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Quantifying Many-Body Effects by High-Resolution Fourier Transform Scanning Tunneling Spectroscopy S. Grothe,1,2 S. Johnston,1,2 Shun Chi,1,2 P. Dosanjh,1,2 S. A. Burke,1,2,3 and Y. Pennec1,2 1
Department of Physics and Astronomy, University of British Columbia, Vancouver British Columbia, Canada V6T 1Z1 2 Quantum Matter Institute, University of British Columbia, Vancouver British Columbia, Canada V6T 1Z4 3 Department of Chemistry, University of British Columbia, Vancouver British Columbia, Canada V6T 1Z1 (Received 8 August 2013; published 9 December 2013) High-resolution Fourier transform scanning tunneling spectroscopy (FT-STS) is used to study manybody effects on the surface state of Ag(111). Our results reveal a kink in the otherwise parabolic band dispersion of the surface electrons and an increase in the quasiparticle lifetime near the Fermi energy Ef . The experimental data are accurately modeled with the T-matrix formalism for scattering from a single impurity, assuming that the surface electrons are dressed by the electron-electron and electron-phonon interactions. We confirm the latter as the interaction responsible for the deviations from bare dispersion. We further show how FT-STS can be used to simultaneously extract real and imaginary parts of the selfenergy for both occupied and unoccupied states with a momentum and energy resolution competitive with angle-resolved photoemission spectroscopy. From our quantitative analysis of the data we extract a Debye energy of @D ¼ 14 1 meV and an electron-phonon coupling strength of ¼ 0:13 0:02, consistent with previous results. This proof-of-principle measurement advances FT-STS as a method for probing many body effects, which give rise to a rich array of material properties. DOI: 10.1103/PhysRevLett.111.246804
PACS numbers: 73.20.r, 68.37.Ef, 71.20.b, 73.40.Gk
Many-body phenomena are ubiquitous in solids, arising from the interactions of the many electrons and a multitude of elementary excitations of the lattice, magnetic, and electronic degrees of freedom. Often, these interactions yield only subtle modifications of the electronic properties in systems ranging from simple metals [1] to exotic materials such as graphene [2,3] and topological insulators [4]. However, these interactions can also underlie transitions into new phases of matter, as in conventional [5] and unconventional superconductors [6–9], heavy fermion systems [10], and other systems of correlated electrons. As no single theoretical approach describes all such phenomena, the development of versatile methods for measuring manybody effects is key for understanding these complex systems. Noninteracting electrons in a crystal occupy quantum states with an infinite lifetime and band dispersion ðkÞ set by the lattice potential. Interactions with the other electrons and elementary excitations of the system scatter the electrons, resulting in an altered dispersion relation EðkÞ and a finite lifetime. These many-body effects are encoded in the complex self-energy ðk; EÞ ¼ 0 ðk; EÞ þ i00 ðk; EÞ. The imaginary part 00 ðk; EÞ determines the lifetime of the state and is related to the scattering rate. The real part 0 ðk; EÞ shifts the electronic dispersion EðkÞ ¼ ðkÞ þ 0 ðk; EÞ. The tools available for studying energy and momentum resolved self-energy are limited [11]. For example, bulk transport and optical spectroscopies provide some access to k-integrated self-energies while ARPES, the current method of choice [2–4,6,7,9], accesses k-resolved information for only the occupied states. 0031-9007=13=111(24)=246804(5)
Scanning tunneling microscopy or spectroscopy (STMSTS), renowned for its real-space atomic resolution capability, can also access the electronic structure in momentum space, of both occupied and unoccupied states, using Fourier transform scanning tunneling spectroscopy (FT-STS) [12–16]. Here, we report a high-resolution FT-STS measurement of the Ag(111) surface state, revealing fine structure in the otherwise parabolic electronic dispersion. Although 00 ðEÞ for the Ag(111) surface state has been studied previously by STS via electron scattering rates [17–19], the many-body effects on 0 ðEÞ have not been previously resolved by any other method. This achievement advances FT-STS as a quantitative probe of many-body interactions encoded in ðk; EÞ. STM-STS accesses electronic structure through realspace maps of modulations in differential conductance (dI=dV), proportional to the local density of states (LDOS). Periodic modulations arising from the interference of electrons scattered elastically by defects contain information about the initial and final momenta, revealed by a Fourier transform of the real space map. As these 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 constant energy segments of the band dispersion. By tracking the energy dependence of these peaks, the electronic dispersion EðkÞ can be obtained. This technique has been used to map coarse dispersions in many materials [20,21] and to examine scattering selection rules [22,23]. While the influence of many-body effects in FT-STS has been
