Ytterbium-driven strong enhancement of electron-phonon coupling in graphene

Ytterbium-driven strong enhancement of electron-phonon coupling in graphene

Choongyu Hwang    Duck Young Kim    D. A. Siegel    Kevin T. Chan    J. Noffsinger    A. V. Fedorov    Marvin L. Cohen    Börje Johansson    J. B. Neaton    A. Lanzara ALanzara@lbl.gov Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Department of Physics, Pusan National University, Busan 609-735, Republic of Korea Geophysical Laboratory, Carnegie Institution of Washington, Washington DC 20015, USA Department of Physics, University of California, Berkeley, CA 94720, USA Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Department of Materials and Engineering, Royal Institute of Technology, Stockholm, SE-100 44, Sweden The Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Kavli Energy Nanosciences Institute at Berkeley, Berkeley, CA 94720, USA.
July 19, 2019
Abstract

We present high-resolution angle-resolved photoemission spectroscopy study in conjunction with first principles calculations to investigate how the interaction of electrons with phonons in graphene is modified by the presence of Yb. We find that the transferred charges from Yb to the graphene layer hybridize with the graphene bands, leading to a strong enhancement of the electron-phonon interaction. Specifically, the electron-phonon coupling constant is increased by as much as a factor of 10 upon the introduction of Yb with respect to as grown graphene (0.05). The observed coupling constant constitutes the highest value ever measured for graphene and suggests that the hybridization between graphene and the adatoms might be a critical parameter in realizing superconducting graphene.

pacs:
71.38.-k,72.10.Di,73.20.-r,79.60.-i

I Introduction

The interaction of electrons with phonons is of practical and fundamental interest in graphene, as it not only affects the transport properties of actual devices Efetov (), but also induces novel phenomena such as charge density waves Rahnejat () and superconductivity Uchoa (). Hence the manipulation of the electron-phonon coupling is an important issue to realize graphene-based electronic and spintronic devices Geim () and to create new strongly correlated electron phases. In fact, several methods have been proposed to modify the electron-phonon coupling constant, , of graphene using charge carrier density Calandra_resolution (), magnetic field Faugeras (), disorder Song (), and adatoms Profeta (). Among them, the change of charge carrier density can tune the strength of electron-phonon coupling up to 0.05 Calandra_resolution (), while electron-electron interactions are efficiently suppressed DavidNJP (). On the other hand, the presence of adatoms is predicted to drastically enhance electron-phonon coupling up to =0.61 Profeta (), so that graphene enters the regime where phonon-mediated superconductivity might exist Savini (); Profeta (). However, experimental evidence of this striking enhancement in graphene has been controversial so far.

The most prominent manifestation of the electron-phonon coupling is a renormalization or kink of the electronic band structure at the phonon energy accompanied by a change in the charge carrier scattering rate. These effects are directly observed using angle-resolved photoemission spectroscopy (ARPES) AM (). However, experimental studies on the role of adatoms for the electron-phonon coupling of graphene via ARPES have been debated due to the hybridization of the adatom band with the graphene bands, referred to as band structure effect, resulting in an apparent enhancement and anisotropy of the electron-phonon coupling strength Jessica (); Calandra_BandStructureEffect (); Park_vanHove (). On the other hand, strong enhancement of the electron-phonon coupling through adatom intercalation have been reported for graphite, and discussed in same cases as the driver for superconductivity Valla2009 (); Yang (). These previous results suggest the importance of combining experimental and theoretical studies to understand the enhancement of electron-phonon coupling in graphene.

Here we present high-resolution ARPES study showing a strong enhancement of the electron-phonon coupling strength in a monolayer graphene sheet via Yb adsorption. A direct comparison with the theoretical band structure determined by first principles calculations show that the Yb 6 electrons transferred to the graphene layer are hybridized with the graphene bands, resulting in an enhanced electron-phonon coupling from =0.05 for as grown graphene to =0.43 for graphene with Yb. This observation constitutes the highest value ever measured for graphene and is in line with the density-functional perturbation theory that predicts an enhancement of from 0.02 to 0.51.

