibit a free-particle-like behaviour on theirsurfaces, which are confined to two-dimensions (2D) up to a good approx-imation. This 2D electron gas hosts one of the most simple scenarios ofmany-body interactions, turning it into an ideal test-bed to examine thepredictions of quantum mechanics. In this work we show that the collec-tive properties of a 2D electron gas can be experimentally measured with ascanning tunneling microscope, and successfully modelled within a relativelysimple scattering theory referred to as the T-matrix formalism.iiiPrefaceThis thesis is original, unpublished and indepenent work by the author, G.Farahi.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Experimental Techniques . . . . . . . . . . . . . . . . . . . . 52.1 The Working Principle . . . . . . . . . . . . . . . . . . . . . . 62.2 Scanning Tunneling Spectroscopy . . . . . . . . . . . . . . . . 82.3 Instrumentation: The Createc STM/AFM . . . . . . . . . . . 92.3.1 The scanning head: Besocke style head design . . . . . 112.3.2 Upgrade to the new head . . . . . . . . . . . . . . . . 112.4 Methods in Spectroscopy . . . . . . . . . . . . . . . . . . . . 142.4.1 Lock-in Technique . . . . . . . . . . . . . . . . . . . . 142.4.2 Numerical dIdV . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 FT-STS analysis of a 2D electron gas . . . . . . . . . 19v3 Characterizing many-body effects in Cu(111) via FT-STS 213.1 The Cu(111) Surface State . . . . . . . . . . . . . . . . . . . . 213.2 Cu(111) Surface State in The Presence of Non-Magnetic Defects 253.2.1 Finding peak positions . . . . . . . . . . . . . . . . . . 283.2.2 The dispersion of the scattering vectors . . . . . . . . 313.2.3 Self-energies . . . . . . . . . . . . . . . . . . . . . . . . 363.2.4 The phase-shift . . . . . . . . . . . . . . . . . . . . . . 383.3 Absence of Spin-Flip Scattering in Dilute Co/Cu(111) . . . . 424 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . 45Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47viList of TablesTable 3.1 Optimized parameters obtained from parabolic (PB) andtight-binding (TB) fits to the whole range of the bandshown in figure 3.8. Experimental uncertainties have notbeen included in the fit uncertainties, but fit uncertaintiesare reported for comparison between the two models. . . . 33Table 3.2 Optimized parameters obtained from parabolic and tight-binding fits to the grid shown in figure 3.10. Extractedchemical potential and effective mass values deviate fromthose obtained in the fits to the whole band shown in table3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35viiList of FiguresFigure 1.1 (a) The first observation of Friedel oscillations in MR re-sponse of a magnetic-transition metal multi-layer material[17].(b) Friedel oscillation probed with an STM. The tip heightprofile around a CO molecule is shown. The decay rateis fitted assuming a 2DEG. (c) Constant current map ofCu(111) showing oscillations in two dimensions[5].(d) FT-STM of Cu(111) showing the integrated DOS to Fermilevel. Radius of the ring is 2k f [16]. . . . . . . . . . . . . . 4Figure 2.1 Gerd Binning(left) and Heinrich Rohrer(right), the devel-opers of the first STM. Source: IBM. . . . . . . . . . . . . 5Figure 2.2 (a) Schematic of a Scanning Tunneling Microscope. Ametallic tip and its trajectory - controlled by a feedbackloop- shown in blue. As the tip scans the surface, thefeedback loop adjusts the height to maintain the set-pointtunneling current. (b) A topography map of Cu(111) withatomic resolution, showing the closed-pack structure ofthis crystal orientation (Vbias = 10mV, It = 24nA). . . . . . 7viiiFigure 2.3 (a) Createc’s Besocke style head, suspended from stain-less steel springs for enhanced vibration isolation. Blocksof magnets attached to the bottom of the head (not vis-ible in this picture) facilitate the magnetic damping ofvibrations. (b) STM head, showing the main piezo usedfor scanning (labelled MP),the three coarse motion piezos(CP) with sapphire balls epoxied on top, allowing for slip-stick motion on the copper ramp (R). The sample andSTM tip are absent from this image. . . . . . . . . . . . . 10Figure 2.4 Createc’s old STM head. Left panel shows the infamous”nest” of wires that complicated fixing the shorts. Theright panel shows the piezo tubes used for scanning, wherediscolored patches are likely caused by high voltage arcs. 12Figure 2.5 Tunneling noise improvement after the upgrade of theSTM head. Obtained with It = 100 pA, on Cu(111) sur-face. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 2.6 dI/dV point spectrum acquired on bare Cu(111). Ran-dom noise is reduced with a Gaussian filter using s=3. . 17Figure 2.7 Illustration of the relationship between momentum andscattering vectors in a single parabolic band. . . . . . . . 19Figure 2.8 (a) Constant energy cuts of dI/dV spectra, showing QPIpatterns as rings centered around defects. (b) The samemaps in Fourier space. (c) The line-cuts obtained from band, (d) the stack of line-cuts in a high resolution grid,mapping the dispersion in q-space. The dots correspondto the two energy levels shown in (a)-(d). . . . . . . . . . 20ixFigure 3.1 (a) The bulk and the surface state band structure ofCu(111) measured with ARPES[10]. (b) Left: one of theearliest FT-STM analyses on Cu(111) showing the inte-grated density of states up to the Fermi level[16]. Right:A recent FT-STS work on Ag(111) with enhanced en-ergy and momentum resolution. The ring corresponds toE=E f [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 3.2 ARPES-resolved SO-splitting of the Cu(111) surface state[20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 3.3 The allowed scattering states of the Shockley surface statewhen the TR-invariance is broken. q3 is the spin-conservingscattering signal. . . . . . . . . . . . . . . . . . . . . . . . 24Figure 3.4 Averaged point spectroscopy over a 245×245 nm2 area.The surface state onset of Cu(111) leads to a sudden in-crease in the signal at 440 mV, which is the onset of thesurface state. The signal strength decays in higher ener-gies, requiring higher setpoint currents for resolving fea-tures near the Fermi level. . . . . . . . . . . . . . . . . . . 26Figure 3.5 The Cu(111) surface state. (a) The raw dI/dV map show-ing the LDOS in real space, (b) Raw dI/dV zoomed on asmaller area, showing the QPI pattern from point defectsas rings. (c) The Fourier transform of the larger dI/dVmap and, (d) The line-cut showing the LDOS at the Fermienergy in q-space. . . . . . . . . . . . . . . . . . . . . . . 