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postulated since early reports [24], observation of their influence on the band dispersion and extraction of the self-energy has been missing. The two-dimensional Shockley surface state of the noble metal silver, Ag(111), was selected as an ideal system for demonstrating the quantitative capabilities of FT-STS. The Ag(111) surface state is well characterized [17–19,25–33], and while it exhibits distinct many-body effects, it lacks the complicated interplay of interactions that appear in more complex materials. The dispersion of the surface state is free-electron-like over a wide energy range given by ðkÞ ¼ @2 k2 =2m , where m is the effective mass and is the chemical potential. The dispersion is further modified by many-body interactions that are accurately described by conventional theory [18,19]. Thus a straightforward comparison with theory can be made, requiring few parameters. The e-e interaction decreases the electron lifetime for energies away from the Fermi level. The e-ph interaction introduces an additional scattering channel for energies greater than the Debye energy @D , decreasing the quasiparticle lifetime for jEj > @D and modifying the bare dispersion in a window @D of the Fermi energy Ef . Measurements were performed in a Createc ultrahigh vacuum STM at a temperature of 4.2 K with a tungsten tip formed by direct contact with the Ag crystal. The Ag(111) surface was cleaned by three cycles of Ar sputtering each followed by thermal annealing to 500 C. The IðVÞ map measured over 80 h consists of 380 380 spectra taken on a 239 239 nm2 area. Each IðVÞ spectrum consists of 512 data points and was Gaussian smoothed, maintaining a thermally limited energy resolution of E ¼ 1:5 meV. dI=dV spectra were acquired by numerical differentiation of the I-V sweep. Separate atomic resolution scans acquired on the same area as the conductance map were used to obtain the x-y calibration. This calibration enables an accurate determination of the spatial resolution of the conductance map [Fig. 1(a)] of 0:58 0:01 nm [34]. STM-STS measurement of the Ag(111) surface yields real-space conductance maps [see Fig. 1(a)] for the map at E ¼ Ef ] with circular LDOS modulations produced by scattering off of pointlike adsorbates from residual CO, and vertical modulations due to reflections from step edges. The small terraces on the surface produce subtle confinement effects not representative of the pristine surface state [31]. In order to access intrinsic surface properties, we removed their contribution by setting dI=dV in this region to the average value over the entire image. (This treatment does not change the QPI peak positions; see Supplemental Material [34].) The ability to isolate regions of interest in this way is unique to STM-STS, as probes such as ARPES would average over these domains. Figure 1(b) shows a typical dI=dV spectrum, averaged over an area free of scattering centers. The momentum space QPI intensity map Sðq; Ef Þ [Fig. 1(c)] exhibits a ring
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FIG. 1 (color online). (a) Conductance map (dI=dV) of a 239 239 nm2 area at E ¼ eV ¼ 0 meV (tip height set at V ¼ 100 meV, I ¼ 200 pA). LDOS modulation due to scattering at step edges and CO adsorbates are visible. The areas around the step edges indicated by white dashed lines were removed as discussed in the main text. The scale ranges from 2.3 to 3.3 pS. (b) Average dI=dV spectrum from a defect free area with a total size of 100 nm2 . The particle-hole symmetric steps at Ef likely originate from an inelastic co-tunneling pathway via phonon modes polarized perpendicular to the surface. (c) Absolute value of the Fourier transform (power spectrum) of the dI=dV map (E ¼ 0, panel a) showing a ring with radius q ¼ 2kf , where q is the scattering vector. The increased intensity along the qx direction originates from the step edge contributions. (d) The QPI line profile Sðjqj; E ¼ 0Þ. The scattering peak is slightly asymmetric with an enhanced intensity at low q, which is more pronounced at higher energies.
of radius qðEf Þ ¼ 2kðEf Þ, as expected for a free-electronlike dispersion where backscattering is dominant [35]. A line profile Sðjqj; Ef Þ of the QPI map is shown in Fig. 1(d), where due to the isotropic nature of ðkÞ we have performed an angular average of Sðq; Ef Þ in the regions above and below the dashed lines. This restriction isolates the contributions of the pointlike CO scatterers and improves our signal to noise. The momentum space resolution 1 is set by the dimension of the map q 0:0026 A (239 239 nm2 ) while the energy resolution is limited by thermal broadening of the tip and sample states. We now examine the detailed electronic structure of the Ag(111) surface state by considering the full energy dependence of the angle-averaged profile Sðjqj; EÞ, as shown in Fig. 2(a). Considering the coarse features of the data, we observe a parabolic band dispersion over a wide energy range, reflecting the nearly free-electron-like dispersion of the surface state. Furthermore, excluding the anomalies in the energy range Ef @D , the QPI intensity exhibits an overall decrease from the onset of the surface state to
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FIG. 2 (color online). (a) The dispersion of Sðq; EÞ, obtained by plotting the line profiles as shown in one dimension for all bias voltages. Overall, the dispersion is parabolic with ¼ 65 1 meV and m =me ¼ 0:41 0:02, obtained by fitting the peak position excluding the energy range ½20; 20 meV. The intensity of the scattering peak generally decreases with increasing energy but has a nonmonotonic increase near Ef . Subtle kinks in the dispersion are observed at @D , labeled 1. An additional scattering intensity below the onset of the surface state (E < 70 meV) is also observed, labeled 3. The inset reveals a subtle renormalization of the dispersion within Ef 14 meV. (b) Calculated QPI intensity for our model which includes e-e and e-ph interactions and assumes the CO adsorbates scatter in the unitary limit. The line of zero intensity (labeled 2) is a consequence of the unitary scattering.