Ii Methods

ii.1 Experimental details

Figure 1: (Color online) (a) Fermi surface of graphene upon the introduction of Yb. Inset shows a zoomed-in view near the K point (, )=(0, 1.7). (b) Schematic drawing of crystal structure of graphene with inhomogeneous contribution of Yb (the buffer layer and the SiC substrate are not drown for simplicity). (c-d) Raw ARPES data of graphene in the presence of Yb (G+Yb/G: panel (c)) and as grown graphene (As grown G: panel (d)), through the K point perpendicular to the K direction as denoted by the red line in the inset. (e) Calculated bands of graphene in the presence of Yb (GYb/G) perpendicular to the K direction. The G and Yb/G bands are purple and red curves, respectively. The graphene bands are denoted by and , and the Yb/G bands by and . Yb 4 and 4 electrons are denoted by and .

Single layer graphene was grown epitaxially on a 6H-SiC(0001) substrate by an e-beam heating method as described elsewhere Rolling (). Yb was deposited on graphene at 100 K, followed by repeated annealing processes from 400 K to 1000 K to find a stable geometric structure. This process is well-known to enhance intercalation of alkali- and alkali-earth metals such as K and Ca Jessica (), Rb and Cs Watcharinyanon2011 (). This is also true for Yb Watcharinyanon2013_a (); Watcharinyanon2013_b () when annealed above 200 C. As a result, the graphene sample in the presence of Yb exhibits a coexisting phase of Yb-intercalated graphene and graphene without Yb, as observed in Fig. 1(a). High-resolution ARPES experiments were performed at beamline 12.0.1 of the Advanced Light Source in ultra-high vacuum maintained below 210 Torr using a photon energy of 50 eV. The energy and angular resolutions were 32 meV and 0.2 , respectively. The measurement temperature was 15 K.

ii.2 Electronic band structure calculations

The electronic band structure of graphene with Yb are obtained for YbC by ab initio total energy calculations with a plane-wave basis set Cohen () performed using the Vienna Ab-initio Simulation Package (VASP) PhysRevB.47.558 (); Kresse1996 (); PhysRevB.54.11169 (). Projector augmented wave (PAW) potentials Blochl1994 (); Kresse1999 () with a plane-wave cutoff of 500 eV are used. The exchange-correlation of electrons was treated within the generalized gradient approximation (GGA) as implemented by Perdew, Burke, and Ernzerhof Perdew (). The comparison between the measured and the calculated bands using GGAU correction to the electrons of Yb bears 2.0 eV for the on-site Coulomb interaction (U) and 0.7 eV for the intra-atomic exchange interaction (J) Liechtenstein1995 (). These values differ from 5.4 eV and 0.7 eV, respectively, expected for Yb-intercalated graphite as extracted from the full potential linear augmented plane wave method (LAPW) with correction Mazinb (). The value calculated within scheme is usually an overestimate due to the confined screening charge in the same atomic sphere Mazinb (). Although it is not straightforward to directly compare values estimated by two different correction methods, Yb/G shows smaller value than that of Yb-intercalated graphite.

Iii Results

Figure 2: (Color online) (a) Energy spectra of the Yb/G band taken perpendicular to the K direction at =2.05 Å denoted in the inset. The intensity spectrum is taken at =0.2 Å denoted by the black solid line. (b-d) Energy spectra of the Yb/G band taken perpendicular to the K direction at =1.85, 1.65, and 1.7 Å, respectively, denoted in the inset of panel (a). (e) Energy spectra of the Yb/G bands parallel to the K direction denoted in the inset of panel (a). represents the hybridization energy between the G () and Yb/G ( and ) bands, denoted by white dashed lines, and is the Dirac energy.

