Giant Kink in Electron Dispersion of Strongly Coupled Lead

Our photoelectron spectroscopy study shows a giant kink in the electron dispersion, a sign of high-energy manybody interactions of electrons, in a wel...
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NANO LETTERS

Giant Kink in Electron Dispersion of Strongly Coupled Lead Nanowires

2009 Vol. 9, No. 5 1916-1920

Keun Su Kim and Han Woong Yeom* Institute of Physics and Applied Physics and Center for Atomic Wires and Layers, Yonsei UniVersity, Seoul 120-749, Korea Received January 7, 2009; Revised Manuscript Received February 18, 2009

ABSTRACT Our photoelectron spectroscopy study shows a giant kink in the electron dispersion, a sign of high-energy manybody interactions of electrons, in a well-ordered Pb nanowire array self-assembled on a silicon substrate. We show that the unique electronic band structure due to the strong lateral coupling and the atomic structure of the nanowires drives an enhanced manybody interaction for kinked electron dispersion. The major giant kink mechanisms discussed previously, the magnetic and plasmonic excitations, are not relevant in the present system, supporting the recent kink theory based purely on electron-electron correlation. This suggests that tailored electronic band structures in nano array systems can provide unprecedented ways to study manybody interactions of electrons.

Physical properties of condensed matters are largely governed by manybody interactions between electrons or between electrons and collective excitations. This is also true for various nano systems, and moreover, the reduced dimension in nano systems tends to enhance manybody interactions1-7 or often leads to unconventional interactions of electrons such as the non-Fermi liquid state of one-dimensional (1D) electrons in carbon nanotubes.8,9 In a nano system, one may also expect different manybody interactions from a bulk system when it has a unique crystal structure imposed by its reduced dimension. Beyond a single nano structure, arrays of nano structures or nano patterned systems can provide the unprecedented possibility to tune band structures, which in turn can result in various unexpected manybody interactions. This possibility, however, has not been fully exploited yet. Kinks in the dispersion of electrons are indicative of manybody interactions of electrons.10 The conventional kinks have been observed at 40-60 meV from Fermi energy by angle-resolved photoelectron spectroscopy (ARPES)11 and debated to originate from the electron-phonon interaction, which could be important for high-temperature superconductivity.12 However, recent ARPES studies discovered unusual “giant kinks” at energies much higher than phonons, above 100 meV, ubiquitously in various cuprate hightemperature superconductors,13-16 SrVO3,17 transition metal surfaces,18 and graphene.19 The origin of the giant kinks is under significant debate with various models in terms of couplings of electrons with excitations other than phonons such as magnetic fluctuations15,18,20 and low-energy plasmons.19,21 On the other hand, a few very recent studies have * Corresponding author. E-mail: [email protected]. 10.1021/nl900052s CCC: $40.75 Published on Web 03/30/2009

 2009 American Chemical Society

introduced purely electronic origins invoking the generic electron-electron interaction of the Fermi liquid.22,23 Since most of the proposed models may reproduce the kinked dispersion itself, an experimental test has been needed to clarify the underlying mechanism. However, the previous experimental studies have been ambiguous mainly because there exist diverse internal degrees of freedom in those materials to allow various excitations. We adopt a different strategy to solve this issue, that is, to look for a simple material with limited degrees of freedom. A heavy nonmagnetic metal would be a good test case because phonons and plasmons can be out of the energy scale and the possibility of the magnetic degree of freedom can be excluded. We indeed find that a simple metal, a Pb nanowire array formed on a stepped Si(111) substrate, exhibits a consistent giant kink. Since no magnetic degree of freedom is allowed and the plasmon energy is far apart in Pb, we can exclude the major kink models involving collective excitations. Furthermore, the overall spectroscopic features of the kink could be explained quantitatively by the electronic kink theory22 based on the intrinsic nature of Fermi liquid with the electron-electron interaction. The unique band structure of this system, as defined by the unique atomic structure of a single nanowire and modulated by the array periodicity, is thought to enhance the corresponding manybody interactions with a strong similarity to high-temperature superconductors and graphene. We conducted ARPES experiments on the Pohang Accelerator Laboratory 8A1 undulator beamline, equipped with a high-performance electron analyzer (Gammadata, Sweden).24 The angular and energy resolution was 0.15° and 15 meV at best. Microscopic structures were characterized using

