Reexamination of the effective fine structure constant of graphene as measured in graphite

We present a refined and improved study of the influence of screening on the effective fine structure constant of graphene, ${\alpha }^{*}$, as measured in graphite using inelastic x-ray scattering. This followup to our previous study [J. P. Reed et al., Science 330, 805 (2010)] was carried out with two times better energy resolution, five times better momentum resolution, and an improved experimental setup with lower background. We compare our results to random-phase approximation (RPA) calculations and evaluate the relative importance of interlayer hopping, excitonic corrections, and screening from high energy excitations involving the $\sigma$ bands. We find that the static, limiting value of ${\alpha }^{*}$ falls in the range 0.25–0.35, which is higher than our previous result of 0.14, but still below the value expected from RPA. We show the reduced value is not a consequence of interlayer hopping effects, which were ignored in our previous analysis, but of a combination of excitonic effects in the $\pi \to {\pi }^{*}$ particle-hole continuum, and background screening from the $\sigma$-bonded electrons. We find that $\sigma$-band screening is extremely strong at distances of less than a few nanometers, and should be highly effective at screening out short-distance, Hubbard-like interactions in graphene as well as other carbon allotropes.

Figure 1

The imaginary part of the experimental density response of graphite and graphene compared to results from RPA calculations. (a) $-Im{\chi }_{3\text{D}}(\mathit{q},\omega )$ from IXS measurements (blue points), RPA calculations including interlayer hopping (green line), and omitting interlayer hopping (red line), plotted against energy for selected in-plane momenta, $q$. The RPA curves had to be multiplied by an arbitrary scale factor to allow visual comparison to the experiment. The black, vertical lines indicate the energy, $\hslash v_{F}q$, at which the onset of the particle-hole continuum is expected in RPA. (b) $-Im{\chi }_{2\text{D}}(\mathit{q},\omega )$ determined from IXS measurements from Eq. (3) (blue points), and RPA calculations including interlayer hopping (red line) and omitting interlayer hopping (orange line), plotted against energy for selected in-plane momenta, $q$. The black line, again, indicates the onset energy of the continuum expected from RPA. Experimental error bars are derived from both Poisson statistics and our fitting parameters (see Appendix pp1).

Figure 2

(a) Crystal structure of ABA-stacked graphite defining the hopping parameters $t, {\gamma }_1$, and ${\gamma }_3$. (b) Brillouin zone of graphene defining the in-plane azimuthal angle of the basal plane. The spectra were found to be independent of this angle in the range of momenta measured in this study.

Figure 3

Imaginary part of the polarization function of graphite, $-Im{\mathrm{\Pi }}_{3\mathrm{D}}(\mathit{q},\omega )$, from experiment (blue points), compared to the results of RPA calculations both with (green line) and without (red line) interlayer hopping, ${\gamma }_{1,3}$. Within the assumptions described in Sec. 3, $Im{\mathrm{\Pi }}_{3\text{D}}(\mathit{q},\omega )$ should be the same as $Im{\mathrm{\Pi }}_{2\text{D}}(\mathit{q},\omega )$ apart from an overall factor of the interlayer spacing, $d$. Note that the magnitude discrepancy observed in Fig. 1 is not observed here, because polarization functions from different excitations combine additively [Eq. (7)].

Figure 4

(a) Asymptotic screening and the effective fine structure constant of graphene. (a) Combined plot of the static ratios $-Im{\chi }_{3\text{D}}(q,0)/q$ and $-Im{\chi }_{2\text{D}}(q,0)/qd$ from experiment and from RPA calculations both with (${\gamma }_{1,3}\ne 0$) and without (${\gamma }_{1,3}=0$) interlayer hopping. The results from our previous study [16] are also plotted, for comparison. This plot shows that all quantities exhibit the proper asymptotic properties in the regime of small $q$. (b) Momentum dependence of the magnitude of the static value of the fine structure constant of graphene. This plot suggests a limiting value in the range 0.25–0.35.

Figure 5

Illustration of data processing steps described in Appendix pp1. (a) Subtraction of the elastic scattering from the IXS spectra. Red points: Raw IXS data for $q=0.212{\text{Å}}^{-1}$. Blue line: Fit to the elastic peak using a pseudo-Voigt function. Black points: Resulting spectrum, showing the onset of $\pi \to {\pi }^{*}$ excitations at $\sim 1$ eV. (b) Matching of this spectrum to wide-range data from our previous study [16], to aid in evaluting the sum rule integral. (c) Same illustration as in (a) for momentum $q=0.282{\text{Å}}^{-1}$. (d) Matching of the $q=0.282{\text{Å}}^{-1}$ spectrum to data from Ref. [16], illustrating the mismatch present in a few of the spectra. This mismatch is incorporated into the determination of the error bars on the quantity $Im{\chi }_{3\text{D}}(\mathit{q},\omega )$.