Electron-Phonon Coupling in Highly-Screened Graphene
David A. Siegel, Choongyu Hwang, Alexi V. Fedorov, Alessandra Lanzara
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug Electron-Phonon Coupling in Highly-Screened Graphene
D. A. Siegel,
1, 2
C. G. Hwang, A. V. Fedorov, and A. Lanzara
1, 2 Department of Physics, University of California, Berkeley, CA 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA (Dated: November 19, 2018)Photoemission studies of graphene have resulted in a long-standing controversy concerning thestrength of the experimental electron-phonon interaction in comparison with theoretical calculations.Using high-resolution angle-resolved photoemission spectroscopy we study graphene grown on acopper substrate, where the metallic screening of the substrate substantially reduces the electron-electron interaction, simplifying the comparison of the electron-phonon interaction between theoryand experiment. By taking the nonlinear bare bandstructure into account, we are able to show thatthe strength of the electron-phonon interaction does indeed agree with theoretical calculations. Inaddition, we observe a significant bandgap at the Dirac point of graphene.
The electron-electron and electron-phonon interactionsare two of the fundamental interactions in many-bodyphysics, giving rise to superconductivity, Mott-insulatingbehavior, and other collective phenomena. These phe-nomena are often studied in association with graphenenot only because graphene is a simple system featuringtwo carbon atoms per unit cell [1] but also due to theunique potential of this material. However, despite thisapparent simplicity, there have been many difficulties inmatching theoretical predictions to experimental studiesof electron-phonon coupling in graphene [2]. The natureof these discrepancies is due to the way the electron-phonon coupling constant λ is extracted from the ex-perimental data, and more specifically the way the barevelocity is determined. Within the Migdal-Eliashbergregime, electron-phonon coupling results in single phononexcitations that can be treated as perturbations to thebare band dispersion and leads to a renormalization ofthe group velocity with respect to the electronic bareband. The relative change of the renormalized velocitywith respect to the bare velocity provides a measure ofthe electron-phonon coupling constant λ . Angle-resolvedphotoemission spectroscopy has shown to be an invalu-able probe to extract this constant since it can directlymeasure the single particle spectral function and hencethe renormalized velocity. However, the correct deter-mination of λ rests on an accurate determination of thebare velocity, which is often done by assuming a linearband approximation between the Fermi energy and highenergy, a procedure that has been found to be grosslyinappropriate for graphene [2].The LDA band velocity is in fact known to change sig-nificantly over the relevant energy scales, which greatlyaffects the measured values of the electron-phonon cou-pling constant λ if a linear band is assumed. Further com-plicating the analysis, electronic correlations are knownto renormalize the bare-band velocity in a nonlinear man-ner, to a degree determined by the dielectric screening ofthe substrate [3–7]. It should also be noted that theelectron-phonon coupling strength may be enhanced by interplay between electron-electron and electron-phononinteractions[8–10]. Since the experimental bare band isso difficult to determine, ARPES studies typically resortto the linear bare band approximation when determin-ing the electron-phonon coupling strength in graphene,resulting in an over- or under-estimate of the actualelectron-phonon coupling constant when extracted fromthe real self-energy[11–17].In light of these difficulties, one way to simplify thestudy of electron-phonon coupling in graphene might beto grow graphene on a metallic substrate, a growth tech-nique that has recently become popular due to its rel-evance for technological applications[18]. On a metallicsubstrate, the electron-electron interaction in graphene isexpected to be highly screened, which would remove ve-locity renormalizations due to electronic correlations andcause the bare dispersion (experimental minus electron-phonon interaction) to converge to the LDA result [19–21]. In this highly screened limit, the curvature of thegraphene LDA band structure may be taken into accountwhen analyzing electron-phonon coupling. Therefore thepresence of a metallic substrate allows us to examine theelectron-phonon interaction with an accuracy unmatchedin other systems, leading to a straightforward