Figure 1. (a) STM topograph of the Pb/Si(557) at 78 K with +0.5 V sample bias over an area of 7.4 × 8.2 nm2. The side view of the schematic structure model is at the bottom. The top (bottom) balls represent the Pb (Si) atoms. (b) The schematic Fermi surface reported previously,24 where six different bands are tagged by numbers. The contours in thick lines are those calculated from a tight binding model for a 2D Pb layer with a quasi square unit cell. (c) Evolution of the pristine 2D energy bands from the electron pocket at a high binding energy (red) to a hole pocket above Fermi energy (blue). There exist saddle-point singularities at the apexes of the diamond-shaped Fermi contour.

commercial scanning-tunneling microscopy (STM) at 78 K.25 Low-energy-electron diffraction (LEED) was available on both experimental stations to check the surface order. The n-type Si(557) wafer with a 9.5° miscut from [111] was thermally cleaned. This vicinal substrate provides a regular step array.26-28 To form a uniform Pb nanowire array, we deposited about 2 monolayer Pb and annealed it at 620-640 K to remove excess Pb as detailed previously.25-27 Figure 1a shows the STM image. Although the atoms near the steps are strongly enhanced in the image, the Pb monolayer covers densely the narrow Si terraces of four Si rows in width as separated by single-atomic steps (schematics in Figure 1a).24-26 The system is, thus, monolayer-thick nanowires (or nanoribbons) arrayed on a uniform step-terrace superlattice with a period of 1.57 nm, which exhibit a strongly anisotropic and metallic conductivity.29 Figure 1b describes its Fermi surface, reported and analyzed previously,24 which is composed of both major quasi-1D bands (open contours of m2, m3, m4, and m5) and minor two-dimensional (2D) (closed contours of m1 and m6) bands. This rather complex Fermi surface can, however, be traced back to simple 2D Fermi contours with the shapes of an ellipse and a rounded diamond (thick lines in Figure 1b).24 The tight binding model reproduces well these pristine 2D Nano Lett., Vol. 9, No. 5, 2009

Figure 2. (a) ARPES intensity map collected along the nanowires at 70 K with a photon energy of 80 eV. The measured k-space cut is indicated by arrows in Figure 1b. Each band is tagged as in Figure 1b. The parabolic lines are the dispersions of m2 and m3 bands calculated from a tight binding model.24 (b) Enlarged map of the boxed area in (a) for the m4 band. The peak positions (dots) extracted from the MDC fits are overlaid together with the tight binding dispersion (solid line). (c) Examples of EDC and MDC to show the quasiparticle peak fitted well with the Lorenztian shape. (d) A series of MDCs to show the kinked dispersion. Peak positions of m4 are guided by straight lines.

Fermi contours (Figure 1c) by the in-plane p-orbitals of Pb atoms in a 2D monolayer with a quasi square lattice.24 The electrons for Pb-Si bonds would have higher binding energies without any contribution around Fermi energy focused here. Note that this quasi square lattice does not occur on a flat Si(111) surface with a triangular symmetry and is thought to be due to the structural strain imposed by the wire dimension.24 Thus, the diamond-shaped Fermi contour has not been observed for 2D adsorbate monolayers of Pb.30 The complex band splitting observed is due to the 1D modulation by the periodic nanowire array (Figure 1a); the 2D pristine bands are periodically translated along ky (perpendicular to the wires) and gapped at crossing points.24,31 In other words, the electrons of neighboring nanowires are strongly coupled to evolve from strictly 1D bands into a complicated mixture of quasi-1D and 2D bands. That is, this system has a unique band structure through the unique atomic structure of a single wire and the modulation by the array potential. Figure 2a shows the electron dispersions for the Pb nanowires, as measured by ARPES. One can readily find four highly dispersive bands crossing the Fermi level. Among them, the m2, m3, and m4 bands originate from the diamondshaped 2D Fermi contours. The dispersions of m2 and m3 bands are in good agreement with the parabolic dispersion expected from the tight binding calculation.24 However, a noticeable deviation from the parabolic dispersion is clear for the m4 band (arrow in Figure 2a and the details in Figure 2b). For a quantitative comparison, we fit the momentumdistribution curves (MDCs) of the ARPES spectra, shown 1917