27Figure 3.6 T-matrix fit to constant energy line-cuts at (a) near theband onset, (b) below Fermi, (c) at Fermi and (d) aboveFermi level. As the applied bias goes further away fromthe surface onset, LDOS peaks appear in the high-frequencyrange of background where it can be approximated by aconstant. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30xFigure 3.7 An intensity map of the dispersion obtained directly fromQPI rings formed at each energy with 3mV energy reso-lution. Bottom arrow points at the bound state below thesurface state onset which can be seen as a slight increasein intensity below the band. Top arrow is pointing at thee-ph kink near Fermi, seen as an increase intensity in theDebye energy range of ±27 mV. . . . . . . . . . . . . . . . 31Figure 3.8 The parabolic and tight-binding fits to the dispersion fitsthe middle of the band quite well, and the deviation inthe upper and lower limits is likely caused by large uncer-tainties in peak positions at low energies. . . . . . . . . . 33Figure 3.9 An intensity map of the dispersion directly from QPI ringsformed at each energy. The significant increase in inten-sity of the LDOS is due to e-ph interaction that renor-malizes the spectrum in the Debye energy range. . . . . . 34Figure 3.10 Parabolic fit to the dispersion. Data points are peak po-sitions obtained from the T-matrix fit. . . . . . . . . . . . 35Figure 3.11 Extracted self-energies directly from TMatrix Fit (a)-(d)and after a parabolic fit to the peak positions (e). De-viations in quasi-particle life-time and the real part ofself energy are clearly extract in the Debye energy range,shown with dashed vertical lines. . . . . . . . . . . . . . 37Figure 3.12 The energy dependence of the quasi-particle lifetime de-viates from the expected parabolic trend. . . . . . . . . . 38Figure 3.13 The scattering phase-shift obtained from the fit. The uni-tary limit corresponds to the strong scattering scenariowith d = p=2. . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 3.14 (a) FT-STS extracted dispersion of Cu(111) (b) Simula-tion of the density of states for three different phases. . . 41Figure 3.15 The Kondo effect seen in the STS of an individual Coatom on Cu(111), showing a decrease in the LDOS nearthe Fermi level. . . . . . . . . . . . . . . . . . . . . . . . . 42xiFigure 3.16 (a) Left panel is the topography of Co/Cu(111) with highdefect density. Due to the mobility of Cobalt atoms dur-ing grids, they do not image perfectly in their sphericals-state. Right panel is the dI/dV map of the same area atFermi. (b) Comparing line-cuts of Co/Cu(111) with bareCu(111). In the presence of Cobalt impurities, only thespin-conserving scattering vector (q3) can be accessed viaFT-STS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44xiiChapter 1MotivationFrom the advent of the first scanning tunneling microscope, the capabili-ties of this measuring technique have expanded vastly from a local probeof the topography of surfaces and defects to a precise technique mappingout the band’s dispersion in constant-energy maps of the density of states.The STM’s ability to manipulate a metallic tip with Ångstrom precisionhas realized the study of surface states by exploiting the quantum tunnelingphenomenon. This in turn allows for locally probing quasi-particle inter-ference (QPI) that results from the wave-like nature of electrons scatteringfrom defects, generating standing waves on the surface.Friedel was the first to formulate the signature of inhomogeneities inmetals[6] which result in asymptotically decaying oscillations at the Fermilevel. The damping factor is proportional to 1=rd , where r is distance fromthe scattering center (i.e. the defect), and d is the dimension of the electrongas.It took more than three decades to directly observe Friedel oscillationsin magnetoresistance (MR) of Co/Cu and Fe/Cu interlayers. With increas-ing Cu thickness, the MR response was sinusoidal and decaying rapidly[17]as shown in figure 1.1(a). It was only two years later when Crommie etal.[5] resolved Friedel oscillations in Cu(111) in real space using a scanningtunneling microscope(see figure 1.1(b) and (c)). The observed decay rate ofscattering amplitude was simulated by scattering of a two dimensional elec-1tron gas(2DEG), indicating that scattering to the bulk is minimal. One ofthe first characterizations of the surface state in Fourier space was done byPeterson et al.[16] via Fourier Transform Scanning Tunneling Microscopy(FT-STM), in which the real-space standing wave patterns were Fouriertransformed.FT-STM however is only capable of extracting the integrated density ofstates from the onset of the energy band to the set-point bias voltage. Inmore recent years, Fourier Transform Scanning Tunneling Spectroscopy (FT-STS) was developed in order to probe the local density of states (LDOS) atenergy levels below and above the Fermi energy. In FT-STS the derivativeof current (the differential conductance or briefly the dI/dV) is acquiredand Fourier-transformed at each energy in order to determine the LDOS inscattering space.Recent FT-STM studies have mostly focused on more complicated sys-tems in an effort to study superconductors [19] and topological insulators(e.g. [8]). Nonetheless, simple free-electron like behaviour in (111) ter-minated surfaces of noble metals provides a relatively simple test-bed fordeveloping and expanding this novel spectroscopic technique. S. Grothe etal[7] in 2013 were able to optimize the FT-STS energy resolution so that itwas limited by the thermal broadening of electronic states. They enhancedthe momentum resolution to nearly thousandths of Å−1 in Ag(111). LaterA. J. Macdonald et al.[13] showed that even in these simple systems thereare secondary features in Fourier space that are a consequence of the mea-surement technique, and can be overcome by measuring at biases lower thanthe surface state onset.In this work, many-body effects in Cu(111) have been studied with high-resolution FT-STS, in the presence of intrinsic defects and Cobalt adatoms.With its sensitivity to local effects, we will show that FT-STS is capableof capturing many-body effects by measuring the self-energies at energiesabove and below the Fermi level, from which the electron-defect scatteringcan be characterized. Chapter 2 presents an overview of the measurementtechnique, the instrument and the Fourier analysis performed on the data.Results are then presented in chapter 3, in which self-energies and the scat-2tering phase-shift of the quasi-particles are extracted. Finally in chapter4 I summarize the main results and the outlook for future theoretical andexperimental efforts.3Figure 1.1: (a) The first observation of Friedel oscillations in MR re-sponse of a magnetic-transition metal multi-layer material[17].