Quantifying many-body effects by high-resolution Fourier transform scanning tunneling spectroscopy
S. Grothe, S. Johnston, Shun Chi, P. Dosanjh, S. A. Burke, Y. Pennec
QQuantifying many-body effects by high-resolution Fouriertransform scanning tunneling spectroscopy
S. Grothe, , , ∗ S. Johnston, , , ∗ Shun Chi, , P. Dosanjh, , S. A. Burke, , , and Y. Pennec , Department of Physics and Astronomy, University of British Columbia, Vancouver BC, CanadaV6T 1Z1 Quantum Matter Institute, University of British Columbia, Vancouver BC, Canada V6T 1Z4 Department of Chemistry, University of British Columbia, Vancouver BC, Canada V6T 1Z1
November 21, 2018
Many-body phenomena are ubiquitous in solids, as electrons interact with one another andthe many excitations arising from lattice, magnetic, and electronic degrees of freedom. Theseinteractions can subtly influence the electronic properties of materials ranging from metals, exotic materials such as graphene,
2, 3 and topological insulators, or they can induce newphases of matter, as in conventional and unconventional superconductors, heavy fermionsystems, and other systems of correlated electrons. As no single theoretical approach de-scribes all such phenomena, the development of versatile methods for measuring many-body effects is key for understanding these systems. To date, angle-resolved photoemis-sion spectroscopy (ARPES) has been the method of choice for accessing this physics bydirectly imaging momentum resolved electronic structure. Scanning tunneling mi-croscopy/spectroscopy (STM/S), renown for its real-space atomic resolution capability, canalso access the electronic structure in momentum space using Fourier transform scanningtunneling spectroscopy (FT-STS).
Here, we report a high-resolution FT-STS measure-ment of the Ag(111) surface state, revealing fine structure in the otherwise parabolic elec-tronic dispersion. This deviation is induced by interactions with lattice vibrations and hasnot been previously resolved by any technique. This study advances STM/STS as a methodfor quantitatively probing many-body interactions. Combined with the spatial sensitivity ofSTM/STS, this technique opens a new avenue for studying such interactions at the nano-scale. a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l on-interacting electrons in a crystal occupy quantum states with an infinite lifetime andband dispersion (cid:15) ( k ) set by the lattice potential. Interactions with the other electrons and ele-mentary excitations of the system scatter the electrons, resulting in an altered dispersion rela-tion E ( k ) and a finite lifetime. These many-body effects are encoded in the complex self-energy Σ( k , E ) = Σ (cid:48) ( k , E ) + i Σ (cid:48)(cid:48) ( k , E ) . The imaginary part Σ (cid:48)(cid:48) ( k , E ) determines the lifetime of thestate and is related to the scattering rate. The real part Σ (cid:48) ( k , E ) shifts the electronic dispersion E ( k ) = (cid:15) ( k ) + Σ (cid:48) ( k , E ) . The tools available for studying energy and momentum resolved self-energy are limited. For example, bulk transport and optical spectroscopies provide some accessto k -integrated self-energies while ARPES accesses k -resolved information for only the occupiedstates. It is therefore important to develop a more extensive suite of versatile techniques, especiallyin the context of complex systems that remain poorly understood from a theoretical perspective.STM/STS accesses momentum space electronic structure by imaging real-space maps of themodulations in differential conductance ( dI/dV ), which is proportional to the local density ofstates (LDOS) of the sample. These modulations arise from the interference of electrons scatteredelastically by defects, and contain information about the initial and final momenta that are acces-sible by a Fourier transform of the real space map. As the electrons are dressed by interactions,the momentum space scattering intensity map is often referred to as the quasiparticle interference(QPI) map. The dominant intensities in a QPI map occur at scattering wave vectors linking con-stant energy segments of the band dispersion. By tracking the energy dependence of these peaks,the electronic dispersion E ( k ) can be obtained. This technique has been used to map coarse dis-persions in many materials
16, 17 and to examine scattering selection rules.