Figure 3. (a) Real part of self-energy ReΣ(ω) obtained by taking the energy difference between the tight binding and the measured dispersions. The line is the linear fit of ReΣ(ω), whose slope corresponds to the coupling constant (λ). (b) Imaginary part of selfenergy ImΣ(ω) derived from the line width of the MDC (dots). A constant offset (0.08 eV) is subtracted. The solid (dashed) line is a fit using a quadratic (linear) function. The thick line is ImΣKK(ω) obtained from ReΣ(ω) through the Kramers-Kronig relation.

partly in Figure 2c, by Lorentzian line shapes. These fits accurately determine the spectral peak positions as given partly in Figure 2b. The experimental dispersion obviously shows a strong kink, indicating the energy renormalization of electrons from the Fermi energy up to a binding energy of ca. -0.3 eV (ω1). From this energy to ca. -0.5 eV (ω2), the dispersion abruptly returns back to the simple parabolic dispersion. Note that the kink energy is as high as 300-500 meV and all above features and more details shown below are consistent with the giant kinks observed previously in various exotic materials.13-19 If the Fermi-liquid scheme is applicable to this giant kink, its dispersion would follow the quasiparticle dispersion; at low energy the electrons are dressed by manybody interactions to have the renormalized energy and velocity. This behavior is contained in the complex self-energy Σ(k, ω) ) ReΣ(k, ω) + iImΣ(k, ω), which determines the quasiparticle spectral function. If the k-independence of self-energy is assumed, the self-energy for the given direction can be extracted from the experiment;32,33 the real part ReΣ(ω) from the energy difference between the bare band (εk) and the renormalized dispersion and the imaginary part ImΣ(ω) from the MDC width at each energy. The bare band dispersion is rather rigorously set by the tight binding calculation with parameters optimized to reproduce the whole experimental Fermi surface as well as the dispersions of nearly unrenormalized m2 and m3 bands. The resulting ReΣ(ω) is shown in Figure 3a. It exhibits a simple linear dependence on energy up to ω1 ) 0.31 eV. We evaluated the dimensionless coupling constant λ ≈ 0.35 from the slope of ReΣ(ω) (the line in Figure 3a). This value is moderate in between those of cuprates (1.1)34 and graphene (smaller than 0.1).19 The corresponding imaginary part ImΣ(ω) is shown in Figure 3b. After subtracting a constant impurity scattering backgroud, ImΣ(ω) can be fitted well with a parabola. ReΣ(ω) and ImΣ(ω) are mutually connected through the so-called Kramers-Kronig causality relation.32 The ImΣKK(ω) (thick line) calculated from ReΣ(ω) through this relation is fully consistent with the above ImΣ(ω) derived independently from MDCs (Figure 3b). This excellent agreement supports the validity of the self-energy analysis adapted here. 1918

