Gauge-phonon dominated resistivity in twisted bilayer graphene near magic angle

Recent experiments on twisted bilayer graphene (tBG) close to magic angle show that a small relative rotation in a van der Waals heterostructure greatly alters its electronic properties. We consider various scattering mechanisms and show that the carrier transport in tBG is dominated by a combination of charged impurities and acoustic gauge phonons. Charged impurities still dominate at low temperature and densities because of the inability of Dirac fermions to screen long-range Coulomb potentials at charge neutrality; however, the gauge phonons dominate for most of the experimental regime because, although they couple to current, they do not induce charge and are therefore unscreened by the large density of states close to magic angle. We show that the resistivity has a strong monotonically decreasing carrier density dependence at low temperature due to charged impurity scattering, and weak density dependence at high temperature due to gauge phonons. Away from charge neutrality, the resistivity increases with temperature, while it does the opposite close to the Dirac point. A nonmonotonic temperature dependence observed only at low temperature and carrier density is a signature of our theory that can be tested in experimentally available samples.

Figure 1

(a) Scalar and gauge deformations of twisted bilayer graphene and (b) how they are screened within the random phase approximation. The uniform stretching of the moiré Brillouin zone (top sketch) couples to both charge and current and is strongly screened by the electrons in the flat moiré bands at low energy. The asymmetric and shear modes (bottom two sketches) are area preserving and therefore act like a gauge potential that couples to current but not charge. As a result, these remain unscreened even as the Fermi velocity vanishes. Wavy, solid, and springy lines refer to Coulomb interaction $V_q$, electrons, and phonons, respectively. ${\mathrm{\Pi }}_C$ is the RPA polarization bubble.

Figure 2

(a) Electronic structure of twisted bilayer graphene. A low-energy effective Dirac Hamiltonian is valid for energies up to the blue line (marked Dirac) which for $\theta =1.3^{\circ }$ is $\sim 25$ percent of the van Hove singularity energy (marked VHS). (b) Gauge phonons dominate the transport at temperatures higher than $T_{\mathrm{cross}}$, while charged impurities dominate at lower temperatures. This crossover occurs within the window probed in recent experiments. (c) Similarly, phonons dominate at high density, while charged impurities dominate at low carrier density.

Figure 3

(a) Total resistivity as a function of twist angle (above magic angle), showing a crossover from phonon to impurity-dominated carrier transport. The black solid, red dashed, and blue dashed lines are total, phonon, and impurity limited resistivity, respectively. Gauge phonons are not screened by the large density of states associated with the flat bands, and dominate close to magic angle. (b) Total resistivity as a function of temperature at fixed density and twist angle. The crossover temperature in monolayer graphene is $\sim 500\mathrm{K}$, while near magic angle, the crossover temperature reduces to 5 K due to the strong screening of charged impurities by the flat bands, and the immunity of gauge phonons towards screening. The nonmonotonicity arises from the charged-impurity scattering component crossing over from the degenerate ($k_{\mathrm{B}}T<{\varepsilon }_F$) to nondegenerate ($k_{\mathrm{B}}T>{\varepsilon }_F$) regime. We use an effective dielectric constant of ${\kappa }_{\mathrm{eff}}=3.5$ appropriate for tBG on hexagonal boron nitride ($h$-BN).

Figure 4

Full effective medium theory for carrier transport in twisted bilayer graphene as a function of (a) temperature and (b) density considering both charged-impurity and gauge-phonon scattering. At high temperature or high density, the gauge phonons dominate and there is a linear-in-T resistivity with negligible density dependence. At low temperature and close to charge neutrality, there is a nonmonotonic temperature dependence arising from charged-impurity scattering. There is an inversion in temperature dependence as we move away from charge neutrality to higher densities because of a crossover from charged-impurity to gauge-phonon scattering. Inset: Charge density fluctuations vs twist angle. While charged impurities are strongly screened by the large density of states, the electron-hole puddles are weakly affected by twist angle.