Figure 1(a) shows a photoelectron intensity map at as a function of two dimensional wave vectors and , for graphene with Yb. Two pieces of Fermi surface can be clearly distinguished: one with a crescent-like shape centered at the Brillouin zone corner K (zoomed-in in the inset), which resembles the one measured for as grown graphene on SiC(0001) Hwang2011 (), and the other with a triangular shape with the apex near the M point, similar to that of highly electron-doped graphene Jessica (). The observation of these two Fermi surfaces suggests a coexistence of graphene with and without Yb, as schematically shown in Fig. 1(b), similar to the case of Rb- and Cs-adsorbed graphene Watcharinyanon2011 () and consistent with previous results on Yb-intercalated graphene Watcharinyanon2013_a (); Watcharinyanon2013_b (). An estimate of the charge doping in the graphene bands introduced by Yb is given by the area enclosed by the Fermi surface. The occupied area for the crescent-like Fermi surface is 0.025 Å, which corresponds to an electron doping of 1.210 cm, similar to the one reported for as grown graphene DavidAPL (). The larger triangular Fermi surface, which corresponds to an area of 0.33 Å, yields a much higher electron doping of 1.710 cm. The electronic band structure of the former crosses at =0.063 Å (spectra with the strongest intensity in Fig. 1(c)) with a Dirac point at 0.4 eV, which resembles as grown graphene shown in Fig. 1(d), except for the observed discontinuities around 0.6 eV and 1.1 eV below as denoted by white arrows in Fig. 1(c). On the other hand, the electronic band structure of the latter crosses at =0.26 Å (spectra with the weakest intensity in Fig. 1(c)) with a Dirac point at 1.6 eV.

Figure 1(e) shows the calculated electronic band structure for the inhomogeneous sample, where closed packed islands of YbC (referred to as “Yb/G” bands) coexist with islands of clean graphene without Yb (referred to as “G” bands). The G bands, shown in purple and denoted by and , are the well known graphene bands obtained within the tight-binding formalism GruneisTB () in the presence of an energy gap of 0.2 eV at  Zhou2007 (); Kim2008 (), while the origin of the gap-like feature is still controversial Kim2008 (); Eli2008_comment (); Zhou2008_comment (); Eli_plasmaron (); Louie (). The Yb/G bands, shown in red and obtained by ab initio pseudopotential total energy calculations with a plane-wave basis set Cohen (), are denoted by , , , , , and . and are the bands of the Yb/G, while and are the Yb 4 and 4 electrons, respectively. The Yb 4 electrons are strongly hybridized with and bands at 0.7 eV and 2.0 eV below , respectively, resulting in a departure of the Yb/G band from to and , and from to and . The observed discontinuities at the crossing points of with and (white arrows in Fig. 1(c)) may indicate that the G and Yb/G are electronically coupled with each other.

The and bands show weak spectral intensity with respect to the other bands near the K point. Their relative intensity is enhanced away from the K point, as shown in Figs. 2(a-d), in which ARPES data were taken perpendicular to the K direction at several values denoted in the inset of Fig. 2(a). The position of and is determined by the intensity spectrum at =0.2 Å denoted by a black solid line in Fig. 2(a). The hybridization between the Yb/G and Yb bands is clear at =1.85 Å as shown in Fig. 2(b). The deformation of the Yb/G band from to and is observed at the crossing points with the band. The band also shows unusual discontinuity at the crossing points with the band as shown in Figs. 2(b) and 2(c). Such a hybridization is not observed between the G band and Yb 4 electrons, e. g. , does not show such a deformation or discontinuity at the crossing point with the band around (, )=(2.0, 0.26) in Fig. 2(d). On the other hand, the hybridization between the Yb/G and G bands is clear from the energy spectra not only along direction (Figs. 2(c) and Fig. 2(d)), but also along direction (Fig. 2(e)). At =1.65 Å and =1.7 Å (Figs. 2(c) and 2(d)), discontinuities of the G band are observed at the crossing points with the Yb/G ( and ) bands around 1.1 eV below . At =0.0 Å (Fig. 2(e)), weak spectral intensity of the G band is observed at the crossing points with the Yb/G bands around 0.5 eV and 1.1 eV below denoted by with white dashed lines.

Iv Discussions

Figure 3: (Color online) (a-b) Raw ARPES data for the as grown G (panel (a)) and Yb/G (: panel (b)) samples near along the direction denoted by the red line in the inset. (c-d) Comparison of measured and calculated bands of as grown G (panel (c)) and Yb/G (panel (d)). The deviation at low energy in the range of 0.2 eV from is denoted by arrows.