higher energy. From the data we extract band parameters ¼ 65 1 meV and m =me ¼ 0:41 0:02, consistent with previous measurements [18,19,25,27]. These values define the bare dispersion in the absence of additional renormalizations due to many-body interactions. On a more detailed level, deviations from the parabolic dispersion, as well as an enhanced QPI signal intensity, are evident in the vicinity of Ef . This can be seen more clearly in the inset of Fig. 2(a). A similar increase in QPI signal intensity was observed near Ef in one of the first FT-STS reports on the Be(0001) surface state [12] and the e-ph interaction was later proposed as a possible origin [24]. We have observed this renormalization of the bare dispersion in four different data sets on Ag(111). To identify the source of these deviations and to assess the QPI signal we modeled the system using the T-matrix formalism, considering scattering from a single CO impurity in the unitary limit [35]. The QPI intensity jðq; EÞj is obtained from the Fourier transform of the impurityinduced LDOS modulations and is given by i X ðq; EÞ ¼ Im½Gðk; EÞTðEÞGðk þ q; EÞ: 2 k Here, T ¼ V0 sinðÞ expðiÞ is the T-matrix with phase shift ¼ =2 (unitary limit) and scattering potential V0 . The ‘‘bare’’ Green’s function in the absence of impurities is given by Gðk; EÞ ¼ ½E ðkÞ ðk; EÞ1 where ðkÞ ¼ @2 k2 =2m is the dispersion of the surface state and ðk; EÞ includes contributions from the e-e and e-ph
interactions, as well as an additional lifetime broadening due to scattering from the step edges. The e-e interaction was handled within Fermi liquid theory while the e-ph interaction was treated within standard Migdal theory [1] where the phonons were described by the Debye model. Under these approximations the self-energy is a function of energy only and is given by ðEÞ ¼ i iE2 =2 þ e-ph ðEÞ, where ¼ 2:5 meV and ¼ 62:7 meV parameterize scattering from the terraces and the e-e interaction, respectively, [18,19]. The imaginary part of e-ph ð!Þ is given by 00e-ph ðEÞ ¼ 2
Z D 0
d!0
!0 2 ½nf ð@!0 EÞ D
þ nf ð@!0 þ EÞ þ nb ð@!0 Þ; where nf and nb are the Fermi and Bose occupation factors, and is the dimensionless e-ph coupling strength. The real part of e-ph ðEÞ is obtained by the usual KramersKronig relations. From our data we extracted (see below) @D ¼ 14 meV and ¼ 0:13. These values, along with our measured values of m and , serve as input for our model. The simulated QPI intensity is shown in Fig. 2(b). The model closely reproduces both the coarse and fine details of the data. Our calculations show that overall QPI intensity is inversely related to the electron group velocity, producing a decrease in peak height as a function of energy if one excludes the increase within @D of Ef . This anomaly in the intensity, and the deviations from the parabolic band dispersion in the same energy range, arise from the e-ph interaction. The unitary nature of the scatterer also leaves a distinct fingerprint in the QPI intensity. In this limit, ðqÞ switches sign for q just above 2kðEÞ, which is a physical consequence of the =2 phase shift of the scattered quasiparticle. This produces a line of zero intensity on the large q side of the maximum QPI intensity (labeled as 2 in Fig. 2) when the absolute value is taken, as well an additional intensity at small q below the onset of the surface state (labeled as 3 in Fig. 2). Examining the data in Fig. 2(a) we see indications of the small q intensity below 65 meV; however, the zero intensity line is difficult to resolve below the noise floor. We now turn to a quantitative analysis of the data. For reference, Fig. 3(a) shows the e-ph self-energy used in Fig. 2(b). To extract ðEÞ, a Lorentzian was fit to the 1 around the peak data within a window of 0:01 A position expected from ðkÞ. [An example is shown as the dashed line in Fig. 1(d).] A plot of the QPI peak height S0 reflects the behavior of 00 ðEÞ as shown in Fig. 3(b), where we compare the data with the model. Both sets of data have been normalized (as described in the figure caption) to eliminate the tunneling matrix element’s role in setting the scale of the experimental data. There is good agreement between the model and experiment apart from a
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FIG. 3 (color online). (a) Calculated real (0 , solid line) and imaginary (00 , dashed line) parts of the self-energy for the parameters used in Fig. 2(b). (b) Scattering peak height S0 ðq; EÞ as a function of energy determined experimentally (dots) and theoretically (solid line). The theory curve was normalized to the value at E ¼ 0 while the experimental data were normalized to the average value over the window ½5; 5 meV. (c) 0 ðEÞ determined from the difference between the scattering peak position EðkÞ and a parabolic fit ðqÞ. The solid line corresponds to the theoretically determined 0 as presented in (a). The dashed lines at 14 meV in (a) to (c) indicate the position of the Debye energy @D .