Nonintrinsic origins of giant kinks, as raised in a few previous works,35 can be ruled out here: (i) We do not observe any significant disorder or defects on the surface in STM.25 (ii) The possibility of a commensurate superperiodicity can also be excluded from LEED and STM.24,25 There exists a ×2 modulation along the step edges,25 which, however, has no observable effect on the band dispersion for the quasi-1D electronic states discussed here.24 (iii) The artifact from the selection rule can be rejected since the kink structure is consistently reproduced for different photon energies and Brillouin zones. (iv) Any mixing with other bands is not expected since there is no other surface states at the given momentum and energy, and the substrate states do not exist due to the Si bulk-band gap; the corresponding part is out of the projected Si bulk bands. (v) The systematic k-dependence of the kink shown below cannot be explained by an extrinsic effect. The observation of the giant kink in the Pb nanowire provides an important insight into its mysterious origin. First, we can rule out the possibility of a non-Fermi-liquid origin36 since (i) we find no signature of non-Fermi-liquid behavior like the power-law dependence of spectral weights near Fermi energy and (ii) the self-energy follows well the Fermiliquid theory expectations as discussed below. Within the Fermi-liquid framework, the energy renormalization of quasiparticles can be realized either by couplings to collective excitations or by the electronic correlation. The most popular collective excitation of phonons is impossible in such a high energy scale for a heavy element such as Pb. Only a marginal contribution of phonons has recently been indicated even for the low-energy kinks observed in other materials.37 The second possibility, the plasmon excitation, as suggested in graphene19 and cuprates,21 is also far out of the energy scale in Pb because of its high electron density. The surface and interface plasmon energies for the Pb/Si(111) system are larger than 7000 meV.38 Although there may be a lower energy plasmon excited perpendicular to wires, the plasmon (or any bonsonic) coupling would yield a linear energy dependence in the imaginary part of self-energy in clear contrast to the experiment.39 The third one discussed actively in literature is the magnetic fluctuations, which simply do not exist in Pb. Thus, a more general mechanism should be considered beyond the coupling with a specific collective mode. In this respect, it is notable that a recent theory has introduced a purely electronic mechanism.22 This theory focuses on one intrinsic property of Fermi liquid, that its validity is limited within an energy range close to Fermi level. Thus, there should be a crossover from the renormalized (ZFL·εk) to the unrenormalized (εk) dispersion at a specific energy. This crossover marks the termination point of the Fermi-liquid regime and naturally produces the kinked dispersion. The energy of electronic kinks is predicted to depend on the coherent factor ZFL ) 1/(1 + λ) and the bandwidth (W).22 We evaluated the kink energy for the present case by adopting the formula given by this theory, ZFL·(2 - 1)·D (D, approximately half of W), with the parameters quantified from the above analysis. This predicts Nano Lett., Vol. 9, No. 5, 2009

Figure 4. (a) A series of dispersions of the m4 band along the nanowires for different momentum cuts as shown in (b). Dots represent the measured dispersions from MDCs, and the lines are those calculated from the tight binding model.24 (b) Schematics of the Fermi contours where the k-space cuts scanned are indicated by markers and lines. (c) Coupling constant (λ) derived from the self-energy analysis as in Figure 3 as a function of ky.

the kink energy of 0.34 ( 0.03 eV in quantitative agreement with the experimental one of 0.31 eV. Moreover, the linear and quadratic behaviors of ReΣ(ω) and ImΣ(ω) observed here are consistently expected by this theory as due to the decay process by electron-hole pair excitations through the electron correlation.23 In contrast, a bosonic coupling, for example, the electron-phonon and electron-plasmon case, induces a linear energy dependence of ImΣ(ω) as also mentioned above.11,39 Within this theory, the energy renormalization mainly comes from the electron correlation. The size of the kink, then, reflects the strength of the electron correlation. From the Born approximation, we evaluated the on-site Coulomb energy U ≈ 2.8 eV using a simple formula11 2β ) (πU2)/ (2W3) where β ) 0.47 from ImΣ(ω) (Figure 3b) and the bandwidth W ) 2.4 eV from the tight binding calculation. This U value is not unusual if one consider that the U values for simple metal adsorbates on Si can be as large as 1.0-4.0 eV.40 Beyond the mere existence of the giant kink, the present result bears further complexity. That is, the kink is observed only for a specific band m4 among multiple branches split from the same 2D Fermi contour (Figure 1b). In addition to such a band selectivity, the giant kink in the m4 band exhibits a strong momentum dependence. We show below that these two aspects are closely related. Figure 4a shows the variation of the kink with changing momentum along the Fermi contour of m4. It is apparent that the strength of the kink diminishes rapidly away from the high symmetry point ky ≈ 0 or Γ, and no clear kink is observable beyond ky ≈ 0.4(2π/L) within the experimental resolution. The plot of coupling constants as a function of ky (Figure 4c) shows a strong, almost exponential, anisotropy. Nano Lett., Vol. 9, No. 5, 2009