The calculated electronic band structure provides another important information on the Yb/G system, i. e. , the bands of the Yb/G crossing () exhibits non-zero contribution from Yb 6 electrons in addition to the heavy carbon character. In order to understand the impact of this hybridization on the electronic properties, we investigate energy spectra measured near in comparison to calculated bands. Figures 3(a) and 3(b) are raw ARPES data of as grown graphene and Yb/G samples, respectively, along the direction denoted by the red line in the inset of each panel. To compare the measured and calculated bands quantitatively, we extract energy-momentum dispersions using the standard method, i. e. , Lorentzian fit to the momentum distribution curves (MDCs). The measured band of as grown graphene is well described by the tight-binding band, the black curve in Figs. 3(a) and 3(c). On the other hand, the measured Yb/G band () shows a clear kinked structure around 0.16 eV below as denoted by an arrow in Fig. 3(d), which is not expected in the GGAU band, the purple curve in Figs. 3(b) and 3(d). A similar structure, although much weaker, is also observed in the G bands (arrows in Fig. 3(c)). Such a kinked structure has been extensively studied in the literature in the context of band renormalization due to the interaction of electrons with phonons Verga (); Reinert (); Hengsberger (); Lanzara (); Gweon ().

Before proceeding to a direct comparison between the effect of such renormalization on G and Yb/G, and the consequent extraction of the electron-phonon coupling constant, it is imperative to establish whether these low energy kinked structures are real manifestation of many body physics or just reflect the bare band structure of this doped sample. Figure 4 shows a comparison of the near band structure for Yb/G along the two directions (KM: panel (a) and KK: panel (b)) with the GGAU bands. Along the KM direction (Fig. 4(a)), the measured band structure clearly shows a kinked structure around 0.16 eV below . However, GGAU calculations (red curves) also show curved band structure near the kink energy, which is not observed from the electronic band structure of clean graphene, but induced due to an hybridzation between the adsorbate electrons and the graphene bands. When the strength of electron-phonon coupling is determined by the slope of the dispersion below and above the kink energy, this curved band structure results in finite strength, despite the theory does not include the electron-phonon coupling. This is the so-called band structure effect Calandra_BandStructureEffect (). In addition, nearness to the van Hove singularity is supposed to spread the measured spectral intensity away from the calculated Fermi momentum, which is beyond the capability of our first principle calculations. This spread out intensity results in the decrease of the slope near and hence the apparent enhancement of electron-phonon coupling Calandra_BandStructureEffect (). Similar band structure effects have been extensively discussed in the literature for Ca/G and K/G Jessica (); Calandra_BandStructureEffect (); Park_vanHove (). In contrast, perpendicular to the K direction (Fig. 4(b)), these non-trivial effects are not observed allowing us to extract information on the electron-phonon coupling.


Figure 4: (Color online) Measured and calculated Yb/G band along the KM direction (panel (a)) and perpendicular to the K direction (panel (b)). The red curves are GGAU bands.
Figure 5: (Color online) (a-b) Zoomed-in view of the electronic band structure with a kink denoted by the arrow. The dashed line is a tight-binding (TB) band fitted to each case. (c) Re for as grown G (upper panel) and Yb/G (lower panel). The dashed line is a linear fit to Re for 0.10 eV0.03 eV. Shaded area 0.160.18 eV below is a peak position of the Re spectrum. (d) MDC width as a function of for as grown G (upper panel) and Yb/G (lower panel).

We now focus on the near dispersion of as grown graphene and Yb/G perpendicular to the K direction in Figs. 5(a) and 5(b), respectively. It is clear that, while the strength of the kink varies considerably, the characteristic energy of the kink, 0.16 eV below , does not change much. This implies a stronger coupling of electrons to the optical phonon of graphene at the K point (A mode with an energy 0.16 eV) rather than the one at the point (E mode with an energy 0.19 eV), in agreement with previous reports for as grown graphene Zhou08 () and as expected in the case of enhanced electronic correlations Basko (). Similar conclusion can be drawn from the real part of the electron self-energy (Re), i. e. , the difference between the measured band and the tight-binding band, and from the imaginary part of electron self-energy (Im) which is proportional to the full width at half maximum (FWHM) of MDCs. In Figs. 5(c) and 5(d), we report the Re and FWHM spectra. Re, in both cases, is dominated by a strong peak at 0.16-0.18 eV (gray shaded area), while the FWHM exhibits an enhanced quasiparticle scattering rate (or increased width) around the same energy. The shape of Re and Im for Yb/G is consistent with the theoretical prediction of the electron-phonon coupling for highly electron-doped graphene Park2007 (). The upturn of the Re spectra close to is a well-known resolution effect, which typically results in the deflection of MDC peaks within a few tens meV near to lower momentum Plumb (); Valla2006 (), and would result in the apparent increase of Re close to .