slight deviation around 20 meV, which we attribute to a set point effect below qF ¼ 2kf [18,36]. The decrease in 00 ðEÞ within @D ¼ 14 meV of Ef produces the nonmonotonic variation in peak height superimposed over the group velocity dependence previously discussed. This is due to the closing of the phonon scattering channel at energies below @D , resulting in longer-lived quasiparticles near Ef . The value @D required to reproduce the data is close to the value for the top of the bulk acoustic branches [33]. The real part 0 ðEÞ can be estimated from the data by taking the difference between the measured peak position and the large scale parabolic dispersion. The result is shown in Fig. 3(c), where peaks in 0 ðEÞ occur in the data at the same energy scale reflected in Fig. 3(b). We estimate the dimensionless strength of the e-ph coupling ¼ d0 =dEjE¼Ef ¼ 0:13 0:02 [1], consistent with previous estimates [18,28]. Finally, the details of the QPI intensity and dispersion are well reproduced without a
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self-consistent treatment of the T matrix, indicating that the CO impurities do not significantly alter the electronic properties of the Ag(111) surface state. The following steps provide a basis for the application of this technique to a new material: (i) Estimate the size of the effect (both the energy scale and coupling strength) to determine the resolution required in both E and k. (ii) The latter will determine the field of view required, while the largest q vector where many-body effects may be observed will set the required density of points over the field of view. (iii) For anisotropic materials, Sðq; EÞ becomes a threedimensional space which can be examined by plotting cuts in q, or by tracking particular scattering vectors qi . (iv) A mapping of the scattering vectors qi onto k must be identified. For complex materials this may require some a priori knowledge of the band structure and theory in order to deconvolve integration over k values connecting equidistant constant energy segments of ðkÞ. For complex materials, (iii) and (iv) will likely present the most significant challenges. Our FT-STS results provide a stunning visualization of the subtle modifications in dispersion and scattering intensity arising from many-body interactions in a simple system. This method provides high resolution in both momentum and energy that is competitive with state-ofthe-art ARPES. Moreover, FT-STS accesses both occupied and unoccupied states opening up the possibility of examining particle-hole asymmetric systems. These aspects give access to many-body features not previously observed in such a direct way. With enhanced stability and lower temperatures, further advancements in the application of FT-STS to quantify many-body interactions in more complex systems can undoubtedly be expected. However, perhaps the most compelling advantage of FT-STS is the prospect of exploiting STM’s unique spatial sensitivity to explore variations in the many-body interactions in nanoscale regions and intrinsically inhomogeneous materials. S. G. and S. J. contributed equally to this work. The authors thank E. van Heumen and G. Levy for useful discussions. This work is supported by NSERC, CFI, CIFAR, the University of British Columbia, and the Canada Research Chairs program (S. B.). S. G. acknowledges support from The Woods fund.
[1] A. B. Migdal, J. Exptl. Theoret. Phys. (U.S.S.R.) 34, 1438 (1958); [Sov. Phys. JETP 34, 996 (1958)]. [2] A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg, Nat. Phys. 3, 36 (2006). [3] A. Bostwick, F. Speck, T. Seyller, K. Horn, M. Polini, R. Asgari, A. H. MacDonald, and E. Rotenberg, Science 328, 999 (2010). [4] X. Zhu, L. Santos, C. Howard, R. Sankar, F. C. Chou, C. Chamon, and M. El-Batanouny, Phys. Rev. Lett. 108, 185501 (2012).
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[5] W. L. McMillan and J. M. Rowell, Phys. Rev. Lett. 14, 108 (1965). [6] A. Lanzara et al., Nature (London) 412, 510 (2001). [7] T. Dahm, V. Hinkov, S. Borisenko, A. Kordyuk, V. Zabolotnyy, J. Fink, B. Bu¨chner, D. Scalapino, W. Hanke, and B. Keimer, Nat. Phys. 5, 217 (2009). [8] K. Byczuk, M. Kollar, K. Held, Y.-F. Yang, I. Nekrasov, T. Pruschke, and D. Vollhardt, Nat. Phys. 3, 168 (2007). [9] J. Graf et al., Phys. Rev. Lett. 98, 067004 (2007). [10] Z. Fisk, H. Ott, T. Rice, and J. Smith, Nature (London) 320, 124 (1986). [11] J. P. Carbotte, T. Timusk, and J. Hwang, Rep. Prog. Phys. 74, 066501 (2011). [12] P. Sprunger, L. Petersen, E. Plummer, E. Laegsgaard, and F. Besenbacher, Science 275, 1764 (1997). [13] L. Petersen, P. T. Sprunger, P. Hofmann, E. Laegsgaard, B. G. Briner, M. Doering, H.-P. Rust, A. M. Bradshaw, F. Besenbacher, and E. W. Plummer, Phys. Rev. B 57, R6858 (1998). [14] L. Bu¨rgi, H. Brune, and K. Kern, Phys. Rev. Lett. 89, 176801 (2002). [15] J. E. Hoffman, K. McElroy, D.-H. Lee, K. M. Lang, H. Eisaki, S. Uchida, and J. C. Davis, Science 297, 1148 (2002). [16] K. McElroy, R. Simmonds, J. Hoffman, D.-H. Lee, J. Orenstein, H. Eisaki, S. Uchida, and J. Davis, Nature (London) 422, 592 (2003) [17] L. Bu¨rgi, L. Petersen, H. Brune, and K. Kern, Surf. Sci. 447, L157 (2000). [18] L. Bu¨rgi, H. Brune, O. Jeandupeux, and K. Kern, J. Electron Spectrosc. Relat. Phenom. 109, 33 (2000). [19] L. Bu¨rgi, Ph.D. thesis, Swiss Federal Institute of Technology, Lausanne, 1999. [20] M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature (London) 363, 524 (1993). [21] Y. Hasegawa and Ph. Avouris, Phys. Rev. Lett. 71, 1071 (1993).