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While the influenceof many-body effects in FT-STS has been postulated since early reports, the direct influence ondispersion and a quantitative account of self-energy effects were missing. Here, we extend thistechnique to the examination of fine structure in the E ( k ) due to many-body interactions.The two-dimensional Shockley surface state of the noble metal silver Ag(111) was selectedas an ideal system for demonstrating the quantitative capabilities of FT-STS. The Ag(111) surfacestate is well characterized, and while it exhibits distinct many-body effects, it lacks the complicated2nterplay of interactions that appear in many complex modern materials. The dispersion ofthe surface state is free-electron-like over a wide energy range given by (cid:15) ( k ) = (cid:126) k / m ∗ − µ , where m ∗ is the effective mass and µ is the chemical potential. This parabolic dispersionis modified by the electron-electron (e-e) and electron-phonon (e-ph) interactions. As both areaccurately described by conventional theory,
23, 24 a straightforward comparison with theory can bemade, requiring few parameters. The e-e interaction decreases the electron lifetime for energiesaway from the Fermi level. The e-ph interaction introduces an additional scattering channel forenergies above the typical phonon energy scale (the Debye frequency (cid:126) Ω D ), decreasing the lifetimeand modifying the bare dispersion near the Fermi energy E F . The latter is a subtle effect in thecase of Ag(111), that had not yet been observed due to the high resolution required in both energyand momentum.STM/STS measurement of the Ag(111) surface yields real-space conductance maps (see Fig.1a for the map at E = E F ) with circular LDOS modulations arising from scattering from point-likeCO adsorbates, and vertical modulations produced by step-edge reflections. The small terraces onthe surface produce subtle confinement effects not representative of the pristine surface state. Inorder to access intrinsic surface properties, we removed their contribution by setting dI/dV in thisregion to the average value over the entire image. The ability to isolate regions of interest in thisway is unique to STM/STS, as probes such as ARPES would average over these domains. Fig.1b shows a typical dI/dV spectra, averaged over a defect free region. The momentum space QPIintensity map S ( q , E F ) (Fig. 1c) exhibits a ring of radius q ( E F ) = 2 k ( E F ) , as expected for afree-electron-like dispersion where back scattering is dominant. A line profile S ( | q | , E F ) of theQPI map is shown in Fig. 1d, where we have performed an angular average of S ( q , E F ) in theregions above and below the dashed lines. This restriction isolates the contributions of the point-like CO scatterers. The momentum space resolution ∆ q ∼ . ˚ A − is set by the dimensionof the map ( × nm ) while the energy resolution ∆ E = 4 k B T = 1 . meV is limited bythermal broadening of the tip and the sample.We now examine the detailed electronic structure of the Ag(111) surface state by consider-3ng the full energy dependence of the angle-averaged profile S ( | q | , E ) , as shown in Fig. 2a. Aparabolic band dispersion is evident over a wide energy range while the QPI signal intensity ex-hibits a monotonic decrease from the onset of the surface state to higher energy. From the data weextract µ = 65 ± meV and m/m e = 0 . ± . , consistent with previous measurements.
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These values define the bare dispersion in the absence of e-ph coupling. Deviations from this dis-persion, as well as an enhanced QPI signal intensity, are evident in the vicinity of E F , shown moreclearly in the inset of Fig. 2a. A similar increase in QPI signal intensity was observed near E F inone of the first FT-STS reports on the Be(0001) surface state and the e-ph interaction was laterproposed as a possible origin. To identify the source of these deviations and to assess the QPI signal we modeled the systemusing the T-matrix formalism, considering scattering from a single CO impurity in the unitarylimit (see methods). The e-e interaction was handled within Fermi liquid theory while the e-phinteraction was treated within standard Migdal theory with the phonons described by the Debyemodel (Debye frequency (cid:126) Ω D = 14 meV, dimensionless e-ph coupling strength λ = 0 . ). Withinthese approximations the self-energy Σ( E ) is a function of energy only. The resulting simulatedQPI intensity is shown in Fig 