The real part of electron self-energy is a direct measurement of the electron-phonon coupling constant, given by =. The dashed line in Fig. 5(c) is a linear fit to Re for 0.10 eV 0.03 eV. We obtain =0.0460.002 for as grown graphene, which is similar to the previously reported theoretical (=0.02) Calandra_resolution () and experimental (=0.14) Zhou08 () values. The difference from the latter might originate from the method to extract . For Yb/G, we obtain =0.4310.004, which exhibits strong enhancement by an order of magnitude compared to the value for as grown graphene. It is important to note that the GGAU band in Fig. 3(d) does not show the decreasing slope of the dispersion near , so the band structure effect is safely excluded as the origin of the enhanced  Calandra_BandStructureEffect (). The self-consistency of the self-energy analysis is obtained via Kramers-Kronig transformation of Im Kordyuk () as shown in Fig. 6(a). The strength of the electron-phonon coupling is obtained by linear fits to Re and Re (brown dashed lines) for 0.10 eV0.03 eV and 0.1 eV0 eV, resulting in =0.4310.004 and =0.3850.011, respectively.

Figure 6: (Color online) (a) Re (red line) and Re (black line) for Yb/G. Dashed lines are linear fits to Re and Re for 0.10 eV0.03 eV and 0.1 eV0 eV, respectively. (b) Calculated electron-phonon coupling spectrum F() (brown shaded area) and the evolution of as a function of phonon energy (navy curve) for Yb/G.

The calculated electron-phonon coupling spectrum and electron-phonon coupling constant for Yb/G (shown in Fig. 6(b)) are obtained from the density-functional perturbation theory using the program Quantum ESPRESSO Baroni (). The electronic orbitals were expanded in a plane-wave basis set with a kinetic energy cutoff of 75 Ry. The Brillouin zone integrations in the electronic and phonon calculations were performed using Monkhorst-Pack Monkhorst () meshes. We refer to meshes of -points for electronic states and meshes of -points for phonons. The electron-phonon coupling matrix elements were computed in the first Brillouin zone on a 18181 -mesh using individual electron-phonon coupling matrices obtained with a 36361 -points mesh. The electron-phonon coupling spectrum, F(), (brown shaded area in Fig. 6(b)), can be divided into three regions: (i) low-energy Yb-related modes up to 0.005 eV; (ii) carbon out-of-plane modes up to 0.09 eV; and (iii) carbon in-plane modes at 0.160.18 eV and 0.19 eV. We find very strong electronic coupling to the phonon mode at 0.160.18 eV in agreement with our observation (see Fig. 5(c)). The coupling strength can be directly determined from the spectra being =2dF()/ (navy curve in Fig. 6(b)). Clearly the electron-phonon coupling constant is drastically enhanced with respect to the as grown sample over the entire range, from =0.02 for as grown graphene Calandra_resolution () to =0.51 for Yb/G, consistent with the observed enhancement from =0.05 to =0.43 (Fig. 5). The difference of the experimental from the theoretical value might be caused by the lack of the exact unrenormalized band in extracting Re, which underestimates experimental  DavidNJP ().

The observed enhancement up to 0.43 (experimental) and 0.51 (theoretical) due to Yb is far greater than the theoretically and experimentally estimated enhancement up to 0.09 by the change of charge carrier density up to =1.710 cm Calandra_resolution (); DavidNJP (), as for the Yb/G sample. This indicates that charge doping alone cannot explain the observed enhancement. Similar enhancement beyond the capability of charge carrier density has been observed for potassium-intercalated graphene on Ir substrate Ref1 (); Ref2 () with =0.20.28. In the case of calcium-intercalate graphene on Au/Ni(111)/W(110) substrate Ref3 (), the anisotropic increase of from 0.17 (along the K direction) to 0.40 (along the KM direction) has been controversial as ascribed to a change of the electron band structure and the van Hove singularity due to the Ca intercalation, which result in apparent enhancement of  Calandra_BandStructureEffect (); Park_vanHove ().