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[22] G. M. Rutter, J. N. Crain, N. P. Guisinger, T. Li, P. N. First, and J. A. Stroscio, Science 317, 219 (2007). [23] P. Roushan, J. Seo, C. V. Parker, Y. Hor, D. Hsieh, D. Qian, A. Richardella, M. Z. Hasan, R. Cava, and A. Yazdani, Nature (London) 460, 1106 (2009). [24] M. Hengsberger, D. Purdie, P. Segovia, M. Garnier, and Y. Baer, Phys. Rev. Lett. 83, 592 (1999). [25] J. Li, W.-D. Schneider, and R. Berndt, Phys. Rev. B 56, 7656 (1997). [26] J. Li, W.-D. Schneider, R. Berndt, O. R. Bryant, and S. Crampin, Phys. Rev. Lett. 81, 4464 (1998). [27] R. Paniago, R. Matzdorf, G. Meister, and A. Goldmann, Surf. Sci. 336, 113 (1995). [28] A. Eiguren, B. Hellsing, F. Reinert, G. Nicolay, E. V. Chulkov, V. M. Silkin, S. Hu¨fner, and P. M. Echenique, Phys. Rev. Lett. 88, 066805 (2002). [29] R. Paniago, R. Matzdorf, G. Meister, and A. Goldmann, Surf. Sci. 331–333, 1233 (1995). [30] J. Kliewer, R. Berndt, E. Chulkov, V. Silkin, P. Echenique, and S. Crampin, Science 288, 1399 (2000). [31] Y. Pennec, W. Auwa¨rter, A. Schiffrin, A. Weber-Bargioni, A. Riemann, and J. Barth, Nat. Nanotechnol. 2, 99 (2007). [32] C. Tournier-Colletta, B. Kierren, Y. Fagot-Revurat, and D. Malterre, Phys. Rev. Lett. 104, 016802 (2010). [33] L. Chen, L. L. Kesmodel, and J.-S. Kim, Surf. Sci. 350, 215 (1996). [34] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.111.246804 for further details regarding the treatment of the data. [35] L. Capriotti, D. J. Scalapino, and R. D. Sedgewick, Phys. Rev. B 68, 014508 (2003). [36] The tip height is set for each spectroscopic pixel at a particular bias Vset , and current set point Iset . The current used for determining the tip height relates the integrated NðEf Þ to eVset , such that any strong features in this range will influence the tip height, including features corresponding to kf and any other wave vectors corresponding to a strong intensity in this energy range.
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Quantifying Many-Body Effects by High-Resolution Fourier Transform Scanning Tunneling Spectroscopy S. Grothe,1,2 S. Johnston,1,2 Shun Chi,1,2 P. Dosanjh,1,2 S. A. Burke,1,2,3 and Y. Pennec1,2 1
Department of Physics and Astronomy, University of British Columbia, Vancouver British Columbia, Canada V6T 1Z1 2 Quantum Matter Institute, University of British Columbia, Vancouver British Columbia, Canada V6T 1Z4 3 Department of Chemistry, University of British Columbia, Vancouver British Columbia, Canada V6T 1Z1 (Received 8 August 2013; published 9 December 2013) High-resolution Fourier transform scanning tunneling spectroscopy (FT-STS) is used to study manybody effects on the surface state of Ag(111). Our results reveal a kink in the otherwise parabolic band dispersion of the surface electrons and an increase in the quasiparticle lifetime near the Fermi energy Ef . The experimental data are accurately modeled with the T-matrix formalism for scattering from a single impurity, assuming that the surface electrons are dressed by the electron-electron and electron-phonon interactions. We confirm the latter as the interaction responsible for the deviations from bare dispersion. We further show how FT-STS can be used to simultaneously extract real and imaginary parts of the selfenergy for both occupied and unoccupied states with a momentum and energy resolution competitive with angle-resolved photoemission spectroscopy. From our quantitative analysis of the data we extract a Debye energy of @D ¼ 14 1 meV and an electron-phonon coupling strength of ¼ 0:13 0:02, consistent with previous results. This proof-of-principle measurement advances FT-STS as a method for probing many body effects, which give rise to a rich array of material properties. DOI: 10.1103/PhysRevLett.111.246804
PACS numbers: 73.20.r, 68.37.Ef, 71.20.b, 73.40.Gk
Many-body phenomena are ubiquitous in solids, arising from the interactions of the many electrons and a multitude of elementary excitations of the lattice, magnetic, and electronic degrees of freedom. Often, these interactions yield only subtle modifications of the electronic properties in systems ranging from simple metals [1] to exotic materials such as graphene [2,3] and topological insulators [4]. However, these interactions can also underlie transitions into new phases of matter, as in conventional [5] and unconventional superconductors [6–9], heavy fermion systems [10], and other systems of correlated electrons. As no single theoretical approach describes all such phenomena, the development of versatile methods for measuring manybody effects is key for understanding these complex systems. Noninteracting electrons in a crystal occupy quantum states with an infinite lifetime and band dispersion ðkÞ set by the lattice potential. Interactions with the other electrons and elementary excitations of the system scatter the electrons, resulting in an altered dispersion relation EðkÞ and a finite lifetime. These many-body effects are encoded in the complex self-energy ðk; EÞ ¼ 0 ðk; EÞ þ i00 ðk; EÞ. The imaginary part 00 ðk; EÞ determines the lifetime of the state and is related to the scattering rate. The real part 0 ðk; EÞ shifts the electronic dispersion EðkÞ ¼ ðkÞ þ 0 ðk; EÞ. The tools available for studying energy and momentum resolved self-energy are limited [11]. For example, bulk transport and optical spectroscopies provide some access to k-integrated self-energies while ARPES, the current method of choice [2–4,6,7,9], accesses k-resolved information for only the occupied states. 0031-9007=13=111(24)=246804(5)
