Quantum Boltzmann equation for bilayer graphene

AB-stacked bilayer graphene has massive electron and holelike excitations with zero gap in the nearest-neighbor hopping approximation. In equilibrium, the quasiparticle occupation approximately follows the usual Fermi-Dirac distribution. In this paper we consider perturbing this equilibrium distribution so as to determine DC transport coefficients near charge neutrality. We consider the regime $\beta \left|\mu \right|\lesssim 1$ (with $\beta$ the inverse temperature and $\mu$ the chemical potential) where there is not a well-formed Fermi surface. Starting from the Kadanoff-Baym equations, we obtain the quantum Boltzmann equation of the electron and hole distribution functions when the system is weakly perturbed out of equilibrium. The effects of phonons, disorder, and boundary scattering for finite-sized systems are incorporated through a generalized collision integral. The transport coefficients, including the electrical and thermal conductivity, thermopower, and shear viscosity, are calculated in the linear response regime. We also extend the formalism to include an external magnetic field. We present results from numerical solutions of the quantum Boltzmann equation. Finally, we derive a simplified two-fluid hydrodynamic model appropriate for this system, which reproduces the salient results of the full numerical calculations.

Figure 1

Sketch of the bilayer graphene lattice used for the tight-binding Hamiltonian. There are two layers 1 and 2 and in each layer there are two inequivalent sites per unit cell labeled A and B. The couplings ${\gamma }_0,{\gamma }_1$, and ${\gamma }_3$ are defined in the figure.

Figure 2

Plot of the slope of the off-diagonal component of the electrical conductivity ${\sigma }_{xy}/B$ from the QBE (solid) and the two-fluid model (dashed) as a function of $\beta \mu$. This plot uses the experimentally motivated value ${\alpha }_{\mathrm{ph}}=0.05$. The two-fluid model agrees perfectly with the full QBE calculation.

Figure 3

Electrical conductivity at CN ${\tilde{\sigma }}_{xx}(\mu =0)$ plotted as a function of temperature for the canonical value ${\alpha }_{\mathrm{ph}}=0.05$. We have also chosen ${\alpha }_L=\frac{\sqrt[]{\beta }}{\sqrt[]{m}L}=0.03$ (at $T=25\mathrm{K})$ using the scale $L\sim 3\mu \mathrm{m}$ set by the sample size in [18].

Figure 4

Plot of the dimensionless viscosity ${\tilde{\eta }}_{\lambda ,{\lambda }^{\prime }}={(N_{f}m/\beta )}^{-1}{\eta }_{\lambda ,{\lambda }^{\prime }}$ plotted for ${\alpha }_{\mathrm{ph}}=0.05$. We plot ${\eta }_{++}$ (solid line), ${\eta }_{--}$ (dotted), ${\eta }_{+-}$ (dashed), ${\eta }_{-+}$ (dashed).

Figure 5

Plot of the Lorenz number in units of ${(k_B/e)}^2$ for ${\alpha }_{\text{s}}=\beta {\tau }_s^{-1}=0.05$. We compare the QBE results (solid) with the two-fluid model results (dashed).

Figure 6

Plots of the dimensionless conductivity defined via ${\sigma }_{ij}=\frac{N_f{e}^2}{2\hslash }\tilde{\sigma }{\delta }_{ij}$ for various values of ${\alpha }_L$ and ${\alpha }_{\text{ph}}$. We compare the QBE results (solid) with the two-fluid model results (dashed). We see that in the case of phonon or impurity scattering (top row), the results for the electrical conductivity are good for all values of the phonon coupling strength. On the other hand, for scattering off the boundary of the sample (bottom row), the agreement gets worse as the scattering off the boundary increases. The notation in the top row such as ${\alpha }_{\text{ph}}/{\alpha }_{\text{imp}}=0.01$ refers to the case where either ${\alpha }_{\text{ph}}=0.01$ or ${\alpha }_{\text{imp}}=0.01$, they both yield the same result.

Figure 7

Plots of the dimensionless thermal conductivity defined via $K_{ij}=\frac{N_f{k}_B^{2}T}{2\hslash }\tilde{K}{\delta }_{ij}$. We compare the QBE results (solid) with the two-fluid model results (dashed) for the cases of phonon or impurity scattering (top row) and for scattering off the boundary of the sample (bottom row). The notation in the top row such as ${\alpha }_{\text{ph}}/{\alpha }_{\text{imp}}=0.1$ refers to the case where either ${\alpha }_{\text{ph}}=0.1$ or ${\alpha }_{\text{imp}}=0.1$, they both yield the same result. We see that the agreement is good for weak-momentum-relaxing scattering. However, as the momentum-relaxing scattering is increased, the agreement gets significantly worse. Note that at large ${\alpha }_{\text{ph}}/{\alpha }_{\text{imp}}$, the QBE predicts $\tilde{K}$ to increase with $\beta \mu$, whereas the two-fluid model predicts a decrease, so even the qualitative behavior is incorrect in this regime.