The observed =0.43 in our work is the highest value ever measured for graphene. It is interesting to note that, for bulk graphite, the electron-phonon coupling in the Yb intercalated sample (Yb-GIC) is estimated to be weaker than that in the Ca intercalated sample (Ca-GIC), because of the slightly larger interlayer separation which leads to a decrease of the interlayer- electron-phonon matrix element and thus smaller superconducting phase transition temperature, (6.5 K for Yb-GIC versus 11.5 K for Ca-GIC Weller ()). This trend is reversed in their graphene counterparts, =0.43 for Yb/G (in this work) versus =0.4 (or 0.17) for Ca/G Ref3 () suggesting that the hybridization between graphene bands and the electrons from adatoms governs the low energy excitations in monolayer graphene. The hybridization induces strong Coulomb interactions, as evidenced by the preeminent role of the K point phonon compared to the point phonon in the electron-phonon coupling Basko () as shown in Figs. 5 and 6, and allows phonons to be strongly coupled to electrons in graphene.

In line with the plausible phonon-mediated superconductivity in Yb-GIC, the strong enhancement of electron-phonon coupling in Yb/G suggests the exciting possibility that the introduction of Yb might induce superconductivity Rose (); McMillan (). The is calculated using the Allen-Dynes equation Allen (),

(1)

The normalized weighting function of the Eliashberg theory McMillan () is

(2)

The parameter is a dimensionless measure of the strength of with respect to frequency :

(3)

and the logarithmic average frequency, in units of K, is

(4)

The predicted and are estimated to be 1.71 K and 168.2 K, respectively. We use =0.115 for proper comparison with another theoretical work Profeta () and it is worth to note that the predicted can range from 2.17 K (=0.10) to 1.33 K (=0.13).

V Summary

We have reported experimental evidence of strong enhancement of electron-phonon coupling in graphene by as much as a factor of 10 upon the introduction of Yb (from 0.020.05 to 0.430.51). Such an enhancement goes beyond what one would expect by charge doping. Our results reveal the important role of the hybridization between electrons from Yb adatoms and the graphene electrons, pointing to such hybridization as a critical parameter in realizing correlated electron phases in graphene.

Acknowledgements.
The experimental part of this work was supported by Berkeley Lab’s program on sp2 bond materials, funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, of the U.S. Department of Energy (DOE) under Contract No. DE-AC02-05CH11231. Work at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. The theoretical part of this work was supported by NSF Grant No. DMR-IO-1006184 and the theory program at the Lawrence Berkeley National Laboratory through the Office of Basic Energy Science, US Department of Energy under Contract No. DE-AC02-05CH11231 (K. T. C. , J. N. , and M. L. C.); and by the Energy Frontier Research in Extreme Environments Center (EFree) under award number DE-SG0001057. B. J. acknowledges financial support from the European Research Council (ERC-2008-AdG-No. 228074).