Scanning tunneling microscopy or spectroscopy (STMSTS), renowned for its real-space atomic resolution capability, can also access the electronic structure in momentum space, of both occupied and unoccupied states, using Fourier transform scanning tunneling spectroscopy (FT-STS) [12–16]. Here, we report a high-resolution FT-STS measurement of the Ag(111) surface state, revealing fine structure in the otherwise parabolic electronic dispersion. Although 00 ðEÞ for the Ag(111) surface state has been studied previously by STS via electron scattering rates [17–19], the many-body effects on 0 ðEÞ have not been previously resolved by any other method. This achievement advances FT-STS as a quantitative probe of many-body interactions encoded in ðk; EÞ. STM-STS accesses electronic structure through realspace maps of modulations in differential conductance (dI=dV), proportional to the local density of states (LDOS). Periodic modulations arising from the interference of electrons scattered elastically by defects contain information about the initial and final momenta, revealed by a Fourier transform of the real space map. As these 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 constant energy segments of the band dispersion. By tracking the energy dependence of these peaks, the electronic dispersion EðkÞ can be obtained. This technique has been used to map coarse dispersions in many materials [20,21] and to examine scattering selection rules [22,23]. While the influence of many-body effects in FT-STS has been
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postulated since early reports [24], observation of their influence on the band dispersion and extraction of the self-energy has been missing. The two-dimensional Shockley surface state of the noble metal silver, Ag(111), was selected as an ideal system for demonstrating the quantitative capabilities of FT-STS. The Ag(111) surface state is well characterized [17–19,25–33], and while it exhibits distinct many-body effects, it lacks the complicated interplay of interactions that appear in more complex materials. The dispersion of the surface state is free-electron-like over a wide energy range given by ðkÞ ¼ @2 k2 =2m , where m is the effective mass and is the chemical potential. The dispersion is further modified by many-body interactions that are accurately described by conventional theory [18,19]. Thus a straightforward comparison with theory can be made, requiring few parameters. The e-e interaction decreases the electron lifetime for energies away from the Fermi level. The e-ph interaction introduces an additional scattering channel for energies greater than the Debye energy @D , decreasing the quasiparticle lifetime for jEj > @D and modifying the bare dispersion in a window @D of the Fermi energy Ef . Measurements were performed in a Createc ultrahigh vacuum STM at a temperature of 4.2 K with a tungsten tip formed by direct contact with the Ag crystal. The Ag(111) surface was cleaned by three cycles of Ar sputtering each followed by thermal annealing to 500 C. The IðVÞ map measured over 80 h consists of 380 380 spectra taken on a 239 239 nm2 area. Each IðVÞ spectrum consists of 512 data points and was Gaussian smoothed, maintaining a thermally limited energy resolution of E ¼ 1:5 meV. dI=dV spectra were acquired by numerical differentiation of the I-V sweep. Separate atomic resolution scans acquired on the same area as the conductance map were used to obtain the x-y calibration. This calibration enables an accurate determination of the spatial resolution of the conductance map [Fig. 1(a)] of 0:58 0:01 nm [34]. STM-STS measurement of the Ag(111) surface yields real-space conductance maps [see Fig. 1(a)] for the map at E ¼ Ef ] with circular LDOS modulations produced by scattering off of pointlike adsorbates from residual CO, and vertical modulations due to reflections from step edges. The small terraces on the surface produce subtle confinement effects not representative of the pristine surface state [31]. In order to access intrinsic surface properties, we removed their contribution by setting dI=dV in this region to the average value over the entire image. (This treatment does not change the QPI peak positions; see Supplemental Material [34].) The ability to isolate regions of interest in this way is unique to STM-STS, as probes such as ARPES would average over these domains. Figure 1(b) shows a typical dI=dV spectrum, averaged over an area free of scattering centers. The momentum space QPI intensity map Sðq; Ef Þ [Fig. 1(c)] exhibits a ring
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FIG. 1 (color online). (a) Conductance map (dI=dV) of a 239 239 nm2 area at E ¼ eV ¼ 0 meV (tip height set at V ¼ 100 meV, I ¼ 200 pA). LDOS modulation due to scattering at step edges and CO adsorbates are visible. The areas around the step edges indicated by white dashed lines were removed as discussed in the main text. The scale ranges from 2.3 to 3.3 pS. (b) Average dI=dV spectrum from a defect free area with a total size of 100 nm2 . The particle-hole symmetric steps at Ef likely originate from an inelastic co-tunneling pathway via phonon modes polarized perpendicular to the surface. (c) Absolute value of the Fourier transform (power spectrum) of the dI=dV map (E ¼ 0, panel a) showing a ring with radius q ¼ 2kf , where q is the scattering vector. The increased intensity along the qx direction originates from the step edge contributions. (d) The QPI line profile Sðjqj; E ¼ 0Þ. The scattering peak is slightly asymmetric with an enhanced intensity at low q, which is more pronounced at higher energies.
