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 ${\alpha }_g$ of the order of unity. The effective interaction between these quasiparticles is, however, better controlled by the coupling parameter ${\alpha }_g^{*}={\alpha }_g/\epsilon$, which accounts for the dynamic screening due to the complex permittivity $\epsilon$ 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 ${\alpha }_g^{*}$ for graphene, over an energy-momentum region on the $\mathrm{eV}\AA {}^{-1}$ scale. Here, an accurate theoretical framework is provided for ${\alpha }_g^{*}$, 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 ${\alpha }_g^{*}$, which tends to a static limiting value of $\sim 0.14$. 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.

Figure 1

Static fine-structure constant of graphene extracted from the PDOS vs the $\pi \text{−}{\pi }^{*}$ energy $E-E_F$ and the inverse group velocities ${\alpha }_{\mathrm{\Gamma }K}=1/v_{\mathrm{\Gamma }K}$(a), ${\alpha }_{\mathrm{\Gamma }M}=1/v_{\mathrm{\Gamma }M}$(b).

Figure 2

Experimentally derived susceptibility $\mathrm{Im}\left({\chi }_{2\mathrm{D}}^{\mathrm{expt}.}\right)$ of [10] and macroscopic susceptibility $\mathrm{Im}\left({\chi }^M\right)$, obtained from Eq. (2) with $v\to v_{\mathbf{G}{\mathbf{G}}^{\prime }}\left(L\right)$ and the KS structure of dataset (iii). $\mathrm{Im}\left({\chi }^M\right)$ is multiplied by a typical length $d_0$ to match the dimensions of $\mathrm{Im}\left({\chi }_{2\mathrm{D}}^{\mathrm{expt}.}\right)$. The closest values of $\mathbf{q}\parallel \mathrm{\Gamma }M$ to the measured in-plane momenta are selected to represent $\mathrm{Im}\left({\chi }^M\right)$, with a broadening lifetime $\eta$ of 0.02 eV (red). The $\mathbf{q}$-shifted spectra (green) are broadened by 0.2 eV to match the experimental uncertanties.

Figure 3

Experimentally derived unperturbed susceptibility of graphene $\mathrm{Im}\left({\chi }_0^{\mathrm{expt}.}\right)$ and measured susceptibility of graphite $\mathrm{Im}\left({\chi }_{3\mathrm{D}}^{\mathrm{expt}.}\right)$ taken from [10] in comparison with their theoretical counterparts $\mathrm{Im}\left({\chi }_0^M\right)$ and $\mathrm{Im}\left({\chi }_{3\mathrm{D}}^M\right)$.

Figure 4

(a),(c) ${\left({\epsilon }^M\right)}^{-1}$ vs $\omega <25\mathrm{eV}$ and $q<1.5\AA {}^{-1}$, with the $\mathbf{q}$ orientation averaged between $\mathrm{\Gamma }K$ and $\mathrm{\Gamma }M$; (b),(d) ${\left(v\chi \right)}_{\mathbf{00}}$ and $v_{2\mathrm{D}}{\chi }_{2\mathrm{D}}^M$ vs $\omega <13\mathrm{eV}$ at fixed $\mathbf{q}$ values along $\mathrm{\Gamma }M$ that match the experimental in-plane momenta of [10]; (e) ${\left({\epsilon }^M\right)}^{-1}$ vs $q<0.1\AA {}^{-1}$ at $\omega \to 0$, with the datasets (ii) and (iii); (f) peak-to-peak ratios $r_{2\mathrm{D}}^{3\mathrm{D}},i_{2\mathrm{D}}^{3\mathrm{D}}$ of the real and imaginary parts of ${\left(v\chi \right)}_{\mathbf{00}}$ and $v_{2\mathrm{D}}{\chi }_{2\mathrm{D}}^M$ vs $\frac{v_{\mathbf{00}}\left(L\right)}{v_{2D}L}$. (g) Dressed fine-structure constant ${\alpha }_g^{*}$ at $\omega \to 0$ in comparison with the experimentally derived data ${\left({\alpha }_g^{*}\right)}_{\mathrm{expt}.}$ of [8, 10], and the corrected data ${\left({\alpha }_g^{*}\right)}_{\mathrm{expt}.}^{\mathrm{c}}$ with the $r_{2\mathrm{D}}^{3\mathrm{D}}$ points of (f).