Anisotropic Eliashberg function and electron-phonon coupling in doped graphene

We investigate, with high-resolution angle-resolved photoemission spectroscopy, the spectral function of potassium-doped quasi-free-standing graphene on Au. Angle-dependent x-ray photoemission and density functional theory calculations demonstrate that potassium intercalates into the graphene/Au interface, leading to an upshift of the K-derived electronic band above the Fermi level. This empty band is what makes this system perfectly suited to disentangle the contributions to electron-phonon coupling coming from the $\pi$ band and K-derived bands. From a self-energy analysis we find an anisotropic electron-phonon coupling strength $\lambda$ of 0.1 (0.2) for the $K\Gamma$ ($KM$) high-symmetry directions in momentum space, respectively. Interestingly, the high-energy part of the Eliashberg function which relates to graphene's optical phonons is equal in both directions but only in $KM$ does an additional low-energy part appear.

Figure 1

(a) UPS spectrum of pristine and K-doped graphene on Au and (b) corresponding C 1$s$ core-level XPS spectra. (c) Normal and grazing emission core level spectra of C 1$s$ and K 2$p$. (d) ARPES overview scan along the $\Gamma KM$ high-symmetry direction obtained by merging scans with $s$- and $p$-polarized light. $\varepsilon =E-E_F$ denotes the binding energy relative to the Fermi energy. (e) Equienergy contour (relative to the $K$ point) at $E_F$ along with the determined photoemission intensity maxima (yellow crosses) using $p$-polarized light.

Figure 2

DFT optimized geometry and band structure for (a) K/graphene/Au/Ni and (b) graphene/K/Au/Ni model interfaces. Projections of electron states on the C 2$p$ and K 4$s$ orbitals are displayed as black and blue lines with the thickness proportional to the weight of the projection. The bands in red come from Ni and Au. The inset shows the Brillouin zone (BZ) of the original and the $2\times 2$ cell with high-symmetry points of the $2\times 2$ cell labeled with a prime.

Figure 3

(a), (b) ARPES scans of the kink region along the $K\Gamma$ and $KM$ directions using $s$- and $p$-polarized light, respectively. Also shown are the intensity maxima obtained from a MDC analysis (black dots) and a polynomial bare band (white line). (c) The measured real part of the self-energy ${\Sigma }^{\prime }$ (black dots) along with the recalculated ${\Sigma }^{\prime }$ from ${\alpha }^{2}F\left(\omega \right)$ (red solid line). The Eliashberg function is depicted as a blue solid line below ${\Sigma }^{\prime }$ for the $K\Gamma$ and (d) $KM$ directions, respectively. (e) and (f) Measured imaginary part of the self-energy ${\Sigma }^{\prime \prime }$ (black dots) together with ${\Sigma }^{\prime \prime }$ derived from the Kramers-Kronig relation and calculated from the corresponding Eliashberg function in $K\Gamma$ and $KM$, respectively.

Figure 4

Eliashberg function and EPC constant $\lambda$ for (a) the $K\Gamma$ and (b) the $KM$ directions. (c) Comparison of the determined Eliashberg functions to the phonon density of states derived from a model phonon dispersion for graphene/Au.