While the Fermi surface singularity was not explicitly discussed, a very similar anisotropy of the giant kink was also observed for cuprates.14-16 In order to explain this anisotropy, we note that the diamond-shaped 2D Fermi contour is very close to those in most of cuprates.12 For such a band topology, the diverging density of states at the saddle points (the apex of the diamond, see Figure 1c) was theoretically predicted to produce an abrupt and anisotropic variation of manybody couplings.41,42 Since the m2 and m3 bands originate from the parts of the Fermi contours (Figure 1b) that deviate largely from the saddle points, the absence of the kinks may be explained. Moreover, the lack of the saddle point singularity in Fermi surfaces could be the reason for the Pb monolayers on flat Si(111) with more circular Fermi contours to exhibit no kink.30 This observation indicates that the occurrence of giant kinks is not simply dictated by the electron correlation, but its size can be modulated by the Fermi surface singularity. This effect is, however, beyond the present electronic kink theory, and the microscopic mechanism for the role of the singularity is not clear at all, calling for a more refined k-resolved theory. The present work indicates that the giant kink can be a generic phenomenon of Fermi liquid through the electron correlation and not explicitly related to specific collective excitations. The electronic kink mechansim, whose experimental evidence is provided for the first time, might be applied as well to other materials systems such as hightemperature superconductors and graphene. The analysis method for the kinked dispersion, verified here, can be applied to various nano systems to quantify the electron correlation. The unique low dimensional band structures of nano arrayed systems brought by their unique atomic structures and tuned by the array potentials can provide unprecedented possibility and flexibility to investigate manybody interactions of electrons. Acknowledgment. This work was supported by KOSEF through the Center for Atomic Wires and Layers of the CRi program. We are grateful to H. Morikawa, W. H. Choi, and C. Kim for useful discussions. References (1) Segovia, P.; Purdie, D.; Hengsberger, M.; Baer, Y. Nature (London) 1999, 402, 504–507. (2) Yeom, H. W.; Takeda, S.; Rotenberg, E.; Matsuda, I.; Horikoshi, K.; Schaefer, J.; Lee, C. M.; Kevan, S. D.; Ohta, T.; Nagao, T.; Hasegawa, S. Phys. ReV. Lett. 1999, 82, 4898. (3) Terada, Y.; Yoshida, S.; Okubo, A.; Kanazawa, K.; Xu, M.; Takeuchi, O.; Shigekawa, H. Nano Lett. 2008, 8, 3577–3581. (4) Zeng, C.; Kent, P. R. C.; Kim, T.-H.; LI, A.-P.; Weitering, H. H. Nat. Mater. 2008, 7, 539–542. (5) Grioni, M.; Pons, S.; Frantzeskakis, E. J. Phys.: Condens. Matter 2009, 21, 023201. (6) Maruccio, G.; Janson, M; Schramm, A.; Meyer, C.; Matsui, T.; Heyn, C.; Hansen, W.; Wiesendanger, R.; Rontani, M; Molinari, E. Nano Lett. 2007, 7, 2701–2706. (7) Park, C.-H.; Giustino, F.; Cohen, M. L.; Louie, S. G. Nano Lett. 2008, 8, 4229–4233. (8) Bockrath, M.; Cobden, D. H.; Lu, J.; Rinzler, A. G.; Smalley, R. E.; Balents, L.; McEuen, P. L. Nature (London) 1999, 397, 598–601. (9) Ishii, H.; Kataura, H.; Shiozawa, H.; Yoshioka, H.; Otsubo, H.; Takayama, Y.; Miyahara, T.; Suzuki, S.; Achiba, Y.; Nakatake, M.; Narimura, T.; Higashiguchi, M.; Shimada, K.; Namatame, H.; Taniguchi, M. Nature (London) 2003, 426, 540–544. 1919

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NL900052S

Nano Lett., Vol. 9, No. 5, 2009