Enhanced hydrodynamic transport in near magic angle twisted bilayer graphene

Using the semiclassical quantum Boltzmann theory and employing the Dirac model with twist angle-dependent Fermi velocity, we obtain results for the electrical resistivity, the electronic thermal resistivity, the Seebeck coefficient, and the Wiedemann-Franz ratio in near magic angle twisted bilayer graphene, as functions of doping density (around the charge-neutrality point) and modified Fermi velocity $\tilde{v}$. The $\tilde{v}$ dependence of the relevant scattering mechanisms, i.e., electron-hole Coulomb, long-range impurities, and acoustic gauge phonons, is considered in detail. We find a range of twist angles and temperatures, where the combined effect of momentum-nonconserving collisions (long-range impurities and phonons) is minimal, opening a window for the observation of strong hydrodynamic transport. Several experimental signatures are identified, such as a sharp dependence of the electric resistivity on doping density and a large enhancement of the Wiedemann-Franz ratio and the Seebeck coefficient.

Figure 1

(a) Electric resistivity ${\rho }_{\mathrm{C}}$ from electron-hole Coulomb scattering at $\overline{\mu }=0$ (Coulomb drag resistivity) approaches a constant value as the modified Fermi velocity $\tilde{v}$ tends to zero. ${\rho }_{\mathrm{C}}$ also has a very weak (logarithmic) dependence on temperature, which is neglected here. (b) Two-dimensional (2D) contour plot of the intrinsic Coulomb thermal resistivity ${\rho }_{\mathrm{C},\mathrm{th}}\equiv {\rho }_{\mathrm{C}}{\left(\alpha \overline{\mu }\right)}^2{e}^2/\left(k_B^{2}T\right)$ obtained from Eq. (2) in the limit of zero disorder ($\mathrm{\Gamma }=0$) as a function of doping density $n$ and $\tilde{v}$ (in units of graphene Fermi velocity $v$) at fixed $T=50$ K. Notice that ${\rho }_{\mathrm{C},\mathrm{th}}$ vanishes at $\overline{\mu }=0$, reflecting the conservation of momentum.

Figure 2

Thermal resistivity associated with the long-range charged impurity (red curves) and acoustic gauge phonons (blue curves) as functions of (a) $\tilde{v}/v$ (at $T=50$ K) and (b) temperature $T$ (at $\tilde{v}/v=0.05$). Both sets of curves are calculated at the charge-neutrality point and the charged impurity density in (a) is set to $n_{\mathrm{imp}}=1\times {10}^{10}{\mathrm{cm}}^{-2}$. ${\rho }_{\mathrm{th},\mathrm{dis}}$ can be linked to the corresponding electric resistivity ${\rho }_{\mathrm{el},\mathrm{dis}}$, through the Wiedemann-Franz law for noninteracting electronic systems, which takes the form ${\rho }_{\mathrm{el},\mathrm{dis}}/{\rho }_{\mathrm{th},\mathrm{dis}}\simeq 2.4{\pi }^2{k}_B^{2}T/\left(3e^2\right)$ for the Dirac model near charge neutrality.

Figure 3

(a) Contour plot of the total electric resistivity, given by Eq. (1)—including the contributions of charged impurities, gauge phonons, and Coulomb scattering—as a function of $\tilde{v}/v$ and doping density $n$ at fixed $T=50$ K. The charged impurity density is set to $n_{\mathrm{imp}}=1\times {10}^{10}{\mathrm{cm}}^{-2}$. (b) Same plot for the noninteracting electric resistivity, including only charged impurity and gauge-phonon scattering: ${\rho }_{\mathrm{el}}^{\mathrm{dis}}={\rho }_{\mathrm{el},\mathrm{imp}}+{\rho }_{\mathrm{el},\mathrm{e}-\mathrm{ph}}$. Observe how the total resistivity in (a) decreases sharply with increasing doping density, in contrast to ${\rho }_{\mathrm{el}}^{\mathrm{dis}}$ in (b), which is weakly density dependent. Thus, a sharp peak of ${\rho }_{\mathrm{el}}^{\mathrm{total}}$ around the charge-neutrality point constitutes experimental evidence of strong hydrodynamic transport.

Figure 4

Contour plot of $1/{\mathrm{\Gamma }}^2$ in the ($T-\tilde{v}$) plane, where $\mathrm{\Gamma }$, defined in Eq. (3), determines the width in density of the hydrodynamic transport region. An enhanced “hydrodynamic window” characterized by exceptionally large values of $1/{\mathrm{\Gamma }}^2\simeq 10$ is clearly visible in a range of twist angles on the left side of the figure.

Figure 5

Contour plot of (a) $ln$(WF) and (b) Seebeck coefficient $Q$ as a function of doping density $n$ and Fermi velocity $\tilde{v}$ at $T=50$ K. WF is scaled with the Lorentz number ${\pi }^2{k}_B^2/3e^2$ and the charged impurity density is set to $n_{\mathrm{imp}}=1\times {10}^{10}{\mathrm{cm}}^{-2}$. The large enhancement of the WF and the peak in the Seebeck coefficient at small $\tilde{v}$ are both signatures of the hydrodynamic regime.