analysis ofthe experimental data.Here we present a high-resolution ARPES study ofgraphene grown on a copper substrate. Starting froma basic characterization of this system, which has neverbeen studied before by photoemission spectroscopy, weobserve sharp dispersions due to the copper substrateand graphene overlayer, including a band gap at theDirac point of graphene. Proceeding to examine themany-body physics in highly screened graphene, we findan overall agreement between the experimental band-structure and the LDA band calculations. Taking thecurvature of the LDA band into account, we find closeagreement between experimentally extracted electron-phonon coupling constants and theoretical calculations,providing the first real measurement of the electron-phonon coupling constant and providing closure to a FIG. 1: (Color online) (a) Partial map of the Fermi surface.Arrows and circles correspond to bands from the copper sub-strate and graphene overlayer, respectively. Dashed blacklines correspond to the hexagonal Brillouin zone of graphene.The K point is labelled, while the Γ-point is not shown, lo-cated at (k x ,k y ) = (0,0). The black horizontal line throughthe K point illustrates the orientation of the data taken inpanels (b) and (c), while the vertical purple line illustrates theorientation of the data in panel (d). (b) ARPES dispersionand (c) EDCs taken along the Γ-K direction at k y = 0 ˚A − ,showing that the graphene bands are n-doped with a bandgapand intensity minimum at the Dirac point. The presence of abandgap creates two peaks in the EDCs at the K-point (peakpositions marked in red). (d) EDCs taken through the Diracpoint along constant k x . In contrast to panel (c), where pho-toemission matrix elements suppress one branch of the cone,the photoemission intensity in panel (d) is symmetric andallows the presence of a bandgap to be easily seen. (e) Angle-integrated spectra of panel (b) and of highly-doped graphene. long-standing debate.Samples were grown on copper films as previously re-ported [18]. High-resolution ARPES data were taken atBL10.0.1 and BL12.0.1 of the Advanced Light Source at atemperature of 15 ◦ K after annealing samples to 1000 ◦ K,using a photon energy of 50eV. The vacuum was betterthan 3 × − Torr. Potassium was deposited in situwith an SAES potassium vapor source.Figure 1(a) shows a Fermi surface map with bands dueto the copper substrate and graphene overlayers. Twosets of copper bands and Dirac cones can be distinguisheddue to the presence of rotated crystallographic domains
FIG. 2: (Color online) (a-c) ARPES dispersions for severaldopings, taken along the Γ-K direction. Arrows indicate theDirac point in each image. Panels (a), (b), and (c) correspondto dispersions α , γ , ǫ , respectively, in panel (d). (d) Exper-imental dispersions (red) for several dopings extracted fromMDC peak positions, and LDA bands (blue) for the samedoping along the Γ-K direction. Greek letters α , β , γ , δ , ǫ label these dispersions in order of increasing doping. Inset:Comparison of dispersions α and ǫ , showing that the electron-phonon kink is stronger for the more highly doped dispersion.FIG. 3: (Color online) (a) Experimental coupling constantsare given as a function of electronic charge density in red.The data agrees with theoretical calculations [30], shown inblack. The error generated when the linear band approxima-tion is applied to the curved (bare) LDA band is given in blue,for comparison. (b) A zoomed-out version of panel (a), alsoshowing results from the cited references in green (differentreferences have different symbols depending on substrate). of the substrate. The Dirac cones of graphene are visible,and are electron-doped due to their proximity to the cop-per substrate. Although doping can change from sampleto sample, typical values are approximately 2 × cm − .ARPES data through a single Dirac cone is shown alongthe Γ-K direction in figure 1(b). In this measurement ge-ometry, the photoemission intensity is suppressed alonghalf of the cone[22].The dispersion in the vicinity of the Dirac point hasbeen the subject of some controversy in the past. The va-lence and conduction bands are not collinear, possessing aregion of vertical intensity between them. Whether this isdue to the presence of a bandgap in the bare dispersion ora many-body effect has been hotly debated[23, 24]. In thepresent case of graphene on a copper substrate, the di-electric screening of the highly conductive substrate rulesout the possibility of electron-plasmon coupling[19, 25],and instead implies the presence of a bandgap at theDirac point[23, 26]. We find the separation betweenvalence and conduction bands to be somewhat sample-dependent, with a typical bandgap of 400 ±