(b) Friedel oscillation probed with an STM. The tip height pro-file around a CO molecule is shown. The decay rate is fittedassuming a 2DEG. (c) Constant current map of Cu(111) show-ing oscillations in two dimensions[5].(d) FT-STM of Cu(111)showing the integrated DOS to Fermi level. Radius of the ringis 2k f [16].4Chapter 2Experimental TechniquesThe scanning tunneling microscope was invented by Binning and Rohrer [2],later bringing them a Nobel Prize in Physics in 1986. The operation of anSTM is based on quantum tunneling through a vacuum barrier, enabling itto probe the electronic properties of pristine surfaces on the atomic scale.Figure 2.1: Gerd Binning(left) and Heinrich Rohrer(right), the devel-opers of the first STM. Source: IBM.5In this chapter I will present a brief description of the components ofa typical STM, and introduce the instrument that was used in this work.In section 2.2, I will review the theory that draws connections between thetunneling current and the local density of states, and in 2.4 elaborate on thedata acquisition techniques used to extract the LDOS.2.1 The Working PrincipleIn scanning tunneling microscopy (STM) a sharp metallic tip is biased withrespect to the sample(figure 2.2(a)). Depending on the sign of the biasvoltage, a tunneling current flows from the tip to the sample or vice versa.The motion of the tip is controlled by applying high voltages to polarizedpiezo-electric tubes, enabling them to move with sub-nanometer precision inX,Y (in plane), and Z (the tip-sample separation) directions. At a constantsetpoint bias, a feedback loop adjusts Z until the setpoint value of the currentis achieved.A profile of the adjusted Z heights is then recorded, which encapsulatestopographic features of the surface such as the atomic configuration2.2(b),impurity concentration, and crystal defects such as screw dislocations aswell as step edges in the topography maps[4].While STM topography data contains useful information about sur-face structure and integrity, and defect concentration, scanning tunnelingspectroscopy reveals information about the LDOS in a range of energies.The next section explains the theory of STS.6(a)(b)Figure 2.2: (a) Schematic of a Scanning Tunneling Microscope. Ametallic tip and its trajectory - controlled by a feedback loop-shown in blue. As the tip scans the surface, the feedbackloop adjusts the height to maintain the set-point tunneling cur-rent. (b) A topography map of Cu(111) with atomic resolu-tion, showing the closed-pack structure of this crystal orienta-tion (Vbias = 10mV, It = 24nA).72.2 Scanning Tunneling SpectroscopyThe Bardeen theory of tunneling [4] explains the tunneling phenomenon -i.e.penetration of quantum particles through potential barriers- by consideringa one-dimensional tunneling barrier between the tip and sample stationarystates, whose spatial components are labelled yi and c j respectively. Thetunneling matrix element Mi j that is a measure of the overlap between thesestates is formulated as:Mi j =h¯22m∫z=z0[yi¶c j∗¶ z−c j∗ ¶yi¶ z]dxdy; (2.1)with z0 being the tip-sample separation. Using the above equation the tun-neling current at each point is:It(x;y;V ) =4peh¯∫ +¥−¥[ f (EF − eV + e)− f (EF + e)]rs(x;y;E f − eV + e)×rt(E f + e)|M|2de;(2.2)where f is the Fermi-Dirac distribution function, rs and rt are the sampleand tip density of states respectively. eV is the difference between the Fermienergy of the tip and sample, and e is the additional energy term determinedby the applied bias. x and y represent spatial position of the tip on thesample. The implied assumptions to obtain the results above are as follows:• Tunneling is elastic, i.e. electrons can only tunnel into and from equal-energy states,• tunneling occurs only in the Z direction and,• the tip and sample wavefunctions are orthogonal.In the framework of the WKB approximation developed by Wentzel,Brillouin and Kramers in 1926[4], the |M|2 term can be replaced by a trape-zoidal transmission coefficient T :T (e;V;z) = exp[−2z√2mh¯2(ϕ +eV2− e))]; (2.3)8with ϕ being the surface workfunction often taken to be an average of thesample and the tip workfunctions.In order to have access to the physics underlying STS measurements,more simplifications are needed to facilitate the interpretation of the mea-sured tunneling current:• Within the applied bias range the transmission coefficient remains con-stant: this assumption holds as long as the probed bias range does notexceed hundreds of mV.• A perfect metallic tip with a flat density of states is used: tip integritycan be experimentally confirmed by acquiring dI/dV curves on thebare metallic sample.• The Fermi-Dirac distribution is a step function: this assumptions holdsin our experiments where temperature does not exceed 5 K.Eventually one can further simplify equation 2.2 toIt µ∫ eV0rs(E f − eV + e)de: (2.4)Therefore the derivative of the tunneling current with respect to energyto first order is proportional to the local density of states of the sample:dItde(e = eVbias) µ rs(E f − eV + eVbias): (2.5)In the following sections, the terms bias, energy and V are used inter-changeably in order to refer to sample’s bias voltage.2.3 Instrumentation: The Createc STM/AFMThe commercial Createc STM/AFM instruments operate at liquid heliumtemperature (4.5 - 5 K) with more than 80 hours of hold-time, during whichinstrument can remain at its base temperature without re-filling the cryostatthat contains cryogenic Helium and Nitrogen to keep the STM cold. Thefollowing section describes the scanning head whose motion is controlled bythe Nanonis SPM Controller software and electronics.9Figure 2.3: (a) Createc’s Besocke style head, suspended from stainless steel springs for enhanced vibrationisolation. Blocks of magnets attached to the bottom of the head (not visible in this picture) facilitatethe magnetic damping of vibrations. (b) STM head, showing the main piezo used for scanning (labelledMP),the three coarse motion piezos (CP) with sapphire balls epoxied on top, allowing for slip-stickmotion on the copper ramp (R). The sample and STM tip are absent from this image.102.3.1 The scanning head: Besocke style head designK. Besocke first patented the design currently referred to as Besocke styleSTM head in 1987[1]. The modern design consists of three outer (coarse)piezos and a center (main) piezo (see figure 2.3(a)) that accommodates thetip. The outer