Bulk and shear viscosities of the two-dimensional electron liquid in a doped graphene sheet

Hydrodynamic flow occurs in an electron liquid when the mean free path for electron-electron collisions is the shortest length scale in the problem. In this regime, transport is described by the Navier-Stokes equation, which contains two fundamental parameters, the bulk and shear viscosities. In this paper, we present extensive results for these transport coefficients in the case of the two-dimensional massless Dirac fermion liquid in a doped graphene sheet. Our approach relies on microscopic calculations of the viscosities up to second order in the strength of electron-electron interactions and in the high-frequency limit, where perturbation theory is applicable. We then use simple interpolation formulas that allow to reach the low-frequency hydrodynamic regime where perturbation theory is no longer directly applicable. The key ingredient for the interpolation formulas is the “viscosity transport time” ${\tau }_{\mathrm{v}}$, which we calculate in this paper. The transverse nature of the excitations contributing to ${\tau }_{\mathrm{v}}$ leads to the suppression of scattering events with small momentum transfer, which are inherently longitudinal. Therefore, contrary to the quasiparticle lifetime, which goes as $-1/[T^{2}ln(T/T_{\mathrm{F}})]$, in the low-temperature limit we find ${\tau }_{\mathrm{v}}\sim 1/T^2$.

Figure 1

(a) The e-e mean free path ${\ell }_{\mathrm{ee}}$ (in $\mu \mathrm{m})$ in a 2D MDF liquid is plotted as a function of temperature $T$ (in $\mathrm{K})$. Different curves refer to different values of the excess carrier density, $n=0.5\times {10}^{12}{\mathrm{cm}}^{-2},n=1.0\times {10}^{12}{\mathrm{cm}}^{-2}$, and $n=2.0\times {10}^{12}{\mathrm{cm}}^{-2}$. (b) The e-e mean free path ${\ell }_{\mathrm{ee}}$ (in $\mu \mathrm{m})$ in a 2D MDF liquid is plotted as a function of the excess carrier density $n$ (in units of ${10}^{12}{\mathrm{cm}}^{-2})$, for three values of $T$, i.e., $T=100$, 200, and $300\mathrm{K}$. In both panels, the e-e dimensionless coupling constant has been set to ${\alpha }_{\mathrm{ee}}=0.5$.

Figure 2

The high-frequency shear viscosity ${\eta }_{\infty }$ (in units of the excess carrier density $n)$ of the 2D MDF liquid in a doped graphene sheet is plotted as a function of the dimensionless parameter ${\alpha }_{\mathrm{ee}}$, which defines the strength of e-e interactions. Note that the vertical axis must be multiplied by a factor ${10}^{-2}$.

Figure 3

(a) The diagrammatic representation of the current-current response function. The black filled circle represents 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-$N_{\mathrm{F}}$ limit, it corresponds to the GW self-energy, which is depicted in (b). Wavy lines represent the RPA screened interaction $W$. Finally, the triangle represents the vertex function ${\mathrm{\Lambda }}^{\beta }$ which is dressed by e-e interactions and satisfies the Bethe-Salpeter equation in (c). Note that the form of the irreducible interaction $I$ is uniquely determined by the choice of the self-energy, provided that ${\mathrm{\Lambda }}^{\beta }$ satisfies the Ward identities [4] (see Fig. 4).

Figure 4

Feynman diagrams that contribute to the irreducible interaction $I$ in Fig. 3.

Figure 5

(a) shows the low-frequency kinematic viscosity ${\nu }_0\equiv {\eta }_0/\left(n{m}_{\mathrm{c}}\right)$ of a 2D MDF liquid (in units of ${\mathrm{m}}^2/\mathrm{s})$ as a function of the excess carrier density $n$ (in units of ${10}^{12}{\mathrm{cm}}^{-2})$. The three curves refer to different values of the temperature, i.e., $T=150$, 200, and $270\mathrm{K}$. In this plot, ${\alpha }_{\mathrm{ee}}=0.5$. Note that in the range of densities shown in this plot the temperature is always smaller than $T_{\mathrm{F}}={\varepsilon }_{\mathrm{F}}/k_{\mathrm{B}}$. This, in turn, implies that the low-temperature approximation $\left(T\ll T_{\mathrm{F}}\right)$ for $1/{\tau }_{\mathrm{v}}$ in Eq. (60) is valid. Indeed, the minimum Fermi temperature in this plot, corresponding to $n=0.2\times {10}^{12}{\mathrm{cm}}^{-2}$, is $T_{\mathrm{F}}\simeq 605\mathrm{K}$. Therefore, in this plot, $T/T_{\mathrm{F}}\lesssim 0.45$. (b) The low-frequency kinematic viscosity is shown as a function of temperature $T$ (in units of $\mathrm{K})$ for an excess carrier density $n=0.4\times {10}^{12}{\mathrm{cm}}^{-2}$ (corresponding to $T_{\mathrm{F}}\simeq 855\mathrm{K})$. Inset: a logarithmic plot of ${\nu }_0$ in the same range of temperatures as in the main panel. Note that ${\nu }_0$ grows like $T^{-2}$ for $T\ll T_{\mathrm{F}}$. A crossover to a different power law, $\sim T^{-3/2}$, is evident as temperature increases beyond the $T\ll T_{\mathrm{F}}$ regime.

Figure 6

(a) shows the kinematic viscosity ${\nu }_{\omega }={\eta }_{\omega }/\left(n{m}_{\mathrm{c}}\right)$ (in units of ${\mathrm{m}}^2/\mathrm{s})$ of the 2D MDF liquid in a doped graphene sheet—Eq. (1)—as a function of $\omega$ (in units of the Fermi energy ${\varepsilon }_{\mathrm{F}})$. In this plot, we set $n=0.2\times {10}^{12}{\mathrm{cm}}^{-2},{\alpha }_{\mathrm{ee}}=0.5$, and we show results for three different temperatures: $T=150$, 200, and $270\mathrm{K}$. All the curves tend to the finite (albeit small) value at large $\omega$, whose magnitude can be extracted from Fig. 2. In this plot, we subtracted the “conductivity term” proportional to ${\mathcal{B}}_{N_{\mathrm{f}}}\left({\alpha }_{\mathrm{ee}}\right)$ from the definition of ${\eta }_{\infty }$ given in Eq. (54). (b) Same as in panel a) but for $n=0.5\times {10}^{12}{\mathrm{cm}}^{-2}$. (c) Same as in (a) and (b) but for $n=1.0\times {10}^{12}{\mathrm{cm}}^{-2}$.