The Electron-Phonon Coupling Constant for Single-Layer Graphene on Metal Substrates Determined from He Atom Scattering

Recent theory has demonstrated that the value of the electron-phonon coupling strength $\lambda$ can be extracted directly from the thermal attenuation (Debye-Waller factor) of atom scattering reflectivity. This theory is here extended to multivalley semimetal systems and applied to the case of graphene on different metal substrates and graphite. It is shown that $\lambda$ rapidly increases for decreasing graphene-substrate binding strength. The method is suitable for analysis and characterization of not only the graphene overlayers considered here, but also other layered systems such as twisted graphene bilayers, a wider and growing class of novel 2D superconductors.

Current interest in single-layer graphene supported on metal substrates has led several experimental groups to investigate a number of such systems with He atom scattering (HAS), [1][2][3][4][5][6][7][8][9][10][11][12] as well as the surface of clean highly ordered pyrolytic graphite (HOPG) graphite, C(0001). [1][2][3] Two of these systems, namely graphene (Gr) on Ni(111) and Gr/Ru(111) have also been investigated with Ne atom scattering. 10,11 In all of these systems high quality data are available for the thermal attenuation of the specular diffraction peak over a large range of temperatures. Such thermal attenuation measurements are interesting because it has been shown that they can be used to extract values of the electron-phonon coupling constant λ (also known as the mass correction factor) in the surface region.
The ability of atom scattering to measure λ relies on the fact that colliding atoms are repelled by the surface electron density arising from electronic states near the Fermi level, and energy is exchanged with the phonon gas primarily via the electron-phonon coupling. The electron-phonon coupling constant λ at the surface is defined as the average λ = λ Q,ν = Q,ν λ Q,ν /N over the contributions λ Q,ν coming from all the individual phonon modes, where Q is the phonon parallel wave vector, ν the branch index, and N is the total number of phonon modes. 13 It has been theoretically demonstrated that the peak features due to specific (Q, ν) phonon modes as observed in inelastic He atom scattering spectra are individually proportional to their corresponding λ Q,ν . 14, 15 This prediction has been verified through detailed comparisons of calculations with experimental He scattering measurements of multiple layers of Pb on a Cu(111) substrate. 14,15 Since the thermal attenuation of any quantum peak feature in the atom scattering spectra is due to an average over the mean square displacement of all phonon modes weighted by the respective electron-phonon coupling, it is not surprising that such attenuation can be related to the average λ HAS = λ Q,ν . [16][17][18][19] This will be discussed in more detail below where the theory is developed for the case of Gr adsorbed on close-packed metal substrates. Values of λ HAS are obtained from the available He atom scattering data on C(0001) and a discussion of Gr/metals and an analysis of λ HAS from the available data for thermal attenuation of the specular He atom scattering peaks is presented.
It is shown that the λ HAS values exhibit an interesting relationship when compared with the relative binding strengths of the graphene to the corresponding metal substrate. Finally, a discussion is presented of the shear-vertical (ZA) mode of substrate-supported thin layers such as Gr/metals, and how it compares to the flexural mode of a thin flake of the same unsupported two dimensional (2D) material with free-boundary conditions. A summary and a few conclusions are drawn at the end of this work.
As a function of the temperature T , the thermal attenuation of quantum features in He atom spectra, such as elastic diffraction observed in angular distributions and the diffuse elastic peak observed in energy-resolved spectra, is given by a Debye-Waller (DW) factor. [20][21][22][23] This is expressed as a multiplicative factor exp{−2W (k f , k i , T )} where k i and k f are the incident and final wave vectors of the He atom projectile. This means that the intensity of any elastic peak is given by where I 0 is the intensity the peak would have at T = 0 in the absence of zero-point motion (rigid lattice limit). In general I 0 > I(0).
The DW exponent is expressed by 2W (k f , k i , T ) = (∆k · u * ) 2 T , where ∆k = k f − k i is the scattering vector, u * is the effective phonon displacement felt by the projectile atom upon collision, and · · · T denotes the thermal average. However, He atom scattering experiments typically use energies below 100 meV. The atoms do not penetrate the surface, and in fact are exclusively scattered by the surface electron density a fewÅ above the first layer of atomic cores. Thus the exchange of energy through phonon excitation occurs via the phonon-induced modulation of the surface electron gas, in other words via the electronphonon (e-ph) interaction. This implies that the effective displacement u * is not that of the atom cores, such as would be measured in a neutron or X-ray diffraction experiment, but is the phonon-induced displacement of the electron distribution outside the surface at the classical turning point where the He atom is reflected. 16,18 However, the effective mean square displacement of the electron density is related directly to that of the atom cores and shares many of its properties. Notably, for a crystal obeying the harmonic approximation and for sufficiently large temperature (typically temperatures comparable to or greater than the Debye temperature) (∆k · u * ) 2 T is to a very good approximation linear in T , and the proportionality between λ HAS and the DW exponent, for the simplest case of specular diffraction, reads as 16 where N (E F ) is the electronic density of states (DOS) per unit cell at the Fermi energy E F , m * e is the effective mass of an electron, φ is the work function, m is the projectile atomic mass and k B is the Boltzmann constant. The quantity E iz = E i cos 2 (θ i ) = 2 k 2 iz /2m is the incident energy associated with motion normal to the surface at the given incident angle θ i .