piezos are mainly used for coarse motion in X, Y, and Zdirections in micrometer scales. Sapphire balls epoxied to the top of thesepiezos facilitate the stick-slip motion on a copper ramp that gives 0.6 mmrange in Z motion during approach to the sample (figure 2.3(b)). Moredetails about the instrument can be found in chapter 2 in [12].The Nanonis SPM Controller allows for two scanning modes with sub-nanometer resolution:• The main piezo is used for scanning in three directions,• The outer piezos are used for fine X,Y motion with the main piezocontrolling the Z height.2.3.2 Upgrade to the new headWith more than a decade of working lifetime, the old STM head (figure2.4)had to be replaced due to chronic appearance of electrical shorts, partialdepolarization of the piezos resulting from high voltage shocks, and possiblepiezo deformations after migration to its current location in the basement ofthe laboratory. The new head is capable of performing STM measurementswith a baseline current noise lower by a factor of three (see figure 2.5), as wellas atomic force microscopy (AFM). The AFM capability was not exploredin this work and is left to be characterized in the future.11Figure 2.4: Createc’s old STM head. Left panel shows the infamous”nest” of wires that complicated fixing the shorts. The rightpanel shows the piezo tubes used for scanning, where discoloredpatches are likely caused by high voltage arcs.12Figure 2.5: Tunneling noise improvement after the upgrade of theSTM head. Obtained with It = 100 pA, on Cu(111) surface.132.4 Methods in Spectroscopy2.4.1 Lock-in TechniqueOne way of acquiring dI/dV spectra is through use of a lock-in amplifier.This section explains the basics of this technique in both point spectroscopyas well as in two dimensional maps.Point SpectroscopyOnce the tip is at the setpoint height -tip height above the surface deter-mined by setpoint bias and current- on top of the point of interest, thetip-sample bias can be modified with a small modulation Vmod sin(wt+ql),where Vmod is a few percent of Vbias. This results in the tunneling currentbeing modulated sinusoidally, leading to a total current of:Itotalt = I(Vbias+Vmod sin(wt+ql))+ Inoise(t): (2.6)The first term in equation 2.6 can be expanded:I(Vbias+Vmodsin(wt+ql)) = I(Vbias)+dIdV:Vmod sin(wt+ql)+ ::: (2.7)so the total tunneling current can be written as:Itotal(t)≃ I(Vbias)+ dIdV :Vmod sin(wt+ql)+ Inoise(t): (2.8)The current pre-amplifier (the first amplifier in the circuit that the signalpasses through) converts the total current into voltage. The first term inequation 2.8 can be referred to as VDC due to its time independence. Thesecond term in the above equation is proportional to the LDOS, as explainedin section 2.2, therefore it can be labelled Vsig sin(wt+ ql). Then the totalvoltage is[11]:V total(t)≃VDC+Vsig sin(wt+ql)+Vnoise(t): (2.9)14In order to extract the signal, the total voltage is multiplied by Vre f (t) =Vl sin(wt). The Vsig term in the above equation will be a multiplication oftwo sine functions, rewriting it in two cosine terms leads to:V total(t):Vre f (t)≃VDC:Vre f (t)+1=2Vsig:Vl cos((w−w)t+ql)+1=2Vsig:Vl cos((w+w)t+ql)+Vnoise(t):Vre f (t):(2.10)Assuming w is chosen such that it has minimum overlap with low-frequency or higher harmonic terms of the noise spectrum, one can writethe noise term as:åh ̸=wVh :sin(ht+ϕh): (2.11)Integrating over a long period of time, which is equivalent to applying alow-pass filter to the signal, we can average out all the terms except for thesecond one that does not have a time dependence:limt→¥∫ toV total(t):Vre f (t) = 1=2Vsig:Vl cos(ql) (2.12)The final step is to convert the integrated voltage back to current, theamplitude of which is proportional to the LDOS.2D Maps2D maps in lock-in mode are essentially a collection of closely spaced ac-quired dI/dV for a single energy (that is the set-point bias) on a larger patchof the surface. One important aspect of this measurement is whether thefeedback loop is on (constant current map) or off (constant height map).With the feedback loop on, the tip-sample distance is constantly adjustedthroughout the scan, keeping the setpoint current relatively constant. How-ever, it has been shown that in noble metals the feedback loop can generateartifacts [13] leading to the appearance of secondary features. This artifactdisappears in constant height maps.Constant height maps, on the other hand, increase the probability ofcrashing the tip on the surface, especially in long slow-speed lock-in mapswhere piezo drifts and sample tilt gain importance in the absence of the15feedback loop’s response.The advantage of the lock-in method is its high signal/noise ratio. Thedrawback is its lower energy resolution (to get a good signal Vmod ≥ 4 meV),as well as the long acquisition time required at each energy that is of theorder of hours per map for a 100×100 nm2 scan size.The numerical approach, even though with lower signal/noise ratio, canmap out hundreds of mV bias range with resolutions as low as the instru-ment’s thermal limit for scan sizes as large as 240× 240 nm2. The totalmeasurement time takes just less than 80 hours, which is the higher limit ofthe instrument’s hold-time. The next section elaborates on this techniqueand ways to optimize the signal/noise ratio.2.4.2 Numerical dIdVPoint SpectroscopyWhen the tip stablizes its height according to the set point bias (Vbias) andcurrent (It), the feedback loop turns off and a fast bias sweep (roughly 450 msper energy) starts. Each sweep consists of settling the height, and changingthe bias according to the user-defined energy resolution and measuring thecurrent as a function of that, from which I-V and subsequently dI/dV vs Vcurves can be obtained. One problem that arises from the fast acquisitionmode is its high random/Gaussian noise, which can be overcome by:• Adjusting the acquisition speed within the bandwidth of the currentamplifier,• performing multiple bias sweeps per pixel and take an average,• applying Gaussian smoothing of the I-V curves.In previous works[7], it has been shown that by carefully selecting the stan-dard deviation of Gaussian filtering, this method can maintain the energyresolution to the point that it can reach the thermal limit of the electronicstates. Figure2.6 shows the raw and smoothed dI/dV curves acquired on apoint on Cu(111).16Figure 2.6: dI/dV point spectrum acquired on bare Cu(111). Ran-dom noise is reduced with a Gaussian filter using s=3.17I-V GridsWith Nanonis’s “Pattern“ module one can collect I-V spectra in two dimen-sions. In this case, the