Drude weight, plasmon dispersion, and ac conductivity in doped graphene sheets

We demonstrate that the plasmon frequency and Drude weight of the electron liquid in a doped graphene sheet are strongly renormalized by electron-electron interactions even in the long-wavelength limit. This effect is not captured by the random-phase approximation (RPA), commonly used to describe electron fluids, and is due to coupling between the center-of-mass motion and the pseudospin degree of freedom of the graphene’s massless Dirac fermions. By making use of diagrammatic perturbation theory to first order in the electron-electron interaction, we show that this coupling enhances both the plasmon frequency and the Drude weight relative to the RPA value. We also show that interactions are responsible for a significant enhancement of the optical conductivity at frequencies just above the absorption threshold. Our predictions can be checked by far-infrared spectroscopy or inelastic light scattering.

Figure 1

Breakdown of Galilean invariance in graphene. Panel (a) shows the occupied electronic states in the upper band of graphene in the ground state. Notice that every state is characterized by a value of momentum (the origin of the arrow) and a pseudospin orientation (the direction of the arrow). Panel (b) shows the occupied states after a Galilean boost. An observer riding along with the boost would clearly see that the orientation of the pseudospins has changed. It looks like the pseudospins are subjected to a “pseudomagnetic field” that causes them to tilt toward the $+\stackrel{\wedge }{\mathit{x}}$ direction. The appearance of this pseudomagnetic field is the signature of broken Galilean invariance. In contrast, in a Galilean invariant system [panels (c) and (d)], the energy eigenstates are characterized by momentum only: an observer riding along with the boost would not see any change in the character of the occupied states.

Figure 2

Irreducible Feynman diagrams for the density-density response function ${\chi }_{\rho \rho }(q,\omega )$ up to first order in the electron-electron interaction. (a) The bare bubble diagram. (b) Vertex correction. (c) and (d) Self-energy diagrams.

Figure 3

The ratio $\mathcal{D}/{\mathcal{D}}_0$ between the interacting value of the Drude weight $\mathcal{D}$, calculated from Eq. (19), and the RPA value ${\mathcal{D}}_0=4{\varepsilon }_{\mathrm{F}}{\sigma }_0$ is plotted as a function of electron density $n$ (in units of ${10}^{12}\hspace{0.5em}{\mathrm{cm}}^{-2}$) for different values of graphene’s fine-structure constant ${\alpha }_{\mathrm{ee}}$. The value ${\alpha }_{\mathrm{ee}}=0.9$ is believed to be appropriate for graphene deposited on ${\mathrm{SiO}}_2$, the other side exposed to air. Note that $\mathcal{D}/{\mathcal{D}}_0>1$ and that it depends weakly on carrier density.

Figure 4

(a) Deviation of the real part of the ac conductivity $\text{Re}\sigma (\omega )$ from the noninteracting universal value ${\sigma }_0=e^2/(4\hslash )$, as a function of frequency $\omega /(2{\varepsilon }_{\mathrm{F}}$), calculated from Eq. (20) for several values of the fine-structure constant ${\alpha }_{\mathrm{ee}}$ and for $n=1.5\times {10}^{12}\hspace{0.5em}{\mathrm{cm}}^{-2}$ ($\Lambda =50$). (b) Same as in the main panel, but for a fixed value of the fine-structure constant (${\alpha }_{\mathrm{ee}}=0.9$) and two different values of doping (corresponding to $\Lambda =50$ and $100$). Notice that the dependence of the conductivity on the value of the cutoff $\Lambda$ is almost invisible at low frequencies and becomes visible only at frequencies several times ${\varepsilon }_{\mathrm{F}}$.

Figure 5

The real part of the ac conductivity (in units of ${\sigma }_0$) as a function of $\omega /(2{\varepsilon }_{\mathrm{F}})$ calculated by including only the vertex correction [i.e., by neglecting self-energy diagrams in Figs. 2c and 2d] for ${\alpha }_{\mathrm{ee}}=0.9$ and different values of doping $n$ (corresponding to $\Lambda =100,200$, and $1000$). These data have been calculated using long-range (unscreened) interactions, i.e., $q_{\mathrm{TF}}=0$ in Eq. (15). We remind the reader that, in the absence of disorder and within first-order perturbation theory, $\text{Re}\sigma (\omega )=0$ for $\omega <2{\varepsilon }_{\mathrm{F}}$. Notice that $\text{Re}\sigma {}^{(\mathrm{V})}(\omega )$ converges rapidly in the limit $\Lambda \to \infty$ [since, as already stated in the main text, the integral in the second line of Eq. (14) converges in the same limit] and that it drops well below the universal value ${\sigma }_0$ at large $\omega$.

Figure 6

Same as in Fig. 3, but for Thomas-Fermi screened interactions.

Figure 7

Same as in Fig. 4, but for Thomas-Fermi screened interactions.

Figure 8

Same as in Fig. 5, but for Thomas-Fermi screened interactions. Note that the scale in the vertical axis in this figure is completely different from that in Fig. 5.