For application to non-specular diffraction peaks or to other elastic features Eq. (2) should be adjusted to account for the correct scattering vector appropriate to the experimental scattering configuration, namely 4k 2 Given the form of Eq. (2) it is useful to define the dimensionless quantity n s as where a c is the area of a surface unit cell. With this definition and the help of Eqs. (1) and (2) the following form for λ HAS is obtained where T 1 and T 2 are any two temperatures in the linearity region. Eq. (4) neatly separates the surface electronic properties from the quantities measured in an actual experiment. The electronic properties such as N (E F ) and m * e are contained in the dimensionless n s , while the readily determined work function of the surface and the experimentally measured slope of the DW exponent are in α.
An interesting example is to consider the DOS for a single layer of a 2D free electron gas, for which it is readily shown that N (E F ) (2D) = m * e a c /π 2 . This leads to n s = 1 which suggests that in a thin film system consisting of a stack of several 2D free electron layers the quantity n s may be considered as indicating the approximate number of layers contributing to N (E F ) and hence to the evaluation of surface e-ph constant.
In fact, such an interpretation has been demonstrated for the case of layer-by-layer growth of alkali metals on metal substrates. 17 Measurements of the specular peak thermal attenuation showed that the slope of the DW plot, and hence α, was directly proportional to the number of layers n up to a saturation thickness n sat . 17  This concept has been extended to layered semimetals and degenerate semiconductors characterized by a conducting surface and quantum-well states within a surface band bending with a Thomas-Fermi (TF) screening length λ T F . 25 In this case n s can be taken as the number of conducting layers (of 2D electron gases) within λ T F , 19 e.g., n s = λ T F /c 0 where c 0 is the layer thickness. Expressing λ HAS in terms of the Thomas-Fermi screening length provides a certain advantage, as for many systems λ T F has been measured or evaluated with theoretical calculations.
Graphite, as the prototype of layered materials, is characterized by a very low carrier mobility in the normal direction, which basically restricts the screening to one single layer. 26 This emphasizes its apparent 2D character. On the other hand, the presence of six Dirac cones (actually two types per Brillouin zone [BZ], K and K , each one shared between three different BZs) makes of the graphite monolayer, from the point of view of electronphonon interaction, a multivalley semimetal, characterized by a 1D multiple Peierls (alias Kelly-Falicov) intervalley coupling mechanism. [27][28][29] In this specific case only nesting between adjacent pairs K-K occurs within the first BZ and in three different directions (multiplic- Clean graphite C(0001) presents a weakly corrugated surface potential to He atom scattering which means that the specular peak is the dominant elastic scattering feature. The thermal attenuation of C(0001) has been measured by three independent groups over temperatures ranging from below 150 K to 500 K. [1][2][3]31 As is apparent from the plots of the DW exponent in Fig. 1a), measured for graphite as a function of T by Oh et al. 2

and by
Vollmer, 31 the slope is essentially linear at lower temperature up to about 350 K. The slopes of the different measurements, and using n s = 4, provide the values of λ HAS listed in Table 1 together with the input parameters and the respective references. The average over the available data gives for graphite λ HAS = 0.62 ± 0.05.
As is seen in Fig. 1a), above 350 K the slope of 2W (T ) clearly decreases, apparently tending to a value about a factor of 2 smaller at high temperature, and so does λ HAS . This behavior is very interesting and is probably the consequence of the gradual transition of graphite from negative to positive in-plane thermal expansion, occurring at ∼ 500 K. 32,33 A bond contraction, like that produced by an external pressure, generally yields a larger with some adjustment as explained in the text.
λ, while a dilation means a smaller λ, with similar consequences on the superconducting T c . 34 This may explain the transition with temperature from a larger to a smaller λ HAS observed in graphite. Interestingly, this is not observed in graphene, where the in-plane thermal expansion is predicted to remain negative up to a considerably larger temperature, well above the temperature range so far considered in HAS experiments. 35 It is seen from the last column in Table 1 that there is a rather large range of reported values of λ for bulk graphite, so that the values of λ HAS at the surface as measured by He atom scattering are well within the range of reported bulk values.
All of the systems of single layer graphene supported on close-packed metal surfaces for which DW plots of the specular diffraction peak have been measured are listed in Table 1.