scan window is divided into square shaped pixels,the size of which depends on the spatial resolution that can be adjusted inthe software. After starting the grid, the tip acquires an I-V curve for eachpixel, moves forward (e.g. in X) to the next pixel, settles the height (which isrecorded for the topography map) and starts another I-V acquisition. Thisprocedure continues until all the lines have been measured.In this case the data set is a three dimensional matrix for X and Ydirections as well as bias. The values of the elements are current (I) values,or its numerical derivative that can be obtained from I-V curves. Figure2.8(a)) shows constant energy cuts of dI/dV maps which can be extractedfrom the 3D matrix in grid experiments or collected individually with thelock-in technique.The spatial STS maps obtained with the above methods contain use-ful information regarding the band structure of the surface. Owing to theirwave-like nature, electrons scattering from defects generate quasi-particle in-terference patterns that can be characterized in Fourier Transform ScanningTunneling Spectroscopy[4]. The next section explains the FT-STS methodin the context of a parabolic band.182.4.3 FT-STS analysis of a 2D electron gasOne has to remember that the scattering vectors, labelled as q, are relatedto the momentum space band structure by:q⃗= k⃗ f − k⃗i; (2.13)that is the difference between the initial and final momenta. Following theirparabolic dispersion in momentum space, the dispersion of the surface stateof the (111) terminated noble metals in q space is also parabolic. It has beenestablished that due to nesting effects, backscattering is the most probablescattering process (figure 2.7), hence:|⃗q|= 2|⃗k|: (2.14)Figure 2.7: Illustration of the relationship between momentum andscattering vectors in a single parabolic band.Figure 2.8(b) shows the dI/dV maps of Cu(111) in scattering space, qxand qy, indicating an isotropic band structure that allows for taking theradial average of the intensity values for each map. Recording the radii in|q|=√qx2+qy2 and their associated intensities, a ”line-cut” for each energy(see figure 2.8(c)) is obtained that shows the position of the surface states aspeaks. In I-V grids where the energy resolution allows for collecting hundredsof these line-cuts within the band, one can stack the spectra and plot thepeaks in a color map (figure 2.8(d)), illustrating the energy dispersion inq-space.19(a) (b)(c) (d)Figure 2.8: (a) Constant energy cuts of dI/dV spectra, showing QPIpatterns as rings centered around defects. (b) The same mapsin Fourier space. (c) The line-cuts obtained from b and, (d)the stack of line-cuts in a high resolution grid, mapping thedispersion in q-space. The dots correspond to the two energylevels shown in (a)-(d).20Chapter 3Characterizing many-bodyeffects in Cu(111) viaFT-STS3.1 The Cu(111) Surface StateSimilar to the other (111) surfaces of noble metals, the surface state ofCu(111) exists in the gap at the G-point in the bulk. Hengsberger et al.[10]acquired the nearest bulk and surface band structure of Cu(111) with AngleResolved Photoemission Spectroscopy (ARPES) (see figure 3.1(a)). Scatter-ing to the bulk is sufficiently small that in the energy ranges less than a feweV [16] the surface state is believed to follow a simple free-electron modelwith Ek = h¯2k22m∗ − m. From most recent photoemission studies m∗ ≃ 0:42me,and m ≃ 437meV [20] which is the difference between the chemical potentialand the onset of the band.Later FT-STS studies of the system in its excited states showed thatat energies above an eV the dispersion most closely follows a tight-bindingmodel[3]:ETB(k) =−m+ g[3− cos(kya)−2cos(kya2 )cos(√3kxa2)]: (3.1)21a is the lattice constant, and g depends on the one-center and nearestneighbour matrix elements. The first FT-STS studies of noble metals,however, lack the momentum and energy resolution of those of ARPESmeasurement[3][16]. Hence until recently a direct comparison between FT-STS and ARPES resolved energy dispersions in these simple systems hadnot been attempted [7][13][18] (see figures3.1(b) left vs right).In 2013 Tamai et al.[20] were able to characterize the fine features in thedispersion of Cu(111) with (ARPES), by enhancing the energy of the photonbeam that influences momentum resolution of the ejected photo-electrons.In fact, they extracted the self-energies near the Fermi level, and resolvedthe spin-orbit splitting of the surface state (figure 3.2(b)).The spin-orbit splitting of the surface state arises from lack of inversionsymmetry on the surface, leading to an extra Rashba term in the surfacestate’s Hamiltonian[15]:HR =aRh¯(zˆ× p⃗):⃗s (3.2)Where zˆ is normal to the surface, p⃗ is the momentum and s⃗ is the Paulispin matrix vector. Adding the Rashba term to the free electron Hamiltoniangives rise to the two parabolas for spins up and down. Accessing both bandswith FT-STS may be possible by turning on a spin-flip scattering channelby breaking time-reversal(TR) invariance of the surface. Figure 3.3 showsthe allowed scattering vectors in the absence of TR-invariance.22(a)(b)Figure 3.1: (a) The bulk and the surface state band structure ofCu(111) measured with ARPES[10]. (b) Left: one of the earliestFT-STM analyses on Cu(111) showing the integrated density ofstates up to the Fermi level[16]. Right: A recent FT-STS workon Ag(111) with enhanced energy and momentum resolution.The ring corresponds to E=E f [7].23Figure 3.2: ARPES-resolved SO-splitting of the Cu(111) surface state[20].Figure 3.3: The allowed scattering states of the Shockley surface statewhen the TR-invariance is broken. q3 is the spin-conservingscattering signal.24Therefore resolving the splitting is possible if:• The required energy and momentum resolution is achieved to resolvethe splitting,• the time-reversal invariance of the system is broken, so that scatteringbetween the two parabolas will be allowed and,• the strength of the spin-flip scattering signal is comparable to thestrength of the spin-conserving scattering signal.In this chapter, we have studied the Cu(111) surface state with high-resolution FT-STS in an attempt to resolve the self-energies and the scat-tering phase-shift induced by scalar defects, and the spin-orbit splitting ofthe dispersion by depositing magnetic Cobalt atoms on the surface.3.2 Cu(111) Surface State in The Presence ofNon-Magnetic DefectsAll the measurements in these experiments took place at T=4.5 K and in ul-tra high vacuum with pressures less than 1×10−10 Torr. Sample preparationconsisted of cycles of Argon sputtering