2b. The model closely reproduces both the coarse and fine detailsof the data. The overall monotonic decrease in QPI intensity is linked to the group velocity ofthe bare electronic dispersion and is not related to many-body effects. However, the increase inthe intensity near E F and the deviations from the parabolic band dispersion arise from the e-phinteraction.We now perform a quantitative analysis of the self-energy. For reference, Fig. 3a shows thee-ph self-energy for the same values of (cid:126) Ω D and λ used in Fig. 2b. To extract Σ( E ) , a Lorentzianwas fit to the data in the vicinity of the peak to obtain both the QPI peak position and height. (Seemethods. An example fit is shown as the dashed line in Fig. 1d.) A plot of the QPI peak heightreflects the behavior of Σ (cid:48)(cid:48) ( E ) as shown in Fig. 3b, where we compare the data with the model.Here we have normalized both sets of data (as described in the figure caption) in order to eliminatethe role of the tunneling matrix element in setting the scale of the experimental data. There is4ood agreement between the model and experiment apart from a slight deviation ∼ meV, whichwe attribute to a set point effect below q F = 2 k F . The decrease in − Σ (cid:48)(cid:48) ( E ) within (cid:126) Ω D = 14 meV of E F produces the non-monotonic variation in peak height superimposed over the groupvelocity dependence. This is due to the closing of the phonon scattering channel at energies belowthe characteristic phonon frequency, resulting in longer-lived quasiparticles near E F . We note thatthe value (cid:126) Ω D required to reproduce the data is close to the value for the top of the bulk acousticbranches. The real part Σ (cid:48) ( E ) can be estimated from the data by taking the difference betweenthe measured peak position and the parabolic dispersion. The result is shown in Fig. 3c, wherepeaks in Σ (cid:48) ( E ) occur in the data at the same energy scale reflected in Fig. 3b. The dimensionlessstrength of the e-ph coupling λ can be estimated from d Σ (cid:48) /dE | E = E F . We obtain λ = 0 . ± . ,consistent with previous estimates.
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Our FT-STS results provide a stunning visualization of the subtle modifications in disper-sion and scattering intensity arising from many-body interactions in a simple system. This methodprovides high resolution in both momentum and energy that is competitive with state-of-the-artARPES. Moreover, FT-STS accesses both occupied and unoccupied states opening up the possi-bility of examining particle-hole asymmetric systems. These aspects give access to many-bodyfeatures not previously observed in such a direct way. With enhanced stability and lower tem-peratures, further advancements in the application of FT-STS to quantify many-body interactionsin more complex systems can undoubtedly be expected. However, perhaps the most compellingadvantage of FT-STS is the prospect of exploiting STM’s unique spatial sensitivity to explorevariations in the many-body interactions in nanoscale regions and intrinsically inhomogeneousmaterials.
MethodsExperiment:
Measurements were performed in a Createc STM in ultra high vacuum at a tempera-ture of 4.2 K with a tungsten tip formed by direct contact with the Ag crystal. The Ag(111) surface5as cleaned by three cycles of Ar sputtering each followed by thermal annealing to 500 ◦ C. The I ( V ) map measured over 80 hours consists of × spectra taken on a × nm area.Each I ( V ) spectrum consists of 512 data points and was Gaussian smoothed maintaining thermallylimited energy resolution of ∆ E = 1 . meV. dI/dV spectra were acquired by numerical differ-entiation of the I - V sweep. Atomic resolution scans on the same area lead to a spatial resolutionof (0.58 ± S O of each line profile S ( q , E ) was determined by fitting aLorentzian within a range of ± . ˚A − peak position expected from (cid:15) ( k ) . The renormalization ofthe bare dispersion has been observed in four different data sets. Theory:
Calculations were performed using the T -matrix formalism for scattering from asingle impurity. The QPI intensity is given by the impurity-induced LDOS modulations δρ ( q , ω ) = − i π (cid:88) k Im G ( k , ω ) T ( ω ) G ( k + q , ω ) . (1)Here T = − V sin( δ ) exp( iδ ) is the T -matrix, where δ = π/ is the phase shift (unitary limit), V is the scattering potential. The Green’s function is given by G ( k , ω ) = [ ω − ξ ( k ) − Σ( ω )] − where ξ = − (cid:126) k / m ∗ − µ ( m ∗ = 0 . m e , µ = 65 meV) is the dispersion of the surface state and Σ( ω ) = − iη − iγω / e − ph ( ω ) is the self-energy. Here η = 2 . meV is the lifetime broadening dueto scattering from the terraces, γ = 62 . meV is the e-e interactions, and Σ e − ph ( ω ) is obtainedfrom a Debye model with (cid:126) Ω D = 14 meV and coupling strength λ = 0 . . cknowledgements ∗ These authors contributed equally to this work. The authors thank E. van Heumen for usefuldiscussions. The authors gratefully acknowledge support by NSERC, CFI, CIFAR, the Universityof British Columbia, and the Canada Research Chairs program (S.B.). S.G. acknowledges supportfrom The Woods fund. 7
00 nm ab d maxmin 0.4-0.400.4-0.4 0 c q x [A ] -1o q y [ A ] - q [A ] -1o S ( q , E = ) [ a r b . un i t s ] −100 0 1000123 Sample bias [meV] < d I/ d V > [ p S ] Figure 1:
A summary of FT-STS of the surface state of Ag(111). (a) Conductance map (dI/dV)of a × nm area at E = eV = 0 meV (tip height set at V = 100 meV, I = 200 pA).LDOS modulation due to scattering at step edges and CO adsorbates are visible.The areas aroundthe step edges were removed as discussed in the text. Furthermore, a tip change induced stripewas corrected by a line-by-line subtraction of the average line value that excludes CO impurities.(b) Average dI/dV spectrum from a defect free area with a total size of 100 nm .The particle-hole symmetric steps at E F likely originate from an inelastic co-tunneling pathway via phononmodes polarized perpendicular to the surface.(c) Absolute value of the Fourier transform (powerspectrum) of the dI/dV map ( E = 0 , panel a) showing a ring with radius q = 2 k F where q isthe scattering vector. The increased intensity along the q x direction originates from the step edgecontributions. (d) The QPI line profile S ( q , E = 0) . This is obtained by integrating (c) within therange above and below the dashed lines in order to isolate contributions from the CO adsorbates.The scattering peak is slightly asymmetric with an enhanced intensity at low q , which is morepronounced at higher energies. Scattering peak positions and heights were obtained by fitting theline profiles as shown in (a) within a range of ± . ˚A − around the peak position (dashed line).8 max E ne r g y [ m e V ] q [A -1 ] q [A -1 ] Figure 2:
Measured and calculated dispersion of the QPI intensity. (a) The measured disper-sion of S ( q , E ) , obtained by plotting the line profiles as shown in 1d for all bias voltages. Overallthe dispersion is parabolic with µ = 65 ± meV and m ∗ /m e = 0 . ± . , obtained by fittingthe peak position excluding the energy range [ − , meV. The intensity of the scattering peakgenerally decreases with increasing energy but has a non-monotonic increase near E F . We observean additional scattering intensity below the onset of the surface state ( E < − meV). The insetreveals a subtle renormalization of the dispersion within E F ± meV.(b) Theoretical calculationof the QPI intensity with µ = 65 meV, m ∗ /m e = 0 . . The model includes electron-electron(Fermi-liquid theory) and electron-phonon (Debye model, (cid:126) Ω D =14 meV, λ = 0 . ) interactionsand assumes the CO adsorbates scatter in the unitary limit.9 −40 −20 0 20 40−2−1012 Energy [meV] S ( E ) [ a r b . un i t s ] ( E ) , - ( E ) [ m e V ] ( E ) [ m e V ] abc - (E) (E) Figure 3:
Quantitative extraction of the real and imaginary part of the self-energy. (a) Calcu-lated real ( Σ (cid:48) , solid line) and imaginary ( Σ (cid:48)(cid:48) , dashed line) parts of the self-energy for the parametersused in Fig. 2b. (b) Scattering peak height S ( q , E ) as a function of energy determined experimen-tally (dots) and theoretically (solid line). The theory curve has been normalized to the value at E = 0 while the experimental data had been normalized to the average value over the window [ − , meV. The overall decrease of the scattering peak height with increasing energy is related tothe energy dependence of the group velocity. The increased intensity at E F ± meV is causedby a dip in − Σ (cid:48)(cid:48) due to the e-ph interaction. This relationship opens up a new way to experi-mentally obtain Σ (cid:48)(cid:48) . (c) Real part of the self energy determined from the difference between thescattering peak position E ( k ) and a parabolic fit (cid:15) ( q ) . The solid line corresponds to the theoreti-cally determined Σ (cid:48) as presented in (a). Calculated and measured self-energy real part agree well,demonstrating that FT-STS is an alternative tool to obtain Σ (cid:48) ( ω ) .The dashed lines at ±
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