Intrinsic charge and spin conductivities of doped graphene in the Fermi-liquid regime

The experimental availability of ultra-high-mobility samples of graphene opens the possibility to realize and study experimentally the “hydrodynamic” regime of the electron liquid. In this regime, the rate of electron-electron collisions is extremely high and dominates over the electron-impurity and electron-phonon scattering rates, which are therefore neglected. The system is brought to a local quasiequilibrium described by a set of smoothly varying (in space and time) functions, i.e., the density, the velocity field, and the local temperature. In this paper, we calculate the charge and spin conductivities of doped graphene due solely to electron-electron interactions. We show that, in spite of the linear low-energy band dispersion, graphene behaves in a wide range of temperatures as an effectively Galilean-invariant system: the charge conductivity diverges in the limit $T\to 0$, while the spin conductivity remains finite. These results pave the way to the description of charge transport in graphene in terms of Navier-Stokes equations.

Figure 1

(a) The diagrammatic representation of the current-current response function. The left dot is the bare vertex ${\mathrm{\Lambda }}^{(0,\alpha )}$ (we suppress the momentum-energy dependence for brevity), while the solid double lines are Green's functions dressed by the self-energy. In the large-$N$ limit (where $N$ is the number of fermion flavors, 4 for graphene) it corresponds to the $GW$ self-energy, which is depicted in panel (b). Wavy lines represent RPA screened interactions. Finally, the triangle represents the vertex function ${\mathrm{\Lambda }}^{\beta }$ which is dressed by e-e interactions and satisfies the Bethe-Salpeter equation in panel (c). Note that the form of the irreducible interaction $I$ is uniquely determined by the choice of the self-energy, provided ${\mathrm{\Lambda }}^{\beta }$ must satisfy the Ward identities [15] (see Fig. 2).

Figure 2

The diagrams that contribute to the irreducible interaction $I$ of Fig. 1.

Figure 3

A pictorial representation of double particle-hole excitations that contribute, to lowest order in the strength of e-e interactions, to the quasiparticle decay rate calculated in Sec. 2a. Note that, since all the states involved in the scattering process live at the Fermi surface, the conservation of momentum constrains the initial states $\mathit{k}$ and ${\mathit{k}}^{\prime \prime }-\mathit{q}$ to be diametrically opposed. The same happens to the final states $\mathit{k}-\mathit{q}$ and ${\mathit{k}}^{\prime \prime }$.

Figure 4

(a) The quasiparticle lifetime of massless Dirac fermions ${\tau }_{\mathrm{ee}}^{\mathrm{qp}}$, as defined in Eq. (21), in units of picoseconds and plotted as a function of the density $n$ in units of ${10}^{12}{\mathrm{cm}}^{-2}$ for three values of the dimensionless coupling constant ${\alpha }_{\mathrm{ee}}$. In this plot we fixed the temperature $T=300\mathrm{K}$. (b) Same as in panel (a) but shown as a function of temperature (in units of $\mathrm{K}$) for a fixed excess carrier density $n={10}^{12}{\mathrm{cm}}^{-2}$.

Figure 5

A comparison between the quasiparticle lifetime calculated from Eq. (21) and the one computed in Ref. [33]. In this figure the temperature is kept fixed at $T=300\mathrm{K}$ and the quasiparticle lifetime is plotted in units of ps as a function of the density (in units of ${10}^{12}{\mathrm{cm}}^{-2}$). Panels (a)–(c) refer to the three values of the coupling constant ${\alpha }_{\mathrm{ee}}=0.5$, ${\alpha }_{\mathrm{ee}}=0.9$, and ${\alpha }_{\mathrm{ee}}=2.2$, respectively.

Figure 6

A comparison between the quasiparticle lifetime calculated from Eq. (21) and the one computed in Ref. [33]. In this figure, the density is kept fixed at $n={10}^{12}{\mathrm{cm}}^{-2}$ and the quasiparticle lifetime is plotted in units of ps as a function of the temperature (measured in K). Panels (a)–(c) refer to the three values of the coupling constant ${\alpha }_{\mathrm{ee}}=0.5$, ${\alpha }_{\mathrm{ee}}=0.9$, and ${\alpha }_{\mathrm{ee}}=2.2$, respectively.

Figure 7

(a) The spin transport time of massless Dirac fermions ${\tau }_{\mathrm{tr}}^{\left(\mathrm{s}\right)}$, as defined in Eq. (37), in units of picoseconds and plotted as a function of the density $n$ in units of ${10}^{12}{\mathrm{cm}}^{-2}$ for three values of the dimensionless coupling constant ${\alpha }_{\mathrm{ee}}$. In this plot, we fixed the temperature $T=300\mathrm{K}$. (b) Same as in (a) but shown as a function of temperature (in units of $\mathrm{K}$) for a fixed excess carrier density $n={10}^{12}{\mathrm{cm}}^{-2}$.