at room temperature and annealingat 550 ◦C, until a clean surface was obtained as measured by STM. Apartfrom CO molecules and other point-like defects that rise from the bulk dur-ing the anneals, the surface is home to extended defects such as step edgesand holes that lead to a background in Fourier space. In order to overcomethis effect these defects have been Gaussian filtered. Upper panel in fig-ure 3.5 shows a step edge and four elongated holes in addition to point-likescatterers whose QPI signal is of the interest of this study.The surface state onset of Cu(111) is identified by an abrupt increase indI/dV spectrum at E = m(see figure 3.4) that gradually decays further awayfrom the onset energy, a behaviour compatible with the negative slope ofthe dI/dV term calculated in [9].25Figure 3.4: Averaged point spectroscopy over a 245×245 nm2 area.The surface state onset of Cu(111) leads to a sudden increasein the signal at 440 mV, which is the onset of the surface state.The signal strength decays in higher energies, requiring highersetpoint currents for resolving features near the Fermi level.In order to avoid artifacts caused by stabilization bias in grids [13], thestabilization bias was set at voltages as low as -550 mV. Higher voltagesrequire a larger tip-sample separation in order to maintain the tunnelingcurrent, which in turn reduces the sensitivity of the STS signal to the elec-tronic states that decay exponentially away from the sample. This effect,along with the gradual decrease in the dI/dV signal away from the surfacestate onset energy were the limiting factors in resolving the electron-phononkink near the Fermi level, from which the electron-phonon contribution tothe self-energies can be obtained.dI/dV spectra in this experiment have been acquired with the numericalapproach with 8-10 bias sweeps per pixel. The average of the I-V curvesobtained from the bias sweeps have been Gaussian smoothed with s = 3 inorder to minimize noise due to fast data acquisition. In figure 3.5 the dI/dVmap at the Fermi energy, as well as its Fourier transform have been shown.The line-cut shown next to the Fourier ring is the result of averaging theintensity over the whole ring, as explained in chapter 2.26Figure 3.5: The Cu(111) surface state. (a) The raw dI/dV map show-ing the LDOS in real space, (b) Raw dI/dV zoomed on a smallerarea, showing the QPI pattern from point defects as rings. (c)The Fourier transform of the larger dI/dV map and, (d) Theline-cut showing the LDOS at the Fermi energy in q-space.273.2.1 Finding peak positionsS. Grothe et al.[7] were not successful in extracting peak positions using theT-matrix formalism, the theory that was incorporated in their simulations.With Cu(111) data however it is possible to fit a T-matrix modulated densityof states in q-space:dr =− 12p åk∈<BZ>Im[G(w ;⃗k)T (w)G(w ;⃗k+ q⃗)]: (3.3)The Green’s function G(w ;⃗k) has the following form:G(w ;⃗k) =1w−E(k)−SSE ; (3.4)where w is the difference between the Fermi energy and the applied bias.E(k) is the free electron spectrum:E(k) =h¯2k22m∗−m; (3.5)assuming an effective mass of m∗ = 0:42me and a chemical potential (m) of440 mV as reported in literature (e.g. [20]).SSE the self energy that has a real ( S′SE) and an imaginary part (S′′SE).The real part shifts the position of the peak, and the imaginary part deter-mines the LDOS line-width. The real part is a measure of the renormalizedeffective mass, and the imaginary part is sometimes referred to as the quasi-particle lifetime. More details on the extracted self-energies are presentedin section 3.2.3.The summation is over the first Brillouin zone in (−p/a, p/a) limit,assuming a square lattice with lattice constant of a = 2.55 Å. T (w) ≃−V0 sin(d )eid , is the strong scattering approximation of the T-matrix, withV0 being the scattering potential and d the scattering phase-shift. The scat-tering phase-shift determines the LDOS line-shape whose role is discussedin section 3.2.4.Fitting for SSE , V0 and d one can extract peak positions at each energy.Since background noise dominates in the signal away from the peak, only28data points in the vicinity of each peak have been considered in the fits(figure 3.6).At bias values below -420 mV, line-shapes are overly broad and immersedin the low-frequency side of the decaying background that could not beincluded in the fit. In higher biases in the range of -420 to -350 mV, fittingcan be achieved but the peaks are more distorted due to high backgroundin low-frequency regime. In this analysis a constant background term isconsidered in each fit, which does not represent background behaviour inlower frequencies, but regardless makes the analysis faster to run (by almosta factor of 10 at lower energies).To see the effect of the number of points and the background term in-cluded in the fit on the extracted peak positions, I tried to estimate the peakshifts at E =−419mV :• In five trials varying the number of points to the left and right of thepeak, the largest deviation was 0:15% of the mean of all trials, and thestandard deviation was nearly 0:075%,• Comparing one trial with a linear background term and the mean ofthe constant background trials, the difference was 1:2% of the meanof the constant background trials.Therefore the most significant contribution to the systematic uncertaintyat low energies comes from our estimate of the background term. Thiscontribution decreases for higher energies, e.g. the Fermi level:• In five trials varying the number of points to the left and right of thepeak, the largest deviation was 1:6% of the mean of all trials, and thestandard deviation was nearly 0:6%,• Comparing one trial with a linear background term and the mean ofthe constant background trials, the difference was only 0:1% of themean of the constant background trials.Which indicates that at higher energies the number of points included inthe fit gain importance.29(a) (b)(c) (d)Figure 3.6: T-matrix fit to constant energy line-cuts at (a) near theband onset, (b) below Fermi, (c) at Fermi and (d) above Fermilevel. As the applied bias goes further away from the surfaceonset, LDOS peaks appear in the high-frequency range of back-ground where it can be approximated by a constant.303.2.2 The dispersion of the scattering vectorsAfter finding peak positions we can fit for the dispersion of q-vectors. Botha parabolic (PB) and a tight-binding (TB) model [3] have been tested inour analysis, where the energy window between -27 to 27 mV was excludedin the fits in order to dismiss the contribution from e-ph interactions to thedispersion (figure 3.7).Figure 3.7: An intensity map of the dispersion obtained directly fromQPI rings formed at each energy with 3mV energy resolution.Bottom arrow points at the bound state below the surface stateonset which can be seen as a slight increase in intensity belowthe band. Top arrow is pointing at the e-ph kink near Fermi,seen as an increase intensity in the Debye energy range of ±27mV.From figure 3.8 it can be seen that neither the single parabolic description(equation 3.4) nor the tight-binding model (equation 3.1) succeed in fitting31the upper and lower limits of the band. However they are equally goodmodels if the bias-window is narrowed down to less than 200mV. This ispartially due to fact that the low-energy part of the spectrum suffers fromlarge low-frequency background discussed above and their position cannotbe determined as precisely as higher energies.The chemical potential and effective mass as well as the gamma param-eter extracted from the fits are shown in table 3.1. The m and m∗ valuesobtained from parabolic fit are compatible with ARPES[20] and FT-STS[5]studies. The g value is also comparable with the fit attempted in [3], wherethey declare g =1.8. Quantitative comparison between these measurementswere not possible since in most works the uncertainties are unreported.32Figure 3.8: The parabolic and tight-binding fits to the dispersion fitsthe middle of the band quite well, and the deviation in the upperand lower limits is likely caused by large uncertainties in peakpositions at low energies.Parameters PB TB Literaturem (mV ) 430:5±0:4 432:0±0:3 437 [20]m∗ (me) 0:4042±0:0006 N/A 0.41 [20]g (V ) N/A 1:9749±0:0026 1.8 [3]Table 3.1: Optimized parameters obtained from parabolic (PB) andtight-binding (TB) fits to the whole range of the band shown infigure 3.8. Experimental uncertainties have not been included inthe fit uncertainties, but fit uncertainties are reported for com-parison between the two models.33Another grid with higher energy resolution of ≃ 2mV is shown in figure3.9. Inelastic electron-phonon scattering causes the enhanced intensity inthe Debye energy range between -27 to 27 mV.It appears that the fit parameters in this grid deviate from the parabolicfit in figure 3.8 that was applied to a larger energy range. It should bementioned that chemical potential cannot be determined precisely in anyof the high-resolution grids due to the short quasi-particle lifetime near theband onset. Table 3.2 summarizes the fit results. Figure 3.10 shows theextracted peaks and the parabolic fit that crosses the data points at Fermilevel. This high-resolution grid near Fermi is used in the next section forextracting the self-energies.Figure 3.9: An intensity map of the dispersion directly from QPI ringsformed at each energy. The significant increase in intensity ofthe LDOS is due to e-ph interaction that renormalizes the spec-trum in the Debye energy range.34Figure 3.10: Parabolic fit to the dispersion. Data points are peakpositions obtained from the T-matrix fit.Parameters PB TBm (mV ) 403:3±1:1 409:5±1:0m∗ (me) 0:443±0:0013 N/Ag (V ) N/A 1:8262±0:0052Table 3.2: Optimized parameters obtained from parabolic and tight-binding fits to the grid shown in figure 3.10. Extracted chemicalpotential and effective mass values deviate from those obtainedin the fits to the whole band shown in table 3.1.353.2.3 Self-energiese-ph interactions give rise to the kink near the Fermi level in the Debyeenergy range h¯ΩD =±27mV [20](figure 3.9). The deviation of the band fromparabolic near the Fermi energy is due to the e-ph renormalization of thedispersion where Etotal = E(k)+S′e−ph. As a result, the real part of the e-ph self energy can be obtained by subtracting the parabolic fit from thedata points. The increase in intensity of the spectra near E f , and the realpart of the electron-phonon self-energy are shown in figures 3.11 (b) and (d)respectively.According to [7], the total self energy takes the following form:SSE(w) =−ih− ig w22+Se−ph (3.6)h is the lifetime broadening imposed by defects, and g by e-e interactions.The electron-phonon self-energy term also has an imaginary and a real part(S′′e−ph and S′e−ph respectively). Outside the Debye energy range, the e-phcontribution to self-energies is constant.Since the T-matrix formalism appears to provide an accurate descriptionof our data, the imaginary and real parts of the total self energy (SSE) canalso be directly obtained from the fit (figure 3.11(a) and (c)). The self-energyterm suggested in equation 3.6 assumes that the real part of the total selfenergy only arises from the electron-phonon term. Comparing figures 3.11(c) and (d), it can be seen that the real part of the total self-energy has adifferent background term than S′e−ph, hence refuting equation 3.6.36Figure 3.11: Extracted self-energies directly from TMatrix Fit (a)-(d)and after a parabolic fit to the peak positions (e). Deviationsin quasi-particle life-time and the real part of self energy areclearly extract in the Debye energy range, shown with dashedvertical lines.37In addition, equation 3.6 suggests that outside the Debye energy-range,the S′′SE term has a parabolic dependence to energy that is concave down.The output of the fit, however, deviates from the expected trend significantly(see figure 3.12). One reason that may be contributing to this discrepancyis the fact that the real and imaginary parts of the self-energy have beenfitted independently, without assuming a Kramers-Kronig relation betweenthe two terms.Figure 3.12: The energy dependence of the quasi-particle lifetime de-viates from the expected parabolic trend.3.2.4 The phase-shiftFigure 3.13 shows the fit result for the scattering phase-shift and its grad-ual decrease with energy. Phase-shift of 2.0 is compatible with real-spacephase analysis in[5], where d and d ±p are equivalent. This result does notreflect the unitary limit assumed in [7], rather it indicates that the effectivescattering potential lies in between the strong and the weak limits.Using the T-matrix formalism, simulations of the modulated density ofstates have been performed in Python for three different scattering scenarios:weak with phase of d = 0, intermediate scattering with phase of 2:0 obtainedfrom the fits to our data, and strong scattering d = p=2 (figure 3.14(b)).38Figure 3.13: The scattering phase-shift obtained from the fit. Theunitary limit corresponds to the strong scattering scenario withd = p=2.The weak scattering scenario gives rise to line-shapes decaying to theleft of the peak. In the strong case the smooth decay is to the