of radius qðEf Þ ¼ 2kðEf Þ, as expected for a free-electronlike dispersion where backscattering is dominant [35]. A line profile Sðjqj; Ef Þ of the QPI map is shown in Fig. 1(d), where due to the isotropic nature of ðkÞ we have performed an angular average of Sðq; Ef Þ in the regions above and below the dashed lines. This restriction isolates the contributions of the pointlike CO scatterers and improves our signal to noise. The momentum space resolution 1 is set by the dimension of the map q 0:0026 A (239 239 nm2 ) while the energy resolution is limited by thermal broadening of the tip and sample states. We now examine the detailed electronic structure of the Ag(111) surface state by considering the full energy dependence of the angle-averaged profile Sðjqj; EÞ, as shown in Fig. 2(a). Considering the coarse features of the data, we observe a parabolic band dispersion over a wide energy range, reflecting the nearly free-electron-like dispersion of the surface state. Furthermore, excluding the anomalies in the energy range Ef @D , the QPI intensity exhibits an overall decrease from the onset of the surface state to
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FIG. 2 (color online). (a) The dispersion of Sðq; EÞ, obtained by plotting the line profiles as shown in one dimension for all bias voltages. Overall, the dispersion is parabolic with ¼ 65 1 meV and m =me ¼ 0:41 0:02, obtained by fitting the peak position excluding the energy range ½20; 20 meV. The intensity of the scattering peak generally decreases with increasing energy but has a nonmonotonic increase near Ef . Subtle kinks in the dispersion are observed at @D , labeled 1. An additional scattering intensity below the onset of the surface state (E < 70 meV) is also observed, labeled 3. The inset reveals a subtle renormalization of the dispersion within Ef 14 meV. (b) Calculated QPI intensity for our model which includes e-e and e-ph interactions and assumes the CO adsorbates scatter in the unitary limit. The line of zero intensity (labeled 2) is a consequence of the unitary scattering.
higher energy. From the data we extract band parameters ¼ 65 1 meV and m =me ¼ 0:41 0:02, consistent with previous measurements [18,19,25,27]. These values define the bare dispersion in the absence of additional renormalizations due to many-body interactions. On a more detailed level, deviations from the parabolic dispersion, as well as an enhanced QPI signal intensity, are evident in the vicinity of Ef . This can be seen more clearly in the inset of Fig. 2(a). A similar increase in QPI signal intensity was observed near Ef in one of the first FT-STS reports on the Be(0001) surface state [12] and the e-ph interaction was later proposed as a possible origin [24]. We have observed this renormalization of the bare dispersion in four different data sets on Ag(111). To identify the source of these deviations and to assess the QPI signal we modeled the system using the T-matrix formalism, considering scattering from a single CO impurity in the unitary limit [35]. The QPI intensity jðq; EÞj is obtained from the Fourier transform of the impurityinduced LDOS modulations and is given by i X ðq; EÞ ¼ Im½Gðk; EÞTðEÞGðk þ q; EÞ: 2 k Here, T ¼ V0 sinðÞ expðiÞ is the T-matrix with phase shift ¼ =2 (unitary limit) and scattering potential V0 . The ‘‘bare’’ Green’s function in the absence of impurities is given by Gðk; EÞ ¼ ½E ðkÞ ðk; EÞ1 where ðkÞ ¼ @2 k2 =2m is the dispersion of the surface state and ðk; EÞ includes contributions from the e-e and e-ph
interactions, as well as an additional lifetime broadening due to scattering from the step edges. The e-e interaction was handled within Fermi liquid theory while the e-ph interaction was treated within standard Migdal theory [1] where the phonons were described by the Debye model. Under these approximations the self-energy is a function of energy only and is given by ðEÞ ¼ i iE2 =2 þ e-ph ðEÞ, where ¼ 2:5 meV and ¼ 62:7 meV parameterize scattering from the terraces and the e-e interaction, respectively, [18,19]. The imaginary part of e-ph ð!Þ is given by 00e-ph ðEÞ ¼ 2
Z D 0
d!0
!0 2 ½nf ð@!0 EÞ D
þ nf ð@!0 þ EÞ þ nb ð@!0 Þ; where nf and nb are the Fermi and Bose occupation factors, and is the dimensionless e-ph coupling strength. The real part of e-ph ðEÞ is obtained by the usual KramersKronig relations. From our data we extracted (see below) @D ¼ 14 meV and ¼ 0:13. These values, along with our measured values of m and , serve as input for our model. The simulated QPI intensity is shown in Fig. 2(b). The model closely reproduces both the coarse and fine details of the data. Our calculations show that overall QPI intensity is inversely related to the electron group velocity, producing a decrease in peak height as a function of energy if one excludes the increase within @D of Ef . This anomaly in the intensity, and the deviations from the parabolic band dispersion in the same energy range, arise from the e-ph interaction. The unitary nature of the scatterer also leaves a distinct fingerprint in the QPI intensity. In this limit, ðqÞ switches sign for q just above 2kðEÞ, which is a physical consequence of the =2 phase shift of the scattered quasiparticle. This produces a line of zero intensity on the large q side of the maximum