For two of these systems, Gr/Ni(111) and Gr/Ru(0001), measurements were made with both He and Ne atom scattering. For Gr/Ni(111), in addition to experiments done with He and Ne atoms, measurements were taken at very low incident energy using the 3 He isotope.
For estimating the values of λ HAS for monolayer graphene on the different metal substrates we use the same value n s = 4 adopted for graphite. Other input parameters needed for evaluating λ HAS , together with the relevant references are given in Table 1   Gr/Ru(0001) should be assigned to the S 2 mode and not to the ZA mode. In Fig. 2 the value ω ZA (0) = 14 meV has been used.
Another reported value of ω ZA (0) that needs to be discussed is that of Gr/Rh (111) for which a value of 7 meV has been reported. 5 This value would appear to be too small when compared with the other strongly bound Gr/metal systems such as Gr/Ni and Gr/Ru where the value is rather in the range of 14 -20 meV. Moreover, no upward dispersion is apparently observed, as would be expected. It is unfortunate that no phonon measurements were reported above 12 meV for this system. Other HAS measurements, for example for Gr/Ru(0001) and Gr/Ni (111) where v SV is the speed of SV waves, κ is the bending rigidity (sometimes called the flexural rigidity), and ρ 2D is the 2-D mass density. However, if the membrane is coupled to the substrate, the coupling force constant will introduce a gap of frequency ω ZA (0). In this case the corresponding dispersion in the region of small Q is often expressed as 44 written without the quadratic term, its effect being negligible with respect to that of the other two terms. It is Eq. (6) that has been used to determine the bending rigidity κ of the ZA mode of graphene supported by the metals considered here 4-8 as well as for the thinnest known layer of vitreous glass, a bilayer of SiO 2 on Ru(0001). 45 It is important to note that most of the modelling of supported graphene dynamics is based on the assumption of a rigid substrate. However, the phonon spectrum of graphene, with its large dispersion of acoustic modes and high-frequency optical modes, covers the whole phonon spectrum of the substrate, and therefore several avoided crossings are expected between graphene and the substrate modes of similar polarization. For the ZA branch the important interactions are with the S 2 optical branch of the metal and the Rayleigh wave.
For this reason it was convenient to replace the carbon dimer mass with the effective mass in the expression of f ⊥ used in Fig. 2. Moreover, supported graphene is no longer a specular plane, and coupled SV phonon modes acquire some elliptical polarization, leading, e.g., to an avoided crossing between the ZA and LA modes near Q = 0 (see, for example, HAS data for Gr/Ru(0001) 7 ). In recent works on free-standing graphene dynamics (and even on graphite dynamics) there is some apparent confusion between SV transverse (ZA) and flexural modes. 6, [46][47][48][49][50] This also appears in the case of other thin film materials such as bilayer SiO 2 . 45,51 The study of the dynamics of elastic plates dates back to works of Euler, Bernouilli, D'Alembert, Sophie Germaine and Lagarange, just to mention some of the pioneers, and the terminology is well established. 52 When transferred to lattice dynamics, transverse modes refer to lattices with cyclic (or fixed) boundary conditions, whereas flexural modes refer to lattices with free boundaries. The difference is illustrated for a three-atom chain in Fig. 3. On the other hand, finite flakes of weakly coupled (quasi-self-standing) supported graphene may approximate the free-boundary condition, thus yielding a mean-square displacement rapidly increasing with temperature, i.e., a steep decrease of the DW factor. In this case the association of DW slopes like those of Fig. 1a) exclusively to e-ph interaction may be incorrect, especially for the weakest graphene-substrate couplings. He atom scattering on graphene and related systems. One such class of experiments would be to measure extensive energy-resolved inelastic scattering spectra. Such spectra exhibit peak-features due to specific phonons, and these peak intensities are directly proportional to λ Q,ν , the mode-selected e-ph contributions to λ HAS . 15 Thus, inelastic atom-scattering can provide unique information on which phonon modes contribute most importantly to λ.
Another class of experiments would be to measure double and multiple layer supported graphene, and in particular it would be interesting to measure λ for twisted bilayer graphene (tBLG) which can be superconducting for specific twist angles, [78][79][80][81] as well as measurements on the related wider and rapidly growing classes of novel 2D superconductors and chargedensity wave systems. In the specific case of tBLG, where its peculiar electronic structure is considered to favor a strong electron correlation as the basic mechanism for pairing, [78][79][80]82,83 the actual value of λ would be rather small. On the other hand, it has been suggested that the very same electronic structure at the Fermi level supports a strong e-ph interaction with λ of the order of 1, 84-87 or even 1.5, 88 so as to consider tBLG as a conventional superconductor.
This would rely on a strong multivalley e-ph interaction as well as on a decoupling between twisted layers, similar to the orientational staking faults in graphene multilayers grown on 4H − SiC(0001) which makes the surface layer behaves as a self-standing doped graphene. 89 This may well correspond to the large λ in the limit f ⊥ → 0 represented in Fig