left of thepeak. In the case of intermediate strong scattering, the line-shape is similarto the unitary limit except that the decrease to the right of the peak is notas steep as in the unitary limit.The trend in background also differs for the three considered scatteringscenarios. The low-frequency divergence occurs in the weak and interme-diate scattering cases, but vanishes in the strong case. Background contri-bution in the proximity of the peak is considerable to the right of the peakin the intermediate and strong scenarios, even though it approaches zero inthe weak case.In addition, the strength of the states below the onset energy depends39on the phase. It is strongest in the unitary limit and vanishes in the weakscattering scenario. Both the line-shape and the strength of the bound statebelow the surface state onset are represented the best in the intermediatescattering scenario(see figure 3.14(a)).40(a)(b)Figure 3.14: (a) FT-STS extracted dispersion of Cu(111) (b) Simula-tion of the density of states for three different phases.413.3 Absence of Spin-Flip Scattering in DiluteCo/Cu(111)Having assured a pristine Cu(111) surface at T=4.5K, Cobalt atoms weredeposited from a 99.99% purity Cobalt rod using an e-beam evaporator.Before each grid, it was assured that the majority of the defects in thescan range that image as protrusions were Co impurities by acquiring pointspectroscopy on a sample of defects showing the Kondo effect near the Fermilevel[14], shown in figure 3.15.Figure 3.15: The Kondo effect seen in the STS of an individual Coatom on Cu(111), showing a decrease in the LDOS near theFermi level.Cobalt atoms were found to be mobile at negative voltages, and thosehigher than 30 mV, therefore they do not image like point defects in gridtopography. Their instability limited the stabilization biases used for eachgrid, which was set to no more than 100 mV. This resulted in a non-dispersing secondary peak in Fourier space that is an artifact of the feedbackloop [13].Figure 3.16(a) shows the topography of a section of a 210×210 nm2 grid.42Defects imaging as protrusions are mainly Cobalt atoms, defects imaging asdepressions are mainly CO molecules, Carbon or Sulfur rising from the bulkduring anneals, or water molecules. The right panel shows the dI/dV mapof the same area, whose rich QPI pattern makes it challenging to identifyindividual defects’ contribution to the scattering signal.Figure 3.16(b) shows the line-cuts at the Fermi level for Cu(111) andCo/Cu(111) systems, showing that the q-resolution is similar in both datasets. The ARPES experiment resolving the spin-orbit split bands had a mo-mentum resolution of about 0.005 Å−1, which is nearly half of our resolutionin q-space. As a result the backscattering signal would have shown as a peakcorresponding to q2 to the left of the spin-conserving scattering vector q3,and another peak to the right of q3 corresponding to q4, shown in figure3.16. q1 would have been immersed in the low-frequency part of the back-ground and hence not detectable. However, from our results on Co/Cu(111)it can be deduced that spin-flip scattering either does not occur, or its signalstrength is much weaker than spin-conserving scattering events. This resulthas been checked for an energy range between -108 to 92 mV.43(a)(b)Figure 3.16: (a) Left panel is the topography of Co/Cu(111) with highdefect density. Due to the mobility of Cobalt atoms duringgrids, they do not image perfectly in their spherical s-state.Right panel is the dI/dV map of the same area at Fermi. (b)Comparing line-cuts of Co/Cu(111) with bare Cu(111). Inthe presence of Cobalt impurities, only the spin-conservingscattering vector (q3) can be accessed via FT-STS.44Chapter 4Conclusions and OutlookThe surface state of noble metals provides one of the most simplified sce-narios for studying many-body scattering in two dimensions, turning theminto a relatively simple platform to test novel spectroscopic techniques. Inthis work I have presented a study of the Cu(111) surface state with high-resolution FT-STS, and for the first time demonstrated that fine many-bodyeffects observed in our data can be accurately modelled within the T-matrixformalism.Specifically, our approach gives access to the self-energies and the scat-tering phase-shifts in both the occupied and unoccupied levels, in additionto the dominant scattering vector and its associated intensity which werepreviously extracted in [7] on Ag(111) by fitting Lorentzians to the line-cuts.The success of our analysis could further expand the capabilities of FT-STSbeyond its conventional application as merely a probe of the energy disper-sion in the scattering space. The observed deviations found in the dispersion,and in the self-energies from the simple free-electron assumption, however,call for more theoretical work in order to understand these discrepancies.We have shown that FT-STS can efficiently probe local effects by treat-ing defects as scattering potentials whose strength dictate the scatteringphase-shift and the line-shape of the quasi-particles. This capability hasbeen accurately confirmed on Cu(111) by comparing real space and q-spaceanalyses, which can be applied in more complex systems where defect den-45sity and the dominant scattering potential gain importance.It is nonetheless worth mentioning that due to the complex nature of theFT-STS signal, experimental uncertainties arising from Gaussian smooth-ing of the curves, the thermal broadening of electron states, as well as thesystematic uncertainty associated with the background signal were not fullycharacterized. Identifying the role of these factors would give a better esti-mate of the goodness-of-fit parameters that could make the analysis morerigorous.Finally, we showed that in dilute Co/Cu(111) the spin-flip scattering sig-nal cannot be accessed via FT-STS in the probed energy range. It is believedthat spin-flip scattering in such systems does not occur outside the so-calledKondo regime near the Fermi level, and even within that regime the spinsof Co atoms are screened by electrons forming a spin-singlet state[14]. Fromour measurements we can qualitatively deduce that spin-conserving scat-tering in Co/Cu(111) is preferred. Increasing the magnetic/non-magneticadsorbate ratio might however alter the line-shape, or increase the spin-flipscattering probability. Within the timeline of this project these effects havenot been studied and are left to be researched into by the interested reader.46Bibliography[1] K. Besocke. 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