50 meV, whendetermined from the separation between the peaks of en-ergy distribution curves (EDCs, intensity profiles at con-stant momentum), as shown in figures 1c and 1d, withtwo peaks visible at the Dirac point momentum. Theangle-integrated intensity (figure 1e) shows a V-shapedintensity profile, with a minimum at the Dirac point en-ergy, and increasing intensity away from the Dirac point.Far from the Dirac point, the valence and conductionbands are not collinear, with an overall offset of 100 ± λ , can be extracted from either part of the self en-ergy, although in practice the real part of the self-energyis often more reliable, since the imaginary part is moresensitive to noise and the influence of impurity broaden-ing. We have therefore focused on the real self-energy inour analysis.Knowing the bare graphene band, ReΣ is given as thedifference between the experimental and bare band po-sitions. From knowledge of ReΣ, the electron-phononcoupling constant λ can be expressed as λ k = − ∂Re Σ k ( E ) ∂E (cid:12)(cid:12)(cid:12)(cid:12) E = E F , (1)or equivalently, λ k = v k ( E F ) v k ( E F ) − , (2)where v k ( E F ) and v k ( E F ) are the bare and renormalizedvelocities at the Fermi level, respectively. However, thebare band of graphene is not linear, so the method ofextracting λ according to the formula λ k = v v − , (3)(where v and v are the band velocities at higher andlower binding energy than the phonon, respectively) doesnot work, nor will any other method that assumes a linearbare band.[2]The extracted coupling constants are compared withelectron-phonon coupling calculations [30] in figure 3.The agreement between experiment and theory is strik-ing, providing the first experimental support of theoret-ical electron-phonon calculations. Having said this, ouranalysis may require a small correction. Ab initio calcu-lations expect a finite el-ph self-energy even at high bind-ing energies[32]. This differs from our results, where theLDA band gives good agreement with experiment at highbinding energies. This discrepancy could correspond to adifference in the measured coupling constant of approx-imately 0.015 ± v k to be the slope of the line that intersects theLDA band at energies E = E F and E = E F -0.4 eV usingEq. 2. The results, shown as the blue “LDA” line in fig-ure 3, correspond to the linear bare band approximationwhen no electron-phonon coupling is taking place. Datafrom previous works are given in panel (b). The errorfrom the bare band approximation is more than twice aslarge as the actual electron-phonon coupling constant[2].On the other hand, in cases where the bare band curvesin the opposite direction (“concave-up”), such as in thepresence of strong electron-electron interactions[3–7, 33],for the LDA band on the opposite side of the Diraccone (along the M-K-Gamma direction)[2], or for bilayergraphene[17], the linear approximation underestimatesthe coupling constant.In conclusion, we have shown for the first time thatthe magnitude of electron-phonon coupling in grapheneagrees with theoretical calculations. These results settlea long-standing controversy in the field, confirming thevalidity of theoretical calculations, and casting doubt onthe conclusions of many experimental works. This workis also generally applicable to future experiments thatrequire studying electron-phonon coupling in materialswith nonlinear bare bands. We have also shown thatthere is a significant bandgap in graphene grown epitax-ially on a copper substrate, a discovery which may pavethe way for future technological applications.We are greatly indebted to Baisong Geng and FengWang for providing us with the high quality graphenesamples that have made this study possible. We wouldalso like to thank Cheol-Hwan Park for enlightening dis-cussions. ARPES work was supported by the Director,Office of Science, Office of Basic Energy Sciences, Ma-terials Sciences and Engineering Division, of the U.S.Department of Energy under Contract No. DE-AC02-05CH11231.Correspondence and requests for materials should beaddressed to [email protected]. [1] P. R. Wallace, Phys. Rev. , 622 (1946).[2] C.-H. Park, F. Giustino, J. L. McChesney, A. Bostwick,T. Ohta, E. Rotenberg, M. L. Cohen, and S. G. Louie,Phys. Rev. B , 113410 (2008)[3] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109(2009).[4] J. Gonz´alez, F. Guinea, and M. A. H. Vozmediano, Nucl.Phys. B , 595 (1994).[5] J. Gonz´alez, F. Guinea, and M. A. H. Vozmediano,
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