References

  • (1) D. K. Efetov and P. Kim, Phys. Rev. Lett.105, 256805 (2010).
  • (2) K. C. Rahnejat et al., Nature Commun. 2, 558 (2011).
  • (3) B. Uchoa and A. H. Castro Neto, Phys. Rev. Lett. 98, 146801 (2007).
  • (4) A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 (2007).
  • (5) M. Calandra and F. Mauri, Phys. Rev. B 76, 205411 (2007).
  • (6) C. Faugeras et al., Phys. Rev. Lett. 103, 186803 (2009).
  • (7) J. C. W. Song, M. Y. Reizer, and L. S. Levitov, Phys. Rev. Lett. 109, 106602 (2012).
  • (8) G. Profeta, M. Calandra, and F. Mauri, Nat. Phys. 8, 131 (2012).
  • (9) D. A. Siegel, C. Hwang, A. V. Fedorov, and A. Lanzara, New J. Phys. 14, 095006 (2012).
  • (10) G. Savini, A. C Ferrari, and F. Giustino, Phys. Rev. Lett. 105, 037002 (2010).
  • (11) N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, New York, 1976).
  • (12) J. L. McChesney et al., e-print, arXiv:0705.3264 (2007).
  • (13) M. Calandra and F. Mauri, Phys. Rev. B 76, 161406(R) (2007).
  • (14) C. -H. Park et al., Phys. Rev. B 77, 113410 (2008).
  • (15) T. Valla et al., Phys. Rev. Lett. 102, 107007 (2009).
  • (16) S. -L. Yang et al., Nat. Commun. 5, 2 (2014).
  • (17) E. Rollings et al., J. Phys. Chem. Solids 67, 2172 (2006).
  • (18) S. Watcharinyanon, C. Virojanadara, and L. I. Johansson, Surf. Sci. 605, 1918 (2011).
  • (19) S. Watcharinyanon, L. I. Johansson, C. Xia, and C. Virojanadara, J. Vac. Sci. Technol. A 31, 020606 (2013).
  • (20) S. Watcharinyanon, L. I. Johansson, C. Xia, J. I. Flege, A. Meyer, J. Falta, and C. Virojanadara, Graphene 2, 66 (2013).
  • (21) M. L. Cohen, Phys. Scr. T1, 5 (1982).
  • (22) G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).
  • (23) G. Kresse and J. Furthmüller, Computational Materials Science 6, 15 (1996).
  • (24) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  • (25) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  • (26) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  • (27) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  • (28) A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467 (1995).
  • (29) I. I. Mazin and S. L. Molodtsov, Phys. Rev. B 72, 172504 (2005).
  • (30) C. Hwang et al., Phys. Rev. B 84, 125422 (2011).
  • (31) D. A. Siegel et al., Appl. Phys. Lett. 93, 243119 (2008).
  • (32) A. Grüneis et al., Phys. Rev. B 78, 205425 (2008).
  • (33) S. Y. Zhou et al., Nat. Mater. 6, 770 (2007).
  • (34) S. Kim et al., Phys. Rev. Lett. 100, 176802 (2008).
  • (35) E. Rotenberg et al., Nat. Mater. 7, 258 (2008)
  • (36) S. Y. Zhou et al., Nat. Mater. 7, 259 (2008).
  • (37) A. Bostwick et al., Science 328, 999 (2010).
  • (38) J. Lischner, D. Vigil-Fowler, and S. G. Louie, Phys. Rev. Lett. 110, 146801 (2013).
  • (39) S. Verga, A. Knigavko, and F. Marsiglio, Phys. Rev. B 67, 054503 (2003).
  • (40) F. Reinert and S. Hüfner, New J. Phys. 7, 97 (2005).
  • (41) M. Hengsberger et al., Phys. Rev. B 60, 10796 (1999).
  • (42) A. Lanzara et al., Nature 412, 510 (2001).
  • (43) G. -H. Gweon et al., Nature 430, 187 (2004).
  • (44) S. Y. Zhou, D. A. Siegel, A. V. Fedorov, and A. Lanzara, Phys. Rev. B 78, 193404 (2008).
  • (45) D. M. Basko and I. L. Aleiner, I. L. Phys. Rev. B 77, 041409(R) (2008), (and references therein).
  • (46) C. -H. Park, F. Giustino, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 99, 086804 (2007).
  • (47) N. C. Plumb et al., Phys. Rev. Lett. 105, 046402 (2010).
  • (48) T. Valla, Phys. Rev. Lett. 96, 119701 (2006).
  • (49) A.A. Kordyuk, S.V. Borisenko, A. Koitzsch, J. Fink, M. Knupfer, and H. Berger, Phys. Rev. B 71, 214513 (2005).
  • (50) S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001).
  • (51) H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 51882 (1976).
  • (52) M. Bianchi et al., Phys. Rev. B 81, 041403(R) (2010).
  • (53) I. Pletikosić, M. Kralj, M. Milum, and P. Pervan, Phys. Rev. B 85, 155447 (2012).
  • (54) A. V. Fedorov et al., Nat. Commun. 5, 3257 (2014).
  • (55) T. E. Weller et al., Nat. Phys. 1, 39 (2005).
  • (56) W. L. McMillan, Phys. Rev. 167, 331 (1968).
  • (57) A. C. Rose-Innes and E. H. Rhoderick, Introduction to Superconductivity (Pergamon Press, 1978).
  • (58) P. B. Allen and R. C. Dynes, Phys. Rev. B 12, 905 (1975).
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
171968
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description