Calibration of the fine-structure constant of graphene by time-dependent density-functional theory

One of the amazing properties of graphene is the ultrarelativistic behavior of its loosely bound electrons, mimicking massless fermions that move with a constant velocity, inversely proportional to a fine-structure constant ${\ensuremath{\alpha}}_{g}$ of the order of unity. The effective interaction between these quasiparticles is, however, better controlled by the coupling parameter ${\ensuremath{\alpha}}_{g}^{*}={\ensuremath{\alpha}}_{g}/\ensuremath{\epsilon}$, which accounts for the dynamic screening due to the complex permittivity $\ensuremath{\epsilon}$ of the many-valence electron system. This concept was introduced in a couple of previous studies [Reed et al., Science 330, 805 (2010) and Gan et al., Phys. Rev. B 93, 195150 (2016)], where inelastic x-ray scattering measurements on crystal graphite were converted into an experimentally derived form of ${\ensuremath{\alpha}}_{g}^{*}$ for graphene, over an energy-momentum region on the $\mathrm{eV}\phantom{\rule{0.16em}{0ex}}\AA{}{}^{\ensuremath{-}1}$ scale. Here, an accurate theoretical framework is provided for ${\ensuremath{\alpha}}_{g}^{*}$, using time-dependent density-functional theory in the random-phase approximation, with a cutoff in the interaction between excited electrons in graphene, which translates to an effective interlayer interaction in graphite. The predictions of the approach are in excellent agreement with the above-mentioned measurements, suggesting a calibration method to substantially improve the experimental derivation of ${\ensuremath{\alpha}}_{g}^{*}$, which tends to a static limiting value of $\ensuremath{\sim}0.14$. Thus, the ab initio calibration procedure outlined demonstrates the accuracy of perturbation expansion treatments for the two-dimensional gas of massless Dirac fermions in graphene, in parallel with quantum electrodynamics.