QPI intensity (labeled as 2 in Fig. 2) when the absolute value is taken, as well an additional intensity at small q below the onset of the surface state (labeled as 3 in Fig. 2). Examining the data in Fig. 2(a) we see indications of the small q intensity below 65 meV; however, the zero intensity line is difficult to resolve below the noise floor. We now turn to a quantitative analysis of the data. For reference, Fig. 3(a) shows the e-ph self-energy used in Fig. 2(b). To extract ðEÞ, a Lorentzian was fit to the 1 around the peak data within a window of 0:01 A position expected from ðkÞ. [An example is shown as the dashed line in Fig. 1(d).] A plot of the QPI peak height S0 reflects the behavior of 00 ðEÞ as shown in Fig. 3(b), where we compare the data with the model. Both sets of data have been normalized (as described in the figure caption) to eliminate the tunneling matrix element’s role in setting the scale of the experimental data. There is good agreement between the model and experiment apart from a
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(a)
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FIG. 3 (color online). (a) Calculated real (0 , solid line) and imaginary (00 , dashed line) parts of the self-energy for the parameters used in Fig. 2(b). (b) Scattering peak height S0 ðq; EÞ as a function of energy determined experimentally (dots) and theoretically (solid line). The theory curve was normalized to the value at E ¼ 0 while the experimental data were normalized to the average value over the window ½5; 5 meV. (c) 0 ðEÞ determined from the difference between the scattering peak position EðkÞ and a parabolic fit ðqÞ. The solid line corresponds to the theoretically determined 0 as presented in (a). The dashed lines at 14 meV in (a) to (c) indicate the position of the Debye energy @D .
slight deviation around 20 meV, which we attribute to a set point effect below qF ¼ 2kf [18,36]. The decrease in 00 ðEÞ within @D ¼ 14 meV of Ef produces the nonmonotonic variation in peak height superimposed over the group velocity dependence previously discussed. This is due to the closing of the phonon scattering channel at energies below @D , resulting in longer-lived quasiparticles near Ef . The value @D required to reproduce the data is close to the value for the top of the bulk acoustic branches [33]. The real part 0 ðEÞ can be estimated from the data by taking the difference between the measured peak position and the large scale parabolic dispersion. The result is shown in Fig. 3(c), where peaks in 0 ðEÞ occur in the data at the same energy scale reflected in Fig. 3(b). We estimate the dimensionless strength of the e-ph coupling ¼ d0 =dEjE¼Ef ¼ 0:13 0:02 [1], consistent with previous estimates [18,28]. Finally, the details of the QPI intensity and dispersion are well reproduced without a
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self-consistent treatment of the T matrix, indicating that the CO impurities do not significantly alter the electronic properties of the Ag(111) surface state. The following steps provide a basis for the application of this technique to a new material: (i) Estimate the size of the effect (both the energy scale and coupling strength) to determine the resolution required in both E and k. (ii) The latter will determine the field of view required, while the largest q vector where many-body effects may be observed will set the required density of points over the field of view. (iii) For anisotropic materials, Sðq; EÞ becomes a threedimensional space which can be examined by plotting cuts in q, or by tracking particular scattering vectors qi . (iv) A mapping of the scattering vectors qi onto k must be identified. For complex materials this may require some a priori knowledge of the band structure and theory in order to deconvolve integration over k values connecting equidistant constant energy segments of ðkÞ. For complex materials, (iii) and (iv) will likely present the most significant challenges. Our FT-STS results provide a stunning visualization of the subtle modifications in dispersion and scattering intensity arising from many-body interactions in a simple system. This method provides high resolution in both momentum and energy that is competitive with state-ofthe-art ARPES. Moreover, FT-STS accesses both occupied and unoccupied states opening up the possibility of examining particle-hole asymmetric systems. These aspects give access to many-body features not previously observed in such a direct way. With enhanced stability and lower temperatures, further advancements in the application of FT-STS to quantify many-body interactions in more complex systems can undoubtedly be expected. However, perhaps the most compelling advantage of FT-STS is the prospect of exploiting STM’s unique spatial sensitivity to explore variations in the many-body interactions in nanoscale regions and intrinsically inhomogeneous materials. S. G. and S. J. contributed equally to this work. The authors thank E. van Heumen and G. Levy for useful discussions. This work is supported by NSERC, CFI, CIFAR, the University of British Columbia, and the Canada Research Chairs program (S. B.). S. G. acknowledges support from The Woods fund.
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